A Discrete Variational Integrator for Optimal Control Problems on SO(3)
Islam I. Hussein, Melvin Leok, Amit K. Sanyal, Anthony M. Bloch
备注：
IEEE Conference on Decision and Control, 2006 6 pages, 2 figures. v3: improved exposition, and shortened paper to 6 pages v2: revised to include numerical examples on SO(3)
In this paper we study a discrete variational optimal control problem for the rigid body. The cost to be minimized is the external torque applied to move the rigid body from an initial condition to a pre-specified terminal condition. Instead of discretizing the equations of motion, we use the discrete equations obtained from the discrete Lagrange--d'Alembert principle, a process that better approximates the equations of motion. Within the discrete-time setting, these two approaches are not equivalent in general. The kinematics are discretized using a natural Lie-algebraic formulation that guarantees that the flow remains on the Lie group SO(3) and its algebra so(3). We use Lagrange's method for constrained problems in the calculus of variations to derive the discrete-time necessary conditions. We give a numerical example for a three-dimensional rigid body maneuver.
A variational formulation of accelerated optimization on normed spaces was recently introduced by considering a specific family of time-dependent Bregman Lagrangian and Hamiltonian systems whose corresponding trajectories converge to the minimizer of the given convex function at an arbitrary accelerated rate of $\mathcal{O}(1/t^p)$. This framework has been exploited using time-adaptive geometric integrators to design efficient explicit algorithms for symplectic accelerated optimization. It was observed that geometric discretizations were substantially less prone to stability issues, and were t…
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 effec…
This paper addresses the problem of efficiently computing higher-order variational integrators in simulation and trajectory optimization of mechanical systems as those often found in robotic applications. We develop $O(n)$ algorithms to evaluate the discrete Euler-Lagrange (DEL) equations and compute the Newton direction for solving the DEL equations, which results in linear-time variational integrators of arbitrarily high order. To our knowledge, no linear-time higher-order variational or even implicit integrators have been developed before. Moreover, an $O(n^2)$ algorithm to linearize the DE…
Optimal control problems for underactuated mechanical systems can be seen as a higher-order variational problem subject to higher-order constraints (that is, when the Lagrangian function and the constraints depend on higher-order derivatives such as the acceleration, jerk or jounces). In this paper we discuss the variational formalism for the class of underactuated mechanical control systems when the configuration space is a trivial principal bundle and the construction of variational integrators for such mechanical control systems. An interesting family of geometric integrators can be defined…
Numerical methods that preserve geometric invariants of the system, such as energy, momentum or the symplectic form, are called geometric integrators. Variational integrators are an important class of geometric integrators. The general idea for those variational integrators is to discretize Hamilton's principle rather than the equations of motion in a way that preserves some of the invariants of the original system. In this paper we construct variational integrators with fixed time step for time-dependent Lagrangian systems modelling an important class of autonomous dissipative systems. These …
High order variational integrators in the optimal control of mechanical systems
Universidad de Valladolid（瓦拉多利德大学）；Instituto de Ciencias Matemáticas（数学科学研究院）；University of Paderborn（帕德博恩大学）；Sorbonne Universités, UPMC Univ. Paris 06（索邦大学，巴黎第六大学）；Institut Universitaire de France（法国高等研究院）
Geometric integrators for higher-order variational systems and their application to optimal control
University of Michigan（密歇根大学）；Universidad Nacional del Sur（国立苏尔大学）；Instituto de Matemática Bahía Blanca（巴亚布兰卡数学研究所）；CONICET（阿根廷国家科学研究委员会）；Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM)（数学科学研究所（西班牙高等科学研究理事会-马德里自治大学-马德里理工大学-马德里 Complutense 大学））
Southwest Research Institute（西南研究院）；Northwestern University（西北大学）
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 lineariz…
The solution of inverse problems in a variational setting finds best estimates of the model parameters by minimizing a cost function that penalizes the mismatch between model outputs and observations. The gradients required by the numerical optimization process are computed using adjoint models. Exponential integrators are a promising family of time discretizations for evolutionary partial differential equations. In order to allow the use of these discretizations in the context of inverse problems adjoints of exponential integrators are required. This work derives the discrete adjoint formulae…
In this paper we study, from a variational and geometrical point of view, second-order variational problems on Lie groupoids and the construction of variational integrators for optimal control problems. First, we develop variational techniques for second-order variational problems on Lie groupoids and their applications to the construction of variational integrators for optimal control problems of mechanical systems. Next, we show how Lagrangian submanifolds of a symplectic groupoid gives intrinsically the discrete dynamics for second-order systems, both unconstrained and constrained, and we s…
We consider the problem of verifying safety for a pair of identical integrator agents in continuous time with compact set-valued input uncertainties. We encode this verification problem as that of certifying or falsifying the intersection of their reach sets. We transcribe the same into a variational problem, namely that of minimizing the support function of the difference of the two reach sets over the unit sphere. We illustrate the computational tractability of the proposed formulation by developing two cases in detail, viz. when the inputs have time-varying norm-bounded and generic hyperrec…
