# Lane Change Trajectory Planning Based on Quadratic Programming in Rainy Weather

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}, the center of mass lateral deflection angle varies within the range of ±0.15°, and the yaw rate remains within the ±0.1°/s range.

## 1. Introduction

## 2. Quantitative Analysis of Rainy Weather Impact on Lane Change Maneuvers

#### 2.1. Coefficient of Friction

_{f}and a

_{r}are the front and rear wheel braking deceleration; ε is the correction factor, generally taking the value of 0.9; g is the acceleration of gravity (9.8 m/s

^{2}).

#### 2.2. Delayed Reaction Time

_{f}represents the delayed reaction time, S

_{d}is the critical value of safe visual distance (m), S

_{f}stands for rainy weather visibility (m), and t

_{r}is the normal reaction time, v

_{0}denotes the initial vehicle speed, with a value of 1 s [19].

#### 2.3. Lane Change Feasible Region in Rainy Conditions

#### 2.3.1. Boundary Conditions for the Current Lane Feasible Region

_{0}denotes the initial vehicle speed; t

_{r}is the normal reaction time; and t

_{dd}signifies driving delay time. As delineated in reference [13], the braking process is divided into phases, where during the initial increase in braking, the distance covered is denoted as L

_{di}, and during the continuous braking phase, the distance is L

_{c}. The sum of the distances for each phase constitutes the total braking distance, denoted as L

_{f}[26].

_{d}, comprising brake delay time t

_{dd}and brake duration t

_{di}(t

_{d}= t

_{dd}+ t

_{di}); typically, t

_{dd}is set at 0.15 s and t

_{di}at 0.1 s [15]. Following the lane change, vehicle A maintains a lower trailing speed. Upon the initiation of braking by vehicle A, vehicle B also engages in braking, with L

_{l}as the braking distance, where L represents the distance maintained between the two vehicles when at a stop, which is usually set as L = 2 m [18]. Thus, the following safe distance under rainy conditions is proposed as:

_{p}

_{1}and y

_{p}

_{1}denote the vehicle coordinate values on the X−axis and Y−axis, t

_{m}is the lane change start time, and t

_{p}

_{1}denotes the time when the vehicle is at the collision point p

_{1}.v

_{A}is the speed of the host vehicle, θ

_{A}is the vehicle yaw angle, D

_{1}is the longitudinal distance between the host vehicle A and the vehicle F in front of it, and b is the lateral distance driven by the host vehicle. v

_{F}(t) is denoted as the speed of vehicle F. Therefore, the longitudinal distance OP

_{1}of the host vehicle should satisfy the minimum safe following distance in rainy weather, and the boundary conditions for the current lane in rainy scenarios are proposed as follows:

#### 2.3.2. Boundary Conditions for the Feasible Region of the Target Lane

_{0}to time te, with a collision occurring at time t

_{c}, the process analysis is outlined as follows:

_{0}represents the initial vehicle separation distance, while the length of vehicle A is denoted as L

_{A}. The distance covered by vehicle A from t

_{0}to t

_{c}is represented by S

_{A}, and the corresponding distance for vehicle B within the same time interval is denoted as S

_{B}. If v

_{A}< v

_{B}, a collision between the vehicles will occur within the time t

_{c}. To avoid such a collision, it is necessary to fulfill the condition stated in Equation (9). Conversely, if v

_{A}> v

_{B}, the target vehicle A will move away from vehicle B. In this situation, a certain safe distance for S

_{0}must be maintained to prevent the trailing vehicle from being unable to avoid a collision through emergency braking.

_{A}, v

_{B}are the speeds of vehicle A and vehicle B; t

_{c}is the collision time, and the minimum safe distance for the rainy weather lane change process is proposed [18]:

_{2}, as shown in Figure 4; similarly, its coordinates (x

_{p2}, y

_{p2}) can be expressed as:

_{p}

_{2}and y

_{p}

_{2}denote the vehicle coordinate values on the X-axis and Y-axis, t

_{m}is the lane change start time, t

_{p}

_{2}denotes the time when the vehicle is at the collision point p

_{2}, V

_{A}is the velocity of the host vehicle, θ

_{A}is the vehicle yaw angle, v

_{R}(t) is denoted as the speed of the rear vehicle B, D

_{2}is the longitudinal distance between the host vehicle A and the rear vehicle B, and W is the lateral distance driven by the host vehicle. Therefore, the longitudinal distance OP

