Stochastic estimation of arm mechanical impedance during robotic stroke rehabilitation

Abstract

This paper presents a stochastic method to estimate the multijoint mechanical impedance of the human arm suitable for use in a clinical setting, e.g., with persons with stroke undergoing robotic rehabilitation for a paralyzed arm. In this context, special circumstances such as hypertonicity and tissue atrophy due to disuse of the hemiplegic limb must be considered. A low-impedance robot was used to bring the upper limb of a stroke patient to a test location, generate force perturbations, and measure the resulting motion. Methods were developed to compensate for input signal coupling at low frequencies apparently due to human-machine interaction dynamics. Data was analyzed by spectral procedures that make no assumption about model structure. The method was validated by measuring simple mechanical hardware and results from a patient's hemiplegic arm are presented.

Fig. 1

Block diagram of linear MIMO impedance structure: C_xFx (_s) and C_xFy (_s) are the transfer functions from the interaction forces, F_x and F_y, to the displacement x: C_xFx (_s) and C_xFy (_s) are the transfer functions from the interaction forces to the displacement the displacement y.

Fig. 2

Block diagram of linear MIMO structures using vector and matrix notation: T(s) is the transfer function matrix from the commanded force perturbations, F_Δ, to the interaction forces, F; C(s) is the transfer function matrix from the interaction forces to the displacements, d.

Fig. 3

Block diagram of linear MIMO structure using vector and matrix notation: R(s) is the transfer function matrix from the commanded force perturbations, F_Δ, to the displacements, d.

Fig. 4

Top view of MIT-MANUS and the mechanical spring array used to validate the impedance spectral estimation method. In this test configuration (sa1), six extension springs are connected between the inner and outer fixtures to generate an elliptical stiffness field whose major and minor axes are different from the x- and y-axes.

Fig. 5

Stiffness ellipses for the nine test configurations (sa1 to sa9) that were defined to validate the analytical and experimental procedures in the impedance spectral estimation method.

Fig. 6

Expected behavior (based on linearized model) and mean spectral estimates for six trials of test configuration sa1. Both sets of spectral analysis parameters result in excellent estimates of the expected impedance frequency response.

Fig. 7

Mean partial and multiple coherence functions for six trials of test configuration sa1 that correspond to the spectral estimates shown in Fig. 6. Decreases in the partial and multiple coherences occur near the system resonances. Decreases in the off-diagonal elements of the partial coherence also occur at higher frequencies (inertia is isotropic and the off-diagonal elements are identically equal to zero).

Fig. 8

Summary of the VAF values for each element of the impedance spectral estimate and the R² values of the entire estimate for test configurations sa1 to sa9. Off-diagonal elements of VAF for sa4 and sa5 are numerically ill-conditioned because the inertia and stiffness matrices are isotropic (ideally, the off-diagonal elements are identically equal to zero).

Fig. 9

Modeled behavior (based on linear second-order model) and mean spectral estimates for six trials of a patient's hemiplegic left arm. The modeled frequency response elements capture the gain of both estimates throughout the frequency range shown, whereas the phase of both estimates deviates from the model for f > 5 Hz.

Fig. 10

Mean partial and multiple coherence functions for six trials of a patient's hemiplegic left arm that correspond to the spectral estimates shown in Fig. 9. Decreases in the partial and multiple coherences occur near the system resonances. The partial coherence functions are close to 1 at higher frequencies.