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Martin Thoma 2013-12-21 22:20:30 +01:00
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documents/math-minimal-distance-to-cubic-function/introduction.tex
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@ -3,7 +3,7 @@ When you want to develop a selfdriving car, you have to plan which path
it should take. A reasonable choice for the representation of
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}.
is by 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 (the position of the car)
to a cubic spline (the prefered path)
@ -13,7 +13,7 @@ be prefered), it is not only necessary to
get the minimal absolute distance, but might also help to get all points
on the spline with minimal distance.
In this paper I want to discuss how to find all points on a cubic
In this paper, I want to discuss how to find all points on a cubic
function with minimal distance to a given point.
As other representations of paths might be easier to understand and
to implement, I will also cover the problem of finding the minimal
@ -21,7 +21,7 @@ distance of a point to a polynomial of degree 0, 1 and 2.
While I analyzed this problem, I've got interested in variations
of the underlying PID-related problem. So I will try to give
robust and easy-to-implement algorithms to calculated the distance
robust and easy-to-implement algorithms to calculate the distance
of a point to a (piecewise or global) defined polynomial function
of degree $\leq 3$.