# Adaptive Visual Servoing Control for Hoisting Positioning Under Disturbance Condition
## 4 Controller Design  
### Adaptive Control Law  
In this paper, the visual error model is defined by the image error of the feedback signal based on dynamic equations containing disturbance. Considering the local asymptotic stability of the visual servo system, the visual error signal is denoted as  
\[\xi=\hat{f}_{c}^{-}e \tag{13}\]  
\(\hat{f}_{c}^{-}\) is the Moore-Penrose pseudo inverse of the estimated Jacobian matrix \(\hat{f}_{c}\). Here, we use an estimation method [34] to obtain the interaction matrix \(\hat{f}_{c}\). It is estimated to satisfy the following equation. \[\dot{s}=\hat{f}_{c}\mathrm{v} \tag{14}\]  
Moreover, \(\hat{f}_{c}\mathrm{v}\) can be written in the following linear form  
\[\dot{s}=\hat{f}_{c}\mathrm{v}=(\eta)\hat{s} \tag{15}\]  
where \((\eta)\) is a matrix without depending on the intrinsic and extrinsic parameters of the camera. \(\hat{s}\) is a vector where the components of \(\hat{f}_{c}\) are listed [34]. In this paper, we adopt the method of image-based visual servoing (IBVS) [35]. In image-based visual servoing (IBVS), the control loop is directly closed in the image space of the vision sensor. Compared with position-based visual servoing and hybrid visual servoing, IBVS schemes get rid of 3-D reconstruction steps to compute the visual features. The derivative of the visual error could be calculated by (13) as  
\[\dot{\xi}=-\hat{f}_{c}^{-}\hat{f}_{c}\mathrm{v}+\dot{\hat{f}}_{c}^{-}e \tag{16}\]  
The estimation error of the Jacobian matrix is defined as  
\[\overline{\hat{f}_{c}}=\hat{f}_{c}-\hat{f}_{c} \tag{17}\]  
If we substitute (17) into (16), we have  
\[\dot{\xi}=-\hat{f}_{c}^{-}\hat{f}_{c}\mathrm{v}+\dot{\hat{f}}_{c}^{-}e=- \mathrm{v}-\xi \tag{18}\]  
where \(\xi=\hat{f}_{c}\overline{\hat{f}_{c}}\mathrm{v}-\dot{\hat{f}}_{c}^{-}e\). The control purpose for the visual servo system is to use visual feedback \(\xi\) to drive the target to the desired pose, like \(\eta\rightarrow\eta^{*}\). At the same time, we should ensure the asymptotic stability of the system in the case of unknown and bounded uncertainties. Proportional control law is a widely used method and can be seen in most visual servo literature. However, it is difficult for a system using only proportional control law to obtain an ideal dynamic response. In this paper, sliding mode control (SMC) is used to compensate for the influence of external disturbance. The velocity control signal could be calculated from the visual system based on the inverse of the estimated Jacobian matrix and the change in the visual frame. We design the sliding surface \(S\) as follows  
\[S=e-\lambda\int edt \tag{19}\]  
Where \(e=s^{*}-s\) is the feature error state, \(s^{*}\) represents the reference state vector. Since \(s^{*}\) is the constant, \(\dot{s}^{*}=0\). \(\lambda\) is the positive definite gain matrix. The derivative of the sliding surface (19) is:  
\[\hat{S}=\hat{e}-\lambda e \tag{20}\]  
We propose a control law combining proportional control with SMC, and it is designed as follows  
\[U_{p}=-\hat{J}_{c}^{-1}k_{p}\lambda e \tag{21}\]  
The proportional control makes the output \(U_{p}\) proportional to the input \(e\). In order to reduce the deviation, improve the response speed and shorten the adjustment time, it needs to increase \(k_{p}\). \(k_{p}\) is the proportional gain matrix. Due to the existence of an external disturbance, the system error cannot converge to zero. Similarly,  
\[U_{s}=-\hat{J}_{c}^{-1}k_{s}\lambda sign(S), \tag{22}\]  
where \(k_{s}\) is the diagonal positive proportional gain matrix. Because the sigum function is discontinuous, which can cause a chattering effect, to eliminate the buffeting effect, we use saturation function \(sat(S_{i})\) instead of the sigum function in the equation.