Inverse dynamics of serial and parallel underactuated multibody systems using a DAE optimal control approach

The inverse dynamics analysis of underactuated multibody systems aims at determining the control inputs in order to track a prescribed trajectory. This paper studies the inverse dynamics of non-minimum phase underactuated multibody systems with serial and parallel planar topology, e.g. for end-effector control of flexible manipulators or manipulators with passive joints. Unlike for minimum phase systems, the inverse dynamics of non-minimum phase systems cannot be solved by adding trajectory constraints (servo-constraints) to the equations of motion and applying a forward time integration. Indeed, the inverse dynamics of a non-minimum phase system is known to be non-causal, which means that the control forces and torques should start before the beginning of the trajectory (pre-actuation phase) and continue after the end-point is reached (post-actuation phase). The existing stable inversion method proposed for general nonlinear non-minimum phase systems requires to derive explicitly the equations of the internal dynamics and to solve a boundary value problem. This paper proposes an alternative solution strategy which is based on an optimal control approach using a direct transcription method. The method is illustrated for the inverse dynamics of an underactuated serial manipulator with rigid links and four degrees-of-freedom and an underactuated parallel machine. An important advantage of the proposed approach is that it can be applied directly to the standard equations of motion of multibody systems either in ODE or in DAE form. Therefore, it is easier to implement this method in a general purpose simulation software.