Efficient G Approximation of Clothoids with Quintic Bézier...
Transcript of Efficient G Approximation of Clothoids with Quintic Bézier...
![Page 1: Efficient G Approximation of Clothoids with Quintic Bézier ...imi.ntu.edu.sg/NewsEvents/Events/PastSeminars/... · Relevant Research 5 Method Cons 1 Continuous function approximation](https://reader036.fdocuments.us/reader036/viewer/2022071212/6024aef88cc3fc035111a6d0/html5/thumbnails/1.jpg)
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
![Page 2: Efficient G Approximation of Clothoids with Quintic Bézier ...imi.ntu.edu.sg/NewsEvents/Events/PastSeminars/... · Relevant Research 5 Method Cons 1 Continuous function approximation](https://reader036.fdocuments.us/reader036/viewer/2022071212/6024aef88cc3fc035111a6d0/html5/thumbnails/2.jpg)
Outline of the Presentation
1. Problem formulation
2. Overview of the relevant research
3. Methodology
4. Results and discussions
2
![Page 3: Efficient G Approximation of Clothoids with Quintic Bézier ...imi.ntu.edu.sg/NewsEvents/Events/PastSeminars/... · Relevant Research 5 Method Cons 1 Continuous function approximation](https://reader036.fdocuments.us/reader036/viewer/2022071212/6024aef88cc3fc035111a6d0/html5/thumbnails/3.jpg)
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)
![Page 4: Efficient G Approximation of Clothoids with Quintic Bézier ...imi.ntu.edu.sg/NewsEvents/Events/PastSeminars/... · Relevant Research 5 Method Cons 1 Continuous function approximation](https://reader036.fdocuments.us/reader036/viewer/2022071212/6024aef88cc3fc035111a6d0/html5/thumbnails/4.jpg)
Problem Formulation
4
Clothoid
Disadvantage: No closed form due to Fresnel integrals
Advantage: Shortest path satisfying Maximum Principle (optimal control theory)
![Page 5: Efficient G Approximation of Clothoids with Quintic Bézier ...imi.ntu.edu.sg/NewsEvents/Events/PastSeminars/... · Relevant Research 5 Method Cons 1 Continuous function approximation](https://reader036.fdocuments.us/reader036/viewer/2022071212/6024aef88cc3fc035111a6d0/html5/thumbnails/5.jpg)
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
![Page 6: Efficient G Approximation of Clothoids with Quintic Bézier ...imi.ntu.edu.sg/NewsEvents/Events/PastSeminars/... · Relevant Research 5 Method Cons 1 Continuous function approximation](https://reader036.fdocuments.us/reader036/viewer/2022071212/6024aef88cc3fc035111a6d0/html5/thumbnails/6.jpg)
6
Methodology
Elementary Clothoid
Basic Clothoid
General Clothoid
Lookup Table
![Page 7: Efficient G Approximation of Clothoids with Quintic Bézier ...imi.ntu.edu.sg/NewsEvents/Events/PastSeminars/... · Relevant Research 5 Method Cons 1 Continuous function approximation](https://reader036.fdocuments.us/reader036/viewer/2022071212/6024aef88cc3fc035111a6d0/html5/thumbnails/7.jpg)
Elementary Clothoid
7
where
Quintic Bézier Curve
G3 Continuity Constraints
![Page 8: Efficient G Approximation of Clothoids with Quintic Bézier ...imi.ntu.edu.sg/NewsEvents/Events/PastSeminars/... · Relevant Research 5 Method Cons 1 Continuous function approximation](https://reader036.fdocuments.us/reader036/viewer/2022071212/6024aef88cc3fc035111a6d0/html5/thumbnails/8.jpg)
Elementary Clothoid Approximation
8
Apply Beta-constraints (BA Barsky, 1989):
![Page 9: Efficient G Approximation of Clothoids with Quintic Bézier ...imi.ntu.edu.sg/NewsEvents/Events/PastSeminars/... · Relevant Research 5 Method Cons 1 Continuous function approximation](https://reader036.fdocuments.us/reader036/viewer/2022071212/6024aef88cc3fc035111a6d0/html5/thumbnails/9.jpg)
Condition & Error Measure
9
![Page 10: Efficient G Approximation of Clothoids with Quintic Bézier ...imi.ntu.edu.sg/NewsEvents/Events/PastSeminars/... · Relevant Research 5 Method Cons 1 Continuous function approximation](https://reader036.fdocuments.us/reader036/viewer/2022071212/6024aef88cc3fc035111a6d0/html5/thumbnails/10.jpg)
10
Transformation:
A reasonable assumption:
![Page 11: Efficient G Approximation of Clothoids with Quintic Bézier ...imi.ntu.edu.sg/NewsEvents/Events/PastSeminars/... · Relevant Research 5 Method Cons 1 Continuous function approximation](https://reader036.fdocuments.us/reader036/viewer/2022071212/6024aef88cc3fc035111a6d0/html5/thumbnails/11.jpg)
