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30 lines
1.6 KiB
TeX
30 lines
1.6 KiB
TeX
\chapter*{Introduction}
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When you want to develop a selfdriving car, you have to plan which path
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it should take. A reasonable choice for the representation of
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paths are cubic splines. You also have to be able to calculate
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how to steer to get or to remain on a path. A way to do this
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is by applying the \href{https://en.wikipedia.org/wiki/PID_algorithm}{PID algorithm}.
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This algorithm needs to know the signed current error. So you need to
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be able to get the minimal distance of a point (the position of the car)
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to a cubic spline (the prefered path)
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combined with sign (which represents the steering direction).
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As one steering direction might be prefered, it is not only necessary to
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get the minimal absolute distance, but might also help to get all points
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on the spline with minimal distance.
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In this paper, I want to discuss how to find all points on a cubic
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function with minimal distance to a given point.
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As other representations of paths might be easier to understand and
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to implement, I will also cover the problem of finding the minimal
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distance of a point to a polynomial of degree 0, 1 and 2.
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While I analyzed this problem, I've got interested in variations
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of the underlying PID-related problem. So I will try to give
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robust and easy-to-implement algorithms to calculate the distance
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of a point to a (piecewise or global) defined polynomial function
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of degree $\leq 3$.
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When you're able to calculate the distance to a polynomial which is
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defined on a closed invervall, you can calculate the distance from
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a point to a spline by calculating the distance to the pieces of the
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spline.
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