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

Figure A.1: Clothoid.
This figure shows the geometric shape known as a clothoid, whose curvature varies linearly with distance along the curve.

The position of points along a clothoid is evaluated with the use of the Fresnel Integrals, shown below:

$$C(t) = \ensuremath\int_{0}^{t} \cos(\frac{\pi}{2} u^{2})\,du$$ (A.1)

$$S(t) = \ensuremath\int_{0}^{t} \sin(\frac{\pi}{2} u^{2})\,du$$ (A.2)

Here $t$ 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[14].

The equations of a clothoid are:

$$x = aC(t)$$ (A.3)

$$y = aS(t)$$ (A.4)

In the above, $a$ is a scaling parameter and $t$ is the implicit variable, in general ranging from zero to $\infty$. The range of $t$ 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[15].

arclength

The arclength, $s$, of a clothoid at a given value of $t$ can be found from the scaling parameter, $a$.

$$s = at$$ (A.5)

curvature

The curvature $k$ at a given value of $t$ is determined from the following.

$$k = \frac{\pi t}{a}$$ (A.6)

tangent angle

The tangent angle $\theta$ is found with this equation.

$$\theta = \frac{\pi t^{2}}{2}$$ (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 $r$ and angle $\gamma$ at the other end. The parameter $a$ and the range of $t$ can be found using these specified values.

$$t_{0} = \sqrt{\frac{2\theta_{t=0}}{\pi}}$$

$$= 0$$ (A.8)

$$t_{f} = \sqrt{\frac{2\theta_{t=t_{f}}}{\pi}}$$

$$= \sqrt{\frac{2\gamma}{\pi}}$$ (A.9)

$$k_{f} = 1/r$$

$$= \frac{\pi t}{a}$$ (A.10)

$$a = \pi r t_{f}$$ (A.11)

The arclength of the clothoid can now be calculated.

$$l = a t_{f}$$ (A.12)