Spacecraft Actuation Using CMGs and VSCMGs

Transcription

1 Spacecraft Actuation Using CMGs and VSCMGs Arjun Narayanan and Ravi N Banavar (ravi.banavar@gmail.com) 1 1 Systems and Control Engineering, IIT Bombay, India Research Symposium, ISRO-IISc Space Technology Cell, IISc, Bengaluru, February 22-27, February 26, 2016

2 Outline 1 References 2 Introduction 3 Spacecraft dynamics and control 4 Reaction wheels 5 CMGs CMG Introduction Singularity Steering laws Manipulator analogy Condition for escapability 6 VSCMGs

3 Outline 1 References 2 Introduction 3 Spacecraft dynamics and control 4 Reaction wheels 5 CMGs CMG Introduction Singularity Steering laws Manipulator analogy Condition for escapability 6 VSCMGs

4 N. Bedrossian. Steering law design for redundant single Gimbal control moment gyro systems. PhD thesis, Massachusetts Inst. of Technology, H. Kurokawa. A geometric study of single gimbal control moment gyros. Report of Mechanical Engineering Laboratory, 175: , G. Margulies and J. N. Aubrun. Geometric theory of single-gimbal control moment gyro systems. Journal of the Astronautical Sciences, 26(2): , H. Schaub and J. L. Junkins. Singularity Avoidance Using Null Motion and Variable-Speed Control Moment Gyros. Journal of Guidance, Control, and Dynamics, 23(1):11 16, January 2000.

5 H. Schaub, S. R. Vadali, and J. L. Junkins. Feedback control law for variable speed control moment gyros. Journal of the Astronautical Sciences, 46(3): , P. Tsiotras and H. Yoon. Singularity Analysis and Avoidance of Variable-Speed Control Moment Gyros Part II : Power Constraint Case. In AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Guidance, Navigation, and Control and Co-located Conferences. American Institute of Aeronautics and Astronautics, August P. Tsiotras and H. Yoon. Singularity analysis of variable speed control moment gyros. Journal of Guidance, Control, and Dynamics, 27(3): , 2004.

6 Outline 1 References 2 Introduction 3 Spacecraft dynamics and control 4 Reaction wheels 5 CMGs CMG Introduction Singularity Steering laws Manipulator analogy Condition for escapability 6 VSCMGs

7 Contents Motivation Spacecraft attitude control Attitude representation Spacecraft dynamics Control strategy Momentum exchange devices Reaction wheels (RW) Control moment gyros (CMG) Variable speed control moment gyros (VSCMG) Why (VS)CMG? Saves fuel Torque amplification Energy storage Challenges Singular states Steering law design Controllability

8 Outline 1 References 2 Introduction 3 Spacecraft dynamics and control 4 Reaction wheels 5 CMGs CMG Introduction Singularity Steering laws Manipulator analogy Condition for escapability 6 VSCMGs

9 Actuation methods using thrusters using reaction wheel β i β i γ i γ i using CMG (initial configuration) using CMG (after gimbal rotation) Figure: Actuation methods

10 CMG Torque Spacecraft β R, Ṙ Steering Law Control Law Figure: Spacecraft attitude control block diagram Spacecraft dynamics Ṙ = RˆΩ Ḣ = Ω H H = IΩ + H CMG where, R SO(3) represents the spacecraft attitude Ω R 3 is the spacecraft angular velocity in body frame H R 3 is the total angular momentum of the space craft in body frame I spacecraft inertia (locked part).

11 Outline 1 References 2 Introduction 3 Spacecraft dynamics and control 4 Reaction wheels 5 CMGs CMG Introduction Singularity Steering laws Manipulator analogy Condition for escapability 6 VSCMGs

12 Reaction wheels s i γ i Figure: Reaction Wheel Device consists of a rotating wheel with inertia I w and an angular velocity γ about its spin axis s. The spin axis is fixed with respect to the body of the spacecraft and rotates along with the body. A motor exerts torque on the wheel resulting in change of its angular velocity. The equal and opposite reaction torque acts on the spacecraft body and this reaction is used to control the attitude and rate of the spacecraft.

