# Cramer-Rao Lower Bounds on 3D Position and Orientation Estimation in Distributed Ranging Systems
## 3 Bounds on Position Estimation
### Geometric Dilution of Precision  
By inspection, the CRLB derived in Equation (20) only depends on the ToF estimator performance (\(\sigma_{\tau}\)) and the geometry of the ground nodes \(g_{i}\). In related estimation problems, the geometric portion of this equation is often referred to as \"geometric dilution of precision\", and effectively represents the penalty incurred by using a noisy estimate to further estimate other quantities, in this case using range to estimate position. We can rewrite Equation (20) in the form \(\sigma_{p}^{2}\geq c^{2}\sigma_{\tau}^{2}\)GDoP, where geometric dilution of precision (GDoP) is a unit-less quantity defined by  
\[\text{GDoP}=\sqrt{\frac{N}{2\sum_{i=1}^{N}\sum_{j=1}^{N}\Delta_{i,j}^{2}}}= \sqrt{\frac{2\,N}{\sum_{k=1}^{M}\,\sin^{2}(\gamma_{k})}}, \tag{24}\]  
where \(\Delta_{i,j}=\frac{1}{2}\sin(\gamma_{i,j})\) is the area of a triangle with unit sides and angle \(\angle\gamma_{i,j}\) between them and \(M=\begin{pmatrix}N\\ 2\end{pmatrix}\) is the number of unique triangles enclosed by \(\overrightarrow{hg_{i}}\) and \(\overrightarrow{hg_{j}}\)\(\forall\,i,j\in\{1,2,\cdots,N\}\). By inspection, observe that maximizing \(\mathcal{M}=\sum_{k=1}^{M}\,\sin^{2}(\gamma_{k})\) minimizes the CRLB for a given ranging precision \(\sigma_{\tau}\). This is a convex problem within the range \(\gamma_{k}\in(0,\pi]\) with unique unambiguous solutions \(\{\gamma_{1},\gamma_{2},\cdots,\gamma_{M}\}\) that yield the best performance for a given number of ground antennae \(N\). This is important because it directly informs the optimal antenna placement for a fixed number of antennas \(N\).  
In Figure 5, the positioning CRLB, from Equation (20) is plotted as a function of the ranging precision \(\sigma_{\tau}\) and the geometric dilution of precision from Equation (24). By reformulating the bound as a function of traditional metrics (ranging precision and GDoP) rather than abstract volumes, this formulation provides a visual reference of the trade space between the quantities that a design engineer actually cares about, namely, the ranging precision, the positioning precision, and the GDoP induced by the platform geometry.  
The ranging precision \(\sigma_{\tau}\) is primarily governed by the choice of ranging technology. If the design constraints limit the available locations of the antennas on an aircraft, then this curve can help inform decisions about which ranging technology is most appropriate to achieve a target positioning precision. In contrast, if the ranging precision is fixed either by design or operating conditions, this curve can instead inform decisions about the optimal placement of antennas on an aircraft or ground installation. Additional insight in this regard is also provided in [43]. Naturally, all of this information is contained in the original formulation presented in Equation (20), this figure is simply included as a visual reference that does not require any calculations.  
In Table 1, the best achievable GDoP value is enumerated as a function of the number of antennas \(N\) between three and eight. This particular positioning method requires at least three antennas to converge on a solution and experiences diminishing returns beyond eight. In Table 2, we include a qualitative description of GDoP values summarized from [51]. In many TWR and other positioning applications--particularly GPS--it is common to summarize the GDoP of an installation without expressly stating the actual geometry, so this information is included as a coarse reference.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**N** & **3** & **4** & **5** & **6** & **7** & **8** \\ \hline Best GDoP & 1.155 & 1.0 & 0.894 & 0.816 & 0.756 & 0.707 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Best achievable GDoP vs. number of antennas.