# Enhancing Autonomous Vehicle Navigation with a Clothoid-Based Lateral Controller
## 2 Overview of the Clothoid-Based Lateral Control  
Clothoids are an effective solution for vehicle path generation due to their ability to maintain continuous curvature. The primary benefit of using clothoids is their capacity to create paths that are smooth in position, direction, and curvature. Our study employs a previously described method [3], which incorporates clothoid construction into the tracking controller's functionality. At each time step, the controller computes the optimal clothoid from the vehicle's current position to a predetermined lookahead point on the reference path, leading the vehicle along a hyper-clothoid trajectory. These generated clothoids facilitate the calculation of the steering wheel angle for the upcoming time step. Furthermore, this method is characterized by its low computational demands and ease of real-time application. Figure 1 graphically depicts the controller's operation, showing vehicle positions over time and the corresponding clothoids generated by the controller. It is evident that as the cross-track error diminishes, the constructed clothoid increasingly aligns with the intended reference path. Regarding the foundational theory of clothoids, these curves are characterized by a curvature that varies linearly with respect to their arc length. Clothoids are formally  
Figure 1: Sequential vehicle positions (\(t_{0}\) to \(t_{4}\)) with the corresponding optimal clothoids generated by the lateral controller. expressed through parametric representations of Fresnel Integrals, specifically \(S(t)\) for the sine component and \(C(t)\) for the cosine component. \[S(t) = \int_{0}^{t}\sin\Bigl{(}\frac{\pi}{2}\tau^{2}\Bigr{)}\mathrm{d}\tau \tag{1}\] \[C(t) = \int_{0}^{t}\cos\Bigl{(}\frac{\pi}{2}\tau^{2}\Bigr{)}\mathrm{d}\tau \tag{2}\]  
Theoretically, fitting clothoids between two points to create a smooth path depends on the specific requirements for the start and end conditions, such as orientation (direction) and curvature at those points, for example, for the following:  
* Two clothoids: Transitioning from one curve of a certain curvature to another curve of a different curvature, requires two clothoid segments--one to decrease the curvature to zero (straighten out) and another to increase the curvature to match the second curve. * Three clothoids: In more complex path-planning scenarios, such as an S-curve where the vehicle needs to transition from a curve in one direction to a curve in the opposite direction, a sequence of three clothoids might be used. The first clothoid segment adjusts the curvature from the initial curve to a straight path, the second segment transitions through zero curvature, and the third segment increases the curvature to match the final curve's requirements. The presented approach for waypoint navigation employs hyper-clothoids, utilizing a single clothoid segment fit from the location of the car to a target point located on the reference path at a certain Euclidean distance in front of the vehicle. Within the context of path planning, the boundary value problem associated with this method is established with the primary objective of approximating the target point as closely as feasible, subject to the constraints imposed by the saturation limits of the curvature rate (\(\kappa^{\prime}\)). Consequently, the boundary conditions are reformulated as detailed below:  
\[ \begin{array}{c}x(0)=x_{\mathrm{car}},\qquad x(L)=x_{\mathrm{target}}\pm \delta_{x}\\ y(0)=y_{\mathrm{car}},\qquad y(L)=y_{\mathrm{target}}\pm\delta_{y}\\ \psi(0)=\psi_{\mathrm{car}},\qquad\kappa(0)=\kappa_{\mathrm{car}}\end{array}  \tag{3}\]  
The clothoid segment from the location of the car is projected such that the path is drivable and is continuous with respect to the curvature and curvature rate. Due to these constraints on the clothoid projection, there may be instances where the segment does not terminate precisely at the target point.