Kinematic Synthesis of Binary and Continuously Actuated Planar Platforms UNIVERSITY OF DAYTON

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1 Kinematic Synthesis of Binary and Continuously Actuated Planar Platforms Thesis Submitted to The School of Engineering of the UNIVERSITY OF DAYTON in Partial Fulfillment of the Requirements for The Degree Master of Science in Mechanical Engineering by Michael Stephen Hanchak UNIVERSITY OF DAYTON Dayton, Ohio May, 000

2 Kinematic Synthesis of Binary and Continuously Actuated Planar Platforms APPROVED BY: Andrew P. Murray, Ph.D. Dr. Reza Kashani Mr. Phil Doepker Date The final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline.

3 ABSTRACT Kinematic Synthesis of Binary and Continuously Actuated Planar Platforms Name: Hanchak, Michael Stephen University of Dayton, 000 Advisor: Andrew P. Murray, Ph.D. A mechanical system defined by a jointed set of rigid bodies can be actuated to perform desired tasks like moving a part along a trajectory or pick and place operations. Examples include gantry systems and six-axis robots. Mechanical systems called platforms, hexapods or parallel manipulators can also be used to perform these robot-like tasks. The primary difference between a robot or gantry system and a (robotic) platform is that the platform has more than one fixed joint producing mechanical closures in the system. To the platform s advantage, the closure tends to yield a faster system that is mechanically simpler, stiffer, more precise, and more repeatable. The disadvantage is that this closure introduces non-trivial joint limitations and singularities, configurations at which the system either wants to seize or becomes uncontrollable. Designing for these disadvantages presents a significant challenge to the existing theory. In this thesis, we present the kinematic design theory for two types of mechanical systems possessing closure, three-legged planar platforms and binary actuated planar mechanisms. Then we review basic design theory for a third type of system, the 4C mechanism, and present its dynamics and controls. Planar platforms can arbitrarily displace iii

4 and orient a part in a plane of motion. The kinematic synthesis of this platform includes specifying an array of part locations and determining the three leg types and their locations such that the resulting platform does not suffer from singularities or extension beyond joint limitations. Binary actuated mechanisms are also designed to solve the planar part location problem for a low number of part locations. Binary actuation of the mechanism provides a controls-free scheme as all of the actuators are used in only two states. The solution to this problem is the leg locations and the required states for the binary actuators. Finally, a 4C mechanism produces spatial motion using only two actuators, three or four less than the standard robots typically used for spatial motion. We investigate the dynamics and controls of a 4C mechanism to evaluate its use in lean manufacturing schemes. iv

5 Contents Chapter 1 Introduction Background and Motivation Literature Survey Binary Actuated Planar Platforms Continuously Actuated Planar Platforms C Mechanism Overview Kinematics Review 14.1 Introduction Coordinate Transformations Planar Quaternions Line Coordinates Resultants Singularity Discussion Binary Actuated Planar Platforms 3.1 Introduction The Binary Crank Constraint v

6 3.3 Kinematic Synthesis of an RBR Chain Parallel Connection of Three RBR Chains Design of an N-Type Chain Example: A Singularity-Free N-Type Mechanism Example: 10 Position Synthesis Conclusions Constraint Manifolds Introduction Constraint Manifolds of PRR and RRR Chains Constraint Manifold of an RPR Chain Continuously Actuated Planar Platforms Introduction PRR Chain Synthesis RRR Chain Synthesis RPR Chain Synthesis Synthesis of the Final Chain Example Conclusions Dynamics and Control of a 4C Mechanism Introduction Three Position Synthesis of a CC Dyad Kinematics of the 4C Mechanism Dynamics of the 4C Mechanism Control of the 4C Mechanism vi

7 6.6 Conclusions Conclusions 67 Bibliography 69 vii

8 Figures Figure 1.1 A planar open-chain device with a single pivot fixed to ground A closed-chain device A 3-RBR planar mechanism An N-type 3-RBR mechanism is capable of reaching four prescribed positions A planar platform composed of three seperate chains A 4C mechanism A planar displacement A robot singularity A four-bar singularity A platform singularity An RBR chain with fixed and moving pivots defined relative to the fixed frame, F, and the moving frame, M i Two lengths that reach the six positions shown A 3-RBR mechanism that reaches the six positions shown. The B-joints lie between the darkened circles representing revolute joints A movement of the mechanism by actuating the chains in the following order: leg 1, leg 3, leg 1, and leg 3. Note that the length of leg is fixed throughout the motion viii

9 3.5 An N-type 3-RBR mechanism is capable of reaching four prescribed positions The shaded regions are acceptable locations of the fixed pivot G 1 for a singularity-free N-type mechanism A serial connection of three N-type mechanisms Acceptable locations of the fixed pivots for each of the three mechanisms that reach four of the ten positions shown A singularity-free 10 position mechanism Kinematic diagram of a PRR serial chain Kinematic diagram of an RRR serial chain Kinematic diagram of an RPR serial chain Graphical representation of the PRR constraint manifold Intersection of the PRR constraint manifolds at each position Graphical representation of the RRR constraint manifold Intersection of the RRR constraint manifolds at each position Graphical representation of the RPR constraint manifold Intersection of the RPR constraint manifolds at each position A Completed Platform A CC dyad is created by preserving the offset and twist between a fixed and moving line A model of the 4C mechanism was created in ADAMS c dynamical simulation software The kinematic diagram of a 4C mechanism shows the joint parameters The desired output versus the actual output of the displacement d The desired output versus the actual output of the angle θ ix

10 6.6 Block diagram of the feed-forward, feedback control scheme The desired output versus the actual output of the displacement d using feedback control The desired output versus the actual output of the angle θ using feedback control The motion of the 4C mechanism x

11 Tables Table 3.1 Potential leg states for the chains of a 3-RBR mechanism The leg states at each position for the N-type 3-RBR mechanism The four desired positions to be reached with the N-type 3-RBR mechanism Leg lengths at each position for the N-type 3-RBR mechanism example The ten desired positions to be reached with the three serially connected 3-RBR mechanisms The 1 positions described by a displacement and a rotation The 1 positions described by planar quaternions Leg design information for a platform that reaches the 1 positions Three spatial positions are reached by the 4C mechanism xi

12 Chapter 1 Introduction 1.1 Background and Motivation Various manufacturing tasks and assembly operations in industry are performed by multi-axis robotic systems. These systems are flexible from the standpoint that they can be reprogrammed to accomplish many different tasks. If they are used in an ever-changing production facility and are often changed to meet the needs of a particular task for a given time period, then the advantages of multi-axis robotic systems are realized through their flexibility. However, if the task requires high-cycle, repeatable motions with high inertial and vibrational forces, then closed-chain mechanisms designed specifically for that task are needed. A typical serial robotic chain is classified as an open-chain device. Open chains are connected to ground at only one place. See Figure 1.1. Open-chains are often susceptible to high inertial forces due to serial nature of their connections and the often far-reaching tasks which place the end-effector far from the base of the robot. In order to make open-chain robotic systems robust, they need to be designed to withstand these forces. This often means larger actuators or smaller members; both tend to increase cost. We seek to replace these robotic systems with closed-chain mechanisms, an example of which appears in Figure 1.. Closed-chain devices have multiple connections to ground increasing their ability to withstand high inertial and vibrational forces. These systems are

13 also little more than a few links connected together with revolute (hinge) joints and prismatic (sliding) joints. Although flexibility from a robotic standpoint is lost in these devices, they are potentially faster and offer lower energy solutions. Separate manufacturing tasks could be accomplished through dedicated mechanisms. Another big difference between open-chain and closed-chain devices is the number of degrees-of-freedom and hence the number of actuators needed. A traditional multi-axis robot may have up to six or more actuators and require control on each one. Closed-chain mechanisms can have fewer. As a result, certain closed-chain mechanisms can only reach a specific number of positions or a limited workspace based on their geometry. However, these closed-chain mechanisms can be designed for a specific planar or spatial task or trajectory, one in which only a few key positions are required. Closed-chain mechanisms are ideally suited for reaching specific positions. Designing closed-chain systems can be difficult and requires advanced kinematic techniques not learned in everyday engineering environments. Kinematics is the study of motion of mechanical systems without regard for the forces that cause them. Kinematic synthesis is the use of algebraic or geometric techniques to design mechanical systems to reach a specified workspace. As will be shown in this thesis, the techniques used in designing closed-chain mechanisms are often mathematically rigorous. Software was developed to incorporate the mathematics and thus create a fast, useable interface for designing these systems. 1. Literature Survey This thesis covers three types of closed-chain mechanisms. We now review some of the literature to provide background information for each subject.

