Motorized FES-cycling and closed-loop nonlinear control for power tracking using a finite-time stable torque algorithm

Abstract

Functional electrical stimulation (FES)-induced cycling is a rehabilitation strategy that activates lower-limb muscles to achieve coordinated pedaling in individuals with movement disorders. An electric motor is included in-the-loop assisting the rider as needed to prolong exercise duration and mitigate muscle fatigue. Power tracking objectives have been prescribed for motorized FES-cycling, where muscles and the electric motor are assigned to track desired cadence (speed) and torque trajectories. However, predetermined desired trajectories can yield poor cycling performance since the functional capacity of each individual is unknown. In particular, when muscles are tasked to track a desired torque, a dynamic approach is well-motivated to adjust the torque demand for the rider in real-time (e.g., a constant torque demand may be unfeasible throughout a cycling session since muscles fatigue). In this paper, input-output data is exploited using a finite-time algorithm to estimate the target desired torque leveraging an estimate of the active torque produced by muscles via FES. The convergence rate of the finite-time algorithm can be adjusted by tuning selectable parameters. The cycle-rider system is modeled as a nonlinear, time-varying, state-dependent switched system to activate lower-limb muscles and an electric motor. To achieve cadence and torque tracking, nonlinear robust tracking controllers are designed for muscles and motor. A robust sliding mode controller is designed for the electric motor to track a desired constant cadence trajectory. Moreover, an integral torque feedback controller is designed to activate quadriceps, hamstrings, and gluteus muscle groups to track the desired torque trajectory computed by the finite-time algorithm. A Lyapunov-based stability analysis is developed to ensure exponential tracking of the closed-loop cadence error system and global uniformly ultimate bounded (GUUB) torque tracking. A discrete-time Lyapunov-based stability analysis leveraging a recent tool for finite-time systems is developed to ensure convergence and guarantee that the finite-time algorithm is Hölder continuous. The developed tracking controllers for the muscles and electric motor and finite-time algorithm to compute the desired torque are implemented in real-time during cycling experiments in seven able-bodied individuals. Multiple cycling trials are implemented with different gain parameters of the finite-time torque algorithm to compare tracking performance for all participants.