## 3 Results and Discussion
### Motion Trajectory Results  
In this study, the path of the platform was a mixture of roll and pitch motions. Therefore, the motion trajectory of the platform would be calculated by mathematical simulation analysis software Matlab. In addition, the motion sensor was used to measure the actual movement displacement, speed, and acceleration of the platform motion to verify the consistency of the trajectory trend and platform movement.  
The roll and pitch motions were analyzed based on harmonic curves, and the single-cycle period of the path was 4 s. In addition, the motion angle was \(\pm 20^{\circ}\). The preset path of roll motion was a cosine curve, as shown in Figure 17a, and the pitch motion was a sine curve, as shown in Figure 17b.  
The trajectory of the crank and linkage positions of the roll and pitch motion were integrated and combined with the motor output angles of each action point to create a multi-axis motion platform trajectory model by Matlab, as shown in Figure 18. The maximum upward motor output angle was \(58.3^{\circ}\), the maximum downward angle was \(58.6^{\circ}\), and the total stroke angle was about \(117^{\circ}\), as shown in Figure 19. In addition, the maximum motor output velocity was about 7.8 rpm, as shown in Figure 20.  
Figure 16: The stress trend chart of the pitch motion.  
Figure 19: Motor output angle diagram, total stroke angle was about 117\({}^{\circ}\).  
Figure 17: The trajectory of **(a)** roll (**b**) pitch.  
Figure 18: Multi-axis motion platform trajectory model.  
Moreover, in order to confirm the error between simulation and test, the roll and pitch motion path were input into the platform and performed for the actual measurement. In roll motion, the maximum angular velocity of the platform simulation was 15.71 rad/s, and the maximum angular velocity of the actual measurement was 14.55 rad/s. The comparison of the velocity for the simulation and the actual measurement is shown in Figure 21a. The maximum angular acceleration of the simulation was 24.67 rad/s\({}^{2}\), and the maximum angular acceleration of the actual measurement was 22.8 rad/s\({}^{2}\), as shown in Figure 21b.  
In pitch motion, the maximum angular velocity of the platform simulation was 15.71 rad/s, and the maximum angular velocity of the actual measurement was 14.36 rad/s. The velocity of the simulation and the actual measurement is shown in Figure 22a. The maximum angular acceleration of the simulation was 24.67 rad/s\({}^{2}\), and the maximum angular acceleration of the actual measurement was 22.46 rad/s\({}^{2}\), as shown in Figure 22b.  
Figure 21: (**a**) Angular velocity (**b**) Angular acceleration comparison chart of the roll motion.  
Figure 20: Motor output speed graph, the maximum velocity was about 7.8 rpm.  
To sum up, the velocity error during the roll motion was about 7.41%, and the acceleration error was about 7.58%. Thus, the velocity error during the pitch motion was 8.6%, and the acceleration error was about 8.96%. Obviously, the overall trend of those motions was in line with the actual measurement. The small error in the value could be judged as the difference from the load above.