On the Structure of the Time-Optimal Path Parameterization Problem with Third-Order Constraints
This work addresses a critical open problem in robotics for motion planning, though it is incremental as it builds on prior second-order methods.
The paper tackles the time-optimal path parameterization problem with third-order constraints, which is largely unsolved in robotics, by proposing the TOPP3 algorithm to address difficulties in connecting optimal profiles and handling singularities.
Finding the Time-Optimal Parameterization of a Path (TOPP) subject to second-order constraints (e.g. acceleration, torque, contact stability, etc.) is an important and well-studied problem in robotics. In comparison, TOPP subject to third-order constraints (e.g. jerk, torque rate, etc.) has received far less attention and remains largely open. In this paper, we investigate the structure of the TOPP problem with third-order constraints. In particular, we identify two major difficulties: (i) how to smoothly connect optimal profiles, and (ii) how to address singularities, which stop profile integration prematurely. We propose a new algorithm, TOPP3, which addresses these two difficulties and thereby constitutes an important milestone towards an efficient computational solution to TOPP with third-order constraints.