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many improvements (theorem-proof-structure for constant function; corrected errors)

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Martin Thoma 2013-12-21 19:10:35 +01:00
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documents/math-minimal-distance-to-cubic-function/introduction.tex
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@ -5,7 +5,9 @@ paths are cubic splines. You also have to be able to calculate
how to steer to get or to remain on a path. A way to do this
is applying the \href{https://en.wikipedia.org/wiki/PID_algorithm}{PID algorithm}.
This algorithm needs to know the signed current error. So you need to
be able to get the minimal distance of a point to a cubic spline combined with the direction (left or right).
be able to get the minimal distance of a point (the position of the car)
to a cubic spline (the prefered path)
combined with the direction (left or right).
As you need to get the signed error (and one steering direction might
be prefered), it is not only necessary to
get the minimal absolute distance, but might also help to get all points
@ -22,3 +24,8 @@ of the underlying PID-related problem. So I will try to give
robust and easy-to-implement algorithms to calculated the distance
of a point to a (piecewise or global) defined polynomial function
of degree $\leq 3$.
When you're able to calculate the distance to a polynomial which is
defined on a closed invervall, you can calculate the distance from
a point to a spline by calculating the distance to the pieces of the
spline.