A Geometric Study of Single Gimbal Control Moment Gyros Singularity Problems and Steering Law

Transcription

1 A Geometric Study of Single Gimbal Control Moment Gyros Singularity Problems and Steering Law Haruhisa Kurokawa Mechanical Engineering Laboratory Report of Mechanical Engineering Laboratory, No. 175, p.18, 1998.

2 A Geometric Study of Single Gimbal Control Moment Gyros Singularity Problems and Steering Law by Haruhisa Kurokawa Abstract In this research, a geometric study of singularity characteristics and steering motion of single gimbal Control Moment Gyros (CMGs) was carried out in order to clarify singularity problems, to construct an effective steering law, and to evaluate this law s performance. Passability, as defined by differential geometry clarified whether continuous steering motion is possible in the neighborhood of a singular system state. Topological study of general single gimbal CMGs clarified conditions for continuous steering motion over a wider range of angular momentum space. It was shown that there are angular momentum vector trajectories such that corresponding gimbal angles cannot be continuous. If the command torque, as a function of time, results in such a trajectory in the angular momentum space, any steering law neither can follow the command exactly nor can be effective. A more detailed study of the symmetric pyramid type of single gimbal CMGs clarified a more serious problem of continuous steering, that is, no steering law can follow all command sequences inside a certain region of the angular momentum space if the command is given in real time. Based on this result, a candidate steering law effective for rather small space was proposed and verified not only analytically, but also using ground experiments which simulated attitude control in space. Similar evaluation of other steering laws and comparison of various system configurations in terms of the allowed angular momentum region and the system s weight indicated that the pyramid type single gimbal CMG system with the proposed steering law is one of the most effective candidate torquer for attitude control, having such advantages as a simple mechanism, a simpler steering law, and a larger angular momentum space. Keywords Attitude control, Singularity, Momentum exchange device, Inverse kinematics, Steering law i

3 Acknowledgments This research work is a result of projects conducted at the Mechanical Engineering Laboratory, Agency of Industrial Technology and Science, Ministry of International Trade and Industry, Japan. Related projects are, Development of Attitude Control Equipment (FY ), Attitude Control System for Large Space Structures (FY ), and High Precision Position and Attitude Control in Space (FY ). I wish to acknowledge my debt to many people. Prof. Nobuyuki Yajima of the Institute of Space and Astronautical Science (ISAS) are earnestly thanked for inspiring me with this theme, as well as for collaborations during his tenure as a division head of our laboratory. I would extend thanks to the late Prof. Toru Tanabe, formerly of the University of Tokyo for his guidance in the culmination of this work into a dissertation. In finishing this work, the following professors guided me, Assoc. Prof. Shinichi Nakasuka of the University Tokyo, Prof. Hiroki Matsuo of ISAS, Prof. Shinji Suzuki, Prof. Yoshihiko Nakamura, Assoc. Prof. Ken Sasaki of the University of Tokyo. Many discussions with Dr. Shigeru Kokaji of our laboratory proved invaluable. He patiently listened to my abstract explanation of geometry and provided valuable suggestions. Furthermore, he assisted me by soldering and checking circuits, and reviewed this paper from cover-to-cover, providing constructive criticism. I would also like to thank my colleague Akio Suzuki who constructed most of the experimental apparatus, and designed and installed controllers for the attitude control. Prof. Tsuneo Yoshikawa of Kyoto University helped me when we started the project of attitude control by CMGs. Discussions held with Dr. Nazareth Bedrossian and Dr. Joseph Paradiso of the Charles Stark Draper Laboratory (CSDL) were invaluable. They gave me valuable suggestions with various research papers in this field. Dr. Mark Lee Ford as a visiting researcher of our laboratory spent his precious hours for me to correct expressions in English. I would like to thank all the above people, other colleagues sharing other research projects, and the Mechanical Engineering Laboratory (MEL) and the directors especially the Director General Dr. Kenichi Matuno and the former Department Head Dr. Kiyofumi Matsuda for allowing me to continue this research. Finally, I thank my wife and daughters for their patience particularly during some hectic months. Haruhisa Kurokawa June 7, 1997 ii

4 Contents Abstract... i Acknowledgments...ii Terms... viii Nomenclature... ix List of Figures... x List of Tables... xiii Chapter 1 Introduction Research Background Scope of Discussion Outline of this Thesis... 4 Chapter 2 Characteristics of Control Moment Gyro Systems CMG Unit Type System Configuration Single Gimbal CMGs Two Dimensional System and Twin Type System Configuration of Double Gimbal CMGs Three Axis Attitude Control Block Diagram CMG Steering Law Momentum Management Maneuver Command Disturbance Angular Momentum Trajectory Comparison and Selection Performance Index Component Level Comparison System Level Comparison Work Space Size and Weight... 9 Chapter 3 General Formulation Angular Momentum and Torque Steering Law Singular Value Decomposition and I/O Ratio Singularity iii

5 3.5 Singularity Avoidance Gradient Method Steering in Proximity to a Singular State Chapter 4 Singular Surface and Passability Singular Surface Continuous Mapping Envelope Visualization Method of the Surface Differential Geometry Tangent Space and Subspace Gaussian Curvature Passability Quadratic Form Signature of Quadratic Form Passability and Singularity Avoidance Discrimination Internal Impassable Surface Impassable Surface of an Independent Type System Impassable Surface of a Multiple Type System Minimum System Chapter 5 Inverse Kinematics Manifold Manifold Path Domain and Equivalence Class Terminal Class and Domain Type Class Connection Type 2 Domain Type 1 Domain Class Connection Rules Continuous Steering over Domains Manifold Selection Discussion of the Critical Point Topological Problem Chapter 6 Pyramid Type CMG System System Definition Symmetry Singular Manifold for the H Origin iv

6 6.4 Singular Surface Geometry Chapter 7 Global Problem, Steering Law Exactness and Proposal Global Problem Control Along the z Axis Global problem Details of the Problem Possible Solutions Steering Law with Error Geometrical Meaning Exactness of Control Path Planning Preferred Gimbal Angle Exact Steering Law Workspace Restriction Repeatability and Unique Inversion Constrained Control Reduced Workspace Characteristics of Constrained Control Chapter 8 Ground Experiments Attitude Control Dynamics Exact Linearization Control Method Experimental Facility and Procedure Facility Design of Control Command Sequence Experimental Procedure Experimental Results Attitude Keeping under Constant Disturbance Rotation About the z Axis Maneuver after Momentum Accumulation Mode Selection and Switching Summary of Experiments Chapter 9 Evaluation Conditions for Comparison Spherical Workspace Evaluation by Weight v

7 9.4 Ellipsoidal Workspace Summary of Evaluation Chapter 1 Conclusions Appendix A Double Gimbal CMG System A.1 General Formulation A.2 Singularity A.3 Steering Law and Null Motion... 8 A.4 Passability... 8 A.4.1 Two Unit System... 8 A.4.2 Three Unit System A.5 Workspace Appendix B Proofs of Theories B.1 Basis of Tangent Spaces B.2 Gaussian Curvature B.3 Inverse Mapping Theory B.4 Impassable condition for two negative signs Appendix C Internal Impassability of Multiple Type Systems C.1 Roof Type System M(2, 2) C.1.1 Evaluation of Singular Surface (2) C.1.2 Evaluation of Singular Surface (3) C.1.3 Evaluation of Singular Surface (5) C.1.4 Evaluation of Singular Surface (7) C.1.5 Conclusion C.2 M(3, 2): M(2, 2) C.2.1 Condition (3) of M(2,2) C.2.2 Condition (5) of M(2,2) C.3 M(3, 3): M(2, 2) C.4 M(2, 2, 1): M(2, 2) C.5 M(2, 2, 2): M(2, 2) C.6 Minimum System Appendix D Six and Five Unit Systems D.1 Symmetric Six Unit System S(6) D.1.1 System Definition vi

8 D.1.2 Fault Management D.1.3 Four out of Six Control D.2 Five Unit Skew System Appendix E Specification of Experimental Apparatus and Experimental Procedure E.1 Experimental Apparatus E.2 Specifications E.3 Attitude Control System E.4 Steering Law Implementation E.5 Code Size and Calculation Time E.6 Parameter Estimation Appendix F General kinematics F.1 Analogy with a Spatial Link Mechanism F.2 Spatial Link Mechanism Kinematics F.3 Singularity F.4 Passability References vii

9 Terms Class : A set of manifolds which correspond to a certain domain and are equivalent to each other. Domain : A region in the angular momentum space which is surrounded by singular surfaces and does not contain any singular surface. Double gimbal CMG : Fig. 2 1 Gimbal vector : A unit vector of gimbal direction. Independent type : A single gimbal CMG system without parallel gimbal direction pair. Manifold : A connected subspace of gimbal angle space whose element corresponds to the same total angular momentum. Manifold equivalence : Two manifolds corresponding to a certain domain are equivalent if there is an angular momentum path which corresponds to a continuous manifold path between these two manifolds. Multiple type : A single gimbal CMG system composed of groups each of which elements possess identical gimbal direction. Null motion : Gimbal angle motion which keeps the angular momentum vector constant. Single gimbal CMG : Fig. 2 1 Singular surface : A surface formed by the total angular momentum vector point, H, which corresponds to singular point. Singular vector : A unit vector to the plane spanned by all torque vectors when the system is singular. Skew type : A single gimbal CMG system with gimbal directions axially symmetric about one direction. Symmetric type : A single gimbal CMG system with gimbal directions arranged normal to surfaces of a regular polyhedron. Torque vector : A unit vector of a component CMG to which direction the CMG can generate an output torque. Workspace: Allowed region of the angular momentum vector of a CMG system. viii

10 Nomenclature Symbol Definition Section number α : Skew angle of the symmetric pyramid type system 6.1 β : Euler parameter of satellite orientation β* : Vector part of β B : Strip like surface of impassable surface called branch 6.4 c* : = cosα 6.1 c i : = g i h i. Torque vector 3.1 C : Jacobian matrix of the kinematic function, H = f (θ) 3.1 dθ S Θ S dθ N Θ N dθ T Θ T D : Domain in the H space surrounded by singular surfaces 5.3 Symbol Definition Section number M(2, 2): Roof type system M i : Manifold 5.1 M Sj : Singular manifold 5.1 n : Number of CMG units in the system 3.1 p i : = 1 / (u h i ) P: Diagonal matrix of p i θ i : Gimbal angle of ith CMG unit 3.1 θ : Θ S : =(θ i.). A state variable of the system. Point of n dimensional torus T (n) 3.1 Singularly constrained tangent space of the θ space (two dimensional) Θ N : Null space of C (n 2 dimensional) Θ T : Complementary subspace of Θ N (two dimensional) r g : Symmetric transformation in the θ space. 6.2 ε : = {ε i }. Sign parameter of the singular surface R g : Symmetric transformation in the H space. 6.2 s* : = sinα 6.1 g i : Gimbal vector 3.1 G : Equivalence class in a domain. 5.3 h i : Normalized angular momentum vector 3.1 S(n) : Symmetric type single gimbal CMG system S ε : A region of the singular surface of sign ε H : κ : L A : = Σ h i.= f (θ). Total angular momentum vector. 3.1 Gaussian curvature of the singular surface Segment included by a manifold of H=(,,) t 6.3 T : Total output torque of the system 3.1 u : Singular vector. Unit vector normal to all torque vectors. 3.4 ω : Gimbal rate vector. Time derivative of θ. 3.1 ω N : Null motion, 3.2 ix

11 List of Figures Chapter Two types of CMG units 2 2 Configurations of single gimbal CMGs 2 3 Twin type system 2 4 Block diagram of three axis attitude control Chapter Orthonormal vectors of a CMG unit 3 2 Gimbal angle and vectors 3 3 Input Output ratio 3 4 Singularity condition and singular vector 3 5 Typical vector arrangement for a 2D system 3 6 Steering at a singular condition Chapter Vectors at a singularity condition 4 2 Examples of the singular surfaces for the pyramid type system. 4 3 Envelope of a roof type system M(2, 2). 4 4 Cross sections of a singular surface of the pyramid type system. 4 5 Infinitesimal motion from a singular point of 2D system. 4 6 Second order infinitesimal motion from singular surface. 4 7 Possible motions in both direction of u at a singular point. 4 8 Local shape of an impassable singular surface. 4 9 Impassable surface of S(6) 4 1 Impassable surface of Skew(5) with skew angle α =.6 rad Impassable surface of another Skew(5), with skew angle α = 1.2 rad Impassable surface of S(4). Chapter Manifolds in the neighborhood of a singular point. 5 2 Continuous change of manifolds. 5 3 An example of a continuous manifold path. 5 4 Relations between H space, manifold space and θ space. 5 5 Domains and manifolds of the pyramid type system 5 6 Class connection graph around domains 5 7 An illustration of class connection rule (1). 5 8 An illustration of class connection rule (2). 5 9 An illustration of motion by the gradient method. 5 1 Manifold relations around critical point Chapter Schematic of a pyramid type system 6 2 Transformation in H space and in θ space 6 3 Line segments for singular manifold 6 4 Definition of the cross sectional plane and the distance d 6 5 Saddle like part of the envelope 6 6 Cross sections of singular surface 6 7 Internal impassable singular surface 6 8 Analytical line on an impassable surface 6 9 Equilateral parallel hexahedron of impassable branches 6 1 Overall structure of impassable branches 6 11 Internal impassable surface with envelope cutaway 6 12 Cross section through the xz plane 6 13 Cross section through the xy plane Chapter Candidate of workspace 7 2 Cross section nearly crossing P x

12 7 3 Manifold bifurcation and termination from D A 7 4 Simplified class connection diagram around domain D A 7 5 Manifolds of eight domains around the z axis 7 6 Singular manifold of a point U on the z axis 7 7 Manifold of H near the origin 7 8 Continuos change of manifold for H nearly along the z axis 7 9 Manifold connection over several domains 7 1 Cross sections of domains 7 11 Possible motion following an example of singular surface 7 12 Illustration of H trajectory of the CMG system for the example maneuver 7 13 Avoidance of an impassable surface 7 14 Problems of movement on an impassable surface 7 15 Change in manifolds for H moving along the x axis 7 16 Estimation of reduced workspace for exact steering 7 17 Discontinuity in the maximum of det(cc t ) 7 18 Cross section of possible workspace by constrained steering law 7 19 Reduced workspace of the constrained system 7 2 Reduced workspace of three modes Chapter Experimental test rig showing the center mount suspending mechanism 8 2 Target trajectory 8 3 Block diagram of the control system 8 4 Results of Experiment A 8 5 Results of Experiment B 8 6 Results of Experiment C 8 7 Results of Experiment D 8 8 Results of Experiment E 8 9 Results by Experiment F 8 1 Results of Experiment G 8 11 Results of Experiment H 8 12 Command sequence of Experiment J 8 13 Results of Experiment J Chapter System configurations for comparison 9 2 Spherical workspace size for various system configurations 9 3 Trade-off between workspace size and system weight 9 4 Definition of ellipsoidal workspace 9 5 Average radius vs. skew angle 9 6 Workspace radius as a function of aspect ratio 9 7 Combined plot of radii as a function of aspect ratio 9 8 Converted weight as a function of aspect ratio 9 9 Radius as a function of aspect ratio for a degraded system with one faulty unit Appendix A A 1 Vectors and variables relevant to a double gimbal CMG A 2 Vectors at singularity conditions A 3 Infinitesimal motion at a singular point of condition (b) Appendix D D 1 Envelopes of S(6) and degraded systems D 2 Four unit subsystem of MIR type system D 3 Restricted workspace of a constrained MIR-type system D 4 Concept of singularity avoidance by an additional torquer Appendix E E 1 Experimental apparatus E 2 Block diagram of experimental apparatus E 3 Three axis gimbal mechanism E 4 Single gimbal CMG E 5 Balance adjuster E 6 Onboard computer E 7 Block diagram of the model matching controller. E 8 Block diagram of the tracking controller. xi

13 E 9 Block diagram of the gradient method. E 1 Block diagram of the constrained method. Appendix F F 1 Analogy to a parallel link mechanism xii

14 List of Tables Chapter Component Level Comparison 2 2 System Level Comparison Chapter Symmetric Transformations 6 2 Segment Transformation Rule Chapter Condition and Results of Experiments (2) Appendix E E 1 Specification of experimental apparatus E 2 Code size and calculation time of process Appendix F F 1 Similarity between CMGs and link mechanism 8 1 Condition and Results of Experiments (1) xiii

15 xiv

16 2. Characteristics of Control Moment Gyro Systems Chapter 1 Introduction A Control Moment Gyro (CMG) is a torque generator for attitude control of an artificial satellite in space. It rivals a reaction wheel in its high output torque and rapid response. It is therefore used for large manned satellites, such as a space station, and is also a candidate torquer for a space robot. There are two types of CMGs, single gimbal and double gimbal. Though single gimbal CMGs are better in terms of mechanical simplicity and higher output torque than double gimbal CMGs, the control of single gimbal CMGs has inherent and serious singularity problem. At a singularity condition, a CMG system cannot produce a three axis torque. Despite various efforts to overcome this problem, the problem still remains, especially in the case of the pyramid type CMG system. This research aims to elucidate this singularity problem. Detailed study of the pyramid type system leads to a global problem of singularity. The final objective of this work involves evaluation of various steering laws and the proposal of an effective steering law. As all the geometric studies are either theoretical or analytical and based on computer calculations, ground experiments were carried out to support those results. 1.1 Research Background Research of CMG systems started in the mid 196s. This was intended for later application to the large satellite of the USA, Skylab, and its high precision component, Apollo Telescope Mount (ATM) 1, 2, 3). The studies included hardware studies of a gyro bearing and gyro motor, and software studies for attitude control and CMG steering control. Evaluation of various types and configurations was made in terms of weight and power consumption 4). At that time, an onboard computer lacked the ability to perform real time matrix inversion calculation. One of the candidates was a twin type system made of two single gimbal CMGs driven in opposite directions. Control of this system requires only simple calculation 5). If another system was chosen, a simple computation scheme was required using an analog circuit. For example, a method using an approximation with some feedback was proposed 6, 7, 8). For the three double gimbal CMG system 9) applied to the Skylab, an approximated inverse using the transposed Jacobian was used 1). This CMG system successfully completed its mission, though one of the CMGs became nonfunctional during the flight 11). After that, studies of double gimbal CMGs have continued for eventual application to the space shuttle and the space station Freedom which is now called ISS 12, 13). Another CMG type, i.e., a single gimbal one, was studied for use in satellites such as the High Energy Astronomical Observatory (HEAO) and the Large Space Telescope (LST). One of the configurations intensively studied was a pyramid type, which consists of four single gimbal CMGs in a skew configuration. Comparing six different independently developed steering laws indicated that an exact inverse calculation was necessary 14). It was also observed from various simulations that the singularity problem could not be ignored. It was concluded that some sort of singularity avoidance control using system redundancy was required for this type system. A roof type system, which is another four unit system of single gimbal CMGs, was also a candidate for the HEAO. As its mathematical formulation is simpler than that of the pyramid type, singularity avoidance was originally included in a steering law 15). An improvement of this law involved a new approach in which the nature of numerical calculation and discrete time control were utilized 16). Singularity avoidance has been studied for all CMG types. This was a simple matter for double gimbal CMG systems 17) 2). Typically used was a gradient method, which maximized a certain objective function by using redundancy 21). While this method was effective in the evaluation of double gimbal CMG systems, it was not successful for single gimbal CMG systems. For 1

17 Technical Report of Mechanical Engineering Laboratory No.175 example, optimization of a redundant variable resulted in discontinuity 16) or an optimized value became singular 15) in the case of a roof type system. In the case of a pyramid type system, various problems were found in computer simulations even when a gradient method was used. Margulies was the first to formulate a theory of singularity and control 22). His paper included geometric theory of a singular surface, a generalized solution of the output equation and null motion, and the possibility of singularity avoidance for a general single gimbal CMG system. Also, some problems of the gradient method were pointed out using an example of a two dimensional system. Works by the Russian researcher, Tokar, were published in the same year, and included a description of the singular surface shape 23), the size of the workspace 24) and some considerations of the gimbal limits 25). In his next paper 26), passability of a singular surface was introduced. It was made clear that a system such as a pyramid type has an impassable surface inside its workspace. Moreover, the problems of steering near such an impassable surface were described. In spite of those important results, his work was not widely received, because the original papers were published in Russian. Even though an English translation appeared, several terms were used for a CMG, such as gyroforce, gyro stabilizer and gyrodyne. His conclusion was that a system with no less that six units would provide an adequately sized workspace including no impassable surfaces. After this work, a six unit symmetric system was designed for the Russian space station MIR 27). Some years later after Tokar s studies, Kurokawa formulated passability again 28) in terms of the geometric theory given by Margulies. Most of these results coincided with Tokar s work. In addition, the existence of impassability in the roof type system was clarified 29) and a discrimination method using the surface curvature was presented 3,31,32,33). In the last paper, the theory was expanded to a general system including a double gimbal CMG system. It was made clear that multiple systems of no less than six units do not have any internal impassable surface, while any system of less that six units must have such a surface. Various configurations, even containing faulty units, were compared with regard to their workspace size as an extension of Tokar s work. Along with these theoretical and general research works, intensive efforts continued to find an effective steering law regarding the passability problem as a local problem. Most of these dealt with the pyramid type system. The reason this type was selected was because a six unit system was considered too large and too complicated. Many proposals suggested a type of gradient method 34, 35, 36). The method utilized for the four unit subsystem of the MIR was also of this kind 27). Another method used global optimization 28), and nearly all methods showed some problems in computer simulations. Passability is defined locally and its problem reported first was a kind of local problem 28). Later, Bauer showed difficulty in steering as a global problem 37). He found two different command sequences, both of which could not be realized by the same steering method. After this, Vadali proposed a method to overcome this problem using a preferred state 38). Finally, the problem by Bauer was formulated exactly, stating that no steering law can follow an arbitrary command sequence inside certain wide region of the workspace 39). Under this limitation, an effective method was proposed. The research described above dealt with exact control, but other research has also been carried out. One research effort permitted an error in the output if required. Generalized inverse Jacobian 22) minimizes the error. Extension of this method, called the SR inverse method, was first proposed for control of a manipulator and later applied to CMG control 4,41). Another research type dealt with path planning. If the command sequence in the near future is given, steering can be planned beforehand which realizes not only singularity avoidance but also some degree of optimization 42, 43, 44). In one of the research papers 43), some paths were chosen by Kurokawa in consideration of impassable surfaces. Since all these tended to take a heuristic approach, evaluation was made by computer simulation considering attitude control of a given satellite. More realistic studies have also been made which dealt with attitude control using a CMG system, considering disturbance and other torquers. The largest problem may be a precision control using a CMG system. Since a CMG system can generate a large output torque and its output resolution is critical for precision control, various analyses and simulations have shown that pointing control by a CMG system can result in a limit cycle because of friction in gimbal motion 45, 46, 47). In spite of efforts such as improvement of motor control 48) and torque cancellation by additional reaction wheels 49), the problem of precision control has not been overcome. For application to the space station, another studies were carried out such as an effective combination of a CMG and RCS 5) and integration of CMGs and power 2

18 2. Characteristics of 1. Control Introduction Moment Gyro Systems storage 51). In order to evaluate its attitude control performance, not only numerical simulations, but also some experiments using real mechanisms have been made, such as a platform supported by a spherical air bearing 44, 52). The author also developed ground test equipment using normal ball bearings 53) and attempted robust attitude control using a CMG system 54,55). The motion of a CMG system with regard to the motion of the angular momentum vector is similar to the motion of a link mechanism 22). Analysis of the motion and control of such a mechanism has been widely studied. Those results were, therefore, used for CMG control 4, 41). On the other hand, some researchers first studied CMG control and then applied their results to a robot control 56, 57, 58). In spite of various researches in robot kinematics 59, 6, 61), generalized theory for singularity and inverse kinematics has not been formulated yet. 1.2 Scope of Discussion This research effort deals with the following subjects: (1) General formulation of an arbitrarily configured CMG system, especially of single gimbal CMGs. (2) Geometric study of the singularity problem of a general single gimbal CMG system. (3) Problem of exact and real-time steering of the pyramid type CMG system. (4) Proposal and evaluation of steering laws for the pyramid type CMG system. (5) Evaluation of various CMG systems. The main purposes of this work are to clarify the singularity problems, to construct an exact and strictly real-time steering law, and to specify and evaluate its performance. Among all, singularity problems are the most important relating to the others. A singularity can degrade a CMG system, even causing the system to loose control, and this situation might be fatal for an artificial satellite. Therefore, a CMG system must have redundancy and it must be controlled to avoid singularities by using an appropriate steering law. Problems include whether such singularity avoidance is globally possible and which steering law can realize such control. Even if a steering law cannot avoid all the singularities, the system s working range of the angular momentum must be specified in which singularity avoidance is strictly guaranteed because such specification is necessary for designing the total attitude control system. Thus, this work deals with CMG systems alone, but it is made in consideration with the attitude control of artificial satellites. Exactness and strict real time feature of steering laws are essential for the realtime attitude control. For this aim, a geometric approach was taken. As described above, there have been various research works dealing with singularity and steering laws. Most used computer simulations to evaluate their steering laws, for lack of other methods. As simulations alone cannot guarantee the performance of a system as nonlinear as a CMG system, it is necessary to clarify the problem of singularity by other means. A geometrical approach is a more effective way of simplification and qualitative comprehension. The theoretical portion of this work aims for general formulation of singularity problems. Under consideration of these general results, extensive study was made for a specific type of system, that is, the pyramid type. The reasons why this system was chosen are: 1) A three-unit system does not need further study because it has no redundancy. Systems with no less than six units also do not need detailed study for singularity avoidance, a fact described in more detail in this work. Thus, four and five unit systems remain for further study. 2) Most previous research works dealt with this pyramid type system. Four units are the minimum having one degree of redundancy. The number of units is important in the real situation. By a simplified evaluation, a system with fewer units is lighter for a given total storage of angular momentum. Also, steering law calculation is less complicated for a system with fewer units. 3) The pyramid type system has symmetry, which enables easier analysis. Numerical data and analytical expression of some geometric characteristics can be reduced by using this symmetry. This fact is useful for actual implementation. As geometric study is more qualitative rather than quantitative, ground experiments were performed to demonstrate the performance of the steering laws. Also for evaluation, various system types are compared in terms of the size of the possible angular momentum vector operational space and the systems weight. As mentioned above, specific studies of an attitude control are beyond the scope of this work. Such studies involve optimal maneuvering and angular momentum management, which are possible only after the 3

