  Continuous formulations of trajectory planning problems have two main
benefits. First, constraints are guaranteed to be satisfied at all times.
Secondly, dynamic obstacles can be naturally considered with time. This paper
introduces a novel B-spline based trajectory optimization method for
multi-jointed robots that provides a continuous trajectory with guaranteed
continuous constraints satisfaction. At the core of this method, B-spline basic
operations, like addition, multiplication, and derivative, are rigorously
defined and applied for problem formulation. B-spline unique characteristics,
such as the convex hull and smooth curves properties, are utilized to
reformulate the original continuous optimization problem into a
finite-dimensional problem. Collision avoidance with static obstacles is
achieved using the signed distance field, while that with dynamic obstacles is
accomplished via constructing time-varying separating hyperplanes. Simulation
results on various robots validate the effectiveness of the algorithm. In
addition, this paper provides experimental validations with a 6-link FANUC
robot avoiding static and moving obstacles.
