  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.
