# Formula for Turn Radius

Okay…  I just had to write this page. I was thinking about turning the robot. We know from the section on turning that when the two motors are traveling at different speeds the robot will turn. Well, I got obsessed with trying to predict exactly how big of turn radius you would get if you are given the different motor speeds.  Or more practically speaking: if I wanted my robot to turn at a very specific turn radius, what speeds to I need to make my motors run at?

First I used my imagination and “zoomed-in” to part of the turn in my mind. I thought about how the wheels would move if they were not on the same axle. The slower wheel would go a shorter distance in the same time as the faster wheel. They started side by side, but now they are at an angle. If you take the line the two motors were on at the beginning of the moment, the line the two motors end up on at the end of the moment and the lines they traveled on in that moment, you get a trapezoid. Now, a trapezoid is like a triangle with a corner cut off so that the cut is parallel to the opposite side.

And that missing corner is similar to the original triangle. Similar triangles have proportional sides. With algebra, we write these as:

(1) BD/BE = AC/AE

and

(2) BD/AC = BE/AE

(3) BA = BE-AE

changing this one around,

we get (4) BE-BA=AE

combining (1) and (4)

we get (5) BD/BE = AC/(BE-BA)

flip both sides over…

(6) BE/BD = (BE-BA)/AC

multiplying both sides by both denominators…

(7) BE*AC=BD*(BE-BA)

(8) BE*AC=(BD*BE)-(BD*BA)

(9) BD*BA=(BD*BE)-(BE*AC)

(10) BD*BA=(BD-AC)*BE

(11) (BD*BA)/(BD-AC)=BE

(12a) BA*(BD/(BD-AC))=BE

Remember that BA is the length of your axle, BE is your turn radius, BD is the distance the faster motor travels at a given amount of time, and AC is the distance the slower motor moves at the same amount of time.

So this gives us:

(12b) axle_width*(speed_of_faster_motor/(speed_of_faster_motor – speed_of_slower_motor)=turn_radius

or

(12c) A(F/F-S)=R

Now lets check to see if this equation holds at a couple of extremes:

1. If S=F (both wheels are at the same speed), R = Infinity! (F/0) or the robot is NOT turning, its going straight!

2. If S=0, R = A or the robot is turning about the stopped wheel.

3. If S = -F, R = A/2 or the robot is turning ‘in place’ about the center of the axle!

I think we have a good general formula to calculate the turn radius.

1. ropa calvin klein
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