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

(2)

An initial track is then given by 

(3)

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.  

 

 

 

 

 

 

 

 

 

 

 

 

 

Pushing the Bicycle Forward:

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

(4)

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. 

 

 

 

 

 

 

 

 

 

 

 

Pushing the Bicycle Backward

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

(5)

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.