14 1..1 Binary Actuated Planar Platforms A binary state prismatic joint (or B-joint), placed between fixed and moving revolute (or R-joint) pivots, produces two precise lengths for the RBR chain. One goal of using binary actuation is to limit control requirements; the B-joint is simply expected to drive between the two lengths. As a result, a device driven by a set of binary actuators has a finite number of fixed states. Prior to the work of Chirikjian (1994) for truss like manipulators and Waldron and Yang (1998) for large parallel arrays, little can be found on the use of binary actuation (Anderson and Horn, 1967; Pieper, 1968; Roth et al. 1973). A B-joint actuated mechanism, used for rigid body guidance, can move the body to a finite number of positions including orientations. In this thesis, we present an algorithm for determining the revolute joint locations and two required lengths for the RBR chain. Additionally, we consider couplings of these chains to define a viable mechanism. A simple diagram of A 3-RBR mechanism is shown in Figure 1.3. Although we deal with low numbers of actuated joints, the kinematic synthesis with large numbers of actuators to achieve large sets of positions without regard to orientation is presented in Chirikjian (1995). The design of mechanisms composed of RBR chains is similar to the design of fourbar mechanisms via classical Burmester Theory. Specifying up to five positions, Burmester Theory determines the fixed and moving pivot locations of a set of two-degree-of-freedom RR chains capable of guiding a body through the positions. Solving the simultaneous equations generated by writing the constant length condition (or crank constraint) for the unknown joint locations in each of the specified positions calculates these pivot locations. Any onedegree-of-freedom mechanism defined by coupling two of these chains can be assembled at the specified positions. The mechanism, however, cannot necessarily move between the positions by actuating a single joint requiring a separate discussion, typically called solution rectification, on the coupling of chains into mechanisms. Consult most machine theory texts 3

15 for discussions of Burmester Theory and solution rectification (for example Erdman and Sandor, 1997; Waldron and Kinzel, 1999). The design of a mechanism defined by a grouping of RBR chains proceeds analogously to the design of a fourbar mechanism. Specifying up to six positions, we determine the pivot locations of a set of RBR chains by solving the simultaneous equations generated by writing the constant length condition for the unknown pivot locations in each of the specified positions. The constant length in this case can be either of two values due to the two possible states of the B-joint. Any mechanism defined by grouping three of these chains can be assembled at the specified positions. The mechanism, however, cannot necessarily move between the positions by actuating the three B-joints requiring a separate discussion on the grouping of chains into mechanisms. In addition to the design of an individual RBR chain to guide a body through six positions, three other design problems are considered. First, the grouping of three of these chains into a single mechanism capable of reaching the six positions is presented. Due to the difficulties in producing viable designs from the first technique, a second method details the design of platforms with an N-type (or truss-like) arrangement of the chains to achieve four positions as shown in Figure 1.4. A common moving pivot location between the first and second chains and a common fixed pivot location between the second and third chains defines this N-type arrangement. The third design problem extends the second to synthesis for ten positions by serially connecting three of the N-type arrangements. All of the design routines mentioned are implemented and verified with Matlab c software available upon request from the authors. 1.. Continuously Actuated Planar Platforms Planar parallel manipulators capable of translating and rotating their moving platforms can be realized by rigidly joining the outermost moving pivots of three serial chains. 4

16 If each chain is either RPR, PRR or RRR in structure, all three-leg combinations form ten possible kinematic topologies: 3 RPR, 3 PRR, 3 RRR, RPR+PRR, RPR+RRR, PRR+RPR, PRR+RRR, RRR+RPR, RRR+PRR, and RPR+PRR+RRR. Note that R represents a revolute joint and P a prismatic joint. Figure 1.5 shows a RPR+PRR+RRR device. The kinematic synthesis of any of these topologies is the specification of the link dimensions and pivot locations defining a manipulator capable of reaching a desired workspace. The kinematic synthesis of nine of these topologies is presented; the 3 RRR manipulator is excluded from this discussion. Given a desired workspace represented by a finite set of positions, this design methodology locates the fixed and outermost moving pivots of three chains defining a kinematically viable manipulator. In addition to reaching the workspace positions, the resulting manipulator has several passive joint limitations imposed and is potentially singularity free. Any RPR and PRR chains are assumed to be actuated at the prismatic joint while RRR chains are assumed to be actuated at the fixed revolute joint. A constraint manifold is the workspace of an individual RPR, PRR, or RRR chain expressed in the space of planar quaternions, first used in kinematics by Blascke (1956). Investigations into algebraic descriptions of constraint manifolds are found in Ravani and Roth (198) for the C spatial chain, Bodduluri (1990) for a wide range of spatial dyads, and Bottema and Roth (1979) for a detailed study of the constraint manifold of a planar 4R linkage. The design methodology presented here uses a manipulation of the parameterized form of these manifolds, calculated here by the prescription presented in McCarthy (1990). To represent the desired workspace of the parallel manipulator, we discretize the workspace and identify positions both along its boundaries and inside the workspace. These positions are mapped to points in planar quaternion space. The requirement that each of these points lies on the constraint manifold of a chain defines a set of constraint equations for the design parameters. We exploit the structure of these equations and use its 5

17 coefficients directly rather than the specified design parameters. The design problem for individual chains reduces to intersecting sections of geometric regions in two different planes. Ravani s (198) numerical method for fitting constraint manifolds to as many as ten points representing rigid body positions is an example of design performed in the space of planar quaternions, see also Ravani and Roth (1983). Others include extensions of Ravani s curve fitting methodology to spherical 4R (Bodduluri and McCarthy, 199) and spatial 4C (Larochelle and McCarthy, 1994) linkages. The creation of designs that are potentially singularity free is done through a manipulation of the locus of singularity configurations (Sefrioui and Gosselin, 1993 & 1995) of the manipulator defined as a manifold in the space of planar quaternions called the Singularity Surface (Collins and McCarthy, 1998) C Mechanism Robots produce complex trajectories that include paths through three-dimensional space with accompanying changes in orientation. An intriguing alternative for generating a robot-like path is a two degree-of-freedom 4C mechanism. A 4C mechanism is the mechanical chain formed by connecting four cylindric joints into a closed chain as seen in Figure 1.6. A cylindric joint allows both translation along and rotation about a single axis. Considering that a robot is a multi-actuator serial chain, 4C mechanisms offer an interesting prospect in lean manufacturing. The ability of the 4C mechanism to produce robot-like motions with far fewer actuators partially arises from the fact that 4C mechanisms are closed chains. In addition to requiring fewer actuators, closed-chains mechanisms can potentially posses higher rigidity than open chain robots. Further potential advantages of closed chains are discussed in detail by Kota and Erdman (1997). Regarding 4C mechanisms, many kinematic analysis and synthesis issues have been addressed in the literature. Several formulations were employed to determine their complete 6

18 kinematic analysis (Duffy, 1980 and Yang et al, 1964). A 4C mechanism can be synthesized to guide a body through five spatial positions (Roth, 1967). Simply put, given five locations along a path (produced by a robot), a 4C mechanism can be designed to reach the five critical locations. Murray and McCarthy (1999) presented a geometric interpretation of the synthesis methodology for designing a 4C mechanism to achieve five spatial positions, the theoretical maximum achievable by one of these mechanisms. Many lean manufacturing tasks are pick and place operations requiring only two or three positions be reached with high accuracy. Beyond five positions though, approximation techniques are available. 4C mechanisms have been designed to approximately reach up to 10 positions on a trajectory (Larochelle and McCarthy, 1994). Lastly, Larochelle (1998) introduced SPADES, a 3D interactive environment to assist in 4C mechanism design. Dynamic analysis of a closed chain manipulator is more complicated than that of an open chain. Both Newton-Euler and Larangian approaches have been proposed by researchers (Guglielmetti, P. and Longchamp, R., 1994, Pang, H. and Shahinpoor, M., 1994, Zhang, C. D., and Song, S. M., 1993) for such analysis. The Largangian approach is used to formulate the inverse dynamics of the 4C mechanism. The closed chain nature of the mechanism requires a set of constraint equations and their derivatives be adjoined to the Lagrange equation (Tsai, 1999). These constraints are derived from the kinematics of the manipulator. Formulation of the inverse dynamics of the manipulator allows for the use of feedforward control (known also as computed torque/force control). The adverse effects of the uncertainties in the inverse dynamics model as well as disturbances entering the system are partially compensated for by augmenting the feed-forward controller by a set of feedback control loops around each active joint. This control strategy is very popular in applications where accurate trajectory following is very important such as robotics (Spong and Vidyasagar, 1989). Modern computers make the real-time computational burden of 7

