Planning smooth trajectories along parametric paths

In this paper we present a flexible method for representing and optimizing smooth trajectories along specified paths for robotic applications. The main idea of this approach consists in defining the geometric path and the kinematics of the trajectory by independent parametric functions. The geometric path in the joint space can be any parametric curve q(u), the only restriction is that q must be twice differentiable with respect to the parameter u. The kinematics of the trajectory is defined by modulating the parameter u with respect to the time by a C2 piecewise cubic spline function. The trajectory is optimized using a non-linear programming approach capable of accommodating various constraints and different optimization criteria.