19 Technical Report of Mechanical Engineering Laboratory No.175 specification of CMG systems are given by using the results of this work. In addition, this work does not treat in detail steering laws of double gimbal CMG systems, combinations of single and double gimbal CMGs, combination of CMGs and other torquers, passive type CMGs 62), and systems with different size or controllable size CMGs 63). Also, the effect of gimbal limit is not considered in general except in the proposed method for the pyramid type system. 1.3 Outline of this Thesis Chapter 2 will represent a general description of a CMG unit and CMG systems. The difference between three types of torquers, reaction wheels, single gimbal CMGs and double gimbal CMGs, will be described. Also an important parameter termed workspace in this paper will be defined in terms of an attitude control system. Chapter 3 will represent a general formulation of an arbitrary system of single gimbal CMGs, which includes the kinematic equation and the torque equation. The general steering law, singularity and singularity avoidance will be outlined. This chapter is analytical while the following chapters, from Chapter 4 to 7, are mainly geometrical. Chapter 4 will detail singularity. A singular surface which includes the angular momentum envelope will be examined. For this surface, passability which is one of the most important characteristics of a singular surface will be defined. Passability and surface geometry will be related by the curvature of the surface. Chapter 5 will introduce a way of understanding the steering motion as to whether continuous control is possible, or how the impassable situation can be avoided, if possible. From Chapter 6 to 8, the pyramid type system will be detailed. Chapter 6 will offer analytical and geometric system results without considering a steering law. The impassable singular surface of this system will be fully defined. Chapter 7 will prove the global problem. After various proposals are evaluated based on this result, a new proposal will be offered. Chapter 8 will demonstrate the performance of the proposed method by using a ground test apparatus. Chapter 9 will offer evaluation not only of the proposed method for the pyramid type system but also of various system configurations. Chapter 1 will conclude this work. Because double gimbal CMG systems and various systems other than the pyramid type system will not be detailed in the main text, Appendices A and D will provide these details. Appendices B and C present detailed proofs of some theories given in Chapter 4. Appendix E will give the specifications and implementation of the ground test apparatus. Appendix F will detail the kinematics of a general spatial link mechanism which is analogous to the CMG kinematics. 4

20 2. Characteristics of Control Moment Gyro Systems Chapter 2 Characteristics of Control Moment Gyro Systems A control moment gyro (CMG) system is a torquer for three axis attitude control of an artificial satellite. There are two types of CMG units and various configurations of three axis torquer systems. Designing a CMG system therefore includes a process of selecting a unit type and a system type defined by configuration. Among two unit types and various system types, a single gimbal CMG system of pyramid configuration is mainly described in this work. For the simple comparison, this chapter gives an outline of CMG system characteristics with consideration paid to its use in an attitude control system. The angular momentum workspace, torque output, steering law and singularity problems are the important factors for evaluation of a CMG system. 2.1 CMG Unit Type A CMG consists of a flywheel rotating at a constant speed, one or two supporting gimbals, and motors which drive the gimbals. A rotating flywheel possesses angular momentum with a constant vector length. Gimbal motion changes the direction of this vector and thus generates a gyro effect torque. There are two types of CMG units, as shown in Fig. 2 1, single gimbal and double gimbal. A single gimbal CMG generates a one axis torque and a double gimbal CMG generates a two axis torque. In both cases, the direction of the output torque changes in accordance with gimbal motion. For this reason, a system composed of several units is usually required to obtain the desired torque. 2.2 System Configuration Typical system configurations will now be discussed. The configuration is defined by a set of principal axes of all the component CMG units, which are the gimbal axes in the case of single gimbal CMGs and the outer gimbal axes in the case of double gimbal CMGs. In the following figures, these principal axes are indicated by arrows denoted by g i. The system of each configuration is named as a system type such as twin type system or the pyramid type system. Gimbal Motor Gimbal Mechanism Outer Gimbal Motor Flywheel Gyro Motor Fig. 2 1 Gyro Effect Torque ω T (a) Single gimbal CMG Gyro Motor Two types of CMG units Flywheel Angular Momentum Vector Inner Gimbal (b) Double gimbal CMG Outer Gimbal Inner Gimbal Motor 5

21 Technical Report of Mechanical Engineering Laboratory No Single Gimbal CMGs Typical single gimbal CMG systems have certain kinds of symmetries, which can be classified into two types, independent and multiple. They are somewhat different in their mathematical description. (1) Independent Type Independent type CMGs have no parallel axis pairs. Two categories of independent type CMGs, symmetric types and skew types, have been mainly studied. Symmetric Type Gimbal axes are arranged symmetrically according to a regular polyhedron. There are five regular polyhedrons with 4, 6, 8, 12 and 2 surfaces. Possible configurations of this type are three, four, six and ten unit systems, because only surfaces not parallel to each other are considered and because a tetrahedron and hexahedron are complementary or dual to each other. The three, four, six, and ten unit systems are denoted as S(3), S(4), S(6) and S(1). The four unit or S(4) system, shown in Fig. 2 2(a), is called the symmetric pyramid type. Most of this work deals with this type of system. An example of the six unit or S(6) system, shown in Fig. 2 2(b), is now in use on the Russian space station MIR. Skew Type All individual units are arranged in axial symmetry about a certain axis as depicted in Fig. 2 2(c). Skew three and four unit systems of certain skew angles are the same as the S(3) and the S(4). (2) Multiple Type In this type some number of individual units possess g 4 θ 3 g 3 h 3 g 4 θ 4 h 4 Z α h 2 θ 2 h 5 g 5 h 4 α h 3 g 3 X h 1 Y g 2 g 6 2π n g 2 h2 g 1 θ 1 h 6 g 1 h 1 (a) Pyramid type S(4) (c) Skew type g 1 θ13 h 2 g 6 g 1 θ12 g 2 g 2 h 6 θ23 h 1 g 1 h 3 g 1 g 5 θ11 g 2 θ22 g 3 h 5 g 2 h 4 g 4 θ21 (b) Symmetric type S(6) (d) Multiple type M(3, 3) Fig. 2 2 Configurations of single gimbal CMGs 6

22 2. Characteristics of Control Moment Gyro Systems identical gimbal directions. These are denoted as M(m 1, m 2,...) hereafter, where m i is the number of the units with the same gimbal direction. As an example, the system in Fig. 2 2(d) is denoted by M(3,3). A similar system called roof type 15, 16) would be denoted as M(2,2) with this notation. Gimbal Motor Two Dimensional System and Twin Type System A single gimbal CMG system of an arbitrary number of units all having a common gimbal direction will be called a two dimensional system in this work. In such a system, the angular momentum vector and output torque vector are always on a certain plane normal to the gimbal direction. Though this type of system is not ordinarily used by itself for attitude control, it can easily be visualized and understood. It is, therefore, used for some examples in this work. If a pair of single gimbal CMG units with a common gimbal direction are driven in opposite directions by the same angle, the direction of the output torque is always kept constant, as shown in Fig This type of system is called a twin type or a V pair system 5). Though a three axis system is easily designed by combining several twin type CMGs, such a system is not so much advantageous. A three or more V pair system is identical to a multiple system, M(2, 2,..., 2), whose state variables are constrained, but its workspace is smaller than that of the original multiple system. Though a V pair system is the easiest to control, a multiple system can be also simply controlled as will be described later. Fig. 2 3 Twin type system Configuration of Double Gimbal CMGs Two typical configurations of double gimbal CMGs are an orthogonal type and a parallel type. The orthogonal type consists of three orthogonally positioned units. This type of system was used for the Skylab space vehicle. The parallel type consists of an arbitrary number of units all having a common axis 2). 2.3 Three Axis Attitude Control The design requirement of a CMG system is determined by the specification of a spacecraft attitude control. There are various kinds of attitude control techniques such as spin stabilization, bias momentum stabilization and zero momentum active control. The last is also called three axis attitude control. Reaction wheels and CMGs are commonly used torquers for this Disturbance Maneuver Command Generator A B C T com + - Vehicle Control Law CMG Steering Law CMG System T CMG Spacecraft D Momentum Management Control Logic Unloading Torquers Attitude & Rate Sensors Fig. 2 4 Block diagram of three axis attitude control 7

23 Technical Report of Mechanical Engineering Laboratory No.175 attitude control Block Diagram A functional block diagram of a three axis attitude control is shown in Fig Most of the blocks are the same when either reaction wheels or CMGs are used. The attitude and rotational velocity commands are generated by a maneuver command generator denoted by A in Fig The command and sensor information are the inputs to the vehicle control law block, B. This block calculates the torque necessary for control. The next block, C, shows the CMG steering law which calculates the CMG motion for the torque calculated by block B. In this manner the actual CMG system is driven and an output torque to the satellite is generated. The blocks relating to CMG control are the CMG steering law, C, and the momentum management block, D. Those two blocks are described first in the following sections. Then, relating subjects, i.e., maneuver commands, disturbances and the motion of angular momentum vector will be explained CMG Steering Law The steering law block computes a set of gimbal angle rates which produce the required torque. The steering law is usually realized in two parts, one being simply a solution to a linear equation and the other for singularity avoidance by using system redundancy. This block is usually designed independent of the particulars of the total attitude control system. This implies that the vehicle control law (B in Fig. 2 4) is designed under the assumption that the output of the CMG system corresponds exactly to the command. The CMG steering law must satisfy this requirement. The meaning of this exactness is described in a later chapter Momentum Management A CMG and a reaction wheel are called momentum exchange devices because they don t actually produce angular momentum but rather exchange it with the satellite. Such torquers have limits to their accumulation of angular momentum, because the rotational speed of a flywheel is limited. Therefore, another type torquer is needed when it becomes necessary to offload excess accumulated momentum. This unloading is usually done by gas jets or magnetic torquers. The unloading process must be carefully managed by the momentum management control block, D, because such torquers have their own limitations, i.e., a gas jet does not have enough resolution and it have a limit of storage, and a magnetic torquer s output depends on orbit position. For effective management of angular momentum, the space of allowed angular momentum of a CMG system must be defined beforehand. This space is termed workspace in this paper. The workspace must be included by the possible angular momentum space of the CMG itself. Moreover, a simple shaped space such as a sphere tends to result in more simplified management Maneuver Command The command issued by a maneuver command generator depends on the mode of operation. Typical operational modes are pointing, maneuvering, scanning and tracking. In the pointing mode, precision is of primary importance and is affected by disturbances, torque response and resolution. The speed of maneuvering as well as momentum accumulation while pointing is a matter of workspace size of the torquer Disturbance The time dependence of disturbances vary according to orbit parameters and a mission type, such as earth pointing or inertial pointing. In any case, a disturbance may have cyclic terms and offset terms. The following function is an example of disturbance used for the simulation of HEAO with a pyramid type CMG system 14) ; T g = (T x sin ωt, T y (cos ωt 1), T z sin ωt) t, where ω denotes orbital angular rate. Because there is an offset in the y direction, angular momentum will be accumulated in this direction while pointing Angular Momentum Trajectory The size and shape of the workspace determines the maximum accumulation of disturbances or the maximum speed of maneuvering. A disturbance or a maneuvering command can be expressed as a function of time by a trajectory of the angular momentum vector of the satellite. Since the total angular momentum of the system is equal to the time integral of the disturbance, the angular momentum trajectory of a CMG system can be expressed using the spacecraft s momentum and disturbance. The 8

24 2. Characteristics of Control Moment Gyro Systems workspace of a CMG system must include any possible angular momentum trajectory when the unloading torquers are not operating. 2.4 Comparison and Selection CMG systems and a reaction wheel system are all examples of the same type of torquers. In order to design an attitude control system, some sort of selection criteria is needed. By using the following performance indices, a brief comparison will be made, first at the component level then at the system level Performance Index The performance of a CMG systems depends not only on elements of hardware design, such as the CMG unit type and the system configuration, but also on the design of the steering law. These factors all affect the maximum workspace and the magnitude of the output torque, two nonscalar performance indices. Another performance index is the steering law complexity, which affects the attitude control cycle time and the capacity of an onboard computer Component Level Comparison Table 2 1 clarifies the main differences among these three torquers 64). A reaction wheel has only one motor which is used not only for accumulation of angular momentum but also for generation of torque. On the other hand, the CMGs use either two or three motors, one for accumulation of angular momentum and the others for torque generation. Since the torque of a motor depends on its speed and the same maximum torque cannot be generated over the motor s working speed range, both angular momentum and output torque of a reaction wheel are much smaller than for CMGs. Size and weight of a CMG depends on the size of the flywheel and complexity of the mechanism. A double gimbal CMG is the most complicated at the unit level, but less so at the system level because this unit generates Table 2 1 Component Level Comparison Angular Momentum Torque Reaction Wheel 1 to 1 1 Double Gimbal CMG 1 to 3 1 Single Gimbal CMG 1 to 2 1 a two axis torque. Maximum output torque is much different. A single gimbal CMG can produce more output torque than a double gimbal CMG. The reason is as follows. The output torque of a single gimbal CMG appears on the flywheel and is then transferred directly to the satellite across the gimbal bearings. The output torque can be much larger than the gimbal motor torque required to drive the gimbal. This is called torque amplification. By contrast, some part of the output torque of a double gimbal CMG must be balanced by the gimbal motors. Thus, in this case, the output toque is limited by the motor torque limit System Level Comparison Table 2 2 shows a system level comparison for the three types of torquers being compared. Difference in the first two indices, torque and weight, are derived from component level differences. The other two indices relate to each other. The steering law of any reaction wheel system is linear and no singularity problems arise. Steering law complexity and singularity problems of CMG systems, especially single gimbal CMGs, can be serious and thus form the main subject of the present work Work Space Size and Weight The size and shape of the maximum workspace are not compared in the above table because they depend on the number of units and system configuration. Workspace size as a scalar value, and the weigh of the CMG system can be roughly evaluated in terms of the number of units. Let s consider similarly shaped Table 2 2 System Level Comparison Torque Weight Steering Law Singularity Reaction Wheel 1 1 simple none Double Gimbal CMG 1 2 not simple slight Single Gimbal CMG 1 2 most complex serious 9

25 Technical Report of Mechanical Engineering Laboratory No.175 flywheels of diameter d and thickness t. Similarity implies t d. Then, the weight and the size of maximum workspace of an n unit system, denoted as W and H, follow the following relation if the rotational rate of the gyro is the same: W n d 2 t n d 3, (2 1) H n (t d d 2 ) dr n d 5. (2 2) If H is set constant, W is given by; W n d 3 n 2/5. (2 3) This implies that the system with fewer units is lighter but can still realize the same workspace size. Despite the fact that other factors are ignored in estimating the weight, it can generally be concluded that the systems of less units have advantages in weight. In this evaluation, it is assumed that the size of the work space is proportional to the number of units by the same multiplier for any system. From the comparison in Chapter 9, this is almost true for systems of no less than 6 units in the case of single gimbal CMGs. This, however, is not true in the case of less that 6 units. Therefore it is better to evaluate some configuration composed of 4 to 6 units. 1

26 3. General Formulation Chapter 3 General Formulation This chapter first defines vectors, variables and parameters of a single gimbal CMG system in an arbitrary configuration, after which a basic mathematical description of several system characteristics are made. These characteristics are the kinematic equation, the steering law, the torque output performance index, and singularity avoidance. The shape of the maximum workspace and singularity problem are described in the next chapter. Similar descriptions for double gimbal systems are given in Appendix A. 3.1 Angular Momentum and Torque A generalized system is considered consisting of n identically sized single gimbal CMG units. The number n is not less than 3 to enable three axis control. The system configuration is defined by the relative arrangement of the gimbal directions. The system state is defined by the set of all gimbal angles, each of which are denoted by θ i. Three mutually orthogonal unit vectors are shown in Fig. 3 1 and defined as follows: g i : h i : c i : where gimbal vector, normalized angular momentum vector, torque vector, c i = h i / θ i = g i h i. (3 1) The gimbal vectors are constant while the others are dependent upon the gimbal angle θ i. Once the initial vectors are defined as in Fig. 3 2, the other vectors are obtained as follows; h i = h i cosθ i + c i sinθ i, c i = h i sinθ i + c i cosθ i. (3 2) The total angular momentum is the sum of all h i multiplied by the unit s angular momentum value which is denoted by h. In this work, H denotes the total angular momentum without the multiplier h: H = Σ h i. (3 3) This relation is simply written as a nonlinear mapping from the set of θ i to H; H = f (θ). (3 4) The variable, θ=(θ 1, θ 2,..., θ n ), is a point on an n dimensional torus denoted by T(n) which is the domain of this mapping. The mapping range is a subspace of the physical Euclidean space and is denoted by H. This space is the maximum workspace. By the analogy of this relation with a spatial link mechanism, this relation will be called kinematics or kinematic equation in this work (see Appendix F). The output torque without the multiplier h is obtained by taking the time derivative as follows. T = dh / dt = Σ h i / θ i dθ i /dt. (3 5) Any additional gyro effect torques generated by the satellite motion are omitted because they are usually θ h i g i h c g θ i h i c i c i Fig. 3 1 Orthonormal vectors of a CMG unit Fig. 3 2 Gimbal angle and vectors 11

27 Technical Report of Mechanical Engineering Laboratory No.175 treated in the overall satellite system dynamics, which includes the CMG system (see Chapter 8). Because the total output torque is a sum of output of each unit, it is also given as, where T = Σ c i ω i = C ω, (3 6) ω i = dθ i /dt, and ω = (ω 1, ω 2,..., ω n ) t. (3 7) The variable ω i is the rotational rate of each gimbal. The vector ω is a component vector of a tangent space of T(n). The matrix C is a Jacobian of Eq. 3 4 and is given by, C = (c 1 c 2... c n ). (3 8) As the unit s angular momentum value is omitted in Eqs. 3 5 and 3 6, the real output is obtained by multiplying h. 3.2 Steering Law The steering law functions to compute the gimbal rates, ω, necessary to produce the desired torque, T com, and is generally given as a solution of the linear equation given in Eq. 3 6: ω = C t (CC t ) 1 T com + (I C t (CC t ) 1 C) k. (3 9) where I is the n n identity matrix and k is an arbitrary vector of n elements. The first term has the minimum norm among all solutions to the equation. The matrix C t (CC t ) 1 is called a pseudo-inverse matrix. The second term, denoted by ω N, is a solution of the homogeneous equation; C ω N =. (3 1) This implies that the motion by this ω N does not generate a torque (T) and keeps the angular momentum (H) constant. In this sense, this term is called a null motion. The null motion has n 3 degrees of freedom because it is an element of the kernel of the linear transformation represented by C. An effective method of calculating a null motion is given in Ref. 22. For example, a null motion of a four unit system is generally given as, ω N = ([c 2 c 3 c 4 ], [c 3 c 4 c 1 ], [c 3 c 1 c 2 ], [c 1 c 2 c 3 ]), (3 11) where [a b c] denotes the vector triple product, a (b c). 3.3 Singular Value Decomposition and I/O Ratio The magnitude of the total output torque is not a simple sum of the output of each unit. An elements of each output, ω i c i, normal to T cancels each other. The ratio of input and output norms, ω / T, can be evaluated by a singular value of the matrix C. The matrix C can be decomposed into a diagonal matrix by two orthonormal matrices, Q (3 3) and R (n n) as follows; σ1.. QCR = σ 2.., (3 12) σ 3.. where σ i is called a singular value of C. As shown in Fig. 3 3, the maximum ratio of the input and output norms is given by the radius of the ellipsoid whose principal diameters are the singular values. Thus, the ω 1 ω 2 (a) Gimbal rate σ 3 ω 3... ω n ω =1 H σ 1 σ 2 (b) Angular momentum ellipsoid Fig. 3 3 Input Output ratio n - sphere 12

28 3. General Formulation size of this ellipsoid represents the performance index of the output torque. The following relations are derived from the fact that all row vectors of C are unit vectors. σ σ σ 3 2 = Trace(CC t ) = n, (3 13) det(cc t ) = (σ 1 σ 2 σ 3 ) 2. (3 14) 3.4 Singularity The steering law function in Eq. 3 9 is invalid at lower ranks of C where the following condition is satisfied: h 3 h 2 H h 1 S Singular Line O A B Angular Momentum Envelope S 1 det(cc t ) =. (3 15) Referring to Fig. 3 4, degeneration of rank implies that all the possible output, T of Eq. 3 6, does not span three dimensional space. Since all the row vectors, c i, of matrix C become coplanar, the output T does not have a component normal to this plane. Let u denote the unit normal vector of this plane and be called a singular vector. It is defined by u c i =, where i = 1, 2,..., n, (3 16) and may also be written in the matrix form as; u t C =. (3 17) O A S B (a) Angular momentum The rank of C does not generally reduce to 1. The rank is unity only when all c i are aligned in the same direction. This can only happen if all the g i are on the same plane, as the case for a roof type system. When the system is singular, one of the singular values reduces to. In this sense, the minimum singular value can be used as a singularity measure. But the determinant, det(cc t ), is also useful as such a measure and is more easily calculated. Figure 3 5 shows two types of singularity of a two dimensional, three unit system. The border of the maximum workspace, termed the angular momentum c 1 c 2 Singular Vector u Fig. 3 4 Singularity condition and singular vector c n S 1 (b) Vector arrangement Fig. 3 5 Typical vector arrangement for a 2D system envelope is clearly singular. The singular H other than this envelope are called internal. 3.5 Singularity Avoidance Any steering law is based on the solution of Eq Among all solutions, the pseudo-inverse solution with no null motion was regarded effective. However, the fact that the pseudo-inverse solution has a minimum norm implies that once the torque vector is nearly normal to the required output then this unit hardly moves. If the required torque maintains its direction, such a unit keeps its state so the system sometimes approaches a singular state. In order to avoid such a situation, singularity avoidance is usually included in the steering law. 13

29 Technical Report of Mechanical Engineering Laboratory No Gradient Method A system containing more than three units possesses null motion redundancy. Freedom in determining null motion can realize singularity avoidance while keeping the output torque exactly equal to the command. The gradient method is a general method in which some objective function is maximized. The following formulation of a gradient method is taken from Ref. 21. The objective function, W(θ), is chosen as a continuous function of θ. It is zero in the singular state and otherwise positive. The dependence of W on CMG motion is: where W = Σ ξ i ω i, (3 18) ξ i = W/ θ i. (3 19) In order to obtain the objective function extremum, the motion ω should be determined so that W is positive. This W has two parts, one given by the pseudo-inverse solution and the other by a null motion. The first depends on the command torque T com, while the latter depends on the selection of a null motion. Though the first part cannot be changed, the latter can be freely determined. The latter part is evaluated as follows; W N = ξ t (I C t (CC t ) 1 C) k. (3 2) It is easily observed that the matrix (I C t (CC t ) 1 C) is semi-positive symmetric. If the vector k is selected as: k = k ξ, where k >, (3 21) then W N becomes a semi-positive quadratic form. Thus, the null motion by this k results in non-negative W N, so it is expected that singularity is avoided. Various objective functions have been proposed, such as: pyramid type single gimbal CMG systems, various simulations showed that a gradient method is not effective. Details of this problem is described in Chapters 4, 5 and Steering in Proximity to a Singular State There is no solution to Eq. 3 6 in a singular state except when T com is orthogonal to the singular vector u. Even when T com is normal to u, the solution is not given by Eq. 3 9 because the linear equation is mathematically singular. A generalized solution can be obtained which is the exact solution when T com is normal to u otherwise minimizes the output error. The minimum error is realized when the output is equal to the projection of the torque command onto the plane normal to the singular direction (Fig. 3 6). Such motion is given as 22) : ω = C t (CC t + k uu t ) 1 T com. (3 22) Derivation of this is explained by supposing that there is a virtual CMG unit whose torque vector c equals u. Another method called the SR (Singularity Robust) inverse steering law is proposed as a smooth extension of this 41). This method minimizes the weighted sum of the input norm, ω, and the norm of the error. The SR solution is given as: ω = C t (CC t + W) 1 T com, where W is a n n matrix. (3 23) In both methods, the solution is zero if the command, T com, is either zero or parallel to the u direction. This method, therefore, cannot always guarantee avoidance of a singular state nor can it escape from one. Moreover, this kind of control is effective only if the attitude control is not totally degraded by the error in torque. Details are described in Section 7.2. Possible Output T com (1) (det(cc t )) 1/2, 21) (2) min(σ i ), 36) (3) min(1/ d i ), where d i is a row vector of the matrix C t (CC t ) 1, 35) (4) Σ i,j c i c j 2, 27). c 1 c 2 c n u This gradient method has been successful for double gimbal CMG systems 21). However, in the case of Fig. 3 6 Steering at a singular condition 14

30 4. Singular Surface and Passability Chapter 4 Singular Surface and Passability Angular momentum vectors in a singular condition form a smooth surface which includes the angular momentum envelope. This chapter first summarizes the geometric theory of the singular surface of a general single gimbal CMG system by following the research work in Ref. 22. It includes a definition of a singular surface, a mapping from a sphere to the surface, and techniques for drawing the surface by computer calculation. By using these techniques, the workspace is visualized for various system configurations. Also, geometric characteristics such as Gaussian curvature of a singular surface is defined. The passability of a singular surface is then defined. The existence of an impassable surface explains why most steering laws fail to generate output starting from certain initial states. A gradient method works well for avoiding passable singular points but not for avoiding impassable ones. The passability can be determined by the curvature of the singular surface. It is demonstrated that any independent type system has an internal impassable surface while multiple type systems of no less than six units have no internal impassable surfaces. 4.1 Singular Surface Continuous Mapping ε i = sign( u h i ). (4 1) Thus there are 2 n combinations of singular points for the given direction u. This combination is denoted by ε or by a set of signs, such as { }. For the given singular direction u and the given set of signs, each torque vector in the singular condition is determined by: c Si = ε i g i u / g i u. (4 2) From this point, variables subscripted by S denote singular point values. The total angular momentum H S is obtained as follows: H S = Σ ε i (g i u) g i / g i u. (4 3) This defines a continuous mapping from u to H S while the ε i are fixed as parameters. The domain of u is a unit sphere except ±g i direction, because the denominator of Eq. 4 3 is zero when u = ±g i. Thus H S with fixed ε i form a two dimensional surface with u covering this sphere. This surface is denoted as S ε. If all the ε i are reversed and the vector u is changed to u, H S remains the same. This implies that the surface of {ε i } and the surface of all the ε i reversed are identical. For example S { + +} is the same as S {+ }. One may thus suppose that no less than half of the ε i are positive. Thus, the number of different surfaces is 2 n 1. In case that u = ±g i, any state of this ith unit satisfies Let s examine all the singular points and their H vectors. First, an independent type system is assumed in the following discussion. The torque vectors, c i, satisfy the condition given by Eq.(3 16) when the system is singular. On each singular point, a singular vector u is defined. As a reverse relation of this, singular points are obtained from a given u vector. Given any singular vector u, there are two possibilities of singularity condition for each unit as h S and h S in Fig The two cases are distinguished by the following sign variable; h S g c S u ε = 1 ε = 1 c S h S Fig. 4 1 Vectors at a singularity condition 15

