Motion Simulation: Dealing with the Ill-Conditioned Equations of Motion
for Articulated Figures

Summary:

This paper considers the inherent mathematical problems with articulated,
joint based movement and discusses several ways to re-define the
constraints of such problems in order to avoid singularities (or at least
find the most reasonable solution near them).

Problem:

This paper attempts to solve, or at least avoid, the mathematical
singularity problems that can arise with complex, articulated movement.

Method:

The author first discusses the existing constraint formulations and tools
for articulated movement. He then reviews the use of singular value
analysis and pseudoinverses to solve the mathematical constraint systems
and the singularities they still suffer. He then presents the damped
least-squares formulation and explains how it can be used in both
kinematic and dynamic simulations.

Key Ideas:

Certain types of articulated movement can exhibit very large changes
overall when one part of the articulation is moved only a short distance.
This causes mathematical singularities which can make it very hard to
calculate interpolated movement in a smooth, human-believable way.
The use of singular value analysis and pseudoinverses can lessen the
singularity problems, but not completely remove them. The use of damped
least-squares can greatly improve the situation and make repeated
iterative solutions feasible.

Contributions:

This paper presents the damped least-squares formulation which
significantly improves the singularity problems of articulated figure
interpolation for both kinematic and dynamic simulations.

Questions:

I unfortunately didn't follow nearly as much of the math involved in this
paper as I would have liked. I am hoping that in the discussion of the
paper a clearer explanation of the use of singular value analysis and
psuedoinverses will be presented.