Extremum-seeking nonlinear controllers for a human exercise machine

In this paper, a next generation exercise machine controller is developed for a single degree of freedom (DOF) system to maximize the user´s power output and ensure passivity with the user. In an effort to optimize the user´s power expenditure, a desired velocity trajectory is developed that seeks the unknown user-dependent optimal velocity setpoint. Two extremum-seeking algorithms are presented (e.g., Kristic and Deng, and Tuekosky et al.) that seek the optimal velocity setpoint while ensuring the trajectory is sufficiently differentiable. To track the reference trajectory and to ensure passivity, two controllers are developed. The first controller is developed based on the assumption that the user´s torque input can be measured. A second controller is designed that estimates the user´s torque input. Both controllers are proven to ensure that the exercise machine remains passive with respect to the user´s power output. The controllers are proven to yield semiglobal tracking through Lyapunov-based analyses. Proof-of-concept experimental results are provided that illustrate the performance of the torque estimation controller.