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# Clothoids

A clothoid is a curve with linearly varying curvature. Since the intention of this research is to create a track with continuous curvature, clothoids are used extensively. An illustration of a clothoid is shown in figure A.1 The position of points along a clothoid is evaluated with the use of the Fresnel Integrals, shown below:   (A.1)   (A.2)

Here is the implicit variable along the curve.

These integrals cannot be solved analytically. Algorithms, as well as C code, have been created to evaluate the Fresnel Integrals.

The equations of a clothoid are:   (A.3)   (A.4)

In the above, is a scaling parameter and is the implicit variable, in general ranging from zero to . The range of determines the variation of curvature within the clothoid, as well as the initial and final tangent angles.

Some useful parameters of clothoids are given below.

arclength
The arclength, , of a clothoid at a given value of can be found from the scaling parameter, . (A.5)

curvature
The curvature at a given value of is determined from the following. (A.6)

tangent angle
The tangent angle is found with this equation. (A.7)

In the methodology being used, the clothoid will always have zero curvature and zero tangent angle at one end, with a specified radius and angle at the other end. The parameter and the range of can be found using these specified values.     (A.8)     (A.9)     (A.10)   (A.11)

The arclength of the clothoid can now be calculated. (A.12)    Next: Rotation Transformations Up: Dynamic Simulation and Analysis Previous: Dynamics   Contents
Darla Weiss 2000-02-13