## 3 Distributed Cubature Kalman Filter
### Distributed Cubature Kalman Filter
The state of the object is completely represented by \(x(t)\in\mathbb{R}^{6}\), which includes its position and velocity. When we describe the satellite moving along an orbit, it is often represented in the form of six orbital parameters, i.e., the semi-major axis \(a_{h}\), the eccentricity \(e\), the inclination \(u\), the argument of perigee \(\gamma\), the longitude of the ascending node \(\Gamma\) and the mean anomaly \(m_{h}\). The nonlinear transformations that converts position and velocity into orbital elements can be found in [39]. ### Measurement Model

The dynamics (47) is measured by a distributed satellites system. In this example, we consider a satellite equipped with an optical sensor that can obtain the azimuth \(\alpha\) or elevation \(\beta\) of the object. Measurement mapping of azimuth \(\alpha\) and elevation \(\beta\) can be expressed as follows,

\[\alpha_{k}=\arctan(\frac{ry_{k}-r\dot{y}_{k}}{rx_{k}-r\dot{x}_{k} })+va_{k}, \tag{53}\] \[\beta_{k}=\arctan(\frac{rz_{k}-r\dot{z}_{k}}{\sqrt{(rx_{k}-r\dot{x }_{k})^{2}+(ry_{k}-r\dot{y}_{k})^{2}}})+vb_{k}, \tag{54}\]

where \([r\dot{x}_{k},r\dot{y}_{k},\dot{r}z_{k}]^{\top}\) is the position of the satellite at time \(k\) and \(va_{k}\) and \(vb_{k}\) are measurement noise at time \(k\). In [26], the authors assume that the measurement equation of different satellites is the same. However, in practice, this does not always hold. In this example, we assume that a satellite can



### Simulation Results

The trajectories of observation satellites and the true object are generated by the six orbital parameters as shown in Table 2. We consider the dynamics (47); the state of the object is \(x=[rx,ry,rz,\dot{rx},\dot{r},\dot{r},\dot{r},\dot{r}]^{\top}\). The noise variance of the process is \(Q_{k}=diag\{10^{-4},10^{-4},10^{-4},10^{-6},10^{-6}\}\). The initial state of each agent was chosen randomly from \(\mathbb{N}(x_{0},P_{0})\), where \(x_{0}=[x_{p0}^{\top}\quad x_{v0}^{\top}]^{\top}\) is the true initial state of the object, \(P_{0}=diag[(10^{10}\ 10^{10}\ 10^{10}\ 100^{2}\ 100^{2}\ 100^{2})]\), and \(x_{p0}=[4.36\times 10^{6}\ \text{m}\ \ 3.13\times 10^{6}\ \text{m}\ \ 6.61\times 10^{6}\ \text{m}]^{\top}\), \(x_{v0}=[-5505.2\ \text{m/s}\ -207.5\ \text{m/s}\ 3954.8\ \text{m/s}]^{\top}\). Numerical simulations are conducted through Monte Carlo experiment, in which 50 Monte Carlo trials are done for each tracking algorithm. The total of the mean square estimation errors (MSE) is considered, which is widely used to indicate the performance of estimates, which is defined as:

\[MSE_{k}=\frac{1}{N}\sum_{i=1}^{N}\left[\frac{1}{50}\sum_{j=1}^{50}(\hat{x}_{i, k}^{(j)}-x_{k})^{T}(\hat{x}_{i,k}^{(j)}-x_{k})\right]. \tag{56}\]

The performance of mean square estimation for each satellite is defined as:

\[MSE_{i,k}=\frac{1}{50}\sum_{j=1}^{50}(\hat{x}_{i,k}^{(j)}-x_{k})^{T}(\hat{x}_{ i,k}^{(j)}-x_{k}). \tag{57}\]

The Runge-Kutta method is used to generate prediction \(\hat{x}_{i,k}\) in DCKF. Namely, the solution of nonlinear propagation \(X_{i,k}^{*}=f(X_{i,k-1})\) is computed by the Runge-Kutta method. Simulation Case 1: We test the performance of the proposed DCKF, where the communication topology is fixed as shown in Figure 4. We assume each satellite can obtain both azimuth \(\alpha\) and elevation \(\beta\), where the measurement mapping is defined in (53) and (54). The measurement noise variances are generated by \(R_{i}=i\cdot diag\{0.001^{2}\ 0.001^{2}\},i=1,\dots,N\). The number of consensus is \(S=1\). For Simulation Case 1, the comparison of MSE curves of different satellites is shown in Figure 5. The filtering precision and stability of the proposed DCKF for different satell