Mixed $\mu$-synthesis tracking control and disturbance rejection in a robotic digit of an impaired human hand for anthropomorphic coordination

Abstract

In a partially impaired anthropomorphic hand, maintaining the movement coordination of the robotic digits with the central nervous system (CNS) and natural digits is crucial for robust performance. A challenge in the control perspective of movement coordination of a human hand is finding methods robust to the disturbances in a well-posed control problem of a biomechanical model. We use visco-elastic dynamics in the human palm frame of reference to explore the biomechanics of movement coordination to solve this control problem. Our biomechanical model incorporates the time delay due to actuation force, parametric uncertainty, exogenous disturbances, and sensory noise to constitute a 21-degree of freedom model. A mixed $\mu$-synthesis controller, considering the real parametric uncertainty, represents the CNS in the control paradigm. We consider the robotic finger’s flexion movement when perturbed from the initial equilibrium. The controller provides feedback force at the joints to regulate the robotic finger movement. The index finger follows a reference trajectory of the joint angular position profile and stabilizes at a flexion angle of 1 rad/s at a time of 1 s. The main control objective is to keep the angular displacement of the finger joint constant when a disturbance force acts. We simulate the modeling scheme in MATLAB/ Simulink. The results demonstrate that our controller scheme is robust against the worst-case disturbance and achieves the desired performance value. Developing a biologically inspired neurophysiological controller with robust performance has many applications, including assistive rehabilitation devices, hand movement disorder diagnosis, and robotic manipulators.

Appendix A: system dynamics

Table 7 Parametric values Of the Anthropomorphic Finger

$$\begin{aligned} B(plant){=} \begin{bmatrix} 0 &{}\quad 0 &{}\quad 0.0016\\ 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 8.0000\\ 0.5000 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 128.0000\\ 0 &{}\quad 0 &{}\quad 0 \end{bmatrix} \end{aligned}$$

$$\begin{aligned} C(plant)= \left[ \begin{array}{lllllllllllllllllllllllll} 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0.3200\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -77.3438 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0.08 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0.3200 \end{array}\right] \qquad \quad \end{aligned}$$

(A1)

$$\begin{aligned} D(plant)= \begin{bmatrix} 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 10.0000 \\ 0 &{}\quad 0.0010 &{}\quad 0 \end{bmatrix} \end{aligned}$$

$$\begin{aligned}{} & {} K(s){=}\frac{\begin{aligned}&0.0002411 s^{24} {+} 0.2911 s^{23} {+} 54.72 s^{22} {+} 4986 s^{21}{+} 2.845e05 s^{20} {+} 1.155e07 s^{19} {+} 3.601e08 s^{18}{+} 8.978e09 s^{17}\\ {}&\quad {+} 1.823e11 s^{16}{+} 3.041e12 s^{15} {+} 4.184e13 s^{14}{+} 4.741e14 s^{13} {+} 4.434e15 s^{12} {+} 3.442e16 s^{11} {+} 2.23e17 s^{10} \\&\quad {+} 1.198e18 s^9{+} 5.321e18 s^8 {+} 1.917e19 s^7 {+} 5.387e19 s^6 {+} 1.141e20 s^5 {+} 1.483e20 s^4{+}\\ {}&\quad 9.557e19 s^3 {+} 2.395e19 s^2 {+} 2.012e18 s {+} 1.817e15\end{aligned}}{\begin{aligned}&s^{25} {+} 427.7 s^{24} {+} 5.783e04 s^{23} {+} 4.203e06 s^{22} {+} 2.016e08 s^{21} {+} 7.148e09 s^{20}{+} 1.992e11 s^{19}{+} 4.489e12 s^{18}\\ {}&\quad {+} 8.3e13 s^{17} {+} 1.269e15 s^{16}{+} 1.607e16 s^{15}{+} 1.694e17s^{14}{+} 1.492e18 s^{13}{+} 1.108e19 s^{12} {+} 6.945e19 \\ {}&\quad s^{11} {+} 3.674e20 s^{10}{+} 1.633e21 s^9 {+} 6.017e21 s^8 {+} 1.809e22 s^7 {+} 4.281e22s^6 {+} 7.579e22 s^5{+}\\ {}&\quad 9.232e22 s^4 {+} 6.795e22 s^3 + 2.391e22 s^2 {+} 1.843e21 s {+} 4.631e18\end{aligned}} \end{aligned}$$