# Angle-Only Cooperative Orbit Determination Considering Attitude Uncertainty
## 3 Attitude and Orbit Determination Method with Angle-Only Measurements
### Case II: The Orbit of \(S_{1}\) Is Known, and the Attitude of \(S_{1}\) Is Unchanged
 \tag{45}\]  
where:  
\[\frac{\partial}{\partial\mathbf{r}_{1}}\bigg{(}\frac{\mathbf{r}_{2}-\mathbf{r}_{1}}{\left\| \mathbf{r}_{2}-\mathbf{r}_{1}\right\|}\bigg{)}=-\frac{1}{\left\|\mathbf{r}_{2}-\mathbf{r}_{1} \right\|}\bigg{[}\mathbf{I}_{3\times 3}-\frac{(\mathbf{r}_{2}-\mathbf{r}_{1})(\mathbf{r}_{2}-\mathbf{r}_{1})^{ \mathrm{T}}}{\left\|\mathbf{r}_{2}-\mathbf{r}_{1}\right\|^{2}}\bigg{]} \tag{46}\]  
\[\frac{\partial}{\partial\mathbf{v}_{1}}\bigg{(}\frac{\mathbf{r}_{2}-\mathbf{r}_{1}}{\left\| \mathbf{r}_{2}-\mathbf{r}_{1}\right\|}\bigg{)}=\mathbf{0}_{3\times 3} \tag{47}\]  
The STM \(\mathbf{\Phi}(t_{k},t_{0})\) in this case has the following format:  
\[\mathbf{\[PERSON]}(t_{k},t_{0})= \begin{bmatrix}\mathbf{\[PERSON]}_{r_{1}r_{1}}(t_{k},t_{0})&\mathbf{ \[PERSON]}_{r_{1}v_{1}}(t_{k},t_{0})&\mathbf{0}_{3\times 3}&\mathbf{0}_{3\times 3}&\mathbf{0}_{3 \times 4}\\ \mathbf{\[PERSON]}_{v_{1}r_{1}}(t_{k},t_{0})&\mathbf{\[PERSON]}_{v_{1}p_{1}}(t_{k},t_{0})&\mathbf{0}_{3 \times 3}&\mathbf{0}_{3\times 3}&\mathbf{0}_{3\times 4}\\ \mathbf{0}_{3\times 3}&\mathbf{0}_{3\times 3}&\mathbf{\[PERSON]}_{r_{2}r_{2}}(t_{k},t_{0})&\mathbf{\[PERSON]}_{r_{2}v_{2} }(t_{k},t_{0})&\mathbf{0}_{3\times 4}\\ \mathbf{0}_{3\times 3}&\mathbf{0}_{3\times 3}&\mathbf{\[PERSON]}_{v_{2}r_{2}}(t_{k},t_{0})&\mathbf{\[PERSON]}_{v_{2}v_{2} }(t_{k},t_{0})&\mathbf{0}_{3\times 4}\\ \mathbf{0}_{4\times 3}&\mathbf{0}_{4\times 3}&\mathbf{0}_{4\times 3}&\mathbf{0}_{4\times 3}&\mathbf{1}_{4 \times 4}\end{bmatrix}  \tag{48}\]  
Case IV: Both the Orbits of \(S_{1}\) and \(S_{2}\) Are Unknown, and the Angular Velocity of \(S_{1}\) Is Known  
In this case, the attitude of \(S_{1}\) is unknown and changed. For spacecraft \(S_{1}\), the angular velocity \(\mathbf{\omega}=\left[\omega_{x}\ \omega_{y}\ \omega_{z}\right]^{\mathrm{T}}\) is foreknown (assume that \(\omega_{x}\), \(\omega_{y}\) and \(\ \omega_{z}\) are constant and satisfy \(\omega_{x}^{2}+\omega_{y}^{2}+\omega_{z}^{2}\
eq 0\)). Hence, the state vector to be estimated is the same as that of case III, with the form of Equation (43). The state model is written as:  
\[\dot{\mathbf{X}}=\mathbf{F}(\mathbf{X})=\left[\dot{\mathbf{X}}_{1}^{\mathrm{T}},\dot{\mathbf{X}}_{2}^ {\mathrm{T}},\dot{\mathbf{q}}^{\mathrm{T}}\right]^{\mathrm{T}}=\left[\mathbf{f}(\mathbf{X} _{1})^{\mathrm{T}},\mathbf{f}(\mathbf{X}_{1})^{\mathrm{T}},\mathbf{g}_{1}(\mathbf{q})^{ \mathrm{T}}\right]^{\mathrm{T}}=\left[\mathbf{f}(\mathbf{X}_{1})^{\mathrm{T}},\mathbf{f} (\mathbf{X}_{1})^{\mathrm{T}},\mathbf{q}^{\mathrm{T}}\mathbf{\Omega}^{\mathrm{T}}(\mathbf{ \omega})/2\ \right]^{\mathrm{T}} \tag{49}\]  
The STM is in the form of:  
\[\mathbf{\Phi}(t_{k},t_{0})=\begin{bmatrix}\mathbf{\Phi}_{r_{1}r_{1}}(t_{k},t_{0})&\mathbf{ \Phi}_{r_{1}v_{1}}(  
The initial position and velocity errors (if unknown) are set to 10 km and \(10^{-3}\) km/s, respectively. The initial attitude of spacecraft \(S_{1}\) is set to be \(\boldsymbol{q}_{13}=\left[0,0,0\right]^{\mathrm{T}}\) and \(q_{0}=\sqrt{1-\boldsymbol{q}_{13}^{\mathrm{T}}\boldsymbol{q}_{13}}=1\). For all the cases above, the initial estimation of the quaternion is given by:  
\[\boldsymbol{\hat{q}}=\left[\cos\frac{\Theta}{2}\,\frac{\sqrt{3}}{3}\sin\frac{ \Theta}{2}\,\frac{\sqrt{3}}{3}\sin\frac{\Theta}{2}\,\frac{\sqrt{3}}{3}\sin \frac{\Theta}{2}\right]^{\mathrm{T}} \tag{52}\]  
where \(\Theta=5^{\circ}\). The initial angular velocity of spacecraft \(S_{1}\), if existing (i.e., for case IV and case V), is set to be \(\boldsymbol{\omega}=\left[0.01,0.01,0.01\right]^{\mathrm{T}}\,\circ/s\). The initial estimation of angular velocity is considered as \(\hat{\boldsymbol{\omega}}=\left[0.09,0.09,0.09\right]^{\mathrm{T}}\,\circ/s\) if \(\boldsymbol{\omega}\) is to be estimated (only for case V).