Algebraic geometric nd geometric modeling 2006 Approximating Clothoids by Bezier curves Algebraic...
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Transcript of Algebraic geometric nd geometric modeling 2006 Approximating Clothoids by Bezier curves Algebraic...
Algebraic geometric nd geometric modeling 2006
Approximating Clothoids by Bezier curves
Algebraic geometric and geometric modeling, September 2006, Barcelona
Nicolás Montés and Josep TorneroDept. of Systems Engineering and Control
Technical University of Valencia
Algebraic geometric nd geometric modeling 2006
Algebraic geometric and geometric modeling, September 2006, Barcelona
Outline
• Generation of a clothoid approximation in a standard CAD/CAD• Least Squares fitting are used to approximate a set of Clothoid points by Bezier curves• Clothoid points are obtained by a more accurate non-polynomial approximation
Algebraic geometric nd geometric modeling 2006
Algebraic geometric and geometric modeling, September 2006, Barcelona
Outline
• Bezier control points are allocated in a straight line for a constant end angle of the clothoid and different constant parameters of the Clothoid.• Bezier equation that represent the clothoids in a selected work range can be generated combining two Bezier equations
Algebraic geometric nd geometric modeling 2006
Algebraic geometric and geometric modeling, September 2006, Barcelona
From spirals to clothoids
¿what is a spiral?
“A planar curve where curvature is continuously changing.
That is, curvature decreases as radius increases”
Algebraic geometric nd geometric modeling 2006
Algebraic geometric and geometric modeling, September 2006, Barcelona
Type of spirals
Uniform spiral or Arquímedes spiral
Arwhere:
A characteristic constant parameter
r radius in a point of the curve
α Angle in a point of the curve
Algebraic geometric nd geometric modeling 2006
Algebraic geometric and geometric modeling, September 2006, Barcelona
Logarithmic spiral or geometric spiral
)cot(BeAr
where:
A,B characteristic constant parameters
r radius in a point of the curve
α Angle in a point of the curve
Type of spirals
Algebraic geometric nd geometric modeling 2006
Algebraic geometric and geometric modeling, September 2006, Barcelona
Fermat’s spiral
22 Arwhere:
A characteristic constant parameter
r radius in a point of the curve
α Angle in a point of the curve
Type of spirals
Algebraic geometric nd geometric modeling 2006
Algebraic geometric and geometric modeling, September 2006, Barcelona
Cornu’s spiral , Euler’s spiral, Clothoid Gomes(1909)
The curvature is proportional to the arc length:
2Alr 2
2
2 r
A
where:
A characteristic constant parameter
r radius in a point of the curve
α angle in a point of the curve
l length followed until a point of the curve
From spirals to clothoids
Algebraic geometric nd geometric modeling 2006
Algebraic geometric and geometric modeling, September 2006, Barcelona
The use of clothoids
In topography:
• It is used to build curves without discontinuities in highways and railways
In mobile robotics, they can be used for:
• Generating continuous paths
• Identifying clothoids in road and highway profiles
Algebraic geometric nd geometric modeling 2006
Algebraic geometric and geometric modeling, September 2006, Barcelona
Path Generation(N. Montes, J. Tornero. WSEAS. December 2004)
(K. Fotiades and J. Siemenis. IEEE Intelligent Vehicles. June 2005)
copied by
Algebraic geometric nd geometric modeling 2006
Algebraic geometric and geometric modeling, September 2006, Barcelona
Path Generation
(N. Montes, J. Tornero. WSEAS 2004)
(K. Fotiades and J. Siemenis. IEEE Intelligent Vehicles. June 2005)
copied by
Algebraic geometric nd geometric modeling 2006
Algebraic geometric and geometric modeling, September 2006, Barcelona
Continuous trajectory to join straight lines and circles with 3 clothoids:
Path Generation
Algebraic geometric nd geometric modeling 2006
Algebraic geometric and geometric modeling, September 2006, Barcelona
Overtaking in highways
Path Generation(N. Montes, J. Tornero and L. Armesto.
International Simulation Conference. June 2005)
Algebraic geometric nd geometric modeling 2006
Algebraic geometric and geometric modeling, September 2006, Barcelona
Avoiding obstacles
Path Generation(N. Montes, J. Tornero and L. Armesto.
International Simulation Conference. June 2005)
Algebraic geometric nd geometric modeling 2006
Algebraic geometric and geometric modeling, September 2006, Barcelona
t
t
d
d
tS
tCB
ty
txtQ
0
20
2
2sin
2cos
)(
)(
22tBtk
tAtBL BA
L
AR
k
21
1.Angle of tangent:
2.Curvature:
3.Arc length L:
B is a positive real number, parameter t is a non-negative real number
where R is the radius of the curvature.