31 Technical Report of Mechanical Engineering Laboratory No.175 z g 1 g 2 x y g 3 g 4 (a) Lattice points of the unit sphere of vector u Unit Circle C 1 g Envelope 1 z x y the singular condition. As the vector h i rotates about g i, these singular H form a unit circle which appears as a hole or a window of the surface S ε as shown in Fig. 4 2 (b). As there is a hole for each g i, or g i, direction, the surface has 2n holes in total. Surfaces of different {ε i } are connected by these unit circles (for example, C 1 in Fig. 4 2 (b) and (c)). Thus all the surfaces form a closed surface. This closed surface is called a singular surface. It may be noted that the same kind of continuous mapping is defined from u to θ S with all the θ S forming a two dimensional surface in the n dimensional torus of θ. Such a surface, however, is not termed a singular surface in this paper. An independent type system is assumed in the above discussion. In the case of a multiple type system, the number of different singular surface is 2 m 1 where m is the number of groups. Each surface has 2m holes of diameter of several values which is determined by the number of units in a group and sign ε. In case that u = ±g i, any state of units of this group satisfies the singular condition. Thus, all singular H of this u form a circular plate which fills the hole. Another singular surface of different sign connects to this plate by a circle of different diameter Envelope (b) Singular surface of all sign positive denoted by S {++++} Unit Circle C 1 Envelope Portion Internal Portion Unit Circle C 2 (c) Singular surface of one minus sign denoted by S { +++} Fig. 4 2 Examples of the singular surfaces for the pyramid type system. Each dot of Figs. (b) & (c) corresponds to the lattice point of Fig. (a). The unit circle indicated by C 1 connects two singular surfaces S { } & S { + + +}. Other circles of the surface S { + + +}, C 2 for example, are connections to other singular surfaces such as S { + +} The angular momentum envelope, which is the border of the maximum workspace, is most definitely singular. The surface corresponding to all ε i positive is clearly a part of the envelope. Surfaces with one negative sign which is connected to this surface by the holes share the envelope surface in the case of an independent type system. The envelope of a multiple type system consists of a singular surface of all positive signs and circular plate which fills 2m holes 22). The one negative sign surfaces do not share the envelope surface and is fully internal. The singular surface of a M(2, 2) roof type system shown in Fig. 4 3 is part of an envelope of all positive signs. There are four circular holes of diameter 2. The circular plates filling these four circles share the envelope. The singular surface of one negative sign is connected at the center of these plates Visualization Method of the Surface The singular surface and envelope are visualized by taking θ at each lattice points of the unit sphere and calculating the angular momentum using Eq

32 4. Singular Surface and Passability z x y Fig. 4 3 Envelope of a roof type system M(2, 2). Dots are obtained from Eq. 4 3 for lattice points of a u sphere with all positive signs. Circles are filled by plates. Figures 4 2 and 4 3 are such examples. A singular surface and an envelope may also be visualized using various cross sections. The following inverse mapping theory 22) is available to obtain a cross section of the singular surface. Inverse Mapping Theory Suppose that θ is constrained singular and V is an arbitrary vector normal to u. If the differential dh along the singular surface satisfies, dh = V u, (4 4) then the differential of u is given by du = κ ( CPC t V) u, (4 5) where κ is the Gaussian curvature of the singular surface, which is described in Section The matrix P is a diagonal matrix whose nonzero element P ii is given by: P ii = p i = 1 / (u h i ). (4 6) Using this theory, a cross section of the singular surface is calculated by the following procedure. First, obtain a singular point on the cross sectional plane and its u vector by some means. Second, obtain dh on the intersection of the surface tangential plane and the cross sectional plane. Third, obtain V by Eq. 4 4 and du by Eq. 4 5 after which dθ is obtained by the relation dθ = p i c i du (Appendix B). Finally, H on the cross sectional plane is obtained by numerical integration of dθ. Figure Unit Circle C 1 Fig. 4 4 Cross sections of a singular surface of the pyramid type system. The outermost unit circle is the same as C 1 in Fig The other lines are cross sections of planes orthogonal to the gimbal axis g is such an example. The proof of this theory is given in Appendix B Differential Geometry Geometric theory presented in Ref. 22 formulated fundamental forms of the singular surface and clarified geometric characteristics. Other than Gaussian curvature, details are given in the original paper Tangent Space and Subspace Suppose that θ is on a singular point. The differential dθ is a tangent vector of the θ space. The following three subspaces are defined in the tangent space of the θ space 32) : Θ S : Singularly constrained tangent space of the θ space (two dimensional). Θ N : Space of null motion, i.e., the null space of C (n 2 dimensional). Θ T : Complementary subspace of Θ N (two dimensional). The solution given by Eq for all T com belongs to this space. The elements of these three subspaces are denoted by dθ S, dθ N and dθ T. These are illustrated in Fig. 4 5 for a two dimensional three-unit system, for example. The general bases of subspaces are given in Appendix B Gaussian Curvature The Gaussian curvature, κ, of a singular surface is 17

33 Technical Report of Mechanical Engineering Laboratory No.175 Singular Direction u h 1 h 2 B (a) Singular state B h 3 dθ S =(,, ) t (c) Θ S B dθ N1 =(,, ) t B dθ N2 =(,, ) t (b) Null motion Θ N B dθ T =(,, ) t (d) Θ T By using the relation as, H / θ i = c i, 2 H / θ i θ j = h i, if i = j otherwise,(4 9) the difference H as H = H(θ S +dθ) H(θ S ), (4 1) is expressed as, H = Σ c i dθ i 1 / 2 Σ i h i (dθ i ) 2, (4 11) where the third order terms are omitted. The first order difference is a linear combination of c i and has no component in the u direction. On the other hand, the second order term may have a component to this direction. More specifically, H u = 1 / 2 u ( Σ i h i (dθ i ) 2 ), = 1 / 2 Σ i (dθ i ) 2 / p i. (4 12) given by: Fig. 4 5 Infinitesimal motion from a singular point of 2D system. Four independent motions, dθ N1, dθ N2, dθ S and dθ T, are members of three subspaces, Θ N, Θ S and Θ T. 1 / κ = 1/2 Σ i Σ j p i p j [c i c j u ] 2. (4 7) The proof of this is detailed in Appendix B.2. The sign of Gaussian curvature has an important role in determining the following passability of the surface. 4.3 Passability Quadratic Form Suppose that the system state is singular, that is, θ S is a singular point and H S is on the singular surface. It is instructive to examine an infinitesimal change in θ from this singular point and the resulting infinitesimal change in H. A second order Taylor s series expansion of H(θ) in the neighborhood of the θ S is given by: H(θ S +dθ) = H(θ S ) + Σ i H / θ i dθ i + 1 / 2 Σ i Σ j ( 2 H / θ i θ j )dθ i dθ j + O(dθ i 3 ). (4 8) This may also be expressed in matrix form as: H u = 1 / 2 dθ t P 1 dθ. (4 13) This is a quadratic form of dθ i. If any dθ are decomposed as follows, dθ = dθ S + dθ N, (4 14) the quadratic form (4 12) is also similarly decomposed: H u = 1 / 2 dθ S t P 1 dθ S 1 / 2 dθ N t P 1 dθ N. (4 15) This is derived by the fact that P 1 dθ S is an element of Θ T hence dθ N t P 1 dθ S = (See Appendix B.1). Let Q S and Q N denote the two quadratic forms on the right of Eq. 4 15: Q S = 1 / 2 dθ S t P 1 dθ S, Q N = 1 / 2 dθ N t P 1 dθ N. (4 16) The vectors, dθ S and dθ N, are elements of the tangent subspace Θ S and Θ N, and they can be represented by using bases of each subspaces: dθ S = φ 1 e S1 + φ 2 e S2, dθ N = ψ 1 e N ψ n-2 e Nn-2, (4 17) where e Si and e Ni are bases of Θ S and Θ N. These expressions are expressed simply as, dθ S = E S φ, where E S : n 2, φ : 1 2, 18

34 4. Singular Surface and Passability dθ N = E N ψ, where E S : n n 2, ψ : 1 n 2. (4 18) Substituting these into two quadratic forms in Eq results in the following: Q S = 1 / 2 φ t E S t P 1 E S φ, Q N = 1 / 2 ψ t E N t P 1 E N ψ. (4 19) The first quadratic form is of order 2 and expresses the curvature of the singular surface, because dθ S represents a motion on the surface. The second quadratic form is of order n 2. By the definition, this Q N is H(dθ N ) u. Therefore, if this quadratic form is not zero, this null motion moves the vector H away from the surface as shown in Fig Note that the decomposition in Eq is not always possible, for example if Cdθ S =. This case, however, can be treated by similarity with another neighborhood. H( θ S ) H Q N Fig. 4 6 Second order infinitesimal motion from singular surface Signature of Quadratic Form Any quadratic form, Σa ij x i x j, can be transformed to Σb i y i 2 by using a regular transformation from {x i } to {y i }. The set of two numbers of positive and negative b i are called signature of the quadratic form. Any quadratic form has a unique signature, that is, the signature does not depend on the transformation, as is Sylvester s law of inertia. By the signature, a quadratic form is categorized as definite, semi-definite or indefinite. Definite form have only the same signs, while an indefinite form has both positive and negative signs. A semi-definite form has only the same sign but their number is less than the order of the form. If the quadratic form, Q N is definite or semi-definite, it implies that any motion away from the surface is u Q S Singular surface H(θ N ) limited to a certain side of the surface. Thus, no motion from this side of the surface to the other side is possible. With indefinite forms, some motions result on one side of the surface while others appear on the other side. The signature is the characteristics of the form itself, independent of the variables which is dθ N in this case. Thus, any singular point is categorized by this characteristics such as definite or indefinite Passability and Singularity Avoidance Since the quadratic form and its derivatives are continuous with respect to θ, its eigenvalues which determine the signature are also continuous along the surface. This implies that if a point has a definite form then its neighborhood likewise does. The points of definite form make up a certain area of the surface, near which it is not possible to pass from one side to the other if θ is in the neighborhood of this singular point. In this sense, such an area is called impassable, while that of an indefinite form is termed passable. This notation follows that of Tokar 26) who pioneered this work. Other notation used in other references are elliptic/ hyperbolic 28, 65) and definite /indefinite 32). Another aspect of this form category is as follows. If the singular point is passable, i.e., having an indefinite form, a certain value of dθ N results in a zero value of the quadratic form. The motion by this dθ N keeps H on the singular surface but θ does not stay singular. This implies that escape from the passable singular point is possible while keeping H the same. On the contrary, no motion can keep H at an impassable singular point. The internal singular point of a two dimensional system is passable. Figure 4 7 shows two motions in opposite directions at the singular point. The null motion B dθ =dθ N1 dθ N2 =(,, ) t H u < Singular direction u B dθ =dθ N1 + dθ N2 =(,, 2 ) t H u > Fig. 4 7 Possible motions in both direction of u at a singular point. This is the case of an internal singular state of a 2D system. Infinitesimal null motions dθ N1 and dθ N2 are defined in Fig

35 Technical Report of Mechanical Engineering Laboratory No.175 in Fig. 4 5 (b) allows the system to escape from the singular point while keeping H the same. From the above discussion, it is clear that no steering law can avoid an impassable singularity if the command is to approach the surface and the initial θ is in the neighborhood of the impassable singular point. On the other hand, a steering law such as the gradient method is effective in avoiding a passable singularity because escape from such a singular point is always possible even when H is on the singular surface Discrimination Passability of a surface is defined by the signature. The following discussion gives a discrimination method of this by the sign {ε i } and the curvature of the surface. Equations 4 18 represents a basis change for each subspace. As mentioned above, the signature is conserved by any basis change. Thus the signature of the total quadratic form is conserved and is simply obtained by the signs of p i, which is ε i, because of Eq Thus, passability which is defined by the signature of Q N, is determined by the total signature and the signature of Q S. The signature of Q S indicates characteristics such as concavity/convexity of the surface because this quadratic form expresses curvature of the singular surface. Thus the following three conditions for an impassable surface are obtained in terms of the sign and the curvature of the surface 32). From these results, passability is discriminated for any singular point as follows. First obtain the sign {ε i }. Reverse all the sign if required so that the number of negative signs is less than that of positive signs. If they are all positive, this surface is impassable. If more than two signs are negative, this surface is passable. The remaining cases may correspond to the above two cases (2) or (3). The next procedure is to calculate the Gaussian curvature κ by Eq If there is only one negative sign, it is impassable when κ is negative otherwise passable. If there is two negative signs and κ is positive, we need additional calculation to determine passability. (a) ε = { } u u Singular Surface Condition (1) ε={ }. Both Q N and Q S have only positive signs. This singular point is on the surface S { } which is a part of the envelope and is trivially impassable. Gaussian curvature is positive and the surface is convex to u. (Fig. 4 8(a)) Condition (2) All the ε i but one are positive and the signature of Q S is { +}. This surface is partially on the envelope and impassable. Some part of this impassable surface is possibly inside the envelope. The Gaussian curvature is negative and the surface is a hyperbolic saddle point. (Fig. 4 8(b)) Condition (3) All the ε i but two are positive and the signature of Q S is { }. The surface is fully inside the envelope. The surface is concave to u and the Gaussian curvature is positive. Note that positive κ is not a necessary condition for this because there is a possibility that the signature of Q S is {+ +}. (Fig. 4 8(c)) (b) ε = { } Fig. 4 8 Local shape of an impassable singular surface. A side of the surface in u direction is allowed H region while another side to u direction is unreachable through the surface. (a) is a concave part of the envelope, (b) is a saddle point of the envelope and internal surface, and (c) is convex and fully internal. u (c) ε = { } 2

36 4. Singular Surface and Passability By removing one unit whose sign is negative and by checking the above condition (2) for this subsystem, passability of the original system is determined. This is proven in Appendix B Internal Impassable Surface Impassable Surface of an Independent Type System In the case of an independent type system, the singular surface with all ε i positive except one joins smoothly into the envelope, as depicted in Fig Because the surface and the curvature are continuous, any impassable portion also goes into the envelope. Thus, any independent type system has an impassable surface distinct from the envelope. Figures 4 9, 4 1, 4 11 and 4 12 show examples of internal impassable surfaces, along with the envelopes. In Fig. 4 9 it is clear that although the symmetric six unit system S(6) has internal impassable surfaces, they are very near the envelope. On the contrary, Figs. 4 1, 4 11 and 4 12 show that the skew five unit system and the S(4) system have internal impassable surface considerably further inside their envelope. If the workspace of these systems is defined such that it does not include these impassable surfaces in order to assure singularity avoidance, it becomes much smaller than the workspace given by the envelope Impassable Surface of a Multiple Type System The analysis of a multiple type system is quite different from the above discussion. For multiple systems, every surface corresponding to all ε i positive except one are totally inside the envelope. It is instructive to examine passability conditions (2) and (3) of section Here, each variable is subscripted by the group number, because all the variables of the same group can be represented by only one member. The number of units in the ith group is denoted by m i. z z x y x (a) Envelope. Singular surface S { } (a) Envelope. Singular surface S { } Envelope Internal Part z Envelope Internal Part z x Unit of H x Unit of H (b) Impassable surface S { } (b) Impassable surface S { } Fig. 4 9 Impassable surface of S(6) Fig. 4 1 Impassable surface of Skew(5) with skew angle α =.6 rad. 21

37 Technical Report of Mechanical Engineering Laboratory No.175 z the right is the Gaussian curvature of the subsystem envelope excluding the ith group, and thus is positive. The first term is zero if m i =2 and otherwise positive. Thus the overall curvature is positive and condition (2) is not satisfied. Envelope x (a) Envelope. Singular surface S { } x Internal Part z Condition (3) Suppose that two signs are negative. If the two units corresponding to these two negative signs belong to different groups, condition (3) simply results in condition (2) of the subsystem by removing one of the two units, so the condition is not satisfied. Suppose then that the two units corresponding to the two negative signs belong to the same group, this being the ith one. If the unit number, m i, is larger than two, the above reasoning is applied and the condition is not satisfied. If m i =2, 1 / κ = 1/2 ( p i p i )Σ j p j [c i c j u ] 2 + Σ j i Σ k i p j p k [c j c k u ] 2. (4 21) If the overall system is definite, the subsystem without one unit of the ith group is also definite and condition (2) is satisfied for this subsystem, so 1 / κ is negative. In this case, Eq in its entirety is also negative, so the condition is not satisfied. (b) Impassable surface S { } Fig Impassable surface of another Skew(5), with skew angle α = 1.2 rad. Condition (2) Suppose that only one of the signs is negative and it is in the ith group. The Gaussian curvature of (4 7) is written as: 1/κ = 1/2( p i +p i +...+p i )Σ j p j [c i c j u ] 2 +Σ j i Σ k i p j p k [c j c k u ]2, (4 2) where p i and all the p j are positive. The second term on The discussion presented here does not hold for systems of fewer units, such as M(2,2). Further details are examined in Appendix C and the results lead to the following conclusion Minimum System The conclusion is as follows. Any multiple type system with no less than six units has no impassable singular surface other than the envelope, while any independent type system has internal impassable surfaces. 22

38 5. Inverse Kinematics Chapter 5 Inverse Kinematics The impassable surfaces defined in the previous chapter cause steering law problems. It is possible to leave this problem unsolved and define a workspace which excludes the impassable surfaces, but the resulting space would be much smaller in the case of a four or five unit system. Impassability, however, is defined locally only in the neighborhood of an impassable singular point. There is a possibility to avoid an impassable situation by using some kind of global control. For this, a geometric approach was taken in order to understand the CMG control qualitatively by ignoring the factor of time. The sequence of torque commands is represented as a trajectory of the angular momentum vector, while the possible gimbal angles are represented by a manifold. By using equivalence relations of manifolds and their connections, conditions necessary for continuous control are formulated. 5.1 Manifold A steering law is a method to obtain gimbal rates which corresponds to a given torque command. If we ignore the factor of time, the steering law is regarded as a method to obtain gimbal angles by a given change of the angular momentum. This is the reverse relation of the kinematic equation 3 4. The (forward) kinematics is a one-to-one mapping but the reverse relation, which is called an inverse kinematics, is generally a one-tomulti mapping. Therefore, possible θ having the same H is given by an inverse image of this mapping. The inverse image from H to θ is a set of sub-spaces disjoint to each other. Supposing that a sub-space has no singular state, an n 3 dimensional tangent space is defined at each point of this space as a linear space of null motion. Thus, this sub-space is a n 3 dimensional manifold. Supposing that a sub-space has singular points, no tangent is defined there, but even in this case, tangent spaces are defined at all other points of this space. Thus, this sub-space is nearly the same as a manifold and in this work will be termed a singular manifold. The inverse image is a sum of manifolds and singular manifolds, which are denoted by M i and M Sj respectively. Note that this manifold should be termed rather null motion manifold or self-motion manifold 6). In this work however, no other manifold is used and it is simply called manifold. The shape of manifolds in the neighborhood of a singular point is characterized by the quadratic relationship given by Eq Suppose that H is in the neighborhood of a singular surface where H = H S + eu. By the same discussion as Eqs. from 4 1 to 4 15, possible θ in the neighborhood of the singular point, θ S, satisfies the following quadratic relation; 1 2 Σ (dθ Ni ) 2 p i e, where θ = θ S + dθ N. (5 1) In this equation, the motions, dθ N, is a tangent vector at the singular point θ S. In the case of an impassable singular state, this quadratic form is definite, so this manifold resembles a super-ellipsoid. The quadratic form of an impassable singular point is indefinite, so the shape of the manifold resembles super-hyperbolic surfaces in the neighborhood of this singular point. This is illustrated in Fig. 5 1 for a four unit system, for which the manifolds are loops in the four dimensional torus. M S (H S ) = {θ S } M (H S +e u) M (H S +2 e u) (a) Ellipsoidal manifolds around an impassable singular point. M 1 (H S e u) θ S M S (H S ) M (H S +e u) M 1 (H S +e u) (b) Manifolds crossing near a passable singular point. Fig. 5 1 Manifolds in the neighborhood of a singular point. These manifolds are one-dimensional loops if the number of units is four. 23

39 Technical Report of Mechanical Engineering Laboratory No Manifold Path H path Domain As H changes continuously, each manifold changes its shape continuously as shown in Fig A manifold may deform its shape into a point when H crosses an impassable surface and may bifurcate when H crosses a passable surface. These continuous change of a manifold can be simplified as a continuous path in the manifold space, where each manifold is regarded as a point (Fig. 5 3). If we need an exact definition of manifold continuity, it can be made based on the distance d between manifolds which is defined as follows; d(m A, M B ) = max( min( θ φ ); θ M A ) ; φ M B ), (5 2) where M A and M B are two manifolds and the norm θ φ is defined appropriately in the gimbal angle space. By this definition, a manifold becomes discontinuous at a bifurcation point. The meaning of a continuous manifold path can be thought of in the following terms. If the manifold (M 1 in Fig. 5 3 for example), including an initial θ, is on a continuous manifold path for a given H path (the path H H 1 in Fig. 5 3), then any θ of the manifold on the other side of the path (M 2 in Fig. 5 3) can be reached by some continuous steering method using an appropriate null motion, while any θ of another manifold (M 4 in Fig. 5 3 for example) cannot be reached. If the manifold path bifurcates, path selection (from M 3 either to M 4 or to M 5 ) depends on the null motion hence on the steering method. If the manifold path including the initial θ terminates somewhere for a given H path (θ S2 for the path H 1 H 2 in Fig. 5 3 for example), no steering method can realize this motion. 5.3 Domain and Equivalence Class The angular momentum space is divided into several domains by one or more continuous singular surfaces. These will hereafter be simply termed domains. Let each domain have no singular surface inside and its border be a set of singular surfaces. Each domain will be denoted by D i. Any continuous path of H inside a domain corresponds to a finite number of continuous manifold paths with neither bifurcation nor termination. Thus, the number of manifolds for each point in the domain is constant. H H S e u H S H S +e u H M M 1 M 2 H S e u H S H S +e u θ θ M M 1 (c) Continuous change of manifolds in the neighborhood of impassable H. (d) Continuous change of manifolds in the neighborhood of passable H. Fig. 5 2 Continuous change of manifolds corresponding to H path across a singular point. H 2 H 1 H M 6 M 2 M 1 M 4 θ S2 M 3 θ S1 M 5 Manifold Paths D 1 D 2 D 3 Fig. 5 3 An example of a continuous manifold path. A passable singular point θ S1 is a bifurcating point and an impassable point θ S2 is a terminal of the path. A manifold equivalence relation is defined as follows: definition: Two manifolds of a domain are considered equivalent when there is a path from one manifold to the other which corresponds to an H path inside the domain. All the equivalent manifolds form a domain in the m = 2 m = 3 m = 2 24

40 5. Inverse Kinematics Singular Surface H path H A Equivalence Class G 1 M A M B Manifold Path θ A M A θ B M B D 2 H B θ B DomainD 1 M B G 2 H space Manifold space θ space Fig. 5 4 Relations between H space, manifold space and θ space. manifold space which is isomorphic to the original domain of H. As the representation of this set of equivalent manifolds, an equivalence class is defined. The number of the classes is called the order of the domain. Let G i and m denote the class and order, respectively. The order of each domain is obtained in a step by step fashion. The outermost domain next to the envelope is of order 1. Two domains facing each other have an order difference of 1, because only one manifold path either bifurcates or terminates. The definition of equivalence class may be extended to different domains connected by a certain H path. Classes of different domains are termed equivalent when the H path connecting the domains corresponds to a continuous manifold path which includes these classes. In Fig. 5 3 for example, the path from M 1 to M 6 via M 2 is continuous through domains D 1, D 2 and D 3, so these manifolds and classes on this path are equivalent. On the other hand, M 3 in domain D 1 is not equivalent to any manifold of the domain D 2. The relationship between manifold and class is illustrated in Fig The equivalence among two domains implies that the manifold path is continuous. If there is bifurcation on the manifold path, classes are not equivalent. In this case, classes can be termed connected, because a continuous θ path can be chosen. 5.4 Terminal Class and Domain Type For continuous steering, it is important to know whether or not each class has equivalent or connected classes for any H path exiting of the domain. The class is called a terminal class if there is an H path exiting to a neighbor domain which results in termination from this class. By this definition, the class of the domain just inside the envelope is not a terminal class even though it terminates on the envelope, because an H path exiting the envelope has no meaning. Each domain is classified into one of the following three types by considering the order and number of terminal classes, k: Type 1: m = k >, Type 2: m > k >, Type 3: k =. The outermost domain nearest the envelope is Type 3 by the above definition. Type 3 domains have no terminal class, and as such, no difficulty arises as far as steering inside of itself and its neighboring domains. 5.5 Class Connection Class connection around Type 1 and 2 domains is described by examples in the following sections. A Type 2 domain is examined first. By introducing a graph of class connection, a Type 1 domain is next examined Type 2 Domain The following examples are obtained by computer calculations for the S(4) system. Figure 5 5(a) shows a part of a cross section of a singular surface near the envelope. The curved triangle is where the surface was cut, and the bold line indicates an impassable edge. This triangle divides the H space into two parts; the domain outside is denoted by D and the domain inside by D 1. Domain D is a Type 3 domain just inside the envelope 25

41 Technical Report of Mechanical Engineering Laboratory No.175 DomainD 1 (Type 2) (a) Cross section of an internal singular surface and H path H H θ 2 Impassable Surface S R Passable Surface S 1 θ 2 θ 1 B Bifurcation at A θ 1 (b) Manifolds for path PQR Impassable Singular Point (c) Manifolds for path QCS C Q S A M 2 M 1 P M 2 G 2 M 1 Bifurcation at B DomainD Fig. 5-5 Domains and manifolds of the pyramid type system A cross section of a Type 2 domain and manifolds for several points are obtained by computer calculation. Manifolds are drawn as a two dimensional projection on (θ 1, θ 2 ) coordinates in a quadrilateral whose right and left edges (space 2π apart) are regarded as the same points. C Q P R Q P S and D 1 is a Type 2 domain of m = 2. Figures 5 5(b) and (c) show manifolds of some points in these two domains. The manifolds are drawn using a two-dimensional projection of the skew coordinates. While the actual θ space is a four-dimensional torus, the illustration is made in a quadrilateral whose edges parallel to each other are regarded as the same points. Consider now the angular momentum path PAQBR, which traverses the domain D 1 in Fig. 5 5(a). There is no equivalent class for this path which traverses three domains, from D back to D via D 1. The manifold path bifurcates at points A and B. Though the classes are not equivalent, they are connected and continuous steering is very much possible by some gradient method because such bifurcation points (passable singular points) are easily avoidable. Consider next the path PAQCS penetrating the impassable surface. Figure 5 5(c) shows that the manifold M 2 is of the terminal class in this domain. Once this class is selected when going into D 1 from P through A, there is no continuous way to reach another manifold, such as M 1, so steering fails. On the contrary, if manifold M 1 is selected, continuous motion egressing this domain along QCS is strictly guaranteed without any special steering methods. In this case, the main question is how to select an appropriate class. The above discussion is more easily understood by utilizing a class connection graph, as shown in Fig. 5 6(a). The jagged lines represent the cross section of a singular surface obtained from numerical computation. Various circles drawn inside each domain represent equivalence classes of domains and the color of the circles (white and gray) indicates whether they are a terminal class or not. Circles drawn on the edge of the domain represent a class of singular manifolds. Curved lines connecting the circles represent class connections. This connection graph makes it easily understood that all classes would be connected even after the omission of the terminal class G 2. Therefore, necessary class selection is unique for any H path crossing this domain Type 1 Domain Figure 5 6(b) shows another cross section of a singular surface and a class connection graph. The triangular domain D 3 is Type 1. In this case, no class remains if we omit the terminal classes of this domain. Thus, there is no contiguous connection of classes for an H path such as FG. This implies that no steering law can realize this angular momentum path. On the other 26

