Equations of Motion Homework: 4-Wheel Car in Computer Game

[sabatier]

Homework Statement

A simple car with 4 wheels for use in a computer game. Front wheels turn only.
Derive the equations of motion for the car.

Car is controlled by setting its speed s, and steering angle phi.
The forward Euler method is used to compute the forward kinematics.

At time t the position vector is x, the orientation is theta.
Given the speed and steering angle, we want to compute
the updated values of x and theta for time t = t + delta t.
The car travels a distance s*(delta t) along the arc of a circle.
The radius of the circle (turning radius) is (steering angle)/4.
(where steering angle is in radians).

Derive the equations for the updated position and orientation
of the car.

Homework Equations

The Attempt at a Solution

 

[jbsteele]

Let x = [x, y] be the position of the car at time t, and theta be the orientation of the car. The updated position and orientation at time t + delta t can be computed as follows:x(t+delta t) = x(t) + s*[cos(theta)*delta t, sin(theta)*delta t] Theta(t+delta t) = theta(t) + (steering angle/4)*(delta t)

 

[nlightner]

To derive the equations of motion for the car, we first need to understand the basic principles of kinematics. Kinematics is the study of motion without considering the forces that cause the motion. In this case, we are dealing with a simple car with 4 wheels, where the front wheels are the only ones that turn.

To begin, we can define the position of the car at time t as x(t) and the orientation as theta(t). The car's position and orientation can be updated at time t + delta t by using the forward Euler method, which is a numerical method for approximating solutions to differential equations.

Using the forward Euler method, we can calculate the updated position and orientation of the car by using the following equations:

x(t + delta t) = x(t) + s*(delta t)*cos(theta(t))
theta(t + delta t) = theta(t) + (s*(delta t)/R)

where s is the speed of the car, delta t is the time interval, and R is the turning radius, which is equal to the steering angle divided by 4.

These equations take into account the fact that the car travels a distance of s*(delta t) along the arc of a circle with a radius of R. The steering angle determines the radius of the circle, which in turn affects the car's orientation.

In summary, the equations of motion for the car can be derived using the forward Euler method and taking into account the car's speed and steering angle. These equations can then be used to update the car's position and orientation in a computer game, allowing for realistic and accurate movement of the car.