Minor Change, Major Gains: The Effect of Orientation Formulation on Solving Time for Multi-body Trajectory Optimization

Many different coordinate formulations have been established to describe the position of multi-body robot models, but the impact of this choice on the tractability of trajectory optimization problems has yet to be investigated. Relative formulations, which reference the position of each link to its predecessor, reduce the number of variables and constraints in the problem, but lead to cumbersome expressions for the equations of motion. By contrast, referencing the positions to an absolute frame simplifies these equations, but necessitates more coordinate variables and connection constraints. In this letter, we investigate whether changing the orientation coordinates of a multi-body system model from relative to absolute angles can reduce the time required to solve the problem. The two approaches are tested on a variety of two- and three-dimensional models, with and without unscheduled unilateral contacts. Across all cases, the absolute formulation was found to be the more successful option. The performance improvements increased with the complexity of the system and task, culminating in the challenging example of a 60-degree turn on a 3D quadruped model, which was only able to converge in the allotted time when absolute angles were used.