  This paper considers the optimal control problem of an extended spring-loaded
inverted pendulum (SLIP) model with two additional actuators for active leg
length and hip torque modulation. These additional features arise naturally in
practice, allowing for consideration of swing leg kinematics during flight and
active control over stance dynamics. On the other hand, nonlinearity and the
hybrid nature of the overall SLIP dynamics introduce challenges in the analysis
and control of the model. In this paper, we first show that the stance dynamics
of the considered SLIP model are differentially flat, which has a strong
implication regarding controllability of the stance dynamics. Leveraging this
powerful property, a tractable optimal control strategy is developed. This
strategy enables online solution while also treating the hybrid nature of the
SLIP dynamics. Together with the optimal control strategy, the extended SLIP
model grants active disturbance rejection capability at any point during the
gait. Performance of the proposed control strategy is demonstrated via
numerical tests and shows significant advantage over existing methods.
