# Mechanical Design of a Bioinspired Compliant Robotic Wrist Rehabilitation Equipment
## 4 Experimental Research
Around this point compliance tends asymptomatically to infinity, what signifies the instability of the entire system. In order to obtain a compliance with finite values, adjustable over the entire operational range of the rehabilitation equipment, the force developed by the muscle must increase permanently, from maximum flexion to maximum extension. Table 3 show the value sets \((p,s,\theta,F)\) that comply with this requirement. For these data the equations were determined that describe the dependency of the force on the axial contraction of the pneumatic muscle, on the flexion/extension angle of the wrist, and on the feed pressure, respectively:  
\[F(s)=0.0396\cdot s^{2}+1.4469\cdot s+18.2544 \tag{5}\]  
\[F(\theta)=0.0033\cdot\theta^{2}+0.8341\cdot\theta+64.187 \tag{6}\]  
\[F(p)=0.8025\cdot p^{2}+16.9292\cdot p+6.5692. \tag{7}\]  
Equation (7) is important for the control of the proportional pressure regulator. A specially developed computer program based on the experimentally determined polynomial Equation (7) is loaded to a PLC. The PLC sends an electric signal to the proportional pressure regulator whose voltage is continuously modified according to the experimentally obtained polynomial function. This affects the desired variation of air pressure in the pneumatic muscle and consequently the desired contractions and forces. Figure 11 shows the three curves that describe the variation of the force developed by the pneumatic muscle versus the modification of the flexion/extension angle of the wrist: The curve of the necessary force (Equation (1)), the theoretical curve (Equation (3)) and the real curve according to that the rehabilitation equipment has to function (Equation (6)). To be noted is the fact that the latter curve ensures in each point values of the force that are greater than the theoretically computed ones, and thus the necessary force for the proper operation of the rehabilitation equipment. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \(p\) [bar] & 0 & 0.2 & 1 & 2 & 3 & 4.4 & 6 \\ \(s\) [mm] & 0 & 0 & 4.2 & 11.4 & 20.4 & 30.5 & 39.25 \\ \(\theta\) [\({}^{\circ}\)] & \(-\)78.2 & \(-\)78.2 & \(-\)62.2 & \(-\)34.7 & \(-\)0.3 & 38.2 & 67.3 \\ \(F\) [N] & 0 & 16.6 & 26.5 & 42.2 & 61.9 & 99 & 136.5 \\ \hline \end{tabular}
\end{table} 
Table 3: Value sets \((p,s,\theta,F)\) that ensure the adjustable compliance of the rehabilitation equipment. Starting from Equation (5) the compliance of the rehabilitation equipment is:  
\[C(s)=\left(\frac{dF}{ds}\right)^{-1}=\frac{1}{0.0792\cdot s+1.4469}. \tag{8}\]  
Figure 12 shows the graphs of compliance variation versus the axial contraction of the pneumatic muscle and the flexion/extension angle of the wrist:  
From Figure 12 follows that charging the pneumatic muscle with compressed air at pressures according with Equation (7) leads to obtaining an adjustable compliance that allows the adaptability of the system to the patient's state, namely to the degree of rigidity of the hand joints. To be noted is the fact that as air pressure in the muscle increases and consequently also its axial contraction, the compliance of the system records a decreasing trend. Compliance has maximum values when the wrist is in flexion and decreases as the transition from flexion to extension occurs. It follows that the adaptability of the rehabilitation equipment to the patient's state is higher at flexion and lower at extension.