Dimensional Synthesis of RPC Serial Robots

1. Dimensional Synthesis of RPC Serial Robots ICAR 2003 The 11th International Conference on Advanced Robotics June 30 - July 3, 2003 University of Coimbra, Portugal Alba Perez (maperez@uci.edu), J.M. McCarthy (jmmccart@uci.edu) Robotics and Automation Laboratory Department of Mechanical and Aerospace Engineering University of California, Irvine

2. Overview Constrained robotic system:A workpiece, or end-effector, supported by one or more serial chains such that each one imposes at least one constraint on its movement. Classification of constrained robotic systems 3-RPS constrained robot (category 3I, 3 degrees of freedom) • The constraints provide structural support in some directions, while allowing movement in the others. • The workspace of a constrained robot has less that six degrees of freedom. Therefore, positions that lie within the physical volume of the system may be unreachable.

3. Overview • Kinematic Synthesis : • Determine the mechanical constraints (i.e., links and joints) that provide a desired movement. • Finite-position Synthesis: • Can be interpreted as the design of constrained robotic systems. • Identify a set of task positions that represent the desired movement of the workpiece. • The methodology is developed for synthesis of serial open chains. The multiple solutions can be assembled to construct parallel chains.

4. Overview • Finite-position Synthesis Methodology: • Given: (a) a constrained serial chain, and (b) a task defined in terms of a set of positions and orientations of a workpiece, • Find: The location of the base, the location of the connection to the workpiece, and the dimensions of each link such the the chain reaches each task position exactly. • A set of design equations evaluated at each of the task positions is used to determine the mechanism. • There exist different methodologies to create the set of design equations.

5. Overview • The Design Equations for Finite Position Synthesis can be obtained in several ways: • Geometric features of the chain are used to formulate the algebraic constraint equations. (distance and angle constraints) • Kinematic geometry based on the screw representation of the composition of displacements. (equivalent screw triangle) • Robot kinematics equations define the set of positions reachable by the end-effector. Equate to each task position to obtain design equations Solve for the base position [G], the connection to the workpiece [H], and the link dimensions (j, aj) and joint parameters (j, dj)j (i positions).

6. Background Geometric features of the chain are used to formulate the algebraic constraint equations. (distance and angle constraints) • Chen, P., and Roth, B., 1969, ‘‘Design Equations for the Finitely and Infinitesimally Separated Position Synthesis of Binary Links and Combined Link Chains,’’ ASME J. Eng. Ind. 91(1):209–219. • Suh, C.H., and RadcliÆe, C.W., 1978, Kinematics and Mechanisms Design. John Wiley. • Innocenti, C., 1994, ``Polynomial Solution of the Spatial Burmester Problem.'' Mechanism Synthesis and Analysis, ASME DE vol. 70. • Murray, A.P., and McCarthy, J.M., 1994, “Five Position Synthesis of Spatial CC Dyads”. Proced. ASME Mechanisms Conference, Minneapolis, MN, Sept. 1994. • Kim, H. S., and Tsai, L. W., 2002, “Kinematic Synthesis of Spatial 3-RPS Parallel Manipulators,” Proc. ASME Des. Eng. Tech. Conf. paper no. DETC2002/MECH-34302, Sept. 29-Oct. 2, Montreal, Canada. CS chain

7. Background • Kinematic geometry based on the screw representation of the composition of displacements. (equivalent screw triangle) • Tsai, L. W., and Roth, B., 1972, “Design of Dyads with Helical, Cylindrical, Spherical, Revolute and Prismatic Joints,” Mechanism and Machine Theory, 7:591-598. • Robot kinematics equations define the set of positions reachable by the end-effector. Equate to each task position to obtain design equations • Mavroidis, C., Lee, E., and Alam, M., 2001, A New Polynomial Solution to the Geometric Design Problem of Spatial RR Robot Manipulators Using the Denavit-Hartenberg Parameters, J. Mechanical Design, 123(1):58-67. • Lee, E., and Mavroidis, D., 2002, “Solving the Geometric Design Problem of Spatial 3R Robot Manipulators Using Polynomial Homotopy Continuation”, ASME J. of Mechanical Design, 124(4), pp.652-661.

8. Features of this Problem • Stating the design equations • Methods based on geometric constraints give simpler equations but lack a general methodology to find the constraints for all kinds of chains. • Methods based on the kinematics equations are general but give a more complicated set of equations with extra variables. • RR chain: • 10 geometric constraints • 5R chain: • geometric constraints? • Using the kinematics equations, we obtain a set of 120 equations in 120 variables, including the joint angles.

