# Path of a single wheel

Hello, I have a simple device : a vertical wheel bound to an horizontal bar. The bar always belongs to the plan of the wheel (the wheel is bounded
to follow the bar direction). The movement of the wheel is intended to perform without slipping. Which is the path taken by the wheel when one pulls or push the bar from the opposite end constrained to follow any arbitrary trajectory ? I can solve the problem when the trajectory is a straight line with a non trivial initial condition corresponding to the wheel contact point not in line whith the linear trajectory (a solution of the differential equation y'(x)^2 + 1 - 1/x^2 = 0). What approach do you suggest for the general case ? Thank you for your help. Claude Animo
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Would you say that this setup is effectivly like a caster wheel, wher eyou can measure the amount of wheel rotation and its vector angle?
I think I have some math kicking around that I was going to use for my lawnmower robot, which uses differential drive, I was going to use the caster for position tracking.
http://eds.dyndns.org:81/~ircjunk/robots/mowerbot/mowbot3.jpg
dan
snipped-for-privacy@gmail.com wrote:

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Hello, in fact it is a one wheel trailer towed by a car. What is it's trajectory according to the towing vehicle path. regards, Claude Animo
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then is will be just like a nice big caster
I'll see if I cant find you the math I worked out, but it looks like you might have a reply there already
dan
snipped-for-privacy@gmail.com wrote:

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snipped-for-privacy@gmail.com wrote:

I'd turn to a computer simulation to plot the path.
Why do you need to know the path? Is this a math problem or an engineering problem?

--
Curt Welch http://CurtWelch.Com /
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I took a quick look at the problem and am thinking in terms of a simpler differential equation when the vehicle is following a straight-line path. I may be wrong and, so, am curious as to how you found yours.
The following is strictly in terms of a kinematics solution with fixed vehicle velocity. Assume that the wheel is centered on the axis of the tow bar and, thus, always points toward the trailer hitch. Assume that the vehicle starts from the origin and is following the positive X axis at constant speed s, so that the vehicle hitch's x position at time t is simply xHitch(t) = s*t. The initial position of the wheel is y0. Given that the length of the tow bar is L, then the angle between the bar and the X axis is alpha where sin(alpha) = -y(t)/L. The component of the wheel's motion in the the direction of the y axis is just s*sin(alpha), so we have an elementary separable differential equation dy/dt = y*(-s/L) and y(t) = y0*exp( (-s/L)*t). Having y, and knowing that the distance between the wheel and the hitch is fixed, we have x = xHitch-sqrt(L*L-y*y).
Assuming the vehicle follows fixed rates (speed and turn rate), it will follow a circular path. I'm wondering if you can't just compute the position of the wheel by constructing a pair of axes through the position of the hitch and oriented in the instantaneous direction of the vehicle's motion. The xand y values become the coordinates of the wheel relative to the translated axes.
Gary
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