Adaptive dynamics for Articulated Bodies

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

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

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

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

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.

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

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

Main Pass For internal nodes For leaf nodes dependent on motion subspace dependent on active forces

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

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

Hybrid body

Hybrid body

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

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

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

Computing In active region compute using:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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.