## 3 Proof of P3P for the Stereo Camera System  
Given its intrinsic parameters and a set of n correspondences between its 3D points and 2D projections to determine the camera's pose is known as the perspective-n-point (PnP) problem. This well-known work [7] has proved that at least four correspondences are required to uniquely determine the pose of a monocular camera, a situation referred to as the P4P problem. To illustrate, consider the P3P case for a monocular camera (Figure 2). Let points A, B, and C exist in space, with \(O_{1}\), \(O_{2}\) and \(O_{3}\) representing different perspective centers. The angles \(\angle\)AOB, \(\angle\)AOC and \(\angle\)BOC remain the same across all three perspectives. Given a fixed focal length, the image coordinates of points A, B, and C will be identical when observed from these three perspectives. In other words, it is impossible to uniquely identify the camera's pose based solely on the image coordinates of three points. Figure 1: The eye-in-hand robot configuration. Figure 2: P3P case of a monocular camera. The PnP problem with a stereo camera has not been thoroughly addressed in prior research. A stereo camera can detect three image coordinates of a 3D point in space. This paper proposes that a complete solution to the PnP problem for a stereo camera can be framed as a P3P problem. Below is the complete proof of this proposition. For a stereo system, if all intrinsic parameters are fixed and given, we can readily compute the 3D coordinates of an object point given the image coordinates of that point. This provides a unique mapping from the image coordinates of a point to its corresponding 3D coordinates in a Cartesian frame. The orientation and position of the camera system uniquely define the origin and axis orientations of this Cartesian coordinate system in space. Consequently, PnP problem can be framed as follows: given n points with their 3D coordinates measured in an unknown Cartesian coordinate system in space, what is the minimum number n required to accurately determine the position and orientation of the 3D Cartesian coordinate frame established in that space? 1. P1P problem with the stereo camera:  
If we know the coordinates of a single point in space, defined by a 3D Cartesian coordinate system, an infinite number of corresponding coordinate systems can be established. Any such coordinate system can have its origin placed on the surface of a sphere centered at this point, with a radius \(R=\sqrt{X^{C2}+Y^{C2}+Z^{C2}}\), where \(\left[X^{C},Y^{C},Z^{C}\right]^{T}\) are the coordinates measured by the Cartesian system (see Figure 3). 2. P2P problem with the stereo camera:  
When the coordinates of two points in space are known, an infinite number of corresponding coordinate systems can be established. Any valid coordinate system can have its origin positioned on a circle centered at point \(O\) with a radius \(R\) as illustrated in Figure 4. This circle is constrained by the triangle formed by points \(O_{C}\), \(P_{1}\), and \(P_{2}\), where the sides of the triangle are defined by the lengths \(R_{1},~{}R_{2}\) and \(D\). Specifically, \(R_{1}=\sqrt{X_{1}^{C2}+Y_{1}^{C2}+Z_{1}^{C2}}\), \(R_{2}=\sqrt{X_{2}^{C2}+Y_{2}^{C2}+Z_{2}^{C2}}\), \(D=\sqrt{\left(X_{1}^{C}-X_{2}^{C}\right)^{2}+\left(Y_{1}^{C}-Y_{2}^{C}\right)^ {2}+\left(Z_{1}^{C}-Z_{2}^{C}\right)^{2}}\)  
Figure 3: P1P Problem with a stereo camera system. The radius of the circle \(R\) corresponds to the height of the base \(D\) of the triangle. The center of the circle \(O\) is located at the intersection of the height and the base. 3. P3P problem with the stereo camera: When three points in space are known, and the lines connecting these points are not collinear, we can uniquely establish one coordinate system. As illustrated in the Figure 5 below, three non-collinear points define a plane in space, which has a uniquely defined normal unit vector \(\overset{\rightarrow}{n}\).