A Smooth Trajectory Generation Algorithm for Addressing Higher-Order Dynamic Constraints in Nanopositioning Systems

The generation of a time-optimal feedrate trajectory has received significant attention in CNC machining and robotics applications. Most of the existing feedrate planning algorithms take velocity and acceleration into the consideration for capability constraints. The introduction of higher order dynamic states, such as jerk and/or jounce into the feedrate scheduling problem makes generating computationally efficient solutions while simultaneously guaranteeing optimality a challenging problem, as the dimension of the planning problem is increased accordingly. This paper proposes a heuristic trajectory planning algorithm that can provide a near optimal trajectory for problems with higher order dynamic states. The algorithm starts with a non-optimal but feasible velocity trajectory, which is interpolated from a number of knot points by piece-wise spline interpolation with high order continuity. Then the trajectory is improved by scanning the interpolating knot points and increasing the velocity at each knot points while maintaining the feasibility of the resulting trajectory. A near optimal trajectory is achieved when the improvement in travel time is neglectable from the last scan iteration. The algorithm supports the incorporation of high order dynamic states (up to fifth order derivative of position) in constraints for optimization without sacrificing the computational efficiency. Examples including linear and curved toolpaths are presented to illustrate the effectiveness of this algorithm for high-speed contouring.