## 3 Improved Strategy for the Uncalibrated Visual Servo of a Hyper-Redundant Space Manipulator
### Improved Homography-Based Task Function  
In the original PHUVS strategy proposed by [PERSON] [23], the homography-based task function is defined as  
\[\hat{h}=\left[ \begin{array}{cccc}\hat{H}_{1}&\hat{H}_{2}&\hat{H}_{3}\\ \hat{h}_{1}&\hat{h}_{2}&\cdots&\hat{h}_{9}\\ \end{array} \right]^{T}_{9\times 1} \tag{9}\]  
where \(\hat{H}_{i}\) is the \(i\)th row of the estimated homography matrix. The error vector of the system is denoted by  
\[e=h_{0}-\hat{h} \tag{10}\]  
where \(h_{0}\) is constructed by stacking the rows of the identity matrix \(I_{3\times 3}\). The purpose of our strategy can be considered to be making the current camera frame \(F\) coincide with \(F^{*}\). To guarantee this, the constraint \(\det(\hat{H})=1\) is defined to obtain the unique projective homography matrix \(\hat{H}\) between the current and desired feature points in each iteration. It can be proved that \(e=0\) if and only if rotation matrix \(R=I_{3\times 3}\) and position vector \(t=0\)[23]. In our strategy, the constraint \(\hat{h}_{9}=1\) is defined, where \(h_{9}\) is the last element of the homography matrix and a direct linear transformation (DLT) is introduced to estimate the homography matrix. First, it is necessary to normalize the two images, respectively, as follows:  
1. Translate the feature points as \(\widetilde{p}_{i}=T_{1}p_{i}\) and \(\widetilde{p}_{i}^{*}=T_{1}^{*}p_{i}^{*}\) to ensure that the centroid of these feature points is at the origin;
2. Scale feature points as \(\widetilde{p}_{i}=T_{2}\widetilde{p_{i}}\) and \(\widetilde{p}_{i}^{*}=T_{2}^{*}\widetilde{p}_{i}^{*}\) to make their average distance from the origin equal to \(\sqrt{2}\). On the basis of the properties of homogeneous coordinates, homogeneous simultaneous equations can be constructed with transformed pixel coordinates as  
\[\underbrace{\left( \begin{array}{ccc}\vdots&\vdots&\vdots\\ \widetilde{p_{i}^{T}}&0^{T}&\widetilde{u_{i}p_{i}^{*}}\\ 0^{T}&\widetilde{p_{i}^{T}}&\widetilde{v_{i}p_{i}^{*}}\\ \vdots&\vdots&\vdots\\ \end{array} \right)}_{A_{2\text{mog}}X=0}\left( \begin{array}{c}\overline{H }_{1}^{T}\\ \overline{H}_{2}^{T}\\ \overline{H}_{3}^{T}\\ \end{array} \right)}=0 \tag{11}\]  
where \(A\) is the coefficient matrix and \(\overline{H}_{i}\) is the \(i\)th row of \(\overline{H}\). The purpose of the above normalization operation is to prevent the coefficient matrix \(A\) from becoming ill-conditioned owing to image noise. This solves the problem of numerical instability. Then, the least squares solution of equation (11) can be calculated via singular value decomposition. Then, we conduct an inverse transform on \(\overline{H}\) as  
\[\widetilde{H}=\left(T_{2}T_{1}\right)^{-1}\overline{h}\left(T_{2}^{*}T_{1}^{ *}\right) \tag{12}\]  
where \(\overline{H}\) is the estimated homography between the current and desired feature points. The homography \(\widetilde{H}\) can be scaled at any scale. Thus, \(\widetilde{H}\) is scaled as \(\hat{H}=\overline{H}/\overline{h}_{9}\) and the new task function is constructed as  
\[\hat{h}=\left[ \begin{array}{cccc}\hat{h}_{1}&\hat{h}_{2}&\cdots&\hat{h}_{8} \\ \end{array} \right]^{T}_{8\times 1} \tag{13}\]where \(\hat{h}\) reflects the eight degrees of freedom of the homography matrix and further reduces the dimensions of the system. The error vector of system is defined as  
\[e= \begin{bmatrix}1&0&0&0&1&0&0&0\\ &-\begin{bmatrix}\hat{h}_{1}&\hat{h}_{2}&\cdots&\hat{h}_{8}\end{bmatrix}^{T} \end{bmatrix}  \tag{14}\]  
where \(e=0\) if and only if the rotation matrix \(R=0\) and the position vector \(t=0\). Its sufficiency and necessity can be proved as follows. 1. Necessity proof: If \(R=I\) and \(t=0\), then \(G=I\). On the basis of the similarity of \(H\) and \(G\), we obtain \(H=I\). From Equation (8), we obtain \(\hat{H}=H/\beta=I/\beta\). The constraint \(h_{9}=1\) gives \(1/\beta=h_{9}=1\); i.e., \(\beta=1\), \(\hat{H}=I\), and \(e=0\). 2. Sufficiency: It is obvious that \(\hat{H}=I\) and \(H=\beta I\) if \(e=0\).