## 3 Distributed Cubature Kalman Filter
### Distributed Cubature Kalman Filter
We proposed a distributed cubature Kalman filter, which does not need linearization and can achieve better performance. On the other hand, we provide a structure for the distributed Bayesian filter with the help of K-L divergence, which could approach the centralized solution more efficiently compared with the one in Euclidean space._

**Remark 6**: _An important feature of proposed algorithm is that the only used global information is the number of sensors \(N\), which is suitable in application. In [26], the author proposed a distributed cubature information filter (DCIF) for the cooperative tracking space object, where the number of sensor \(N\) and the maximum degree \(\Delta_{\max}\) of the network are needed. However, in practice, \(\Delta_{\max}\) may change and is not easy to obtain in time._

## 4 Numerical Simulations

In this section, we illustrate the effectiveness of the proposed DCKF for the space object tracking problem, where the scenario is shown in \[X_{i,t,k-1}= S_{i,k-1}\xi_{t}+\hat{x}_{i,k-1},\ \ X_{i,t,k}^{*}=f(X_{i,t,k-1}),\ \ \hat{x}_{i,k}=\frac{1}{m}\sum_{t=1}^{m}X_{i,t,k}^{*},\]1 A distributed satellite system is observing a non-cooperative object, where the bearing-only measurement information is considered. The number of observation satellites is \(N=6\). In what follows, we first give the dynamics of the space object and measurement mapping by a distributed satellite system, then we solve the cooperative space object tracking problem by the proposed DCKF. ### Dynamics of Space Target

The dynamics of space object can be described as follows,

\[X_{i,t,k-1}= S_{i,k-1}\xi_{t}+\hat{x}_{i,k-1},\ \ X_{i,t,k}^{*}=f(X_{i,t,k-1}),\ \ \hat{x}_{i,k}=\frac{1}{m}\sum_{t=1}^{m}X_{i,t,k}^{*},\]2

where \(r=[rx\;ry\;rz]^{\top}\) represents the position of the object in the Earth-centered inertial (ECI) coordinate frame, \(\mu\) is the gravitational constant, \(J_{2}\) stands for perturbations and \(w\) is Gaussian noise with zero mean. Denote \(x=[rx,ry,rz,rx,ry,rz]^{\top}\) as the state variables; we can rewrite (46) in state-space description as follows,

\[\dot{x}= \begin{bmatrix}r\dot{x}\\ r\dot{y}\\ \dot{r}\dot{z}\\ -\frac{\mu}{|r|^{3}}r\dot{x}+J_{2,1}+w_{1}\\ -\frac{\mu}{|r|^{3}}r\dot{y}+J_{2,2}+w_{2}\\ -\frac{\mu}{|r|^{3}}r\dot{z}+J_{2,3}+w_{3}\end{bmatrix} =f(x), \tag{47}\]

Figure 1: Scenario of cooperative space object tacking. where \(w=[w_{1},w_{2},w_{3}]^{\top}\) is process noise, and the perturbation has the following form,

\[J_{2}=\frac{3}{2}a_{I}(\frac{E_{r}}{\parallel r\parallel^{3}})^{2}\frac{\mu}{ \parallel r\parallel^{3}} \begin{bmatrix}rx(5\frac{r^{2}}{|r|^{2}}-1)\\ ry(5\frac{r^{2}}{|r|^{2}}-1)\\ rz(5\frac{r^{2}}{|r|^{2}}-3)\end{bmatrix} , \tag{48}\]

where \(E_{r}\) is the Earth radius, \(a_{I}\approx 0.00108263\). Dynamics (47) is a continuous time model, thus (47) should be discretized in order to apply the EKF algorithm. Let \(T=t_{k+1}-t_{k}\) be the sampling period, then the discrete model of (47) is described in the following,

\[x_{k+1}=x_{k}+\int_{t_{k}}^{t_{k+1}}f(x(t))dt. \tag{49}\]

When \(T=t_{k+1}-t_{k}\) is sufficient small, the Taylor expansion of \(f(x(t))\) is:

\[f(x(t))\approx f(x_{k})+A(x_{k})f(x_{k})(t-t_{k}), \tag{50}\]

where \(A(x_{k})\) has the following form:

\[A(x_{k})=\frac{\partial f(x_{k})}{\partial x}|_{t=t_{k}}= \begin{bmatrix} \frac{\partial f}{\partial x}&\frac{\partial f}{\partial x}\\ \frac{\partial f}{\partial x}&\frac{\partial f}{\partial x}\\ \frac{\partial f}{\partial x}&\frac{\partial f}{\partial x}\end{bmatrix} = \begin{bmatrix}0_{3\times 3}&I_{3}\\ A_{21}&0_{3\times 3}\end{bmatrix} . \tag{51}\]

Combine (49)-(51), the discrete model can be given as:

\[x_{k+1}=x_{k}+f(x_{k})T+A(x_{k})f(x_{k})\frac{T^{2}}{2}, \tag{52}\]

where \(x_{k}\in\mathbb{R}^{n_{x}}\) is the state that needs to be estimated. The discretized dynamics (52) is used in the prediction step of the EKF. It can be seen that the higher order terms are ignored in (50), which may enlarge the estimate errors.