42 5. Inverse Kinematics Domain D 3 (Type 1) G 2 G 1 Domain D 1 (Type 2) Domain D 2 (a) Type 2 domain Domain D 3 (Type 1) (c) Another type 1 domain Domain D 4 (Type 1) D D' Impassable surface (Bold) E E' Passable surface Class of manifold F G Class connection Bifurcation Termination H path (b) Type 1 domain Envelope part Fig. 5 6 Class connection graph around domains hand, continuous motion along the path DE is possible by selecting the appropriate class prior to bifurcation. This selection, however, is not always effective. If another path, D E for example, is taken, the class to be selected is different. This implies that there is no unique selection rule for entering a Type 1 domain. Figure 5 6(c) shows another Type 2 domain example including various class connections. The above discussion also holds in this case Class Connection Rules The class connections in Fig. 5 6 can be derived without calculation of manifolds but by considering continuity. In the cross section shown in Fig. 5 7, the sharp point R represents the borderline between an impassable and a passable sides of the singular surface. Since the singular point θ along the surface is continuous, passable points θ P and θ Q smoothly change to impassable points θ S and θ T. This implies that after bifurcation by the passable surface, one of the two manifolds must terminate at the impassable surface. Thus, class connections such as those drawn in Fig. 5 6(a) are general for this type of domain even if its order is not 2. Suppose that two impassable surfaces cross as shown in Fig Singular points are continuous along each surface. Manifolds M P and M Q which are the terminal manifolds of each surface are equivalent and manifolds M R and M S also. Therefore, there can be no connection between two manifolds of the different group, M P and M R for example. Thus, class connections such as those drawn in Fig. 5 6(b) are derived. With increased complexity in surface crossings however, finding class connections is more difficult. 27

43 Technical Report of Mechanical Engineering Laboratory No.175 Impassable Surface Continuous Steering over Domains (a) H space (b) θ space T A Passable Surface M R θ R M S M Q M T M A1 M P S Q P θ Q θ S θ T θ P R M A2 From the above two examples, the following general facts are observed. (1) Continuous steering around a Type 2 domain depends upon manifold selection prior to bifurcation. If any class other than a terminal class is selected, continuous control is guaranteed. (2) Selecting a manifold other than the terminal classes is impossible while entering a Type 1 domain. (3) Some paths of H which cross a Type 1 domain do not have a connected manifold path. The item (3) implies that there is no continuous θ path for a certain H path. The item (2) implies that even if a continuous θ path exists for any given H path, real-time steering is not guaranteed when H path is not given beforehand. Those two items, therefore, implies that no steering law can maintain continuous steering over the entire work space if the system contains Type 1 domains. On the other hand, an impassable surface of a Type 2 domain does not cause any problem if an appropriate manifold is selected before bifurcation as the item (1). Fig. 5 7 An illustration of class connection rule (1). M P M Q R Impassable Surface Q T S (a) H space P Lines of Singular Point θ T1 (b) θ space M R θ T2 M S Fig. 5 8 An illustration of class connection rule (2). If two impassable surfaces cross each other, both terminal classes are different Manifold Selection There has been no simple method to select an appropriate manifold prior to bifurcation. Although a gradient method avoids a passable singular point (a bifurcation point), judicious manifold selection depends on the control values of θ before bifurcation. The gradient method is unsuitable because of the following reason. The objective function of a gradient method is defined zero at a singular point and otherwise positive. Thus the singular point is the minimum of the objective function along the manifold. There must therefore be local maxima on both side of the singular point, such as at A and B in Fig Knowing that the objective function is continuous, the manifolds before and after bifurcation have local maxima A, B, A, and B in the neighborhood of A and B. The gradient method only maintains the local maximum and its motion may be either A AA or B BB. Even if one of the manifolds after bifurcation belongs to a terminal class, this method can not move θ from one maximum point to the other. If sufficient time computing power were available, a method like a path planning 42) could be utilized (See Chap. 8). If a number of possible H paths of a certain length are assumed, calculations along those paths may then be carried out in order to determine whether there 28

44 5. Inverse Kinematics Passable Singular Point Terminal Class A Impassable B" Passable V A" B A A' Local Minimum B' B (a) Cross section orthogonal to g i direction. C Fig. 5 9 An illustration of motion by the gradient method. A, A', A", B, B' and B" are the local maxima of an objective function. Motion may be either A'AA" or B'BB" in the neighborhood of the passable singular surface. θ 2, θ 3, θ 4 Manifolds A θ 1 2 are any bifurcations or intersections with impassable surfaces. It would then be possible to determine an appropriate motion. For this strategy, a question still remains whether such manifold selection can be consistent for all possible H path. This will be discussed in Chapter Discussion of the Critical Point The above discussion pertains to manifolds and classes within domains. An arbitrary angular momentum point not on a singular surface is discussed. It was assumed that anything on the singular surface and on the singular manifold is qualitatively the same as that in the H neighborhood and in the neighbor manifold. There is however some exceptions. If the H path starts at an intersection of singular surfaces, there is a problem of manifold selection. Three triangular domains in Fig. 5 1 are Type 2 of order 2. Having the same kind of class relations as the Type 2 domain in Fig. 5 6(a) means any trajectory across one of them can be continuously realized by an appropriate control. However, if one were to commence at the crossing point V, the possibility of selecting a terminal class cannot be omitted once the initial θ is selected on the manifold. This situation is depicted in Fig. 5 1(b). 5.6 Topological Problem A steering law can be represented by a mapping from the H space to the θ space. It would be nice if there is a continuous mapping which uniquely determines θ from H. However, it is clearly observed from the examples B Terminal Manifold (b) Manifold of points A, B and C Fig. 5 1 Manifold relations around critical point. The distance between the origin and the cross sectional plane of (a) is.875 of the maximum distance. of Type 1 domains that there is no such mapping. This fact is explained directly by the topology of kinematic mapping. Consider the circle on the envelope which corresponds to the case u=g i. Suppose there is a candidate mapping from H to θ. The image of the circle by this mapping is a loop on the torus where θ i changes from to 2π. Consider a deformation of the circle in the H space and the image in the θ space. The circle can be deformed to a point in the H space but their image in the θ space cannot, because continuity requires θ i to vary from to 2π. (Note: A similar statement for robot kinematics was generally proven by topological theories in Ref. 59.) The above is true as long as the mapping covers the H space in its entirety. If a small enough region of H space, a domain for example, is considered, any C Singular Manifold M S V 29

45 Technical Report of Mechanical Engineering Laboratory No.175 continuous steering law can obviously realize such a continuous mapping. Even if such a region includes a Type 2 domain, we can realize such a mapping by using an appropriate manifold selection. Thus, the question is how large a region of H space can be covered by such a continuous mapping, and what steering law actually can realize such a mapping. It is to be noted that the examples of Type 1 domains and critical points in the previous section are only found near the envelope by various computer calculations of the S(4) system. Therefore, the above problems may not be serious. On the other hand, the discussion in this chapter only suggested that there is a possibility of continuous steering in the presence of a Type 2 domain. The discussion in the above pertains to the local area around one domain. In actuality, the candidate workspace may involve various domains, so a study of global class connections is necessary. In the following chapters, a relatively specific problem will be studied for the symmetric pyramid type system using geometric tools given in this chapter. 3

46 6. Pyramid Type CMG System Chapter 6 Pyramid Type CMG System This chapter and the next two deal with a symmetric pyramid type system of single gimbal CMGs. This chapter describes system characteristics obtained by analysis and computer calculations. These are kinematic equations, system symmetry, expression of the gimbal angles when the angular momentum is at its origin, internal singular surface details, and impassable surface geometry. All of them will be utilized for the analysis of steering motion in the next chapter. 6.1 System Definition The pyramid type CMG system consists of four single gimbal CMGs in a skew configuration, as depicted in Fig An example of this type is the S(4) symmetric system, where each gimbal axis lies in the direction normal to each surface of a regular octahedron. The pyramid shown in Fig. 6 1 is the upper half of an octahedron. The skew angle of this type denoted by α is given as cosα = 1 3, and is about 53.7 degrees. It is expedient to define additional parameters: s* = sinα = 2 3, c* = cosα = 1 3. (6 1) Angular momentum vectors and torque vectors of all the units are given as: c * sinθ1 h1 = cosθ 1, s * sinθ1 c * sinθ3 h3 = cos θ3, s * sinθ3 c * cosθ1 c1 = sinθ 1, s * cosθ1 c * cosθ3 c3 = sinθ3, s * cosθ3 cosθ2 h2 = c * sinθ 2, s * sinθ2 cosθ4 h4 = c * sinθ 4, s * sinθ4 sinθ2 c2 = c * cosθ 2, s * cosθ2 sinθ4 c4 = c * cosθ 4, s * sinθ4 (6 2) (6 3) g4 c 4 θ 3 g 3 h 3 c 3 where the origin and the direction of each gimbal angle are defined by Fig θ 4 h 4 c 1 g 1 X θ 1 Z α Y h 1 c 2 h 2 θ 2 Fig. 6 1 Schematic of a pyramid type system. The origin of each θ i is defined when h i is on the square in the xy plane. The symmetric type S(4) is the case where the skew angle α is set as cosα = 1/ 3. g Symmetry The pyramid type CMG system has symmetry in its kinematics. This symmetry is useful for understanding the geometry of the singular surface and will be used for deriving a global problem in the next chapter. This symmetry is derived by the rotational transformations of regular octahedron, which is a well-known example of finite group theory. There are 48 symmetric transformations of an octahedron including the mirror transformations. The symmetry of the pyramid type system is represented by two groups of transformations and an 31

47 Technical Report of Mechanical Engineering Laboratory No.175 Z θ 3 X Z θ 4 θ 4 θ 1 θ 2 θ 3 X θ 1 Y Y θ 2 (a) Rotation about z axis Z θ 3 θ 2 Y' θ 4 X' θ 2 Z θ 1 θ 3 θ 1 New Coordinate System Y' (b) Rotation about g 1 axis θ 4 X' Fig. 6 2 Transformation in H space and in θ space. A new coordinate system (X', Y', Z) is defined in (b) for a simple expression. equivalence relationship of those. Transformations in the H space represent rotation of a vector, while the others in the θ space represent permutation with translation, i.e., a kind of an affine transformation. The meaning of the equivalence is as follows. As all four CMG units are arranged on the surfaces of the hexahedron, a certain rotation in the H space preserves the hexahedron and thus results in the exchange of four units. Therefore such a rotation in the H space is equivalent to the transformation in the θ space. An example of the H transformation is a 1 4 reverse rotation about the z axis shown in Fig. 6 2 (a), after which CMG unit i is replaced by unit i +1. This transformation of θ is expressed by: r z (θ=(θ 1, θ 2, θ 3, θ 4 )) = (θ 2, θ 3, θ 4, θ 1 ),(6 4) and the transformation of the angular momentum vector by: R z ((x, y, z) t ) = (y, x, z) t. (6 5) By those two transformations, the following equivalence relationship is satisfied. R z (H(θ)) = H(r z (θ)). (6 6) Another example is a 1 3 rotation about a gimbal axis shown in Fig. 6 2 (b). Two transformations are as follows: r g (θ) = (θ 1 +2 π 3, θ 3 2 π 3, θ 4 +2π 3, θ 2 2 π 3), (6 7) R g ((x, y, z) t ) = (z, x, y ) t. (6 8) The latter transformation is expressed in a different coordinate system from the original; one which is rotated 45 about the z axis as in Fig. 6 2 (b). This is because expressions based on these new coordinates are simpler than those based on the original coordinates. The equivalence Eq. 6 6 is also maintained by the transformations r g and R g. Applicable notation of all will now be defined. Rotations about g 1 are first defined. The identical transformation and the g 1 -z plane reflection are denoted by R e1 and R E1, respectively. A 1 3 rotation about the g 1 axis after R e1 (or R E1 ) is denoted by R r1 (or R R1 ). A reverse 1 3 rotation after R e1 (or R E1 ) is denoted by R q1 (or R Q1 ). Thus six transformations are defined. Successive 1 4 rotations about the z axis are simply denoted by increasing the indices. For example, R e2 is a 1 4 rotation about the z axis and R r3 is a 1 2 rotation 32

48 6. Pyramid Type CMG System Table 6 1 Symmetric Transformations Notation* H θ H Transformation (A) Transformation Transformation ** of Mirror (MA e1 ( x, y, z) ( θ 1, θ 2, θ 3, θ 4 ) ( x, y, z) E1 ( y, x, z) ( π θ 1, π θ 4, π θ 3, π θ 2 ) ( y, x, z) r1 ( z, x, y) (2 σ+θ 1, 2σ θ 3, 2σ θ 4, 2σ +θ 2 ) ( z, x, y) R1 ( z, y, x) ( σ θ 1, σ+θ 3, σ+θ 2, σ θ 4 ) ( z, y, x) q1 ( y, z, x) ( 2σ+θ 1, 2σ+θ 4, 2σ θ 2, 2σ θ 3 ) ( y, z, x) Q1 ( x, z, y) ( σ θ 1, σ θ 2, σ+θ 4, σ+θ 3 ) ( x, z, y) e2 ( y, x, z) ( θ4, θ 1, θ 2, θ 3 ) ( y, x, z) E2 ( x, y, z) ( π θ 2, π θ 1, π θ 4, π θ 3 ) ( x, y, z) r2 ( x, z, y) ( 2σ+θ 2, 2σ+θ 1, 2σ θ 3, 2σ θ 4 ) ( x, z, y) R2 ( y, z, x) ( σ θ 4, σ θ 1, σ+θ 3, σ+θ 2 ) ( y, z, x) q2 ( z, y, x) (2 σ θ 3, 2σ+θ 1, 2 σ+θ 4, 2σ θ 2 ) ( z, y, x) Q2 ( z, x, y) ( σ+θ 3, σ θ 1, σ θ 2, σ+θ 4 ) ( z, x, y) e3 ( x, y, z) ( θ 3, θ 4, θ 1, θ 2 ) ( x, y, z) E3 ( y, x, z) ( π θ 3, π θ 2, π θ 1, π θ 4 ) ( y, x, z) r3 ( z, x, y) (2 σ θ 4, 2σ+θ 2, 2 σ+θ 1, 2σ θ 3 ) ( z, x, y) R3 ( z, y, x) ( σ+θ 2, σ θ 4, σ θ 1, σ+θ 3 ) ( z, y, x) q3 ( y, z, x) ( 2σ θ 2, 2σ θ 3, 2σ+θ 1, 2σ+θ 4 ) ( y, z, x) Q3 ( x, z, y) ( σ+θ 4, σ+θ 3, σ θ 1, σ θ 2 ) ( x, z, y) e4 ( y, x, z) ( θ 2, θ 3, θ 4, θ 1 ) ( y, x, z) E4 ( x, y, z) ( π θ 4, π θ 3, π θ 2, π θ 1 ) ( x, y, z) r4 ( x, z, y) ( 2σ θ 3, 2σ θ 4, 2σ+θ 2, 2σ+θ 1 ) ( x, z, y) R4 ( y, z, x) ( σ+θ 3, σ+θ 2, σ θ 4, σ θ 1 ) ( y, z, x) q4 ( z, y, x) (2 σ+θ 4, 2σ θ 2, 2 σ θ 3, 2σ+θ 1 ) ( z, y, x) Q4 ( z, x, y) ( σ θ 2, σ+θ 4, σ+θ 3, σ θ 1 ) ( z, x, y) Each transformation is represented only by its suffix. σ = π 3 about z after a 1 3 rotation about g 1. So far, these total 24 transformations. Subsequent point symmetric transformations by the origin are denoted by adding M to the left of the original notation, MR e1 for example. After including these, all 48 transformations are defined. Before continuing, it should be noted that R r2 is not a simple rotation about the g 2 axis. Table 6 1 presents a list of all 48 symmetric transformations. The first row shows the notation, the second row gives the H transformation, the third shows the θ transformation and the last row gives the H transformation of the point symmetric image. Both the H transformation and the θ transformation are expressed as the right hand side only. So, for example, the expressions 6 4 and 6 5 are given by R e2 and 6 7 and 6 8 by R r1 in Table 6 1. Note again that all H transformations are expressed in the new coordinate system rotated 45 for simplicity. Τransformation of all the point symmetric images in θ space is omitted in Table 6 1 but is simply accomplished by adding π to each θ i. 6.3 Singular Manifold for the H Origin The origin of H, (,, ) t, is used as a nominal state of control. This H corresponds to one singular manifold with 6 singular points. The 6 singular points divides the singular manifold into 12 line segments. The 12 segments are classified into two groups. These segments and groups are used to explain the global problem and a steering law in the next chapter. The singular manifold for this H origin has analytical expressions. It consists of four lines, which are straight lines but closed in the torus space. Two of them are given by, (1) θ = ( φ, φ, φ, φ), where π < φ π, (6 9) (2) θ = ( φ+π 2, φ π 2, φ+π 2, φ π 2) t, where π < φ π. (6 1) 33

49 Technical Report of Mechanical Engineering Laboratory No.175 θ 2, θ 4 π e Singular f a b θ j π k l (g) (h) (k) g π π c d π θ 1, θ 3 (a) (θ i, θ j ) =(φ, φ) π (l) π θ i π (b) (θ i, θ j )=(φ+ψ, φ ψ) φ π, ψ= π 2, 5π 6 h 3 Line segments for singular manifold. Arrows a to l are parametric line segments of a pair of coordinates. The remaining two lines can be obtained as symmetric images of the latter. Let s define notation of total 12 line segments and derive their symmetric relations. The first line by Eq. 6 9 includes 6 singular points hence 6 line segments. Let line segment of θ be expressed by the combination of two coordinate sets. Referring to Fig. 6 3 (a), each segment will have a parameter along it and the direction in which the parameter increases will be expressed by an arrow. Six line segments from L A to L F in a four-dimensional torus Table 6 2 Segment Transformation Rule Α G H K L M N G H K L M N e1 G H K L M N H G L K N M E1 H G M N K L G H N M L K r1 L K N M G H K L M N H G R1 K L G H N M L K H G M N q1 M N H G L K N M G H K L Q1 N M L K H G M N K L G H e2 H G M N L K G H N M K L E2 G H L K M N H G K L N M r2 N M Κ L H G M N L K G H R2 M N H G K L N M G H L H q2 L K G H N M K L H G M N Q2 K L N M G H L K M N H G e3 G H L K N M H G K L M N E3 H G N M L K G H M N K L r3 K L M N G H L K N M H G R3 L K G H M N K L H G N M q3 N M H G K L M N G H L K Q3 M N K L H G N M L K G H e4 H G N M K L G H M N L K E4 G H K L N M H G L K M N r4 M N L K H G N M K L G H R4 N M H G L K M N G H K L q4 K L G H M N L K H G N M Q4 L K M N G H K L N M H G Note:Each transformation and each segment are represented by their suffices. 34

50 6. Pyramid Type CMG System are then given by a pair of those segments (from a to f) as follows: {L A, L B, L C, L D, L E, L F } = { {a,a}, {b,b}, {c,c}, {d,d}, {e,e}, {f,f}; {(θ 1, θ 2 ),(θ 3, θ 4 )} }. (6 11) Segment L A in this expression, for example, is given as follows: L A ={θ:=(ϕ, ϕ, ϕ, ϕ), π 6 ϕ < π 6}. (6 12) The line by Eq. 6 1 orthogonally crosses the first line at two singular points. The two singular points divide the line into two segments. Referring to Fig. 6 3 (b), the two segments are defined as follows: L G = {(θ 1, θ 2 )=g, (θ 3, θ 4 )=g}, transformation of L A, while any segment from L G to L N by some transformation of L G. This implies that any characteristics of the segments must accordingly be derived from characteristics of either L A or L G. 6.4 Singular Surface Geometry Singular surface has been described by its curvature or by an example of a cross section in the previous chapters. Now the total geometry of the singular surface especially of the impassable surface will be examined by using a series of cross sections 66). Here, cross sectional planes orthogonal to the g 1 axis are mainly used as shown in Fig Each plane has a parameter d which is a distance from the H origin to the plane. All distances will henceforth be normalized by its maximum value, which is the distance to the unit circle L H = {(θ 1, θ 2 )=h, (θ 3, θ 4 )=h}. (6 13) Other two lines, i.e., four line segments are defined similarly: Sectional Plane Orthogonal to g 1 A d : Distance from the H Origin L K = {(θ 1, θ 4 )=k, (θ 3, θ 2 )=l}, L L = {(θ 1, θ 4 )=l, (θ 3, θ 2 )=k}, L M = {(θ 2, θ 1 )=k, (θ 4, θ 3 )=l}, L N = {(θ 2, θ 1 )=l, (θ 4, θ 3 )=k}. (6 14) The set of segments from L A to L F are transformed to the same set of the segments by any symmetric transformation. The followings are examples of transformed results, where a minus sign before the segment implies that the direction is reversed. R E1 (L A, L B, L C, L D, L E, L F ) =( L D, L C, L B, L A, L F, L E ), R r1 (L A, L B, L C, L D, L E, L F ) = (L C, L D, L E, L F, L A, L B ), R e2 (L A, L B, L C, L D, L E, L F ) =( L A, L F, L E, L D, L C, L B ), MR e1 (L A, L B, L C, L D, L E, L F ) = (L D, L E, L F, L A, L B, L C ). (6 15) Similarly, the other segments from L G to L N are also transformed to the same segments. The results of all such transformations are listed in Table 6 2 in which each segment is represented by its suffix. From the above analysis, it is observed that any segment from L A to L F can be obtained by some Portion of Internal Impassable Surface (a) Envelope and sectional plane g 4 g 2 g 1 g 3 (b) Cross section on plane A Envelope Fig. 6 4 Definition of the cross sectional plane and the distance d. The distance d is divided by 2 2 for normalization such that d=1 for the unit circle on the envelope. 35

51 Technical Report of Mechanical Engineering Laboratory No.175 A Internal impassable part Domain D 1 D 1 A B A B d = 1. Unit Circle d =.99 d =.92 Envelope part (a)d=.9 A D 1 B Internal passable part A (b)d=.8919 B Fig. 6 5 Saddle like part of the envelope. Curves are various cross sections of S { + + +}. For d<.94, some portions of surfaces are inside the envelope. (c)d=.885 D 2 (d)d=.88 on the envelope connecting two singular surface of different signs. Figure 6 5 illustrates a singular surface of one negative sign, a portion of which is shared with the envelope. The outermost unit circle is the case of u=g i and d=1. Other curves are cross sections with d<1. It is of interest to observe what happens to this surface when d is made smaller and smaller. As d decreases in size, the curve deforms like a triangle. After sharp edges appear, folding and then small triangles appear on the curve as shown by A in Fig As a result of this folding, each curve is divided into six smoothly curved segments. As the curvature changes sign at the folding point, the Gaussian curvature also changes its sign. This folding is therefore a connecting point between a passable and an impassable surface. From continuity to the envelope, passability of each curve is determined as shown in Fig Observing the cross sections of various d in this figure, it can be seen that both passable and impassable surfaces penetrate the envelope like strips or belts. Folding also appears in the cross sectional line at d.65 and as d becomes smaller, the strip bifurcates. By repetitive calculation for various d values, impassable portions of the singular surface are obtained as in Fig This figure does not show all of the internal impassable surfaces. All surfaces are obtained by successive 1 4 rotations about the z axis. Let s make a simplification of these impassable surfaces. The following analytical expressions are found 29) which corresponds to a smooth line shown A (e)d=.87 A D 4 D 3 D 3 (g)d=.85 (i)d=.6 A B B D 2 A D 3 (f)d=.857 Fig. 6 6 Cross sections of singular surface. Bold curves are impassable and thin curves are passable. All are drawn at the same scale except the last two, (h)d=.64 and (i)d=.6. The impassable curve segments AB have envelope parts as shown in (a), are totally internal as shown in (b) and are divided into two as shown in (h). A (h)d=.64 B B B 36

52 6. Pyramid Type CMG System A Z Z g 1 Q B C P (,,2s*) X Y X A F 4c* α O D E Y D P C B F Q Fig. 6 7 Internal impassable singular surface. Cross sections orthogonal to g 1 are drawn at a d step size of.5.the detail of the region indicated by A is in Fig Fig. 6 8 Analytical line on an impassable surface. The surface near this line is called a branch. (4) Straight line DE in Fig This line is on the impassable surface shown in Fig. 6 7 hence can represent the surface. This line has the following four parts: (1) Elliptic arc AB H = (2(c* cosφ ),, 2s*sinφ ) t, (6 16) where: θ = ( π 2, φ, π 2, π φ ), π φ π 2 (B in Fig. 6 8), u = ( s*cosφ,, sinφ ) t. (2) Straight line BC H = (2c*sinφ,, 2s*) t, (6 17) where: θ = (φ, π 2, φ, π 2 ), (B) π 2 φ 5π 6 (C), u = (, s*cosφ, sinφ ) t. (3) Circular arc CD H =(c* cosφ, c*(1 sinφ ), c*(1+sinφ )) t, (6 18) where: θ = ( 5π 6, φ, 5π 6, π 2 ), (C) π 2 φ π 6 (D), u = g 2 = (, s*, c*) t. H = ( c*, c*, s*) t + c*sinφ(1, 1, 2 ) t, where: θ = (φ +π 3, π 6, 5π 6, φ π 3 ), (D) 5π 6 φ π (E), u = ( ( c*cosφ + sinφ ), ( c*cosφ s*cosφ ), c*cosφ ) t. (6 19) Let point F on the arc AB be the location at which the line AB is divided to an envelope side (AF) and an internal side (FB). The line FBCDE is connected and continuous both in the H space and in the θ space. By the transformation MR R1 in the notation of Sec. 6.2, a line F B C D E can be obtained which is continuously connected to the original line. Thus, referring to Fig. 6 8, the line FBCDED C B F connects two points on opposite sides of the envelope. The surface of Fig. 6 7 is composed of several strips of impassable surface and a portion of it is represented by this line. This particular strip will be called an impassable branch and be denoted by B e1. In the following figures, the analytical line FBCDED C B F is simplified by using a broken line QPP Q, as shown in Fig This branch, along with its symmetric images by transformations R r1, R q1, R E1, R R1 and R Q1, form a frame of parallel hexahedron shown in Fig. 6 9, which fits all the surface shown in Fig Each branch is denoted by B and the subscript denotes the 37

