# Trajectory Optimisation for Cooperative Target Tracking with Passive Mobile Sensors
## 3 Reward Functions
### Expected RID  
The Renyi information divergence [21] is the information divergence for probability densities \(f_{1}(x)\) and \(f_{2}(x)\), for \(\alpha\geq 0\),  
\[I_{\alpha}(f_{1},\ f_{2})=\frac{1}{\alpha-1}\ln\left[\int f_{1}(x)^{\alpha}f_ {2}(x)^{\alpha-1}dx\right]. \tag{6}\]  
For the underlying system with one-step ahead sensor scheduling, \(f_{1}=p(\mathbf{x}_{k+1}|\mathbf{\beta}_{1:k+1})\) signifies the posterior density at \(k+1\) and \(f_{2}=p(\mathbf{x}_{k+1}|\mathbf{\beta}_{1:k})\) signifies the predicted density at \(k+1\), where \(\mathbf{\beta}_{k+1}\) is the future measurement at time \(k+1\) given by (2). Thus, (6) becomes  
\[I_{\alpha}\big{(}p(\mathbf{x}_{k+1}|\mathbf{\beta}_{1:k+1}),\ p( \mathbf{x}_{k+1}|\mathbf{\beta}_{1:k})\big{)}\] \[=\frac{1}{\alpha-1}\ln\left[\int p(\mathbf{x}_{k+1}|\mathbf{\beta}_{1 :k+1})^{\alpha}p(\mathbf{x}_{k+1}|\mathbf{\beta}_{1:k})^{\alpha-1}d\mathbf{x}_{k+1 }\right]. \tag{7}\]The evaluation of the integral (7) requires the future measurement \(\mathbf{\beta}_{k+1}\), which may be obtained numerically via Monte Carlo simulations [15, 16]. A statistical solution in a closed-form can be derived for the linear Gaussian case [14], where the dependence on the future measurement \(\mathbf{\beta}_{k+1}\) is removed by taking the expectation w.r.t. \(p(\mathbf{\beta}_{k+1}|\mathbf{\beta}_{1:k})\). The linear Gaussian case is described by the transition  
\[\mathbf{x}_{k}|\mathbf{x}_{k-1}\sim\mathcal{N}(\mathbf{x}_{k};\mathbf{F}\mathbf{x} _{k-1},\mathbf{Q}), \tag{8}\]  
and measurement  
\[\mathbf{y}_{k}|\mathbf{x}_{k}\sim\mathcal{N}(\mathbf{y}_{k};\,\mathbf{H}\mathbf{x }_{k},\mathbf{R}). \tag{9}\]  
The predicted measurement distribution is then given by  
\[\mathbf{y}_{k-1}|\mathbf{y}_{k}\sim\mathcal{N}(\mathbf{y}_{k+1};\,\hat{ \mathbf{y}}_{k+1},\mathbf{S}_{k+1}). \tag{10}\]  
where \(\mathbf{S}_{k+1}=\mathbf{H}\mathbf{P}_{k|k-1}\mathbf{H}^{\prime}+\mathbf{R}\). Assume that the sequence of measurements \(\mathbf{y}_{1:k}\) has been collected so that the posterior density is given by  
\[p(\mathbf{x}_{k}|\mathbf{y}_{1:k})=\mathcal{N}(\mathbf{x}_{k};\hat{\mathbf{x} }_{k|k},\mathbf{P}_{k|k}), \tag{11}\]  
where \(\mathbf{P}_{k|k}\) is the updated covariance at \(k\). For the properties of the Kalman filter, the divergence between the prior PDF and posterior PDF at \(k+1\) is given by  
\[I_{\alpha}= \frac{1}{\alpha-1}\ln\left[\frac{|2\pi R/\alpha|^{1/2}\mathcal{N }(\mathbf{y}_{k+1};\hat{\mathbf{y}}_{k+1},\mathbf{S}_{k+1}+\mathbf{R}(1-\alpha)/ \alpha)}{|2\pi R|^{\alpha/2}\mathcal{N}(\mathbf{y}_{k+1};\hat{\mathbf{y}}_{k +1},\mathbf{S}_{k+1})^{\alpha}}\right]\] \[= \frac{0.5}{\alpha-1}\ln\biggl{(}\frac{|\mathbf{R}|^{1-\alpha}|\mathbf{ S}_{k+1}|^{\alpha}}{|\mathbf{\alpha}\mathbf{S}_{k+1}+(1-\alpha)\mathbf{R}|}\biggr{)}\] \[+ 0.5\int(\mathbf{y}_{k+1}-\hat{\mathbf{y}}_{k+1})^{\prime}[\mathbf{I} -\mathbf{S}_{k+1}^{-1}(\alpha\mathbf{R}^{-1}+(1-\alpha)\mathbf{S}_{k+1}^{-1})^{-1}]\] \[I_{\alpha}\big{(}p(\mathbf{x}_{k+1}|\mathbf{\beta}_{1:k+1}),\ p( \mathbf{x}_{k+1}|\mathbf{\beta}_{1:k})\big{)}\]0  
Taking the expectation of (12) with respect to \(p(\mathbf{y}_{k+1}|\mathbf{y}_{1:k})\) and using the formula  
\[I_{\alpha}\big{(}p(\mathbf{x}_{k+1}|\mathbf{\beta}_{1:k+1}),\ p( \mathbf{x}_{k+1}|\mathbf{\beta}_{1:k})\big{)}\]1 \[I_{\alpha}\big{(}p(\mathbf{x}_{k+1}|\mathbf{\beta}_{1:k+1}),\ p( \mathbf{x}_{k+1}|\mathbf{\beta}_{1:k})\big{)}\]2  
we obtain the closed-form Expected RID function [22]:  
\[I_{\alpha}\big{(}p(\mathbf{x}_{k+1}|\mathbf{\beta}_{1:k+1}),\ p( \mathbf{x}_{k+1}|\mathbf{\beta}_{1:k})\big{)}\]3 \[I_{\alpha}\big{(}p(\mathbf{x}_{k+1}|\mathbf{\beta}_{1:k+1}),\ p( \mathbf{x}_{k+1}|\mathbf{\beta}_{1:k})\big{)}\]4  
where \(\mathbf{\Psi}_{k+1}=\mathbf{R}^{-1}\mathbf{H}\mathbf{P}_{k+1|k}\mathbf{H}^{\prime}\) and \(\mathbf{P}_{k+1|k}\) is the covariance matrix associated with the predicted state estimate \(\hat{\mathbf{x}}_{k+1|k}\).