Structured Linearization of Discrete Mechanical Systems for Analysis and Optimal Control
This work addresses the problem of integrating mechanical system simulation with control design for researchers in robotics and dynamics, though it is incremental as it builds on existing variational integrator methods.
The paper tackles the challenge of applying control analysis and synthesis tools to variational integrators by formulating their first- and second-order linearizations, enabling the solution of a discrete LQR problem for a 40 DOF system with 6 constraints.
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.