Energy-Optimal Hopping in Parallel and Series Elastic One-Dimensional Monopeds

In this paper, we examine the question of whether parallel elastic actuation or series elastic actuation is better suited for hopping robots. To this end, we compare and contrast the two actuation concepts in energy optimal hopping motions. To enable a fair comparison, we employ optimal control to identify motion trajectories, actuator inputs, and system parameters that are optimally suited for each actuator concept. In other words, we compare the best possible hopper with parallel elastic actuation to the best possible hopper with series elastic actuation. The optimizations are conducted for three different cost functions: positive mechanical motor work, thermal electrical losses, and positive electrical work. Furthermore, we look at three representative cases for converting rotary motor motion to linear leg motion in a legged robot. Our model featured an electric DC-motor model, a gearbox with friction, damping in the leg spring, and contact collisions. We find that the optimal actuator choice depends both on the cost function and on the conversion of motor motion to leg motion. When considering only thermal electrical losses, parallel elastic actuation always performs better. In terms of positive mechanical motor work and positive electrical work, series elastic actuation is better when there is little friction in the gear-train. For higher gear-train friction parallel elastic actuation is more economical for these cost functions as well.