19 feed-forward, feedback control less problematic than a decade ago. Moreover, since inverse dynamics only needs the values of the desired trajectory, the feed-forward control part of the computed torque/force can be evaluated off-line and implemented in real-time via either look up tables or accurate curve fits. 1.3 Overview In order to effectively speak about kinematics as it applies to closed-chain mechanisms, a review of relevant kinematic theory is presented in Chapter. Planar coordinate transformations used to generate the equations of motions for any kinematic chain are discussed. These are extended to homogeneous transformations and spatial transformations. Also detailed are line coordinates (Plucker coordinates) that uniquely describe lines in space. Planar quaternions are defined as an alternative for describing the kinematics of mechanical systems. Resultants are presented as a way to solve complex equations in two variables. Chapter ends with a basic discussion on singularities. Binary actuators are attractive since they have only two states in which to reside. The transient mode is not considered in their use. They are also easier to control than continuous actuators. Chapter 3 explores the use of binary actuators in the design of planar parallel mechanisms for six, four, and ten position synthesis tasks. The different number of positions for each mechanism type is dependent on the geometry and arrangement of the actuators and physical links. Planar quaternions are used to develop the kinematics of several kinematic chains. When planar quaternions are used to formulate the kinematics of a particular mechanism, a constraint manifold is generated. Constraint manifolds of planar serial chains are discussed in Chapter 4. Using the information in Chapter 4, the design of continuously actuated planar parallel platforms is presented in Chapter 5. These systems are designed to account for joint 8

20 limitations, a real world obstacle faced by these types of mechanisms. These systems can potentially be designed for an infinite number of positions. Also explored is the design of these mechanisms to be potentially free of singularities. Chapter 6 presents the design and control of a spatial 4C mechanism composed of four cylindric joints. It is a two degree-of-freedom device and is designed for a specific pickand-place task. This chapter shows not only the kinematic design of such a mechanism, but also the dynamic equations of motion. Finally, feed-forward, feedback control is used to achieve precise trajectory following characteristics. Conclusions and proposed future work finishes this thesis with Chapter 7. 9

21 Figure 1.1: A planar open-chain device with a single pivot fixed to ground. Figure 1.: A closed-chain device. 10

22 Figure 1.3: A 3-RBR planar mechanism. 11

23 Figure 1.4: An N-type 3-RBR mechanism is capable of reaching four prescribed positions. Figure 1.5: A planar platform composed of three seperate chains. 1

24 Figure 1.6: A 4C mechanism. 13

25 Chapter Kinematics Review.1 Introduction Kinematics is the study of motion without regard to the forces that create the motion. This chapter will present basic kinematic theory.. Coordinate Transformations The foundation of kinematics lies in relating information between multiple reference frames. Often times information is known with respect to only one reference frame. Using coordinate transformations, that information can be viewed in many different reference frames. A planar displacement, shown in Figure.1, consists of a displacement and a rotation from a fixed (F) reference frame. To relate coordinates in the displaced (M, moving) frame to coordinates in the fixed frame, X = Ax + d, (.1) where A = cos θ sin θ sin θ cos θ (.) is a x rotation matrix and d is the displacement of the moving frame relative to the fixed

26 Figure.1: A planar displacement. frame. A rotation matrix is one which has the following two properties: det(a) = 1 and A T A = I, (.3) where I represents the corresponding identity matrix. Equation.1 is not linear. In order to obtain a linear transformation, the information is packaged into a single homogeneous transform. It is now easier to produce the forward kinematics of a chain by multiplying the homogeneous transforms between successive reference frames. The forward kinematics relates joint variables to mechanism position, i.e. it is possible to input both the physical parameters and joint variables to determine accurately where the end-effector lies. Starting from the fixed frame, homogeneous transforms are written from one frame to the next and multiplied together. The planar homogeneous transform has the form H = cos θ sin θ d x sin θ cos θ d y (.4) This form is linear and invertible to allow for back and forth relations between the coordinates of various frames. These ideas are extended to space by realizing that an arbitrary spatial reference 15

27 frame has six degrees of freedom. In order to represent the possible three rotations, separate 3x3 rotation matrices are formulated for each. The displacement vector is similar to the planar case. The three rotation matrices around the X, Y, and Z axes respectively are as follows: [X(θ)] = cos φ 0 sin φ 0 cos θ sin θ, [Y (φ)] = 0 1 0, 0 sin θ cos θ sin φ 0 cos φ cos ψ sin ψ 0 [Z(ψ)] = sin ψ cos ψ 0. (.5) Mulitplying these together and assembling both the displacment and rotation information into one matrix yields the general form of the 4x4 spatial homogeneous transform: [H(θ, φ, ψ, x, y, z)] = cos φ cos ψ cos φ sin ψ sin φ x sin θ sin φ cos ψ + cos θ sin ψ sin θ sin φ sin ψ + cos θ cos ψ sin θ cos φ y.(.6) cos θ sin φ cos ψ + sin θ sin ψ cos θ sin φ sin ψ + sin θ cos ψ cos θ cos φ z Again, a sequence of these relating successive reference frames may be multiplied together to obtain the forward kinematics of a spatial robot or mechanism. 16

28 .3 Planar Quaternions Another way to formulate the forward kinematics of a chain is to represent it with planar quaternions. The planar quaternion is defined by the following formula: ( ) 1 p 1 x cos θ + y sin θ ( ) p 1 y cos θ P (x, y, θ) = = x sin θ, (.7) p 3 sin θ p 4 cos θ where x, y, and θ describe the displacment information. This is a 4x1 vector and has the property p 3 + p 4 = 1. Given a second quaternion {q 1, q, q 3, q 4 }, the multiplication of planar quaternions follows a specific procedure: P Q = p 1 q 4 + p q 3 p 3 q + p 4 q 1 p 1 q 3 + p q 4 + p 3 q 1 + p 4 q p 3 q 4 + p 4 q 3 p 3 q 3 + p 4 q 4. (.8) This procedure is simplified by formulating a matrix out of one of the quaternions and then using matrix multiplication. This is shown below. P Q = p 4 p 3 p p 1 p 3 p 4 p 1 p 0 0 p 4 p p 3 p 4 q 1 q q 3 q 4. (.9).4 Line Coordinates In certain spatial mechanisms it is desirable to represent joint axes as lines in space. However, lines are infinitely long objects and require parameterization. In order to capture 17

29 the information of a line in space without worrying about parameterizing that line with respect to some arbitrary starting point, one must employ the idea of line coordinates, also known as Plücker coordinates. The Plücker coordinates of a line are a 6x1 vector composed of a direction and a cross product between a point on the line and the direction: s S =, (.10) p s where s is the direction and p is a point on the line. The Plücker coordinate of a line is a unique representation regardless of which point on the line is choosen. In order to manipulate these lines to gain information about the mechanisms that they represent, certain properties and procedures involving line coordinates must be introduced. The dot product and cross product of lines are defined by s w s w S L = = and (.11) p s q w s q w + p s w s w s w S L = =. (.1) p s q w s (q w) + (p s) w The transformation of line coordinates is given by Z i A i 0 = Q i Z i D i A i A i z q z, i = , (.13) where D i is a skew-symmetric matrix composed of the elements of d i and A i is a rotation matrix. These ideas wll be implemented in Chapter 6 on a spatial mechanism..5 Resultants Often times solving the kinematics of mechanisms can lead to multiple non-linear equations in the desired variables. Two non-linear equations in two unknowns, where the 18

30 unknowns have only integer powers, can be simplified to a single polynomial equation in one unknown by using the resultant. A simple example follows. Given two equations 3x y + y + 3x + 5 = 0 (.14) xy + 5xy + y + 7 = 0, (.15) solve for the x s and y s which make the equations true. In order to accomplish this, choose one of the variables to supress, x, and write the equations in the following matrix form 3y 3 y + 5 x 0 0 y + 5y y + 7 x = 0, (.16) y + 5y y where the third row is equation (.15) mutliplied by x. By definition, the 3x3 matrix has lost rank. So, by taking its determinate and setting it equal to zero, one is left with a single polynomial equation in y. Roots are found and substituted into both equations. They together are used to solve for the x s..6 Singularity Discussion During the motion of both open and closed chain mechanisms and robots, there are places where the mechanism will either seize or become uncontrollable. More specifically, these places occur when the mechanism Jacobian(s) become singular. These are called singularities. For example, the planar robot in Figure. is actuated by its two revolute joints. As one can see, the instantaneous motion of the end point of the second link is either in an up or down direction. Solving for the joint velocities required to produce a motion other than what is possible results in infinite values. Without a sophisticated controller, this mechanism could cease to function if it is given an order to move in a direction other than what it can achieve at that moment. 19