9. Features of the problem • Solving the design equations • Set of polynomial equations of high degree in several variables. • The joint variables can be eliminated to reduce the dimension of the problem. • Due to internal structure, they could be much simplified. • Some sample cases: • RR chain (2 dof robot): • Initial total degree: 210 = 1024. • Final solution: six roots, with only two real solutions. • RPR chain (3 dof robot): • Initial total degree: 23*46 = 32768. • Final solution: 12 roots. • RPS chain (5 dof robot): • Initial total degree: 262144. • Final solution: 1020 roots.

10. Our Approach • Stating the design equations : • Use dual quaternion synthesis: systematic way of creating the design equations that allows elimination of the joint variables. • Solving the design equations: • For those cases where it is possible, algebraic elimination leads to a close solution: • Resultant methods to create a univariate polynomial. • Matrix eigenvalue methods. • For those cases that are too big for algebraic elimination, numerical methods to find all solutions: • Polynomial continuation methods.

11. Dual Quaternion Synthesis of Constrained Robots • The robot kinematics equations of the chain are used to formulate design equations. • The set of displacements of the chain are written as a product of coordinate transformations, • Formulate the kinematics equations of the robot using dual quaternions,

12. Dual Quaternion Synthesis of Constrained Robots • From the dual quaternion kinematics equations, • Create the design equations: equate the kinematics equations to each task position written in dual quaternion form: • We obtain a set of vector equations where the variables to solve for are the Plucker coordinates of the axes Sj in the reference position. • The equations are parameterized by the joint variables j, j=1,…k.

13. Dual Quaternion Synthesis of Constrained Robots • How many task positions can we define? • Consider a serial chain with r revolute joints and t prismatic joints. • Number of task positions, n. • Dual Quaternion design equations, 6(n-1) • Parameters: • R joint-- 6 components of a dual vector, 6j. • P joint-- 3 components of a direction vector, 3k. • Associated constraint equations: • R joint-- 2 constraints (Plucker conditions), 2j. • P joint-- 1 constraint (unit vector), k . • Imposed constraint equations, c. • Joint variables, (r+t)(n-1) (measured relative to initial configuration). Equations: 6(n-1)+2r+t+c. Unknowns: 6r+3t+(r+t)(n-1). n = (4r +2t -c)/(6-r-t) + 1. (note r+t < 6 for constrained robotic systems)

14. Dual Quaternion Synthesis of Constrained Robots • Systematic methodology to create design equations for any constrained robot. • A formula for counting the maximum task positions we can define for each robot topology. • Solve the parameterized design equations for both dimensions and inverse kinematics, or • Create reduced algebraic equations that may be further reduced to find closed solutions.

15. Design Example: RPC Robot • Design equations • We can define a maximum of n=5 task positions if we impose g.h=0, w.h=0.

16. Design Example: RPC Robot • Parameterized design equations: • 31 equations in 31 unknowns, 16 of them are joint variables. • each equation is at most of multi-degree 6. • Reduced design equations: • Eliminate the joint variables to obtain a set of 15 equations in 15 parameters. • Separation of rotations and translations and further resultant eliminations lead to a set of linear equations plus a 6th degree polynomial. We obtain at most 6 RPC robots.

17. Numerical Example RPC Robot Task definition: 5 positions Software: Synthetica 1.0, developed by Hai Jun Su, Curtis Collins and J.M. McCarthy

18. Numerical Example RPC Robot Dual quaternion synthesis: 4 soutions

19. Numerical Example RPC Robot

20. Conclusions • The dual quaternion synthesis procedure uses the kinematics equations of an open chain to formulate the design equations. • Multiple solutions can be assembled to create parallel robots. • The synthesis procedure can be applied to general 2-5 degree of freedom serial chains. • For simple cases, algebraic simplification is performed to obtain a closed solution. More complicated cases require polynomial continuation algorithms. • The synthesis procedures are being implemented in the spatial design software SYNTHETICA 1.0. The java applet can be downloaded at http://synthetica.eng.uci.edu/~mccarthy/Synthetica1.0/Synthetica.htm • Work needed: • Strategies for specifying spatial linkage tasks. • Numerical solutions that are robust relative to local minima. • Conditions for branching, joint limits and self-intersection are required for general parallel systems.