13 Reaction wheels s i The equation is usually simplified as* γ i Torque i = I w γ i s i Figure: Reaction Wheel h i = I w γ i s i Torque i = dhi dt d = I w γ i s i +I w γ i s i dt The total torque is the linear combination of torques of the individual reaction wheels. Torque generated per unit velocity can be seen to be I w. A minimum of three reaction wheels with a linearly independent set of spin axes can generate torques in arbitrary directions in R 3. *The second term may be accounted separately or neglected for slow satellite rotations

14 Outline 1 References 2 Introduction 3 Spacecraft dynamics and control 4 Reaction wheels 5 CMGs CMG Introduction Singularity Steering laws Manipulator analogy Condition for escapability 6 VSCMGs

15 Outline 1 References 2 Introduction 3 Spacecraft dynamics and control 4 Reaction wheels 5 CMGs CMG Introduction Singularity Steering laws Manipulator analogy Condition for escapability 6 VSCMGs

16 Control moment gyro - mechanism β Satellite Wheel Body frame B γ Gimbal frame Gimbal frame G Inertial frame I Figure: CMG The gimbal frame and wheel can rotate (β) about g i. The wheel spin axis is orthogonal to the gimbal frame and changes direction when the gimbal rotates. The wheel speed is fixed. The torque is generated by rate of change of direction of the wheel angular momentum rather than magnitude.

17 Dynamics of a single CMG γ g β h i = I w γ i s i Torque i = dhi dt d = I w γ i s i +I w γ i s i dt = I w γ i t i βi s ( s i t i ) = d s i= dt Figure: CMG ( ) ( ) Cβi Sβ i s i0 Sβ i Cβ i t i0 d s i βi = t i βi dβ i t Torque generated per unit velocity of β i can be seen to be I w γ i. This torque acts along t i and hence can be larger than the specifications of the wheel and gimbal motors. However the the direction of the torque is not fixed with respect to the body of the spacecraft. *For a CMG, γ = 0.

18 Multiple CMGs The total torque is a linear combination of the individual torques which are along the direction t i. N Torque = I w γ i t i i=1 βi = linear combination of t i C (β) β C (β) I w[ γ 1 t 1,..., γ N t N ] 3 N β [ β 1,..., β N ] T Torques can be generated in arbitrary directions only if the set { t i} has at least three vectors which are linearly independent. Since the direction of the vector t i is a function of the gimbal angle β i, this cannot be guaranteed.

19 Torque generation equation is T x γ 1 t 1x γ N t Nx β 1 T y = I w γ 1 t 1y γ N t Ny. T z γ 1 t 1z γ N t Nz β N T = γ 1 t 1 β1 + + γ N t N βn It is a system of three linear equations with N gimbal velocities as independent variables. Also the RHS of the equation evaluates to a vector in the column space of C, i.e, a linear combination of t i.

20 Torque generation equation is T x γ 1 t 1x γ N t Nx β 1 T y = I w γ 1 t 1y γ N t Ny. T z γ 1 t 1z γ N t Nz β N T = γ 1 t 1 β1 + + γ N t N βn It is a system of three linear equations with N gimbal velocities as independent variables. Also the RHS of the equation evaluates to a vector in the column space of C, i.e, a linear combination of t i. If there are more than three CMGs, the number of independent variables is more than the number of equations. Then there may exist multiple solutions which produce the same torque and the steering law has to generate a continuous profile for β.

21 Torque generation equation is T x γ 1 t 1x γ N t Nx β 1 T y = I w γ 1 t 1y γ N t Ny. T z γ 1 t 1z γ N t Nz β N T = γ 1 t 1 β1 + + γ N t N βn It is a system of three linear equations with N gimbal velocities as independent variables. Also the RHS of the equation evaluates to a vector in the column space of C, i.e, a linear combination of t i. In case C is not maximum rank (three), then the RHS spans only a two (or one) dimensional subspace of the three dimensional space. This condition is called singularity. If the required torque is not an element of this subspace, then the steering law should generate a gimbal rate profile which minimises the error between required torque and produced torque.

22 Null motion When the number of CMGs is more than three or if the matrix C is rank deficient (singular condition), the null space of C 3 N is non trivial. There exist gimbal rate combinations β null such that R N β null 0, C β null = 0 These gimbal velocities do not produce torque. Also note that β null null(c) β null row(c) Such gimbal velocities can be used to re-configure the CMG gimbal angles without introducing disturbing torques on the spacecraft.