31 Figure.: A robot singularity A planar four-bar mechanism is shown in Figure.3 is currently at a joint limitation, a type of singularity. The mechanism cannot move since the force produced at the actuated joint, φ, is in line with the three lined-up joints. Mechanism workspaces are bounded by joint limits and therefore are also bounded by singularities. A platform has the added difficulty of possessing platform singularities in addition to its individual leg singularities. This is shown in Figure.4. The 3-RPR platform is in a configuration where the lines of the three legs intersect at a single point. No leg actuation is capable of producing torques on the platform and thus this mechanism is at a platform singularity. This thesis is centered on designing mechanisms to avoid these very singularities. Viable mechanisms must not contain singularities while moving through the desired motions, however they may (and often will) contain singularities that do not lie in the desired workspace. 0

32 Figure.3: A four-bar singularity Figure.4: A platform singularity 1

33 Chapter 3 Binary Actuated Planar Platforms 3.1 Introduction The following section outlines the methodology to design a single RBR (revolutebinary prismatic-revolute) chain capable of reaching six positions. Three of these RBR chains, each reaching the six positions, can be assembled to create a planar parallel platform. Unfortunately, this does not guarantee a singularity-free mechanism. To simplify this procedure and create more rigid platforms, an N-type arrangement of the chains is considered. An N-type arrangement has the first and second chains sharing their moving pivots and the second and third chains sharing their fixed pivots. This arrangement, because of the extra constraints, may be designed to reach four positions. Lastly, this chapter shows a serial connection of three separate N-type mechanisms that is capable of reaching ten positions. Examples illustrate the various mechanisms. 3. The Binary Crank Constraint Let A i and d i be the rotation matrix and displacement vector, respectively, of the moving frame M i relative to the fixed frame F. With G being the coordinates of a fixed pivot in F, and g i being the coordinates of the the pivot in M i, G = A i g i + d i. (3.1)

34 See Figure 3.1. Likewise, the location of a moving pivot z of an RBR chain in M i is related to its location in F by Z i = A i z + d i. (3.) Figure 3.1: An RBR chain with fixed and moving pivots defined relative to the fixed frame, F, and the moving frame, M i. Given that the distance between G and Z i is l i, (Z i G) T (Z i G) = (z g i ) T (z g i ) = l i (3.3) If we constrain all locations of M i to have equal values of l i, equation (3.3) is called the crank constraint. For an RBR chain, however, the distance between the two pivots can be one of two lengths, l 1 or l. If we let positions 1 through j be reachable at length l 1 of the RBR chain, then by subtracting equation (3.3) evaluated at the first position from equation (3.3) evaluated at positions through j, (Z i Z 1 ) G 1 (Z i Z i Z 1 Z 1 ) = 0, i =,..., j. (3.4) Alternately, (g i g 1 ) z 1 (g i g i g 1 g 1 ) = 0, i =,..., j (3.5) 3

35 in moving frame coordinates. If we let positions j + 1 through n be reachable at length l of the RBR chain, then by subtracting equation (3.3) evaluated at position j + 1 from equation (3.3) evaluated at positions j + through n, (Z i Z j+1 ) G 1 (Z i Z i Z j+1 Z j+1 ) = 0, i = j +,..., n. (3.6) Alternately, (g i g j+1 ) z 1 (g i g i g j+1 g j+1 ) = 0, i = j +,..., n (3.7) in moving frame coordinates. 3.3 Kinematic Synthesis of an RBR Chain We now present the method for designing an RBR chain to reach six positions. The design problem has six unknowns: G x, G y, z x, z y, l 1, and l. We write equation (3.3) at all six positions and recognize that the positions must be grouped into two sets. The first set is reached with the length l 1 between the fixed and moving pivots, the second set with l between pivots. Grouping positions 1,, and 3 into the first set, we see from equation (3.4) that (Z Z 1 ) G 1 (Z Z Z 1 Z 1 ) = 0 and (Z 3 Z 1 ) G 1 (Z 3 Z 3 Z 1 Z 1 ) = 0. (3.8) Grouping positions 4, 5, and 6 into the second set, we see from equation (3.6) that (Z 5 Z 4 ) G 1 (Z 5 Z 5 Z 4 Z 4 ) = 0 and (Z 6 Z 4 ) G 1 (Z 6 Z 6 Z 4 Z 4 ) = 0. (3.9) Equations (3.8) and (3.9) are four bilinear equations in the four unknowns G and z. Using a resultant, equations (3.8) and (3.9) reduce to a single fourth order equation in G x (due to 4

36 coefficients being equal). Hence, we determine four solutions to this synthesis problem. For the arbitrary case, the solution of four general bilinear equations is sixth order (Wampler, 1996). Because of the potential for pairs of roots to be imaginary, there are 0,, or 4 real RBR chains that result from this procedure. Each grouping of the six positions produces a different set of up to four chains. For example, grouping positions 1, 3, and 5 in the first set and positions, 4, and 6 in the second set results in a different set of chains as compared to those determined above. An example of a grouping and the two leg lengths which reach the positions is shown in Figure 3.. Note that there are ten potential groupings of the six positions for this procedure. Figure 3.: Two lengths that reach the six positions shown. 3.4 Parallel Connection of Three RBR Chains Three RBR chains designed via the previous procedure define a mechanism capable of being assembled at the six positions. Note that each chain must be designed from a different grouping of the six positions (see Table 3.1 for example). Table 3.1 also suggests the actuation scheme to move the mechanism through the six positions if the mechanism is a kinematically viable one. For example, to move from position to position 5, all three legs must change length. To move from position 3 to position 1, leg must be actuated. 5

37 Figure 3.3 illustrates one of these mechanisms designed via the grouping presented in Table 3.1. Table 3.1: Potential leg states for the chains of a 3-RBR mechanism position leg leg leg Testing of the method has revealed that designing a kinematically viable mechanism that reaches the six positions using the grouping and actuation scheme shown in Table 3.1 is unlikely. Two issues need to be addressed. A mechanism, when located at each of the six positions, must assemble to the same side of a singularity. The joint velocities are related to the operational velocities by J x ẋ = J d ḋ, (3.10) where ḋ, the vector of joint velocities, is the rates of change of the lengths of the RBR chains and ẋ is the vector of operational velocities. Each RBR chains varies between two lengths greater than 0. By virtue of this fact, the Jacobian J d can never be singular. We recognize singularities when the Jacobian J x loses rank or det (J x ) = 0. Thus, we seek mechanisms in which either det (J x ) > 0 at every position or det (J x ) < 0 at every position. In our testing, only a small percentage of mechanisms pass this criteria. As we now argue, passing this criteria is necessary but not sufficient. Although the grouping of chains dictates a potential actuation scheme to move between two positions, this order is not unique. Further, the obvious order of actuation may not work while a far less trivial ordering succeeds. As long as the chains that need to change length from one position to another are actuated an odd number of times and the chains that do not need to change length from one position to another are actuated an even 6

38 number of times, many different actuation schemes are possible. For example, consider the two actuation schemes: (1) leg 1, leg3, leg, leg 1, leg 3 and () leg 1, leg, leg 1. Both of these result in a mechanism with the same leg lengths, however they may be in different configurations. This results from the ability of the mechanism to possess singularity-free paths from one configuration to another (Innocenti and Parenti-Castelli, 199). See Wenger and Chablat, 1998 for a descriptive study of this notion. In Figure 3.4, we illustrate this fact by actuating two of the legs of the mechanism an even number of times. The mechanism completes the motion in a different configuration from its starting location (where the leg lengths are the same in both). Searching all possible actuation schemes for successful paths between positions is the challenge. 3.5 Design of an N-Type Chain In order to design singularity-free RBR platforms, we connect the chains in an N- type truss as seen in Chirikjian, This constrains the first and second chains to share moving pivots and the second and third chains to share fixed pivots (see Figure 3.5). With these simplifications, the N-type configuration is capable of reaching four desired positions. There are also two free parameters that can be specified in the design procedure, in our case the coordinates of the unshared fixed pivot, G 1. We choose a unique grouping of the positions for each of the three chains (see Table 3.). Table 3.: The leg states at each position for the N-type 3-RBR mechanism. position leg leg leg Since we are dealing with two distinct fixed pivots in this procedure, G 1 and G, we 7

