Optimal design of nonlinear springs in robot mechanism: simultaneous design of trajectory and spring force profiles

In this paper, we aim at minimizing the actuator torques of robots working on production lines by adding to the mechanism dynamic equilibrators based on nonlinear springs, that work in parallel with the joints. We propose a method to simultaneously optimize the trajectory of the robot and the force profiles of the nonlinear springs to minimize the actuator torques. First, we express the trajectory and force profiles of the springs as a Hermite interpolation of a finite number of nodes, and then we show that the cost function of the optimization problem is a quadratic function of the springs design parameters. We derive a closed-form solution of the optimal spring parameters as a function of the trajectory parameters. As a consequence, the initial optimization problem is reduced to a trajectory optimization problem, solved with a sequential quadratic programming algorithm. We explain how the cost function can be modified to tune the nonlinearity of the springs and impose constraints on the stiffness. We show an example of optimal design of a threedegrees-of-freedom serial manipulator. Finally, we show that the nonlinear springs calculated for this manipulator can be technically realized by a noncircular cable spool mechanism.