Approximating Clothoids by Bezier curves

1
Approximating Clothoids by Bezier curves
Nicolás Montés and Josep Tornero Dept. of Systems
Engineering and Control Technical University of
Valencia
Algebraic geometric and geometric modeling,
September 2006, Barcelona
2
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 and geometric modeling,
September 2006, Barcelona
3
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 and geometric modeling,
September 2006, Barcelona
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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 and geometric modeling,
September 2006, Barcelona
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Type of spirals
Uniform spiral or Arquímedes spiral  
where A ? characteristic constant parameter r ?
radius in a point of the curve a ? Angle in a
point of the curve
Algebraic geometric and geometric modeling,
September 2006, Barcelona
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Type of spirals
Logarithmic spiral or geometric spiral
where A,B ? characteristic constant parameters r
? radius in a point of the curve a ? Angle in a
point of the curve
Algebraic geometric and geometric modeling,
September 2006, Barcelona
7
Type of spirals
Fermats spiral
where A ? characteristic constant parameter r ?
radius in a point of the curve a ? Angle in a
point of the curve
Algebraic geometric and geometric modeling,
September 2006, Barcelona
8
From spirals to clothoids
Cornus spiral , Eulers spiral, Clothoid
Gomes(1909)
The curvature is proportional to the arc length

• where
• A ? characteristic constant parameter
• r ? radius in a point of the curve
• a ? angle in a point of the curve
• l ? length followed until a point of the curve

Algebraic geometric and geometric modeling,
September 2006, Barcelona
9
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 and geometric modeling,
September 2006, Barcelona
10
Path Generation
(N. Montes, J. Tornero. WSEAS. December 2004)
copied by
(K. Fotiades and J. Siemenis. IEEE Intelligent
Vehicles. June 2005)
Algebraic geometric and geometric modeling,
September 2006, Barcelona
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Path Generation
(N. Montes, J. Tornero. WSEAS 2004)
copied by
(K. Fotiades and J. Siemenis. IEEE Intelligent
Vehicles. June 2005)
Algebraic geometric and geometric modeling,
September 2006, Barcelona
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Path Generation
Continuous trajectory to join straight lines and
circles with 3 clothoids
Algebraic geometric and geometric modeling,
September 2006, Barcelona
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Path Generation
(N. Montes, J. Tornero and L. Armesto.
International Simulation Conference. June 2005)
Overtaking in highways
Algebraic geometric and geometric modeling,
September 2006, Barcelona
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Path Generation
(N. Montes, J. Tornero and L. Armesto.
International Simulation Conference. June 2005)
Avoiding obstacles
Algebraic geometric and geometric modeling,
September 2006, Barcelona
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Mathematical definition of clothoids
B is a positive real number, parameter t is a
non-negative real number
Properties of the clothoid

• Angle of tangent

2.Curvature
where
3.Arc length L
The most attractive property of the clothoid is
that
where R is the radius of the curvature.
Algebraic geometric and geometric modeling,
September 2006, Barcelona
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Approach of Clothoids
Non-Polynomial functions
Approaching a clothoid in a selected t point
(Boresma, 1960), (Cody, 1986), (Heald, 1985),
(Klaus, 1997, 2000)
(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)
Algebraic geometric and geometric modeling,
September 2006, Barcelona
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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 and geometric modeling,
September 2006, Barcelona
18
Clothoid to Bezier curve
Bezier curves have the formulation
where
Bezier control points
Intrinsic parameter.
Order of the Bezier equation
Bezier equation can be rewired to represent a
clothoid in the interval
Tangent angle are linearly distributed along the
clothoid, avoiding iterative methods. (Borges,
2002)
Algebraic geometric and geometric modeling,
September 2006, Barcelona
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Clothoid to Bezier curve
Bezier equation can be expressed as a lineal
equation
where
is the kth Bernstein basis function, which is
A set of linear equations can be expressed in the
next matrix form
Algebraic geometric and geometric modeling,
September 2006, Barcelona
20
Clothoid to Bezier curve
This representation permits the use of least
squares
Variance of the approximation can be obtained as
Also a percentage in the point of maximum
variance is obtained as
Algebraic geometric and geometric modeling,
September 2006, Barcelona
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Clothoid to Bezier curve
Example 1 tangent angle interval 0, p/2,
A300
5th order
7th order
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September 2006, Barcelona
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Clothoid to Bezier curve
Example 2 tangent angle interval 0, p, A300
11th order
15th order
Algebraic geometric and geometric modeling,
September 2006, Barcelona
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Clothoid to Bezier curve
Example 3 tangent angle interval 0, p/2,
A500,3000. 7th order
Algebraic geometric and geometric modeling,
September 2006, Barcelona
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Clothoid to Bezier curve
Control points are approximated by least squares
with a1st order Bezier curve
Algebraic geometric and geometric modeling,
September 2006, Barcelona
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Clothoid to Bezier curve
It permits to rewrite a Bezier equation that
represents the clothoids in a selected interval
where
Bezier Control points of the straight line
Bernstein basis functions for A and
,
Algebraic geometric and geometric modeling,
September 2006, Barcelona
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Clothoid to Bezier curve
Example of road design tangent angle interval
0, p/2, A30,3000
Error in the approximation for a limit cases
Algebraic geometric and geometric modeling,
September 2006, Barcelona
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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
  4.5410-4.
• 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 and geometric modeling,
September 2006, Barcelona
28
Approximating Clothoids by Bezier curves
Nicolás Montés and Josep Tornero Dept. of Systems
Engineering and Control Technical University of
Valencia
Algebraic geometric and geometric modeling,
September 2006, Barcelona