39 use G j = A i g ji + d i, i = 1,, 3, 4, j = 1,. (3.11) to solve for the fixed pivot location g ji in each desired position. Rewriting equation (3.5) and having selected the first fixed pivot G 1, we solve two linear equations in two unknowns, (g 1 g 11 ) z 1 1 (g 1 g 1 g 11 g 11 ) = 0 and (g 14 g 13 ) z 1 1 (g 14 g 14 g 13 g 13 ) = 0, (3.1) for the first moving pivot z 1. We then use Z ji = A i z j + d i, i = 1,, 3, 4, j = 1,. (3.13) to obtain the moving pivot s location in the fixed frame at each position. Manipulating equation (3.4), (Z 13 Z 11 ) G 1 (Z 13 Z 13 Z 11 Z 11 ) = 0 and (Z 14 Z 1 ) G 1 (Z 14 Z 14 Z 1 Z 1 ) = 0, (3.14) yields the secong fixed pivot G. Finally, using equations (3.11) and (3.5) we find the second moving pivot z : (g 4 g 1 ) z 1 (g 4 g 4 g 1 g 1 ) = 0 and (g 3 g ) z 1 (g 3 g 3 g g ) = 0. (3.15) 3.6 Example: A Singularity-Free N-Type Mechanism To generate singularity-free designs, we grid the workspace for possible fixed pivot locations. A mechanism is designed at each fixed pivot location G 1. Each platform is tested 8

40 at every position using two singularity conditions: Let G 1 = {G x, G y }, G = {H x, H y }, Z 1i = {Z xi, Z yi }, and Z i = {W xi, W yi }, then C i = (Z xi G x )(H y G y ) (Z yi G y )(H x G x ) (3.16) and D i = (Z xi H x )(W yi H y ) (Z yi H y )(W xi H x ) (3.17) for positions i = 1,, 3, 4. Geometrically, these conditions describe both the cross product of the line of leg 1 with the line between the fixed pivots and the cross product of the line of leg with the line of leg 3. If the cross products change sign between the four positions, then the legs have crossed each other alerting a singularity. Equations (3.16) and (3.17) are evaluated at all four positions for each possible design. A viable mechanism will have either C i > 0, i = 1,, 3, 4 or C i < 0, i = 1,, 3, 4, and either D i > 0, i = 1,, 3, 4 or D i < 0, i = 1,, 3, 4. Note that the sign need not be the same between the two conditions C i and D i. All platforms that meet these criteria are singularityfree and we plot the associated fixed pivots as shown in Figure 3.6. Figure 3.6 also shows a viable binary actuated mechanism that reaches the four positions found in Table 3.3 (θ in radians). Table 3.4 lists the various leg lengths at the four positions. The fixed pivots are G 1 = { 11.7, 7.8} and G = { 13.3, 13.5}. The moving pivots are z 1 = { 4.9, 6.} and z = { 5, 9.}. Table 3.3: The four desired positions to be reached with the N-type 3-RBR mechanism. position x y θ

41 Table 3.4: Leg lengths at each position for the N-type 3-RBR mechanism example. position leg leg leg Example: 10 Position Synthesis We reach ten positions by serially connecting three N-type mechanisms as shown in Figure 3.7. The first mechanism is designed to reach four of the positions. The second mechanism is designed to reach four positions, having one position in common with the first mechanism. The third is designed to reach four positions with one position in common with the second mechanism. Applying the same singularity check as previously required generates Figure 3.8. There are three separate regions dictated in Figure 3.8 (dots, crosses, and asterisks). Each of these correspond to a single four position synthesis problem. A manipulator capable of reaching ten positions results from rigidly joining the fixed pivots of the first mechanism to the moving pivots of the second mechanism and likewise the fixed pivots of the second mechanism to the moving pivots of the third mechanism. Potentially, one could keep joining mechanisms in this fashion, each time adding three positions to the design task. Figure 3.9 shows the mechanism designed to reach the ten desired positions given in Table 3.5 (θ in radians). 3.8 Conclusions This chapter presented the kinematic synthesis of mechanisms driven by binary actuated prismatic joints for rigid body guidance. Several synthesis problems were addressed. First, six positions, divided into two groups, were shown to define up to four RBR chains capable of guiding the body. Different groupings of the positions produced additional sets 30

42 Table 3.5: The ten desired positions to be reached with the three serially connected 3-RBR mechanisms. position x y θ of four chains. Second, a collection of three of these chains define a mechanism that can be assembled at the six positions. Satisfactory mechanisms prove elusive, however, due to singularity conditions and non-trivial actuation schemes. Third, a simplification from the general to an N-type mechanism with shared fixed and moving pivots allowed for four position synthesis in a singularity-free fashion. Finally, serially connecting three of these N-type mechanisms solved a ten position synthesis problem using only binary actuated joints. All of the design routines were implemented in Matlab c to verify the results. 31

43 Figure 3.3: A 3-RBR mechanism that reaches the six positions shown. The B-joints lie between the darkened circles representing revolute joints. 3

44 Leg Leg Leg 4 6 Leg 3 Leg Leg Leg Leg Leg 1 Leg Leg Leg Leg 0 Leg 3 Leg Figure 3.4: A movement of the mechanism by actuating the chains in the following order: leg 1, leg 3, leg 1, and leg 3. Note that the length of leg is fixed throughout the motion. 33

45 Figure 3.5: An N-type 3-RBR mechanism is capable of reaching four prescribed positions Leg 3 10 Leg 5 Leg Figure 3.6: The shaded regions are acceptable locations of the fixed pivot G 1 for a singularity-free N-type mechanism. 34

46 Figure 3.7: A serial connection of three N-type mechanisms. 35

47 Figure 3.8: Acceptable locations of the fixed pivots for each of the three mechanisms that reach four of the ten positions shown Figure 3.9: A singularity-free 10 position mechanism. 36

48 Chapter 4 Constraint Manifolds 4.1 Introduction The constraint manifold is the sub-manifold of positions available to a constrained moving body in the space of planar quaternions. The constraint manifold of the moving platform of a parallel manipulator is the intersection of the constraint manifolds of the individual chains connecting the platform to the base. We now derive the constraint manifolds of the PRR, RRR, and RPR chains shown in Figures 4.1, 4., and 4.3, respectively. For additional details on this derivation, see Murray and Hanchak (1999) and Murray and Pierrot (1998). 4. Constraint Manifolds of PRR and RRR Chains The PRR and RRR chains have the same form of the forward kinematics. They differ only in their assignment of joint variables and physical parameters. We derive the constraint manifolds of an arbitrary PRR chain shown in Figure 4.1 and an arbitrary RRR chain shown in Figure 4. by writing their forward kinematics using planar quaternions. Q P RR,RRR = Q(r, s, 0) Q(0, 0, φ 1 ) Q(d 1, 0, 0) Q(0, 0, φ )... Q(d, 0, 0) Q(0, 0, ψ) Q( ρ, σ, 0)

49 = 1 cos φ 1 r d s sin φ sin φ cos φ d ρ σ 0 sin ψ 0 1 cos ψ 1 (4.1) where designates quaternion multiplication (McCarthy, 1990). Expanding equation (4.1), Q P RR,RRR = {p 1, p, p 3, p 4 } = ( ) ( ) σ+s sin φ1 +φ +ψ + r ρ cos φ1 +φ +ψ + d 1 ( ) ( ) s σ cos φ1 +φ +ψ r+ρ sin φ1 +φ +ψ + d 1 sin cos cos sin ) ( φ1 +φ +ψ ( ) φ1 +φ +ψ ( ) φ1 φ ψ + d ( φ1 φ ψ cos ) + d sin Let {p 1, p, p 3, p 4 } T be a point on the constraint manifold Q P RR,RRR. Then ( ) φ1 +φ ψ ( ) φ1 +φ ψ.(4.) p 3 = sin φ 1 + φ + ψ and p 4 = cos φ 1 + φ + ψ. (4.3) Substituting eq. (4.3), the angular relationship φ 1 ψ = φ 1 + φ θ, and the following change of variables A 1 A = A 3 A ρ σ r s (4.4) into equation (4.) yields the sought parameterized form of the constraint manifold: ( ) ( ) p 1 A 1 A cos φ 1 θ cos φ 1 + φ θ + p 3 + p 4 = d 1 ( ) + d ( ). (4.5) p A 4 A 3 sin φ 1 θ sin φ 1 + φ θ For the PRR chain, φ 1 is fixed and d 1 is a joint variable. For the RRR chain, d 1 is fixed and φ 1 is the joint variable. 38