Mathematical definition of clothoids
Properties of the clothoid:
where
The most attractive property of the clothoid is that:
Algebraic geometric nd geometric modeling 2006
Algebraic geometric and geometric modeling, September 2006, Barcelona
Approach of Clothoids
(Boresma, 1960), (Cody, 1986), (Heald, 1985), (Klaus, 1997, 2000)
Approaching a clothoid in a selected t point
(Klaus, 1997, 2000) : Approach a selected point of the
Fresnel integrals with an accuracy of 1x10-9
Non-Polynomial functions are ruled out, because they cannot be expressed in standard CAD/CAM, (Sanchez Reyes and Chacon, 2003)
Non-Polynomial functions:
Algebraic geometric nd geometric modeling 2006
Algebraic geometric and geometric modeling, September 2006, Barcelona
Approach of Clothoids
Polynomial functions:
(Wang et Al., 2001): The clothoid is approximated by a Bezier form using Taylor expansion. The order of the resulting Bezier curve is 23 with an error order of 1x10-6
(Sanchez Reyes and Chacon, 2003): the clothoid is approximated by an s-power series. The coefficients can be translated to a Bezier form between a transformation matrix. The calculus of the coefficients is complicated
(Meek and Walton, 2004): the clothoid is approximated by a set of arc Splines. The selected piecewise clothoid is converted in a discrete clothoid and each part is represented with an arc spline. The disadvantage is that it is only tangent vector continuous between arcs.
Algebraic geometric nd geometric modeling 2006
Algebraic geometric and geometric modeling, September 2006, Barcelona
Clothoid to Bezier curve
Bezier curves have the formulation:
N
k
knkk uu
kNk
NCuP
01
!!
!)(
where:
kC
kCuN
1...0
: Bezier control points
: Intrinsic parameter.
: Order of the Bezier equation
Bezier equation can be rewired to represent a clothoid in the interval fi
N
k
kN
if
f
k
if
fk kNk
NCP
01
!!
!)(
Tangent angle are linearly distributed along the clothoid, avoiding iterative methods. (Borges, 2002)
Algebraic geometric nd geometric modeling 2006
Algebraic geometric and geometric modeling, September 2006, Barcelona
Clothoid to Bezier curve
Bezier equation can be expressed as a lineal equation:
NN BCBCBCP ...1
10
0
kBwhere is the kth Bernstein basis function, which is:
kN
if
f
k
if
fk
kNk
NB
1!!
!
A set of linear equations can be expressed in the next matrix form:
N
k
N
N
C
C
BB
BB
p
p
ff
ii
f
i
.
.
..
...
...
..
.
.
0
0
CBP
Algebraic geometric nd geometric modeling 2006
Algebraic geometric and geometric modeling, September 2006, Barcelona
Clothoid to Bezier curveThis representation permits the use of least squares:
PBBBC TT 1ˆ
Variance of the approximation can be obtained as:2
0
2
f
i
N
k
kkBCP
Also a percentage in the point of maximum variance is obtained as:
1001 0
P
BCN
k
kk
Algebraic geometric nd geometric modeling 2006
Algebraic geometric and geometric modeling, September 2006, Barcelona
Clothoid to Bezier curve
N σ2x σ2
y Max(σ2x) Max(σ2
y) |εx| (%) |εy| (%)
5 0.2245 0.0751 0.0103 0.0046 0.03 0.46
6 0.0057 0.0011 2.9·10-4 6.2·10-5 0.029 0.0044
7 1.8·10-5 1.8·10-4 1.8·10-6 8.8·10-6 1.5·10-4 8.8·10-4
8 1.3·10-6 2.5·10-6 5.1·10-8 1.5·10-7 5.1·10-6 1.5·10-5
9 8.4·10-8 1.4·10-9 4.5·10-9 1.4·10-10 4.5·10-7 1.4·10-8
5th order 7th order
Example 1: tangent angle interval [0, π/2], A=300
Algebraic geometric nd geometric modeling 2006
Algebraic geometric and geometric modeling, September 2006, Barcelona
Clothoid to Bezier curve
Example 2: tangent angle interval [0, π], A=300
11th order 15th order
N σ2x σ2
y Max(σ2x) Max(σ2
y) |εx| (%) |εy| (%)
8 0.0109 0.72 0.0089 0.0276 0.88 2.75
