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.
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.
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
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.
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