Efficient G 3 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 1
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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
1
Outline of the Presentation
1. Problem formulation
2. Overview of the relevant research
3. Methodology
4. Results and discussions
2
Problem Formulation
3
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
4
Clothoid
Disadvantage: No closed form due to Fresnel integrals
Advantage: Shortest path satisfying Maximum Principle (optimal control theory)
Relevant Research
5
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
6
Methodology
Elementary Clothoid
Basic Clothoid
General Clothoid
Lookup Table
Elementary Clothoid
7
where
Quintic Bézier Curve
G3 Continuity Constraints
Elementary Clothoid Approximation
8
Apply Beta-constraints (BA Barsky, 1989):
Condition & Error Measure
9
10
Transformation:
A reasonable assumption:
Optimization via Numerical Search
11
Elementary Clothoid Approximation
12
Divergence Problem
13
k should be limited within
Basic Clothoid Approximation
14
Piecewise approximation:
Accuracy Improvement
15
The accuracy can be significantly improved by adjusting the segment lengths
in the lookup table.
General Clothoid Approximation
16
For a general clothoid with positive initial conditions:
Figure: Compared with sharpness, winding
angle plays a more important role.
Error Analysis
17
Errors are always within (0, 0.01)
as parameters are limited within
allowable region.
Winding angle contributes most in
the obtained curvature error.
Comparison
18
(a) C2 Hermite approximation. (b) G3 approximation with numerical search. (c) G2+
approximation. (d) Proposed G3 approach.
with unit length
Comparison
19
(a) C2 Hermite approximation. (b) G3 approximation with numerical search. (c) G2+
approximation. (d) Proposed G3 approach.
Non-unit length clothoid approximation
Comparison
20
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!
21
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