# A Linear Strong Constraint Joint Solution Method Based on Angle Information Enhancement
## 4 The Joint Positioning Method of Angle and Distance
### A Linear Strong Constraint Joint Solution Method  
In the case of good 5G wireless signal coverage performance, there are more base stations with line-of-sight paths, and the corresponding angle and distance observation information is more accurate. In addition, the angle measurement method based on fourth-order cumulants can effectively suppress Gaussian noise and has good noise resistance performance. Based on this, this section assigns higher weight to angle observation information. This not only avoids nonlinear errors caused by constructing observation equations solely based on distance measurement information, but also improves the robustness of the positioning system. Simultaneously performing tangent calculations on both sides of the direction angle and pitch angle equations, and then shifting the terms, can convert the angle measurement equation into:  
\[(x_{i}-u_{x})sin\theta_{i}-\big{(}y_{i}-u_{y}\big{)}cos\theta_{i}=0 \tag{28}\]  
\[(x_{i}-u_{x})cos\theta_{i}sin\varphi_{i}+\big{(}y_{i}-u_{y}\big{)}sin\theta_{i }sin\varphi_{i}-(z_{i}-u_{z})cos\varphi_{i}=0 \tag{29}\]  
Moving the unknown terminal coordinates to one side of the equation yields  
\[sin\theta_{i}x_{i}-cos\theta_{i}y_{i}=sin\theta_{i}u_{x}-cos\theta_{i}u_{y} \tag{30}\]  
\[cos\theta_{i}sin\varphi_{i}x_{i}+sin\theta_{i}sin\varphi_{i}y_{i}-cos\varphi_ {i}z_{i}=cos\theta_{i}sin\varphi_{i}u_{x}+sin\theta_{i}sin\varphi_{i}u_{y}-cos \varphi_{i}u_{z} \tag{31}\]  
Linearize it as  
\[\mathbf{A}_{1}\mathbf{u}^{T}=\mathbf{h}_{1} \tag{32}\]  
\[\mathbf{A}_{1}=\left[ \begin{array}{ccc}sin\theta_{1}&-cos\theta_{1}&0\\ cos\theta_{1}sin\varphi_{1}&sin\theta_{1}sin\varphi_{1}&-cos\varphi_{1}\\ &\cdots&\\ sin\theta_{N}&-cos\theta_{N}&0\\ cos\theta_{N}sin\varphi_{N}&sin\theta_{N}sin\varphi_{N}&-cos\varphi_{N}\\ \end{array} \right]\in\mathbb{R}^{2N\times 3} \tag{33}\]  
\[\mathbf{h}_{1}=\left[ \begin{array}{ccc}sin\theta_{1}x_{1}-cos\theta_{1}y_{1} \\ cos\theta_{1}sin\varphi_{1}x_{1}+sin\theta_{1}sin\varphi_{1}y_{1}-cos\varphi_ {1}z_{1}\\ &\cdots&\\ sin\theta_{N}x_{N}-cos\theta_{N}y_{N}\\ cos\theta_{N}sin\varphi_{N}x_{N}+sin\theta_{N}sin\varphi_{N}y_{N}-cos\varphi_ {N}z_{N}\\ \end{array} \right]\in\mathbb{R}^{2N\times 1} \tag{34}\]  
In practical scenarios, due to hardware limitations such as antenna arrays, RF front-end, and baseband processors, not all base stations can accurately measure angles. This section discusses situations where only one channel of accurate angle information can be obtained and situations where multiple channels of accurate angle information can be obtained. When only one accurate angle observation can be obtained through measurement, its corresponding base station is used as the differential station of the TDOA. The distance difference formula is rewritten as \(d_{i,1}+\|\mathbf{u}-\mathbf{s}_{1}\|=\|\mathbf{u}-\mathbf{s}_{i}\|\), and the square of both sides is obtained:  
\[d_{i,1}{}^{2}+\|\mathbf{s}_{1}\|^{2}-\|\mathbf{s}_{i}\|^{2}=2(\mathbf{s}_{1}-\mathbf{s}_{i}) \mathbf{u}^{T}-2d_{i,1}d_{1} \tag{35}\]  
Among them, \(d_{1}\) and \(\mathbf{u}\) are unknown variables with correlation. The commonly used methods for solving such functions are two-step weighted least squares and semi definite relaxation algorithms that simultaneously estimate unknown parameters and auxiliary variables. These methods have high complexity.