The operational stability of unmanned vehicles requires a comprehensive consideration of vehicle yaw stability and roll stability. Therefore, a vehicle dynamics model with comprehensive equivalent constraints of yaw, sideslip and roll is established.
1. Kinetic model derivation
The vehicle dynamics model is extended on the basis of the kinematics model, the vehicle speed is increased, combined with the pure cornering characteristics of the tire, and the coupling relationship of the transverse and longitudinal forces is ignored, and the vehicle dynamics model is obtained.
In the vehicle coordinate system, the components of the center-of-mass velocity v on the x-axis and y-axis are pictures and pictures. When the vehicle is driving, it is not only doing translation, but also rotating around a certain instantaneous axis. At the time of the picture, the vehicle coordinate system The magnitude and direction of the center-of-mass velocity v change relative to time t, and at this time the position and direction of the vehicle coordinate system relative to the earth coordinate system also change. Under the vehicle coordinate system at the time of the picture relative to the time t, the changes of the picture and the picture within the picture time are respectively:
Considering that the picture is very small, make the following assumptions (within 15°, it can be enlarged according to actual calculation):
At the same time ignoring the second-order trace, the changes of the picture and the picture within the picture time are respectively:
Therefore, in the vehicle coordinate system, the rate of change of velocity along the x-axis and y-axis, that is, the acceleration along the x-axis and y-axis is:
The picture and the picture are an inertial acceleration, also called Coriolis acceleration, and the corresponding force is also called Coriolis force. Therefore, in the vehicle coordinate system, the balance equations along the x-axis, y-axis and the balance equation around the z-axis are:
Decompose the force of the front and rear wheels along the x-axis and y-axis into the lateral force and longitudinal force of the front and rear tires, and get:
The component of the front wheel lateral force on the x-axis is always negative, so add a negative sign in front to correct the positive and negative influence of the wheel rotation angle, and the parameters in the above formula are all vectors.
The lateral force and longitudinal force of the tire are related to the lateral/longitudinal stiffness, side slip angle/slip rate, ground friction coefficient, and vertical load of the tire. In the determined vehicle and ground, the lateral force and longitudinal force can be considered as follows relation:
The front and rear wheel slip angles can be obtained according to the tangent of the speed. Since the front wheels are steering wheels, the slip angle is actually the difference between the front wheel rotation angle and the tangent of the speed:
Since the vehicle is a whole, the speed of each point in the x-axis direction is the same. In the y-axis direction, according to the direction of the vehicle yaw angular velocity, the y-axis velocity of the front and rear wheels relative to the center of mass is the product of the angular velocity and the distance from the center of mass, and the velocity of the y-axis direction relative to the instantaneous center is plus the center of mass The velocity in the y-axis direction relative to the instant center:
Putting the above formula into the side slip angle formula, the front and rear wheel side slip angles can be obtained as:
Combining the above relations, the following relations can be obtained:
In the earth coordinate system, the speed of the vehicle along the X-axis and Y-axis is:
Let the control quantity and the state quantity be respectively:
can get:
So far, the derivation of the vehicle dynamics model is completed.
Second, small angle simplification
At small angles, the above formula can be simplified as:
3. Kinetic Model Validation
Carsim outputs the vehicle's lateral and longitudinal velocity and yaw rate to the simulink module to verify its accuracy.
The Carsim output is as follows:
The longitudinal speed is maintained at 60km/h, and the corner input is as follows:
The simulink model is as follows:
The s-function code is as follows:
function sys=mdlUpdate(t,x,u)
%% 车辆参数
tt=0.05; % 采样时间
cfc=66900; % 此处轮胎刚度借用《无人驾驶车辆模型预测控制》中的刚度参数
cfl=66900;
crc=62700;
crl=62700;
sf=u(7); % 前后轮滑移率
sr=u(8);
a=1.015; % 前轮到质心的距离
b=1.895; % 后轮到质心的距离
iz=1536.7; % 转动惯量
m=1270; % 质量
sw=u(9)/18*pi/180;
%% 动力学模型
Y=x(1);
X=x(2);
phi=x(3);
y_dot=u(4)/3.6;
x_dot=u(5)/3.6;
phi_dot=u(6);
Y_dot=x_dot*sin(phi*pi/180)+y_dot*cos(phi*pi/180);
X_dot=x_dot*cos(phi*pi/180)-y_dot*sin(phi*pi/180);
x(1)=Y+Y_dot*tt;
x(2)=X+X_dot*tt;
x(3)=phi+phi_dot*tt;
y_dot=x(4)/3.6;
x_dot=x(5)/3.6;
phi_dot=x(6)*pi/180;
dy_dot=-x_dot*phi_dot+2/m*(cfc*(sw-(y_dot+a*phi_dot)/x_dot)+cfl*sf*sw-crc*(y_dot-b*phi_dot)/x_dot);
dx_dot=y_dot*phi_dot+2/m*(-cfc*sw*(sw-(y_dot+a*phi_dot)/x_dot)+cfl*sf+crl*sr);
dphi_dot=2/iz*(a*cfc*(sw-(y_dot+a*phi_dot)/x_dot)+a*cfl*sf*sw+b*crc*(y_dot-b*phi_dot)/x_dot);
x(4)=x(4)+dy_dot*tt*3.6;
x(5)=x(5)+dx_dot*tt*3.6;
x(6)=x(6)+dphi_dot*tt*180/pi;
sys = x;
The simulation results are as follows (the blue line is the carsim output value, and the red line is the calculated value):
4. Linearization
Similar to the linearization of the kinematic model, consider that a certain working point of the system is a picture, and the picture is the state quantity of the system obtained after applying picture control, and there is the following first-order linear differential equation:
Carry out Taylor expansion at the point picture, ignoring the second-order and higher-order terms, get:
make:
Available:
in:
5. Discretization
6. Extension
Pull motion, Coriolis force, Coriolis acceleration, tire transverse model, tire longitudinal model, magic formula.
7. Personal understanding
Judging from the simulation results, the dynamic model is not very accurate in calculating the lateral and longitudinal velocities. From the carsim animation, the attitude of the vehicle changes relatively little. Consider trying to correct the tire stiffness and slip rate. In addition, when turning at a large angle, the posture of the car body in the animation changes more obviously, and the above model is no longer applicable at this time. Therefore, the simplified model actually has stricter requirements on the size and change rate of the front wheel angle.
When calculating the tire stiffness, the tire curve in carsim is used to invert the tire stiffness using the magic formula, and the deviation is relatively large. Therefore, the tire size and stiffness in the book are directly borrowed, and the follow-up plan is to expand the tire separately. The subsequent update plan is to apply the LQR and MPC algorithms on the basis of the above model, followed by the "episode" of the EPS system, and then the development of LKA and so on.