_{2}from the host vehicle should satisfy the minimum lane change safety distance constraint in rainy weather, and the proposed lane change boundary condition for the target lane in a rainy weather scenario is:

_{1}and P

_{2}, can be determined based on Equations (7) and (12). The constraints that P

_{1}and P

_{2}need to satisfy can be derived from Equations (8) and (13). Consequently, the feasible region for rainy day lane changing can be ascertained, as illustrated in Figure 4. The trajectories of OP

_{1}and OP

_{2}serve as boundary constraints, and the spatial region between OP

_{1}and OP

_{2}constitutes the viable rainy day lane change domain.

## 3. Lane Change Trajectory Planning

#### 3.1. Dynamic Programming

#### 3.1.1. Discrete Space Based on Rainy Day Lane Change Feasible Region

_{ij}denotes the vertices, i and j denote the row and column numbers, Δs denotes the S−direction unit spacing, and offset Δl is the sampling lateral offset.

_{ij}, with identical longitudinal distances as layer

_{i}, and employ it to construct the search space for generating preliminary lane change trajectories. This search space encompasses a series of contiguous trajectory clusters, serving as potential lane change candidates. Through dynamic programming computations, we identify sampling points that fulfill the minimum cost criterion. Subsequently, accounting for the vehicle’s smooth operation, the first and second derivatives of the trajectory are continuous and smooth. Therefore, we employ quintic polynomials to connect adjacent sampling points, producing rough lane change trajectory candidates. Ultimately, we formulate a cost function to assess the quality of these trajectory candidates.

_{0}, a

_{1}, a

_{2}, a

_{3}, a

_{4}, a

_{5}] is the longitudinal trajectory function coefficients, and lat = [b

_{0}, b

_{1}, b

_{2}, b

_{3}, b

_{4}, b

_{5}] is the lateral trajectory function coefficients; the rough trajectory of the lane change can be obtained as shown in Figure 9. It is represented in the Frenet coordinate system, as shown in Figure 10.

#### 3.1.2. Cost Function

_{DP}. Herein, w

_{vis}, w

_{obs}, and w

_{ref}, respectively, represent the corresponding weight coefficients for the cost terms.

_{n}represents the sampling points, S

_{f}denotes the visibility (m) during rainy weather conditions, and k

_{1}stands for the proportional coefficient. Under rainy conditions, due to the decrease in visibility, it becomes necessary to reduce the scope of the trajectory planning to enhance driving safety. Additionally, when the range of sampling points exceeds the current visibility limitations, the visibility cost function J

_{vis}is introduced to impose penalties. Similarly, during favorable visibility conditions, it becomes feasible to expand the trajectory planning scope, thereby enabling the dynamic adjustment of trajectory ranges. The parameter values are detailed in Table 1.

_{obs}, imposes penalties based on the distance between the ego vehicle and the obstacles. With the slippery road conditions and the reduced coefficient of friction due to rainy weather, the braking performance deteriorates, leading to an increase in braking distance. Therefore, we introduce an exponential function, as defined in Equations (6) and (11) to establish a safety distance constraint and to apply penalties to sampling points. Here, d represents the distance between the ego vehicle and the obstacles; k

_{1}, k

_{2}, k

_{3}, and k

_{4}are proportionality coefficients, and d

_{n}signifies the rainy weather nudge safety distance. When d

_{c}≤ d ≤ d

_{n}, the exponential function J

_{nudge}is introduced, representing a sharp increase in collision cost as the distance decreases. When d becomes smaller than the minimum collision distance d

_{c}, the cost function J

_{collision}reaches a maximum value. The parameter values are detailed in Table 1.

_{ref}, is designed to encourage the vehicle to travel along the centerline of the lane. Here, f

_{ref}represents the lane centerline, g

_{l}signifies a sampling point, and J

_{ref}imposes penalties on sampling points g

_{l}that deviate significantly from the lane centerline. According to the cost function, a traversal of the sampling points is performed to search for the sampling point with the minimum cost. Subsequently, by connecting the sampling points using a fifth−degree polynomial, a preliminary lane change trajectory is obtained, as illustrated in Figure 11.

