Multi-directional Dynamic Mechanical Impedance of the Human Ankle; A Key to Anthropomorphism in Lower Extremity Assistive Robots

Abstract

The mechanical impedance of the human ankle plays a central role in lower-extremity functions requiring physical interaction with the environment. Recent efforts in the design of lower-extremity assistive robots have focused on the sagittal plane; however, the human ankle functions in both sagittal and frontal planes. While prior work has addressed ankle mechanical impedance in single degrees of freedom, here we report on a method to estimate multi-variable human ankle mechanical impedance and especially the coupling between degrees of freedom. A wearable therapeutic robot was employed to apply torque perturbations simultaneously in the sagittal and frontal planes and record the resulting ankle motions. Standard stochastic system identification procedures were adapted to compensate for the robot dynamics and derive a linear time-invariant estimate of mechanical impedance.

Applied to seated, young unimpaired human subjects, the method yielded coherences close to unity up to and beyond 50 Hz, indicating the validity of linear models, at least under the conditions of these experiments. Remarkably, the coupling between dorsi-flexion/plantar-flexion and inversion/eversion was negligible. This was observed despite strong biomechanical coupling between degrees of freedom due to musculo-skeletal kinematics and suggests compensation by the neuro-muscular system. The results suggest that the state-of-the-art in lower extremity assistive robots may advance by incorporating design features that mimic the multi-directional mechanical impedance of the ankle in both sagittal and frontal planes.

Appendix

Elements of the mechanical impedance matrix were found as described in [38].

$$ \mathbf{Z}=\frac{1}{1-{\gamma^2}_{\theta_{dp}{\theta}_{ie}}}\left[\begin{array}{ll}\hfill \frac{P_{\theta_{dp}{\tau}_{dp}}}{P_{\theta_{dp}{\theta}_{dp}}}\left(1-\frac{P_{\theta_{dp}{\theta}_{ie}}{P}_{\theta_{ie}{\tau}_{dp}}}{P_{\theta_{ie}{\theta}_{ie}}{P}_{\theta_{dp}{\tau}_{dp}}}\right)\hfill & \frac{P_{\theta_{ie}{\tau}_{dp}}}{P_{\theta_{ie}{\theta}_{ie}}}\left(1-\frac{P_{\theta_{ie}{\theta}_{dp}}{P}_{\theta_{dp}{\tau}_{dp}}}{P_{\theta_{dp}{\theta}_{dp}}{P}_{\theta_{ie}{\tau}_{dp}}}\right)\hfill \\[8pt] \frac{P_{\theta_{dp}{\tau}_{ie}}}{P_{\theta_{dp}{\theta}_{dp}}}\left(1-\frac{P_{\theta_{dp}{\theta}_{ie}}{P}_{\theta_{ie}{\tau}_{ie}}}{P_{\theta_{ie}{\theta}_{ie}}{P}_{\theta_{dp}{\tau}_{ie}}}\right)\hfill & \frac{P_{\theta_{ie}{\tau}_{ie}}}{P_{\theta_{ie}{\theta}_{ie}}}\left(1-\frac{P_{\theta_{ie}{\theta}_{dp}}{P}_{\theta_{dp}{\tau}_{ie}}}{P_{\theta_{dp}{\theta}_{dp}}{P}_{\theta_{ie}{\tau}_{ie}}}\right)\hfill \end{array}\right] \\[3pt] $$

(6.A.1)

where P _xy denotes the cross-power spectral density of two data sequences x and y (time sequences of input torques and output angles in the DP and IE directions) and γ ² _xy (f) is the ordinary coherence function between two data sequences x and y defined as

$$ {\gamma^2}_{xy}(f)=\frac{\left|{P}_{xy}(f)\right|{}^2}{P_{xx}(f){P}_{yy}(f)} $$

(6.A.2)

A coherence function indicates linear dependency between two time sequences. In the multivariable case, the appropriate measure is partial coherence which measures the linear dependency between each input and output after removing the effects of the other inputs. The partial coherence matrix is defined as

$$ \begin{aligned}[b]{\boldsymbol{\Omega}}^2&=\left[\begin{array}{ll}\hfill {\Omega^2}_{11}\hfill & \hfill {\Omega^2}_{12}\hfill \\ {\Omega^2}_{21}\hfill & \hfill {\Omega^2}_{22}\hfill \end{array}\right]\\ &=\left[\begin{array}{ll}\hfill \frac{\left|{P}_{\theta_{dp}{\tau}_{dp}}{P}_{\theta_{ie}{\theta}_{ie}}-{P}_{\theta_{ie}{\tau}_{dp}}{P}_{\theta_{dp}{\theta}_{ie}}\right|{}^2}{P_{\theta_{ie}{\theta}_{ie}}{P}_{\theta_{ie}{\theta}_{ie}}{P}_{\theta_{dp}{\theta}_{dp}}{P}_{\tau_{dp}{\tau}_{dp}}\left(1-{\gamma^2}_{\theta_{ie}{\theta}_{dp}}\right)\left(1-{\gamma^2}_{\theta_{ie}{\tau}_{dp}}\right)}\hfill & \hfill \hfill \\ \frac{\left|{P}_{\theta_{dp}{\tau}_{ie}}{P}_{\theta_{ie}{\theta}_{ie}}-{P}_{\theta_{ie}{\tau}_{ie}}{P}_{\theta_{dp}{\theta}_{ie}}\right|{}^2}{P_{\theta_{ie}{\theta}_{ie}}{P}_{\theta_{ie}{\theta}_{ie}}{P}_{\theta_{dp}{\theta}_{dp}}{P}_{\tau_{ie}{\tau}_{ie}}\left(1-{\gamma^2}_{\theta_{ie}{\theta}_{dp}}\right)\left(1-{\gamma^2}_{\theta_{ie}{\tau}_{ie}}\right)}\hfill & \hfill \hfill \end{array}\right.\\ & \left.\begin{array}{ll}\hfill \hfill & \hfill \frac{\left|{P}_{\theta_{ie}{\tau}_{dp}}{P}_{\theta_{dp}{\theta}_{dp}}-{P}_{\theta_{dp}{\tau}_{dp}}{P}_{\theta_{ie}{\theta}_{dp}}\right|{}^2}{P_{\theta_{dp}{\theta}_{dp}}{P}_{\theta_{dp}{\theta}_{dp}}{P}_{\theta_{ie}{\theta}_{ie}}{P}_{\tau_{dp}{\tau}_{dp}}\left(1-{\gamma^2}_{\theta_{dp}{\theta}_{ie}}\right)\left(1-{\gamma^2}_{\theta_{dp}{\tau}_{dp}}\right)}\hfill \\ \hfill & \hfill \frac{\left|{P}_{\theta_{ie}{\tau}_{ie}}{P}_{\theta_{dp}{\theta}_{dp}}-{P}_{\theta_{dp}{\tau}_{ie}}{P}_{\theta_{ie}{\theta}_{dp}}\right|{}^2}{P_{\theta_{dp}{\theta}_{dp}}{P}_{\theta_{dp}{\theta}_{dp}}{P}_{\theta_{ie}{\theta}_{ie}}{P}_{\tau_{ie}{\tau}_{ie}}\left(1-{\gamma^2}_{\theta_{dp}{\theta}_{ie}}\right)\left(1-{\gamma^2}_{\theta_{dp}{\tau}_{ie}}\right)}\hfill \end{array}\right]\end{aligned} $$

(6.A.3)