Efficient G Approximation of Clothoids with Quintic Bézier...
Transcript of Efficient G Approximation of Clothoids with Quintic Bézier...
Efficient G3 Approximation of Clothoids with Quintic
Bézier Curves for Path Smoothing
CHEN YONG - School of Mechanical & Aerospace Engineering
- Institute for Media Innovation
Supervisors:
Assoc Prof. Cai Yiyu - School of Mechanical & Aerospace Engineering
Prof. Daniel Thalmann - Institute for Media Innovation
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Outline of the Presentation
1. Problem formulation
2. Overview of the relevant research
3. Methodology
4. Results and discussions
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Problem Formulation
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Planar Path A piecewise curve as a superset of n segments
Clothoid A curve whose curvature changes linearly with its curve length (Euler Spiral)
Problem Formulation
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Clothoid
Disadvantage: No closed form due to Fresnel integrals
Advantage: Shortest path satisfying Maximum Principle (optimal control theory)
Relevant Research
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Method Cons
1 Continuous function approximation (Wang, Lazhu Z.,
2001) Degree can be 26th order
2 C2 Hermite interpolation via s-power series (Sánchez-
Reyes, 2003)
Complicated coefficients
calculation
3 G3 Bézier approximation with numerical search (Cross,
2012; L Lu., 2013)
Numerical search procedure
is expensive; not robust
4 G2+ deterministic approximation (Cross, 2015) Not accurate due to linear
approximation
Pointwise approximation Circular interpolation between points (Brezak, M., 2014): No geometric
property reserved
Use other curves for clothoidal approximation
Note: 1, 2, 3, 4 can only deal with unit-lenth clothoids
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Methodology
Elementary Clothoid
Basic Clothoid
General Clothoid
Lookup Table
Elementary Clothoid
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where
Quintic Bézier Curve
G3 Continuity Constraints
Elementary Clothoid Approximation
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Apply Beta-constraints (BA Barsky, 1989):
Condition & Error Measure
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Transformation:
A reasonable assumption:
Optimization via Numerical Search
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Elementary Clothoid Approximation
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Divergence Problem
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k should be limited within
Basic Clothoid Approximation
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Piecewise approximation:
Accuracy Improvement
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The accuracy can be significantly improved by adjusting the segment lengths
in the lookup table.
General Clothoid Approximation
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For a general clothoid with positive initial conditions:
Figure: Compared with sharpness, winding
angle plays a more important role.
Error Analysis
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Errors are always within (0, 0.01)
as parameters are limited within
allowable region.
Winding angle contributes most in
the obtained curvature error.
Comparison
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(a) C2 Hermite approximation. (b) G3 approximation with numerical search. (c) G2+
approximation. (d) Proposed G3 approach.
with unit length
Comparison
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(a) C2 Hermite approximation. (b) G3 approximation with numerical search. (c) G2+
approximation. (d) Proposed G3 approach.
Non-unit length clothoid approximation
Comparison
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Compared with modified quadratic
polynomial interpolation
(UY Huh, 2014) and clamped B-spline (M
Elbanhawi, 2015), the proposed path
smoothing method has the shortest path length
with smallest curvature maxima.
Thank You!
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