# 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
To minimize the error of joint angle and distance calculation, it is necessary to minimize \(e^{T}\mathbf{e}\), that is:  
\[\min_{\mathbf{u}}\left(\mathbf{A}\mathbf{u}^{T}-\mathbf{\hat{h}}\right)^{T}\left(\mathbf{\hat{A}} \mathbf{u}^{T}-\mathbf{\hat{h}}\right) \tag{51}\]  
In the above optimization problem, there are three types of observations: the TDOA, direction angle, and pitch angle, all of which can be assumed to be independent, identically distributed, and zero mean Gaussian distributions. Since the TDOA is based on PRS estimation, angle information is based on SRS estimation, and distance and angle are measured using different techniques, it can be assumed that the observation errors of the TDOA and angle are independent and have different variances. But the direction angle and pitch angle affect each other during the search process, so they have correlation. Therefore, the variance of the error vector \(\mathbf{e}\) is not constant and there is correlation between the error terms, which does not satisfy the Gaussian Markov hypothesis. The covariance matrix of the error terms needs to be corrected. The traditional weighted least squares method can only adjust the values of diagonal elements through the weight matrix, ignoring the processing of data on non-diagonal elements, and cannot solve the problem of correlation between observations. In this section, the generalized least squares method is used for estimation, while solving the problems of heteroscedasticity and correlation. The estimated value of generalized least squares can be expressed as:  
\[\overline{u}=\left(\mathbf{\hat{A}}^{T}\mathbf{W}^{-1}\mathbf{\hat{A}}\right)^{-1}\mathbf{ \hat{A}}^{T}\mathbf{W}^{-1}\mathbf{\hat{h}} \tag{52}\]  
Among them, \(\mathbf{W}\) is the covariance matrix of error \(\mathbf{e}\). In actual measurement, error \(\mathbf{e}\) cannot intuitively reflect the relationship between measurement error and positioning. Therefore, the error vectors of the TDOA and angle observation should be taken as the research objects. Therefore, Equation (51) can be expanded by first-order Taylor expansion to obtain a pseudo linear equation:  
\[\mathbf{\hat{A}}\mathbf{u}^{T}-\mathbf{\hat{h}}\cong\mathbf{G}\mathbf{n}^{T} \tag{53}\]Among them, \(\mathbf{n}\in\mathbb{R}^{(2N+M-1)\times 1}\) is the error vector of the joint positioning of the TDOA and AOA for mesoscale measurements, and \(\mathbf{G}\) is the partial derivative of the observation values corresponding to each error vector  
\[\mathbf{G}\mathbf{n}^{T}=\begin{cases} \begin{bmatrix}\mathbf{G}_{1}\\ \mathbf{G}_{2}\\ \mathbf{G}_{1}\\ \mathbf{G}_{3}\end{bmatrix} \mathbf{n}^{T},N=1\\  
Among them, \(\mathbf{W}=\left(E\left(\mathbf{G}\mathbf{n}^{T}\mathbf{n}\mathbf{G}^{T}\right)\right)^{-1}=\left(\mathbf{G} \mathbf{Q}_{\mathbf{m}}\mathbf{G}^{T}\right)^{-1}\), where \(\mathbf{Q}_{\mathbf{m}}\) is the covariance matrix of the observed noise \(\mathbf{n}\). In the calculation process, the weight matrix \(\mathbf{W}\) is unknown and depends on the terminal position calculated in the previous epoch. In the first calculation process, the identity matrix is used instead of \(\mathbf{W}\) to obtain the initial position of the terminal, and then the more accurate weighting matrix is iteratively calculated to obtain the final accurate solution. Due to the insensitivity of algorithm performance to the approximation of the weight matrix, accurate terminal location information can be obtained in just a few iterations, greatly reducing computational complexity. Compared with traditional nonlinear observation models, this chapter constructs a linear observation model by using angle information as an intermediate variable, which improves computational efficiency and numerical stability.