53 Technical Report of Mechanical Engineering Laboratory No.175 Q B r1, B R1 X B e1, B E1 Bq1, B Q1 B R1 B r1 B E1 B Q Be1 BE1, B r1 B q1 BR1, Bq1 Fig. 6 9 Equilateral parallel hexahedron of impassable branches. Z P Y B e1, B Q1 transformation, B r1 for example. In this figure, some lines are shared by two branches. This is partially the result of simplification of the curved line. The other reason is that two branches share the same surface near the envelope and this surface bifurcates as depicted in Fig By the successive rotation about the z axis, the stellar hexahedron in Fig. 6 1 is obtained. In this figure, only the suffix is shown for each branch. The same stellar hexahedron is drawn in Fig with a cut envelope to reveal the size and the shape of the internal impassable surface. In summary, all impassable surfaces of the pyramid type CMG system are described by the envelope r1, R1 q4, Q4 r4, R4 q1, Q1 e1, E1 r2, Q4 e4, E4 E2, e3 Q3, r1 e4, E3 Z E1, e2 r2, R2 e3, E3 e2, E2 E4, e1 Q2, r4 E1, R1 q3, Q3 q2, Q2 r3, R3 g 3 Z O g 1 X X e3, Q3 E2, r2 E3, r3 Q2, e2 q4, R1 r2, q2 q3, R3 R2, q1 R4, q3 r3, Q1 q2, R3 R4, q4 r1, q1 e4, Q4 r4, E4 e1, Q1 Y Envelope Fig. 6 1 Overall structure of impassable branches Cutaway of Envelope Fig Cross section through the xz plane. Z P Envelope Passable P X Y Y Impassable Simplified Branches Fig Internal impassable surface with envelope cutaway. Fig Cross section through the xy plane. X 38

54 6. Pyramid Type CMG System and the frame like structure of the branches. Figures 6 12 and 6 13 are examples of other cross sections which are not orthogonal to the gimbal axes. In these figures, all singular surface, passable and impassable, are drawn. Two figures show that there are relatively large region with no singular surface and impassable surfaces are narrow strips compared with the maximum workspace. Nevertheless, impassable surfaces cannot be ignored because they surround the origin which is the nominal point for the control. Moreover, some of the impassable surfaces crosses z axis and others lie on the x-y plane. As the CMG system s axes coincide with those of an attitude control system, angular momentum of the CMG system tend to travel near such axes. 39

55 Technical Report of Mechanical Engineering Laboratory No.175 4

56 7. Global Problem, Steering Law Exactness and Proposal Chapter 7 Global Problem, Steering Law Exactness and Proposal This chapter deals with the question whether it is possible to steer the pyramid type CMG system to avoid any terminal class of Type 2 domains. In Chapter 5, it is clarified that any steering law will fail if it aims to cover the workspace in its entirety. Moreover in this chapter, it will be shown by examples that any steering law fails continuous and real-time control for the wide variation of command inputs even if an appropriate manifold selection is tried. Based on this global problem, various steering laws are evaluated. In so doing, the CMG motion by each steering law is analyzed geometrically. Three groups of steering laws are examined and their performance and problems are clarified. The first of those permits errors in the output. The second is realized as a path planning. The third one is effective for a certain fixed direction. By those evaluation, importance of steering law exactness is clarified. Finally, a new type steering law will be proposed which assures exact and real time control inside a reduced workspace. This steering law uses a simple constraint and determines uniquely the system state from the angular momentum. The reduced workspace is larger than the spherical one which excludes all impassable surfaces, but has the same length in one direction as the original maximal workspace. 7.1 Global Problem Following the discussion of Section 5.7, the workspace size must be slightly reduced from the maximum to enable continuous control. Here, it is shown that continuous control is impossible if the workspace includes certain domains. An example of the workspace is a sphere around the H origin O in Fig. 7 1 including the hexahedron made of impassable branches which is a part of the stellar hexahedron in Fig Domains around two vertices, P and Q, are relevant for the following discussion. First, an H path and its deviations along the z axis from O to P are considered. For continuous control on these paths, the necessary condition of θ at the origin O is obtained. Then, by consideration of an H path from O to Q, the impossibility of continuous control is derived Control Along the z Axis Let s find necessary conditions for continuous realtime steering along the z axis as shown by OP in Fig Consider now only H in the neighborhood of P. Fig. 7 2 (a) shows a cross section of the singular surface by a plane orthogonal to z axis and which crosses near the point P. In the close-up view of Fig. 7 2(b), it can be seen that there are eight domains around the center and four pairs of impassable branches cross each other. One of the eight domains, the domain D A in Fig. 7 2(b), has 2 equivalence classes whose elements (which are manifolds of those classes) are M A and M A1 in Fig The class G A including M A bifurcates into two classes G and G 1 when entering the neighbor domain D 1 in Fig. 7 2(b). Two classes are represented by two manifolds M and M 1 in Fig The classes G Candidate workspace X Q O R Fig. 7 1 Candidate of workspace Z P T S Y 41

57 Technical Report of Mechanical Engineering Laboratory No.175 DomainD 1 Branch B E4 H path 1 Envelope (a) Cross section normal to the z axis. Y Y Branch B e3 H path 22 Passable Impassable H = 1 X X DomainD A (b) Magnified view inside the dotted square shown in (a). Fig. 7 2 Cross section nearly crossing P. The distance from O to the plane is 1.4 (not normalized). This plane crosses the z axis nearer the origin than P, as OP = 2 s* and G 1 are equivalent to the terminal classes of the impassable branches B E4 and B e3 respectively. This implies that M continuously changes to a singular point (θ E4 in Fig. 7 3(c)) when H follows the path 1 in Fig. 7 2, and M changes to another singular point (θ e3 ) by the H path 2. Thus the class containing M A1 in Fig. 7 3 should be selected in the domain D A for continuous steering in consideration of these H paths. This is more easily understood by making a simplified class connection map around these domains, as in Fig The situation is similar to that around a Type 2 domain described in Chapter 5 and hence, only one class G A must be selected in this domain D A. As illustrated in Fig. 7 5, domains and manifolds symmetric about the z axis are obtained by successive 1/4 rotations. Though there are two equivalence classes to each of the four domains, i.e., D A to D D, one of them must be selected with the consideration above. Manifolds to be selected are M A1, M B1, M C1, and M D1 in Fig All the four domains are connected with each other by a line segment lying on the z axis which is shown as a cross point U in Fig. 7 5 (b). Figure 7 6 shows a singular manifold for this point U. By comparing the manifolds M A1, M B1, M C1, and M D1 in Fig. 7 5 and the singular manifold M U in Fig. 7 6, it is observed that all the manifolds are continuously connected by two curved line segments, φ 1 φ 2 and φ 3 φ 4, shown bold in Fig Thus, the necessary condition for continuous steering from the point U to either D A, D B, D C or D D is controlling θ on one of the two segments, either φ 1 φ 2 or φ 3 φ 4, when H is at U. If a cross sectional plane orthogonal to the z axis is moved towards the H origin, the topology of domain connections, the intersection of the singular surface and class connections over domains become different from those in Figs. 7 2 and Fig However, two segments of the singular manifold in Fig. 7 6 are analytically defined for any H= (,,H z ) t on the z axis as follows: θ = (φ+ψ, φ ψ, φ+ψ, φ ψ), where H z = 4 s* cosψ sinφ. (7 1) One of the segments includes the point (φ, φ, φ, φ) where φ = sin -1 (H z (4 s*)) and both of its edges are singular. The other segment can be obtained as a mirror image of this. Because the segment (and its edges) given by Eq. 7 1 are continuous with respect to H z, θ must be located on this segment, or on its mirror image, for any H z. At the H origin, this segment and its mirror image take the following simplified form: θ(h=(,,)) = (ψ, ψ, ψ, ψ), where π/6 ψ π/6 and 5π/6 ψ 7π/6. (7 2) 42

58 7. Global Problem, Steering Law Exactness and Proposal 3π 2 3π 2 M A G A θ 2 θ 2 M A1 π 2 π 2 θ 1 3π 2 π 2 π 2 θ 1 3π 2 (a) Two manifold of H = (.2,.2, 1.4) t in domain D A. 3π 2 3π 2 M 1 θ 2 M G G 1 θ 2 θ e3 θ E4 π 2 π 2 θ 1 3π 2 π 2 π 2 θ 1 3π 2 (b) Two manifolds of H = (.5,.5, 1.4) t in domain D 1. Both are connected with M A in (a). Another manifold connected with M A1 is not drawn. (c) Impassable singular points of branches B E4 and B e3. θ E4 is connected with M in (b) through Path 1 in Fig. 7 2 and θ e3 is connected with M 1 in (b) through Path 2. Fig. 7 3 Manifold bifurcation and termination from D A. Manifolds are drawn using (θ 1, θ 2 ) coordinates. The θ origin is not on the center to avoid a manifold drawn separately. G 1 G A B E4 D 1 Domain D A G A1 G 12 G B e3 Fig. 7 4 Simplified class connection diagram around domain D A. For clarity, this figure of domains has been simplified by omitting some singular surfaces. 43

59 Technical Report of Mechanical Engineering Laboratory No.175 Y Ε3 e1 D D D C e2 Ε2 U D A D B X U(,, 1.4) Ε4 e4 (b) Four domains around the point U e3 D A Ε1 (a) Branches and domains M D1 M C G C M D G D M C1 D D D C D A DB M A1 M A G A M B1 M B G B (c) Manifolds of eight domains. Fig. 7 5 Manifolds of eight domains around the z axis. In Fig. (a), each branch is indicated by its suffix such as E 1 for B E1. In Fig. (c), each domain, from D A to D D, has two manifolds. They are drawn using (θ 1, θ 2 ) coordinates. Though four manifolds, from M A1 to M D1, are congruent, they look different in two dimensional projections. 44

60 7. Global Problem, Steering Law Exactness and Proposal These crossings are not singular points but only due to the 2D projection. θ 2 θ 1 φ 1 Segment Global problem φ 2 Fig. 7 6 Singular manifold of a point U on the z-axis. Two curved line segments drawn bold, φ 1 φ 2 and φ 3 φ 4, are the segments continuously connected to manifolds, M A1 to M D1 in Fig The same discussion can be made for the H path from O to Q. All θ and H are simply transformed by a rotational transformation such as the 1 3 rotation about the g 1 axis. This transformation is denoted by R r1 in the notation of the previous chapter. By the corresponding θ transformation, the above conditions, Eq. 7 2, is now transformed to the following segment: θ(h=(,,)) = (ψ, ψ, ψ, ψ), where π/2 ψ 5π/6 and π/2 ψ π/6. (7 3) Since the two sets of segments given by Eqs. 7 2 and 7 3 have no common θ, continuous control from O to P and from O to Q cannot be satisfied simultaneously. It is clear that once the condition imposed by Eq. 7 3 is satisfied, the system will meet an impassable singularity on the H path nearly along the z axis while crossing some of the impassable branches B Ei and B ei in Fig Thus it is concluded that continuous steering over this entire workspace, including O, P and Q is not possible. These two sets of segments defined by the above two equations are (L A, L D ) and (L C, L F ) in the notation of Section 6.3. The remaining segments L B and L E are the condition for the continuous control in the OR direction. φ 4 φ 3 M U Singular Point This is a more general conclusion than the difficulty reported in the former research work 37), which dealt only with the specific examples of motion along the z axis. By the geometric analysis made above, it is understood that not only the H path on the z axis but also a variety of other paths cannot be realized by any steering law Details of the Problem Suppose that θ is on the latter segment of Eq. 7 3, when H =. This segment is denoted by L F after the notation of Section 6.3. Referring to Fig. 7 7, an infinitesimal motion of H towards (1, 1, 1) t moves H away from a singular surface and θ moves onto a manifold which is originally a rectangle outlined by segments L F, L M, L C and L H when H =. As H moves closely along the z axis in the same domain, the manifold changes equivalently as shown in Fig. 7 8, with neither bifurcation nor termination. Finally, near the point U in Fig. 7 5 (b) the manifold connects with either M A, M B or M D (Fig. 7 9). Because the impending H path is not given, there is a possibility of these manifolds being selected once θ is determined on manifold M V in Fig The three manifolds inevitably bifurcate into terminal classes if the H path crosses certain branches, for example branches B E4 and B e3 if manifold M A is selected, branches B E1 and B e4 for M B and branches B E3 and B e2 for M D. The manifolds in Fig. 7 8 are all inside one domain, denoted by D V. Though Fig. 7 1 in the next page indicates that this domain is not large by itself, some V L F L H M V (H=(.2,.2,.2) t ) θ 2 S U L M T L C Fig. 7 7 Manifold of H near the origin. M V is one of the manifolds for H = (.2,.2,.2) t which continuously deformed from the rectangle STUV. Four edges of the rectangle are L F, L M, L C and L H by the notation of Section 6.4. θ 1 45

61 Technical Report of Mechanical Engineering Laboratory No.175 a =.2 Y a =.25 D V a =.5 a =.75 a = 1. X a = 1.3 H =1 Fig. 7 8 Continuous change of manifold for H nearly along the z axis. These six manifolds are for H=(.2,.2, a) t where a =.2,.25,.5,.75, 1. and 1.3. The last manifold for a = 1.3 is M V of the next figure. Filled circles and blank circles are maximum and minimum of det(cc t ) along the manifolds. (a) H Z =.1 Envelope D V M V θ 2 θ 1 (b) H Z =.5 Envelope (a) Manifold M V D V M D M B θ 2 θ 1 M A (c) H Z = 1. Envelope (b) Manifolds in the neighborhood Fig. 7 9 Manifold connection over several domains. The manifold M V connects partially with M A, M B and M D. Fig. 7 1 Cross sections of domains. Domain D V corresponding to manifolds in Fig These domains have manifolds equivalent to M V. 46

62 7. Global Problem, Steering Law Exactness and Proposal neighboring domains have equivalent classes to this manifold. Once the segments at H = given by Eq. 7 3 are selected, H then moves inside these domains, there is no way to escape from the manifold equivalent to this M V. The branches mentioned above pass along edges of the top half of an octahedron, namely PQ, PR, PS and PT in Fig Since it is safe to assume continuity of the surfaces and manifolds, it can be expressed that some parts of this manifold connect to manifolds belonging to terminal classes of these branches. Therefore there is a possibility of termination for any H path crossing such branches. Moreover, these domains are so large that this problem cannot be neglected. Manifold M V, however, is not connected to M C so it does not present a problem with respect to branches B E2 and B e1 when this manifold is selected Possible Solutions The above discussion is made without consideration for any specific steering law. The problem applies to any steering law which aims an exact and strictly real time control. Exactness implies that an output is always equal to the command input. Strict real time feature implies that information of future command is not used. Possible methods to overcome the problem could involve either of the following. 1) relaxing an exactness condition. 2) relaxing a real time condition. 3) restricting the workspace. In the following sections, from Section 7.2 to 7.4, various proposals of the former two kinds will be evaluated. By these evaluations, importance of steering law exactness and real time feature will be clarified. Then, a new steering law using workspace restriction will be proposed in Section 7.5. surface. Because a passable surface is generally avoided by steering laws using a gradient method, such a steering motion will take place on an impassable surface. Its solution is obtained so that the output torque lies on a plane tangential to the singular surface. Therefore, this steering motion can be imagined as a sliding motion of H on the singular surface. As depicted in Fig. 7 11, there are four possibilities of motion along the singular surface when the command, T com, is fixed. The case (d) can be ignored straight off because it is not stable. The case (c) is possible when the surface is convex to u and this is the case of an envelope (see Section and Fig. 4 8). For an internal surface, only the cases (a) and (b) are possible. Thus, a motion is always possible in response to the command as long as the command is fixed. This discussion ignores how large the torque error is. If the area of the impassable surface is excessively large, this steering law is not effective in practical use. As shown previously in Figs. 4 1, 11 and 12, the impassable surface of a 4 or 5 unit CMG system is shaped like a narrow strip and the curvature of the surface is negative to its narrow direction. Because of this, such Singular Surface (1) (2) (a) Smooth Break Away (3) (1) (2) (b) Folding (3) 7.2 Steering Law with Error The steering laws described in Section enable calculation of the inverse Jacobian even on a singular point. This is made possible by permitting a minimum error in output torque. This kind of method has so far been evaluated only by a limited number of simulations. (2) Stop (1) (c) Stop at Convex Stop (d) Unstable Stop Geometrical Meaning The CMG motion by a steering method accompanied by error is understood from the shape of an impassable Fig Possible motion following an example of singular surface. Motions may be (1) reach the singular surface, (2) go along the surface, then (3) break away from the surface, if possible. 47

63 Technical Report of Mechanical Engineering Laboratory No.175 escape motion as in the cases (a) and (b) will take place approximately to the narrow direction. Possible error, therefore, can be roughly estimated by the width of the singular suface stip. If a faster sliding motion is required, judicious knowledge of the direction of narrow width is very useful. This direction can be approximately obtained as an eigenvector of the negative curvature of the surface. Supposing that this direction is obtained as v, the θ motion, dθ S, that will realize this sliding motion is obtained by movement along the singular surface as follows: dθ S = PC t (CC t + k u u t ) 1 v. (7 4) H Trajectory Deceleration H motion due to Vehicle's Rotation Acceleration Fig Illustration of H trajectory of the CMG system for the example maneuver. This is derived by Eq and Eq. B 9 in Appendix B Exactness of Control As mentioned previously, the steering of a CMG system is similar to the kinematic control of a multijoint manipulator (also see Appendix F). If a CMG is used by itself and the objective is to realize a certain H trajectory, the steering law problem is analogous to the kinematic control of a manipulator. In this situation, the above method gives a possible solution whose H deviates slightly from the desired trajectory. The difference between CMG control and control of a manipulator is that a CMG is used for the attitude control. The objective is not to control the actual CMG but rather to control the vehicle s attitude. If there is an output torque error, not only is there deviation in the path of H but also the attitude of the satellite changes from that intended. This attitude error changes the command issued by the feedback control and then the desired H path also changes. The above method should therefore be evaluated in consideration of the attitude control. Suppose that the angular momentum of the satellite is zero and the angular momentum of the CMG system on board is not zero. Suppose further that the control command is to maneuver the satellite and finally stop the rotation. This implies that the final angular momentum of the satellite is zero and the angular momentum of the CMG system is not zero. By the conservation law, the initial and final angular momentum must be the same in the inertial coordinates. Since the coordinate frame of the CMG system rotates, the initial and final angular momentum will be different in the rotating coordinate frame. Let H and H 1 denote the initial and final H in Desired Path H 1 Impassable Surface H Detour Fig Avoidance of an impassable surface coordinates fixed on the CMG system. The H trajectory of the CMG system will be some path from H to H 1, as depicted in Fig Though the exact path varies for different control methods, H and H 1 will not vary if the control is successful and if there are no disturbances. Suppose that there is an impassable surface somewhere along this path. If this surface is located sufficiently far from the goal and the surface are small enough, it might be possible to make a detour as shown in Fig. 7 13, and reach the goal by the above steering law. If the surface is near enough the goal, H will stay on the singular surface (at the point A in Fig. 7 14(a)) despite the negative surface curvature, because the point A is the nearest to H 1. Since the residual angular momentum H H 1 of the spacecraft is not zero, there remains some rotation of the spacecraft when H is on this surface. Though this rotation depends on the inertia matrix of the body and the direction of the residual angular momentum, H of the CMG may stay inside some area of the impassable surface, as shown in Fig. 7 14(b). 48

64 7. Global Problem, Steering Law Exactness and Proposal No sliding motion possible (a) Impassability A H 1 Impassable Surface H Trajectory This residual angular momentum causes rotation of spacecraft H 1 Possible H Trajectory 43, 44). This problem is similar to the path planning and its realization of a robot manipulator. In Section 7.1, it was concluded that some of the various possible command sequences cannot be realized simultaneously by the same steering law. This is true as long as the future H path is not specified. On the contrary, manifold selection and continuous θ path is possible for a given H path, even for one of the two paths in Fig. 7 2 for example. Of course, continuous control is not possible if the H path starts from the singular surface as described in Section or if the H path crosses a Type 2 domain as described in Section 5.5. Thus, conditions of successful path planning can be clarified by this geometric study. If we permit a minimum error in the solution 42,43), path planning is always possible because of the nature of such motions. Geometric study also reveals some problem of this manner of path planning. Since different manifolds may be selected for different H paths, the θ path may be completely different even though the H path is very similar, which may degrade the robustness of the control system. Moreover, optimization is limited only to the given H path but no future situations are considered. Moreover, this method is too complicated for actual implementation. (b) Motion on the impassable surface Fig Problems of movement on an impassable surface. In this case, it is impossible to reach the goal and the spacecraft will continue its rotation. If an attitude keeping problem under some disturbance is considered, such a situations as in Fig is not avoided by this steering law. Because of these, it is better not to use the above steering law and better to keep steering law exactness. 7.3 Path Planning Another steering law approach takes advantage of rapid maneuvering to enable off-line planning. If the maneuver occurs fast enough, the period of maneuver will be short enough that any disturbance can be neglected and hence the maneuver trajectory and H path can be designed beforehand. For this given path, the CMG motion can be planned by off-line calculations 42, 7.4 Preferred Gimbal Angle Another method 38) is similar to path planning but supposes that the direction of a near future maneuver can be known, and this direction is one of certain predefined possibilities. This method introduces preferred gimbal angles from the maneuver direction and adjusts the system to the preferred angles before the maneuver motion. An examples of preferred gimbal angles and their corresponding maneuver directions are given as follows 38) : Direction of Maneuver Preferred Gimbal Angles z-axis, (1, 1, 1) t direction (,,, ) x-axis, (4, 2, ) t direction ( π 3, π 3, 2π 3, 2π 3 ) y-axis, (2, 4, )t direction ( 2π 3, π 3, π 3, 2π 3 ) It is guaranteed from the discussion of Section that an initial gimbal angle of (,,, ) is suitable for z-axis maneuvers. However, evaluation of the others are not simple. 49

65 Technical Report of Mechanical Engineering Laboratory No.175 θ 3 A(θ ), B θ 4 θ 1 H X =1. θ 2 H X =1. H x = H X = A(θ ) B (a). H X 1. B, C H X =1. H X =1.2 H X =1. H X =1.2 H X =1.4 H X =1.4 C Connection B (b) 1. H X 1.4 H X =1.4 D H X =2.6 H X =1.4 H X =2.6 C (c) 1.4 H X 2.6 Fig Change in manifolds for H moving along the x axis. (H = (H X,, ) t, H X step size =.2). Manifolds are drawn in (θ 1, θ 3 ) and (θ 2, θ 4 ) coordinates as shown in (a). Manifold bifurcations are observed in (b). Motion of θ from the preferred angles θ =( π 3, π 3, 2π 3, 2π 3) follows the line ABCD, which is a trace of the maxima of det(cc t ) and is indicated by dots. In the (1, 1, 1) t direction, there are two impassable branches, i.e., B E2 and B e1. The discussion in Section suggests that the θ on the segments L F, L M, L C or L H may be better than preferred angle (,,, ). The second and third θ are on segments L L and L M. The second preferred θ in the above list will next be evaluated by observing the change in manifolds as H moves along the x axis. Figure 7 15 shows manifolds corresponding several H on the x axis. A gradient method is used and θ is maintained at the local maximum of det(cc t ). Starting with θ set to the values in line 2 of the above list, θ subsequently follows the path ABCD. There are only two bifurcations in this manifold path, as shown in Fig. 7 15(b), and these correspond to passable surfaces in the neighborhood of two impassable branches. The bold curve in the left plot of Fig. 7 15(b) indicates a connection, and this shows that the manifolds are 5

66 7. Global Problem, Steering Law Exactness and Proposal connected by this part before and after the bifurcations. The robustness of this motion is evaluated by the length of this part, which is more than π 2 in this case. The preceding discussion verifies the performance of this steering method but also clarifies its limitations. This method is valid only when H is initially on the origin. No method was specified to obtain θ when H is not zero. If a gradient method is applied, this method is valid as long as the maneuver is carried out exactly along the defined direction. But if the maneuvering path deviates, various gradient method problems may occur, which will be described in the next section. Conceptually, this method may be effective for exact and real-time steering as long as the system does not meet impassable singularity before H returns to the origin. The main question is whether this can be assured. By extension of the first set of preferred angles and by making the motion exact, a more effective method will be proposed in the following section. Possibly Passable Impassable 2 c* X Estimated Workspace Q R Z P T 4 s* S Y 7.5 Exact Steering Law The evaluation above clarified that steering law exactness and real time feature are important for the real usage of the CMG system in the attitude control of a satellite. In this section, a new steering law is proposed which assures its exactness and real time feature 39). Though, the idea of manifold selection prior to bifurcation is important for analyzing continuous control, suitable algorithms for actually doing this have not been developed. Segments defined by Eqs. 7 1 and 7 2 only defines θ where H is on the z axis. While geometric concepts such as class connection around domains are useful for evaluating the steering law, the actual steering law algorithm must determine the θ value at any H point so that the desired manifold selection is made Workspace Restriction Because of the problem in Section 7.1, the workspace must be restricted in order to keep exact steering. One way of workspace restriction is to exclude all impassable surfaces from the workspace. This however is a too strict way of restriction. The condition imposed by Eq. 7 1 is effective with regard to motion nearly along the z axis, while it is not applicable for control in the neighborhood of Q. Thus, a new workspace of a two-lobe shape may be obtained by excluding some of the impassable branches crossing near Q, R, S and T as shown in Fig. Fig Estimation of reduced workspace for exact steering. Branches drawn by bold lines are impassable but those drawn by thin lines may be made passable The shape of this workspace is not defined by the discussion in Section 7.1, however it does include the z axis and its neighborhood, and an area on the xy plane slightly smaller than the square QRST as shown in Fig Repeatability and Unique Inversion It is required that θ remains on the segment given by Eq. 7 1 whenever H is on the z axis. This is a matter of repeatability of inverse kinematics. Generally, repeatability is realized only when the system has an inverse mapping 67). The following example illustrates that an ordinary gradient method does not possess repeatability over the workspace in Fig Figure 7 17 shows manifolds of several H points on the line from O to Q. Each jagged-edge rounded rectangle is a computer output of the manifold drawn in (θ 1, θ 2 ) coordinates. Dots on the manifolds indicate local maxima of det(cc t ) along each manifold. This implies that θ may be controlled on these points by a gradient method. If the initial θ meets the condition of Eq. 7 2, θ will follow line AB for commands on the H path approaching Q from O. The line linking the local 51

