# Motion Estimation of Non-Cooperative Space Objects Based on Monocular Sequence Images
## 2 Proposed Method  
### The Relationship between Object Motion and Camera Equivalent Motion  
When the spacecraft is controlled to state at a non-cooperative object, the object is in the field of vision of the spacecraft camera. As the motion of the spacecraft is known, the problem of estimating the object motion can be simplified as how to use fixed monocular camera sequence images to obtain the motion of the object. This section provides the model for motion estimation of a non-cooperative object. First, we define the following reference frames (Figure 1) to make them useful for understanding the problem:  
1. Camera-fixed frame (\(\{O_{o}\)-\(X_{c}Y_{c}Z_{c}\}\)): A fixed camera frame centred at the position of the camera, and the frame motion with the camera. 2. Object-fixed frame (\(\{O_{b}\)-\(X_{b}\)\(Y_{b}\)\(Z_{b}\}\)): A fixed object frame centred at the centroid of the object, and the frame motion with the object. A pinhole projection model of the camera is assumed, and a feature point \(P_{b}\) on the object is photographed to the point \(P_{c}\) on the focal plane; a single feature's measurement is expressed as:  
\[ \begin{bmatrix}X_{c}\\ Y_{c}\\ Z_{c}\end{bmatrix} =R \begin{bmatrix}X_{b}\\ Y_{b}\\ Z_{b}\end{bmatrix} +T \tag{1}\]  
where \([X_{b},Y_{b},Z_{b}]^{T}\), \([X_{c},Y_{c},Z_{c}]^{T}\), \(R\) and \(T\) are the coordinates in the object-fixed frame (\(\{O_{b}\)-\(X_{b}\)\(Y_{b}\)\(Z_{b}\)) of \(P_{b}\), the coordinates in the camera-fixed frame (\(\{O_{c}\)-\(X_{c}\)\(Y_{c}\)\(Z_{c}\)\(\}\)) of \(P_{c}\), the rotation matrix and the translation vector from the object-fixed frame to the camera-fixed frame. The pinhole projection model of the camera is expressed as:  
\[Z_{c}\cdot \begin{bmatrix}u\\ v\\ 1\end{bmatrix} = \begin{bmatrix}\frac{f}{\Delta a}&0&u_{0}\\ 0&\frac{f}{\Delta b}&v_{0}\\ 0&0&1\end{bmatrix} \cdot \begin{bmatrix}X_{c}\\ Y_{c}\\ Z_{c}\end{bmatrix}  \tag{2}\]  
where \(u\) and \(v\) are the two direction image pixel coordinates of feature point \(P_{b}\), \(u_{0}\) and \(v_{0}\) are the center pixel coordinates of the image photographed by the camera, and \(\Delta a\), \(\Delta b\) and \(f\) are the two direction cell sizes of the focal plane and the focal length of the camera, respectively. For a translating and rotating space non-cooperative object, we use a monocular camera fixed with an inertial reference frame to take pictures of the object. The position and attitude, relative to the camera-fixed frame \(\{O_{c}\cdot X_{c}Y_{c}Z_{c}\}\), are represented by translation vector \(T\) and attitude rotation matrix \(R\); then, the different \(T\) and \(R\) of the object motion, relative to the camera, can be obtained at different times. Similarly, the object position and attitude between two moments can be represented by another translation vector and attitude rotation matrix (Figure 2). Figure 1: Definition of the reference frames. The line with the arrow represents the position vector of a point. A feature point \(P_{b}\) on the object is photographed to point \(P_{c}\) on the focal plane. Figure 2: The relative motion of the object and camera. The curve with the arrow represents the changes in the position and attitude between frames, and 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 \(\Delta G_{i}\) and \(\Delta L_{i}\) represent the rotation and translation of the object-fixed frame from moment \(i\) to moment \(i\) + 1. At the two different moments, \(i\) and \(i+1\), the coordinates of the feature point P on the non-cooperative object in the frame \(\{O_{c}\)-\(X_{c}\)\(Y_{c}\)\(Z_{c}\)\(\}\) are (\(X_{c,i}\), \(Y_{c,i}\), \(Z_{c,i}\)) and (\(X_{c,i+1}\), \(Y_{c,i+1}\), \(Z_{c,i+1}\)). The attitude rotation matrices from \(\{O_{b}\)-\(X_{b}\)\(Y_{b}\)\(Z_{b}\}\) to \(\{O_{c}\)-\(X_{c}\)\(Y_{c}\)\(Z_{c}\}\) are \(R_{i}\) and \(R_{i+1}\), and the translation vectors are \(T_{i}\) and \(T_{i+1}\).