## 3 Modelling and Simulation of the Actuation System of the Rehabilitation Device
### Modelling and Simulation with MATLAB-Simulink  
The modelling and simulation of dynamic systems by means of MATLAB(r) are specific Model-Based Design techniques that can reduce the duration of the product development cycle by up to 50% [19]. For the modelling and simulation of systems, MATLAB(r) includes a graphic extension called Simulink(r) by means of which systems can be modelled with the help of block type elements grouped into categories, such as transfer functions, summing junctions, etc. as well as function generators and oscilloscopes [20]. The Simulink(r) approach is based on a time-based and multi-rate system. On its turn, Simulink(r) includes Simscape(tm), a graphical programming environment for modelling, simulating and analysis of dynamic systems. Simscape(tm) Fluids(tm) is used for the modelling and simulation fluid systems that provide a set of components specific for this field. The modelling and simulation of the actuation system of the hand joints rehabilitation equipment by means of MATLAB(r) R2021b-Simulink(r)-Simscape(tm) Fluids(tm) starts with establishing beforehand a recovery exercise based on a sequence of motions recommended by the physical therapist. Figure 5 presents and illustrates such an exercise that further serves as the basis for conceiving the MATLAB(r) functional model consisting of blocks (Figure 6). The duration of the exercise is of 60 s. The first half of the cycle is designed to generate extension, more accelerated in the beginning and slower as the upper limit of this motion is approached (+70\({}^{\circ}\)) when the onset of pain is possible. Flexion is conducted during the time interval of t \(\in\)[30, 60] seconds, the last part of this motion being also slowed down for the same reasons. Figure 5: Sequence of recovery motions to be generated by the equipment. The central block of the model is that of the pneumatic muscle (Air Muscle Actuator), its characteristics and analysis pattern being based on the [PERSON] model [21]. Port A is the gas conserving port associated with the actuator inlet. Port H is the thermal conserving port associated with the temperature of the gas inside the actuator. Ports R and C are the mechanical translational conserving ports associated with the two end caps of the actuator. The main dimensional characteristics of the pneumatic muscle were introduced as input parameters of the block. The pneumatic muscle is fed air from the Pressure source block, wherefrom the gas passes to a 3-Way Directional Valve. Ports P, T and A are associated with feeding, airing and the connection to the equipment's actuator, respectively. The valve opening fraction between \(-1\) and \(1\) is set by the physical signal port S. Port S receives a voltage signal by means of a Proportional Valve Actuator. A complete rehabilitation cycle of the wrist (an extension and a flexion) conducted in 60 s was simulated imposing a signal of the form shown in Figure 7 as the input of the Proportional Valve Actuator. The Proportional Valve Actuator converts the control input into the valve actuator position, the signal sent to the 3-Way Directional Valve having the form shown in Figure 8.  
Figure 6: Functional model of the rehabilitation equipment actuation system. Figure 7: Characteristics of the signal provided by the signal generator (Voltage Signal). During a cycle of rehabilitation exercising (60 s), the feed pressure of the pneumatic muscle and its axial contraction vary according to the graphs in Figures 9 and 10. Figure 8: Variation versus time of the output signal of the Proportional Valve Actuator. Figure 10: Variation versus time of the pneumatic muscle axial contraction. Figure 9: Variation versus time of the pneumatic muscle feed pressure. The graphs shown in the figures above were generated by an Ideal Translational Motion Sensor block. This block measures velocity and/or displacement of the free and of the pneumatic muscle. Figure 10 shows a maximum stroke carried out by the rack of the mechanism (the free end of the pneumatic muscle) of 39.486 mm.