67 Technical Report of Mechanical Engineering Laboratory No.175 D Local Maxima of det(cc t ) θ 2 C B E Manifold of H=(,,) t A θ 1 θ 1 θ 2 + θ 3 θ 4 =, (7 5) because of the following reasons. When H is on the z axis, this condition gives the center of the segment described by Eq. 7 1 which has the maximum det(cc t ) on the segment. This condition preserves some of the system s z axis symmetry. Moreover, it is simple and the constrained kinematics are also simple. Finally, it will be shown in the next section that the workspace by this constraint is an appropriate realization of the expected workspace in Section The constrained kinematics has a following analytical form; maxima is discontinuous at point B where H (.3,.3, ) t. After passing this H, θ approaches another maximum, either C or E. Suppose the case of C here. If after this motion the command path of H is reversed back to O, θ never goes back to B but follows CD, the other line of maxima. Finally, θ does not satisfy Eq. 7 2 when H returns to O. Thus, such a method is not a possible candidate. This problem is derived from the fact that an equilibrium point by a gradient method, i.e., nominal θ for a given H is neither unique nor continuous. The condition of Eq. 7 1 requires that θ must be uniquely determined by an inversion from H to θ. Thus, this unique inversion is required for the exact steering law Constrained Control Manifold of H=(.5,.5,) t Fig Discontinuity in the maximum of det(cc t ). Rounded rectangles are parts of the manifold for H on the z axis. Dots indicate the local maxima of det(cc t ). In kinematics, characteristics of unique inversion are realized by utilizing direct constraints of variables 68). By using some algebraic relation of variables as constraints, inversion of the constrained kinematics becomes a one-to-one and continuous inside of some range, which specifies the workspace. For a four dimensional system, it is adequate to constrain one degree of freedom. Though various constraining conditions were possible, the following was applied 39) : c * cosφsinψ + sinφsinγ H = 2 sinφsin ψ c * cosφsinγ, s * sin φ(cosψ + cos γ) where θ = (φ+ψ, φ+γ, φ ψ, φ γ). (7 6) Reduced Workspace The allowed workspace of this system is defined by keeping unique inversion feature within the domain of three variables, φ, ψ, γ, to [ π 2, π 2]. Figure 7 18 shows possible regions of H in several cross sectional plane orthogonal to the z axis. As an envelope of each region is not simple enough to be handled by the momentum management procedure of a controller, an approximation is required. An example approximation is made where H z 2s*, which is shown by rounded squares in the same figure. These rounded squares are defined by the following equation; where 2( c* cp sq) H = 2( sp c * cq) H z, (7 7) Hz 2s*, s = Hz (2s*), c = (1 s 2 ) 1 2, p = sin 1 2 s and q = cos 1 2 s. By using this approximation, a workspace of the constrained steering law is defined as illustrated in 3- dimension in Fig The reduced workspace has the same maximum length as the maximum workspace in the z axis, but the 52

68 7. Global Problem, Steering Law Exactness and Proposal y Angular Momentum Envelope Internal Singular Surface y 2c* x x 2(1+c*) Cross Section of Allowed (a) H z =. Workspace (b) H z =.4 y Approximated Workspace by Eq. 7 7 y x x (c) H z =.75 (d) H z =1. y y y x x x (e) H z =1.4 (f) H z =2. (g) H z =2.6 Fig Cross section of possible workspace by constrained steering law. Possible region of angular momentum given by Eq. 7 5 is drawn by two parameter net of (ψ, γ) under condition that H z is constant and the determinant of the Jacobian is positive. Approximated workspace is defined by Eq

69 Technical Report of Mechanical Engineering Laboratory No.175 z 4s* type of mechanism but flexible wires can be used for power supply. x 2c* y 2c* (3) Simplicity of Calculation Even though the proposed steering law involves calculation of an inverse Jacobian, which in turn involves inversion of a 3 3 matrix, this is simpler than calculation of the pseudo-inversion of a 3 4 matrix. Also, no complex calculations for the gradient method are needed. The actual implementation is also simple, though it does use some feedback. Implementation details are described in Appendix E. Because of the simplicity, this method actually installed on the experimental computer needed about 2 3 memory storage and about 1 2 calculation time compared with the gradient method with an objective function det(cc t ) (see Appendix E.5). Fig Reduce workspace of the constrained system. minor diameter on the xy plane is only about 1 3 that of the maximum workspace Characteristics of Constrained Control The followings are characteristics of this method. Most of all are useful for the real usage in the attitude control system. (1) Exactness and Repeatability The inversion of Eq. 7 6 is not exactly one-to-one. In order to maintain a one-to-one feature, H must be kept inside the previously defined workspace. By adhering to this limitation, continuous control over this space is strictly guaranteed. Moreover, unique inversion characteristic of the steering law assures repeatability. (2) Gimbal Limits Because of the uniqueness, each gimbal angle is exactly within a certain domain. The domains of φ, ψ, and γ are included in the domain [ π 2, π 2]. Each gimbal angle, θ i, is therefore inside the domain [ π, π]. This is very advantageous compared with other steering laws. With a gradient method for example, the domain of θ is not defined. Gimbal angles greater than one revolution are observed in results of some computer simulations. As a result, mechanisms such as a slip ring is needed to permit free rotation of the gimbal. In contrast, the method described here does not require this (4) Modes and Mode Changing The constraint of Eq. 7 5, the kinematics of Eq. 7 6 and the workspace of Eq. 7 7 defines one constrained system. As the original unconstrained system has symmetry, this constrained system can be symmetrically transformed. There are six possible transformations whose representations in the H space are the identical transformation, a mirror transformation about x-z plane and ±2/3π rotation about g 1 with or without the mirror transformation. By those transformations, six constrained systems are defined which have their own constraint condition and own workspace, and have the similar properties such as exactness. The six constrained systems makes three pairs. These pairs are called modes and termed M1, M2 and M3. The workspace of each pair has a shape similar that shown in Fig. 7 19, and the dominant direction lies along the z-axis for the M1 mode, along (1, 1, ) t for the M2 mode, and along (1, 1, ) t for the M3 mode as shown in Fig The nominal gimbal angles, which correspond to H=(,, ) t are of the form of (ψ, ψ, ψ, ψ), where ψ= or π for the M1 mode, 1/3π or 2/3π for the M2 mode, and 1/3π or 2/3π for the M3 mode. Since the dominant directions of all the workspaces are orthogonal to each other, attitude control performance will be improved by introducing mode switching. Different modes share a region in H space inside of which we can select and change modes. When it is required to change the steering law mode, gimbal angles must be changed to satisfy another constraint while keeping the same H. There is, however, no continuous path from θ of one mode to θ of another mode without a 54

70 7. Global Problem, Steering Law Exactness and Proposal z z (1, -1, ) t z x y x y x y (1, 1, ) t (a) M1 (b) M2 (c) M3 Fig. 7 2 Reduce workspaces of three modes. change in H, except for H=. (When H=, there exists a mode connection path given by θ (1, 1, 1, 1).) Therefore, operations like feedback attitude control should be deferred until the switching process is completed. In the experiments, the following simple method was applied. Here, one specifies a condition such that H is on the dominant direction of the newer mode (e.g. the z- axis of the M1 mode). The gimbal angle for H along the z-axis is acquired by a direct inverse calculation of Eq. 7 6, as follows: φ = sin 1 (H z 2s*), ψ = γ =. (7 8) The simplest way of changing θ from the current to the above is a motion along a line. This gimbal motion causes undesired torque but its effect can be made small. Since H is the same for the initial and the final θ in the CMG coordinate frame, the initial and the final angular momentum of the spacecraft alone may be similar when this motion is made fast enough. Thus, this motion will result in small deviation of the spacecraft s orientation and this deviation can be easily corrected by the feedback attitude control once the mode is changed. Though mode changing while H is not on any principal axes cannot be specified by an analytical solution, iterative numerical solution can be applied to find a goal gimbal angles by using the solution 7 8. (5) General Skew Case and the Maximum Spherical Workspace The proposed method does not depend on a specific configuration symmetry. Equation 7 6 is satisfied with the constraint of Eq. 7 5 for any value of the skew angle α as long as the four units are set symmetrically about the z axis. For any α, the workspace size to the z direction is 4 sinα, while that of x or y direction is a little less than 2 cosα as shown in Fig If a smaller skew angle α is used, the workspace becomes shorter in the z direction and wider in the x and y directions. In this manner an arbitrary design of the workspace shape can be obtained. Of course, the original symmetry of the regular octahedron is lost in an arbitrary skew angle α and only one mode in the item (4) is available. If the skew angle α = tan 1 (1 2), the size of the workspace along the x, y and z axes is almost identical. This configuration therefore gives the maximum unidirectional workspace size. If a spherical workspace is desired for convenience of the attitude control, this is the best configuration of four unit systems. Application of the constrained control is not limited to skew type systems. Any four unit system can be controlled using one constraint. If the system does not have symmetry, however, a simple constraint as Eq. 7 5 may not be effective. In Appendix D, the same constraint as Eq. 7 5 will be applied to the four unit subsystem of the MIR-type, i.e., S(6) system 69). (6) Performance Performance of the proposed steering law was demonstrated by using ground-based test equipment. The results are detailed in the following chapter. Also, the pyramid type system controlled by this steering law was evaluated by comparing with other type CMG systems in terms of the workspace size. The results are detailed in Chapter 9. 55

71 Technical Report of Mechanical Engineering Laboratory No

72 8. Ground Experiments Chapter 8 Ground Experiments The previous chapters dealt only with CMG systems while aiming primarily at a geometric understanding of the CMG motion and leaving quantitative matters neglected. In this chapter, a CMG-based total attitude control system is briefly formulated and in order to quantitatively verify the steering law performance, ground experiments were carried out. Test results showed clearly the problems found with typical steering laws and the performance of the steering law proposed in the previous chapter. 8.1 Attitude Control Dynamics The attitude, namely the orientation of a body can be represented by various ways 7) such as the direction cosines, the Euler angles, Roll-Pitch-Yaw angles, Rodrigues parameters and Euler parameters. In this work, representation is made using Euler parameters. Any attitude is defined as is caused by a single rotation. The Euler parameters β = (β, β 1, β 2, β 3 ) represent an attitude caused by a single rotation of angle φ about the axis e = (e 1, e 2, e 3 ) t : β = cos(φ / 2), (8 1) β i = e i sin(φ / 2), where i = 1, 2, 3 and e =1. As the rotation has three degree of freedom, there is a constraining condition that Σ β i 2 = 1. Any attitude, which is the result of a rotation a = (a, a 1, a 2, a 3 ) after a rotation b = (b, b 1, b 2, b 3 ), is expressed as a multiplication by a b in the sense of a Hamiltonian quarternion as follows: a b = (a b a 1 b 1 a 2 b 2 a 3 b 3, a b 1 + a 1 b + a 2 b 3 a 3 b 2, a b 2 a 1 b 3 + a 2 b + a 3 b 1, a b 3 + a 1 b 2 a 2 b 1 + a 3 b ). (8 2) The time derivative of β, i.e., dβ dt is expressed by angular velocity denoted by ω V : where dβ* dt = 1 2 (β + β* ω V ), (8 3) β* = (β 1, β 2, β 3 ) t. (8 4) This β* is called a vector part of β and regarded as a usual vector in three-dimensional physical space. The attitude dynamics of a rigid body is represented in the body s coordinates by the following Euler equation: I V dω dt = τ ω V p, (8 5) where I V denotes satellite s moment of inertia and τ denotes the torque applied to the satellite. This torque comes from both the outside as a disturbance torque and from the inside by the CMG system. The vector p is the total angular momentum of both the satellite and the CMG system and is given by: p = I V ω V + H CMG. (8 6) This total angular momentum is conserved in the inertial coordinates if there is no disturbance torque. By substituting Eq. 8 6 into Eq. 8 5, the ω V H CMG term appears. This term is omitted in Eq. 3 5, i.e., the output equation of the CMG but is evaluated here. Both the kinematic equation and the dynamic equation, Eqs. 8 3 and 8 5, are the describing functions of the system Exact Linearization The system has six independent variables, three components of β* and three components of ω V. The dynamics is nonlinear as seen by the term ω V (I V ω V ) when Eq. 8 6 is substituted into Eq Nevertheless, it is well known that this nonlinear system can be exactly linearized with a suitable feedback 71, 72). Let (β*, dβ* dt) be state variables and let v be a new input variable. If a real input τ is given as: 57

73 8. Ground Experiments Chapter 8 Ground Experiments The previous chapters dealt only with CMG systems while aiming primarily at a geometric understanding of the CMG motion and leaving quantitative matters neglected. In this chapter, a CMG-based total attitude control system is briefly formulated and in order to quantitatively verify the steering law performance, ground experiments were carried out. Test results showed clearly the problems found with typical steering laws and the performance of the steering law proposed in the previous chapter. 8.1 Attitude Control Dynamics The attitude, namely the orientation of a body can be represented by various ways 7) such as the direction cosines, the Euler angles, Roll-Pitch-Yaw angles, Rodrigues parameters and Euler parameters. In this work, representation is made using Euler parameters. Any attitude is defined as is caused by a single rotation. The Euler parameters β = ( β, β 1, β 2, β 3 ) represent an attitude caused by a single rotation of angle φ about the axis e = (e 1, e 2, e 3 ) t : β = cos(φ / 2), (8 1) β i = e i sin(φ/ 2), where i = 1, 2, 3 and e =1. As the rotation has three degree of freedom, there is a constraining condition that Σβ i 2 = 1. Any attitude, which is the result of a rotation a = (a, a 1, a 2, a 3 ) after a rotation b = (b, b 1, b 2, b 3 ), is expressed as a multiplication by a b in the sense of a Hamiltonian quarternion as follows: a b = (a b a 1 b 1 a 2 b 2 a 3 b 3, a b 1 + a 1 b + a 2 b 3 a 3 b 2, a b 2 a 1 b 3 + a 2 b + a 3 b 1, a b 3 + a 1 b 2 a 2 b 1 + a 3 b ). (8 2) The time derivative of β, i.e., dβ dt is expressed by angular velocity denoted by ω V : where dβ* dt = 1 2 β ω V + β* ω V, (8 3) β* = (β 1,β 2,β 3 ) t. (8 4) Thisβ* is called a vector part of β and regarded as a usual vector in three-dimensional physical space. The attitude dynamics of a rigid body is represented in the body s coordinates by the following Euler equation: I V dω dt = τ ω V p, (8 5) where I V denotes satellite s moment of inertia and τ denotes the torque applied to the satellite. This torque comes from both the outside as a disturbance torque and from the inside by the CMG system. The vector p is the total angular momentum of both the satellite and the CMG system and is given by: p = I V ω V + H CMG. (8 6) This total angular momentum is conserved in the inertial coordinates if there is no disturbance torque. By substituting Eq. 8 6 into Eq. 8 5, the ω V H CMG term appears. This term is omitted in Eq. 3 5, i.e., the output equation of the CMG but is evaluated here. Both the kinematic equation and the dynamic equation, Eqs. 8 3 and 8 5, are the describing functions of the system Exact Linearization corrected July, 211 The system has six independent variables, three components of β* and three components of ω V. The dynamics is nonlinear as seen by the term ω V (I V ω V ) when Eq. 8 6 is substituted into Eq Nevertheless, it is well known that this nonlinear system can be exactly linearized with a suitable feedback 71, 72). Let (β*, dβ* dt) be state variables and let v be a new input variable. If a real input τis given as: 57

74 Technical Report of Mechanical Engineering Laboratory No.175 where τ = ω V I V ω V + (2 β ) I V (β 2 µ + β*β* t µ β β* µ), (8 7) µ = v ω V t ω V β*, (8 8) Three Axis Gimbal CMG3 Support Rod g 3 Rate Gyroscope g 2 CMG2 the system becomes a set of second order linear systems described by: 75mm d 2 β* dt 2 = v. (8 9) Control Method Several controllers given by linear control theory can be applied to the above linearized system. In the ground tests done in the experimental portion of this work, a model matching controller and a tracking controller were used. Design of the model matching controller was made by applying a model transfer function in order to satisfy specified steady and transient properties 54). The tracking controller allowed an appropriate motion of the angular momentum vector to be designed. Tracking PD control of a given trajectory is realized by the following input: g 4 h 4 CMG4 Onboard Computer Balance Adjusters Rotary Encoder Fig. 8 1 Experimental test rig showing the center-mount suspending mechanism. This figure shows that the pyramid configuration can be realized so that all four units fit the surfaces of a rectangular parallelepiped. z x g 1 CMG1 y v = f 1 (β* r) + f 2 (dβ* dt r 1 ) + r 2,(8 1) where r(t) is the trajectory to be followed and r 1 and r 2 are its time derivatives, defined by: r 1 (t)=dr dt, r 2 (t)=d 2 r dt 2. (8 11) Details of both are described in Appendix E. 8.2 Experimental Facility and Procedure In order to quantitatively demonstrate the problems and performance of steering laws, a ground test facility was constructed 54, 55) and a set of ground tests was carried out Facility The ground test facility shown in Fig. 8 1 simulates the attitude dynamics of a spacecraft. The main structure is a cubic frame made of steel pipes and joints. Triangular plates on several surfaces holds such devices as the CMG units, the CMG driver circuits, balance adjusters and a computer. The system in its entirety was suspended from the ceiling by a three axis gimbal mechanism in its center. If the center of the gimbal coincides with the body s center of gravity, no torque appears at any orientation due to the gravitational force. In this way motion in space can be simulated. This situation was realized by using three balance adjusters, which could control their weight along three orthogonal axes, which allowed the center of gravity to be controlled. These mechanisms were also used for initial set up without CMG control, and generation of disturbance torque and unloading. The orientation and angular rate of the body were measured by rotary encoders at the three axis gimbal and rate gyroscopes. In actual satellites these quantities are measured by various sensors such as star/sun/earth sensors and rate gyroscopes. The main torquer was a pyramid type single gimbal CMG system. All attitude control and the steering law processes were installed in an onboard computer. A wireless link was used for command transfer from the stationary computer. All power was supplied from the laboratory by a pair of thin wires which caused little disturbance force. Additional details are presented in Appendix E. 58

75 8. Ground Experiments Design of Control Command Sequence Figure 8 2 shows a typical tracking control trajectory. This function is continuous with regard to the first time derivative. It consists of a constant acceleration, a constant speed rotation, a constant deceleration, a constant attitude and then the same sequence in reverse. The maximum rotation of this trajectory was set in consideration with the limit of rotation of the supporting gimbal. The magnitude of the acceleration and the deceleration which are almost proportional to the CMG output torque was set as large as possible so that the friction torque of the supporting gimbal can be neglected Experimental Procedure Tests were conducted using the software whose block diagram is shown in Fig Additional details of each block are described in Appendix E. The control command sequence for each experiment was a sequence of a number of maneuver motions given either as reference attitude or as a trajectory in Fig After each maneuver, the attitude returned to the original position and rotation of the body ceased. To allow comparison, three types of steering laws were tested: a gradient method (abbreviated to GM hereafter), a SR inverse method (abbreviated to SR) and the constrained method (abbreviated to CM) proposed in Chapter 7. The GM uses procedure described in Section Its objective function is det(cc t ) and the free parameter is the gain k defined in Eq The Rotational angle φ (1) (2) (3) (4) (5) t 1 t 2 t 1 t 3 t 1 t 2 t 1 t 3 time Fig. 8 2 Target trajectory. This trajectory has eight parts as, (1) constant acceleration by d 2 φ dt 2 = a, (2) constant rate rotation, dφ dt=at 1, (3) constant deceleration by d 2 φ dt 2 = a, (4) pointing control at φ =at 1 2 +at 1 t 2, (5) to (8) are the reverse of (1) to (4). SR is represented by Eq The CM method was introduced in Section 7.5 and allows one free parameter, a feedback gain denoted by k. Further details of the GM and the CM implementations are given in Appendix E. The test procedure can be outlined as follows: First, the body was controlled at a nominal attitude of β*=(,, ) t by using only the balance adjusters along with the PID controller. This control mode was done in order to wait until the pendulum motion of the body and the supporting rod stopped. Then the CMG control started with a sequence of attitude commands consisting of either an attitude reference in the model matching controller case, or a trajectory in the tracking controller case. The balance adjusters were controlled so that they generated an expected disturbance torque during the experiment, but which was zero otherwise. (6) (7) (8) Reference Attitude or Target Trajectory Attitude Command Generator Attitude Controller Torque Command T COM CMG Steering Law Momentum and Disturbance Management Desired Motion ω Proportional Limiter Balance Adjusters Pyramid Type CMG System Torque Output T Body Attitude and Rotational Rate Fig. 8 3 Block diagram of the control system. Proportional limiter is used to limit the gimbal rate, ω com, to the maximum gimbal rate so that the real rate vector is proportional to the desired vector. By using this, the real output becomes proportional to the torque command, i.e., T com T. 59

76 Technical Report of Mechanical Engineering Laboratory No Experimental Results The experiments were carried out to demonstrate the following problems and performance characteristics of the CMG steering laws 69). (1) Performance and problems of a singularity avoidance steering law such as SR-inverse law. (2) Problem of z axis rotation and advantage of preferred gimbal angles (3) The gradient method s inability to keep a nominal condition (4) Performance in various modes of constrained control (5) Advantages of mode switching. The results of the following experiments are drawn in five graphical forms with time on the horizontal axis. The first graph shows the ideal and measured attitude variation on the vertical axis. Other variables on the vertical include the measured gimbal angles, the output torque level and the angular momentum of the CMG system. Also shown is the determinant det(cc t ), which provides information regarding the proximity of the system to a singular point. Note that the output torque T here is an actual value with the multiplier h but the angular momentum H is still without the multiplier (see Section 3.1). The third experiment was made from another initial gimbal angles, which are one of the preferred angles for this direction. The results in Fig. 8 6 shows that this initial angles are appropriate for this situation. Table 8-1 Condition and Results of Experiments (1) Experiment Initial θ Steering Results law Experiment A (,,, ) GM Fig. 8 4 Experiment B (,,, ) SR Fig. 8 5 Experiment C ( π/3, π/3, π/3, π/3) GM Fig Attitude Keeping under Constant Disturbance By the following three experiments, the item (1) was demonstrated. The H path along ( 1, 1, ) t direction was planned and pointing control under the constant disturbance about this direction was carried out by the model matching controller. The conditions of the three experiments are listed in Table 8 1. The reference attitude and the disturbance were kept constant and the angular momentum of the CMG system, i.e., H continuously increased along ( 1, 1, ) t direction when the pointing control was successful. The first two experiments clarifies the impassable singularity problem of the gradient method and performance of singularity avoidance by the SR-inverse steering law. By the GM, the system became singular (AB in Fig. 8 4 (f)), after that no torque was generated and the body rotated by the disturbance torque (AA in Fig. 8 4 (a)). On the contrary, the SR worked well with slight degradation of pointing accuracy (BA in Fig. 8 5). 6

77 det H T (N m) θ β Ground Experiments β 2 β 3 β (a) attitude 1 time(s) 2 3 θ 4 θ 1 θ 3 θ 2 1 time(s) 2 3 (b) gimbal angle T com T out 1 time(s) 2 3 H y H z H x (c) torque command & output -2 1 time(s) 2 3 (d) CMG momentum (normalized) AB (e) determinant time(s) Fig. 8 4 Results of Experiment A. The attitude keeping by the gradient method under constant disturbance torque about ( 1,1,) t direction from initial θ of (,,, ). 61

78 det H T (N m) θ β Technical Report of Mechanical Engineering Laboratory No θ 4 1 θ 1 θ2-1 θ (b) gimbal angle time(s) β β 3 2 β (a) attitude time(s) T out T com time(s) H y H z H x (c) torque command & output -2 1 (d) CMG momentum (normalized) time(s) (e) determinant time(s) Fig. 8 5 Results by Experiment B. The attitude keeping by the SR method under constant disturbance torque about ( 1,1,) t direction from initial θ of (,,, ). 62

79 det H T (N m) θ β 8. Ground Experiments (a) attitude β 2 β 3 β time(s) θ θ 2 4 θ 1 θ (b) gimbal angle time(s) T com T out 1 time(s) 2 3 (c) torque command & output H y H z H x -2 1 time(s) 2 3 (d) CMG momentum (normalized) (e) determinant time(s) Fig. 8 6 Results of Experiment C. The attitude keeping by the gradient method under constant disturbance torque about ( 1,1,) t direction from initial θ of ( π 3, π 3, π 3, π 3). 63

80 det H T (N m) θ β Technical Report of Mechanical Engineering Laboratory No Rotation About the z Axis By the next three experiments, the above item (2) was demonstrated. The H path along the z axis was planned which is symmetric to the H path of the previous three experiments. This time, maneuvering motion was performed. For attitude control about the z axis, the H trajectory is also on the z axis. This H path intersects a singular surface. There is no singularity problem from (a) attitude 1-1 the nominal θ = (,,, ), while there is a problem of impassability from another initial θ. The conditions of the three experiments are listed in Table 8 2. In the experiments, a command sequence consisting of two maneuver motions was used when the model matching controller was operating. The reference attitude to the controller was changed twice, at first to an orientation rotated 3 degrees about the z axis for t 1 seconds then to the initial orientation for 1 seconds 1 2 (b) gimbal angle time(s) β 1, β 2 β 3 Reference 1 2 time(s) T com T out 1 2 time(s) H z H x H y (c) torque command & output 1 2 time(s) (d) CMG momentum (normalized) (e) determinant 1 time(s) 2 Fig. 8 7 Results of Experiment D. The z-axis maneuver from initial θ of (,,, ) by the gradient method. 64

