A family of rigid body models: connections between quasistatic and dynamic multibody systems

1. A family of rigid body models: connections between quasistatic and dynamic multibody systems Jeff Trinkle Computer Science Department Rensselaer Polytechnic Institute Troy, NY 12180 Jong-Shi Pang, Steve Berard, Guanfeng Liu

2. Parts Feeder Design • Parts feeder design goals: • Exit orientation independent of entering orientation • High throughput • Design geometry of feeder to guarantee 1) and maximize 2). • Feeder geometry has 12 design parameters • Evaluate feeder design via simulation Part enters cg down Part enters cg up Motivation Dexterous Manipulation Planning Valid quasistatic plan exists No quasistatic plan found, but dynamic plan exists

3. Simulation of Pawl Insertion

4. Past Work in Quasistatic Multibody Systems Grasping and Walking Machines – late 1970s. Used quasistatic models with assumed contact states. Whtney, “Quasistatic Assembly of Compliantly Supported Rigid Parts,” ASME DSMC, 1982 Caine, Quasistatic Assembly, 1982 Peshkin, Sanderson, Quasistatic Planar Sliding, 1986 Cutkosky, Kao, “Computing and Controlling Compliance in Robot Hands,” IEEE TRA, 1989 Kao, Cutkosky, “Quasistatic Manipulation with Compliance and Sliding,” IJRR, 1992 Peshkin, Schimmels, Force-Guided Assembly, 1992

5. Past Work in Quasistatic Multibody Systems Mason, Quasistatic Pushing, 1982 - 1996 Brost, Goldberg, Erdmann, Zumel, Lynch, Wang Trinkle, Hunter, Ram , Farahat, Stiller, Ang, Pang, Lo, Yeap, Han, Berard, 1991 – present Trinkle Zeng, “Prediction of Quasistatic Planar Motion of a Contacted Rigid Body,” IEEE TRA, 1995 Pang, Trinkle, Lo, “A Complementarity Approach to a Quasistatic Rigid Body Motion Problem,” COAP 1996

6. Model Space Dynamic Quasistatic Kinematic Geometric Compliant Rigid Hierarchical Family of Models • Models range from pure geometric to dynamic with contact compliance • Required model “resolution” is dependent on design or planning task • Approach: • Plan with low resolution model first • Use low resolution results to speed planning with high resolution model • Repeat until plan/design with required accuracy is achieved

7. Quasistatic model: time-scale the Newton-Euler equation. Components of a Dynamic Model Newton-Euler Equation Defines motion dynamics Kinematic Constraints Describe unilateral and bilateral constraints Normal Complementarity Prevents penetration and allows contact separation Friction Law Defines friction force behavior: Bounded magnitude Maximum Dissipation Leads to tangential complementarity Maintains rolling or allows transition from rolling to sliding

8. Linear Complementarity Problem of size 1. Given constants and , find such that: Complementarity Problems Let be an element of and let be a given function in . Find such that:

9. - configuration - generalized velocity - symmetric, positive definite inertia matrix - non-contact generalized forces • Jacobian relating generalized velocity and time rate of change of configuration where Newton-Euler Equation Non-contact forces

10. Locally, C-space is represented as: Kinematic Quantities at Contacts Normal and tangential displacement functions:

11. Normal Complementarity Define the contact force Normal Complementarity where

12. Assume a maximum dissipation law where is the contact slip rate Coulomb Linearized Coulomb Friction Friction Friction Slip Slip Slip Dry Friction

13. Non-contactforces Instantaneous-Time Dynamic Model

14. Scale the Times of the Input Functions Scale the driving inputs. Replace with in the driving input functions.

15. Time-Scaled Dynamic Model Change variables Application of chain rule and algebra yields:

16. Time Stepping Methods Approximate derivatives by: where is the time step, , and is the scaled time at which the state of the system was obtained.

17. Constraint Stabilization Kinematic Control LCP Time-Stepping Problem

18. Example: Fence and Particle Assume: Particle is constrained from below Non-contact force: Fence is position-controlled Wall is fixed in place Expected motion: Quasistatic: no motion when not in contact with fence. Dynamic: if deceleration of paddle is large, then particle can continue sliding without fence contact

19. Time-Scaled Fence and Particle System Dynamic Quasistatic Boundary

20. Time-Scaled Fence and Particle System Dynamic Quasistatic

21. Cast Model as Convex Optimization Problem Introduce the friction work rate value function: Linear in Introduce the friction work rate minimum value function:

22. Equivalent Convex Optimization Problem OPT := Hypograph of is convex. Therefore is concave and is convex. KKT conditions are exactly the discrete-time model.

23. Theorem If solves the model with quadratic friction cone, then is a globally optimal solutions of OPT corresponding to . Conversely, if is a globally optimal solution to OPT for a given and if is equal to an optimal KKT multiplier of the constraint in OPT, then defining as below, the tuple solves the model with quadratic friction cone.

24. Proposition: Solution Uniqueness Corresponding to the solution of the discrete-time model with quadratic friction cone, is the unique solution of OPT, if and only if the following implication holds: Added motion does not decrease work Added motion does not change friction work. Added motion does not cause penetration where is a small change in

25. Friction Slip Slip Friction Example Solution is unique with non-zero quadratic friction on plane Solution is not unique without friction Solution is not unique with linearized friction on plane

26. Future Work Convergence analysis Experimental validation Design applications

27. Fini

28. Friction Impulse Limit Curve where Relative Velocity Maximum Work Boundary or Interior Maximum Work Inequalty: Unilateral Constraints is the vector of the components of relative velocity at the contact in the directions. Linearize the limit curve at contact where the columns of are the vectors transformed into C-space.

29. Friction Impulse Limit Curve Relative Velocity Tangential Complementarity: Example

30. - diagonal matrix of friction coefficients at rolling contacts Instantaneous Rigid Body Dynamics in the Plane

31. Example: Sphere initially translating on horizontal plane.

32. Simulation with Unilateral and Bilateral Constraints

33. With Constraint Stabilization Admissible Configurations Admissible Configurations Current implementation uses stabilization and the “path” algorithm (Munson and Ferris). Time-Stepping with Unilateral Constraints Without Constraint Stabilization Solution always exists and Lemke’s algorithm can compute one (Anitescu and Potra).

34. Solution Non-uniqueness:LCP Non-Convexity Two Solutions

35. Solution Non-Uniqueness:Contact Force Null Space Both friction cones can “see” the other contact point. Assume: Blue discs are fixed in space Red disc is initially at rest Solution 1 – disc remains at rest Solution 2 – disc accelerates downward External Load