A Numerical Algorithm
for Creating a Unicycle Track with a Bicycle
To create a unicycle track with a bicycle, we first create
an initial track segment
as described in Creating
the Initial Track Segment. Then, we sample the initial track
segment and information that describes the initial track segment. Specifically,
we need at each point on the curve the unit tangent vector, the principal
unit normal vectors, and the signed curvature. From the signed curvature,
we can then compute the angle
by
where
is the signed curvature. We then use the process of pushing the
bicycle forward and pushing the bicycle backward to extend the initial track
segment to
a unicycle track that can be created by a bicycle.
Sampling Curve: Let be the initial track segment. To sample this segment, we choose M+1
points, for instance set
with i=0,1,2, ... ,M then define
. We then symbolically (or numerically)
calculate the unit tangent vector
and the curvature
for each sampled point
.
From this information, we can determine the principal unit normal vector
and the angle
at each sample point, by using the equation
and the trick for computing the principal
unit normal from the unit tangent vector
if
Forward Direction: The data points from the initial segment can be used to generate new data points on the unicycle track following the method for pushing the bicycle forward. Following the method for pushing the bicycle forward (for details see Pushing the Bicycle Forward), we get the iterative formulas
|
(6) |
|
|
which will produce an extension of the initial track segment in the forward direction, that is for i>M.
Backward Direction The data points from the initial segment can be used to generate new data points on the unicycle track following the method for pushing the bicycle backward. Following the method for pushing the bicycle backward (for details see Pushing the Bicycle Backward), we get the iterative formulas
|
(7) |
|
|
which will produce an extension of the initial track segment in the backward direction, that is for i<0
For more details, on the equations used see the page Deriving the Equations.