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
Handle B A
Handles: positions where external forces can be applied

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

5. The complete articulated body
Rigid bodies
The complete articulated body C A B
Tree representation of an articulated body

6. Inverse inertias and cross-inertias
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
The dynamics of an articulated body can be described by an articulated-body equation, which gives the relationship between the forces applied to the articulated bodies and their 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 subspace
dependent 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

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

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

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

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

24. Adaptive joint selection Acceleration simplification
= 96 -3
= 81
= 6 -6
We thus compute the joint acceleration of this node. We find -6.
Compute its joint acceleration

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

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

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

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

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

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 velocities to 0 Constraint:
Solve for
Compute by computing
Compute
Apply to the hybrid body
basis vector for

32. video

33. References
FEATHERSTONE, R 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):
S. Redon, N. Galoppo, and M. Lin. Adaptive dynamics of articulated bodies: ACM Trans. on Graphics (Proc. of ACM SIGGRAPH), 24(3), 2005.