# Trajectory Tracking of Autonomous Vehicle Using Clothoid Curve
## 2 Related Work
### Control Curve  
The reference path obtained from the path planning module is used as the target curve of the trajectory tracking module. The general approach of the geometric controller is to select a preview point on the reference path, plan a curve from the current vehicle position to the preview point, and calculate the target angle of the steering wheel according to the geometric model of the vehicle, so as to realize the lateral control of the vehicle in the path tracking process. Pure pursuit as the most popular geometric trajectory tracking method uses an arc to connect the current position of the vehicle and the preview point on the reference path as the control curve, ignoring the constraint of the vehicle's path on the curvature continuity, which will lead to tracking deviation in theory. [PERSON] et al. proposed a trajectory tracking method based on the spiral theory, which considered both the position and direction of the vehicle at the target point [15], but this method is not widely used due to its complexity. [PERSON] et al. proposed using a quintic polynomial to draw the planned control curve, taking into account the position, direction, and curvature constraints of the starting and ending points [16]. This method does not guarantee a feasible solution and has high computational complexity, so it is not widely used. [PERSON] et al. [17] and [PERSON] et al. [18] both mentioned to use a clothoid curve to fit the control curve between the current position of the vehicle and the preview point. The most important feature of a clothoid curve is that its parameter function satisfies the continuity on the second derivative; that is, the curvature change in the curve is continuous. In fact, the curvature of the clothoid curve is proportional to the arc length of the curve, making it highly popular in highway or railway design to ease the curvature change between straight and circular routes. The general form of the clothoid curve parameter equation can be described as [18].  
\[\left\{\begin{aligned} x(s)&=x_{0}+\left[\int_{0}^{s} \cos(\tfrac{1}{2}k^{\prime}\tau^{2}+\kappa\tau+\theta_{0})d\tau\right]\\ y(s)&=y_{0}+\left[\int_{0}^{s}\sin(\tfrac{1}{2}k^{ \prime}\tau^{2}+\kappa\tau+\theta_{0})d\tau\right]\end{aligned}\right., \tag{1}\]  
where \(s\) represents the arc length of the curve, \(x_{0}\) and \(y_{0}\) represent the starting point coordinates of the curve, \(\theta_{0}\) represents the deflection angle of the starting point of the curve, \(\kappa\) represents the curvature at the starting point of the curve, and \(\kappa^{\prime}\) represents the curvature change rate of the curve.  
Figure 2: Geometric model based on Ackerman steering configuration.  
Compared with other curve fitting methods, using a clothoid curve to plan the vehicle control path has the advantages as follows.
* When the driver controls the steering wheel, in order to ensure lateral stability, the steering wheel is generally controlled at a constant speed, which means the angular speed of the steering wheel is constant. When the speed and front wheel angle change are very small in a tiny period of time and can be ignored, the change rate of the path curvature is constant, which is consistent with the characteristics of the clothoid curve. Therefore, the planned control path constructed by the clothoid curve can ensure that the vehicle control process is stable and the jitter is tiny.
* The clothoid curve satisfies the G2 continuity conditions. Compared with the arc curve constructed by the pure pursuit tracking algorithm, the curvature of the starting point of the curve is consistent with the initial turning curvature of the vehicle, which can reduce tracking deviation at the beginning period of the path.
* Compared with the polynomial curve, because its curvature changes linearly and continuously, it is easier to control and verify the curvature and curvature change rate of the planned control path and other constraints or limitations.