Unparameterized Optimization of the Spring Characteristic of Parallel Elastic Actuators

In electrically actuated robots most energy losses are due to the heating of the actuators. This energy loss can be greatly reduced with parallel elastic actuators, by optimizing the elastic element such that it delivers most of the required torques. Previously used optimization methods relied on parameterizing the spring characteristic, thereby limiting the set of spring characteristics optimized over and with that the loss reduction that can be obtained. This letter shows that such parametrization is not necessary; a method is presented to compute the optimal characteristic as an analytic function of the trajectory. The efficacy of this method is demonstrated using two examples. The first example considers the optimal spring characteristic for a parallel elastic actuator supporting the human ankle during walking. The second example applies the method in combination with trajectory optimization on a single degree of freedom robot performing a specific pick-and-place task. The task at hand has a height difference between the pick and the place location. With the analytical optimal spring, it is shown that the robot can recover enough of the energy released by the package to function without external electric energy supply.