Inverse dynamics of a flexible 3D robotic arm for a trajectory tracking task

In the aim of improving safety and efficiency of robotic manipulators, links with reduced bulkiness and weight can be integrated to the robot. As a result, the manipulator becomes more flexible and might encounter vibration issues that have to be well controlled. Such flexible manipulators are said to be underactuated since they potentially have an infinite number of degrees of freedom (dof) and a finite number of actuators. Flexibility can be dealt with by acting on the control system of the manipulator. Feedback action can be implemented to compensate for vibrations see, e.g., [1]. A second possibility is to model such flexible multibody system (MBS) in order to compute an input feedforward control signal that results in a vibration-free motion of the robot. Both the feedforward and the feedback control methods can be combined to achieve robust performances as presented in [2, 3]. To perform an end-effector trajectory tracking task, examples of feedforward commands for the manipulator would be the torques or the angular position of each of its joints. To find those inputs, the inverse dynamics of the MBS needs to be solved. In the case of a flexible system, some internal dynamics remains when the output trajectory is prescribed. The system is said to be non-minimum phase when this internal dynamics is unstable. If the inverse dynamics of a non-minimum phase system is simply solved using time integration algorithms, the resulting input control can be unbounded. In order to obtain a bounded solution, a non-causal solution must be considered. A time domain inverse dynamics method is presented and tested for a linear system in [4]. For flexible nonlinear systems, a stable inversion method is presented in [5] and is applied in [6, 7]. An optimal control approach is proposed in [8] for 2D multibody systems. The present work extends this last method to solve the inverse dynamics of flexible 3D systems. The flexible MBS is modeled using nonlinear beam finite elements [9], rigid bodies and kinematic joints [10] formulated on a Lie group. The inverse dynamics is then stated as an optimal control problem where the amplitude of the internal dynamics has to be minimized. The prescribed end-effector trajectory is defined as an additional servo constraint of the optimization problem.