## 2 Projective Homography  
The current frame \(F\) and the desired frame \(F^{*}\) of an eye-in-hand camera are defined for problems with an eye-in-hand system in which the camera moves with the manipulator. Figure 1 shows the geometric relationship between these two camera frames. Suppose \(M_{i}=[X_{i},Y_{i},Z_{i}]^{\prime}\) is the coordinate of the feature point \(P_{i}\) in \(F\) and \(M_{i}^{*}=\left[X_{i}^{*},Y_{i}^{*};Z_{i}^{*}\right]^{\prime}\) is the coordinate described by the desired camera, while \(p_{i}=\left[\begin{array}{cc}u_{i}&v_{i}&1\end{array}\right]^{\prime}\) and \(p_{i}^{*}=\left[\begin{array}{cc}u_{i}^{*}&v_{i}^{*}&1\end{array}\right]^{\prime}\) are the homogeneous pixel coordinates of feature point \(P_{i}\) captured by the current and desired cameras, respectively. The geometric relationship between \(M_{i}\) and \(M_{i}^{*}\) can be expressed as  
\[M_{i}=RM_{i}^{*}+t \tag{1}\]where \(R\in\mathbb{R}^{3\times 3}\) and \(t\in\mathbb{R}^{3\times 1}\) are the orientation matrix and displacement vector between \(F\) and \(F\) expressed in \(F\), respectively.  
Suppose each target point \(P_{i}\) is on the plane \(\pi\), \(d^{*}\) is the distance between the desired projection center and plane \(\pi\) and \(n^{*}\) is the normal vector of \(\pi\) expressed in the desired camera frame \(F^{*}\) with length \(\|n^{*}\|=1/d^{*}\). Based on Equation (1), we can obtain  
\[M_{i}=\Big{(}R+tn^{*T}\Big{)}M_{i}^{*} \tag{2}\]  
\[G=R+tn^{*T} \tag{3}\]  
where \(G\) is the Euclidean homography matrix. In normalized coordinates, (2) can be expressed as  
\[m_{i}=\frac{Z_{i}^{*}}{Z_{i}}\Big{(}R+tn^{*T}\Big{)}m_{i}^{*} \tag{4}\]  
where \(m_{i}=[X_{i}/Z_{i},Y_{i}/Z_{i},1]^{\prime}\) and \(m_{i}^{*}=\left[X_{i}^{*}/Z_{i}^{*},Y_{i}^{*}/Z_{i}^{*},1\right]^{\prime}\). The depth scale is denoted as  
\[\alpha_{i}=\frac{Z_{i}^{*}}{Z_{i}}. \tag{5}\]  
The following relationship can be obtained from Equations (2), (3) and (5):  
\[p_{i}=\alpha_{i}\Big{(}KGK^{-1}\Big{)}p_{i}^{*} \tag{6}\]  
where \(K\) is the internal parameter matrix of the eye-in-hand camera [36]. The homography matrix can be obtained as  
\[H=KGK^{-1} \tag{7}\]  
which refers to the mapping between different images projected from the same object to two different image planes. It has been applied in many domains such as motion estimation, 3-D reconstruction [49] and of course the visual servo. Even in an uncalibrated situation, this matrix can be estimated up to a scale \(\beta\) with more than four corresponding image features in different images via  
\[\hat{H}=\beta H \tag{8}\]  
where \(\beta\) is independent of the depth ratio \(\alpha_{i}\) in (5).  
Figure 1: Projective homography model.  
Because the images are always disturbed by noise during transmission and processing, the exact solution of \(H\) cannot be obtained via the noisy measurement image point \(p_{i}\). However, it is possible to obtain the optimal estimated homography matrix \(\hat{H}\) in the sense of least squares. The estimated image point of \(P_{i}\) in the current image can be obtained via \(\hat{p}_{i}=\hat{H}p_{i}^{*}\). The root-mean-squared (RMS) error between \(\hat{p}_{i}\) and the true points can be obtained via \(\hat{\varepsilon}\approx\varepsilon(4/n)^{1/2}\)[38], where \(n\) is the number of image features and \(\varepsilon\) is the RMS error between the measurement point \(p_{i}\) and the true point. It is worth noting that the estimated homography matrix \(\hat{H}\) can be equivalently considered as being estimated by the matching points \(\hat{p}_{i}\leftrightarrow p_{i}^{*}\). Obviously, if \(n>4\), building the task function with the elements of the homography matrix will have a certain compensation effect for image noise.