Control of a spring-mass hopper

This dissertation examines control of a spring-mass hopper, a mathematical model of a robotic pogo stick. The model is widely used in biomechanics and robotics to describe legged locomotion. We focus on determining the model's optimal performance and its limitations.; To address optimal performance we study deadbeat control strategies , strategies that achieve a desired motion exactly in a finite number of strides. We ask the question: what is the minimum number of strides required to stabilize hopping? We find that the answer depends upon the constraints we place on the gait. For example, if we only impose a direction on the hopper, a perturbation can be corrected exactly in only one stride. However, if we impose a line of motion on the hopper, perturbations take two strides to correct.; To address the model's limitations, we identify tasks that are impossible for the hopper. For example, we find that it is impossible to change lanes (move from one line of motion to a parallel one) in one stride. In addition we find that it is impossible to take a step to the side then resume motion in the original line without taking two correction steps. More generally, we determine which sequences of footprints are hard to follow by looking at the sensitivity of the hopper's footprints to the applied control. We present preliminary data to test whether human hoppers can perform the tasks that a spring-mass hopper cannot.; Though evaluating the deadbeat control law (the function relating control outputs to inputs) is very expensive, we show it can be approximated in real time by interpolating previously computed values. The number of precomputed values needed greatly diminishes if the speed and height is approximately the same at every apex. We study what is required to produce and stabilize hopping of this sort.; Finally we show that the optimal performance of a hopper improves if it has lateral control over the location of its foot (or center of pressure) during stance. Such control can be achieved if the hopper has a wide foot and an ankle joint.