23 Outline 1 References 2 Introduction 3 Spacecraft dynamics and control 4 Reaction wheels 5 CMGs CMG Introduction Singularity Steering laws Manipulator analogy Condition for escapability 6 VSCMGs

24 Singularity û ĝ j ± ĝj û ĝj û locus of ˆt j Figure: Singularity. u denotes the singular vector normal to the plane on which all the t i lie. For any chosen plane and any gimbal axis direction g i, there exists at least two values of β i which puts the corresponding torque vector on the plane. As gimbal angles evolve, the torque vectors of all the individual CMGs can happen to lie on a plane (or a line). When this happens, the linear combinations of the individual torques (total torque) that can be produced by the system at that instant cannot be perpendicular to the said plane. This cannot be avoided by choosing some gimbal axis directions or increasing the number of CMGs. (See figure) In fact, for any chosen plane, there exists atleast 2 N combinations of β i which result in all torque vector lying on that plane.

25 When the CMGs are in a singular configuration, the rank of C (β) drops below 3. The { t i} do not span the full three dimensional space. For a desired torque, a solution might not exist for the equation Desired Torque = C (β) β Hence it is desirable to keep the rank of C (β) at three.

26 Measure of singularity To avoid singular gimbal angle configurations, it is first necessary to measure how close the current gimbal angle configuration (or the matrix C (β) ) is to being singular. When a matrix is singular, its determinant is zero, at least one eigenvalue is zero, rows are linearly dependent, at least one singular value σ is zero, condition number κ = σ max/σ min is infinity. The first two measures are applicable for square matrices. Due to the existence of fast algorithms to compute the singular value decomposition of a matrix and availability of analytical expression to find its Jacobian / derivative, the condition number is a good choice to measure the singularity.

27 Singular Value Decomposition Any matrix can be decomposed into a product of two orthogonal and one diagonal matrices. In our case, condition number C = UΣV R 3 N U = U 1 R 3 3 V = V 1 R N N Σ = [diag(σ i) 0] R 3 N κ = σmax σ min (well conditioned matrix) 1 κ (singular matrix)

28 Outline 1 References 2 Introduction 3 Spacecraft dynamics and control 4 Reaction wheels 5 CMGs CMG Introduction Singularity Steering laws Manipulator analogy Condition for escapability 6 VSCMGs

29 Steering laws The spacecraft attitude control law generates the required torque profile based on the attitude and rate information. This required torque value is fed to the CMG steering law. The CMG steering law has two functions. Generate the gimbal rates which will produce the required torque. Prevent / avoid the gimbal angle configuration from reaching a singular configuration. Note that the matrix C is not necessarily square or maximum rank. This means that the torque equation can be solved for gimbal velocities in the least square or least norm sense. An example of a steering law is β = W C T (CW C T ) 1 T where W is a suitable weighting matrix.

30 T = C (β) β, C(β) R 3 N, β R N Null motion When the number of CMGs is more than three or if the matrix C is rank deficient, there exist gimbal rate combinations β null which do not produce torque. β null null(c) β null row(c) Avoidance Consider a measure of singularity such as the condition number κ(c) of C (β). The gradient of κ(c (β) ) indicates the direction in the N dimensional gimbal angle space which result in increase of κ. ( β avoidance, κ(c(β) ) ) < 0 β Escapable singularity For a given singular configuration, if the gimbal rate can be chosen such that it belongs to both the sets, then a possibility exists to escape without introducing unwanted torques on the spacecraft.

31 Outline 1 References 2 Introduction 3 Spacecraft dynamics and control 4 Reaction wheels 5 CMGs CMG Introduction Singularity Steering laws Manipulator analogy Condition for escapability 6 VSCMGs

32 Manipulator analogy for CMG Before proceeding further with steering laws and singularity avoidance techniques, it helps to build some intuition about the problem. The CMG system under consideration is analogous to a robotic manipulator whose arm is made of multiple links connected in a chain like fashion. Since the links are connected in a chain, the position of the tip / end effector of the manipulator is the vector sum of individual links. This is similar to how the total angular momentum of the CMG is the vector sum of the individual CMG angular momenta. It can also be demonstrated that some singular configurations are not escapable by null motion techniques.

