# Lane Detection Algorithm Using LRF for Autonomous Navigation of Mobile Robot
## 4 Lane Recognition and Prediction
### Curvature Algorithm  
When a point moves along a curve at a given velocity, the moving direction changes according to \(s\), the distance covered (the length of the arc of the curve). Therefore, the rate of change is called the curvature of the curve. Figure 7 illustrates the degree of rotation required to follow the curve while moving along the curve from the start point to the end point. where \(\Delta s\) denotes the point-to-point infinitesimal distance and \(\Delta\theta\) represents the infinitesimal angle formed by the tangent lines of two points. \[k=\lim_{\Delta s\to\infty}\left|\frac{\Delta\theta}{\Delta s}\right|=\left| \frac{d\theta}{ds}\right| \tag{5}\]  
where \(\Delta s\) denotes the infinitesimal distance covered from a start point on a curve to the end point and \(\Delta\theta\) denotes the angle formed by two tangent lines at two points. Further, the multiplicative inverse of curvature is the curvature radius. The curvature radius is the radius of a circle having a certain curvature and is also referred to as the rotational radius. Curvature radius \(\rho\) is defined as Equation (6). \[k>0\Rightarrow\rho=1/k \tag{6}\]  
Equation (5) shows that \(\Delta s\), which denotes the total length of the arc moved along the curve, is proportional to the curvature radius. On the whole, curvature \(k\) is shown to be inversely related to \(\Delta s\). If \(k=0\), the curvature radius is infinity, and thus, the curve becomes a straight line. In other words, a straight line can be thought to be a circle having an infinite radius. If the robot moves rotationally, a rotational centroid is generated, and this point is the instantaneous center of rotation (ICR). ICR is a point at which the extension lines of the rotational axes of the wheels meet [34; 35; 36; 37]. Since the axis of the two wheels of the mobile robot analogous to the vehicle wheels' moving tendency is positioned along the same line, ICR can be positioned anywhere on the wheel axis. Further, the relationship between the wheel velocity values and the robot-to-ICR distance can be expressed by a proportional equation such as Equation (7). \[v_{L}:v_{R}=\rho-\frac{l}{2}:\rho+\frac{l}{2} \tag{7}\]  
Figure 7: Curvature of the curve. where \(v_{L}\) denotes the velocity of the left side wheels; \(v_{R}\), that of the right side wheels; \(\rho\), the curvature radius; and \(l\), the lane between two wheels of the robot. The curvature radius can be expressed by Equation (8). \[\rho=\frac{l}{2}\bigg{(}\frac{v_{R}+v_{L}}{v_{R}-v_{L}}\bigg{)} \tag{8}\]  
In other words, the rotational radius is determined by the velocity of the robot's left and right wheels. According to the above equation, if the robot navigates along a straight line, \(\rho=\infty(v_{R}=v_{L})\), and if \(v_{R}\
eq v_{L}\), the robot implements a circular motion. In Figure 8, if the mobile robot moves from point A to point B, let the time at point A be \(t_{0}\) and its location \(\big{(}x_{p},y_{p},\theta_{p}\big{)}\), then the time at point B can be expressed as \(t_{1}=t_{0}+\Delta t\) and its location as \(\big{(}x^{\prime},y^{\prime},\theta^{\prime},\theta^{\prime}\big{)}\). ICR's coordinates can be defined as in Equation (9). \[ICR=\big{[}x_{R}-R\sin(\theta_{R}),y_{R}+R\cos(\theta_{R})\big{]} \tag{9}\]  
The robot's location \((x^{\prime},y^{\prime},\theta^{\prime},\theta^{\prime})\) at the instant of time lapse of \(\delta t\) from \(t_{0}\) can be expressed by Equation (10). \[ \begin{array}{c}\left[\begin{array}{c}x^{\prime}_{p}\\ y^{\prime}_{p}\\ \theta^{\prime}\end{array}\right]\quad=\left[\begin{array}{ccc}\cos(\psi \bullet\Delta t)&-\sin(\psi\bullet\Delta t)&0\\ \sin(\psi\bullet\Delta t)&\cos(\psi\bullet\Delta t)&0\\ 0&0&1\end{array}\right]\left[\begin{array}{c}x_{p}-ICR_{x}\\ y_{p}-ICR_{y}\\ \theta\end{array}\right]+\left[\begin{array}{c}ICR_{x}\\ ICR_{y}\\ \psi\bullet\Delta t\end{array}\right]\\ =\left[\begin{array}{c}R\sin(\psi\bullet\Delta t)+ICR_{x}\\ -R\cos(\psi\bullet\Delta t)+ICR_{y}\\ \theta+\psi\bullet\Delta t\end{array}\right]\end{array}