Efficient Online Trajectory Planning for Integrator Chain Dynamics using Polynomial Elimination

TL;DR

This paper introduces an algebraic method using polynomial elimination to efficiently compute exact time optimal trajectories for integrator chain systems, enhancing control performance with reduced vibrations.

Contribution

It presents a novel algebraic precomputation and symbolic polynomial elimination approach for online trajectory planning in integrator chains, avoiding numerical iterations.

Findings

Exact time optimal trajectories computed efficiently.

Applicable to various trajectory types and boundary conditions.

Reduces overshoot and vibrations in control applications.

Abstract

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 controller canonical form. We present an efficient method to plan time optimal 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 Gr\"obner 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 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.