#### 3.2. Quadratic Programming

_{m}signifies the total number of trajectory nodes in the optimization stage, ω

_{sta}represents the weight for the stability cost, ω

_{acc}denotes the weight for the acceleration cost, ω

_{jerk}stands for the weight for the jerk cost, and ω

_{smo}indicates the weight for the smoothness cost.

_{stability}, is defined using parameters representing lateral acceleration and yaw angle, which characterize stability. Here, φ represents the yaw angle; φ

_{des}signifies the desired minimum yaw angle; and a

_{l}and a

_{ldes}, respectively, denote lateral acceleration and desired minimum lateral acceleration along the l direction, while k

_{4}and k

_{5}are proportionality coefficients. By imposing penalties on the trajectory points with significant deviations in lateral acceleration and yaw angle, manipulation stability and passenger comfort are improved.

_{s}

_{,i}and a

_{l}

_{,i}denote the acceleration along the s and l directions; this penalty function makes the curvature and longitudinal acceleration of the exact trajectory relatively flat. J

_{jerk}is denoted as the rate of change of the acceleration and is defined as:

_{s}

_{,i}and j

_{l}

_{,i}, respectively, denote the rates of acceleration variation along the s direction and l direction.

_{smooth}, which introduces penalties for trajectories with higher curvature to reduce the degree of bending. Here, f′(s) represents heading error, f″(s) is related to curvature, and the derivative of curvature, denoted as f‴(s), ensures minimal variation in trajectory curvature. The coefficients k

_{6}, k

_{7}, and k

_{8}are proportionality factors. The parameter values are detailed in Table 1.

_{s},

_{i}is the longitudinal acceleration, a

_{l}

_{,i}is the lateral acceleration, and a

_{max}is the maximum acceleration.

## 4. Trajectory Tracking Control

_{y}is the acceleration along the body coordinate y direction at the center of mass of the vehicle; m is the mass of the vehicle; F

_{yf}and F

_{yr}are the combined lateral forces on the front and rear axle tires, respectively; I

_{z}is the rotational moment of inertia of the vehicle around the z-axis of the center of mass; ω is the yaw rate of the vehicle; l

_{r}and l

_{f}are the distances from the vehicle’s center of mass to the front and rear axles of the vehicle; c

_{f}and c

_{r}are the lateral deflection stiffnesses of the tires on the front and rear axles of the vehicle, respectively; and δ

_{f}is the front wheel angle.

_{y}is the lateral error; ${\stackrel{\xb7}{e}}_{y}$ is the lateral velocity error; e

_{φ}is the heading angular error; and ${\stackrel{\xb7}{e}}_{\phi}$ is the heading angular rate error. By designing a control step of T and utilizing a discrete LQR controller, the system is controlled based on its state−space equations [12]:

_{d}= (I − TA/2)

^{−1}(I + TA/2); B

_{d}= TB; x(k) represents the system state at time k; and u(k) denotes the control input at time k [17]. When performing tracking control, the controller’s objective is not only to reduce trajectory tracking errors but also to minimize the control effort, thus ensuring stable vehicle operation. Therefore, the objective function of the LQR controller is defined as follows:

^{T}PB(R + B

^{T}PB)

^{−1}B

^{T}PA + A

^{T}PA + Q is the positive definite solution of the Riccati equation; the longitudinal trajectory tracking control is mainly based on the literature [20] on the design of the longitudinal dual PID controller, the position PID controller, and the velocity PID controller for the transverse and longitudinal synergistic tracking control of the change in track trajectory.

## 5. Discussion

_{f}= 1.221 s. For the simulation setup, the ego vehicle is represented by a red rectangular block, designated with a velocity of v

_{ego}= 10 m/s. The surrounding obstacle vehicles are depicted as blue rectangular blocks, with a set velocity of v

_{obs}= 5 m/s.

^{2}. Thorough comparative analysis reveals that, with the introduction of a stability objective function during the quadratic programming phase, which is notably evident in the left/right lane change scenarios, the vehicle’s lateral acceleration curve exhibits a more pronounced reduction trend. Specifically, the peak lateral acceleration experiences a decrease of approximately 1 m/s

^{2}compared to the scenario where the stability objective function is not considered.

