# Remote-Controlled Method with Force and Visual Assists Based on Time to Collision for Mobile Robot
## 3 Proposed Method
### Visual Assist  
A visual assist is used for detecting the trajectories of the mobile robot while the driver is checking the monitor. The predicted trajectory on the monitor is generated by transforming the coordinates from the camera coordinates to the monitor coordinates. On the camera coordinate, the predicted trajectory is drawn using \(v^{cmd}\), \(\omega^{cmd}\), and with regard to the TTC, as shown in Figure 8.  
This predicted trajectory is drawn from the mobile robot to the predicted end position after the TTC, assuming uniform motion of the mobile robot. The maximum TTC is set by the user as \(T_{max}\). If the TTC exceeds \(T_{max}\), the predicted end position is set to the position after \(T_{max}\). The predicted trajectory of the mobile robot is represented by the dots. The \(k\)th positions for each dot of the predicted trajectory \({}^{C}x_{k}\), \({}^{C}y_{k}\), and \({}^{C}z_{k}\) are determined using the circle equation. \(k\)\((k=1,2,\cdots,n^{s})\) is the coefficient for each dot. \(n^{s}\) is the number of dots determined by the TTC between the mobile robot and obstacle.  
\[\left({}^{C}x_{k}-r^{cmd}\right)^{2}+{}^{C}z_{k}^{2}=r^{cmd} \tag{28}\]  
In addition, as shown in Figure 8, the coordinate transformation is expressed using Equation (13). The color of the predicted trajectory is used as a gradation color to support remote-controlled operation. The details of this gradation color are explained later in this paper.  
Figure 9 shows an example of the predicted trajectory on the monitor. The blue wall is an obstacle. The gradation of the dots is expressed by the colors between green and yellow. The three lines consist of dots for the predicted trajectory. These lines are generated from the lines on the camera coordinates using Equation (28). The center line expresses the center position of the mobile robot. Both side lines indicate the edges of the mobile robot and is taken as the width.  
As shown in Figure 10, the distance \(l^{s}\) from the mobile robot to the obstacle is used to thin dots using a quadratic function. \(L\) and \(N\) are the axes of the distance from the mobile robot to the obstacle and the number of dots, respectively. \(l^{s}\) is calculated using Equation (15). \(N_{max}\) represents the maximum number of dots determined from the user setting. The number of dots \(n^{s}\) determined by the TTC between the mobile robot and obstacle is calculated as follows:  
\[n^{s}=\left\lceil\sqrt{\frac{l^{s}}{L_{max}}}\cdot N_{max}\right\rceil \tag{29}\]  
where \(\left\lceil\bigcirc\rceil\right\rceil\) is the ceiling function that assigns the smallest integer greater than or equal to each real number. \(L_{max}\) is the maximum distance of the trajectory from the mobile robot. The angular velocity of the mobile robot does not exceed the value. The distance \(l_{k}\)\((k=1,2,\cdot\cdot\cdot,n^{s})\) of each dot from the mobile robot is calculated as follows:  
\[l_{k}=\frac{L_{max}}{N_{max}^{2}}\cdot k^{2} \tag{30}\]  
where \(k\) is the coefficient of each dot.  
Figure 8: Predicted trajectory on the camera coordinate.  
Figure 9: Example of visual assist.