Orthotic body-weight support through underactuated potential energy shaping with contact constraints

Abstract

Body-weight support is an effective clinical tool for gait rehabilitation after neurological impairment. Body-weight supported training systems have been developed to help patients regain mobility and confidence during walking, but conventional systems constrain the patient's treatment in clinical environments. We propose that this challenge could be addressed by virtually providing patients with body-weight support through the actuators of a powered orthosis (or exoskeleton) utilizing potential energy shaping control. However, the changing contact conditions and degrees of underactuation encountered during human walking present significant challenges to consistently matching a desired potential energy for the human in closed loop. We therefore introduce a generalized matching condition for shaping Lagrangian systems with holonomic contact constraints. By satisfying this matching condition for four phases of gait, we derive control laws to achieve virtual body-weight support through a powered knee-ankle orthosis. We demonstrate beneficial effects of virtual body-weight support in simulations of a human-like biped model, indicating the potential clinical value of this proposed control approach.

Experimental implementation of underactuated potential energy shaping on a powered ankle-foot orthosis

“…In this section we will express the equations of motion as equivalent constrained dynamics in order to derive an underactuated control law that achieves the desired potential energy for a given contact condition [12], [13]. For the sake of generality we drop the subscripts associated with specific contact conditions.…”

“…Based on the results in [12] and [13], the desired dynamics (9) can be achieved in closed loop if the following matching condition is satisfied: $$B_{\lambda }^{\perp }false(N_{\lambda }-{\stackrel{\sim }{N}}_{\lambda }false)=0.$$…”

“…We will simplify the multiplication between A heel and $M_{\text{st}}^{-1}$ with blockwise inversion as in [12]. Define the top-left, top-right, and bottom-right submatrices of M st as M 1 ∈ ℝ 2×2 , M 2 ∈ ℝ 2×3 , and M 4 ∈ ℝ 3×3 , respectively.…”

“…Define the top-left, top-right, and bottom-right submatrices of M st as M 1 ∈ ℝ 2×2 , M 2 ∈ ℝ 2×3 , and M 4 ∈ ℝ 3×3 , respectively. Following the derivation in [12], [13]: $$\begin{array}{ccc}left2ptleft{B}_{\lambda 1}& left=\left[\begin{array}{c}{V}_1{B}_{\text{st}(3,5)}\\ B_{\text{st}false(3,5false)}\end{array}\right]& left=\left[\begin{array}{c}{V}_{12}P\\ 0\\ P\end{array}\right],\\ left{N}_{\lambda 1}& left=\left[\begin{array}{c}{V}_1{N}_{\text{st}(3,5)}\\ N_{\text{st}false(3,5false)}\end{array}\right]& left=\left[\begin{array}{c}{V}_{11}{N}_{\text{st}(3,3)}+V_{12}{N}_{\text{st}(4,5)}\\ N_{\text{st}false(3,3false)}\\ N_{\text{st}false(4,5false)}\end{array}\right],\end{array}$$ where $V_1=false[V_{11},V_{12}false]=M_2{M}_4^{-1}$, V 11 ∈ ℝ 2×1 , V 12 ∈ ℝ 2×2 , and P = [1, 0] T . The subscript ( k , z ) indicates rows k through z of a matrix.…”

“…However, the changing contact conditions and degrees of underactuation encountered during human walking present significant challenges to consistently matching a desired potential energy for the human in closed loop. Therefore, we are investigating contact-invariant ways of matching desired dynamics to enable exoskeletal BWS, and we have demonstrated beneficial effects of this control methodology in simulations of a powered knee-ankle orthosis on a human-like biped model [12], [13]. This feedback control strategy is fundamentally task-invariant, and its parameterization allows systematic adjustments for patient-specific therapy.…”