# Cramer-Rao Lower Bounds on 3D Position and Orientation Estimation in Distributed Ranging Systems
## 4 Bounds on Orientation Estimation
### Three-Dimensional CRLB on Orientation  
Assume the 3D observations \(\overline{b}_{k}\) follow a [PERSON] distribution [52] whose probability density function is given by  
\[p_{\overline{b}}(\overline{b};\overline{v},\theta,\kappa)=\left(\prod_{k=1}^{ M}C_{3}(\kappa_{k})\right)\exp\left\{\sum_{k=1}^{M}\kappa_{k}\,\overline{b}_{k}^{T} \mathbf{R}(\theta)\overline{v}_{k}\right\} \tag{31}\]  
where concentrations \(\kappa_{k}\geq 0\), means \(\|\bar{\mu}_{k}\|=1\), \(\bar{\mu}_{k}=\mathbf{R}(\theta)\overline{v}_{k}\), and \(C_{3}(\kappa)\) is a normalized constant  
\[C_{3}(\kappa)=\frac{K}{4\pi\sinh\kappa}=\frac{K}{4\pi(e^{\kappa}-e^{-\kappa})} \tag{32}\]  
The relative rotation between two 2D vectors can be sufficiently defined with a single angle, whereas at least three are required to do the same in a 3D space. For convenience, we chose Tait Bryan angles--roll (\(\alpha\)), pitch (\(\beta\)) and yaw (\(\gamma\))--to represent the relative attitude of the target \(H\) with respect to the observing node \(G\). Therefore, the rotation matrix \(\mathbf{R}(\theta)\) can be written as the product of three independent rotations by \(\alpha,\beta,\gamma\) about \(x,y,z\) axis, i.e., \(\mathbf{R}(\theta)=\mathbf{R}_{z}(\gamma)\)\(\mathbf{R}_{y}(\beta)\)\(\mathbf{R}_{x}(\alpha)\) where \(\theta=[\alpha,\beta,\gamma]\), which becomes  
\[\mathbf{R}(\theta)=\begin{bmatrix}\cos(\beta)\cos(\gamma)&\begin{matrix}-\cos( \alpha  
The FIM \(\mathbf{I}(\theta)=-\mathbb{E}\bigg{[}\frac{\partial^{2}}{\partial\theta^{2}}\text{ ln }p_{\overline{b}}(\overline{b};\overline{v},\theta,\kappa)\bigg{]}\) reduces to  
\[\mathbf{I}(\theta)= \begin{bmatrix}-\sum_{i=1}^{N}\kappa_{i}\,b_{i}^{\,T}\frac{ \partial^{2}}{\partial\alpha^{2}}\mathbf{R}(\theta)\tilde{v}_{i}&-\sum_{i=1} ^{N}\kappa_{i}\,b_{i}^{\,T}\frac{\partial^{2}}{\partial\alpha\partial\overline {\gamma}}\mathbf{R}(\theta)\tilde{v}_{i}&-\sum_{i=1}^{N}\kappa_{i}\,b_{i}^{\, T}\frac{\partial^{2}}{\partial\alpha\partial^{2}\gamma}\mathbf{R}(\theta) \tilde{v}_{i}\\ -\sum_{i=1}^{N}\kappa_{i}\,b_{i}^{\,T}\frac{\partial^{2}}{\partial\alpha\partial \overline{\gamma}}\mathbf{R}(\theta)\tilde{v}_{i}&-\sum_{i=1}^{N}\kappa_{i} \,b_{i}^{\,T}\frac{\partial^{2}}{\partial\beta\partial^{2}\gamma}\mathbf{R}( \theta)\tilde{v}_{i}&-\sum_{i=1}^{N}\kappa_{i}\,b_{i}^{\,T}\frac{\partial^{2 }}{\partial\beta\partial^{2}\gamma}\mathbf{R}(\theta)\tilde{v}_{i}\\ -\sum_{i=1}^{N}\kappa_{i}\,b_{i}^{\,T}\frac{\partial^{2}}{\partial\alpha \partial\overline{\gamma}}\mathbf{R}(\theta)\tilde{v}_{i}&-\sum_{i=1}^{N} \kappa_{i}\,b_{i}^{\,T}\frac{\partial^{2}}{\partial\beta\partial^{2}\gamma} \mathbf{R}(\theta)\tilde{v}_{i}&-\sum_{i=1}^{N}\kappa_{i}\,b_{i}^{\,T}\frac{ \partial^{2}}{\partial\beta\partial^{2}\gamma}\mathbf{R}(\theta)\tilde{v}_{i} \end{bmatrix} \]  
The CRLB is then computed by inverting the FIM,  
\[\mathbf{I}(\theta)^{-1}=\frac{\text{adj}\{\mathbf{I}(h)\}}{|\mathbf{I}(h)|}. \tag{34}\]  
It is important to note that the structure of the rotation matrix provides simplifications that can be leveraged to derive the CRLB in closed form. The partial derivatives in the rotation matrix significantly complicate the computation of Equation (34), which can be mitigated by the following simplifications. Let \(S[a,b,c]\) represent a skew-symmetric matrix of the form  
\[S[a,b,c]= \begin{bmatrix}0&-c&b\\ c&0&-a\\ -b&a&0\end{bmatrix} . \tag{35}\]  
Useful properties of \(S\) include \(\mathbf{S}[1,0,0]^{2}=\text{diag}[0,-1,-1]\), \(\mathbf{S}[0,1,0]^{2}=\text{diag}[-1,0,-1]\), and \(\mathbf{S}[0,0,1]^{2}=\text{diag}[-1,-1,0]\).