Kinematic Synthesis of Robotic Manipulators from Task Descriptions

1. Kinematic Synthesis of Robotic Manipulators from Task Descriptions June 2003 By: Tarek Sobh, Daniel Toundykov

2. Envisioning Optimal Geometry

3. Objectives • Parameters considered in this work: • Coordinates of the task-points • Spatial constraints • Restrictions (if any) on the types of joints • Goals • Simplified interface • Performance • Modular architecture to enable additional optimization modules (for velocity, obstacles, etc.)

4. Optimization Techniques • Minimization of cost functions • Stochastic algorithms • Parameters space methods • Custom algorithms developed for specific types of robots

5. Steepest Descent Method {fi(x)=0} → S(x)=∑fi(x)2 • System of equations is combined into a single function whose zeroes correspond to the solution of the system • Algorithm iteratively searches for local minima by investigating the gradient of the surface S(x). • Points where S(x) is small provide a good approximation to the optimal solution.

6. Manipulability Measure w=√det(J∙JT) • For performance purposes the manipulability measure was the one originally proposed by Tsuneo Yoshikawa • Singular configurations are avoided by maximizing the determinant of the Jacobian matrix

7. Optimization Measure

8. Single Target Problem Cost = [b + Manipulability]-1 + p [Distance to target] b := bias to eliminate singularities p := precision factor • Parameters that minimize the cost yield larger manipulability and small positional error • Increase of the precision factor forces the algorithm to reduce the positional error in order to compensate the overall cost growth

9. Optimization for Multiple Targets • Several single-target cost functions are combined into a single expression • Single-target cost functions share the same set of invariant DH-Parameters; however, each of these functions has its own copy of the joint variables

10. Invariant DH-Parameters • Invariant parameters depend on the types of joints • When no joints are specified, the algorithm compares all possible configurations based on the average manipulability value • Invariant DH-parameters have a dumping factor. If dumping is large, the dimensions of the robot must decrease to keep the total cost low

11. Results of Optimization Geometry that maximizes manipulability at each target Shared DH-parameters → Joint Vector for Target 1 → Inverse Solution for Target 1 … … Joint Vector for Target N → Inverse Solution for Target N

12. Powerful mathematical and graphics tools for scientific computing Flexible programming environment Availability of enhancing technologies: Nexus to Java-based applications via J/Link interface Flexible Web-integration provided by webMathematica®software Potential access to distributed computing systems, such as gridMatematica® Mathematica® (Wolfram Research Inc )

13. CAD Module Structure

14. Input Data • The set of task points • Configuration restrictions: • DOF value if the system should determine optimal types of joints by itself • or a specific configuration, such as Cartesian, articulated etc. • Precision and size-dumping factors • Output file name

15. Screenshots

16. Sample I • Design a 3-link robot for a specific parametric trajectory • No configuration was given, so the software had to choose the types of joints • Dimensions of the robot were severely restricted

17. Sample I : Trajectory

18. Sample I : DH-Table (PRP)

19. Sample I : Manipulability Ellipsoids

22. Sample II • The trajectory has been changed • This time we require a spherical manipulator • No significant spatial constraints have been provided

23. Sample II : Trajectory

24. Sample II : DH-Table (RRR)

25. Sample II : Manipulability Ellipsoids

28. Further Research • Work has been done to account for robot dynamics and velocity requirements • Online interface to the design module • Future research may include obstacle avoidance and integration with distributed computing architectures