9 0.0231 0.0367 8.2·10-4 0.0021 0.02 0.2144
10 0.0038 1.6·10-4 1.7·10-4 6.7·10-6 0.01 0.001
11 7.4·10-5 1.6·10-4 4.6·10-6 5.6·10-6 4.8·10-4 9.8·10-4
12 1.9·10-6 1.1·10-5 1·10-7 5.3·10-7 2.3·10-4 5.3·10-5
Algebraic geometric nd geometric modeling 2006
Algebraic geometric and geometric modeling, September 2006, Barcelona
Example 3: tangent angle interval [0, π/2], A=[500,3000]. 7th order
Clothoid to Bezier curve
A σ2x σ2
y Max(σ2x) Max(σ2
y) |εx| (%) |εy| (%)
200 8·10-6 8·10-5 6.75·10-7 3.9·10-6 6.7·10-5 3.9·10-4
400 3.2·10-5 3.2·10-4 2.7·10-6 1.5·10-5 2.7·10-4 1.5·10-3
800 1.2·10-4 1.2·10-3 1·10-5 6.28·10-5 1·10-3 6.2·10-3
1500 4.5·10-4 4.5·10-3 3.8·10-5 2.2·10-4 3.8·10-3 0.022
3000 1.8·10-3 0.018 1.5·10-4 8.8·10-4 0.015 0.088
Algebraic geometric nd geometric modeling 2006
Algebraic geometric and geometric modeling, September 2006, Barcelona
Clothoid to Bezier curve
Control points are approximated by least squares with a1st order Bezier curve
C σ2x σ2
y Max(σ2x) Max(σ2
y) |εx| (%) |εy| (%)
1 5·10-23 8·10-24 1·10-23 2·10-24 4·10-8 7·10-9
2 4·10-25 1·10-22 1·10-26 2·10-23 6·10-14 2.1·10-9
3 8·10-24 9·10-22 5·10-25 1·10-22 3 ·10-13 1·10-9
4 3·10-24 1·10-21 4·10-25 2·10-22 4·10-14 2.8·10-11
5 1·10-23 9·10-22 4·10-24 1·10-22 1·10-13 7·10-12
6 1·10-23 2·10-22 1·10-24 5·10-23 6·10-14 1.4·10-12
7 3·10-23 4·10-23 1·10-24 6·10-24 2·10-13 3.6·10-13
8 2·10-23 2·10-23 3·10-24 5·10-24 7·10-14 1.5·10-13
Algebraic geometric nd geometric modeling 2006
Algebraic geometric and geometric modeling, September 2006, Barcelona
Clothoid to Bezier curve
It permits to rewrite a Bezier equation that represents the clothoids in a selected interval
k
N
kK
kA
kA
A
ACAP
0
1
0),(
where:kAC
kN
if
f
k
if
fkA AA
AA
AA
AA
kkA
1!1!
!1
kN
if
f
k
if
fk
kNk
N
1
!!
!
: Bezier Control points of the straight lineAkA
k, : Bernstein basis functions for A and
Algebraic geometric nd geometric modeling 2006
Algebraic geometric and geometric modeling, September 2006, Barcelona
Clothoid to Bezier curve
Example of road design: tangent angle interval [0, π/2], A=[30,3000]
kAC
k
A σ2x σ2
y Max(σ2x) Max(σ2
y) |εx| (%) |εy| (%)
30 6.1·10-8 6·10-7 7.33·10-9 4.54·10-8 7.3·10-7 4.5·10-6
3000 6.1·10-4 6·10-3 7.33·10-5 4.54·10-4 0.0073 0.0454
Error in the approximation for a limit cases:
Start control point End control point
X Y X Y
1 -8.5·10-5 -2.1·10-4 -8.5·10-3 -2.1·10-2
2 5.967 2.7·10-3 596.7 0.2703
3 11.926 -1.2·10-2 1192.6 -1.21
4 17.916 0.636 1791.6 63.699
5 23.82 2.362 2382 236.29
6 29.46 6.141 2946.5 614.1
7 32.781 11.974 3278.1 1197.4
8 33.072 17.93 3307.2 1793
Algebraic geometric nd geometric modeling 2006
Algebraic geometric and geometric modeling, September 2006, Barcelona
Conclusions
•A strategy to approximate a selected piecewise clothoid by Bezier curves is presented.
• This approximation is based on least squares fitting. The points of the clothoid to fit are obtained by more accurate non-polynomial functions.
• The resulting approximation is an accurate approximation with a low degree Bezier order.
• In the interval of road design, 7th order Bezier curve is used. The variance in the worst case is 4.54·10-4.
• This representation can be easily introduced in CAD/CAM fields because it is expressed in Bezier form.
• These approximation can also be used other application requiring parametric curves such as mobile robots and control systems.
Algebraic geometric nd geometric modeling 2006
Approximating Clothoids by Bezier curves
Algebraic geometric and geometric modeling, September 2006, Barcelona
Nicolás Montés and Josep TorneroDept. of Systems Engineering and Control
Technical University of Valencia