Trajectory Planning for Kinematically Redundant Systems via a B-Spline Based Optimis0061tion Approach

The combination of highly dynamic systems with a limited work envelope, with a less dynamic system which has a larger working envelope, promises to combine the advantages of both systems while eliminating the disadvantages. Due to the created kinematic redundancy it is not trivial to generate trajectories that make use of both systems, while considering limits and allowing for parallel motion. There are several approaches to creating trajectories for such systems. One of which is to formulate an optimisation problem that directly considers the stroke and dynamic limits of each axis to plan trajectories. However, the dimension of the resulting optimisation problems can quickly make this approach infeasible, as they result in long computation times. The objective of this contribution is to develop an optimisation based approach for trajectory planning of redundant systems, which scales well with the problem size. Furthermore, the approach aims at avoiding tunable parameters which can not be physically linked to the machine or the process. The primary method of achieving that objective is the incorporation of b-spline functions into the optimisation problem. These represent the position, velocity, acceleration and jerk of each axis over time. By only having to solve the optimisation problem for the b-spline coefficients, the dimension of the problem is reduced without sacrificing temporal resolution. Constraints are formulated to ensure that axis bounds are respected and an objective function is designed to minimise the jerk of the base axes. In the presented non-redundant test case, the optimisation approach generated valid trajectories comparable to those generated by the TwinCAT CNC kernel, confirming the plausibility of the approach. For the redundant system, the algorithm effectively resolved the kinematic redundancy by generating synchronised trajectories that respected the dynamic constraints of both systems.