50 4.3 Constraint Manifold of an RPR Chain The derivation of the constraint manifold of a general RPR chain, as shown in Figure 4.3, can be found in Murray and Pierrot (1998). Multiplying out the forward kinematics of the RPR chain using planar quaternions yields: Q RP R = {p 1, p, p 3, p 4 } = σ+s sin φ 1+ψ + r ρ cos φ 1+ψ + d 1 cos φ 1 ψ s σ cos φ 1+ψ r+ρ sin φ 1+ψ + d 1 sin φ 1 ψ sin φ 1+ψ cos φ 1+ψ. (4.6) Let {p 1, p, p 3, p 4 } T be a point on the constraint manifold Q RP R. Then p 3 = sin φ 1 + ψ and p 4 = cos φ 1 + ψ. (4.7) Substituting eq. (4.7), the angular relationship φ 1 ψ = φ 1 θ, and the change of variables, equation (4.4), into equation (4.6) yields the sought parameterized form of the constraint manifold: ( ) p 1 A 1 A cos φ 1 θ + p 3 + p 4 = d 1 ( ). (4.8) p A 4 A 3 sin φ 1 θ 39

51 Figure 4.1: Kinematic diagram of a PRR serial chain Figure 4.: Kinematic diagram of an RRR serial chain 40

52 Figure 4.3: Kinematic diagram of an RPR serial chain 41

53 Chapter 5 Continuously Actuated Planar Platforms 5.1 Introduction This chapter provides a methodology for designing planar parallel platforms composed of RPR, PRR, and RRR chains to reach an arbitrary number of positions. Furthermore, the platforms will be designed to move inside joint limits placed on the individual chains and be potentially free of platform singularities. Constraint manifolds, discussed in the previous chapter, will be used to design the individual chains. An example illustrates the procedure. 5. PRR Chain Synthesis For each position {p 1, p, p 3, p 4 } j in the desired workspace, the values {A 1, A, A 3, A 4 } define an acceptable PRR chain design when 0 < d 1 d 1,max, 90 < φ < 90. (5.1) The desired workspace is assumed, without loss of generality, to contain positions with θ = 0. At θ = 0, p 3 = 0 and p 4 = 1. Substituting this simplification into the design form of the constraint manifold, equation (4.5), and rewriting yields the following equation. A p 1 cos φ 1 cos(φ 1 + φ ) = + d 1 + d. (5.) A 3 p sin φ 1 sin(φ 1 + φ )

54 At the specified joint limits, this equation has a graphical representation as shown in Figure 5.1. Since the physical parameters are contained in the A and A 3 terms, we will solve for them in the following manner. Writing equation (5.) at each position, we realize that the {A, A 3 } pairs which solve the design problem at the θ = 0 positions lie in the intersection of these graphical regions. This is shown in Figure 5.. Since there is an infinite number of possible pairs, we grid this intersection which determines the {A, A 3 } pairs that define the chain. Now we rewrite equation (4.5) noting that the A and A 3 terms are known: A 1 p 1 = A 4 p 3 p ( ) d 1 cos φ 1 θ p ( ) 3 sin φ 1 θ p 4 p 3 + d p 3 A +... A 3 ( ) cos φ 1 + φ θ ( ), (5.3) sin φ 1 + φ θ Using similar graphical representations, this time in the A 1 and A 4 terms, we grid the intersection of the manifolds in the θ 0 positions. We are left with many values of {A 1, A, A 3, A 4 } and thus this results in many complete chains that solve the position synthesis problem. 5.3 RRR Chain Synthesis For each position {p 1, p, p 3, p 4 } j in the desired workspace, the values {A 1, A, A 3, A 4 } define an acceptable RRR chain design when φ 1,min < φ 1 φ 1,max, 180 < φ < 0. (5.4) The synthesis procedure is very similar to that of the PRR chain. Since the constraint manifold is identical, we only notice a difference in the assignment of joint variables versus physical parameters. 43

55 Figure 5.1: Graphical representation of the PRR constraint manifold Figure 5.: Intersection of the PRR constraint manifolds at each position 44

56 The graphical representation of the RRR constraint manifold is shown in Figure 5.3. Using the same intersection technique, the regions for the RRR case look like those in Figure RPR Chain Synthesis For each position {p 1, p, p 3, p 4 } j in the desired workspace, the values {A 1, A, A 3, A 4 } define an acceptable RPR chain design when 0 < d 1,min d 1 d 1,max and φ 1,min φ 1 φ 1,max. (5.5) Again, the desired workspace is assumed, without loss of generality, to contain positions with θ = 0. At θ = 0, p 3 = 0, p 4 = 1, and equation (4.8) simplifies to A A 3 = p 1 cos φ 1 + d 1, (5.6) p sin φ 1 The graphical representation for the RPR constraint manifold is shown in Figure 5.5. Again the intersection technique is employed resulting in Figure 5.6. For the θ 0 positions, the RPR constraint manifold has the form: ( ) A 1 = p 1 p A 4 + d cos φ 1 1 θ A 4 p 3 p p 3 A 3 p ( ). (5.7) 3 sin φ 1 θ These procedures conclude with a set of values {A 1, A, A 3, A 4 } defining the joint locations of serial chains capable of reaching all positions in the desired workspace and satisfying the joint limits specified on both the actuated and passive joints. 5.5 Synthesis of the Final Chain An acceptable final chain must satisfy eqs. (4.5) or (4.8) for all positions in the workspace within the corresponding limits, given in eqs. (5.1) and (5.5), and must define a 45

57 Figure 5.3: Graphical representation of the RRR constraint manifold Figure 5.4: Intersection of the RRR constraint manifolds at each position 46

58 Figure 5.5: Graphical representation of the RPR constraint manifold Figure 5.6: Intersection of the RPR constraint manifolds at each position 47

59 platform free of singularities. We have implemented methodologies for the last chain to be either an RPR or a PRR. We now seek the parameters of this third chain, {A 13, A 3, A 33, A 43 }. Step 1: A 3 and A 33 are determined in an identical procedure as for the first two legs. Step : Given a value for A 3, limits on A 13 are defined by the first row of eq. (4.5) for the PRR and eq. (4.8) for the RPR and their respective joint limits. For all positions, a range on A 13 is determined. Potential values must lie within this range. At this point, we have obtained 11 design parameters of the 1 we seek: 4 for each of the first two chains, and 3 for the third chain. Step 3: Given a value for A 33, limits on A 43 are defined by the second row of eq. (4.5) for the PRR and eq. (4.8) for the RPR and their respective joint limits. For all positions, a range on A 43 is determined. Potential values must lie within this range. Step 4: The singular positions can be determined from the relationship between the operational velocities (in quaternion space) and the joint velocities. The relationship between the velocities is J p ṗ = J d ḋ, (5.8) where ṗ is the vector of operational velocities and ḋ is the vector of joint velocities. Note from the quaternion identity, p 3 + p 4 = 1, p 3 p 3 + p 4 p 4 = 0. (5.9) Singularities result from either one or both matrices J p and J d becoming singular. Due to the design requirements placed on the joint limits for both types of chains (equations (5.1) and (5.5)), J d is nonsingular by definition. Thus, we recognize singularities when det (J p ) = 0. We seek a platform that either has det (J p ) > 0 when evaluated at every position of the desired workspace or det (J p ) < 0 when evaluated at every position. Assembling the operational velocity Jacobian at position h, h = 1,..., number of 48

60 positions, in terms of the 11 known design parameters, the unknown parameter A 43, and the joint variable d 1 corresponding to the third chain results in K 11 K 1 K 13 K 14 K 1 K K 3 K 4 J p,h =, (5.10) K 31 g 1,h (A 43, d 1,h ) g,h (A 43, d 1,h ) g 3,h (A 43, d 1,h ) 0 0 p 3,h p 4,h Rows 1 and correspond to the completely designed chains and row 3 to the unfinished chain. This is a 4x4 matrix where all rows but the third contain known values. If the third chain is an RPR, the three functions g 1,h, g,h, g 3,h are at most quadratic in the unknown A 43 and independent of the joint variable d 1. Therefore, det (J p,h ) = α h A 43 + α 1h A 43 + α 0h (5.11) where the α i,h s, functions of the position and the 11 design parameters already determined, are known values. If the third chain is a PRR, then det (J p,h ) = α 1h d 1h + α h d 1h + α 3h d 1h A 43 + α 4h A 43 + α 5h A 43 (5.1) where the α i,h s are known values. The algebraic form of the constraint manifold of the PRR chain (see McCarthy, 1990) is β 1h d 1h + β h d 1h + β 3h d 1h A 43 + β 4h A 43 + β 5h A 43 = 0 (5.13) where the β ih s are known values. Eliminating d 1 between equations (5.1) and (5.13) yields δ 4 A δ 3 A δ A 43 + δ 1 A 43 + δ 0 = 0 (5.14) where the δ i s are known. 49

