## 4 Testing and Validation  
The experimental results concerning the performance of the equipment were presented in detail in [14]. Of interest for the present study is to identify dependencies of form \(p=p(t)\) and \(\theta=\theta(t)\) that meet the specifics of a rehabilitation exercise such as the one proposed in Figure 5 (\(p=\) air pressure; \(\theta=\) wrist rotation angle). Further on, the sequence of the steps of such a rehabilitation exercise is presented. At rest, in the absence of compressed air, the pneumatic muscle is relaxed, and the rack is displaced maximum to the right (Figure 4). The position of the rehabilitation mechanism in this case corresponds to a maximum flexion of the wrist (lever AB turned to \(-80^{\circ}\)). In order to initiate a rehabilitation exercise, the first step is to set the palm support to zero position with a horizontal lever AB. This motion is meant to allow fixing the patient's hand on the palm support. For this, the pneumatic muscle is charged with air to a pressure at which it is caused to shorten and consequently to generate a sufficient displacement of the rack so that lever AB turns by \(80^{\circ}\). Experimentally, it was determined that charging the pneumatic muscle with air at a pressure of 2.72 bar will shorten and displace the rack by 19.62 mm, and consequently lever AB will reach its horizontal position (Figure 13a). Figure 13: Positions of lever AB while the pneumatic muscle is charged with air: (**a**) Starting position; (**b**) Maximum wrist flexion; (**c**) Maximum wrist extension. Decreasing the air pressure in the muscle to 0 bar causes the flexion of the wrist (Figure 13b), while increasing the pressure leads to extension (Figure 13c). Compressed air is fed to the muscle through a proportional pressure regulator controlled by a reference module, produced by Festo AG & Co Esslingen, Germany (Figure 14). The continuous modification of the pressure fed to the proportional pressure regulator causes the permanent variation in time of the air pressure in the pneumatic muscle and allows for modifying the compliance of the entire system. This is a pressure-based control scheme where the pressure within the pneumatic muscle is measured and controlled in a closed loop. Figure 15 shows the experimentally determined variation curves of the feed pressure versus time. The variation in time of the pneumatic muscle feed pressure was controlled so that the form of the resulting curves is as close as possible to that obtained consequently via modelling and simulation by means of MATLAB(r) (Figure 9). Figure 14: Control of the rehabilitation equipment by means of a proportional pressure regulator. Figure 15: Charging/discharging curves of the pneumatic muscle with air during a rehabilitation exercise cycle. Equations (1) and (2) represent the functions that describe the variation of pressure at the charging of the pneumatic muscle (for conducting extension) and at the discharging of air (for conducting flexion), respectively:  
\[p(t)=-2.59\cdot 10^{-5}\cdot t^{4}+0.0023\cdot t^{3}-0.076\cdot t^{2}+1.1058\cdot t \qquad t\in[0,\ 30) \tag{1}\]  
\[p(t)=-0.0003\cdot t^{3}+0.0479\cdot t^{2}-2.7003\cdot t+51.6 \qquad t\in[30,\ 60] \tag{2}\]  
Knowing these functions is necessary for controlling the proportional pressure regulator. A specially developed computer program based on the above polynomial equations is loaded to a PLC. The PLC sends an electric signal to the proportional pressure regulator whose voltage is continuously modified according to the polynomial function. The charging/discharging of the pneumatic muscle with pressure according to the curves in Figure 15 allowed the determining of the variation versus time of the muscle's axial contraction (\(\Delta l\)). This quantity was measured by a linear potentiometer provided by German company Festo and which has the following characteristics: measured stroke: 200 mm; electrical resistance: 10 k\(\Omega\pm 20\%\); supply voltage: 13-30 V DC; output voltage: 0-10 V DC.