81 det H T (N m) θ β 8. Ground Experiments t. As the initial gimbal angles of Experiment D is preferred gimbal angles for this direction, the system did not approach any singular point as shown in Fig. 8 7 and smooth maneuvering was performed as is the analytical result of Section 7.1. On the contrary, in Experiment E, the CMG system became singular at t 2.3 second (EC in Fig. 8 8 (e)). For this reason, GM Table 8-2 Condition and Results of Experiments (2) Experiment Initial θ Steering Results law Experiment D (,,, ) GM Fig. 8 7 Experiment E ( π/3, π/3, π/3, π/3) GM* Fig. 8 8 Experiment F ( π/3, π/3, π/3, π/3) SR Fig. 8 9 (GM* is a modified GM).1 β -.1 1, β 2 β (a) attitude time(s) 4 θ 4 2 θ 2 θ3 θ (b) gimbal angle time(s) T com T out 1 2 time(s) H 2 z H x H y 2-2 (c) torque command & output (d) CMG momentum (normalized) time(s) (e) determinant time(s) Fig. 8 8 Results of Experiment E. The z-axis maneuver from initial θ of ( π 3, π 3, π 3, π 3) by the modified gradient method. 65

82 det H T (N m) θ β Technical Report of Mechanical Engineering Laboratory No.175 was modified so that the transpose Jacobian was used on behalf of the pseudo-inverse Jacobian when the system was nearly singular. By this, the above singular state was avoided but the element of H to the z axis, i.e., H z once saturated at smaller value than that of the maximum in Experiment D (EB in Fig. 8 8 (d) and DB in Fig. 8 7 (d)). As the result, the transient motion of the control was degraded compared with the Experiment D (EA in Fig. 8 8 (a) and DA in Fig. 8 7 (a)). This time, the SR was not effective as shown in Fig The CMG system approached a singular point similar to the above experiment (FC in Fig. 8 9 (e)). Moreover, no singularity avoidance motion was realized because the command changed faster than that in Experiment B. β β 1, β (a) attitude time(s) θ 2 θ 4 θ 3 θ time(s) T com T out (b) gimbal angle 1 2 (c) torque command & output time(s) H z H x H y (d) CMG momentum (normalized) time(s) time(s) (e) determinant Fig. 8 9 Results of Experiment F. The z-axis maneuver from initial θ of ( π 3, π 3, π 3, π 3) by the SR method. 66

83 det H T (N m) θ β 8. Ground Experiments Maneuver after Momentum Accumulation In Section 7.1, it was concluded that a gradient method cannot always keep the nominal θ given by Eq. 7 2 after H travels around and back to the origin. In order to demonstrate this problems, i.e., the above item (3), a control sequence including maneuvering motions as well as momentum accumulation was used while the system was controlled by the model matching controller. Two experiments were carried out, i.e., Experiment G by CM and Experiment H by GM for the same control sequence. The results of these are shown in Fig. 8 1 and Fig The control sequence consists of five parts. The reference orientation for the regulator is shown in Fig. 8 1(a). The first part of the sequence is a maneuver about the z axis, shown as the part A in Fig. 8 1(a). This command is similar to that of Section In the second part B, a disturbance torque was.1 β reference(β 3 ) GA time(s) (a) attitude θ 3 θ 4 GB θ θ time(s) T com T out (b) gimbal angle time(s) (c) torque command & output A B C D E H x H z H y GC (d) CMG momentum (normalized) time(s) 2 4 time(s) (e) determinant Fig. 8 1 Results of Experiment G. Control by the proposed constrained method. 67

84 det H T (N m) θ β Technical Report of Mechanical Engineering Laboratory No.175 applied to move H to (.6,.4,.) t while the body s orientation was fixed (see GC in Fig. 8 1(d)). This motion is similar to that of Section The third part, indicated by C, is the same as the first part. Then in the part D a reversed disturbance was applied to move H back to its origin similarly to the part B. Both processes B and D resulted in a similar H path as described in Section Finally, the same maneuver as the part A was tried (E in Fig. 8 1(a)). Those results in Fig. 8 1 and Fig show that the motions of the first three parts of the two experiments are almost the same. This implies that both the steering laws have similar control performances. In the fourth part, however, motions of θ are different. The gimbal angle θ returned to the original in the case of CM (GB in Fig. 8 1) but to the different point in the case of GM (HB in Fig. 8 11). This is because GM controlled the CMGs to another local maximum of det(cc t ) as described in As the result of this, the last part was successful in the case of CM (GA in Fig. 8 1), while it A B C D E β 3 reference HA time(s) (a) attitude θ HB 4 θ 3 θ 2 θ time(s) T T com out (b) gimbal angle time(s) (c) torque command & output H x H z HC H y 2 4 time(s) (d) CMG momentum (normalized) HD HE 2 4 time(s) (e) determinant Fig Results of Experiment H. Control by the gradient method. 68

85 β 8. Ground Experiments was not in the case of GM (HA in Fig. 8 11). In the case of GM, the determinant went to zero and H saturated (HC in Fig. 8 11) by hitting or approaching some singular surface (HD and HE in Fig. 8 11). These two results clarifies that the unique inversion characteristic of CM is important even for such a simple maneuver Mode Selection and Switching The next test was carried out using only the CM in order to demonstrate the above items (4) and (5). The different modes of the constrained method have different workspaces of H, as described in Section 7.5.4(4). In Experiment J, a maneuver motion resulting in two kinds of the H paths were planned for which different control modes were required and mode switching was performed. In this experiment, trajectory tracking control was used for the attitude control. The target trajectory for this tracking control is shown in Fig This consisted of the following motions: A maneuver about the z axis shown by the part A in Fig was planned by a trajectory defined in Section After this maneuver, a disturbance was applied in the part B so that the angular momentum was accumulated to H=(.5,.5,.) t. Then in the part C, the z axis maneuver was repeated. As the H path for these three parts are in the workspace of the M1 mode, this mode is selected. The maneuver command of the latter parts was designed so that the resulting H path went out of the M1 mode workspace but was inside the M2 mode one. For this reason, mode switching in the part D was conducted before the next maneuver. Then, the new principal axis was in the (1,1,) t direction. In the part E, a maneuver was performed about this new direction, then in the part F another disturbance was applied to accumulate H to (1.2,.8,.) t and finally in the part G the maneuver about the (1,1,) t direction was repeated. The results are shown in Fig The first three parts A, B and C, show almost the same motion as in the previous experiment. In the part C there was a slight degradation of control compared with the response in the part A. In the part D, the mode of the CM was changed directly, without considering the attitude control and by using the fastest direct path to the other mode as described in Section (4). This motion inevitably generated an undesired torque and there was some attitude deviation of the body, as indicated by JA in Fig However, because the CMG was moved as fast as possible, this deviation was somewhat minimized. The attitude control immediately following this motion easily corrected such a deviation. As the theory in item (4) of Section predicts, the two maneuver motions in the M2 mode, shown in the parts E and G of Fig. 8 13, were successful and did not meet a singularity. This experiment demonstrated the advantages of the constrained method proposed in the previous chapter. In addition, trajectory tracking control was used this time. The results in Fig. 8 13(c) show that almost constant torque was realized during the period of constant acceleration or deceleration. The proposed method can also cope with a change of maneuver direction by switching between modes. Even by using a direct change of the mode, deviation in attitude can be made small enough to be corrected by the attitude control. 8.4 Summary of Experiments From experiments A to F, it is observed that an appropriate combination of the initial gimbal angles and the maneuver direction (or momentum accumulation A B C D E F G.4.3 β 3 β 3.2 β 1, β 2 β 1, β β 2 1, β 2.1 β time(s) Fig Command sequence of Experiment J. Maneuver motions in A and C are the same rotations about the z axis. Maneuver motions in E and G are the same rotation about the (1,1,) t axis. In periods B and F, a disturbance torque was applied so that the final H becomes (.5,.5,.) t in B and (1.2,.8,.) t in F. The CM mode is changed at D. 69

86 det H T (N m) θ β Technical Report of Mechanical Engineering Laboratory No.175 direction) is the most important. This implies that the method of preferred gimbal angles described in Section 7.4 is effective. Though the experiments showed the capability of singularity avoidance of the SR inverse method, its performance was worse than that when the initial gimbal angles were set appropriately. Moreover, its performance of singularity avoidance depends on the speed of momentum accumulation and it sometimes fails. Experiments G and H clarified the advantage of the proposed constraint method. This method not only preserves the merit of the above preferred gimbal angle method but also realizes continuous steering and repeatability. Though the workspace of each mode is restricted, it can maintain nearly continuous steering for various directional maneuver and/or momentum accumulation events by using the proposed mode switching operation. This was successfully demonstrated by Experiment J. A β B C D E F G 3 JA β 1, β (a) attitude time(s) θ θ 3 1 θ θ (b) gimbal angle 4 6 time(s) T com 2 1 T out time(s) (c) torque command & output H x H z H y time(s) (d) CMG momentum (normalized) (e) determinant time(s) Fig Results of Experiment J. Tracking control for the command given in Fig. 8 7, illustrating use of the proposed constrained method. 7

87 9. Evaluation Chapter 9 Evaluation The previous chapters dealt mainly with a specific pyramid type CMG system. In order to evaluate its performance, comparison with regard to the workspace and weight was made for various system configurations. 9.1 Conditions for Comparison The previous chapters revealed that it is generally difficult to avoid impassable singularities. Most steering laws have various problems. The only exception is the constrained steering law proposed in Section 7.5, whose performance is verified within a certain workspace. But this method is only effective for the pyramid type CMG system, and if another configuration is used, a gradient method is the only candidate. Thus, evaluation of various systems was made under the assumption that a gradient method is used and the work space is determined so that it does not include any impassable surfaces. S(4), S(6) and S(1). (b) Skew Type of 5 and 6 units, denoted by Skew(n). (c) Multiple Type M(m,m) and M(m,m,m) with orthogonal gimbal axes. In addition to those, the following systems were selected for comparison. A system denoted by 2 Skew(n) is a doubled skew configuration. The system denoted by 1+Skew(n) and shown in Fig. 9 1 (a) indicates that one unit is added to the symmetric axis of a skew type system. The system denoted by S(3,4) is a combination of two symmetric configurations, S(3) and S(4) as shown in Fig. 9 1 (b). The units are arranged in the surface directions and the vertex directions of a regular octahedron. g 5 g 4 g n+1 α g Spherical Workspace g 6 2π n g 2 Several CMG systems were examined including double gimbal CMGs 32, 73) with the following criteria r and χ : r : χ : Maximum radius of a sphere in the angular momentum space, centered on the H origin, and including no impassable surfaces. = r n. g 1 (a) Example of 1+Skew(n). g 7 g 4 g 3 Obviously, r = n for all double gimbal CMG systems because their work space is a unit sphere of radius n (Appendix A). The radius of any multiple type system is obtained simply, because its envelope has circular plates which the maximum sphere touches. Thus, the radius r of M(m, m) with orthogonal axes is m and that of M(m, m, m) with orthogonal axes is 2m. Various CMG configurations of up to ten units were examined. These included: (a) Symmetric Type, S(3), g 6 g 2 g 1 g 5 (b) S(3, 4). Fig. 9 1 System configurations for comparison. 71

88 Technical Report of Mechanical Engineering Laboratory No.175 r : Radius of Spherical Workspace SKEW(6) M(2,2,2) Double Gimbal CMG S(6) 1+SKEW(6) S(3,4) 2 S(4) Envelope of S(4) M(4,4) M(3,3). S(4) 1+SKEW(4) SKEW(5) 5 1 n : Number of Units 2 SKEW(5) χ=.75 χ=.67 S(1) M(3,3,3) M(5,5) 1+SKEW(7) Fig. 9 2 Spherical workspace size for various system configurations. Filled circles indicates the workspace size of the original system, while attached open circles indicate the workspace size of degraded systems. As a reference, a square indicates the envelope size of the S(4) system. The radius r was commonly obtained by computer calculation. It was searched by calculating H for impassable surfaces of all lattice points of u on the unit sphere at a given increment. In the case of skew type system, calculations were made using various values of the skew angle and the maximum radius r is sought. The various possibilities of unit s breakdown are also considered and the worst cases are taken, except in the case of multiple systems for which all possible cases are considered. The results are shown in Fig. 9 2 as a graph of the number of units n versus workspace radius r. The ratio χ is the slope of the line from the origin. Filled circles indicate values of original systems, while blank circles connected to them by straight lines indicate the performance of degraded systems. Conclusion from these results are as follows: a) As the number of units increases, the shape of the angular momentum envelope approaches a sphere and χ also approaches a limiting value given by: χ = ( h u) ds u S 2 S u S 2 ds = π 2 sin 2 φφ d = π / 4.765, (9 1) which is for an infinite number of units arranged equally in all directions. b) Most systems of no less than six units have respectable χ values ranging from.67 to.75. This is because although such systems have internal impassable surfaces, they are only near the envelope. On the contrary, four and five unit systems have smaller χ values because they have internal impassable surfaces much further inside. c) Although any multiple type system composed of no less than six units has no internal impassable surfaces, its radius r is considerably smaller than that of other independent systems. d) Degradation of the system due to unit s break down becomes smaller as the number of units increases. 9.3 Evaluation by Weight The workspace size and system weight will be evaluated in light of the preceding results. Suppose that the work space size H and system weight W satisfy the similar relation as Eqs. 2 1 and 2 2, which are W n d 3 and H r d 5 where d is the diameter of the flywheel. Then, the following relationship is obtained by setting 72

89 9. Evaluation W as a parameter while H is set constant: Z r 3 W 5 n 5 = constant. (9 2) Figure 9 3 shows results of this comparison. Dotted curves indicate the relationship of n and r which satisfy the condition 9 2. While the W = a curve passes the S(6) point, the other points are under this curve. This implies that the S(6) is the lightest for the same spherical workspace size among all systems evaluated above. As the W = 1.5a curve passes the S(4) point, the S(4) system is 5% heavier than the S(6) system. The data point labeled Skew(4, α opt ) shows the workspace of a skew type four unit system with the skew angle α = tan -1 (1 2) described in Section 7.5.4(5). As the W = 1.15a curve passes this data point, this particular system with the proposed steering law can realize Skew(4) system only 15% heavier than S(6). 9.4 Ellipsoidal Workspace An ellipsoidally shaped workspace may be required when the attitude control has a certain principal axis. In this section, skew type CMG systems of 4, 5 and 6 units are evaluated in terms of their workspace size. Systems of more than 6 units are omitted because they have disadvantages in weight as the results above. An evaluation similar to that done in Section 9.2 was made under the same condition that the workspace does not include any impassable surface. The shape of the workspace is defined axially symmetric with a fixed aspect ratio µ as shown in Fig Evaluation criteria r : Radius of Spherical Workspace 1 5 r 3 W 5 n 5 = const W = a W = 1.15a W = 1.5a S(6) SKEW(5) SKEW(4, α S(4) opt ) n : Number of Units Fig. 9 3 Trade-off between workspace size and system weight. Dotted curves indicate equal workspaces with equal weight (W). X Fig. 9 4 Definition of ellipsoidal workspace. Aspect ratio µ is defined as r 2 r 1. r 1 and r 2 (=µr 1 ) are the minimum and maximum radii of the ellipsoid. Skew type systems of 4, 5 and 6 units were examined at various skew angle values. As we have two parameters, i.e., the skew angle α and the aspect ratio µ, comparison will be made by keeping one parameter constant. By keeping the aspect ratio constant, the radii r 1 and r 2 were calculated with respect to the skew angle α as shown in Fig In these figures, the radii are represented by an average radius defined as follows; r A = r 1 r 2 = µr 1 3 r1 2 r2. (9 3) This value represents the radius of a sphere which has the same volume as the ellipsoid. For four unit systems, each resulting curve has a maximum at a different skew angle, as shown in Fig. 9 5 (a). Noteworthy is the system corresponding to the maximum of the µ=1. case, the S(4) symmetric type system. For the other aspect ratios, the optimum skew angle increases as the ratio increases. The results of five unit skew systems are different. Though there are local maxima, the global maxima for any aspect ratio are given when the skew angle is π 2. At this skew angle, all gimbal axes are on the same plane. In the case of six unit systems, we can find two groups of candidates result in the largest workspace, one has a skew angle of π 2 and the other.26 α.33. Next, the radii r 1 and r 2 were calculated with respect to the aspect ratio by keeping the skew angle constant (Fig. 9 6). In each figure, the maximum and minimum radii are plotted with respect to the aspect ratio, for various skew angles. Fig. 9 6 (a), for the Skew(4) arrangements, shows that there is an optimal skew angle r1 Y 73

90 radius r radius r radius r Average Radius r A Technical Report of Mechanical Engineering Laboratory No.175 µ=1.5 µ=1. µ=2. µ= Skew Angle α (radian) (a) Skew(4, α) Average Radius r A µ=2., 2.5 µ=1. µ=1.5 Skew Angle α (radian) (b) Skew(5, α) Average Radius r A 4 2 µ=2.5 µ=1. µ=2. µ=1.5 Skew Angle α (radian) (c) Skew(6, α) 2 α= r r Aspect Ratio µ (a) Skew(4, α).45 6 α=.5 r r Aspect Ratio µ (b) Skew(5, α) Fig. 9 5 Average radius vs. skew angle. (c) Skew(6,α) r 2 6 α= r Aspect Ratio µ giving the largest workspace for each value of the aspect ratio. For example, the optimal skew angle is.35 π at an aspect ratio of On the contrary, a π 2 skew angle is optimum for any aspect ratio in the case of Skew(5). For Skew(6), a.32 π skew angle is optimum for aspect ratio less than 1.4, while π 2 is optimum for larger aspect ratios. These optimum values are selected and plotted in Fig Radius values in this figure can be converted to indicate the system weight as discussed in Section 9.3. This converted weight W is equivalent to the weight of a system whose workspace size is a certain fixed value. By a relationship similar to Eq. 9 2, this converted weight is defined as follows; radius r 1 and r 2 Fig. 9 6 Workspace radius as a function of aspect ratio. α=.5 Skew(6, α) 6 α=.32 Skew(5, π 2) r 2 4 r 1 M(2, 2)=Roof(4) Skew(4, α) α= Aspect Ratio µ Fig. 9 7 Combined plot of radii as a function of aspect ratio. Envelope size of multiple systems, M(2, 2), is drawn in addition to the results in Fig W = n (r 1 2 r 2 ) 1 / 5. (9 4) The results are plotted in Fig The converted weight of the Skew(4, α) system, controlled by the proposed constrained method, is also included. In this evaluation, the workspace of this system is approximated by an ellipsoid whose radii are given by 2cosα and 4sinα (See Fig. 7 19). The results in Fig. 9 8 show that the weight is much larger in the case of the original Skew(4) system than the other systems. All the other systems, including the 4-unit skew system with the constrained steering law, have similar weight values. 74

91 Weight 9. Evaluation 4 2 Skew(6) Skew(5) Skew(4) Aspect Ratio µ radius r 4 Skew(6, π 2) 3 Skew(4,.4 π ) Skew(5, π 2) 2 Skew(4, α) Aspect Ratio µ Fig. 9 8 Converted weight as a function of aspect ratio. The original Skew(4) system is shown by the solid line and the Skew(4) system controlled by the proposed constrained method is shown by the heavy dashed line of Skew(4)'. The next figure, Fig. 9 9, shows a relation similar to Fig. 9 7, when one unit becomes nonfunctional. This figure shows that degradation of the Skew(6) system is much less than the others. On the contrary, degradation of the Skew(4, α) with the constrained method is serious. 9.5 Summary of Evaluation (1) For the spherical workspace, the S(6) system is superior in terms of the system weight. The Skew(4) system driven by the constrained method is only 15 % heavier in the simplified comparison. (2) For the ellipsoidal workspace, skew type systems Fig. 9 9 Radius as a function of aspect ratio for a degraded system with one faulty unit. of 4, 5 and 6 units with optimal skew angle, and with constrained control in the case of the 4 unit system, have similar workspace size with the same weight. (3) If fault tolerance is required, the Skew(6) system is much better than Skew(4) and Skew(5) in terms of degradation of the workspace due to loss of one unit. In the evaluation of this chapter, the three modes of a symmetric pyramid type system were not considered. Since the workspace of each mode is a similarly shaped ellipsoid, this pyramid type system becomes more promising by considering the three modes. In addition to this, other factors are also important, such as mechanical complexity and steering law complexity. By considering these, the Skew(4) system with the proposed constrained method is advantageous for actual use, especially the S(4) system. 75

92 Technical Report of Mechanical Engineering Laboratory No

93 9. Evaluation Chapter 1 Conclusions In this paper, the control of a single gimbal CMG system has been investigated, with emphasis on the symmetric pyramid type. Specifically, the singularity problems have been examined using geometric theories and computer calculations. The global problem of the pyramid type system has been clarified, and a new steering law approach has been proposed and verified using ground experiments. In Chapter 2, single gimbal CMGs were described in comparison with double gimbal CMGs and reaction wheels. Then in Chapter 3, an analytical formulation of general single gimbal CMGs was presented. In Chapter 4, the singularity problem was described. Methods for obtaining singular surfaces, especially the envelope, were presented. The passability of a singular surface was defined. Then, examples of some impassable surfaces of various CMG systems were given. The results showed that impassable singularity is a serious problem for the steering law of 4 and 5 unit CMG systems. In Chapter 5, continuous steering under the existence of an impassable singular surface has been generally examined by using a topological study. A method to overcome some types of impassable singularities were described in a geometric manner. Some example conditions were presented in which no steering law can realize continuous motion. In Chapter 6, the symmetric pyramid type CMG system was defined. Analytical results including symmetry and singular surface structure were presented. Chapter 7 clarified the global steering problem that continuous real-time steering cannot be realized over most of the workspace. By the consequences of this and by geometric theories, typical steering laws were evaluated. This evaluation showed that steering law exactness is the most important. An alternative steering law was proposed which maintains exactness but which is valid in a restricted workspace. Then in Chapter 8, this proposed method was evaluated using ground experiments. First, the problems described in Chapter 7 were demonstrated. Then, the proposed method and other steering laws were tested using some attitude control sequences. The performance of the proposed method was verified, especially for a realistic sequence including maneuvering and pointing under a specified directional disturbance. In Chapter 9, a pyramid type system with the proposed steering law was compared with other types of CMG systems. Evaluation according to workspace size showed that the symmetric six unit system was superior in terms of weight. However, the proposed method, with spherically shaped workspace, showed significant improvement. Moreover, it was shown that the workspace size was almost equal to that of the five or six unit skew system when an ellipsoidal workspace is considered. Because of this result and the fact that a symmetric pyramid type system has three modes, as well as mechanical and steering law simplicity, it was concluded that the pyramid type CMG system with the proposed steering law would be an ideal candidate for three axis attitude control. This paper does not include studies of more realistic attitude control problems, which should be investigated in consideration with the results of this paper. Evaluation may require more detailed characteristics with regard to mission requirement, disturbance profiles and unloading torquer specifications. 77

94 Technical Report of Mechanical Engineering Laboratory No

95 Appendix A Double Gimbal CMG System Formulation of an arbitrarily configured double gimbal CMG system is made in accordance with the formulation presented in Chapters 3 and 4. A.1 General Formulation Consider a system of n equally sized double gimbal CMGs in an arbitrary configuration. For each unit, one fixed vector, two variable gimbal angles, and four other vectors are defined. These are diagrammed in Fig. A 1 and defined by: g Oi : Fixed unit vector along the outer gimbal axis. θ Oi : Outer gimbal angle with origin located as shown in Fig. A 1 θ Ii : Inner gimbal angle with origin located as shown in Fig. A 1 g Ii : Unit vector along inner gimbal axis (a function of θ Oi ) h i : Angular momentum vector, normalized to h i =1 c Oi : Outer gimbal torque vector c Ii : Inner gimbal torque vector where both torque vectors are defined as follows: c Oi = h i / θ Oi = g Oi h i, c Ii = h i / θ Ii = g Ii h i, Note that the vector c Ii is a unit vector while the vector c Oi is not. The system configuration is then defined by the set of {g Oi }. The dependent system variables, namely the total angular momentum H and the output torque T are given by: H = Σ i h i = H(θ) T = Σ i (c Oi ω Oi + c Ii ω Ii ) = C ω, where θ is a point on a 2n dimensional torus whose coordinates are given by: θ = ( θ O1 θ I1 θ O2 θ I2... θ On θ In ). The 2n dimensional vector ω is defined by: ω = (dθ i /dt). The difference between these expressions and those for the single gimbal system are that the vector c Oi is not a unit vector and that some vector variables are not always independent to another θ variable as follows: c Ii / θ Ii = h i, c Ii / θ Oi = (g Oi g Ii ) h i + g Ii (g Oi h i ), c Oi / θ Ii = g Oi (g Ii h i ), c Oi / θ Oi = g Oi (g Oi h i ). g I θ O c I The last expression implies that the vector c Oi / θ Oi is parallel to g Ii. c O g O A.2 Singularity θ I h When the system is singular, the following relation is satisfied. Fig. A 1 Vectors and variables relevant to a double gimbal CMG det(cc t ) = 79

96 Technical Report of Mechanical Engineering Laboratory No.175 g I h = u g O previous section, suppose further that some unit is in condition (b), then any dθ Oi results in the same H. This implies that outer gimbal motion of this unit alone is a null motion. When there are two units both in condition (a) but different signs of ε, the angular momentum vectors of these two units are in opposite directions and they move on the unit sphere. Motion of the two units can be chosen exactly canceling each other. Thus, this motion is a null motion. (a) Condition (a). A.4 Passability g I u Passability of a singular surface can be defined by a quadratic form similar to Eq However, another approach is also possible. h = g O A.4.1 Two Unit System Geometric comprehension of this fact is easier than the case of a single gimbal system. Since both torque vectors are orthogonal to each other and to h, a singular vector u is determined as u parallels h. An exception arises when the vector h i is parallel to the vector g Oi, where c Ii is a zero vector. Thus, there are four possible conditions of singularity for each unit when u is specified: (a) h i = ε i u, where ε i =1 or 1 (b) h i = ε i g Oi, and (g Ii g Oi ).u =, where εi=1 or 1. (b) Condition (b). Fig. A 2 Vectors at singularity conditions. First, a two unit system will be considered. For an arbitrary configuration of g O1 and g O2, there are five spherical singular surfaces. One surface is of diameter 2 and is an angular momentum envelope. The remaining four are unit spheres with their centers at g O1, g O1, g O2 and g O2. It is adequate to check the following two cases: The first case, CASE I, is that one unit satisfies condition (b). Singular surface for this case is a unit sphere. The second case, CASE II, is that both units satisfy condition (a). In this case H is at its origin. CASE I Infinitesimal motion of the unit satisfying (a) is exactly on the singular surface. On the other hand, infinitesimal motion of another unit, say unit 1 for example, includes a motion out of the surface with regard to the second order differential. Second order differentials such as dθ I1 dθ O1 are orthogonal to g O1 but These are drawn in Fig. A 2. Condition (b) is called gimbal lock because such a unit looses one degree of freedom autonomously. Because the domain of u is a unit sphere, the total angular momentum H forms a number of spheres in accordance with the set of ε i. g I u h = g O A.3 Steering Law and Null Motion dθ I dθ O dθi Any steering law has the same expression as in Eq When the system is singular, some null motions are easily obtained as follows. Referring back to the Fig. A 3 Infinitesimal motion at a singular point of condition (b). 8