61 Solving equations (5.11) or (5.14) for A 43 produces values at which the det (J p,h ) = 0. These values divide the range of A 43 into usable and unusable regions. Checking the midpoints of these ranges at every position determines those manipulators with configurations all lying to the same side of a singularity. We focus this process by only checking values of A 43 that lie within the range determined in the previous step. 5.6 Example Software to aid in the design process was created in Matlab c. The user enters the desired positions and chooses which types of legs will define the platform. Limits are entered and the software presents the user with multiple choices for each leg. Any of these can be animated to verify its ability to reach the positions and obey the joint limits. Figure 5.7 illustrates a completed platform. Table 5.1 contains the position information, Table 5. contains the position quaternions, and Table 5.3 contains the design parameters and joint limits for each leg. 5.7 Conclusions We presented a methodology for designing nine of the ten planar platforms defined by combinations of RPR, RRR and PRR chains. The platforms are designed to achieve a set of desired positions, move within joint limits on all actuated joints and several passive joints, and reach every position without crossing singularities. Table 5.1: The 1 positions described by a displacement and a rotation. X Y θ rad

62 Table 5.: The 1 positions described by planar quaternions. p p p p End Effector 10 Workspace 5 Leg 3: RPR Leg 1: RPR 15 Leg : PRR Figure 5.7: A Completed Platform Table 5.3: Leg design information for a platform that reaches the 1 positions. A 1 A A 3 A 4 Leg 1: RPR Leg : PRR Leg 3: RPR Leg 1 φ 1,min = 0.5, φ 1,max = 0.5 d 1,min = 5, d 1,max = 5 Leg d = 1, φ 1 = 1.05, d 1,max = 5 Leg 3 φ 1,min = , φ 1,max = 0 d 1,min = 5, d 1,max = 5 51

63 Chapter 6 Dynamics and Control of a 4C Mechanism 6.1 Introduction This chapter presents the design, kinematics, dynamics, and control of a spatial 4C mechanism. Not unlike a planar four-bar mechanism, a 4C mechanism is composed of four joints connected to form a closed-chain loop. However, the 4C contains cylindrical joints while the four-bar contains revolute joints. This mechanism s workspace is the intersection of the workspaces of the two CC dyads composing it. Each dyad may be designed for up to five positions. Here, a design problem is formulated composed of three positions in a pick-and-place type operation. The two dyads are designed, rigidly connected, and observed for any link collisions or singularities throughout the motion. After the inverse kinematics are solved for the input joint values at each end-effector position, a cubic curve is fitted to their motion to provide zero velocity and acceleration at the first and last positions. This is done to allow this mechanism to be implemented for cyclical tasks at one or two Hertz frequencies. The dynamics were solved analytically using the constrained Lagrangian formulation. The result is an explicit representation of the actuator forcings needed to accomplish the task. This mechanism is implemented in the MDI ADAMS c software to verify the dynamics. Feed-forward, feedback control is utilized to compensate for model error arising from the use of lumped parameters in the analytical model versus the distributed parameters

64 inherent in the ADAMS c model. Desirable results are shown in the output trajectory graphs. 6. Three Position Synthesis of a CC Dyad Three spatial positions are defined by 3x3 rotation matrices, A i, and 3x1 displacement vectors, d i, i = 1,, 3, that relate coordinates in the moving frame relative to the fixed frame. We define the fixed and moving lines of a CC dyad in normalized Plücker coordinates (see Figure 6.1). The fixed line is given as G = (G, P G), where G is the direction of the line and P is a point on the line. Similarily, we represent the moving line as z = (z, q z), where z is the direction of the line and q is a point on the moving line both represented in the moving frame. The transformation of line coordinates is Z i A i 0 z =, i = , (6.1) Q i Z i D i A i A i q z where D i is a skew-symmetric matrix composed of the elements of d i, to calculate the coordinates of the moving line in each of the three positions, Z i = (Z i, (Q Z) i ). Note that the directions G and z are of unit length. Using the dot product for Plücker coordinates, McCarthy (1990) observes the constraint equations Z i G = cos ρ, i = 1,, 3, (6.) where ρ is the angle of twist between G and Z along the common normal, and Z i P G + Q i Z i G = r sin ρ, i = 1,, 3, (6.3) where r is the offset distance between G and Z i along the common normal. Subtracting equation (6.) at i = 1 from equation (6.) at i = and i = 3 yields the equation (Z Z 1 ) T (Z 3 Z 1 ) T G = 0. (6.4) 53

65 Without loss of generality, we choose the third element of G. Equation (6.4) then defines G completely. Let P G = {u, v, w}. Using this and subtracting equation (6.3) at i = 1 from equation (6.3) at i = and i = 3 yields the equations (Z x Z 1x )u + (Z y Z 1y )v + (Z z Z 1z )w + c c 1 = 0, (Z 3x Z 1x )u + (Z 3y Z 1y )v + (Z 3z Z 1z )w + c 3 c 1 = 0, (6.5) where the c 1, c, and c 3 are known values. Note that G (P G) = 0 G x u + G y v + G z w = 0 (6.6) Equations (6.5) and (6.6) produce three equations in the unknowns {u, v, w}. We now have the fixed line completely. Two different CC chains are designed to reach the three positions shown in Table 6.1 (in inches and degrees). For the first chain, we chose the moving line, Z = { 1, 0, 0}, and a point on the line, Q = {0, 0, 0}. This produced the fixed line, G = {.645,.4091,.645} and P = { 11.98, 13.83, 3.1}. For the second chain, we chose Z = {0, 0, 1} and Q = {0, 3, 0}. This produced G = {.3780,.6547,.6547} and P = { 16.7, 3.76, 5.64}. The two chains are coupled to produce a 4C mechanism. Table 6.1: Three spatial positions are reached by the 4C mechanism. Position 1 3 X Y Z 0 18 Longitude Latitude Roll

66 6.3 Kinematics of the 4C Mechanism The two chains designed in the previous step were modeled in the MDI ADAMS c software, shown in Figure 6., to verify the ability of the 4C mechanism to reach the desired positions. Connecting links preserve the relationships between the fixed and moving lines of each dyad. The location of the cylindric joints along these lines does not affect these relationships. At first, we connected the cylindric joints directly to the locations of P and q for each dyad. Unfortunately, a collision occurred between two of the links during the motion. To solve this, the location of cylindric joint along the fixed line of second CC dyad was changed to { 18.54,.17, 1.70} by moving it in the G direction six inches. All joint parameters, seen in Figure 6.3, are zero in the first position. Solving the inverse kinematics at the last position yields the final joint parameters. Joint 1 is both the actuated and measured joint. The trajectory of joint 1 is represented with a cubic curve with zero velocity at the first and third positions. The trajectories are as follows: d(t) = 5.t 3 7.8t (inches) and θ(t) = 4.71t t (radians) where t varies from 0 to 1 second. The kinematics of the 4C mechanism were solved by formulating the foward kinematics of each CC dyad. Equating these two expressions yields the following twelve implicit equations of motions (S = sin, C = cos): 0 =.97CζCψ +.694CζSψ.95SζCψ +.040SζSψ.49CφCθ.749CφSθ +.131Cφ +.444SφCθ.570SφSθ.08Sφ (6.7) 0 =.36Cψ.717Sψ.49SφCθ.749SφSθ +.131Sφ.444CφCθ +.570CφSθ +.08Cφ (6.8) 0 =.97SζCψ.694SζSψ.95CζCψ +.040CζSψ 55

67 +.716Cθ.095Sθ +.08 (6.9) 0 =.654Cζ +.378Sζ.7CφCθ.178CφSθ.38Cφ +.051SφCθ.67SφSθ +.60Sφ (6.10) 0 =.654.7SφCθ.178SφSθ.38Sφ.051CφCθ +.67CφSθ.60Cφ (6.11) 0 =.654Sζ +.378Cζ +.4Cθ.154Sθ.60 (6.1) 0 =.97CζSψ +.694CζCψ +.95SζSψ +.040SζCψ.757CφCθ +.488CφSθ +.06Cφ.619SφCθ.43SφSθ.098Sφ (6.13) 0 =.36Sψ.717Cψ.757SφCθ +.488SφSθ +.06Sφ +.619CφCθ +.43CφSθ +.098Cφ (6.14) 0 =.97SζSψ.694SζCψ +.95CζSψ +.040CζCψ.138Cθ.741Sθ (6.15) 0 =.36f Cψ +.717f Sψ Cψ CφCθ.48CφSθ +.395Cφ SφCθ 1.71SφSθ.64Sφ.716cCθ +.095cSθ.08c Cθ 7.389Sθ +.3d (6.16) 0 = f + e.817cφcθ.535cφsθ 1.146Cφ +.155SφCθ.803SφSθ Sφ.4cCθ +.154cSθ +.60c 6.65Cθ +.40Sθ.933d (6.17) 0 =.36f Sψ +.717f Cψ Sψ CφCθ CφSθ +.187Cφ 1.857SφCθ 56

