[Script Info]
Title:
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:04.00,Default,,0000,0000,0000,,In homework 3.4, you're asked to simulate circular robotic motion.
Dialogue: 0,0:00:04.00,0:00:08.00,Default,,0000,0000,0000,,We gave you some equations to help you along in your simulations.
Dialogue: 0,0:00:08.00,0:00:12.00,Default,,0000,0000,0000,,I want to give you those formulas again and explain where some of them came from.
Dialogue: 0,0:00:12.00,0:00:17.00,Default,,0000,0000,0000,,The first equation I want to talk about is this one.
Dialogue: 0,0:00:17.00,0:00:23.00,Default,,0000,0000,0000,,The radius of curvature is equal to the length of the vehicle over the tangent of alpha
Dialogue: 0,0:00:23.00,0:00:27.00,Default,,0000,0000,0000,,where alpha is our steering angle. Let me write that up here.
Dialogue: 0,0:00:27.00,0:00:30.00,Default,,0000,0000,0000,,So where does this equation come from.
Dialogue: 0,0:00:30.00,0:00:36.00,Default,,0000,0000,0000,,To derive it, the key realization is that the front and rear tire do not travel along the same circle.
Dialogue: 0,0:00:36.00,0:00:40.00,Default,,0000,0000,0000,,Here's my rear tire, and here's my front tire.
Dialogue: 0,0:00:40.00,0:00:45.00,Default,,0000,0000,0000,,They are, of course, separated by a distance that we called "L."
Dialogue: 0,0:00:45.00,0:00:49.00,Default,,0000,0000,0000,,Let's draw the circles that these tires travel along.
Dialogue: 0,0:00:49.00,0:00:56.00,Default,,0000,0000,0000,,Well, this rear tire is actually going to travel along a smaller inner circle
Dialogue: 0,0:00:56.00,0:00:59.00,Default,,0000,0000,0000,,while this tire is going to travel along a larger outer circle.
Dialogue: 0,0:00:59.00,0:01:04.00,Default,,0000,0000,0000,,Since we defined our radius of curvature as the distance from the back tire to the center,
Dialogue: 0,0:01:04.00,0:01:11.00,Default,,0000,0000,0000,,Let's label this r, and we can see that the line connecting these tires defines an axis,
Dialogue: 0,0:01:11.00,0:01:16.00,Default,,0000,0000,0000,,and here we have our steering angle, alpha, from here.
Dialogue: 0,0:01:16.00,0:01:18.00,Default,,0000,0000,0000,,Now we can do a little bit of geometry.
Dialogue: 0,0:01:18.00,0:01:20.00,Default,,0000,0000,0000,,Let's make a right triangle.
Dialogue: 0,0:01:20.00,0:01:26.00,Default,,0000,0000,0000,,Well, if this angle here is alpha, then this much be a 90 degree angle,
Dialogue: 0,0:01:26.00,0:01:30.00,Default,,0000,0000,0000,,because a radius intersecting with a tangent line always forms a right angle.
Dialogue: 0,0:01:30.00,0:01:37.00,Default,,0000,0000,0000,,That means that that this angle here must be equal to 90 degrees minus alpha,
Dialogue: 0,0:01:37.00,0:01:43.00,Default,,0000,0000,0000,,which means this angle, since this is a right triangle must be alpha.
Dialogue: 0,0:01:43.00,0:01:47.00,Default,,0000,0000,0000,,Well, we're almost there. The tangent of this angle is equal to the opposite side,
Dialogue: 0,0:01:47.00,0:01:51.00,Default,,0000,0000,0000,,which is the length, over the adjacent side, which is the radius.
Dialogue: 0,0:01:51.00,0:01:56.00,Default,,0000,0000,0000,,So tangent of alpha is equal to L over r.
Dialogue: 0,0:01:56.00,0:02:00.00,Default,,0000,0000,0000,,We manipulate this equation a little bit, and we find that the radius of curvature
Dialogue: 0,0:02:00.00,9:59:59.99,Default,,0000,0000,0000,,is equal to the length of the vehicle over the tangent of the steering angle.