Discrete mechanics and optimal control for constrained systems
Discrete mechanics and optimal control for constrained systems
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DOI：
10.1002/oca.912
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发表时间：
2010-11
影响因子：
1.8
通讯作者：
S. Leyendecker;S. Ober-Blöbaum;J. Marsden;Magdalena Ortiz
中科院分区：
文献类型：
--
作者：
S. Leyendecker;S. Ober-Blöbaum;J. Marsden;Magdalena Ortiz
受完整约束的受控机械系统的运动方程可以通过应用拉格朗日-达朗贝尔原理的约束形式，用状态和控制来表示。本文导出了一个结构保持方案，用于此类系统的最优控制，作为关键成分之一，该原理的离散模拟。当系统通过离散零空间方法降低到其最小维数时，继承了该性质。连同初始和最终条件的配置和共轭动量，减少离散方程作为一个给定的目标功能的最小化的非线性等式约束。该算法产生一个序列的离散配置连同一个序列的驱动力，最佳地引导系统从初始到所需的最终状态。特别地，对于多体系统的最优控制，引入了与关节约束一致的力公式。这使得人们能够证明动量映射演化的一致性。使用一个双连杆摆，该方法与现有的方法进行了比较。并将其应用于卫星重定向机动和生物运动问题。版权所有© 2009约翰威利父子有限公司。
The equations of motion of a controlled mechanical system subject to holonomic constraints may be formulated in terms of the states and controls by applying a constrained version of the Lagrange‐d'Alembert principle. This paper derives a structure‐preserving scheme for the optimal control of such systems using, as one of the key ingredients, a discrete analogue of that principle. This property is inherited when the system is reduced to its minimal dimension by the discrete null space method. Together with initial and final conditions on the configuration and conjugate momentum, the reduced discrete equations serve as nonlinear equality constraints for the minimization of a given objective functional. The algorithm yields a sequence of discrete configurations together with a sequence of actuating forces, optimally guiding the system from the initial to the desired final state. In particular, for the optimal control of multibody systems, a force formulation consistent with the joint constraints is introduced. This enables one to prove the consistency of the evolution of momentum maps. Using a two‐link pendulum, the method is compared with existing methods. Further, it is applied to a satellite reorientation maneuver and a biomotion problem. Copyright © 2009 John Wiley & Sons, Ltd.