# Monovision End-to-End Dual-Lane Overtaking Network without Map Assistance
## 2 Overtaking Guide Net
### Gaussian Differential Control Planner  
This paper adopts the primitive function of the Gaussian distribution as the trajectory for overtaking by an AV. It takes the form of a smooth S-shaped curve, as shown in the first part of Figure 2. The first derivative of this function represents the angle between the vehicle's body and the road in the vertical direction at the current moment, while the second derivative precisely corresponds to the steering wheel angle at the vehicle's position. The aim of this study was to design a _GDCP_ that can generate corresponding control signals by incorporating the current safety probability, ultimately achieving lane-changing overtaking. The overall model flow is illustrated in Figure 1. This paper proposes the overtaking phase parameter \(o_{p}\), combined with the vehicle speed \(v\), to replace the coordinate information input in the control planning function. When the vehicle's speed changes, it leads to variations in the longitudinal displacement of the vehicle. To address this, a velocity adaptation coefficient \(\alpha\) and a reference speed \(v_{0}\) are introduced, as shown in Equation (1). The overtaking trajectory, vehicle heading angle, and steering wheel angle at \(v_{0}~{}=~{}20\) m/s are depicted in Figure 2.  
As the vehicle speed increases, a higher value of \(\alpha\) results in the reduced curvature of the overtaking curve. The smoothing parameter \(\sigma_{s}\) of the overtaking trajectory function increases, and the stretching factor \(\sigma_{s}\) is defined as shown in Equation (1). \[\sigma_{s}=\alpha\sigma_{s_{0}}=\frac{v}{v_{0}}\sigma_{s_{0}} \tag{1}\]  
where \(\sigma_{s_{0}}\) is the smoothing parameter that can complete overtaking when \(v=v_{0}\). Since the variation range of \(x\) is different when the vehicle speed is different, this paper defines a range\([-\varepsilon\alpha,\varepsilon\alpha]\), and the AV only performs overtaking operations within this range, as shown in Equations (2) and (3). \[t^{\prime}=-\varepsilon\alpha+\frac{2\varepsilon\alpha}{o_{p}}t, \ t\in[0,o_{p}] \tag{2}\] \[S(-\varepsilon\alpha)<\xi \tag{3}\]  
Figure 2: Vehicle overtaking trajectory, heading angle, and steering wheel angle. During the entire overtaking phase, the size of the blue shaded area is the degree to which the vehicle changes its orientation every time the steering wheel controls the vehicle. The black pattern in the lower right corner of the image is the logo of the steering wheel. where \(\xi\) is a hyperparameter, which is a number close to 0. If \(\xi\) is too large, it could result in the vehicle not making smooth turns at the beginning and end of the overtaking maneuver, and it could also amplify the yaw angle error caused by differential operations. On the other hand, if \(\xi\) is too small, it might lead to a delay in transitioning to the opposite lane after receiving the overtaking signal, potentially causing a missed opportunity for optimal overtaking. When \(t\) is equal to \(o_{p}\), the vehicle has occupied the opposite lane and complete the lane change. \(\varepsilon\) is a hyperparameter that is utilized to control the range of lateral displacement in the transition lane. After putting Equation (2) into \(S(x)\) and calculating its second-order derivative, the output value theta is the rotation angle s of the steering wheel at this moment, as shown in Equation (4). \[\theta(t^{\prime})=\frac{t^{\prime}}{k\alpha_{s}^{3}}e^{\frac{-t^{\prime 2}}{2 \alpha_{s}^{2}}} \tag{4}\]  
where \(k\) is a fixed value that can affect the maximum lateral displacement of the vehicle during overtaking and is determined through repeated experiments. Once the vehicle completes the maneuver of overtaking the target vehicle in the opposite lane, the system reverses the past direction control sequence to return to the main lane. When the output probability is low, our system gradually reduces the vehicle's speed.