# Motion Estimation of Non-Cooperative Space Objects Based on Monocular Sequence Images
## 2 Proposed Method
### The Model for Estimating the Object Rotation State  
At the initial moment of observation, the object-fixed frame \(\{O_{b}\)-\(X_{b}Y_{b}Z_{b}\}\) parallel to the camera-fixed frame \(\{O_{c}\)-\(X_{c}Y_{c}Z_{c}\}\) is established on the object; that is, the attitude rotation matrix is a unit matrix. Then, the matrix \(\Delta Rs_{i}\) is equal to \(R_{i}\), and the transpose matrix \(R_{i}^{T}\) of \(R_{i}\) is the attitude rotation matrix of the object, which represents the attitude rotation of the object. Therefore, it can be concluded that the rotation of the object from the observation start time to any time can be determined.  
Taking the derivation of the attitude rotation matrix \(R_{i}^{T}\), the object angular velocity relative to the object-fixed frame \(\{O_{b}\)-\(X_{b}Y_{b}Z_{b}\}\) can be obtained:  
\[\vec{\omega}_{b,i}=-\vec{R_{i}^{T}}R_{i} \tag{15}\]  
where \(\omega_{b,i}=\begin{bmatrix}\omega_{bx,i}\\ \omega_{by,i}\\ \omega_{bz,i}\end{bmatrix}\) and \(\omega_{b,i}=\begin{bmatrix}0&-\omega_{bz,i}&\omega_{by,i}\\ \omega_{bz,i}&0&-\omega_{bx,i}\\ -\omega_{by,i}&\omega_{bx,i}&0\end{bmatrix}\).  
Furthermore, through the projection transformation of the object angular velocity in the object-fixed frame by \(R_{i}\), the object angular velocity relative to the camera-fixed frame can be obtained:  
\[\omega_{c,i}=R_{i}\cdot\omega_{b,i} \tag{16}\]  
Similarly, the attitude quaternion of the object can be obtained through the attitude rotation matrix \(R_{i}\):  
\[\mathbf{q}_{e,i}=quat(R_{i}) \tag{17}\]  
Then, the rotation axis direction and rotation angle are:  
\[\left\{\begin{array}{l}\theta_{i}=\pi-2\  
position of the translation vector \(\Delta T_{i}\), we discuss the equivalent translation process of the camera in detail, as shown in Figure 4.  
From the decomposition of the essential matrix, \(\lambda\cdot\Delta T_{i}\) can be obtained. This means that the direction is known, but the distance scale is missing. The triangle constituted by \(\Delta T_{i}\), \(T_{i}-\Delta R_{i}\cdot T_{i}\) and \(-\Delta L_{i}\) cannot be determined by only one side direction. To know what information is needed to determine the triangle, step by step, some assumptions are made:  
1. If the size and direction of \(\Delta T_{i}\) and \(T_{i}-\Delta R_{i}\cdot T_{i}\) are known, \(\Delta L_{i}\) can be determined.
2. If the size and direction of \(T_{i}\) and \(T_{i+1}\) are known, \(\Delta T_{i}\) and \(T_{i}-\Delta R_{i}\cdot T_{i}\) can be determined.
3. If only the direction of \(T_{i}\) and \(T_{i+1}\) is known, the ratio of \(\Delta T_{i}\), \(T_{i}\) and \(T_{i+1}\) can be determined, that is, unified to the same scale. Then, \(T_{i}-\Delta R_{i}\cdot T_{i}\) can be unified to this \(T_{i}\) scale, and the direction of \(\Delta L_{i}\) can be determined.
4. On the basis of 3, if the size of \(T_{i}\) is also known, \(\Delta L_{i}\) can be determined. This means that if the position and direction at the initial moment and the direction at any moment of the object centroid are known, the translation of the object can be solved.
5. If only the direction of \(T_{i}\) is known, the direction of \(T_{i}-\Delta R_{i}\cdot T_{i}\) can be determined. If the directions of \(T_{i}-\Delta R_{i}\cdot T_{i}\) and \(\Delta T_{i}\) are not parallel, the object must have translation. It should be pointed out that this is a sufficient and unnecessary condition.  
The above assumptions can guide us to obtain the relevant data from other sensors or the configuration of sensors during the application of the proposed method.