Essential Properties of Numerical Integration for Time-optimal Trajectory Planning Along a Specified Path
This work provides incremental theoretical foundations for trajectory planning in robotics, benefiting researchers and practitioners in the field.
The paper presents rigorous mathematical proofs for known and new properties of the Numerical Integration method for time-optimal trajectory planning, including failure conditions when kinematic constraints are considered, and verifies these with simulations on a unicycle vehicle.
This letter summarizes some known properties and also presents several new properties of the Numerical Integration (NI) method for time-optimal trajectory planning along a specified path. The contribution is that rigorous mathematical proofs of these properties are presented, most of which cannot be found in existing literatures. We first give some properties regarding switch points and accelerating/decelerating curves of the NI method. Then, for the fact that when kinematic constraints are considered, the original version of NI which only considers torque constraints may result in failure of trajectory planning, we give the concrete failure conditions with rigorous mathematical proof. Accordingly, a failure detection algorithm is given in a run-and-test manner. Some simulation results on a unicycle vehicle are provided to verify those presented properties. Note that though those known properties are not discovered first, their mathematical proofs are given first in this letter. The detailed proofs make the theory of NI more complete and help interested readers to gain a thorough understanding of the method.