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



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



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



From spirals to clothoids

¿what is a spiral?

“A planar curve where curvature is continuously changing.

That is, curvature decreases as radius increases”



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



Logarithmic spiral or geometric spiral

)cot(BeAr

where:

A,B characteristic constant parameters



Type of spirals



Fermat’s spiral

22 Arwhere:




Type of spirals



Cornu’s spiral , Euler’s spiral, Clothoid Gomes(1909)

The curvature is proportional to the arc length:

2Alr 2

2

2 r

A

where:



α angle in a point of the curve

l length followed until a point of the curve

From spirals to clothoids



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



Path Generation(N. Montes, J. Tornero. WSEAS. December 2004)

(K. Fotiades and J. Siemenis. IEEE Intelligent Vehicles. June 2005)

copied by



Path Generation

(N. Montes, J. Tornero. WSEAS 2004)

(K. Fotiades and J. Siemenis. IEEE Intelligent Vehicles. June 2005)

copied by



Continuous trajectory to join straight lines and circles with 3 clothoids:

Path Generation



Overtaking in highways

Path Generation(N. Montes, J. Tornero and L. Armesto.

International Simulation Conference. June 2005)



Avoiding obstacles

Path Generation(N. Montes, J. Tornero and L. Armesto.

International Simulation Conference. June 2005)



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:



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:



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.



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)




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



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




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




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



Example 3: tangent angle interval [0, π/2], A=[500,3000]. 7th order


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




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




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




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



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.


Approximating Clothoids by Bezier curves


Nicolás Montés and Josep TorneroDept. of Systems Engineering and Control

Technical University of Valencia

algebraic geometric nd geometric modeling 2006 approximating clothoids by bezier curves algebraic...

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