# Trajectory Optimisation for Cooperative Target Tracking with Passive Mobile Sensors
## 3 Reward Functions
### Determinant of Fisher Information Matrix  
In general, it is non-trivial to derive the FIM or its determinant in a closed-form. For this particular case, the measurement model (4) is independent of target velocity and the resulting FIM is only of rank 2. In consequence, we only need to consider the target position components in derivation of the FIM, and therefore the target state vector \(\mathbf{x}_{k}\) in derivation will denote just the target position components (\([x_{k},\,y_{k}]^{\prime}\)). A closed form expression for the determinant of the FIM is analytically tractable as follows.  
Suppose that the posterior at \(k-1\) is \(p(\mathbf{x}_{k-1}|\boldsymbol{\beta}_{1:k-1})\). Using the current sensor locations and a proposed action \(\boldsymbol{n}_{k}\), one can compute the one-step ahead sensor locations \((x_{k}^{s_{i}},y_{k}^{s_{i}})\) for \(i=1,2\). The FIM \(G_{k}\) at time \(t_{k}\) for the measurement (2) is defined to be  
\[G_{k}\stackrel{{\Delta}}{{=}}E\Big{[}\big{(}\
abla_{\mathbf{x}} \ln p(\boldsymbol{\beta}|\mathbf{x})\big{)}\big{(}\
abla_{\mathbf{x}}\ln p( \boldsymbol{\beta}|\mathbf{x})\big{)}^{\prime}\Big{]}_{\mathbf{x}=\mathbf{x}_{ k}}\, \tag{16}\]  
where \(\ln p(\boldsymbol{\beta}|\mathbf{x})\) is the log-likelihood, which is obtained from Equation (2), and \(G_{k}\) is a \(2\times 2\) matrix with entries  
\[G_{k}=\left[\begin{array}{cc}g_{11}&g_{12}\\ g_{21}&g_{22}\end{array}\right] \tag{17}\]  
Under the Gaussian measurement noise assumption, we have  
\[g_{ij}=\bigg{[}\frac{\partial\boldsymbol{h}(\mathbf{x}_{k})}{ \partial(\mathbf{x}_{k})_{i}}\bigg{]}^{T}\Sigma_{k}^{-1}\bigg{[}\frac{\partial \boldsymbol{h}(\mathbf{x}_{k})}{\partial(\mathbf{x}_{k})_{j}}\bigg{]}\] \[\qquad\qquad\qquad\qquad+\frac{1}{2}\text{tr}\bigg{[}\Sigma_{k}^ {-1}\frac{\partial\boldsymbol{\Sigma}(\mathbf{x})}{\partial(\mathbf{x}_{k})_ {i}}\Sigma_{k}^{-1}\frac{\partial\boldsymbol{\Sigma}(\mathbf{x})}{\partial( \mathbf{x}_{k})_{j}}\bigg{]}. \tag{18}\]  
Therefore,  
\[g_{11} =\sum_{i=1}^{2}\frac{1}{\sigma_{i}^{2}}\frac{(y_{k}-y_{k}^{s_{i}} )^{2}}{[(x_{k}-x_{k}^{s_{i}})^{2}+(y_{k}-y_{k}^{s_{i}})^{2}]^{2}}, \tag{19}\] \[g_{12} =g_{21}=\sum_{i=1}^{2}-\frac{1}{\sigma_{i}^{2}}\frac{(x_{k}-x_{k} ^{s_{i}})(y_{k}-y_{k}^{s_{i}})}{[(x_{k}-x_{k}^{s_{i}})^{2}+(y_{k}-y_{k}^{s_{i} })^{2}]^{2}},\] (20) \[g_{22} =\sum_{i=1}^{2}\frac{1}{\sigma_{i}^{2}}\frac{(x_{k}-x_{k}^{s_{i}} )^{2}}{[(x_{k}-x_{k}^{s_{i}})^{2}+(y_{k}-y_{k}^{s_{i}})^{2}]^{2}}. \tag{21}\]  
The determinant of \(G_{k}\) is a function of \(\sigma_{1},\sigma_{2},\mathbf{x}_{k},\mathbf{x}_{k}^{s_{1}}\) and \(\mathbf{x}_{k}^{s_{2}}\). From the definition and using (19)-(21), we have  
\[\text{Det}\{G_{k}\}\stackrel{{\Delta}}{{=}}\frac{1} {\sigma_{1}^{2}\sigma_{2}^{2}}\] \[\times\frac{[(x_{k}-x^{s_{1}})(y_{k}-y^{s_{2}})-(x_{k}-x^{s_{2}}) (y_{k}-y^{s_{1}})]^{2}}{[(x_{k}-x^{s_{1}})^{2}+(y_{k}-y^{s_{1}})^{2}]^{2}[(x_{ k}-x^{s_{2}})^{2}+(y_{k}-y^{s_{2}})^{2}]^{2}}. \tag{22}\]  
Calculation of the FIM requires knowledge of the target location at future time \(k\), which in practice is approximated using the predicted target state \(\hat{\mathbf{x}}_{k}\) based on the point estimate from \(p(\mathbf{x}_{k-1}|\boldsymbol{\beta}_{1:k-1})\) and the system dynamics (1).