The adjoint method for optimal control of multibody systems for free end time and final constraints

This article presents a novel adjoint approach for solving optimal control problems in multibody dynamics. The equations of motion are governed by a set of differential-algebraic equations leading to a classical two-point boundary value problem. One solution strategy for the underlying boundary value problem is offered by gradient-based methods. However, the numerical evaluation of a gradient with respect to a high number of time steps is a time-consuming task. The approach presented in this article employs the adjoint method to compute the gradient of a cost functional, which decouples the boundary conditions by integrating sequentially the equations of motion and the adjoint equations, by a forward and a backward time integration. In order to satisfy a set of final conditions the system of equations is extended by the influence equations to associate the change of the controls and the change of the final conditions. By solving all three systems of equations, the gradient of the cost functional can be computed efficiently. This proposed method impacts the free final time while approaching the prescribed final conditions. Moreover, the paper presents an iterative optimization approach finding the optimal solution and is evaluated in the application of a three-arm multibody robot.