Scalable Variational Integrators for Constrained Mechanical Systems in Generalized Coordinates

Johnson, Elliot R.; Murphey, Todd D.

doi:10.1109/tro.2009.2032955

Cited by 65 publications

(65 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Rather than using a discrete approximation of the Euler-Lagrange equation, variational integrators are created by approximating the Lagrange-d'Alembert principle [12], [20], [21]. The resulting approximation gives an implicit two-step mapping from two consecutive discrete configurations to the next (q k−1 , q k ) → (q k+1 ).…”

Section: A Overview Of Variational Integratorsmentioning

confidence: 99%

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

Torre

Murphey

2016

2016 IEEE 55th Conference on Decision and Control (CDC)

View full text Add to dashboard Cite

Abstract-This paper explores the relationship between numerical integrators and optimal control algorithms. Specifically, the performance of the differential dynamical programming (DDP) algorithm is examined when a variational integrator and a newly proposed surrogate variational integrator are used to propagate and linearize system dynamics. Surrogate variational integrators, derived from backward error analysis, achieve higher levels of accuracy while maintaining the same integration complexity as nominal variational integrators. The increase in the integration accuracy is shown to have a large effect on the performance of the DDP algorithm. In particular, significantly more optimized inputs are computed when the surrogate variational integrator is utilized.

show abstract

Section: A Overview Of Variational Integratorsmentioning

confidence: 99%

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

Torre

Murphey

2016

2016 IEEE 55th Conference on Decision and Control (CDC)

View full text Add to dashboard Cite

show abstract

“…and, in the unforced case, p k is the generalized momentum quantity conserved by the integrator as discussed in Johnson & Murphey (2009). Furthermore, the derivative of (1.6) with respect to…”

Section: Stochastic Variational Integratorsmentioning

confidence: 99%

“…) and is solved using Algorithm 1 as shown in Johnson & Murphey (2009). The selection of tol not only determines the accuracy of the variational integrator, but also the computational time in implementation.…”

Section: Stochastic Variational Integratorsmentioning

confidence: 99%

“…(2010), and Bou-Rabee & Owhadi (2008) studies their long-run accuracy. Scalable variational integrators were implemented to generic mechanical systems in generalized coordinates by Johnson & Murphey (2009). First-and second-order linearizations of a deterministic variational integrator was formulated by Johnson et.…”

Section: Stochastic Variational Integrators and Structured Linearizationmentioning

confidence: 99%

“…(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.…”

Section: Stochastic Variational Integrators and Structured Linearizationmentioning

confidence: 99%

See 2 more Smart Citations

Stochastic Variational Integrators for System Propagation and Linearization

Torre¹,

Theodorou²

2015

Proceedings of the IMA Conference on Mathematics of Robotics

View full text Add to dashboard Cite

In this paper we present a stochastic variational integrator and its linearization. In order to motivate the use of the proposed variational integrator the Stochastic Differential Dynamical Programming (S-DDP) algorithm is considered as a benchmark for comparison. Specifically, we are interested in investigating if it is advantageous to utilize the variational integrator to propagate system trajectories and linearize system dynamics. Through numerical experiments we show that the Stochastic Differential Dynamical Programming algorithm becomes less dependent on the discretization time step and more predictable when it utilizes the proposed integrator. Furthermore, we show that a significant reduction in computational time can be achieved without sacrificing algorithm performance. Therefore, the proposed variational integrator can be used to enable real-time implementation of nonlinear optimal control algorithms. Stochastic Variational Integrators and Structured LinearizationThis section presents a stochastic variational integrator used to obtain accurate discretized trajectories of stochastic dynamical Hamiltonian systems represented as d ∂L(q,q) ∂q = ∂L(q,q) ∂q dt + F 0 (q,q, u)dt + where L(q,q) is the mechanical system's Lagrangian given as the difference between the system's kinetic energy, T (q,q), and potential energy, V (q), such that L(q,q) = T (q,q) − V (q), q is the state configuration vector,q is the time derivative of the state configuration vector, dω i , i = 1, . . . , m are Brownian noises, F i (q, u), i = 1, . . . , m are the diffusion vector fields, F 0 (q,q, u) is the forcing function representing deterministic non-conservative external forces, and indicates the stochastic integral is evaluated in the Stratonovich sense Kloeden & Platen (1992). Note equation (1.1) provides the fundamental characteristics of a dynamical system and describes how the system configuration vector propagates through time. However, it does not provide any method or algorithm that can be used to solve for the trajectory of the system. Simply using numerical integration schemes developed for general second order stochastic differential equations will result in numerical errors since the system's fundamental characteristics are ignored. A variational integrator computes a discretized trajectory q = {q 0 , . . . , q N } that approximates the systems trajectory q(t) such that q k ≈ q(t k ), where t 0 = 0, t N = t f and t k+1 − t k = ∆t. The derivation of variational integrators relies on the discretization of the classical Hamilton's variational principle and, in forced systems, the Lagrange-d'Alembert principle. As a result, variational integrators are able to ensure (or strongly enforce) the conservation of fundamental quantities such as momentum and energy Marsden & West (2001). Furthermore, since the derivation IMA Conference on Mathematics of Robotics 9 -11 September 2015, St Anne's College, University of Oxford Stochastic Variational IntegratorsTo being our review, the discrete Lagrangian, L d (q k , q k+1 ),...

show abstract

Low‐Infrastructure Real‐Time Embedded Control via Variational Integrators

Schultz

Murphey

2016

Proc Appl Math and Mech

View full text Add to dashboard Cite

This paper presents a discrete time receding horizon control scheme that leverages the numerical properties of a variational integrator to facilitate real‐time control generation on an embedded system. The variational integrator employed is well‐suited to classical estimation and control algorithms, e.g. LQR, extended Kalman filters, and particle filters. The structure‐preserving properties of this variational integrator lead to increased performance of estimation and control routines, especially in low‐bandwidth applications. Several experimental examples are presented that illustrate the features of this receding horizon control scheme when leveraging the desirable numerical properties of the variational integrator. © 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

show abstract

Scalable Variational Integrators for Constrained Mechanical Systems in Generalized Coordinates

Cited by 65 publications

References 16 publications

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

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

Stochastic Variational Integrators for System Propagation and Linearization

Low‐Infrastructure Real‐Time Embedded Control via Variational Integrators

Contact Info

Product

Resources

About