Minor Change, Major Gains II: Are Maximal Coordinates the Fastest Choice for Trajectory Optimization?

It has been shown that changing the coordinates describing a multi-body system to use absolute rather than relative angles produces a significant improvement in the tractability of trajectory optimization problems. This simplifies the equations of motion when modelling long kinematic chains. In this paper, we extend this idea by investigating whether a maximal coordinate system, which also describes the translational position of bodies using absolute coordinates, might lead to further performance improvements. We compare it to the relative translation, absolute orientation (RTAO) coordinate scheme using a batch of trajectory optimization trials selected with contact-implicit legged locomotion applications in mind. We find that maximal coordinates tend to shorten solving times for spatial problems, while the RTAO formulation still performs best in the case of planar motion.

I. Introduction

Trajectory optimization of whole-body models has exciting potential as a method of generating high-level motion plans for challenging locomotion tasks, but its usefulness is hampered by time-consuming and unreliable solving. Even without computationally-demanding additions, such as realistic muscle models [1] and contact-implicit formulations [2], [3], the equations of motion needed to model systems consisting of many interlinked rigid bodies are cumbersome. This is still a problem for those willing to tolerate a long wait for a solution, as it also potentially affects the quality of the result: if a model is difficult to solve, it is also difficult to find a good search direction or extract it from a poor local minimum.