# Cramer-Rao Lower Bounds on 3D Position and Orientation Estimation in Distributed Ranging Systems
## 3 Bounds on Position Estimation
### Three-Dimensional CRLB on Position  
Reconsider Equation (9) with the 3D definitions \(d_{i}=\sqrt{(x-x_{i})^{2}+(y-y_{i})^{2}+(z-z_{i})^{2}}\), \(g_{i}=[x_{i},y_{i},z_{i}]\), and \(h=[x,y,z]\). The FIM now takes the form  
\[\mathbf{I}(h)=\frac{1}{c^{2}\sigma_{\mathrm{T}}^{2}}\begin{bmatrix}\sum_{i=1}^ {N}\frac{(x-x_{i})^{2}}{d_{i}^{2}}&\sum_{i=1}^{N}\frac{(x-x_{i})(y-y_{i})}{d_{i }^{2}}&\sum_{i=1}^{N}\frac{(x-x_{i})(z-z_{i})}{d_{i}^{2}}\\ \sum_{i=1}^{N}\frac{(x-x_{i}(y-y_{i})}{d_{i}^{2}}&\sum_{i=1}^{N}\frac{(y-y_{i}) ^{2}}{d_{i}^{2}}&\sum_{i=1}^{N}\frac{(y-y_{i})^{2}(z-z_{i})}{d_{i}^{2}}\\ \sum_{i=1}^{N}\frac{(x-x_{i})(z-z_{i})}{d_{i}^{2}}&\sum_{i=1}^{N}\frac{(y-y_{i} )(z-z_{i})}{d_{i}^{2}}&\sum_{i=1}^{N}\frac{(z-z_{i})^{2}}{d_{i}^{2}}\end{bmatrix}. \tag{15}\]  
The CRLB is again expressed as \(\sigma_{h}^{2}\geq\mathbf{I}^{-1}(h)=\operatorname{adj}\{\mathbf{I}(h)\}/| \mathbf{I}(h)|\), where the adjuate now takes the form \(\operatorname{adj}\{\mathbf{I}(h)\}=\frac{2}{(c^{2}\sigma_{\mathrm{T}}^{2})^{ 2}}\mathbf{A}\), where  
\[\mathbf{A}=\begin{bmatrix}\sum_{i=1}^{N}\sum_{j=1}^{N}\frac{V_{ij}(y,z)^{2}}{( d_{i},d_{i})^{2}}&\sum_{i=1}^{N}\sum_{j=1}^{N}\frac{V_{ij}(x,y)V_{ij}(z,y)}{(d_{i},d _{i})^{2}}&\sum_{i=1}^{N}\sum_{j=1}^{N}\frac{V_{ij}(x,y)V_{ij}(y,z)}{(d_{i},d _{i})^{2}}\\ \sum_{i=1}^{N}\sum_{j=1}^{N}\frac{V_{ij}(x,y)V_{ij}(z,y)}{(d_{i},d_{i})^{2}}& \sum_{i=1}^{N}\sum_{j=1}^{N}\frac{V_{ij}(x,z)^{2}}{(d_{i},d_{i})^{2}}&\sum_{i= 1}^{N}\sum_{j=1}^{N}\frac{V_{ij}(y,x)V_{ij}(x,z)}{(d_{i},d_{i})^{2}}\\ \sum_{i=1}^{N}\sum_{j=1}^{N}\frac{V_{ij}(x,y)V_{ij}(y,z)}{(d_{i},d_{i})^{2}}& \sum_{i=1}^{N}\sum_{j=1}^{N}\frac{V_{ij}(y,x)V_{ij}(x,z)}{(d_{i},d_{i})^{2}} \end{bmatrix}, \tag{16}\]  
and the determinant takes the form  
\[|\mathbf{I}(h)|=\frac{6}{(c^{2}\sigma_{\mathrm{T}}^{2})^{3}}\sum_{i=1}^{N} \sum_{j=1}^{N}\sum_{k=1}^{N}\Biggl{(}\frac{V_{ij,k}}{d_{i}\,d_{j}\,d_{k}} \Biggr{)}^{2}, \tag{17}\]  
where \(V_{i,j,k}\) is the volume of a tetrahedron formed by points \(h\), \(g_{i}\), \(g_{j}\), and \(g_{k}\), as depicted in Figure 4. When represented as the Cayley-Menger determinant, this takes the form  
\[V_{i,j,k}=\frac{1}{6}\Biggl{|}\begin{matrix}x-x_{i}&y-y_{i}&z-z_{i}\\ x-x_{j}&y-y_{j}&z-z_{j}\\ x-x_{k}&y-y_{k}&z-z_{k}\end{matrix}, \tag{18}\]  
Thus, the closed-form CRLB is written as  
\[\mathbf{I}(h)^{-1}=\frac{\operatorname{adj}\{\mathbf{I}(h)\}}{|\mathbf{I}(h) |}=\frac{c^{2}\sigma_{\mathrm{T}}^{2}}{3}\mathbf{A}\left[\sum_{i=1}^{N}\sum_{j= 1}^{N}\sum_{k=1}^{N}\left(\frac{V_{i,j,k}}{d_{i}d_{j}d_{k}}\right)^{2}\right]^ {-1} \tag{19}\]  
As in Equation (14), we write the \"position variance\" \(\sigma_{p}^{2}=\sigma_{\mathrm{T}}^{2}+\sigma_{\mathrm{T}}^{2}+\sigma_{ \mathrm{T}}^{2}\geq\operatorname{tr}\{\mathbf{I}^{-1}(h)\}\) as  
\[\sigma_{p}^{2}\geq\frac{c^{2}\sigma_{\mathrm{T}}^{2}}{3}\frac{\left[\sum_{i=1}^ {N}\sum_{j=1}^{N}\frac{V_{ij}(x,y)^{2}+V_{ij}(y,z)^{2}+V_{ij}(x,z)^{2}}{(d_{i},d _{j})^{2}}\right]}{\left[\sum_{i=1}^{N}\sum_{j=1}^{N}\sum_{k=1}^{N}\left(\frac {V_{ij,k}}{d_{i}d_{j}d_{k}}\right)^{2}\right]} \tag{20}\]  
where \(V_{i,j}(\cdot,\cdot)\) is the area of the triangle in three-dimensional Cartesian space with vertices \(h\), \(g_{i}\), and \(g_{j}\) when projected onto the \((\cdot,\cdot)\) plane, i.e.,  
\[V_{i,j}(x,y)=\frac{1}{2}\[PERSON]{|}\begin{matrix}x-x_{i}&y-y_{i}\\ x-x_{j}&y-y_{j}\end{matrix}\[PERSON]{|}\begin{matrix}V_{i,j}(y,z)=\frac{1}{2} \[PERSON]{|}\begin{matrix}y-y_{i}&z-z_{i}\\ y-y_{j}&z-z_{j}\end{matrix}\[PERSON]{|},\begin{matrix}V_{i,j}(x,z)=\frac{1}{2} \[PERSON]{|}\begin{matrix}x-x_{i}&z-z_{i}\\ x-x_{j}&z-z_{j}\end{matrix}\[PERSON]{|}\end{matrix}.\]  
To the best of our knowledge, this is the first closed-form, three-dimensional CRLB on position estimation in a TWR system.