Structured Linearization of Discrete Mechanical Systems for Analysis and Optimal Control

TL;DR

This paper develops a method to linearize variational integrators for mechanical systems, enabling control analysis and optimal control design while preserving key mechanical properties over long simulations.

Contribution

It introduces the first- and second-order linearizations of variational integrators, facilitating their use in control and analysis of complex mechanical systems.

Findings

Linearizations enable control design for variational integrators.

Application to a 40 DOF system demonstrates practical effectiveness.

Method preserves mechanical quantities like momentum over long-term simulations.

Abstract

Variational integrators are well-suited for simulation of mechanical systems because they preserve mechanical quantities about a system such as momentum, or its change if external forcing is involved, and holonomic constraints. While they are not energy-preserving they do exhibit long-time stable energy behavior. However, variational integrators often simulate mechanical system dynamics by solving an implicit difference equation at each time step, one that is moreover expressed purely in terms of configurations at different time steps. This paper formulates the first- and second-order linearizations of a variational integrator in a manner that is amenable to control analysis and synthesis, creating a bridge between existing analysis and optimal control tools for discrete dynamic systems and variational integrators for mechanical systems in generalized coordinates with forcing and holonomic constraints. The forced pendulum is used to illustrate the technique. A second example solves the discrete LQR problem to find a locally stabilizing controller for a 40 DOF system with 6 constraints.