Forward euler calculations for position and orientation

[sabatier]

Hi, I'm trying to compute the equations of motion for a car as shown
in the attached image.

α = steering angle
θ = orientation of the car relative to the world coordinate system

Say you're given the linear velocity v and the steering
angle α. How do you compute the position and angle θ for a
particular time?

Any help appreciated.

 

[Cantab Morgan]

I find the problem difficult because a real car has a differential. Without it, a car wouldn't change directions when you turned the wheel, the front tires would just skid. My off the cuff guess is that the angle of steering doesn't give you enough information, but you might have to know something about how the car's differential worked.

In any case, coordinate systems are under your control so you should make them do your bidding. Consider transforming to a frame where the car is at rest relative to the origin, and $$\theta$$ is zero. Try solving the problem for a motorcycle first, or even a unicycle.

 

[astrokreng]

I understand your interest in computing the equations of motion for a car. To answer your question, the position and orientation of the car can be calculated using the Forward Euler method, which is a numerical method for solving ordinary differential equations.

To begin, we can use the equation of motion for linear velocity, v = rω, where r is the radius of the car's wheel and ω is the angular velocity of the wheel. We can then use the steering angle α to calculate the angular velocity of the car, ω = v/r * tan(α).

Next, we can use the Forward Euler method to calculate the position and orientation of the car at a particular time. This method involves using the current position and orientation, along with the calculated velocities, to predict the position and orientation at a future time. The equations for position and orientation using the Forward Euler method are as follows:

x(t+Δt) = x(t) + v * cos(θ(t)) * Δt
y(t+Δt) = y(t) + v * sin(θ(t)) * Δt
θ(t+Δt) = θ(t) + ω * Δt

Where x and y are the coordinates of the car's position, θ is the orientation, and Δt is the time step. By repeatedly applying these equations, we can calculate the position and orientation of the car at each time step.

It is important to note that the accuracy of the Forward Euler method depends on the size of the time step. Smaller time steps will result in a more accurate prediction of the car's position and orientation, but will also require more calculations. It is also important to consider any external forces or factors that may affect the car's motion, such as friction or wind resistance.

I hope this explanation helps you in your calculations. As always, it is important to carefully consider all variables and assumptions in your equations to ensure the accuracy of your results. Happy computing!