# Design and Simulation of a Vision-Based Automatic Trout Fish-Processing Robot
## 2 Material and Methods
### Designing the Mechanical Parts
#### 2.3.2. Calculations Related to Pneumatic Jacks of Head-Cutting and Cleaner Subsystems  
Two pneumatic jacks were applied in a trout-processing system in head-cutting and cleaning subsystems. As the piston moves downwards during the extraction step, the jack is expected to prevent the subset from free falling. To know whether the pneumatic jack will empower or prevent free fall, we investigated the piston velocity in free-fall conditions from a height of 180 mm. The falling situation is presented in Figure 13. In this figure, IP, SP, and FP are the initial, start, and final positions of the head-cutting subsystem. The following are the equations for calculating the final velocity of the head-cutting subset in a free-fall condition. According to these relationships, considering the weight of all stimulus parts was regarded as a single mass. By simplifying the falling conditions, the penetration speed in the initial contact and termination step was calculated. \[\mathrm{U_{0}+K_{0}=U_{2}+K_{2}} \tag{2}\]  
\[\mathrm{m\cdot g\cdot h_{0}+\frac{1}{2}\cdot m\cdot{v_{0}}^{2}=m\cdot g\cdot h_{ 2}+\frac{1}{2}\cdot m\cdot{v_{2}}^{2}} \tag{3}\]  
Figure 13. Free falling condition of the head-cutting subset: (**a**) distance from the home position of the head-cutting subset to the end of the cutting path, (**b**) the length from the home position to the initial contact of the blade and trout (80 mm), and (**c**) head-cutting width. In this figure, IP, SP, and FP are the initial, start, and final positions. \[\mathrm{g\cdot h_{0}}=\frac{1}{2}\cdot\mathrm{v_{2}}^{2} \tag{4}\]  
\[9.81\times 0.18\times 2=\mathrm{v_{2}}^{2} \tag{5}\]  
\[\mathrm{v_{2}}=1.88 \tag{6}\]  
In these relations, 'U' and 'K' are potential and kinetic energies in terms of (J), m is body mass (kg), 'v' represents body velocity (m s\({}^{-1}\)), 'h' is the falling height (m), and 'g' represents gravitational acceleration (m s\({}^{-2}\)). According to the energy conservation law, the sum of kinetic and potential energy at two points at the beginning and end of the motion path must be equal (Equations (2) and (3)). After simplifying the relationship, the velocity value at the end of the path resulted in 1.88 m s\({}^{-1}\). Assuming that the head-cutting subset has a free fall, by substituting the velocity values for 'v\({}_{1}\)' and 'v\({}_{2}\)', the required time for complete cut will be obtained using Equations (7) and (8). \[\mathrm{v_{1}}=\mathrm{g\cdot t_{1}}+\mathrm{v_{0}} \tag{7}\]  
\[\mathrm{v_{2}}=\mathrm{g\cdot t_{2}}+\mathrm{v_{0}} \tag{8}\]  
By setting the values of 1.25 m s\({}^{-1}\) and 1.89 m s\({}^{-1}\) for 'v\({}_{1}\)' and \(\mathrm{v_{2}}\) in Equations (7) and (8), \(\mathrm{t_{1}}\) and \(\mathrm{t_{2}}\) were calculated as 0.13 s and 0.19 s, respectively. The blades will travel the distance from IP to FP (180 mm) in 0.19 s in the vertical direction. Also, the head-cutting blades will travel 'a' distance 'b' in 0.13 s. At this distance, the cutting blades will descend freely to reach the fish in the SP position. Considering the fish's average width of 100 mm, the cutting time from SP to FP resulted in 0.06 s. It should be noted that the penetration velocities in the SP and FP region are high values; this value is also higher than the maximum speed provided by the pneumatic jack. Comparing the velocity values in different states, it is not allowed to consider different speed ranges for the piston. Obviously, the least amount of longitudinal displacement of the fish during the cutting process will be achieved at the highest blade penetration speed and the lowest processing capacity. Therefore, a penetration velocity of 500 mm s\({}^{-1}\) was considered as the final velocity for cutting blade. Under these conditions, as this speed is considered maximum, the other feeding rates will be processed even better than the maximum feeding rate of 10 fish per minute. By comparing the velocity and time of the free fall with the maximum speed piston speed, it is expected that the piston must exert the opposite force against the free fall.