# Design and Optimization of Non-Coplanar Orbits for Orbital Photovoltaic Panel Cleaning Robots
## Appendix A  
The translation matrix for the robot moving from State 1 to State 2 is as follows:  
\[Trans(x_{2},y_{2},z_{2})= \begin{bmatrix}1&0&0&x_{2}\\ 0&1&0&y_{2}\\ 0&0&1&z_{2}\\ 0&0&0&1\end{bmatrix} , \tag{1}\]  
The rotation matrix of the robot moving from State 1 to State 2 is as follows:  
\[R(\phi,\theta,\psi)=Rot(z,\phi)Rot(y,\theta)Rot(x,\psi), \tag{2}\]  
where \(Rot(x,\psi)\) denotes the rotation of the cleaning robot around the \(X\) axis of the local coordinate system by an angle \(\psi\). The specific equation is as follows:  
\[Rot(x,\psi)= \begin{bmatrix}1&0&0&0\\ 0&c\psi&-s\psi&0\\ 0&s\psi&c\psi&0\\ 0&0&0&1\end{bmatrix} , \tag{3}\]  
where \(Rot(y,\theta)\) denotes the rotation of the cleaning robot around the \(Y\) axis of the local coordinate system by an angle \(\theta\). The specific equation is as follows:  
\[Rot(y,\theta)= \begin{bmatrix}c\theta&0&s\theta&0\\ 0&1&0&0\\ -s\theta&0&c\phi&0\\ 0&0&0&1\end{bmatrix} , \tag{4}\]  
* _[PERSON] et al. (2023)_[PERSON]; [PERSON]; [PERSON] Design and development of cleaning solar plants robots. _Energy Sources Part A Recovery Util. Environ. Eff._**2023**, _45_, 7031-7049. [CrossRef]
* _[PERSON] et al. (2020)_[PERSON]; [PERSON]; [PERSON] Research on Design of Intelligent Cleaning Robot for Solar Panel. 2020. Available online: [[https://aisel.aisnet.org/iceb2020/22](https://aisel.aisnet.org/iceb2020/22)]([https://aisel.aisnet.org/iceb2020/22](https://aisel.aisnet.org/iceb2020/22)) (accessed on 1 July 2024). * _[PERSON] et al. (2020)_[PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Smart solar photovoltaic panel cleaning system. In _IOP Conference Series: Earth and Environmental Science_, IOP Publishing: Bristol, UK, 2020. [CrossRef]
* _[PERSON] et al. (2023)_[PERSON]; [PERSON] Speed adaptive robot trajectory generation based on derivative property of b-spline curve. _IEEE Robot. Autom. Lett._**2023**, \(8\), 1905-1911. [CrossRef]
* _[PERSON] et al. (2023)_[PERSON]; [PERSON]; [PERSON]; [PERSON] A novel cartesian trajectory planning method by using triple NURBS curves for industrial robots. _Robot. Comput. Integr. Manuf._**2023**, _83_, 102576. [CrossRef]
* _[PERSON] et al. (2023)_[PERSON]; [PERSON]; [PERSON] Research on obstacle avoidance path planning of UAV in complex environments based on improved Bezier curve. _Sci. Rep._**2023**, _13_, 16453. [CrossRef] [PubMed]
* _[PERSON] (2020)_[PERSON] Path planning of mobile robots in dynamic environment based on analytic geometry and cubic Bezier curve with three shape parameters. _Expert Syst. Appl._**2023**, _233_, 120942. [CrossRef]
* _[PERSON] et al. (2023)_[PERSON]; [PERSON] [PERSON]; [PERSON]; [PERSON]; [PERSON]. Novel Triangular-Based Estimation Technique for Bezier Curve Control Points Generation on Autonomous Vehicle Path Planning at the Roundabout Intersection. _J. Intell. Robot. Syst._**2023**, _109_, 89. [CrossRef]
* _[PERSON] et al. (2024)_[PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] High-Fidelity and Curvature-Continuous Path Smoothing with Quadratic Bezier Curve. _IEEE Trans. Intell. Veh._**2024**, \(9\), 3796-3810. [CrossRef]
* _[PERSON] et al. (2024)_[PERSON]; [PERSON]; [PERSON] Three dimensional path planning for flying car based on improved A* algorithm and Bezier curve fusion. _Proc. Inst. Mech. Eng. Part D. Automoh. Eng._**2024**, 09544070241227094. [CrossRef]
* _[PERSON] et al. (2019)_[PERSON]; [PERSON]; [PERSON] Optimization of aeroassisted rendezvous and interception trajectories between non-coplanar elliptical orbits. _Acta Astronaut._**2019**, _163_, 190-200. [CrossRef]
* _[PERSON] et al. (2023)_[PERSON]; [PERSON]; [PERSON]; [PERSON] Morphological Design of a Bicycle Propulsion Component Using the Hierarchical Analysis Process (AHP). _Appl. Sci._**2023**, _13_, 7792. [CrossRef]  
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