The control points for the B-Spline are chosen such that the resulting curve
is an intuitive fit to the expert??™s data points. A B-Spline order of 3 proved
to be su?±cient to exceed the requirements of the CAD and the trajectory.
The method used to form the B-Spline can be found in detail at [26]. The
algorithm used generates a set of control points. The corresponding B-Spline
passes through all of the data points and the velocity is equal at the knot
points and the middle points between the knots. This does not guarantee that
376
Chapter 15 Information Forwarding and Tra?±c Engineering
Fig. 3. B-Spline Basis Functions For p = 3 and n = 6.
the velocity is constant throughout the curve but it approximates it very well.
We select all the data points as control points and then generate two new
control points between each of the data points. Equation 7 can be used to
calculate the extra control points, where Qk are the data points and Tk are
the tangent vectors.
P1,k = Qk + 1
3?®Tk,
P2,k = Qk+1 ??’ 1
3?®Tk+1.
(7)
An example of a B-Spline trajectory generated from the set of control
points is shown in Figure 4. For every data point, except the first and last, two
extra checkpoints are generated on the line tangent to the B-Spline at the data
point. For the first and last, only one extra control point is generated. Since
this is done before the actual B-Spline is generated, the tangent vectors need
to be approximated using the surrounding data points.
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