## 2 Problem Formulation  
The mobile robot with a limited field-of-view camera system and the coordinates are described in Figure 1. The frame \(\mathcal{F}_{c}\) of the fixed onboard camera is consistent with the robot frame \(\mathcal{F}_{r}\). So the camera is regarded as a part of the mobile robot. Consider a fixed frame \(\mathcal{F}_{w}\) which is consistent with the robot frame \(\mathcal{F}_{r}^{*}\) when the mobile robot is in the desired configuration. A right-handed frame is defined with origin \(O\) and \(x\) axis is parallel to the robot wheel axis, \(z\) axis is along the camera optical axis, \(y\) axis is orthogonal to the motion plane, respectively. The posture state of the mobile robot is described by \(X(t)=\left(x(t),z(t),\theta(t)\right)^{T}\), where \(\left(x(t),z(t)\right)\) are the position coordinates of the mobile robot in the Cartesian plane, and \(\theta(t)\) is the orientation of the mobile robot with respect to \(z\) axis as shown in Figure 1. The kinematics of the mobile robot can be expressed as  
\[\left\{\begin{array}{l}\dot{x}=v\sin\theta\\ \dot{z}=v\cos\theta\\ \dot{\theta}=w,\end{array}\right. \tag{1}\]  
where \(v,w\) are the linear and angular velocities that are used as the control inputs of the mobile robot.  
\(\mathcal{P}_{i}\in\mathcal{R}^{3},i=1,2,3,4\) are the static points in the world frame \(\mathcal{F}_{w}\). In order to focus on the proposed method and simplify the system modeling, four marker points are used in this work while at least two points have the same height in the \(y\) axis direction but not in the camera optical axis plane to avoid the singular structure. In the actual scenario application, the extraction and matching of image points could be carried out using SIFT, SURF and ORB features instead of the marker points. Then the perspective projection geometry for point \(\mathcal{P}_{i}\) is derived,  
\[\left[\begin{array}{c}s_{i}(t)\\ 1\end{array}\right]=\frac{1}{Z_{oi}}\big{[}\begin{array}{cc}K&\mathbf{0}_{3 \times 1}\end{array}\big{]}\bigg{[}\begin{array}{cc}R^{T}&-R^{T}t_{r}\\ \mathbf{0}_{1\times 3}&1\end{array}\bigg{]}\bigg{[}\begin{array}{c} \mathcal{P}_{i}\\ 1\end{array}\bigg{]}, \tag{2}\]  
Figure 1: Coordinate relationship for a mobile robot with a limited field-of-view (dashed lines) camera.  
where \(s_{i}(t)=(u_{i},v_{i}),i=1,2,3,4\) are the image coordinates. \(Z_{ci}\) is the depth of the point \(\mathcal{P}_{i}\) in the frame \(\mathcal{F}_{c}\). \(R\) and \(t_{r}\) defined as  
\[R=\left[\begin{array}{ccc}c\theta&0&s\theta\\ 0&1&0\\ -s\theta&0&c\theta\end{array}\right],t_{r}=(x,0,z)^{T} \tag{3}\]  
are the rotation matrix and translation vector of the mobile robot with respect to the world frame, respectively. \(s\theta\) and \(c\theta\) are the abbreviations of \(\sin\theta\) and \(\cos\theta\), respectively. The camera intrinsic matrix \(K\) is  
\[K=\left[\begin{array}{ccc}f_{u}&0&u_{0}\\ 0&f_{v}&v_{0}\\ 0&0&1\end{array}\right],\]  
where \(f_{u}\) and \(f_{v}\) are the scaled focal length in \(x\) and \(y\) directions, respectively. \((u_{0},v_{0})\) are the coordinates of the principle point in pixels. Calculating the derivative of the image point in equation (2) with respect to time yields  
\[\dot{s}_{i}(t)=\underbrace{\left[\begin{array}{ccc}\frac{u_{i}-u_{0}}{Z_{ci }}&-f_{u}-\frac{(u_{i}-u_{0})^{2}}{f_{u}}\\ \frac{v_{i}-v_{0}}{Z_{ci}}&-\frac{(u_{i}-u_{0})(v_{i}-v_{0})}{f_{u}}\end{array} \right]}_{J(s_{i},Z_{ci})}\left[\begin{array}{c}v\\ w\end{array}\right], \tag{4}\]  
where \(J(s_{i},Z_{ci})\) is the interaction matrix of an image point.  
The desired image is taken at the desired configuration \(X^{*}=(0,0,0)^{T}m\) in advance. The desired image is used to guide the mobile robot from the current configuration to the desired configuration.