Polynomial mechanics and optimal control

TL;DR

This paper introduces a novel trajectory optimization algorithm for mechanical systems that combines pseudo-spectral methods with variational discretization to ensure accurate, momentum-preserving solutions with spectral convergence.

Contribution

It presents a new algorithm that integrates pseudo-spectral methods with variational principles to preserve mechanical invariants during trajectory optimization.

Findings

The method achieves spectral convergence rates.

It preserves momentum and symplectic structure after discretization.

Outperforms existing methods in example systems.

Abstract

We describe a new algorithm for trajectory optimization of mechanical systems. Our method combines pseudo-spectral methods for function approximation with variational discretization schemes that exactly preserve conserved mechanical quantities such as momentum. We thus obtain a global discretization of the Lagrange-d'Alembert variational principle using pseudo-spectral methods. Our proposed scheme inherits the numerical convergence characteristics of spectral methods, yet preserves momentum-conservation and symplecticity after discretization. We compare this algorithm against two other established methods for two examples of underactuated mechanical systems; minimum-effort swing-up of a two-link and a three-link acrobot.