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

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.

Overview Kinematic Synthesis : Finite-position 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.

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.

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).

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

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.

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.

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.

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.

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,

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.

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)

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.

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.

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.

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

Numerical Example RPC Robot Dual quaternion synthesis: 4 soutions

Numerical Example RPC Robot

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.