Path of a single wheel

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

claude.animo@gmail.com wrote:

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

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

claude.animo@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?


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Curt Welch                                            http://CurtWelch.Com/
curt@kcwc.com                                        http://NewsReader.Com/

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




claude.animo@gmail.com wrote: