Enforced symmetry of the stance phase for the Spring-Loaded Inverted Pendulum

The Spring-Loaded Inverted Pendulum (SLIP) is considered the simplest model to effectively describe bouncing gaits (such as running and hopping) for many legged animals and robots. For this reason, it is has often been used as a model for robot design. A key challenge in using this model, however, is the lack of a closed-form solution for the equations of motion that define the stance phase of its dynamics. This results in the impossibility of analytically predicting its trajectory. Consequently, developing a practical control strategy to operate on the model is computationally intensive, because accurately predicting the step-to-step dynamics is still an unsolved problem. By adding an actuator in series with the spring, we can develop a control law for actuator displacement which enforces a desired trajectory during stance. In particular, for our specific chosen control law, we can compute an analytical solution for the stance phase trajectory. Furthermore, we give examples of higher level control strategies for foothold placement and for keeping the forward velocity or the apex height constant on rough terrain that employ our low-level control laws, and we illustrate through simulations the performance typical of our strategy.