## 3 Dynamic Error Modeling and Uncertainty Analysis  
### Dynamic Error Modeling  
In Section 2.2 we have analyzed the dynamic error mechanism. The measurement error in scanning angles causes the coordinate error of the moving receiver. The error propagation model is included in the following equations \(F(\mathbf{P},\mathbf{\theta})=0\):  
Figure 5: Transmitter model and speed decomposition. \[\left\{\begin{array}{l}F_{1}=\left[ \begin{array}{ccccc}a_{11}&b_{11}&c_{11}&d_{11} \end{array} \right]\left[ \begin{array}{ccccc}\cos\theta_{11}&-\sin\theta_{11}&0 &0\\ \sin\theta_{11}&\cos\theta_{11}&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array} \right]\left[ \begin{array}{ccccc}R_{1}&T_{1}\\ 0&1\end{array} \right]\left[ \begin{array}{ccccc}x_{g}&y_{g}&z_{g}&1\end{array}  \right]^{T}=0\\ F_{2}=\left[ \begin{array}{ccccc}a_{12}&b_{12}&c_{12}&d_{12}\end{array}  \right]\left[ \begin{array}{ccccc}\cos\theta_{12}&-\sin\theta_{12}&0&0\\ \sin\theta_{12}&\cos\theta_{12}&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array} \right]\left[ \begin{array}{ccccc}R_{1}&T_{1}\\ 0&1\end{array} \right]\left[ \begin{array}{ccccc}x_{g}&y_{g}&z_{g}&1\end{array}  \right]^{T}=0\\ F_{2n-1}=\left[ \begin{array}{ccccc}a_{n1}&b_{n1}&c_{n1}&d_{n1}\end{array}  \right]\left[ \begin{array}{ccccc}\cos\theta_{n1}&-\sin\theta_{n1}&0&0\\ \sin\theta_{n1}&\cos\theta_{n1}&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array} \right]\left[ \begin{array}{ccccc}R_{n}&T_{n}\\ 0&1\end{array} \right]\left[ \begin{array}{ccccc}x_{g}&y_{g}&z_{g}&1\end{array}  \right]^{T}=0\\ F_{2n-2}=\left[ \begin{array}{ccccc}a_{n2}&b_{n2}&c_{n2}&d_{n2}\end{array}  \right]\left[ \begin{array}{ccccc}\cos\theta_{n2}&-\sin\theta_{n2}&0&0\\ \sin\theta_{n2}&\cos\theta_{n2}&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array} \right]\left[ \begin{array}{ccccc}R_{n}&T_{n}\\ 0&1\end{array} \right]\left[ \begin{array}{ccccc}x_{g}&y_{g}&z_{g}&1\end{array}  \right]^{T}=0\]  
The sources of coordinate uncertainty are scanning angles \(\theta_{1},\theta_{2},\cdots\theta_{2n-1},\theta_{2n}\). They are independent of each other and generally obey the normal distribution in Figure 6. The distribution center is determined by the dynamic error and the variance is determined by the static error. The covariance matrix \(\theta\), also seen as the input, can be expressed as:  
\[\theta=\left[ \begin{array}{ccc}u^{2}(\theta_{1})&&0\\ &\ddots&\\ 0&&u^{2}(\theta_{2n})\end{array} \right] \tag{8}\]  
According to the derivative rule for implicit functions [21],  
\[\frac{d\mathbf{P}}{d\theta}=-\left(\frac{d\mathbf{F}}{d\mathbf{P}}\right)^{+} \frac{d\mathbf{F}}{d\theta} \tag{9}\]  
\[P=\left[ \begin{array}{ccccc}x&y&z&1\end{array} \right]\text{ and it is the homogeneous coordinate of the receiver in the global coordinate system. }\frac{d\mathbf{F}}{d\mathbf{P}}\text{ and }\frac{d\mathbf{F}}{d\Theta}\text{ are the Jacobian matrices of }\mathbf{F}\text{ relative to }\mathbf{P}\text{ and }\theta\text{. }\left(\frac{d\mathbf{F}}{d\mathbf{P}}\right)^{+}\text{ is the generalized inverse matrix of }\frac{d\mathbf{F}}{d\mathbf{P}}\text{. }\]  
Figure 6: Scanning angle error distribution. In simulations we found that the numerical stability of \(\theta\) is unsatisfactory and it is highly sensitive to round-off error in programming, so the other algorithm is tried which depends on the following steps [22]:  
1. Divide \(\theta\) into a lower triangular matrix \(\lambda_{C}\) and its transposed matrix: \(\theta=\lambda_{C}^{T}\lambda_{C}\). 2. Divide \(\frac{d\mathbf{F}}{d\theta}\) into the product of an orthogonal matrix \(\mathbf{Q}_{C}\) and an upper triangular matrix \(\mathbf{W}_{C}\): \(\frac{d\mathbf{F}}{d\theta}=\mathbf{Q}_{C}\mathbf{W}_{C}\). 3. Divide \(\frac{d\mathbf{F}}{d\theta}\) into the product of a lower triangular matrix \(\mathbf{L}_{P}\) and an upper triangular matrix \(\mathbf{W}_{P}\): \(\frac{d\mathbf{F}}{d\mathbf{P}}=\mathbf{L}_{P}\mathbf{W}_{P}\). 4.