# Design and Optimization of Non-Coplanar Orbits for Orbital Photovoltaic Panel Cleaning Robots
## 2 Construction of Models
### Effectiveness Evaluation and Analysis
#### 3.4.3 Simulation Data  
Dynamics simulations were conducted using ADAMS to measure the x, y, and z coordinates of the center of mass of the four walking wheels of the PV panel cleaning robot, both for the pre-optimized and optimized orbits. As illustrated in Figure 10, the trajectory component plots for the centers of mass of the four walking wheels during Stage B are presented. From the figure, it is evident that the trajectory data trends of the four walking wheels, both before and after optimization, are nearly identical, with only a slight variation observed during a specific time segment. This indicates that the cleaning robot can operate smoothly on the non-coplanar orbit, thereby validating the effectiveness of the orbital design. In order to more intuitively illustrate the differences between the pre-optimization and post-optimization B phases, the simulation error curves for the four walking wheels are presented in Figure 11. Error 1 denotes the simulation error prior to optimization, while error 2 indicates the simulation error following optimization. From the figure, it is evident that there are peaks and valleys in both the pre- and post-optimization simulation errors. Additionally, there is an irrational optimization of the simulation errors for wheels 1 and 3 during a specific time period. This phenomenon is attributed to the deflection of the robot's position caused by the differential movement of the traveling wheels during the robot's spanning motion. In addition, it is evident that the simulation errors for the four wheels achieve optimal performance at 15 s, 57 s, 50 s, and 59 s, respectively, with reductions of 8.79 mm, 23.78 mm, 10.11 mm, and 14.97 mm. The data and simulation results confirm that the theoretical model design and optimization are reasonable to a significant extent. In this study, we use Bezier curves with scaling parameters to design non-coplanar trajectories. This approach improves the adaptability and scalability of photovoltaic (PV) panel cleaning robots in variable installation environments and relative positions. Compared to traditional Bezier curves, our approach shows significant advantages in dealing with the dynamic changes of PV panels. Although B-spline curves offer a high degree of flexibility in trajectory design, they can be computationally more complex and it can be difficult for the method to accurately model the desired shape in 3D space, as pointed out by [PERSON] et al. (2016). To overcome these limitations, our approach reduces the computational burden by simplifying the mathematical model, thereby increasing its feasibility in practical applications. Although the NURBS curves used by [PERSON] et al. (2017) in industrial robot trajectory planning provide a powerful tool for Cartesian trajectory planning, the method has a complicated decomposition process, including the calculation of tangent and curvature, which can lead to a cumbersome operation. In contrast, our proposed method exploits the \begin{table}
\begin{tabular}{c c c c} \hline
**Bezier Curve** & **Maximum Curvature** & **Minimum Curvature** & **Curvature Threshold** \\ \hline Unoptimized curve 1 & \(4.8696\times 10^{-4}\) & \(6.6388\times 10^{-8}\) & \\ Optimized curve 1 & \(1.0472\times 10^{-3}\) & \(3.5739\times 10^{-8}\) & \\ Unoptimized curve 2 & \(5.7751\times 10^{-4}\) & \(1.0840\times 10^{-7}\) & \(1.6\times 10^{-2}\) \\ Optimized curve 2 & \(1.4655\times 10^{-3}\) & \(5.6493\times 10^{-8}\) & \\ \hline \end{tabular}
\end{table} 
Table 4: The curvature of the pre-optimized and post-optimized Bezier curves. The method proposed in this paper not only improves the flexibility and adaptability of trajectory design for photovoltaic (PV) panel cleaning robots but also reduces the complexity of practical applications by simplifying the computational process. This provides an efficient and reliable solution for the automated operation and maintenance of PV power plants.