  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.
