Derivations of Equations

 

Recall that the question that we are studying is:

Do there exist any curves  parameterized with respect to arclength such that the positions of the front tire and back tire of a bicycle at time , respectively satisfy   and  for some positions  and  of the curve ?

 

We will assume that the bicycle is ridden on a flat surface, and that the bicycle is not banked into the ground, that is the plane of each tire meet the ground in a right angle. We suspect that one can create a unicycle track without these assumptions, but the equations become much more complicated.  In particular, using these we have that

(1)

where L is the length of the bicycle and  is the unit tangent vector to the back tire  at time t.  Equation (1) arises from two simple observations concerning the manner in which a bicycle is built.  The first observation is that the back tire is fixed in the frame, meaning that the back tire and the frame are aligned.  This means that the tangent line to the back tire track at time t is equal to the secant line through the front and back tire positions at time t, that is

(2)

The second observation is that the frame is rigid, meaning that the distance between  the points of contact between the tires and the ground is a constant  independent if the position of the tires.

 

Rather than express all the equations for describing the tracks of a bicycle in terms of the tracks  and , we describe the motion of the bicycle in terms of the angle  between the unit tangent vectors of the tracks, the angle between the planes of the tires.  We use this angle because when riding a bicycle, we have direct control of this angle since by controlling the angle  and the speed of the bicycle the position of the bicycle.  We require the angle  be signed and satisfy  to distinguish between a right hand turn and a left hand turn.  A left-hand turn will have a positive sign and right-hand turn will have a negative sign, see diagram below.  This convention allows us to quantify left and right by using the principal unit normal vector N(t) for the curve, which is given by

If   is positive then  lies on the same side of  as , while if   is negative then  lies on the same side of  as , which from standard convention respectively corresponds to a left hand and a right hand turn.

A principal use of the angle  is to convert between the two sets of orthogonal basis vectors  and .  Using trigonometry and basic vector arithmetic, we have

(3)

By differentiating equations (1) to (3) with respect to time, and using the Frenet frame equations, we derive the main equations needed to construct a unicycle track with a bicycle.

 

Differentiating equation (1) respect to time, we get

(4)

where  is the signed curvature of the back tire track, and  and  are the arclength parameters of the curves  and  respectively.  There are a couple of important conclusions that we can make from equation (4). Namely, that

(5)

and therefore

(6)

Equating the expression for  in (3) and (6), we find that

(7)

from which we get

(8)

Differentiating the equation for  in (3) with respect to t, we have

 which yields

Using that , we have

(9)

and thus implies that

(11)

The equations (8), (10), (11) allow us to construct a bicycle track knowing the angle  as a function either arclength parameter  or  since with this information we can solve the Frenet frame equations to construct either the back tire track or the front tire track.  We could also solve the Frenet frame equations knowing the angle  and the speed  of the back tire, which are of course the physical controls of a bicycle.

 

It is also useful to have the curvature of the front tire in terms of the curvature for the pact tire.  Differentiating (6) we have

thus

(12)

Equations (1), (6), (12) are useful for determining the conditions needed on a bicycle track for it to be possible to create a unicycle track with a bicycle.  We need every point on the front tire track to be a point on the back tire track.  The equations (1), (6), (12) provide useful relations between two points on the unicycle track, say  and , that correspond to  and .  These equations say that for a unicycle track to be created by a bicycle we need

where s is the arclength parameter on  and  is the parameter for .  Repeatedly differentiating (12) with respect to time or arclength we get conditions specifying the derivatives of the curvature of  when  in terms of the curvature and the derivatives of the curvature when .  A cursory examination of these conditions reveals that they are satisfied by having the curvature and the derivatives of the curvature identically equal to zero when  and .

 

To create a unicycle track with a bicycle, we first create a track segment  with , , and the curvature and the derivatives of the curvature identically equal to zero when  and . Pushing the bicycle forward on the initial track segment, we can extend the segment to the interval  by using the formula

The conditions on the curvature imply that the curvature and the derivatives of the curvature when  are identically equal to zero, and therefore we can extend the unicycle track forward indefinitely.  We can also push the bicycle backwards on the initial unicycle track to extend the track for .  However, we can only guarantee that the bicycle can be pushed for a short amount of time and arclength in the backwards direction at present time, because we generate the backward direction for the unicycle track by solving a differential difference equation. The differential difference equation comes from noticing that for a unicycle track that can be created with a bicycle there exists for every time  a time  with , and therefore at every point we have two methods for calculating the curvature.  Thus by equating the curvature of the back tire track at time  to the curvature of the front tire track at time , we have that the curvature of the unicycle track must satisfy

.

In terms of the angle  which controls the construction of the bicycle, this is equivalent to the equation

.

Given an initial segment, , we can solve these differential difference equations for  only.   If we attempt to solve for  we would overwrite the initial segment and the extension of the initial segment by .