线性二自由度模型

该模型将车辆简化为具有侧向与横摆两个自由度的刚体。设质心位置为 \( G \)，纵向速度为 \( u \)，侧向速度为 \( v \)，横摆角速度为 \( \omega_r \)。运动方程可通过以下Python代码（使用SymPy库）表达：

from sympy import symbols, Function, Derivative

t_var = symbols('t')
long_vel = symbols('u')
lat_vel = Function('lat_vel')(t_var)
yaw_rate = Function('yaw_rate')(t_var)
steer_angle = symbols('steer_angle')
mass = symbols('mass')
inertia_z = symbols('inertia_z')
front_axle_dist = symbols('front_axle_dist')
rear_axle_dist = symbols('rear_axle_dist')
front_stiff = symbols('front_stiff')
rear_stiff = symbols('rear_stiff')

front_lateral_force = -front_stiff * (lat_vel + front_axle_dist * yaw_rate) / long_vel
rear_lateral_force = -rear_stiff * (lat_vel - rear_axle_dist * yaw_rate) / long_vel

lateral_eq = mass * (Derivative(lat_vel, t_var) + long_vel * yaw_rate) - front_lateral_force - rear_lateral_force
yaw_eq = inertia_z * Derivative(yaw_rate, t_var) - front_axle_dist * front_lateral_force + rear_axle_dist * rear_lateral_force

其中，lateral_eq为侧向力平衡方程，yaw_eq为横摆力矩平衡方程。此模型忽略轮胎非线性和悬架影响。

Carsim模型

Carsim为专业整车动力学仿真软件，整合轮胎、悬架、传动系等子系统。其模型采用魔术公式等轮胎模型，精确描述非线性特性。

用户可设置几何参数、质量分布及工况（如转向角、路面条件），输出高精度动力学响应。

运动学模型

此模型描述车辆位置变化，忽略力学因素。基于自行车假设，实现轨迹预测：

import numpy as np
import matplotlib.pyplot as plt

def simulate_trajectory(init_x, init_y, init_heading, velocity, steer_angle, wheelbase, time_step, steps):
    x_pos = np.zeros(steps)
    y_pos = np.zeros(steps)
    heading_angle = np.zeros(steps)
    x_pos[0] = init_x
    y_pos[0] = init_y
    heading_angle[0] = init_heading
    
    for step in range(1, steps):
        dx_dt = velocity * np.cos(heading_angle[step - 1])
        dy_dt = velocity * np.sin(heading_angle[step - 1])
        dtheta_dt = velocity * np.tan(steer_angle) / wheelbase
        x_pos[step] = x_pos[step - 1] + dx_dt * time_step
        y_pos[step] = y_pos[step - 1] + dy_dt * time_step
        heading_angle[step] = heading_angle[step - 1] + dtheta_dt * time_step
    
    return x_pos, y_pos

start_x = 0
start_y = 0
start_heading = 0
speed = 5  
steer_input = np.deg2rad(5)  
base_length = 2.5  
delta_t = 0.1  
iterations = 100

x_coords, y_coords = simulate_trajectory(start_x, start_y, start_heading, speed, steer_input, base_length, delta_t, iterations)

plt.plot(x_coords, y_coords)
plt.xlabel('X坐标 (m)')
plt.ylabel('Y坐标 (m)')
plt.title('基于运动学模型的车辆轨迹')
plt.grid(True)
plt.show()

模型计算量小，适用于路径规划。