# 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 this section, the relationship between base stations and terminals is constructed by enhancing the weight of angle observation information, it can be obtained that:  
\[\mathbf{u}-\mathbf{s}_{1}=d_{1}\mathbf{b_{1}} \tag{36}\]  
Among them, \(\mathbf{b_{1}}=[cos\theta_{1}cos\varphi_{1},sin\theta_{1}cos\varphi_{1},sin\varphi _{1}]\), is the unit vector of the terminal relative to the angle measuring base station, that is:  
\[{d_{i,1}}^{2}+\|\mathbf{s_{1}}\|^{2}-\|\mathbf{s_{i}}\|^{2}=2(\mathbf{s_{1}}-\mathbf{s_{i}}) \mathbf{u}^{T}-2d_{i,1}d_{1}\mathbf{b_{1}}\mathbf{b_{1}}^{T} \tag{37}\]  
\[{d_{i,1}}^{2}+\|\mathbf{s_{1}}\|^{2}-\|\mathbf{s_{i}}\|^{2}-2d_{i,1}\mathbf{b_{1}}\mathbf{s_{1 }}^{T}=2(\mathbf{s_{1}}-\mathbf{s_{i}}-d_{i,1}\mathbf{b_{1}})\mathbf{u}^{T} \tag{38}\]  
Linearize it as  
\[\mathbf{A_{2}}\mathbf{u}^{T}=\mathbf{h_{2}} \tag{39}\]  
\[\mathbf{A_{2}}= \begin{bmatrix}2(\mathbf{s_{1}}-\mathbf{s_{2}}-d_{2,1}\mathbf{b_{1}})\\ \ldots\\ 2(\mathbf{s_{1}}-\mathbf{s_{M}}-d_{M,1}\mathbf{b_{1}})\end{bmatrix} \in\mathbb{R}^{(M-1) \times 3} \tag{40}\]  
\[\mathbf{h_{2}}= \begin{bmatrix}{d_{2,1}}^{2}+\|\mathbf{s_{1}}\|^{2}-\|\mathbf{s_{2}}\|^{2}- 2d_{2,1}\mathbf{b_{1}}\mathbf{s_{1}}^{T}\\ \ldots\\ d_{M,1}^{2}+\|\mathbf{s_{1}}\|^{2}-\|\mathbf{s_{M}}\|^{2}-2d_{M,1}\mathbf{b_{1}}\mathbf{s_{1}} ^{T}\end{bmatrix} \in\mathbb{R}^{(M-1)\times 1} \tag{41}\]  
When the wireless signal propagation environment is good and there are no hardware limitations on multiple base stations, assuming that there are N base stations that can obtain accurate angle observations, we can obtain:  
\[\mathbf{u}-\mathbf{s}_{i}=d_{i}\mathbf{b_{i}},i=2,\ldots N \tag{42}\]  
Among them, \(\mathbf{b_{i}}=[cos\theta_{i}cos\varphi_{i},sin\theta_{i}cos\varphi_{i},sin\varphi _{i}]\). Since both \(\mathbf{b_{1}}\) and \(\mathbf{b_{i}}\) are unit norm vectors, it can be concluded that \(\mathbf{b_{i}}-\mathbf{b_{1}}\) and \(\mathbf{b_{i}}+\mathbf{b_{1}}\) are orthogonal to each other, resulting in:  
\[d_{i}(\mathbf{b_{i}}-\mathbf{b_{1}})(\mathbf{b_{i}}+\mathbf{b_{1}})^{T}=0 \tag{43}\]  
\[2\mathbf{u} = \mathbf{s_{1}}+d_{1}\mathbf{b_{1}}+\mathbf{s}_{i}+d_{i}\mathbf{b_{i}}=\mathbf{s_{1}}+ \mathbf{s}_{i}+d_{1}\mathbf{b_{1}}+\big{(}d_{1}+d_{i,1}\big{)}\mathbf{b_{i}} \tag{44}\] \[= \mathbf{s_{1}}+\mathbf{s_{i}}+d_{i,1}\mathbf{b_{i}}+d_{1}\mathbf{b_{1}}+d_{1}\mathbf{ b_{i}}\] \[2(\mathbf{b_{i}}-\mathbf{b_{1}})\mathbf{u}^{T}=(\mathbf{b_{i}}-\mathbf{b_{1}})(\mathbf{s_ {1}}+\mathbf{s}_{i}+d_{i,1}\mathbf{b_{i}})^{T} \tag{45}\]  
Linearize it as  
\[\mathbf{A_{3}}\mathbf{u}^{T}=\mathbf{h_{3}} \tag{46}\]  
\[\mathbf{A_{3}}= \begin{bmatrix}2(\mathbf{b_{2}}-\mathbf{b_{1}})\\ \ldots\\ 2(\mathbf{b_{N}}-\mathbf{b_{1}})\end{bmatrix} \in\mathbb{R}^{(N-1)\times 3} \tag{47}\]  
\[\mathbf{h_{3}}= \begin{bmatrix}(\mathbf{b_{2}}-\mathbf{b_{1}})(\mathbf{s_{1}}+\mathbf{s_{2}}+d_{2,1 }\mathbf{b_{2}})^{T}\\ \ldots\\ (\mathbf{b_{N}}-\mathbf{b_{1}})(\mathbf{s_{1}}+\mathbf{s}_{N}+d_{N,1}\mathbf{b_{N}})^{T}\end{bmatrix}  \in\mathbb{R}^{(N-1)\times 1} \tag{48}\]Integrating the above three linear equations yields:  
\[\mathbf{A}\mathbf{u}^{T}=\mathbf{h}= \begin{cases}\begin{bmatrix}\mathbf{A}_{1}\\ \mathbf{A}_{2}\\ \begin{bmatrix}\mathbf{A}_{1}\\ \mathbf{A}_{3}\end{bmatrix}\mathbf{u}^{T}=\begin{bmatrix}\mathbf{h}_{1}\\ \mathbf{h}_{2}\\ \begin{bmatrix}\mathbf{h}_{1}\\ \mathbf{h}_{3}\end{bmatrix},N=1\\ \end{bmatrix}\\ \end{cases}  \tag{49}\]  
In the above derivation, it is assumed that the measurement of angle and distance information is accurate. However, in practice, measurement errors are inevitable. The linear equation with errors can be expressed as:  
\[\mathbf{\hat{A}}\mathbf{u}^{T}-\mathbf{\hat{h}}=\mathbf{e} \tag{50}\]  
Among them, \(\mathbf{\hat{A}}\) and \(\mathbf{\hat{h}}\) respectively represent \(\mathbf{A}\) and \(\mathbf{h}\) with observation errors, and e represents the error between \(\mathbf{\hat{A}}\mathbf{u}^{T}\) and \(\mathbf{\hat{h}}\).