Observer Design for a Nonlinear Neuromuscular System with Multi-rate Sampled and Delayed Output Measurements

Robotic devices and functional electrical stimulation (FES) are utilized to provide rehabilitation therapy to persons with incomplete spinal cord injury. The goal of the therapy is to improve their weakened voluntary muscle strength. A variety of control strategies used in these therapies need a measure of participant's volitional strength. This informs the robotic or an FES device to modulate assistance proportional to the user's weakness. In this paper we propose an observer design to estimate ankle kinematics that are elicited volitionally. The observer uses a nonlinear continuous-time neuromuscular system, which has multi-rate sampled output measurements with non-uniform and unknown delays from various sensing modalities including electromyography, ultrasound imaging, and an inertial measurement unit. We assume an allowable maximum value of unsynchronized sampling intervals and nonuniform delays. By constructing a Lyapunov-Krasovskii function, sufficient conditions are derived to prove the exponential stability of the estimation error. Numerical simulations are provided to verify the effectiveness of the designed observer.