Numerical methods for the optimal control of constrained mechanical systems

最优控制问题在多体动力学领域有许多有趣的应用，从轻量级机器人的能量有效控制到人体运动的生物力学。只要能找到描述多体系统的最小坐标，运动方程就可以采用常微分方程的形式，在最优控制问题中，常微分方程起着状态方程的作用。在这种情况下，可以应用成熟的数值方法来解决最优控制问题。然而，最小坐标的使用仅限于具有树结构的相对简单的多体系统。多体动力学的一般方法依赖于冗余坐标的使用。相应地，最优控制问题的状态方程现在假定为微分代数方程的形式。对于这类问题仍然有很大的研究需要，因为常见的方法并不能保证正确的必要最优性条件得到解决。本提案的目标是开发解决现有方法的这一缺点的新方法。

Optimal control problems have many interesting applications in the field of multibody dynamics ranging from the energy efficient control of lightweight robots to the biomechanics of human motion. Provided that minimal coordinates can be found for the description of the multibody system, the equations of motion assume the form of ordinary differential equations which play the role of state equations in the optimal control problem. For this case well-established numerical methods can be applied to solve the optimal control problem. However, the use of minimal coordinates is confined to comparatively simple multibody systems with tree structure. The general approach to multibody dynamics relies on the use of redundant coordinates. Correspondingly, the state equations of the optimal control problem now assume the form of differential-algebraic equations. For this class of problems there is still great need for research, since the common approach does not guarantee that the correct necessary optimality conditions are solved. The goal of the present proposal is the development of new methods which resolve this shortcoming of existing methods.