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The popular approach of assuming a control policy and then finding the largest region of attraction (ROA) (e.g., sum-of-squares optimization) may lead to conservative estimates of the ROA, especially for highly nonlinear systems. We present a sampling-based approach that starts by assuming a ROA and then fi nds the necessary control policy by performing trajectory optimization on sampled initial conditions. Our method works with black-box models, produces a relatively large ROA, and ensures exponential convergence of the initial conditions to the periodic motion. We demonstrate the approach on a model of hopping and include extensive verification and robustness checks. 

Legged robots with point or small feet are nearly impossible to control instantaneously but are controllable over the time scale of one or more steps, also known as step-to-step control. Previous approaches achieve step-to-step control using optimization by (1) using the exact model obtained by integrating the equations of motion, or (2) using a linear approximation of the step-to-step dynamics. The former provides a large region of stability at the expense of a high computational cost while the latter is computationally cheap but offers limited region of stability. Our method combines the advantages of both. First, we generate input/output data by simulating a single step. Second, the input/output data is curve fitted using a regression model to get a closed-form approximation of the step-to-step dynamics. We do this model identification offline. Next, we use the regression model for online optimal control. Here, using the spring-load inverted pendulum model of hopping, we show that both parametric (polynomial and neural network) and non-parametric (gaussian process regression) approximations can adequately model the step-to-step dynamics. We then show this approach can stabilize a wide range of initial conditions fast enough to enable real-time control. Our results suggest that closed-form approximation of the step-to-step dynamics provides a simple accurate model for fast optimal control of legged robots.