MATLAB Car-Like Model Kinematics: Calculating Body Velocity and Angular Velocity

This article provides MATLAB code for a car-like model's kinematics, outlining the equations for calculating body velocity and angular velocity based on vehicle state and input variables.

Model Assumptions:

• Vehicle State: x = [x, y, theta], where x and y represent the vehicle's position in the global coordinate frame, and theta represents the vehicle's heading.
• Input: u = [v, delta], where v is the vehicle's speed, and delta is the steering angle.

Kinematics Equations:

The car-like model's kinematics are defined by the following equations:

• Velocity in Global Coordinates:

dx/dt = [v * cos(theta), v * sin(theta), v * tan(delta) / L]

where L is the vehicle's wheelbase (distance between the wheels).

• Body Velocity:

v_body = v * cos(beta)

• Body Angular Velocity:

w_body = v_body * tan(delta) / L

where beta = atan2(sin(theta), cos(theta)) represents the vehicle heading angle in the body coordinate frame.

MATLAB Code:

% Vehicle state
x = [1, 2, pi/4]; % x, y, theta

% Input
v = 10; % Vehicle speed
delta = 0.1; % Steering angle

% Wheelbase
L = 2; % Distance between wheels

% Calculate beta
beta = atan2(sin(x(3)), cos(x(3)));

% Calculate body velocity
v_body = v * cos(beta);

% Calculate body angular velocity
w_body = v_body * tan(delta) / L;

% Display results
disp(['Body Velocity: ', num2str(v_body)])
disp(['Body Angular Velocity: ', num2str(w_body)])

This code will calculate and display the body velocity and angular velocity for the given vehicle state and input values. You can modify the code to experiment with different values and observe the effects on the output.

Note: This code assumes that the vehicle is moving in a 2D plane. For a 3D model, the kinematics would need to be adjusted to account for the additional degrees of freedom.