## 2 Theoretical Description of Uncalibrated IBVS  
IBVS mainly relies on the variation of features in image space to guide robot actions. In order to realize the manipulator control directly in the image space, it is first necessary to establish the mapping relationship between the manipulator's joint vectors to the feature (image object feature vectors) and further establish the IBVS controller.  
First, define the velocity vector \(V_{c}=\left[v_{c}^{T},w_{c}^{T}\right]^{T}=\left[v_{x},v_{y},v_{z},\omega_{x}, \omega_{y},\omega_{z}\right]^{T}\) of the hand-eye camera in the camera coordinate system. Then, the relationship between the camera velocity \(V_{c}\) and the speed vector \(\dot{s}_{i}\) of the image feature changes \(s_{i}\) is given as follows:  
\[\dot{s}_{i}=I_{x}^{i}V_{c} \tag{1}\]  
where \(J_{x}^{i}\in R^{2\times 6}\) denotes the Jacobian matrix of the image feature \(s_{i}\).  
For a visual servo with \(n\) features, the problem of image-based closed-loop visual servo control can be translated into minimizing feature errors in the image plane. Set \(e(t)=s(t)-s^{*}\). \(s(t)\in R^{n\times 1}\) is the current image features at time \(t\), and \(s^{*}\in R^{n\times 1}\) is the desired image feature. Establish the relationship between the error \(e(t)\) and the camera speed \(V_{c}\in R^{6\times 1}\) through Equation (2):  
\[\left\{\begin{array}{l}e(t)=I_{x}V_{c}\\ V_{c}=-\hat{I}_{x}^{+}e(t)\end{array}\right. \tag{2}\]  
Further, the IBVS servo controller with proportional \(k\) control is constructed by establishing the relationship between the manipulator joint speed \(\dot{\theta}\) and the image feature error \(e(t)\) through Equation (3):  
\[\dot{\theta}=\hat{I}_{\theta}^{+}V_{c}=-k\hat{I}_{\theta}^{+}\hat{I}_{x}^{+}e(t) \tag{3}\]  
where \(\hat{I}_{x}^{+}=\left(\hat{I}_{x}^{T}\hat{I}_{x}\right)^{-1}\hat{I}_{x}^{T} \in R^{6\times 2n}\) is the Moore-Penrose pseudoinverse of the matrix \(\hat{I}_{x}\in R^{2n\times 6}\). In the 6-DOF manipulator used in this paper, \(\hat{I}_{\theta}^{+}=\left(I_{\theta}\right)^{-1}\in R^{6\times 6}\) is the inverse of the robot Jacobian matrix \(I_{\theta}\in R^{6\times 6}\), and \(\dot{\theta}\in R^{6\times 1}\) is the manipulator joint velocity vector.  
Equation (3) can be further expressed as follows:  
\[\dot{\theta}=-k\hat{I}_{s}^{+}e(t) \tag{4}\]  
In Equation (4), \(\hat{I}_{s}^{+}=\hat{I}_{\theta}^{+}\hat{I}_{x}^{+}\), \(\hat{I}_{s}^{+}\in R^{6\times 2n}\) is called the Moore-Penrose pseudoinverse of the feature-motion mapping matrix \(\hat{I}_{s}\in R^{2n\times 6}\). For 6-DOF manipulator IBVS tasks, in order to reflect the overall relative motion relationship between the manipulator and the object, at least three features are generally required, hence the matrix (\(n\geq 3\)).  
To implement image-based uncalibrated visual servoing control, a Kalman state estimation model is required to solve the feature-motion mapping matrix \(J_{s}\) in real time [34; 35]. The Kalman state estimation model, established based on the auxiliary dynamic system, is primarily divided into two parts, an estimation of the matrix \(J_{s}\) and a correction of the matrix \(J_{s}\), based on the feedback of feature \(S\) changes from the observer. In uncalibrated IBVS, when using Kalman estimation to solve the matrix \(J_{s}\), estimation errors and the limited FOV of the camera may cause the features to deviate from the camera's FOV, leading to feature loss, especially in unknown noisy environments. To ensure the visibility of visual features in uncalibrated IBVS, specific feature constraints are required. A detailed analysis of this will be provided in the following sections.