# Angle-Only Cooperative Orbit Determination Considering Attitude Uncertainty
## 2 Attitude and Orbit Determination Model
### State Model  
Two spacecraft, defined as \(S_{1}\) and \(S_{2}\), are considered and shown in Figure 1. Assume that \(S_{1}\) is able to observe the line-of-sight angle between \(S_{1}\) and \(S_{2}\). Note that \(S_{1}\) is not considered as the attitude-reference spacecraft, which is different from recent works. The attitude-orbit determination developed in this paper aims to determine the absolute states (i.e., the absolute orbit of both \(S_{1}\) and \(S_{2}\), together with the absolute attitude of \(S_{2}\)) of the spacecraft system, which for Earth orbiting bodies are usually described in the Earth-centered inertial (ECI) frame (coordinate system \(O_{e}-X_{e}Y_{e}Z_{e}\) in Figure 1). The analysis developed in this paper builds primarily on the two-body dynamics of the spacecraft with Earth as the primary body. Define the state of the \(i\)-th spacecraft in the ECI frame as:  
\[\mathbf{X}_{i}=\left[\mathbf{r}_{i}^{\mathrm{T}},\mathbf{v}_{i}^{\mathrm{T}}\right]^{ \mathrm{T}}=\left[\mathbf{x}_{i},\mathbf{y}_{i},\mathbf{z}_{i},\mathbf{\dot{x}}_{i},\mathbf{\dot{y }}_{i},\mathbf{\dot{z}}_{i}\right]^{\mathrm{T}} \tag{1}\]  
The state equation of the spacecraft orbit under the two-body dynamics can be expressed in general form as:  
\[\dot{\mathbf{X}}_{i}=\mathbf{f}(\mathbf{X}_{i})=\left[ \begin{array}{c}\dot{\mathbf{r}}_{i}\\ \ddot{\mathbf{r}}_{i}\end{array} \right] \tag{2}\]  
In the case of ideal two-body dynamics (i.e., particle dynamics model), Equation (2) is given by:  
\[\ddot{\mathbf{r}}_{i}=-\frac{\mu_{e}}{\|\mathbf{r}_{i}\|}\mathbf{r}_{i} \tag{3}\]  
In the case of considering \(J_{2}\), \(J_{3}\), and \(J_{4}\) perturbations, Equation (3) is rewritten as:  
\[\ddot{\mathbf{r}}_{i}=-\frac{\mu_{e}}{\|\mathbf{r}_{i}\|}\mathbf{r}_{i}+\mathbf{a}_{I_{2}}+\bm {a}_{I_{3}}+\mathbf{a}_{I_{4}} \tag{4}\]  
Figure 1: Diagram of a two-spacecraft system. where \(\mu_{e}\) is the Earth's gravitational parameter, and \(\mathbf{a}_{l_{2}}\), \(\mathbf{a}_{l_{3}}\) and \(\mathbf{a}_{l_{4}}\) are the perturbation acceleration of \(J_{2}\), \(J_{3}\), and \(J_{4}\), which can be obtained by derivative of potential function with respect to position. The potential function is given by:  
\[U=\frac{\mu}{r}\left[1-J_{2}\bigg{(}\frac{R_{e}}{r}\bigg{)}^{2}P_{2}(\sin\varphi )-J_{3}\bigg{(}\frac{R_{e}}{r}\bigg{)}^{3}P_{3}(\sin\varphi)-J_{4}\bigg{(}\frac {R_{e}}{r}\bigg{)}^{4}P_{4}(\sin\varphi)\right] \tag{5}\]  
where \(\varphi\) is the latitude of the spacecraft's ground trace; \(P_{2}(\sin\varphi)=\frac{3}{2}(\sin\varphi)^{2}-\frac{1}{2}\), \(P_{3}(\sin\varphi)=\frac{5}{2}(\sin\varphi)^{3}-\frac{1}{2}(\sin\varphi)\) and \(P_{4}(\sin\varphi)=\frac{35}{8}(\sin\varphi)^{4}-\frac{30}{8}(\sin\varphi)^{2 }+\frac{3}{8}\). The attitude of spacecraft \(S_{1}\) is represented through the quaternion, defined by:  
\[\mathbf{q}=\left[ \begin{array}{c}q_{\mathbf{0}}\\ q_{13}\end{array} \right] \tag{6}\]  
where:  
\[q_{13}=\left[ \begin{array}{c}q_{1}\\ q_{2}\\ q_{3}\end{array} \right]=\hat{\mathbf{n}}\sin(\alpha/2)=\hat{n}\sin(\alpha/2) \tag{7}\]  
\[q_{0}=\cos(\alpha/2) \tag{8}\]  
where \(\hat{\mathbf{n}}\) is a unit vector corresponding to the principal axis of rotation and \(\alpha\) is the angle of rotation. The quaternion kinematics are derived through the spacecraft's angular velocity as follows:  
\[\dot{\mathbf{q}}=g_{1}(\mathbf{q})=\frac{1}{2}\mathbf{\Omega}(\mathbf{\omega})\mathbf{q} \tag{9}\]  
\[\dot{\mathbf{\omega}}=g_{2}(\mathbf{\omega})=\mathbf{J}^{-1}\big{(}\mathbf{\omega}^{\times}\bm {J}\mathbf{\omega}+\mathbf{M}\big{)} \tag{10}\]  
where \(\mathbf{\omega}=[\omega_{x}\ \omega_{y}\ \omega_{z}]^{\rm T}\) denotes the angular velocity, \(\mathbf{J}\) represents the moment of inertia of the spacecraft, \(\mathbf{M}\) is the sum of external moments, and matrix \(\mathbf{\Omega}(\mathbf{\omega})\) is defined as:  
\[\mathbf{\Omega}(\mathbf{\omega})=\left[ \begin{array}{cc}0&-\mathbf{\omega}^{\rm T}\\ \mathbf{\omega}&-\mathbf{\omega}^{\times}\end{array} \right],\omega^{\times}=\left[