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Discrete mechanics and optimal control for constrained systems

机译：约束系统的离散力学和最优控制

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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.
机译：可以通过应用Lagrange-d'Alembert原理的约束形式，根据状态和控制来表达受完整力学约束的受控机械系统的运动方程。本文使用关键原理的离散类似物作为关键要素之一，得出了一种用于此类系统的最佳控制的结构保留方案。通过离散零空间方法将系统缩小到最小尺寸时，将继承此属性。连同构型和共轭动量的初始条件和最终条件，简化后的离散方程充当非线性等式约束，以最小化给定的目标函数。该算法产生一系列离散的配置以及一系列致动力，以最佳方式将系统从初始状态引导到所需的最终状态。特别是，为了最佳控制多体系统，引入了与关节约束一致的力公式。这使人们能够证明动量图演化的一致性。使用双链接摆，将该方法与现有方法进行比较。此外，它被应用于卫星定向操纵和生物运动问题。

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