
In homework 3.4, you're asked to simulate circular robotic motion.

We gave you some equations to help you along in your simulations.

I want to give you those formulas again and explain where some of them came from.

The first equation I want to talk about is this one.

The radius of curvature is equal to the length of the vehicle over the tangent of alpha

where alpha is our steering angle. Let me write that up here.

So where does this equation come from.

To derive it, the key realization is that the front and rear tire do not travel along the same circle.

Here's my rear tire, and here's my front tire.

They are, of course, separated by a distance that we called "L."

Let's draw the circles that these tires travel along.

Well, this rear tire is actually going to travel along a smaller inner circle

while this tire is going to travel along a larger outer circle.

Since we defined our radius of curvature as the distance from the back tire to the center,

Let's label this r, and we can see that the line connecting these tires defines an axis,

and here we have our steering angle, alpha, from here.

Now we can do a little bit of geometry.

Let's make a right triangle.

Well, if this angle here is alpha, then this much be a 90 degree angle,

because a radius intersecting with a tangent line always forms a right angle.

That means that that this angle here must be equal to 90 degrees minus alpha,

which means this angle, since this is a right triangle must be alpha.

Well, we're almost there. The tangent of this angle is equal to the opposite side,

which is the length, over the adjacent side, which is the radius.

So tangent of alpha is equal to L over r.

We manipulate this equation a little bit, and we find that the radius of curvature

is equal to the length of the vehicle over the tangent of the steering angle.