Unified phase variables of relative degree two for human locomotion

A starting point to achieve stable locomotion is synchronizing the leg joint kinematics during the gait cycle. Some biped robots parameterize a nonlinear controller (e.g., input-output feedback linearization) whose main objective is to track specific kinematic trajectories as a function of a single mechanical variable (i.e., a phase variable) in order to allow the robot to walk. A phase variable capable of parameterizing the entire gait cycle, the hip phase angle, has been used to control wearable robots and was recently shown to provide a robust representation of the phase of human gait. However, this unified phase variable relies on hip velocity, which is difficult to measure in real-time and prevents the use of derivative corrections in phase-based controllers for wearable robots. One derivative of this phase variable yields accelerations (i.e., the equations of motion), so the system is said to be relative degree-one. This means that there are states of the system that cannot be controlled. The goal of this paper is to offer relative degree-two alternatives to the hip phase angle and examine their robustness for parameterizing human gait.