1. Algebraic geometric nd geometric modeling 2006 Approximating Clothoids by Bezier curves Algebraic geometric and geometric modeling, September 2006, Barcelona Nicolás Montés and Josep Tornero Dept. of Systems Engineering and Control Technical University of Valencia

2. 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

3. 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

4. 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”

5. Algebraic geometric nd geometric modeling 2006 Algebraic geometric and geometric modeling, September 2006, Barcelona Type of spirals Uniform spiral or Arquímedes spiral where: A  characteristic constant parameter r  radius in a point of the curve α  Angle in a point of the curve

6. Algebraic geometric nd geometric modeling 2006 Algebraic geometric and geometric modeling, September 2006, Barcelona Logarithmic spiral or geometric spiral 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

7. Algebraic geometric nd geometric modeling 2006 Algebraic geometric and geometric modeling, September 2006, Barcelona Fermat’s spiral where: A  characteristic constant parameter r  radius in a point of the curve α  Angle in a point of the curve Type of spirals

8. 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: 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

9. 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

10. 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

11. 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

12. 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

13. 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)

14. 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)

15. Algebraic geometric nd geometric modeling 2006 Algebraic geometric and geometric modeling, September 2006, Barcelona 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:

16. 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:

17. 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.

18. Algebraic geometric nd geometric modeling 2006 Algebraic geometric and geometric modeling, September 2006, Barcelona 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)

19. 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: whereis the kth Bernstein basis function, which is: A set of linear equations can be expressed in the next matrix form:

20. Algebraic geometric nd geometric modeling 2006 Algebraic geometric and geometric modeling, September 2006, Barcelona 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:

21. Algebraic geometric nd geometric modeling 2006 Algebraic geometric and geometric modeling, September 2006, Barcelona Clothoid to Bezier curve Nσ2xσ2x σ2yσ2y Max(σ 2 x )Max(σ 2 y )|ε x | (%)|ε y | (%) 50.22450.07510.01030.00460.030.46 60.00570.00112.9·10 -4 6.2·10 -5 0.0290.0044 71.8·10 -5 1.8·10 -4 1.8·10 -6 8.8·10 -6 1.5·10 -4 8.8·10 -4 81.3·10 -6 2.5·10 -6 5.1·10 -8 1.5·10 -7 5.1·10 -6 1.5·10 -5 98.4·10 -8 1.4·10 -9 4.5·10 -9 1.4·10 -10 4.5·10 -7 1.4·10 -8 5 th order 7 th order Example 1: tangent angle interval [0, π/2], A=300

22. 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 11 th order 15 th order Nσ2xσ2x σ2yσ2y Max(σ 2 x )Max(σ 2 y )|ε x | (%)|ε y | (%) 80.01090.720.00890.02760.882.75 90.02310.03678.2·10 -4 0.00210.020.2144 100.00381.6·10 -4 1.7·10 -4 6.7·10 -6 0.010.001 117.4·10 -5 1.6·10 -4 4.6·10 -6 5.6·10 -6 4.8·10 -4 9.8·10 -4 121.9·10 -6 1.1·10 -5 1·10 -7 5.3·10 -7 2.3·10 -4 5.3·10 -5

23. 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]. 7 th order Clothoid to Bezier curve Aσ2xσ2x σ2yσ2y Max(σ 2 x )Max(σ 2 y )|ε x | (%)|ε y | (%) 2008·10 -6 8·10 -5 6.75·10 -7 3.9·10 -6 6.7·10 -5 3.9·10 -4 4003.2·10 -5 3.2·10 -4 2.7·10 -6 1.5·10 -5 2.7·10 -4 1.5·10 -3 8001.2·10 -4 1.2·10 -3 1·10 -5 6.28·10 -5 1·10 -3 6.2·10 -3 15004.5·10 -4 4.5·10 -3 3.8·10 -5 2.2·10 -4 3.8·10 -3 0.022 30001.8·10 -3 0.0181.5·10 -4 8.8·10 -4 0.0150.088

24. 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 a1 st order Bezier curve Cσ2xσ2x σ2yσ2y Max(σ 2 x )Max(σ 2 y )|ε x | (%)|ε y | (%) 15·10 -23 8·10 -24 1·10 -23 2·10 -24 4·10 -8 7·10 -9 24·10 -25 1·10 -22 1·10 -26 2·10 -23 6·10 -14 2.1·10 -9 38·10 -24 9·10 -22 5·10 -25 1·10 -22 3 ·10 -13 1·10 -9 43·10 -24 1·10 -21 4·10 -25 2·10 -22 4·10 -14 2.8·10 -11 51·10 -23 9·10 -22 4·10 -24 1·10 -22 1·10 -13 7·10 -12 61·10 -23 2·10 -22 1·10 -24 5·10 -23 6·10 -14 1.4·10 -12 73·10 -23 4·10 -23 1·10 -24 6·10 -24 2·10 -13 3.6·10 -13 82·10 -23 3·10 -24 5·10 -24 7·10 -14 1.5·10 -13

25. 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 where: : Bezier Control points of the straight line, : Bernstein basis functions for A and

26. 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] Aσ2xσ2x σ2yσ2y Max(σ 2 x )Max(σ 2 y )|ε x | (%)|ε y | (%) 306.1·10 -8 6·10 -7 7.33·10 -9 4.54·10 -8 7.3·10 -7 4.5·10 -6 30006.1·10 -4 6·10 -3 7.33·10 -5 4.54·10 -4 0.00730.0454 Error in the approximation for a limit cases: Start control pointEnd control point XYXY 1-8.5·10 -5 -2.1·10 -4 -8.5·10 -3 -2.1·10 -2 25.9672.7·10 -3 596.70.2703 311.926-1.2·10 -2 1192.6-1.21 417.9160.6361791.663.699 523.822.3622382236.29 629.466.1412946.5614.1 732.78111.9743278.11197.4 833.07217.933307.21793

27. 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, 7 th 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.

28. Algebraic geometric nd geometric modeling 2006 Approximating Clothoids by Bezier curves Algebraic geometric and geometric modeling, September 2006, Barcelona Nicolás Montés and Josep Tornero Dept. of Systems Engineering and Control Technical University of Valencia