97 A. Double Gimbal CMG System not always to u in Fig. A 3. This motion therefore can realize a second order motion in both directions away from the singular surface. Thus, internal part of this surface is passable. The exception is the case that u = ±g O1 but this is either the case that H is on the envelope or it is regarded as the following CASE II. CASE II This condition is simply expressed as: h 1 = h 2 = u or u. Any null motion satisfies the following: dh 1 = dh 2, so the differential is exactly zero for any null motion. This implies that the quadratic form is exactly zero here. This is similar to the H origin of the roof type system M(2,2) (see Section C.1). A.4.2 Three Unit System The internal singular surface of an arbitrarily configured three unit system consists of a unit sphere whose center is on the origin and additional spheres of diameter 2. The unit sphere corresponds to the case that all the units satisfy condition (a). In this case, the following motion of h vectors is realized by null motions. (1) dh 1 = dh 2 = dh 3 /2, (2) dh 1 = dh 2, and dh 3 =, where it is supposed that the third unit s ε is negative. Clearly, the second order motion by (1) is in the direction of u and that by (2) is in the direction of u. Therefore, this singular point is passable. The sphere of diameter 2 represents the case that one of the units satisfies condition (b). In this case, the two unit subsystem is equivalent to CASE I in the above and this singular point is passable. Thus, a three (or more) unit system has no internal impassable surface. A.5 Workspace There is no internal impassable surface for a double gimbal CMG system of no less than three units. The available work space is a sphere of diameter n. 81

98 Technical Report of Mechanical Engineering Laboratory No

99 Appendix B Proofs of Theories A full description and proof of the formulation given in Chapter 4 is made here. B.1 Basis of Tangent Spaces For an independent type system in a singular state, there are two independent torque vectors. Suppose that they are described by c 1 and c 2. The three sets of bases, e i, of the subspaces Θ S, Θ N and Θ T, are obtained as follows. Basis of Θ S e Si = ( p 1 q i, 1, p 2 q i, 2,...,p n q i, n ) t, where i = 1 or 2. Basis of Θ N (B 1) e N1 = ( q 2, 3, q 3, 1, q 1, 2,,...,) t, e N2 = ( q 2, 4, q 4, 1,, q 1, 2,,...,) t,... e Nn 2 = ( q 2, n, q n, 1,,...,, q 1, 2 ) t, (B 2) Basis of Θ T e T2 = ( q i, 1, q i, 2,...,q i, n ) t, (B 3) where q i, j is defined as the following vector triple product; q i, j = [ c i c j u ]. (B 4) These are derived as follows. The definition of dθ S is given by differentiation of the singularity relation (Eq. 3 16) as, c Si du = dc Si u = dθ Si p i. (B 5) Since the vector du lies on a plane orthogonal to u, there are two independent vectors du 1 and du 2. These can be defined as: du i = c i u, where i = 1 and 2. (B 6) The two candidates of e Si given by Basis B 1 satisfy Eq. B 5 with the above du i for i value in the same order. Thus, the basis candidates B 1 constitutes the bases of Θ S. The following is a general relationship for four arbitrary vectors in three dimensional space: a[ b c d ] b[ c d a ] + c[ d a b ] c[ a b c ]=, (B 7) where [ ] is a box product of three vectors. By substituting c 1, c 2, c i and u into these four vectors, the following relationship is obtained: q 1, 2 c i + q 2, i c 1 + q i, 1 c 2 =. (B 8) Hence, the candidates for e Ni in Eq. B 2 satisfy the definition of null motion since Ce Ni =. Furthermore, they are independent, because q 1, 2 is not zero by definition. The scalar product of a null motion and n dimensional vector in Eq. B 5 is described by: Σ i dθ Ni (dθ Si p i ) = Σ i dθ Ni (c Si du) = (dθ N ) t C t du = (C dθ N ) t du =. (B 9) Thus, the n dimensional vector dθ Si p i is orthogonal to the null motion and belongs to Θ T. Basis B 3 is obtained by simple substitution. B.2 Gaussian Curvature Gaussian curvature of a surface in three dimensional space is defined by the second fundamental form of a surface, for which there are various definitions. Among them is the area ratio of the surface and a Gaussian sphere. By this ratio, the Gaussian curvature κ is defined as: 83

100 Technical Report of Mechanical Engineering Laboratory No κ = [u dh(du 1 ) dh(du 1 )] [u du 1 du 2 ], (B 1) where du 1 and du 2 are selected such that they are independent, as Eq. B 6, for example. The two differentials dθ S1 and dθ S2 corresponding to these du are defined by Eq. B 5 as: paper 22) requires knowledge of the theory of dyadics. Here this is proven in vector form. The differential of H by du given by Eq. 4 5 is obtained by substituting this du into Eq. B 12, giving: dh(du) = CPC t du = κ CPC t ((CPC t V) u). (B 17) dθ Sj = {p i (c Si du j )}, = P C t du j, where j = 1 or 2. Therefore: dh(du i ) = C dθ Si (B 11) (B 12) A part of the term on the right-hand side can be rewritten as follows: C t ((CPC t V) u) = (c i ) t ((CPC t V) u) = ( c i (CPC t V) u ) t = ( c i u ) t (CPC t V) = CPC t du i = Σ j p j (c Sj du i )c j, where i = 1 or 2. Substituting this into the first triple product on the right-hand side of Eq. B 1 results in the following: [u dh(du 1 ) dh(du 2 )] = Σ i Σ j p i p j (c i du 1 )(c j du 2 )[c i c j u], (B 13) From the four vector relationship, the following is obtained; (c i du 1 )(c j du 2 ) (c i du 2 )(c j du 1 ) = (c i c j ) (du 1 du 2 ). (B 14) As vectors such as c i and du i are orthogonal to the unit vector u, the expression on the right with both two terms multiplied by u is unchanged. (c i c j ) (du 1 du 2 ) = (c i c j ) u (du 1 du 2 ) u = [c i c j u] [u du 1 du 2 ]. (B 15) By substituting this into Eq. B 13 and using [c i c j u] = [c j c i u], the following result is obtained: [u dh(du 1 ) dh(du 2 )] = 1 2 Σ i Σ j p i p j [u du 1 du 2 ][c i c j u] 2.(B 16) Thus, the expression of Gaussian curvature in Eq. 4 7 is obtained. B.3 Inverse Mapping Theory Proof of the inverse mapping theory in the original = ( c i u ) t C (PC t V), (B 18) where expressions (x i ) and (v i ) denote a row vector and a matrix, (x i ) = (x 1 x 2... x n ), (B 19) (v i ) = (v 1 v 2... v n ). This notation as well as a matrix notation such as (x ij ) are used from this point in this section. The first term of Eq. B 18, ( c i u ) t C, is the following matrix: ( c i u ) t C =(( c i u ) t (c j )) = ([c i c j u]), (B 2) Multiplying both sides by P, a new matrix R is defined. R = P([c i c j u])p = (p i p j [c i c j u]). (B 21) Thus, the right hand side of Eq. B 17 can be rewritten as: κ CPC t ((CPC t V) u) = κ CRC t V, = κ C(r ij )C t V. = κ ( c i )(r ij )(c j V)) = κ Σ ij r ij (c j V) c i, (B 22) where r ij is the element of R given by the preceding equation. Taking advantage of the fact that matrix B 21 is skew symmetric (r ji = r ij ): κ Σ ij r ij (c j V) c i = 1 2 Σ ij r ij ( (c j V) c i (c i V) c j ).(B 23) From the vector product rule and the fact that u is normal 84

101 B. Proofs of Theories to (c i c j ), (c j V) c i (c i V) c j = (c j c i ) V = [c j c i u] (u V ). (B 24) From Eqs. B 21 and 4 7, κ CPC t ((CPC t V) u) = κ 1 2 Σ ij r ij [c i c j u] ( V u ) = κ 1 2 Σ ij p i p j [c i c j u] 2 ( V u ) From Eq. B 17, = V u. (B 25) dh = V u. (B 26) Thus, Eq. 4 4 is derived and the theory is proven. B.4 Impassable condition for two negative signs Passability is defined by the signature of the quadratic form, Q N, in Eq Let A N denote the matrix of this quadratic form as; A N = E N t P 1 E N. (B 27) Let s find a condition of impassable state in case that two of the signs, ε i, are negative and remaining signs are positive. We have the following linear algebra theory for definite matrix. Theory Consider a symmetric m m matrix denoted by A and its sub-matrix A k. a11 a12... a1m a a a m A = : : : :, am1 am2... amm and A k a11 a12... a1k a a a k = : : : : a a... a k1 k2 kk A is positive definite if and only if the minor, i.e., deta k is positive for all k = 1, 2,..., m. Though all the minors must be examined in general, in case of checking passability, only two of them should be examined because we can suppose that ε n and ε n 1 are negative without loosing generality,. Suppose that the condition (3) in Section is satisfied, that is, the Gaussian curvature κ is positive. In this case, there are two possibilities where the quadratic form Q N is positive definite or its signature has two negative terms. In both cases, determinant of the matrix A N is positive because this determinant is a product of all eigenvalues and its sign is determined by the signature. Consider the subsystem without the nth unit which has the negative sign. For this subsystem, A Nn 3 is the matrix of the quadratic form, because the original matrix A N is n 2 n 2. As there is one negative sign, the condition (2) in Section that the Gaussian curvature is negative is equal to that the matrix A Nn 3 is positive definite. Thus, only this condition is enough to assure that minors for all k = 1, 2,..., n 3 are positive. From the above theory, we have the following conclusion. Conclusion A singular surface of all the ε i but two are positive is impassable if and only if κ > and κ <, where κ is the Gaussian curvature of the subsystem without one unit of negative sign.. 85

102 Technical Report of Mechanical Engineering Laboratory No

103 Appendix C Internal Impassability of Multiple Type Systems In this appendix, the minimum systems with no internal impassable surfaces are found which are M(3, 3) and M(2, 2, 2). First, discrimination of singularity for the four unit roof type system is made. Then, the minimum systems are searched by adding units to this system. Though the subspace Θ N of the null motion is generally two-dimensional, it is one-dimensional when u lies on the plane spanned by the two gimbal vectors g 1 and g 2. In this case, there are three independent null vectors. From this point, each variable p i is defined for each group so that the condition p i is satisfied. C.1 Roof Type System M(2, 2) The four unit roof type system is shown in Fig All singular surfaces are given as follows. u g 1, u g 2 (1) θ 11 = θ 12, θ 21 = θ 22, ε={ }: Envelope (2) θ 11 = θ 12, θ 21 = θ 22, ε={+ + } or { + +}: Internal Surface (3) θ 11 = θ 12 +π, θ 21 = θ 22, ε={+ + +}: Circle orthogonal to g 1 centered on H origin. (4) θ 11 = θ 12, θ 21 = θ 22 +π, ε={+ + + }: Circle orthogonal to g 2 centered on H origin. (5) θ 11 = θ 12 +π, θ 21 = θ 22 +π, ε={+ + }: H origin u = g 1 (6) θ 21 = θ 22 : Envelope (7) θ 21 = θ 22 +π: Inside Circle (3) u = g 2 (8) θ 11 = θ 12 : Envelope (9) θ 11 = θ 12 +π: Inside Circle (4) It is only necessary to evaluate internal surfaces (2), (3), (5) and (7), because pairs such as (3) & (4) or (7) & (9) are identical when the group numbers are changed. C.1.1 Evaluation of Singular Surface (2) is; Independent null motions are; φ 1 = (1, 1,, ), φ 2 = (,, 1, 1). (C 1) The quadratic form dθ N t P 1 dθ N by dθ N = a 1 φ 1 +a 2 φ 2 dθ N t P 1 dθ N = 1 2(a 1 2 p 1 +a 1 2 p 1 a 2 2 p 2 a 2 2 p 2 ) = a 1 2 p 1 + a 2 2 p 2, (C 2) Thus, this is indefinite and passable. C.1.2 Evaluation of Singular Surface (3) is; Independent null motions are; φ 1 = (1, 1,, ), φ 2 = (,, 1, 1). (C 3) The quadratic form dθ N t P 1 dθ N by dθ N = a 1 φ 1 +a 2 φ 2 dθ N t P 1 dθ N = 1 2(a 1 2 p 1 a 1 2 p 1 +a 2 2 p 2 + a 2 2 p 2 ) = a 2 2 p 2, (C 4) Thus, this is semi-definite and impassable. If u happens to be on the plane spanned by g 1 and g 2, another null vector φ 3 (dependent on φ 1 and φ 2 ) is given 87

104 Technical Report of Mechanical Engineering Laboratory No.175 by; φ 3 = (1,, 1, ) (C 5) The quadratic form by dθ N = a 1 φ 1 +a 2 φ 2 +a 3 φ 3 is; dθ t N P 1 dθ N = 1 2{(a 1 +a 3 ) 2 p 1 a 2 1 p 1 +(a 2 a 3 ) 2 p 2 + a 2 2 p 2 } = 1 2{ a 2 3 p 1 + 2a 1 a 3 p 1 + 2a 2 2 p 2 2a 2 a 3 p 2 + a 2 3 p 2 } = 1 2(ra rsa 1 a 3 + 2(a 2 a 3 2) 2 p 2 ) = 1 2{r(a sa 1 ) 2 rs 2 a (a 2 a 3 2) 2 p 2 }, where r = 1 p 1 +3 (4p 2 ) >, 2rs = 2 p 1. (C 6) Thus, this is indefinite and passable. C.1.4 Evaluation of Singular Surface (7) Independent null motions are; φ 1 = (,, 1, 1), (C 11) φ 2 = (a, b, 1, 1). The quadratic form by dθ N = a 1 φ 1 +a 2 φ 2 is; dθ t N P 1 dθ N = 1 2{(a 1 +a 2 ) 2 p 2 (a 1 a 2 ) 2 p 2 }, (C 12) Thus, this is indefinite and passable. C.1.5 Conclusion Thus, this is indefinite and passable. C.1.3 Evaluation of Singular Surface (5) Independent null motions are; φ 1 = (1, 1,, ), (C 7) The roof type system M(2, 2) has internal impassable surfaces given by (3) and (5) with u not on the plane spanned by g 1 and g 2. Both are not fully definite but the quadratic form of (3) is semi-definite and that of (5) is zero. C.2 M(3, 2): M(2, 2)+1 φ 2 = (,, 1, 1). The quadratic form by dθ N = a 1 φ 1 +a 2 φ 2 is; dθ N t P 1 dθ N = 1 2(a 1 2 p 1 a 1 2 p 1 +a 2 2 p 2 a 2 2 p 2 ) =. (C 8) Thus, this is zero for any null motion and impassable. If u is on the plane spanned by g 1 and g 2, another null vector φ 3 (dependent on φ 1 and φ 2 ) is given by; φ 3 = (1,, 1, ). The quadratic form by dθ N = a 1 φ 1 +a 2 φ 2 +a 3 φ 3 is; dθ N t P 1 dθ N (C 9) = 1 2{(a 1 +a 3 ) 2 p 1 a 2 1 p 1 +(a 2 a 3 ) 2 p 2 a 2 2 p 2 } = 1 2 {(1 p 1 +1 p 2 )a (a 1 p 1 a 2 p 2 )a 3 } = 1 2 [(1 p 1 +1 p 2 ) {a 3 +(a 1 p 1 a 2 p 2 ) (1 p 1 +1 p 2 ) } 2 (a 1 p 1 a 2 p 2 ) 2 (1 p 1 +1 p 2 ) ]. (C 1) The system M(3, 2), which results from adding an additional unit to M(2, 2), will now be analyzed. The passability of sub-system M(2, 2) was evaluated above, and so only those impassable conditions should be tested. C.2.1 Condition (3) of M(2,2) System M(3, 2) is asymmetric and so both conditions (3) and (4) should be tested. The singular points are given by; (3') θ 13 = θ 11 = θ 12 +π, θ 21 = θ 22, ε={ }, (4') θ 13 = θ 11 = θ 12, θ 21 = θ 22 +π, ε={ }. Independent null motions are then; φ 1 = (1, ±1,,, ), φ 2 = (1, ±1, 1,, ), φ 2 = (,,, 1, (±1)). (C 13) where upper and lower sign of the multiple sign ± correspond to (3') and (4') respectively. The quadratic 88

105 C, Internal Impassability of Multiple Type Systems form by dθ N = a 1 φ 1 +a 2 φ 2 +a 3 φ 3 is; (3') (4') dθ N t P 1 dθ N = 1 2{a 1 2 p 1 (a 1 +a 2 ) 2 p 1 + a 2 2 p 1 + a 2 3 p 2 + a 2 3 p 2 } = 1 2( 2a 1 a 2 p 1 + 2a 2 2 p 2 ) = 1 2{ 1 2(a 1 +a 2 ) 2 p (a 1 a 2 ) 2 p 1 dθ N t P 1 dθ N + 2a 2 2 p 2 }, (C 14) = 1 2{a 1 2 p 1 + (a 1 +a 2 ) 2 p 1 + a 2 2 p 1 + a 3 2 p 2 a 2 3 p 2 } = 1 2(a 2 1 p 1 + (a 1 +a 2 ) 2 p 1 + a 2 2 p 1 ). (C 15) Thus, (3') is passable and (4') is impassable. The singular H of (4') forms a circle centered on the origin and having a diameter of 3. C.2.2 Condition (5) of M(2,2) The singular points are given by; (5') θ 13 =θ 11 = θ 12 +π, θ 21 = θ 22 +π, ε={+ + + }. Independent null motions are then; φ 1 = (1, 1,,, ), φ 2 = (, 1, 1,, ), φ 3 = (,, 1, 1). (C 16) The quadratic form by dθ N = a 1 φ 1 +a 2 φ 2 +a 3 φ 3 is; dθ N t P 1 dθ N = 1 2 {a 1 2 p 1 (a 1 +a 2 ) 2 p 1 + a 2 2 p 1 + a 3 2 p 2 a 3 2 p 2 } = 1 4( (a 1 +a 2 ) 2 + (a 1 a 2 ) 2 ) p 1, (C 17) Thus, this is passable. It is concluded that M(3, 2) has an impassable surface corresponding to condition (4'). C.3 M(3, 3): M(2, 2)+2 Even if a certain sub-system M(3,2) of M(3, 3) satisfies condition (4'), the same condition will be (3') in the other sub-system. This implies that there is no internal impassable surface. C.4 M(2, 2, 1): M(2, 2)+1 This configuration is not a multiple system, so there is an internal impassable surface. C.5 M(2, 2, 2): M(2, 2)+2 If u is not parallel to g i, all internal surfaces are passable from the discussion of Section If u is parallel to g i, condition (7) is satisfied by sub-system M(2, 2) which includes the ith group. Thus all internal surfaces are passable. C.6 Minimum System The multiple systems M(3, 3) and M(2, 2, 2) are the minimum systems with no internal impassable singular surfaces. Both systems consist of six units. 89

106 Technical Report of Mechanical Engineering Laboratory No.175 9

107 Appendix D Six and Five Unit Systems D.1 Symmetric Six Unit System S(6) D.1.1 System Definition z The S(6) system is a symmetric type with six units arranged in the surface directions of a regular dodecahedron. Its work space possesses symmetry and can be approximated by a sphere. Being an independent type, the system has internal impassable surfaces, which are very near the envelope. Control over most of the entire workspace shown in Fig. D 1 (a) can be accomplished using a gradient method. The diameter of the controllable spherical workspace is about 4.27 times larger than the angular momentum of the unit (see Fig. 9 2 in Chapter 9). D.1.2 Fault Management y x (a) Original S(6) system z (1) Loss of One Unit The S(6) system without any one unit is a congruent five unit skew type system with a different major axis direction. Thus, we need only one steering law for this type of failure. The original and degraded system envelopes are shown in Fig. D 1. The original envelope is similar to a sphere but that of the degraded system is more similar to an ellipsoid. Figure D 1 (b) corresponds to the failure of the unit arranged in the z direction. If another unit fails, the envelope has the same shape but its major axis is different. As the skew angle of this system is not optimized, there is a more serious internal impassable surface problem than the optimized system described in Section 9.4 has. Moreover, even though we can use this workspace, its major axis is unknown before the accident, and there are six possibilities. Thus, it is safe to consider all possible situations and to evaluate a spherical workspace which is included by all six possible envelopes (see Fig. 9 2). y x (b) After loss of one unit z y x (c) After loss of two units Fig. D 1 Envelopes of S(6) and degraded systems. All are drawn in the same scale. 91

108 Technical Report of Mechanical Engineering Laboratory No.175 (2) Loss of Two Units Any failure of two units also results in a congruent configuration of four units. However, the system is not at all symmetric and the envelope is like a skew ellipsoid, as shown in Fig. D 1 (c). Because of reasons similar to those given above, the workspace size must be evaluated by a spherical workspace. D.1.3 Four out of Six Control The CMG system installed on the space station MIR is a S(6) system but only four units out of six are operating simultaneously. The subsystem of four units is the same as that in the above section. As mentioned there, any four unit subsystem is geometrically congruent. Therefore, any fault up to two units can be simply covered by exchange of faulty unit with a backup unit without change of the steering law. Fig. D 2 g 1 g 4 Z α α 1 2 X Y g 2 g 5 g 6 Four unit subsystem of MIR type system. Original Workspace g 3 Constrained workspace A gradient method applied to the MIR system uses the following objective function 27) : W = Σ ij c i c j 2. (D 1) As described in Section 7.5.5, the concept of the constrained control can be applied to this subsystem 69). The four unit subsystem can be regarded as a deformed pyramid configuration in which two units have a skew angle α1 (= 1 sin 1 / 2 ( 1 + cos( π / 5 )) ) and the other two, a skew angle α2 (= π/2 α1) (Fig. D 2). As the kinematic equation of this system is similar to that of the pyramid type system, the same constraining condition as Eq. 7 5 can be applied and nonredundant kinematics similar to Eq. 7 6 can be obtained. This constrained system has a restricted workspace (Fig. D 3), inside which exact steering is assured. Of course, this configuration is not rotationally symmetric about any gimbal axis, so there is no additional mode. D.2 Five Unit Skew System As described in Chapter 9, various ellipsoidal workspaces can be designed by selecting the skew angle. The workspace size is given in the figures of Chapter 9. The fault management is similar to that of the S(6) system described in Appendix D.1.1, except that the skew angle is different. As this type of systems have not studied well, no effective steering laws have been proposed except the gradient method. Application of the constraint method is possible with two independent constraining equations, Impassable Surface Actual Motion of H H' Desired Motion of H H Additional Angular Momentum Imposed by Another Torquer H path Fig. D 3 Restricted workspace of a constrained MIR-type system. Fig. D 4 Concept of singularity avoidance by an additional torquer 92

109 D. Six and Five Unit Systems but the symmetry of this system cannot be preserved by using any two linear equations. Therefore, finding appropriate constraints may need exhaustive calculations and evaluations with some criteria of work space size and shape. Here, a potential steering law with an additional torquer will be briefly outlined, which may be effective for four or five unit system. Impassable surfaces of these systems are shaped like surface strips as shown in Figs. 4 1 to 4 12 and 6 7 to 6 1. In Section 6.4, these strips were called impassable branches for the S(4) system. They can be approximated by the analytical expression given by Eqs to Though there is no such expression for 5 unit systems, they can be expressed by some numerical lookup table. This look-up table can be reduced in its size with the aid of system symmetry. By this knowledge, we can distinguish whether H is approaching an impassable surface. If appropriate angular momentum is added using another torquer when H is nearly crossing some impassable surface strip, the system can avoid the surface, as shown in Fig. D 4. This mechanism is very simple and has the following characteristics. (1) The avoidance movement can be planned so that it is towards the narrower direction of the impassable strip. Thus, the required angular momentum for avoidance can be minimized. Moreover, this motion only aims to avoid the surface, therefore an ON/OFF type torquer such as a gas jet would be adequate. (2) This motion must take place only once to avoid a singular surface. Thus it may be accomplished by a gas jet system. (3) Sometimes, no singularity avoidance is necessary when H is approaching an impassable surface, as is described in Sections 7.1 and 7.4 on manifold connections. The method described here, however, can not use this knowledge of manifold connections, and hence is very simple but requires additional torquing more often than necessary. 93

110 Technical Report of Mechanical Engineering Laboratory No

111 Appendix E Specification of Experimental Apparatus and Experimental Procedure E.1 Experimental Apparatus The experimental apparatus, as shown in Figs. 8 1 and E 1, is composed of a body structure, a three axis gimbal, attitude sensors and related circuitry, a CMG system, balance adjusters and an onboard computer. The block diagram is shown in Fig. E 2. The body is a truss structure made of steel pipes. It is designed sufficiently stiff so that deformation caused by its weight can be neglected when the system changes orientation. The three axis gimbal, which uses normal ball bearings, permits free rotation of the body (Fig. E 3). A precision rotary encoder is installed on each gimbal axis. The encoder s output pulses are converted to an angle value by a decoder circuit, then supplied to the onboard computer. The rotational speed of the body is measured by rate gyroscopes, whose outputs are analog signals that are converted to digital values by an Analog to Digital (A/D) conversion circuit. Fig. E 1 Experimental apparatus Three Axis Gimbal Rate Gyroscope RE P/D A/D Wireless Modem TG DCM Balance Adjusters RE Rate Servo Circuit P/D D/A Onboard Computer Wireless Modem TG DCM CMGs RE P/D Rate Servo Circuit D/A RE: Rotary Encoder P/D: Pulse Decoder A/D: Analog to Digital Converter Stationary Computer D/A: Digital to Analog Converter DCM: DC Servo Motor TG: Tachogenerator Fig. E 2 Block diagram of experimental apparatus 95

112 Technical Report of Mechanical Engineering Laboratory No.175 Fig. E 3 Three axis gimbal mechanism Fig. E 4 Single gimbal CMG Fig. E 5 Balance adjuster Fig. E 6 Onboard computer The CMG system is a S(4) type system composed of four single gimbal CMGs and a unit CMG is shown in Fig. E 4. A wheel motor and a driver circuit installed inside the casing drives the flywheel at constant speed. Slip rings installed through the gimbal axis enable free rotation of the gimbal. The gimbal angle is measured by a rotary encoder. The rotational speed of each gimbal motor is controlled by a servo driver circuit. The balance adjusters are composed of a moving weight driven by a linear ball screw mechanism as shown in Fig. E 5. The speed of the moving weight is controlled by a DC motor and a rate servo circuit, and 96