# Mechanical Design of a Bioinspired Compliant Robotic Wrist Rehabilitation Equipment
## 4 Experimental Research  
The first tests of the rehabilitation equipment were aimed at identifying the motion limits of the Fin-Ray type structures that support the hand. By repeated charging/discharging of the pneumatic muscle with compressed air it could be established that the limits of flexion/extension of the radiocarbon joint carried out by means of the devised equipment are very close to the theoretically obtained values. The maximum obtained flexion angle, measured by means of an incremental rotation encoder placed in joint A, was of \(-78.2^{\circ}\) (theoretical value: \(-80^{\circ}\)). The maximum obtained extension angle was of \(67.3^{\circ}\) (theoretical value: \(70^{\circ}\)). These small deviations are caused by the torsion springs in joints B and C. For the metacarpophalangeal joint the angles that were obtained are of \(-87.6^{\circ}\) for flexion and of \(8.3^{\circ}\) for extension (this angle was measured by a second rotary encoder, similar to the first one, placed in the axis of joint B). The following experiments were aimed at demonstrating the compliance of the rehabilitation equipment. Compliance, namely the flexibility of physical structures in response to an external force manifests if the dependency between the force developed by the pneumatic muscle and the displacement caused by this is of nonlinear type. Typical advantages of compliance are the possibility of rapidly and efficiently storing and releasing energy or reacting instantaneously to sudden impacts, with positive effects on stability and thereby safety [27]. Determining the compliance of the rehabilitation equipment entailed measuring the forces developed by the pneumatic muscle along its entire useful stroke. Measurements were performed by means of a force transducer supplied by Festo, with the following characteristics: Measuring range: 0-1 kN; supply voltage: 24 V DC; output voltage: 0-10 V. The experimental data were processed by FluidLab\({}^{\text{0}}\)-P V1.0 dedicated software (Festo). Table 2 presents the value sets of the forces, charging pressures, axial contractions and flexion/extension angles for that concur with the shape of the curve in Figure 8.  
The following functions describe with the best fidelity the evolution of the measured force in dependence on the axial contraction of the muscle and the flexion/extension angle of the wrist (functions obtained by means of the Graph v.4.4.2 software application):  
\[F(s)=-0.095\cdot s^{2}+4.0578\cdot s+4.2568 \tag{2}\]  
\[F(\theta)=-0.0069\cdot\theta^{2}+0.0112\cdot\theta+47.256. \tag{3}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \(F\) [N] & 0 & 8.3 & 22.2 & 38.3 & 47.6 & 46.5 & 16.7 \\ \(p\) [bar] & 0 & 0.1 & 1 & 2 & 2.72 & 3 & 3.4 \\ \(s\) [mm] & 0 & 0 & 4.6 & 12 & 19.62 & 24 & 39.25 \\ \(\theta\) [\({}^{\circ}\)] & \(-78.2\) & \(-78.2\) & \(-60.7\) & \(-32.4\) & \(-3.3\) & 13.4 & 67.3 \\ \hline \end{tabular}
\end{table} 
Table 2: Force \(F\) developed by the pneumatic muscle versus the stroke \(s\) (axial contraction) and the flexion/extension angle of the wrist \(\theta\). Figure 10: Construction of the rehabilitation equipment of the wrist. The compliance of the analyzed system, which is the inverse of rigidity, represents the system's capacity of adapting to a given operational situation. In case the affected joints of the patient do not allow complete angular displacements, exceeding certain values of such motions may cause pain. A compliant rehabilitation system adapts to such situations by absorbing the shock generated by the increased resisting forces, thus favoring pain-free exercising. Equation (2) reveals that the dependency between the force developed by the pneumatic muscle and the stroke is not linear, what is indicative of the presence of variable compliance. Compliance is defined by Equation (4):  
\[C(s)=\left(\frac{dF}{ds}\right)^{-1}=\frac{1}{-0.19\cdot s+4.0578} \tag{4}\]  
According to the equation above the denominator of the fraction becomes equal to 0 when the axial contraction \(s=4.0578/0.19=21.35\) mm.