Structured Linearization of Discrete Mechanical Systems for Analysis and Optimal Control

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 energypreserving 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 secondorder 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).

On the benefits of surrogate Lagrangians in optimal control and planning algorithms

“…As discussed in [13] and [15] a first-order linearization of the discrete system dynamics can be obtained from the derived implicit one-step mapping (7)- (8). The linearization is given in the following form…”

“…The remaining derivatives are found by explicitly differentiating equation (8). The same linearization procedure is applicable to constrained and stochastic systems [10], [13], [15].…”

Stochastic Variational Integrators for System Propagation and Linearization

“…(2015). The presented stochastic variational integrator and its first-order linearization are based on the deterministic variational integrator and its linearization reported in Murphey (2009) andJohnson et. al.…”

Variational integrator for constrained mechanical systems with pulsed disturbances and optimal feedback control

“…To describe their discrete dynamics, a constrained variational integrator [1] is used. Using a discrete version of the Lagrange-d'Alembert principle yields a forced constrained discrete Euler-Lagrange equation in a position-momentum form that depends on the current and future time steps [2]. The desired optimal trajectory (qopt, popt) and according control input uopt is determined solving the discrete mechanics and optimal control (DMOC) algorithm [3] based on the variational integrator.…”

“…The configuration q(t) is approximated by a linear polynomial and the velocity by finite differences. Applying a discrete variational principle δS d ({q k } N k=0 ) = 0, see [1], and an approximation of the virtual work F ± d (q k , q k+1 , u k ) with control sequence {u k } N −1 k=0 , the Lagrange-d'Alembert principle yields a discrete Euler-Lagrange equation in a "position-momentum form that only depends on the current and future time steps" [2]. Part of the virtual work is the discretisation of a pulsed disturbance force F z (t), see [4], acting at time node t ∈ [t k , t k+1 ) and being defined as F z (t) = F t δ (t − t ), outlined in Figure 1.…”