# Inverse Solution Error Analysis and Correction of Beam Steering System Based on Risley Prisms
## 2 Error Model of Risley Prisms and Forward Solution Error Correction Method
### Error Model of Risley Prisms  
The error model of the Risley prisms is obtained by introducing systemic errors into the mathematical model of the Risley prisms. The systemic errors of Risley prisms can be divided into component errors and assembly errors [14]. Component errors refer to the deviation of the prism wedge angle and the refractive index from the design value, which may change with temperature or wavelength. Assembly errors include prism rotation  
Figure 3: Diagrams illustrating the notation of the normal vector for each prism and the incident ray vector. The black arrows represent the surface normal vectors, and the blue arrows represent the incident and refracted light vectors. axis tilt, prism tilt, incident light tilt, and prism rotation angle error. Their description and notation are shown in Figure 4.  
Take prism 1 as an example. Ideally, the prism rotation axis coincides with the \(Z\) axis, \(Z_{1}\) is the actual rotation axis which deviates from the \(Z\) axis, and the error between the two is represented by \(H_{R1},V_{R1}\). \(V_{R1}\) is the angle between the \(Z_{1}\) axis and plane OZX, which represents the vertical error of the \(Z_{1}\) axis. \(H_{R1}\) is the angle between the projection of the \(Z_{1}\) axis on the plane OZX and the \(Z\) axis, and represents the horizontal error of the \(Z_{1}\) axis. The coordinate system \(X_{2}Y_{2}Z_{2}\) denotes the coordinate system of the tilting prism; the tilt error of the prism in the vertical and horizontal directions are denoted by \(H_{P1},V_{P1}\), respectively. In the same way, the tilt errors of the incident light in the vertical and horizontal directions are represented by \(H_{I},V_{I}\), respectively. In Figure 2, the configuration when the rotation angle of the prisms is \(\theta_{1}=\theta_{2}=0\) is defined as the zero position of the Risley prisms. The analytical model of the Risley prisms is derived when the prisms are in the zero position; therefore, the rotation angle of the prisms is calculated from the zero position. In practice, the prisms cannot be precisely mounted at the zero position, as the prism rotation angle error refers to the angle between the actual initial mounting position of the prism and the zero position. The symbols \(\delta\theta_{1}\) and \(\delta\theta_{2}\) are used to denote the rotation angle errors of prism 1 and prism 2, respectively. According to the errors defined in Figure 4, the incident light vector \(\overset{\rightarrow}{A}_{1}\), the normal vectors of the prisms' four surfaces \(\overset{\rightarrow}{n}_{110},\overset{\rightarrow}{n}_{120},\overset{ \rightarrow}{n}_{210},\overset{\rightarrow}{n}_{220}\), and the prism rotation axis vectors \(\overset{\rightarrow}{k}_{1},\overset{\rightarrow}{k}_{2}\) can be obtained. Let \(\overset{\rightarrow}{k}_{1}=(k_{x},k_{y},k_{z})\); the transformation matrix of any vector rotating around \(\overset{\rightarrow}{k}_{1}\) can be obtained by using Rodrigues' rotation formula:  
\[R_{1}=E\cos\theta_{1}+(1-\cos\theta_{1})\left( \begin{array}{c}k_{x}\\ k_{y}\\ k_{z}\end{array} \right)\left( \begin{array}{ccc}k_{x}&k_{y}&k_{z}\end{array}  \right)+\sin\theta_{1}\left( \begin{array}{ccc}0&-k_{z}&k_{y}\\ k_{z}&0&-k_{x}\\ -k_{y}&k_{x}&0\end{array} \right) \tag{7}\]  
where \(E\) is the unit matrix, and \(\theta_{1}\) is the rotation angle of prism 1. When prism 1 rotates around \(\overset{\rightarrow}{k}_{1}\), the normal vectors of its two surfaces are \(\overset{\rightarrow}{n}_{11}=R_{1}\cdot\overset{\rightarrow}{n}_{110}\) and \(\overset{\rightarrow}{n}_{12}=R_{1}\cdot\overset{\rightarrow}{n}_{120}\). In the same way, the normal vectors of the two surfaces of prism 2 can be obtained: \(\overset{\rightarrow}{n}_{21}^{\rightarrow}=R_{2}\cdot\overset{ \rightarrow}{n}_{210}\) and \(\overset{\rightarrow}{n}_{22}^{\rightarrow}=R_{2}\cdot\overset{ \rightarrow}{n}_{220}\), where \(R_{2}\) the transformation matrix of any vector rotating around \(\overset{\rightarrow}{k}_{2}\).