# Trajectory Optimisation for Cooperative Target Tracking with Passive Mobile Sensors
## 4 Geometric Interpretation on Reward Function  
The above argument on the selection of a reward function in cooperative sensing can be described using information geometry. It is well known that the FIM can be used as a metric tensor defining a Riemannian manifold, called the statistical manifold [25]. Riemannian geometrical concepts, thereby, can inform in problems such as ours. For instance, the Fisher Information distance, defined in terms of the Riemannian metric, can be used to compare parameterised distributions. For the problem at hand, the family of probability distributions \(S=\{p(\mathbf{\beta}|\mathbf{x})\}\), parameterised in the target location space \(\mathbf{x}\in\mathbb{R}^{2}\), forms a 2D statistical manifold where \(\mathbf{x}\) plays the role of a coordinate system of \(S\). The Fisher information distance (FID) between \(p(\mathbf{\beta}|\mathbf{x}(t_{1}))\) and \(p(\mathbf{\beta}|\mathbf{x}(t_{2}))\) is defined as the integral along the curve \(\mathbf{x}(t)\):  
\[\mathcal{D}_{F}\big{(}\mathbf{x}(t_{1}),\,\mathbf{x}(t_{2})\big{)} \stackrel{{\triangle}}{{=}}\] \[\min_{\mathbf{x}}\int_{t_{1}}^{t_{2}}\left(\sqrt{\left(\frac{d \mathbf{x}(t)}{dt}\right)^{T}\mathbf{G}\big{(}\mathbf{x}(t)\big{)}\left(\frac{d \mathbf{x}(t)}{dt}\right)}\right)dt, \tag{24}\]  
where the minimizing curve is a geodesic in the Riemannian manifold. In local coordinates, the geodesic equations are given by the Euler-Lagrange equations as [25]  
\[\frac{d^{2}\mathbf{x}_{l}}{dt^{2}}+\sum_{i=1}^{n}\sum_{j=1}^{n}\Gamma_{ij}^{k} \frac{d\mathbf{x}_{i}}{dt}\frac{d\mathbf{x}_{j}}{dt}=0,\qquad\quad l=1,2.\;\;n =2. \tag{25}\]  
For convenience, we use the subscript \(i\) of \(\mathbf{x}\) to represent its \(i\)th component and thus \(\mathbf{x}_{i},\;i=1,2\) are the coordinates of the curve \(\mathbf{x}(t),\Gamma_{ij}^{l},\,i=1,2\) are the _Christoffel symbols of the second kind_, and we can use the Levi-Civita connection coefficients,  
\[\Gamma_{ij}^{l}=\frac{1}{2}\sum_{k=1}^{2}g^{lk}\left(\frac{\partial g_{ik}}{ \partial\mathbf{x}_{j}}+\frac{\partial g_{jk}}{\partial\mathbf{x}_{i}}-\frac {\partial g_{ij}}{\partial\mathbf{x}_{k}}\right),\;\;\;i,j,l=1,2. \tag{26}\]  
where \([g^{lk}]\) signifies the inverse of \(\mathbf{G}=[g_{lk}]\) and [PERSON] notation for summation is used. The geodesic equations in (25) are ordinary differential equations for the coordinates \(\mathbf{x}_{i},\;i=1,2\). A unique solution \(\mathbf{x}(t)\) can be found for given initial conditions \(\mathbf{x}(0)\) and \(\dot{\mathbf{x}}\), which is analogous to an initial position \(\mathbf{x}(0)\) and the \"speed\" \(\mathbf{v}\in\mathbf{\mathcal{T}}_{\mathbf{x}}S\) in the sense of the classical mechanics, where \(\mathbf{\mathcal{T}}_{\mathbf{x}}\) denotes the tangent vector of \(S\) at \(\mathbf{x}\). Assume that a geodesic is projected onto the parameter space \(\mathbb{R}^{2}\) with a starting point \(\mathbf{x}(0)\) and a tangent vector \(\boldsymbol{\
u}\). The exponential map of the starting point is then defined as [26]  
\[\exp_{\boldsymbol{\
u}}[\mathbf{x}(0)]\stackrel{{\triangle}}{{=} }\boldsymbol{\Psi}\big{(}1;\,\mathbf{x}(0),\,\boldsymbol{\
u}\big{)}. \tag{27}\]  
where the notation \(\boldsymbol{\Psi}\big{(}t;\,\mathbf{x}(0),\,\boldsymbol{\
u}\big{)}\) is used to signify a geodesic with a starting point \(\mathbf{x}(0)\), a tangent vector \(\boldsymbol{\
u}\) and end point \(\mathbf{x}(t)\). It can be shown that the length along the geodesic between \(\mathbf{x}(0)\) and \(\boldsymbol{\Psi}\big{(}1;\,\mathbf{x}(0),\,\boldsymbol{\
u}\big{)}\) is \(|\boldsymbol{\
u}|\)[26, 27]. Thus, for a fixed \(\boldsymbol{\
u}\), the plot of geodesics along all directions at \(\mathbf{x}\) produces an FID circle centered at \(\mathbf{x}\) in \(S\). Intuitively, the projection of an equal FID circle from statistical manifold to the parameter space, which is generally not isotropic in magnitude, reflects the observability of the underlying sensors at given states.