Details on Creating a Unicycle Track with a Bicycle
Creating the Initial Track Segment:
To create an initial track, we perturb a straight line with a smooth (infinitely differentiable) curve that passes through the points [0,0] and [L,0] and is infinitely flat (all derivatives are equal to zero) at the points [0,0] and [L,0].. Such a curve can be constructed using cut-off functions, non-analytic functions such as
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(2) |
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An initial track is then given by
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(3) |
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where f is any function on the interval . These conditions are sufficient for creating
a unicycle track with a bicycle. We
need all derivatives of the curve well-defined and the relationships between
the front tire position and the back tire position compatible.
To create the forward direction of the unicycle track, we push the bicycle forward along the initial curve segment in such a manner that the back tire remains in contact with the initial segment. This means that the back tire's unit tangent line agrees with the unicycle track's tangent line at the point of contact. From an analysis of the geometry of bicycle tracks, there are equations relating the position, tangent vector and curvature of the front tire track in terms of the back tire track. (see Derivations of Equations). From these equations, we have that the front tire track is determined by the back tire track. Thus, we can define a continuation of the initial segment by pushing the bicycle forward and creating the next part of the curve as the front tire track.
In mathematical terms, let be an initial curve segment of form (3). Then, extend
by the formula
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(4) |
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where is the unit tangent vector of c at t.
The algorithm used to compute the extension is slightly more complicated
because of technicalities in approximating the unit tangent vector
without differentiating.
To create the backward direction of the unicycle track, we
push the bicycle backward along the initial curve in such a manner that the
front tire remains in contact with the initial segment. This means that
the front tire's tangent line agrees with the unicycle track's tangent line
at the point of contact. From an analysis of the geometry of bicycle tracks
(see Derivations of Equations),
we can create the back tire track from the front tire track by solving a differential
difference equation for the angle
between the front tire track's unit tangent
and the back tire track's unit tangent.
Once we know the angle
,
we can create the back tire track by solving the Frenet frame equations, since
the signed curvature of both the front tire track and the back tire track can
be computed directly from knowing the angle
.
The differential difference equation in fact arises from equating
the curvature of the unicycle track as computed from the front tire perspective
and the back tire perspective (see Derivations
of Equations). This equation is given, using parameterizations
respect to time for the front tire track
and the back tire track
with respect to time, by
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(5) |
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where and
with
representing the arclength parameter of the
curve
. We solve (5) as an ordinary differential
equation, by integrating. There are a
few major differences. The first being
that to guarantee the existence and uniqueness of solution we need to know an
initial segment of data for
. The second being that we are only guaranteed
differentiability for
because we can only integrate in the
decreasing direction. (We can not overwrite the initial data!!) The third being to guarantee the
differentiability for all t, we need compatibility conditions on the
initial segment of data. In fact, the conditions on an initial segment of form
(3) are exactly what is needed for smoothness of the extension.
Given a function satisfying (5), we construct an extension of
the initial segment c(t) by solving the Frenet frame
equations. The angle
determines the curvature of the curve, and
knowing the signed curvature
is enough to determine the curve by the
fundamental theorem of plane curves.