## 3 Improved Strategy for the Uncalibrated Visual Servo of a Hyper-Redundant Space Manipulator
### Online Estimation of Total Jacobian  
The kinematic parameters of the space manipulator change easily in the harsh space environment. To avoid the repeated calibration of the complex hyper-redundant manipulator, the total Jacobian between the task function and the joints of the manipulator is used to construct the controller. The expression of the total Jacobian is  
\[J_{t}= \begin{bmatrix}\partial\hat{h}_{1}/\partial q_{1}&\cdots&\partial\hat{h }_{1}/\partial q_{9}\\ \vdots&\ddots&\vdots\\ \partial\hat{h}_{8}/\partial q_{1}&\cdots&\partial\hat{h}_{8}/\partial q_{9} \end{bmatrix} _{\otimes\otimes\otimes} \tag{18}\]Therefore, the accuracy of the estimation of the total Jacobian is directly related to the system performance. The uncalibrated visual servo system of the nine degree-of-freedom (9 DOF) manipulator is modeled as  
\[X_{K+1}=X_{K} \tag{19}\]  
\[Z_{K+1}=C_{K+1}X_{K+1} \tag{20}\]  
where \(X_{K}=\left[ \begin{array}{cccc}\left(I_{11}\right)_{K}&\left(I_{22}\right)_{K }&\cdots&\left(I_{8}\right)_{K}\end{array} \right]^{T}\) and \(J_{li}\) is the \(i\)th row of the estimated total Jacobian \(J_{t}\). \(C_{K}=diag\Big{(} \begin{array}{cccc}\hat{q}_{K}&\cdots&\hat{q}_{K}\end{array}  \Big{)}_{8,\times 72}\) is constructed by the joint velocity of the manipulator, and \(Z_{K}\) can be considered as the change rate of task function \(\hat{h}\). It is worth noting that this model is a linear approximation of the visual servo system, and it causes modeling uncertainty. Meanwhile, because the homography-based task function is used to compensate for the image noise, the statistical characteristics of the measurement noise of the task function become difficult to obtain as well. Therefore, the more robust KF-SVSF method [46; 47; 48] is introduced into the visual servo to improve the system accuracy. This method combines the optimality of the KF method with the robustness of the SVSF method. As shown in Figure 3, it introduces the concept of an optimal smoothing variable boundary layer, which is an estimation of the impact of modeling uncertainty and noise. The KF-SVSF method seeks a balance between accuracy and robustness by introducing a saturation limit into the calculated optimal boundary layer. By switching between KF gain and SVSF gain, the optimality of KF is maintained within the limit, while the robustness and stability of SVSF remain outside the limited range. First, the smooth variable boundary layer can be obtained as follows:  
1. Calculate the a priori state space: \[\hat{X}_{K+1|K}=\hat{X}_{K|K}\] (21)
2. Calculate the a priori state error covariance matrix: \[P_{K+1|K}=P_{K|K}+Q_{K}\] (22)
3. Calculate the residuals vector: \[\hat{Z}_{K|K}=C\hat{X}_{K|K}\] (23) \[\hat{Z}_{K+1|K}=C\hat{X}_{K+1|K}\] (24) \[e_{Z,K+1|K}=Z_{K+1}-Z_{K+1|K}\] (25)  
Figure 3: Concept of boundary layers of the Kalman filter–smooth variable structure filter (KF-SVSF) method. \[e_{Z,K|K}=Z_{K}-\hat{Z}_{K|K} \tag{26}\]
4. Calculate the smoothing boundary layer \(\psi_{K+1}\): \[S_{K+1|K}=CP_{K+1|K}C^{T}+R_{K+1}\] (27) \[E_{K+1}=\left|e_{Z,K+1|K}\right|_{abs}+\gamma\left|e_{Z,K|K}\right|_{abs}\] (28) \[\psi_{K+1}=\left(\overline{E}_{K+1}^{-1}CP_{K+1|K}C^{T}S_{K+1}^{-1} \right)^{-1}\] (29) where \(Q_{K}\) and \(R_{K+1}\) are the process noise covariance matrix and measurement noise covariance matrix, respectively; \(\overline{E}_{K+1}\) is the diagonal matrix constructed with \(E_{K+1}\); and the calculated optimal boundary layer \(\psi_{K+1}\) of the uncalibrated visual servo system is the diagonal matrix \(\psi_{K+1}=diag\Big{(} \begin{array}{cc}\left(\psi_{1}\right)_{K+1}&\left( \psi_{2}\right)_{K+1}&\cdots&\left(\psi_{8}\right)_{K+1}\end{array} \Big{)}\). 5.