Real-time estimate of period derivatives using adaptive oscillators: Application to impedance-based walking assistance

Inferring temporal derivatives (like velocity and acceleration) from a noisy position signal is a well-known challenge in control engineering, due to the intrinsic trade-off between noise filtering and estimation bandwidth. To tackle this problem, in this paper we propose a new approach specifically designed for periodic movements. This approach uses an adaptive oscillator as fundamental building block. It is a tool capable of synchronizing to a periodic input while learning its features (frequency, amplitude, ...) in dedicated state variables. Since the oscillator's input and output are perfectly synchronized during steady-state regime, a non-delayed estimate of the input temporal derivatives can be obtained simply by deriving the output analytical form. Pending a (quasi-)periodic input signal, these temporal derivatives are thus synchronized with the actual kinematics, while the signal bandwidth can be arbitrarily tuned by the intrinsic dynamics of the oscillator. We further validate this approach by developing an impedance-based strategy for assisting human walking in the LOPES lower-limb exoskeleton. Preliminary results with a single participant give rise to three main conclusions. First, our method indeed provides velocity and acceleration estimates of the participant's joint kinematics which are smoother and less delayed with respect to the actual kinematics than using a standard Kalman filter. Second, closing the human-robot loop with a high-gain impedance field depending on the acceleration is not possible with a Kalman filter approach, due to unstable dynamics. In contrast, our approach tolerates high gains (up to 70% of the nominal walking torque), showing its intrinsic stability. Finally, no clear benefit of the acceleration-dependent field with respect to a simpler position-dependent field is visible regarding the reduction of metabolic cost. This last result illustrates the challenge of designing sound assistive strategies for complex tasks like walking.