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

3. Computer Science LIGA Tribology Test “Vehicle” LIGA – German acronym for process for making micro- scale parts from metals, ceramics, and plastics. Typical dimensions are on the order of Sandia wants to understand function, efficiency, robustness before building. Optimal design.

4. Computer Science Micro-Machine Assembly Pawl (2.3 mm) and washer (1.0 mm) subassembly. Pins (0.169 mm) in holes (0.165 mm). Need fixture to hold and align washer and pawl. Fixture should guarantee unique positions and orientations of parts. Tweezers

5. Computer Science Pawl in Fixture

6. Computer Science Simulation of Pawl Insertion

7. Computer Science 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

8. Computer Science 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

9. Computer Science 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 Model Space Rigid Compliant Dynamic Quasistatic Geometric Kinematic

10. Computer Science 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 Quasistatic model: time-scale the Newton-Euler equation.

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

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

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

14. Computer Science Normal Complementarity where Define the contact force Normal Complementarity

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

16. Computer Science Instantaneous-Time Dynamic Model Non-contact forces

17. Computer Science Scale the Times of the Input Functions Scale the driving inputs. Replace with in the driving input functions.

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

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

20. Computer Science LCP Time-Stepping Problem Constraint Stabilization Kinematic Control

21. Computer Science 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 Example: Fence and Particle

22. Computer Science Time-Scaled Fence and Particle System Dynamic Quasistatic Boundary

23. Computer Science Time-Scaled Fence and Particle System Dynamic Quasistatic

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

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

26. Computer Science 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 Theorem solves the model with quadratic friction cone.

27. Computer Science where is a small change in 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: Proposition: Solution Uniqueness Added motion does not decrease work Added motion does not change friction work. Added motion does not cause penetration

28. Computer Science Example Slip Friction 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 Friction Slip

29. Computer Science Future Work Convergence analysis Experimental validation Design applications

30. Computer Science Fini

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

32. Computer Science Tangential Complementarity: Example Friction Impulse Relative Velocity Limit Curve

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

34. Computer Science Example: Sphere initially translating on horizontal plane.

35. Computer Science Simulation with Unilateral and Bilateral Constraints

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

37. Computer Science Solution Non-uniqueness: LCP Non-Convexity Two Solutions

38. Computer Science 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