# Motion Estimation of Non-Cooperative Space Objects Based on Monocular Sequence Images
## 2 Proposed Method
### Camera Translation and Rotation from Sequence Images  
A sequence image is constructed by observing the object with a monocular camera, and the same feature point in two adjacent image frames conforms to the epipolar constraint:  
\[\big{[}X_{c,i+1}\quad Y_{c,i+1}\quad Z_{c,i+1}\big{]}\cdot\Delta\widetilde{T}_{i }\cdot\Delta R_{i}\cdot\begin{bmatrix}X_{c,i}\\ Y_{c,i}\\ Z_{c,i}\end{bmatrix}=0 \tag{11}\]  
\[\text{where }\Delta\widetilde{T}_{i}=\begin{bmatrix}0&-T_{z,i}&T_{y,i}\\ T_{z,i}&0&-T_{x,i}\\ -T_{y,i}&T_{x,i}&0\end{bmatrix}\text{ and }\Delta T_{i}=\begin{bmatrix}T_{x,i}\\ T_{y,i}\\ T_{z,i}\end{bmatrix}.\]  
Bringing Equation (2) into the above equation, the following can be obtained:  
\[\big{[}u_{i+1}\quad v_{i+1}\quad 1\big{]}\cdot A^{-T}\cdot\Delta\widetilde{T}_{i }\cdot\Delta R_{i}\cdot A^{-1}\cdot\begin{bmatrix}u_{i}\\ v_{i}\\ 1\end{bmatrix}=0 \tag{12}\]  
\[\text{where }A=\begin{bmatrix}\frac{f}{\Delta a}&0&u_{0}\\ 0&\frac{f}{\Delta b}&v_{0}\\ 0&0&1\end{bmatrix}\text{ is the camera internal parameter matrix, and }\begin{bmatrix}u_{i}\\ v_{i}\\ 1\end{bmatrix}\text{ is the pixel coordinates at the moment }i.\]  
We denote that \(E=\lambda\cdot\Delta\widetilde{T}_{i}\cdot\Delta R_{i}\) and \(F=A^{-T}\cdot\lambda\cdot\Delta\widetilde{T}_{i}\cdot\Delta R_{i}\cdot A^{-1}\), which are the essential matrix and the fundamental matrix of the stereo vision, respectively, and the eight-points method is often used to obtain their values, where \(\lambda\) is a constant scale factor. After obtaining the essential matrix and the fundamental matrix by feature matching, the \(\lambda\cdot\Delta T_{i}\) and \(\Delta R_{i}\) can be obtained by performing singular value decomposition (SVD) decomposition on \(E\).  
Assume that the rotation attitude matrix is \(R_{0}\) and the translation vector is \(T_{0}\) at the initial moment. Then, according to Equation (7), the attitude rotation matrix \(\Delta Rs_{i}\) and  
Figure 3: The equivalent movement of the camera. The curve with the arrow represents the changes in the position and attitude between frames, the line with the arrow represents the position vector of a point. \(R_{i}\) and \(T_{i}\) represent the rotation and translation from the object-fixed frame to the camera-fixed frame at moment \(i\), and \(\Delta R_{i}\) and \(\Delta T_{i}\) represent the rotation and translation of the camera-fixed frame from moment \(i\) to moment \(i+1\).  
translation vector \(\Delta Ts_{i}\) describing the position and attitude change of \(\{O_{c}\)-\(X_{c}Y_{c}Z_{c}\}\) from the initial moment to moment \(i\) are obtained:  
\[\left\{\begin{array}{l}T_{i}=\left(\begin{array}{c}i-1\\ s=1\end{array}\Delta R_{s}\right)\cdot T_{0}+\left(\begin{array}{c}i-1\\ s=2\end{array}\Delta R_{s}\right)\cdot\Delta T_{1}+\cdots+\left(\begin{array}[] {c}i-1\\ \prod\limits_{s=i-1}^{i-1}\Delta R_{s}\end{array}\right)\cdot\Delta T_{i-2}+ \Delta T_{i-1}\\ R_{i}=\Pi\Delta R_{i-1}\cdots\Delta R_{2}\cdot\Delta R_{1}\cdot R_{0}\\ \left\{\begin{array}{l}\Delta TS_{i}=T_{i}-\left(\begin{array}{c}i-1\\ \prod\limits_{s=i-1}^{i-1}\Delta R_{s}\end{array}\right)\cdot T_{0}=\sum \limits_{j=1}^{i-2}\left(\begin{array}{c}i-1\\ \prod\limits_{s=j+1}^{i-1}\Delta R_{s}\end{array}\right)\cdot\Delta T_{j}+ \Delta T_{i-1}\\ \Delta Rs_{i}=\sum\limits_{s=1}^{i-1}\Delta R_{s}\end{array}\right. \tag{14}\]  
where \(\Pi\) is the continuous multiplication notation, and \(\Sigma\) is the continuous addition notation.