## 2 Theoretical Derivation of Methods  
### Principle of the Absolute Pose Measurement for a Rigid Body  
As shown in Figure 1, body and world rigid bodies move relative to one another in space. For the convenience of expression, the body coordinate system \(\rm{O_{B}}\)-\(\rm{X_{B}}\rm{Y_{B}}\rm{Z_{B}}\), world coordinate system \(\rm{O_{W}}\)-\(\rm{X_{W}}\rm{Y_{W}}\rm{Z_{W}}\), and camera coordinate system \(\rm{O_{C}}\)-\(\rm{X_{C}}\rm{Y_{C}}\rm{Z_{C}}\) are, respectively, denoted B, W, and C hereafter. We let \(P\) and \(Q\) denote physical points on the body and world rigid bodies, respectively. We use superscripts to distinguish the coordinate values of the physical points \(P\) and \(Q\) in the different coordinate systems and subscripts to distinguish different points on the rigid body. For example, \(P_{3}^{W}\) represents the coordinate values of the \(P_{3}\) point on the body rigid body in the W system. We use minuscule letters to represent pixel points, e.g., \(p\) represents the pixel points projected by physical point \(P\) on the image. \(R\) and \(T\) represent conversion relationships between coordinate systems, where the superscript is used to represent the target coordinate system in the conversion and the subscript is used to represent the starting coordinate system, e.g., \(R_{C}^{W}\) is the rotation matrix for conversion from the C system to the W system, and \(T_{B}^{W}\) is the translation matrix for the conversion from the B system to the W system.  
The aim of this article is to measure the motion pose of a rigid body in the B system relative to the W system, and the real-time spatial coordinate \(P^{W}\) of the center of mass \(P\) of the B-system rigid body in the W system.  
The imaging projection formulas are  
\[P^{W}=T_{\rm C}^{W}+mR_{C}^{W}p^{C} \tag{1}\]  
\[P^{B}=T_{\rm C}^{B}+mR_{C}^{B}p^{C} \tag{2}\]  
Figure 1: Principle of the absolute pose measurement for a rigid body using a monocular camera.  
Here, \(m=\left\|\frac{P^{C}}{\left\|\frac{P^{C}}{\left\|P^{C}\right\|}}\right\|\) is the ratio of two vector modes, representing the spatial coordinates of the pixel point \(p\) in the C system.  
It follows from (1) and (2) that  
\[P^{W}=T_{C}^{W}+R_{C}^{W}\Big{(}R_{C}^{B}\Big{)}^{T}\Big{(}P^{B}-T_{C}^{B}\Big{)} \tag{3}\]  
\[R_{B}^{W}=R_{C}^{W}\Big{(}R_{C}^{B}\Big{)}^{T} \tag{4}\]  
\[T_{B}^{W}=T_{C}^{W}-R_{C}^{W}\Big{(}R_{C}^{B}\Big{)}^{T}T_{C}^{B} \tag{5}\]  
The motion pose of the rigid body can be solved by \(R_{B}^{W}\), if  
\[R_{B}^{W}=\begin{bmatrix}r_{11}&r_{12}&r_{13}\\ r_{21}&r_{22}&r_{23}\\ r_{31}&r_{32}&r_{33}\end{bmatrix}\]  
According to reference [31], the attitude angle of a rigid body is  
\[\left\{\begin{aligned} Q_{z}=atan2(r_{21},r_{11})\\ Q_{y}=atan2\Big{(}-r_{31},\sqrt{r_{32}^{2}+r_{33}^{2}}\Big{)}\\ Q_{x}=atan2(r_{32},r_{33})\end{aligned}\right. \tag{6}\]  
Here, \(Q_{z}\), \(Q_{y}\) and \(Q_{x}\) are the rotation angles around the Z, Y, and X axes, with the order of rotation being Z, Y, and X.  
It is seen from (3) and (4) that for \(P^{W}\) and \(R_{B}^{W}\), \(R_{C}^{W}\), \(T_{C}^{W}\), \(R_{C}^{B}\) and \(T_{C}^{B}\) need to be determined. \(R_{C}^{W}\) and \(T_{C}^{W}\) are determined from the control points on the W-system rigid body whereas \(R_{C}^{B}\) and \(T_{C}^{B}\) are determined from the control points on the B-system rigid body. The absolute pose problem of a moving rigid body is thus transformed into a problem of solving the rotation and translation matrices.