Analysis of open-loop control design and parallel computation for underactuated manipulators

Trajectory tracking problems for underactuated manipulators represent an actual research topic, which is sustained by the interest in a lighter and faster automatic class of mechanical systems, bringing some benefits as lower energy consumption, higher productivity and a more convenient human–machine interaction. This work presents a feedforward design method, which is based on the inverse dynamics of underactuated manipulators. To achieve this aim, an optimal control problem is formulated based on a direct multiple shooting method and the mechanical model is formulated according to the nonlinear finite element approach. Based on the proposed formulation, a bounded solution of the inverse dynamics problem can be achieved. The main contributions of this work are: (1) the development of an alternative formulation for the trajectory tracking problem of underactuated manipulators; (2) a parallel computation method for the inverse dynamics of underactuated manipulators; (3) a comparison/validation with the direct transcription method and the stable inversion method. A planar underactuated manipulator with one passive joint is considered as an illustrative example, but the methodology is general and could be applied to more complex systems, e.g. with flexible bodies, with parallel kinematics and in 3D.