## 3 Improved Strategy for the Uncalibrated Visual Servo of a Hyper-Redundant Space Manipulator
### Controller Design  
Several advanced controllers based on the estimated Jacobian can be used to realize a well-behaved uncalibrated visual servo; for example, a nonlinear controller with a disturbance observer and a disturbance-rejection controller with an extended-state observer [25]. However, the proportional controller is still the most widely used and developed method in visual servos [22; 23; 26; 29; 31; 32; 33; 34; 35; 36; 37; 38]. In addition, its simplicity is beneficial for us to verify this robust strategy. Before constructing the proportional controller, two problems need to be solved. The first is the disturbance caused by the unstable conditional number of the estimated total Jacobian. Through a large number of simulations, it is found that the estimated Jacobian of the homography-based task function tends to present an unstable conditional number and causes drastic changes of the manipulator velocity in certain directions, resulting in severe vibrations and serious errors. Therefore, the SVF method is introduced to deal with this problem [45]. Singular value decomposition is performed on the estimated total Jacobian \(J_{t}\) of the hyper-redundant manipulator according to  
\[(J_{t})_{m\times n}=U_{m\times n}\sum_{m\times n}V_{m\times n}^{T}=\sum_{i=1}^ {m}\sigma_{i}u_{i}v_{i}^{T} \tag{35}\]  
where \(\sigma_{i}\) is the singular value of the total Jacobian \(J_{t}\) and \(u_{i}\) and \(v_{i}\) are the \(i\)th columns of \(U\) and \(V\), respectively. The filtering function \(h_{\sigma_{0},v}(\sigma)\) of the singular value \(\sigma_{i}\) proposed in the SVF method can maintain the stability of the smallest singular value of the estimated total Jacobian. It is defined as  
\[h_{\sigma_{0},v}(\sigma_{i})=\frac{\sigma_{i}^{3}+v\sigma_{i}^{2}+2\sigma_{i} +2\sigma_{0}}{\sigma_{i}^{2}+v\sigma_{i}+2} \tag{36}\]  
where \(\sigma_{0}\) is the minimum acceptable singular value and \(v\) is a scaling factor that regulates the filtering shape of function \(h_{\sigma_{0},v}(\sigma)\). If \(\sigma_{0}<v\) and \(v\sigma_{0}<2\), \(h_{\sigma_{0},v}(\sigma)\) can stay monotonic. The pseudo-inverse of the total Jacobian \(J_{t}\) can be developed via the SVF method as  
\[J_{S}^{+}=\left\{ \begin{array}{l}J_{t}^{T}\big{(}J_{t}^{T}J_{t}\big{)}^{-1}, \min\lvert\sigma_{i}\rvert>\sigma_{0}\\ \sum\limits_{i=1}^{m}\frac{1}{h_{\sigma_{0},v}(\sigma_{i})}v_{i}u_{i}^{T}, else.\end{array} \right. \tag{37}\]  
The restriction on the minimum singular value guarantees the stability of the conditional number of the estimated total Jacobian. Thus, the disturbance caused by the unstable conditional number can be reduced without generating much additional error in the algorithms. Another problem is that the manipulator joint angles may overrun. If the joint angle is not limited, it may cause a failure of the visual servo or even cause damage to the space manipulator. At this time, the redundancy of the joint space of the hyper-redundant manipulator resulting from the improved homography-based task function can be used to prevent this from happening. The gradient projection based on the redundant Jacobian can be adopted to avoid the joint limit [41; 42]. When the joints move away from the limit of joint, allowing the joints to move freely can reduce unnecessary energy consumption. The gradient function is defined as  
\[\
abla H(q)=\left[ \begin{array}{cc}\partial H(q)/\partial q_{1}&\cdots& \partial H(q)/\partial q_{n}\end{array} \right]^{T} \tag{38}\]  
\[\left\{ \begin{array}{c}\frac{\partial H(q)}{\partial q_{i}}=0,if\Delta\Big{|} \frac{\partial H(q)}{\partial q_{i}}\Big{|}\leq 0\\ \frac{\partial H(q)}{\partial q_{i}}=\frac{(q_{i,\max}-q_{i,\min})^{2}(q_{i}-q_ {i,\max}-q_{i,\min})}{4(q_{i,\max}-q_{i})^{2}(q_{i}-q_{i,\min})^{2}}, else\end{array} \right. \tag{39}\]  
On the basis of these optimizations, the controllers for static positioning and dynamic tracking are designed, respectively, as follows:  
1.