68 1.96SφSθ.96Sφ +.138cCθ +.741cSθ.098c 6.86Cθ Sθ +.153d (6.18) With d and θ as inputs, we use equation (6.11) to solve for φ. Equations (6.10) and (6.1) yield ζ. Equations (6.8) and (6.14) yield ψ. With these angles, the joint displacements can be determined from equations (6.16), (6.17), and (6.18). Furthermore, by writing the forward kinematics up one chain to an end-effector, shown in Figure 6.3, we have three equations specifying its location ({p x, p y, p z }): 0 = 1.556Cθ Sθ CφCθ CφSθ.01Cφ.893SφCθ +.019SφSθ +.893Sφ +.583cCθ.000cSθ +.416c.645d px (6.19) 0 = Cθ Sθ CφCθ +.613CφSθ +.007Cφ.38SφCθ +.996SφSθ.566Sφ +.63cCθ.645cSθ.63c +.409d py (6.0) 0 = 3.61Cθ 14.4Sθ CφCθ.996CφSθ.01Cφ +.651SφCθ +.61SφSθ +.893Sφ.416cCθ.408cSθ +.416c.645d pz (6.1) The numerical values in equations (6.7) through (6.1) arise as a result of the physical parameters of each dyad and are unique to this mechanism. 57

69 6.4 Dynamics of the 4C Mechanism The constrained Lagrangian formulation was used to solve the inverse dynamics of the 4C mechanism. An application of this method to a three degree-of-freedom, spatial, closedchain manipulator is shown in Tsai (1999). The inverse dynamics solves for the forcing needed at the input joint(s) to achieve a given trajectory at the output. The constrained Lagrangian technique seeks to simplify the individual equations of motion by adding more generalized coordinates in the form of constraint equations. Partial derivatives are taken of every equation with respect to the generalized coordinates. Since the kinematics produced implicit constraint equations, this method is advantageous. The result is linear equations in both the Lagrange multipliers and the input forcing. The Lagrange equations of motion are given by d dt ( ) L q j L q j = Q j + k i=1 λ i Γ i q j, j = 1... n, (6.) where Γ i is the ith constraint equation, k is the number of constraint equations, L is the Langrangian function, and λ i is the Lagrange multiplier. The number of generalized coordinates is n. In our case, we have a two degree-of-freedom mechanism in d and θ. We formulate eleven dynamical equations, one for each of the generalized coordinates. Therefore, we need nine constraint equations: (6.7), (6.11), and (6.15) through (6.1). The vector of generalized coordinates is q = {ψ, φ, ζ, c, e, f, p x, p y, p z, θ, d}. (6.3) Tsai (1999) divides equation (6.) into two sets. One in which the Lagrange multipliers are unknowns and one in which the generalized forces are unknowns. These sets of equations are linear in their respective unknowns. The first set has the form k i=1 λ i Γ i q j = d dt ( ) L q j L q j ˆQ j, j = 1... k, (6.4) 58

70 where ˆQ j represents externally applied forces. These are zero in our formulation. We solve equation (6.4), where the right hand side is known, for the Lagrange multipliers. The second set has the form Q j = d dt ( ) L q j where Q j is the actuator forcing. L q j k i=1 λ i Γ i q j, j = k n, (6.5) We have three moving links: link 1, between joints 1 and ; link, between joints 3 and 4; and the coupler, between joints and 3. Our coupler, seen in Figure 6., has the shafts of the cylindric joints and 3 rigidly attached to it. This is different than the mechanism shown in Figure 6.3. This does not change the abilities of the mechanism, but it does simplify the analysis. Links 1 and are now modeled simply as cylinders. The coupler, however, is more complicated. Its physical parameters, mass, moment of inertia, etc., were determined by modeling the part geometry with specific material properties. The software then calculated all of the physical parameters. These parameters were used directly in the dynamic formulation. The Lagrangian function, L, is the total kinetic energy of the system, K total, minus the total potential energy of the system, U total, both in terms of the generalized coordinates. We formulate L in the follow manner. The kinetic energy of the three links are given as follows: K 1 = 1 m 1 d + 1 I 1 θ (6.6) K = 1 m ė + 1 I ψ (6.7) K c = 1 m c( p x + p y + p z ) + 1 (ωt I c ω) (6.8) where I c is the 3x3 moment of inertia matrix of the coupler and ω is the angular velocity 59

71 vector of the coupler: ω =.645 θ + φ.645 sin φ θ cos φ θ. (6.9).645 cos φ θ.409 sin φ θ The total kinetic energy of the mechanism is the sum of the kinetic energies for each link: K total = K 1 + K + K c. The potential energy of the links requires us to know the position vector of each center of mass throughout the motion of the mechanism. We then use the y component of these vectors as heights for the potential energy due to gravity. U 1 = m 1 g(6.915 cos θ sin θ d) (6.30) U = m g(.737 cos ψ 6.39 sin ψ e) (6.31) U c = m c gp y. (6.3) The total potential energy of the mechanism is the sum of the potential energies for each link: U total = U 1 + U + U c. The lagrangian function is given by: L = K total U total. We now have expressions for the Lagrangian function and the nine constraint equations. Following the methodology in Tsai (1999), we must take partial derivatives of each of these with respect to the generalized coordinates and the time derivatives of the generalized coordinates. This provides us with the required information needed to formulate the dynamics. We assemble the information in the following fashion. Equation (6.4) becomes Γ 1 q 1. Γ 1 q 9 Γ 9 q Γ 9 q 9 where the c j s and the 9x9 matrix are knowns. λ 1 c. = 1. λ 9 c 9, (6.33) We left-multiply both sides of equation (6.33) by the inverse of the 9x9 matrix to get 60

72 the instantaneous values of λ. Using these λ j s and rewriting equation (6.5) yields T orque F orce = c 10 c 11 Γ 1 q 10 Γ 1 q 11 Γ 9 q 10 Γ 9 q 11 λ. (6.34) With c 10, c 11, and the x9 matrix being completely known, we determine the instantaneous actuator forcing (T orque and F orce). The dynamics were solved numerically in Matlab c and curves were fitted to the resulting actuator forcing to allow for easy implementation into MDI ADAMS c, where the mechanism was simulated. The result of the curve fitting yielded the following functions: T orque(t) = 8103t t t t +567t 180 (inch-pounds) F orce(t) = 83t t 4 18t 3 9t + 5.4t (pounds) This forms the basis of feed-forward control. After modeling and simulating the mechanism, encouraging results were observed. Figures 6.4 and 6.5 shows the desired and actual trajectories of joint 1 throughout the motion. The variations between the desired and actual trajectories are due to the use of lumped parameters rather than distributed parameters. As will be discussed in the following section, trajectory error could be minimized by augmenting feed-forward control with feedback control. 6.5 Control of the 4C Mechanism A feed-forward, feedback control scheme was utilized to achieve precise trajectory following characteristics. See Figure 6.6. This is perhaps the most applicable technique given our type of application and the availability of the actuator forcing explicitly. Since feed-forward control was already accomplished in ADAMS c, only a feedback system would need to be designed and implemented. Using the controls package in ADAMS c, we were 61

73 able to incorporate PI control by comparing the desired trajectory of joint 1 with the actual, real-time measure of joint 1. The error between these two values was fed into a PI controller and then added to the actuator forcing. Both d and θ were compared and controlled with separate PI controllers. For the controller on the angle θ, a P gain of 10 and an I gain of 5 is used. For the controller on the displacement d, only a P gain of 70 is used. The mechanism followed the desired joint 1 trajectories with very little error. This can be seen in Figures 6.7 and 6.8. Also, a progression of the mechanism as it moves between the two positions is shown in Figure Conclusions The ability of low degree-of-freedom mechanisms performing rigorous spatial tasks was confirmed through the design and control of a 4C mechanism for rigid body guidance. Both the kinematics and dynamics were formulated to calculate the required input motions and forcing. A feed-forward, feedback control scheme was utilized to achieve precise trajectory following characteristics. The design was implemented in Matlab c and ADAMS c software packages for verification of the results. 6

74 ρ Z G r Q P Figure 6.1: A CC dyad is created by preserving the offset and twist between a fixed and moving line. Figure 6.: A model of the 4C mechanism was created in ADAMS c dynamical simulation software. 63