Optimization via Numerical Search
11
![Page 12: Efficient G Approximation of Clothoids with Quintic Bézier ...imi.ntu.edu.sg/NewsEvents/Events/PastSeminars/... · Relevant Research 5 Method Cons 1 Continuous function approximation](https://reader036.fdocuments.us/reader036/viewer/2022071212/6024aef88cc3fc035111a6d0/html5/thumbnails/12.jpg)
Elementary Clothoid Approximation
12
![Page 13: Efficient G Approximation of Clothoids with Quintic Bézier ...imi.ntu.edu.sg/NewsEvents/Events/PastSeminars/... · Relevant Research 5 Method Cons 1 Continuous function approximation](https://reader036.fdocuments.us/reader036/viewer/2022071212/6024aef88cc3fc035111a6d0/html5/thumbnails/13.jpg)
Divergence Problem
13
k should be limited within
![Page 14: Efficient G Approximation of Clothoids with Quintic Bézier ...imi.ntu.edu.sg/NewsEvents/Events/PastSeminars/... · Relevant Research 5 Method Cons 1 Continuous function approximation](https://reader036.fdocuments.us/reader036/viewer/2022071212/6024aef88cc3fc035111a6d0/html5/thumbnails/14.jpg)
Basic Clothoid Approximation
14
Piecewise approximation:
![Page 15: Efficient G Approximation of Clothoids with Quintic Bézier ...imi.ntu.edu.sg/NewsEvents/Events/PastSeminars/... · Relevant Research 5 Method Cons 1 Continuous function approximation](https://reader036.fdocuments.us/reader036/viewer/2022071212/6024aef88cc3fc035111a6d0/html5/thumbnails/15.jpg)
Accuracy Improvement
15
The accuracy can be significantly improved by adjusting the segment lengths
in the lookup table.
![Page 16: Efficient G Approximation of Clothoids with Quintic Bézier ...imi.ntu.edu.sg/NewsEvents/Events/PastSeminars/... · Relevant Research 5 Method Cons 1 Continuous function approximation](https://reader036.fdocuments.us/reader036/viewer/2022071212/6024aef88cc3fc035111a6d0/html5/thumbnails/16.jpg)
General Clothoid Approximation
16
For a general clothoid with positive initial conditions:
Figure: Compared with sharpness, winding
angle plays a more important role.
![Page 17: Efficient G Approximation of Clothoids with Quintic Bézier ...imi.ntu.edu.sg/NewsEvents/Events/PastSeminars/... · Relevant Research 5 Method Cons 1 Continuous function approximation](https://reader036.fdocuments.us/reader036/viewer/2022071212/6024aef88cc3fc035111a6d0/html5/thumbnails/17.jpg)
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.
![Page 18: Efficient G Approximation of Clothoids with Quintic Bézier ...imi.ntu.edu.sg/NewsEvents/Events/PastSeminars/... · Relevant Research 5 Method Cons 1 Continuous function approximation](https://reader036.fdocuments.us/reader036/viewer/2022071212/6024aef88cc3fc035111a6d0/html5/thumbnails/18.jpg)
Comparison
18
(a) C2 Hermite approximation. (b) G3 approximation with numerical search. (c) G2+
approximation. (d) Proposed G3 approach.
with unit length
![Page 19: Efficient G Approximation of Clothoids with Quintic Bézier ...imi.ntu.edu.sg/NewsEvents/Events/PastSeminars/... · Relevant Research 5 Method Cons 1 Continuous function approximation](https://reader036.fdocuments.us/reader036/viewer/2022071212/6024aef88cc3fc035111a6d0/html5/thumbnails/19.jpg)
Comparison
19
(a) C2 Hermite approximation. (b) G3 approximation with numerical search. (c) G2+
approximation. (d) Proposed G3 approach.
Non-unit length clothoid approximation
![Page 20: Efficient G Approximation of Clothoids with Quintic Bézier ...imi.ntu.edu.sg/NewsEvents/Events/PastSeminars/... · Relevant Research 5 Method Cons 1 Continuous function approximation](https://reader036.fdocuments.us/reader036/viewer/2022071212/6024aef88cc3fc035111a6d0/html5/thumbnails/20.jpg)
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.
![Page 21: Efficient G Approximation of Clothoids with Quintic Bézier ...imi.ntu.edu.sg/NewsEvents/Events/PastSeminars/... · Relevant Research 5 Method Cons 1 Continuous function approximation](https://reader036.fdocuments.us/reader036/viewer/2022071212/6024aef88cc3fc035111a6d0/html5/thumbnails/21.jpg)
Thank You!
21