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In homework 3.4, you're asked to simulate circular robotic motion.
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We gave you some equations to help you along in your simulations.
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I want to give you those formulas again and explain where some of them came from.
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The first equation I want to talk about is this one.
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The radius of curvature is equal to the length of the vehicle over the tangent of alpha
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where alpha is our steering angle. Let me write that up here.
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So where does this equation come from.
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To derive it, the key realization is that the front and rear tire do not travel along the same circle.
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Here's my rear tire, and here's my front tire.
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They are, of course, separated by a distance that we called "L."
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Let's draw the circles that these tires travel along.
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Well, this rear tire is actually going to travel along a smaller inner circle
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while this tire is going to travel along a larger outer circle.
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Since we defined our radius of curvature as the distance from the back tire to the center,
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Let's label this r, and we can see that the line connecting these tires defines an axis,
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and here we have our steering angle, alpha, from here.
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Now we can do a little bit of geometry.
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Let's make a right triangle.
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Well, if this angle here is alpha, then this much be a 90 degree angle,
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because a radius intersecting with a tangent line always forms a right angle.
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That means that that this angle here must be equal to 90 degrees minus alpha,
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which means this angle, since this is a right triangle must be alpha.
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Well, we're almost there. The tangent of this angle is equal to the opposite side,
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which is the length, over the adjacent side, which is the radius.
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So tangent of alpha is equal to L over r.
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We manipulate this equation a little bit, and we find that the radius of curvature
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is equal to the length of the vehicle over the tangent of the steering angle.