Design and Comparative Analysis of 1D Hopping Robots

Abstract

Hopping is a highly dynamic motion requiring precise input over brief moments of ground contact in order to achieve desired performance. While this problem has been approached from multiple perspectives, this work provides a comparative analysis of two robot models. The first model uses an actuator to store energy in a spring and release it during the ground phase, while the second uses an actuator to move an additional mass vertically to generate force on the spring. In the first model, analytic expressions are used to find the desired controllers, while trajectory optimization is used in the latter. Orbital stability of each model under the conditions of uncertain damping and poor estimation of the hop height is examined. To this end, Poincaré analysis is used to give a metric of stability in the presence of different initial conditions and parameter uncertainty. Simulations show that the first model converges quickly to a point near the desired height determined by the amount of uncertain damping present. The second model is less robust to uncertainty, but is be made to converge to a desired height with the addition of PD control around the optimal trajectory. This robustness is improved with different gains in the controller. In experiments performed on hardware for the second model, stability is observed through convergence to a periodic orbit within several hops.