## 2 Problem Formulation  
### Models  
Consider \(N\) homogeneous cooperative robots, denoted as \(i\in\mathcal{V}\), performing collaborative self-localization and distributed tracking of a target of interest, denoted as \(t\). The dynamics of the cooperative robots and the target are expressed respectively with the following nonlinear process models:  
\[\mathbf{x}_{i,k+1} =f_{v}(\mathbf{x}_{i,k},\mathbf{u}_{i,k}), \tag{1}\] \[\mathbf{x}_{t,k+1} =f_{t}(\mathbf{x}_{t,k},\mathbf{w}_{t,k}), \tag{2}\]  
where \(\mathbf{x}_{i,k}\in\mathbb{R}^{n_{v}}\) and \(\mathbf{x}_{t,k}\in\mathbb{R}^{n_{t}}\) respectively denote the state of robot \(i\) and target \(t\) at time \(k\). \(\mathbf{u}_{i,k}\in\mathbb{R}^{n_{u}}\) is the control input of robot \(i\), which is assumed to be subject to a Gaussian distribution \(\mathbf{u}_{i,k}\sim\mathcal{N}\left(\bar{\mathbf{u}}_{i,k},\mathbf{Q}_{i}\right)\). \(\bar{\mathbf{u}}_{i}\) denotes the control command, and \(\mathbf{Q}_{i}\) is the control input covariance. \(\mathbf{w}_{t,k}\in\mathbb{R}^{n_{u}}\) is the process noise of the target and is assumed to be drawn from a zero-mean Gaussian distribution \(\mathbf{w}_{t,k}\sim\mathcal{N}\left(\mathbf{0},\mathbf{Q}_{t}\right)\). It is assumed that all robots \(i\in\mathcal{V}\) share the same nonlinear transformation \(f_{v}:\mathbb{R}^{n_{v}}\times\mathbb{R}^{n_{u}}\), and the moving target nonlinear transformation \(f_{t}:\mathbb{R}^{n_{t}}\times\mathbb{R}^{n_{u}}\) is known to all robots.  
In the cooperative localization and target tracking scenario, each robot \(i\) is able to measure three pieces of information: its absolute state and the relative pairwise measurements to neighboring robots and to the target. The measurements at time instance \(k\) are denoted respectively as \(\mathbf{z}_{i,k}^{a}\in\mathbb{R}^{n_{sa}}\), \(\mathbf{z}_{i,k}^{c}\in\mathbb{R}^{n_{sc}},j\in\mathcal{V}\backslash\{i\}\), and \(\mathbf{z}_{it,k}^{t}\in\mathbb{R}^{n_{sa}}\). The corresponding measurement functions are listed below:  
\[\mathbf{z}_{i,k}^{a} =h^{a}(\mathbf{x}_{i,k},\mathbf{v}_{i,k}^{a}), \tag{3}\] \[\mathbf{z}_{i,k}^{c} =h^{c}(\mathbf{x}_{i,k},\mathbf{x}_{j,k},\mathbf{v}_{i,k}^{c}),\] (4) \[\mathbf{z}_{it,k}^{t} =h^{t}(\mathbf{x}_{i,k},\mathbf{x}_{t,k},\mathbf{v}_{i,k}^{t}), \tag{5}\]  
where \(\mathbf{v}_{i,k}^{a},\mathbf{v}_{i,k}^{c}\) and \(\mathbf{v}_{i,k}^{t}\) are the measurement noise of the above three measurement processes and assumed to be drawn from zero-mean Gaussian distributions, i.e., \(\mathbf{v}_{i,k}^{a}\sim\mathcal{N}(\mathbf{0},\mathbf{R}_{i}^{a})\), \(\mathbf{v}_{i,k}^{c}\sim\mathcal{N}(\mathbf{0},\mathbf{R}_{i}^{c})\), and \(\mathbf{v}_{i,k}^{t}\sim\mathcal{N}(\mathbf{0},\mathbf{R}_{i}^{t})\).  
Defining the CLAT as a graph \(G(V,E)\), the node set \(V=\mathcal{V}\cup\{t\}\cup\{0\}\), and the special node \(0\) denotes the absolute position origin. The edge set is denoted as \(E\subseteq\mathcal{V}\times V\). For a robot \(i\in\mathcal{V}\), an edge \((i,0)\) denotes a robot that can access its own absolute position. In this paper, it is assumed that a subset of the robots \(\mathcal{V}\) can obtain the absolute measurement \(\mathbf{z}_{k}^{a}\). An edge \((i,t)\) indicates that a robot \(i\) is able to detect a target \(t\). Since the sensing range of an uncooperative sensor is limited, the availability of an uncooperative target measurement depends on the relative positions of the robot and the target. An edge \((i,j),j\in\mathcal{V}\backslash i\) denotes that a pairwise measurement between robots \(i\) and \(j\) is obtained. Similarly, a cooperative relative measurement is available when two robots are within the cooperative sensing range. Moreover, in this paper, it is assumed that whenever a relative measurement \(\mathbf{z}_{i,j,k}^{c}\) is taken, a communication link is established simultaneously between robots \(i\) and \(j\) so that they can share information.