# Trajectory Tracking of Autonomous Vehicle Using Clothoid Curve
## 3 Tracking Control Method Using Clothoid Curve
### Preview Point Selection
When the vehicle speeds up, the maximum motion curvature is limited by limiting the lateral acceleration. \[\kappa_{max}=\left\{ \begin{array}{l}\frac{tan(\delta_{max})}{L}(v<v_{slow}) \\ \frac{d_{y}}{(v*b)}(v\geq v_{slow})\end{array} \right., \tag{10}\]

where \(\delta_{max}\) is the maximum turning angle of the front wheel, \(L\) is the wheelbase, \(v\) is the vehicle speed, \(a_{y}\) is the lateral acceleration, and \(v_{slow}\) is the low-speed threshold. The parameter used to determine whether the vehicle stops or not can be set to 0.1 m/s in practical application

#### 3.5.2 Maximum Curvature Change Rate Constraint

According to Formula (6), the curvature change rate of the curve is limited by the maximum angular velocity of the steering wheel. When the speed is very low or the vehicle is stationary, the maximum curvature change rate of the limited curve is set to an experience value. The larger this value is, the more obvious the vehicle jitters when starting. To sum up, the calculation method of the curvature change rate of the curve is as follows. \[k^{\prime}_{max}=\left\{ \begin{array}{l}\frac{\omega}{k*Ls+v*cos^{2}(\delta )}(v>v_{slow})\\ experienced\_value(v\leq v_{slow})\end{array} \right. \tag{11}\]

The meaning of the characters in the upper formula are consistent with that in Formulas (7) and (10). In later experiments, the \(experienced\_value\) is set to 0.5. #### 3.5.3 Clothoid Curve Length Constraint

The length of the planned control path designed by the clothoid curve is required to be long enough for two reasons. On the one hand, the algorithm proposed in this paper needs to preview a point at a distance on the clothoid curve according to the vehicle speed and the servo response delay time of the steering system. The length of the planned control path must be greater than the preview length; on the other hand, if the length of the planned control path is too short, cross error may be raised because of the large change in the curvature of the clothoid curves calculated successively. It is determined that the clothoid curve length is not less than one meter, which can achieve good tracking performance. The clothoid curve control follow path length constraint proposed in this paper is defined as follows. \[\left\{ \begin{array}{l}length\Big{(}s_{[0]}\Big{)}>v*0.5(v>2\ \text{m/s})\\ length\Big{(}s_{[0]}\Big{)}>1.00(v<=2\ \text{m/s})\end{array} \right., \tag{12}\]

where \(length\Big{(}s_{[0]}\Big{)}\) is the length of the \(S[0]\) curve or the planned control path, and \(v\) is the current vehicle speed. ### Preview Control Based on Steering System Response Hysteresis

We propose two methods to obtain the target angle of the steering wheel. The first method is to calculate the target steering angle by previewing the curvature of a point on the planned control path, as shown in the following formula. \[\left\{ \begin{array}{l}\kappa_{preview}=\kappa_{v}+\kappa^{\prime}*v*t_{2} \\ \delta=\tan^{-1}(L*\kappa_{preview})\\ angle=\delta*k\end{array} \right., \tag{13}\]

where \(\kappa_{v}\) is the current steering curvature of the vehicle, \(\kappa^{\prime}\) is the curvature change rate of the planned control path, \(v\) is the current speed, \(t_{2}\) is the steering system response hysteresis time, \(\delta\) is the front wheel angle, \(L\) is the wheelbase, \(k\) is the steering system transmission ratio, \(\kappa_{preview}\) represents the target curvature of the preview, and \(angle\) is the target steering angle. The second method is to calculate the target turning rate of the steering wheel using the curvature change rate of the planned path \(k^{\prime}\) according to Formula (6). Then, the target angle is calculated by previewing the steering wheel angle by the steering system response hysteresis time, as shown in the following formula. \[\left\{ \begin{array}{l}\omega=\kappa^{\prime}*k*L*v*\cos^{2}(\delta_{v})\\ angle=angle_{v}+\omega*t_{2}\end{array}