Derivation and Application of a Conserved Orbital Energy for the Inverted Pendulum Bipedal Walking Model

We present an analysis of a point mass, point foot, planar inverted pendulum model for bipedal walking. Using this model, we derive expressions for a conserved quantity, the "orbital energy", given a smooth center of mass trajectory. Given a closed form center of mass trajectory, the equation for the orbital energy is a closed form expression except for an integral term, which we show to be the first moment of area under the center of mass path. Hence, given a center of mass trajectory, it is straightforward and computationally simple to compute phase portraits for the system. In fact, for many classes of trajectories, such as those in which height is a polynomial function of center of mass horizontal displacement, the orbital energy can be solved in closed form. Given expressions for the orbital energy, we can compute where the foot should be placed or how the center of mass trajectory should be modified in order to achieve a desired velocity on the next step. We demonstrate our results using a planar biped simulation with light legs and point mass body. We parameterize the center of mass trajectory with a fifth order polynomial function. We demonstrate how the parameters of this polynomial and step length can be changed in order to achieve a desired next step velocity.