Structured linearization of discrete mechanical systems for analysis and optimal control

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 Linear Quadratic Regulator (LQR) problem to find a locally stabilizing controller for a 40 DOF system with six constraints. Note to Practitioners - The practical value of this work is the explicit derivation of recursive formulas for exact expressions for the first- and second-order linearizations of an arbitrary constrained mechanical system without requiring symbolic calculations. This is most applicable to the design of computer-aided design (CAD) software, where providing linearization information and sensitivity analysis facilitates mechanism analysis (e.g., controllability, observability) as well as control design (e.g., design of locally stabilizing feedback laws).