A convex relaxation for the time-optimal trajectory planning of robotic manipulators along predetermined geometric paths

In this paper, we deal with the problem of time‐optimal trajectory planning and feedforward controls for robotic manipulators along predetermined geometric paths. We propose a convex relaxation to generate time‐optimal trajectories and feedforward controls that are dynamically feasible with respect to the complete nonlinear dynamic model, considering both Coulomb friction and viscous friction. Even though the effects of viscous friction for time‐optimal motions become rather significant due to the required large speeds, in previous formulations, viscous friction was ignored. We present a strategic formulation that turns out non‐convex because of the consideration of viscous friction, which nonetheless leads naturally to a convex relaxation of the referred non‐convex problem. In order to numerically solve the proposed formulation, a discretization scheme is also developed. Importantly, for all the numerical instances presented in the paper, focusing on applying the algorithm results to a six‐axis industrial manipulator, the proposed convex relaxation solves exactly the original non‐convex problem. Through simulations and experimental studies on the resulting tracking errors, torque commands, and accelerometer readings for the six‐axis manipulator, we emphasize the importance of penalizing a measure of total jerk and of imposing acceleration constraints at the initial and final transitions of the trajectory