## 6. Conclusions

^{2}, the sideslip angle fluctuates within the range of ±0.15°, and the yaw rate is kept within the range of ±0.1°/s, fulfilling the requirements of comfort and stability. In this paper, although the environment and other influencing factors are considered, there is no real vehicle test under severe weather, and there are still complex weather environment situations that have not been considered. In the future, the lane change problem under different severe weather conditions can be investigated and more parameters and strategies can be explored to maintain the adaptability of the algorithm in diverse weather conditions. Additionally, in this paper, we only consider the host vehicle and the surrounding obstacle vehicles at a constant speed; in the next stage of research, we will consider the surrounding obstacle vehicles as having a variable speed in the speed planning stage of the host vehicle in order to adapt to more complex driving scenarios.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 7.**Transformation to Frenet coordinate system, Where the red line indicates the reference line and the orange line indicates the lane center line.

**Figure 10.**Lane change trajectory clusters in Frenet coordinates, where the red lines are lane change rough trajectory candidates and the blue points are sampling points.

**Figure 11.**Coarse lane change trajectory obtained via dynamic programming. In the illustration, the red lines depict the cluster of lane change trajectories, the green lines represent the coarse lane change trajectory, the blue points are sampling points, while the grey rectangles represent obstacle vehicles driving at a constant speed.

**Figure 12.**Trajectory optimization process based on quadratic programming. (The process begins with a coarse trajectory as the initial solution and utilizes the lane centerline S as a reference line. The spatial coordinates are discretized into a coordinate system with a resolution of Δs. The upper and lower bounds of L are determined based on road boundaries and obstacle information. By solving for each L

_{i}, the precise trajectory is obtained. Where the orange line forms the discretised sampling space, the indigo curve is the rough trajectory, the magenta curve is the precise trajectory, and the green rectangle is the obstacle vehicle).

**Figure 15.**State profiles of vehicle maneuverability ((

**a**,

**c**,

**e**) are the vehicle state profiles for the left lane change scenario with or without considering the maneuverability objective function, and (

**b**,

**d**,

**f**) are the vehicle state profiles for the right lane change scenario with or without considering the maneuverability objective function).

Parameters | Value |
---|---|

Vehicle weight, m | 2.02 t |

Sampling time, t | 0.05 s |

Wheelbase, L | 2.947 m |

w_{obs}, w_{ref}, w_{acc}, w_{jerk}, w_{smo} | 300, 100, 20, 80, 60 |

k_{1}, k_{2}, k_{3}, k_{4}, k_{5}, k_{6}, k_{7}, k_{8} | 40, 20, 15, 30, 20, 20, 10, 5 |

Moment of inertia, I_{z} | 4.095 t/rad |

Distance constant d_{n}, d_{c} | 4, 2 |

Sampling lateral distance, Δl | 1 m |

Longitudinal sampling distance, Δs | 10 m |

Front wheel cornering stiffness, c_{f} | 175.016 kN/rad |

Rear wheel cornering stiffness, c_{r} | 130.634 kN/rad |

Distance from center of mass to rear axis, b | 1.682 m |

Distance from center of mass to front axle, a | 1.265 m |

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## Share and Cite

**MDPI and ACS Style**

Deng, C.; Qian, Y.; Dong, H.; Xu, J.; Wang, W.
Lane Change Trajectory Planning Based on Quadratic Programming in Rainy Weather. *World Electr. Veh. J.* **2023**, *14*, 252.
https://doi.org/10.3390/wevj14090252

**AMA Style**

Deng C, Qian Y, Dong H, Xu J, Wang W.
Lane Change Trajectory Planning Based on Quadratic Programming in Rainy Weather. *World Electric Vehicle Journal*. 2023; 14(9):252.
https://doi.org/10.3390/wevj14090252

**Chicago/Turabian Style**

Deng, Chengzhi, Yubin Qian, Honglei Dong, Jiejie Xu, and Wanqiu Wang.
2023. "Lane Change Trajectory Planning Based on Quadratic Programming in Rainy Weather" *World Electric Vehicle Journal* 14, no. 9: 252.
https://doi.org/10.3390/wevj14090252