33 CMG Robotic Manipulator The length of the angular momentum vector is fixed for a CMG wheel spinning at constant speed. The total angular momentum of the CMG cluster is the vector sum of the angular momentum of the individual wheels. The torque produced is the linear combination of rate of change of angular momentum vectors. Under null motion, the angular momentum remains unchanged. Under singular combination, torque cannot be produced in certain directions. The length of an arm of a manipulator is fixed. The end effector position of the manipulator is the vector sum of the individual arm vectors. The end effector velocity is the linear combination of rate of change of individual joint rotations. Under null motion, the end effector position remains unchanged. End effector velocity cannot be produced in certain direction.

34 Figure: 2 link manipulator in non singular configuration. Blue lines indicate movable directions due to each link. Figure: 3 link manipulator in non singular configuration.

35 Figure: 3 link manipulator in singular configuration. Motion of any of the links results in an end effector velocity in 135 degree direction only. Velocity in radial direction cannot be achieved. Instantaneous movement of the end effector in the radial direction is not possible in this maximally stretched configuration. This is a physical limitation of the device and cannot be remedied by any control algorithm. This singularity is inescapable.

36 Figure: 2 link manipulator in a singular configuration where end effector can have velocity only in vertical direction. Figure: 2 link manipulator after performing null motion. The end effector position is preserved. The singularity still persists. It is inescapable.

37 Figure: 3 link manipulator in a singular configuration. End effector velocity possible only in vertical direction. Figure: Manipulator escaping singular configuration without changing end effector position using null motion.

38 Outline 1 References 2 Introduction 3 Spacecraft dynamics and control 4 Reaction wheels 5 CMGs CMG Introduction Singularity Steering laws Manipulator analogy Condition for escapability 6 VSCMGs

39 Condition for escapability Consider a singular gimbal angle configuration β s Let u be the direction along which torque cannot be produced instantaneously. i.e, dh = 0 in the direction of u. dt Given N dimensional subspace of gimbal velocities The gimbal velocities which preserve angular momentum β null (produce no torque) form a subspace of V such that t i u β V dh dβ β s u= 0 V = V null V null Consider an (orthonormal?) basis of V null. M = {m i} V null

40 Objective is to change the gimbal angle configuration such that angular momentum can be changed in the u direction also. For that, consider the Taylor series of H upto second order. H(β s + dβ) =H(β s) + dh dβ β s dβ+ 1 d 2 H 2 dβ 2 β s dβ 2 + H.O.T Since we are interested in seeing if the angular momentum can be changed in the singular direction, the u component of the above expression is considered further. T u (H(βs + dβ) H(β s)) =u T dh dβ β s dβ + u T 1 2 =0 + 1 u i 2 = 1 2 jk i jk d 2 u T H dβ idβ j d 2 H dβ 2 β s dβ 2 d 2 H i dβ idβ j dβ idβ j dβ idβ j

41 However, we are interested only in those gimbal angle velocities which do not produce torque since we do not want to disturb the system while we reconfigure the it away from the singular configuration. So, out of all the possible vectors dβ, we want to choose the ones from V null. Such vectors are of the form dβ null = Mλ λ V null Then, the change in angular momentum along u using only null motion is given by λ T M T d u T H Mλ dβ 2 If the matrix in the quadratic form is sign indefinite or semi definite, then null motion may eventually result in a gimbal configuration with the same angular momentum from which change is possible in the singular direction also. Then the singular combination under consideration is escapable.

42 Outline 1 References 2 Introduction 3 Spacecraft dynamics and control 4 Reaction wheels 5 CMGs CMG Introduction Singularity Steering laws Manipulator analogy Condition for escapability 6 VSCMGs

43 VSCMG If the wheel speed is allowed to vary, then the torque equation is N Torque = I w γ i s i +I w γ i t i βi i=1 β 1 = I w γ 1 t 1 I w γ N t N I w s 1 I w s N β N γ 1 γ N [C (β, γ), D (β) ][ β, γ] T = linear combination of t i and s i [C, D] R 3 2N Since the s i are orthogonal to t i, [C, D] will always be maximum rank.

44 Power tracking If the rotating wheels are used to store / release energy in the form of kinetic energy, then the torque producing equation has to be augmented by a power producing equation. ( ) Torque Power ( C D = γt I w ) ( ) β γ To generate torques in arbitrary directions and follow the power profile, the augmented matrix must have rank 4. To satisfy this, singularity avoidance techniques can be used for VSCMGs also.