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.
課題の4問目はロボットの
環状動作のシミュレーションでした
課題のヒントとして方程式をいくつか用意しました
もう一度方程式を見ながら説明したいと思います
まずはこの方程式です
曲率半径=車体の長さ÷正接α
αは操舵角を表します ここに書きましょう
この方程式が何だか分かりますか?
理解するカギはタイヤの前輪と後輪が
同じ円をたどらないという点です
こちらが後輪で こちらが前輪です
前輪と後輪はLという距離で離れています
ではタイヤの通る円を描きましょう
後輪はこのように小さい内円を描きます
そして前輪は大きい外円を描きます
曲率半径を後輪から
円の中心までの距離として定義しました
この距離はrとします
そして前後のタイヤをつなぐ線を軸にすると
ここが操舵角のαになります
では幾何学の分野へ移り直角三角形を作りましょう
この角度がαであるならここは90度になるはずです
半径と接線が交差すると直角を形成するからです
つまりこの角度は90度から
αを引いたものと等しいということになります
これは直角三角形なのでこの角度はαです
もう少しですね
角度αの正接は対辺であるLを
隣辺であるrで割った値と等しいと考えられます
よって正接α=L÷rです
この方程式を少し変えて曲率半径を求めます
曲率半径rは車体の長さL÷操舵角の正接αです