Efficient Online Trajectory Planning for Integrator Chain Dynamics Using Polynomial Elimination

Providing smooth reference trajectories can effectively increase performance and accuracy of tracking control applications while overshoot and unwanted vibrations are reduced. Trajectory planning computations can often be simplified significantly by transforming the system dynamics into decoupled integrator chains using methods such as feedback linearization, differential flatness or the Brunovsky normal form. We present an efficient method to plan trajectories for integrator chains subject to derivative bound constraints. Therefore, an algebraic precomputation algorithm formulates the necessary conditions for time optimality in form of a set of polynomial systems, followed by a symbolic polynomial elimination using Gröbner bases. A fast online algorithm then plans the trajectories by calculating the roots of the decomposed polynomial systems. These roots describe the switching time instants of the input signal and the full trajectory simply follows by multiple integration. This method presents a systematic way to compute fast, for low system orders even time optimal trajectories exactly via algebraic calculations without numerical approximation iterations. It is applied to various trajectory types with different continuity order, asymmetric derivative bounds and non-rest initial and final states.