G1fitting

G1 fitting with clothoids

by Enrico Bertolazzi and Marco Frego

The script buildClothoid implements the algorithm described in the paper

G1 fitting with clothoids, Mathematical Methods in the Applied Sciences, John Wiley & Sons, (2014), Ltd,. http://onlinelibrary.wiley.com/doi/10.1002/mma.3114/abstract

Description: Given two points and two direction associated with the points, a clothoid, i.e. a curve with linear varying curvature is computed in such a way it pass to the points with the prescribed direction. The solution in general is not unique but chosing the one for which the angle direction variation is less than 2*pi the solution is unique. The sofware solves the nonlinear system associated to the fitting problem computing initial curvature and its derivative with the lenght of the curve. An additional routine for the computation of the points along a clothoid curve is added for convenience.

Usage: To compute curvature and length of a clothoid passing throught points P0=(x0,y0), P1=(x1,y1) with angles theta0 and theta1 use the function buildClothoid

[k,dk,L] = buildClothoid( x0, y0, theta0, x1, y1, theta1, tol ) ;

The parameter tol (usually 1e-10) is a tolerance parameter used to stop Newton iteration. The resulting curve can be described by the 5 parameters

• (x0,y0) initial point
• theta0 initial direction (angle)
• k initial curvature of the curve
• dk derivative of the cuvature along arc length

plus a 6th parameter L

• L total length of the curve connecting P0 and P1

to compute points along a clothoid curve use the function pointsOnClothoid

XY = pointsOnClothoid( x0, y0, theta0, k, dk, L, npts ) ;

This function uses the 5 parameters x0, y0, theta0, k, dk which indentify the curve. The parameter L is used to determine length of the portion of the curve to compute. The parameter npts is the number of points computed along the curve.

XY is a 2 x npts matrix whose columns are the points along the curve.

To plot the computed curve use MATLAB plot command as usual:

plot( XY(1,:), XY(2,:), '-r' ) ;

Four sample scripts: TestN0, TestN1, TestN2 and TestN3 shows how to use the functions.

Mex files for fast computation:

In directory src_mex you find a C++ implementation of the proposed algorithm with mex interface. To compile run Compile from MATLAB window. After compilation the compiled version of the scripts

• buildClothoid
• evalClothoid
• FresnelCS
• GeneralizedFresnelCS
• pointsOnClothoid

are available in the G1fitting directory.

Additional mex

The mex implementation of the script:

• intersectClothoid
• TriTriOverlap

are added to the library. The first compute all the intersections between two clothoids. The second check if two triangles (planar) overlap (used in the intersection computation).

Authors:

Enrico Bertolazzi and Marco Frego
Department of Industrial Engineering
University of Trento
enrico.bertolazzi@unitn.it
m.fregox@gmail.com