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This paper considers the optimal control problem of an extended spring-loadedinverted pendulum (SLIP) model with two additional actuators for active leglength and hip torque modulation. These additional features arise naturally inpractice, allowing for consideration of swing leg kinematics during flight andactive control over stance dynamics. On the other hand, nonlinearity and thehybrid nature of the overall SLIP dynamics introduce challenges in the analysisand control of the model. In this paper, we first show that the stance dynamicsof the considered SLIP model are differentially flat, which has a strongimplication regarding controllability of the stance dynamics. Leveraging thispowerful property, a tractable optimal control strategy is developed. Thisstrategy enables online solution while also treating the hybrid nature of theSLIP dynamics. Together with the optimal control strategy, the extended SLIPmodel grants active disturbance rejection capability at any point during thegait. Performance of the proposed control strategy is demonstrated vianumerical tests and shows significant advantage over existing methods.

Optimal Control of a Differentially Flat 2D Spring-Loaded Inverted Pendulum Model