1. Adaptive dynamics for Articulated Bodies

2. Articulated Body dynamics Optimal forward dynamics algorithm –Linear time complexity –e.g. Featherstone’s DCA algorithm –Not efficient enough for many DoF systems

3. Articulated body A B Handles: positions where external forces can be applied Handle

4. Articulated body Created recursively by joining two articulated bodies C Principal joint

5. Articulated body Tree representation of an articulated body Rigid bodies The complete articulated body C AB

6. Featherstone’s DCA Articulated-body equation Change of in causes a change of in Body Accelerations Inverse inertias and cross-inertias Applied Forces Bias accelerations

7. Articulated body equations Kinematic constraint force at the principal joint of C

8. Featherstone’s DCA Algorithm Update body velocity and position Main pass: Compute –Bottom-up pass Solve articulated body equation by back substitution –Top down pass

9. Main Pass For internal nodes For leaf nodes dependent on motion subspacedependent on active forces

10. Back substitution Receive from parent Compute joint acceleration and using Send to A and to B

11. Adaptive Dynamics Simulate n most “important” joints Sacrifice amount of accuracy Other joints are rigidified “Important” and “accuracy” measures based on some motion metric

12. Hybrid body

14. Multilevel forward dynamics algorithm Compute body velocity and position only in active region Compute –Same as DCA for active nodes –Do not recompute for rigid nodes – (*) Compute in force update region using Back substitute only in active region Recompute hybrid body (at a different rate than the simulation timestep) * For the metric we discuss later, this step is not performed

15. Motion metrics Acceleration metric Velocity metric are SPD matrix i.e. metrics correspond to weighted sum of squares

16. Theorem The acceleration metric value of an articulated body can be computed before computing its joint accelerations Computing motion metric

17. Computing In active region compute using:

18. Computing Do not recompute at passive nodes At passive nodes compute ( velocity dependent coefficients ) using linear coefficient tensors ( not dependent on velocity ) –Constant time

19. Computing the hybrid body Compute in fully articulated state Determine transient hybrid body based on acceleration metric Recompute acceleration for transient hybrid body Compute velocity metric to determine hybrid body Rigidification

20. Adaptive joint selection Acceleration simplification = 96 Compute the acceleration metric value of the root

21. = 96 -3 Compute the joint acceleration of the root Adaptive joint selection Acceleration simplification

22. = 96 = 6 = 81 Compute the acceleration metric values of the two children -3

23. Adaptive joint selection Acceleration simplification = 96 Select the node with the highest acceleration metric value -3 = 6 = 81

24. Adaptive joint selection Acceleration simplification = 96 Compute its joint acceleration -3 -6 = 81 = 6

25. Adaptive joint selection Acceleration simplification = 96 = 9= 36 Compute the acceleration metric values of its two children -3 -6 = 6 = 81

26. Adaptive joint selection Acceleration simplification = 96 = 9= 36 -3 -6 = 6 = 81 Select the node with the highest acceleration metric value = 36

27. Adaptive joint selection Acceleration simplification = 96 = 9= 36 -3 -6 = 6 = 81 Compute its joint acceleration 6

28. Adaptive joint selection Acceleration simplification -3 -6 6 Stop because a user-defined sufficient precision has been reached = 96 = 9 = 6

29. Adaptive joint selection Acceleration simplification -3 -6 6 Four subassemblies with joint accelerations implicitly set to zero = 96 = 9 = 6

30. Velocity simplification Compute joint velocities in the transient active region ( computed using acceleration metric) Compute metric in a bottom up manner from the transient rigid front using Compute rigid front like for acceleration metric

31. Rigidification Aim: Rigidify the joint velocitiesto 0 Constraint: Solve for –Compute by computing –Compute Apply to the hybrid body basis vector for

32. video

33. References FEATHERSTONE, R. 1999. A divide-and-conquer articulated body algorithm for parallel o(log(n)) calculation of rigid body dynamics. part 1: Basic algorithm. International Journal of Robotics Research 18(9):867- 875. S. Redon, N. Galoppo, and M. Lin. Adaptive dynamics of articulated bodies : ACM Trans. on Graphics (Proc. of ACM SIGGRAPH), 24(3), 2005.