Curve Elements

A curve is used to turn the track in the $x$-$y$ plane. It can also have a vertical component which moves the rider up or down while traversing the curve. As explained in section 2.1, a curve is a clothoidal element, meaning that it is made up of an entrance clothoid, a circular region, and an exit clothoid. These regions can be seen in figure 2.2. A curve can be specified with six parameters which are described below. Many of these parameters are illustrated in the figure.

Figure 2.2: Curve Element Parameters.
The projection of the curve in the $x$-$y$ plane is shown, illustrating the circular and clothoidal regions. The minimum radius of a curve, as well as the circular and clothoid angles, are depicted here.

direction

Specifies whether the track is to turn left or right relative to the forward direction upon entering the curve.

radius $(r)$

The minimum radius of the curve. This is the constant radius of curvature in the circular region of the $x$-$y$ projection of the curve. ($r > 0$)

entrance clothoid angle $(\gamma_{1})$

Clothoid 1 is the region in which the curvature varies from zero to $1/r$. The azimuth angle change experienced by the car as it traverses clothoid 1 is $\gamma_{1}$. This angle must be greater than zero to guarantee curvature continuity. ( $\gamma_{1} > 0$)

circular curve angle $(\delta)$

The angle of the curve with radius $r$. The car undergoes an azimuth angle change of $\delta$ in this region. No circular region is required for a curve element. ($\delta \geq 0$)

exit clothoid angle $(\gamma_{2})$

Clothoid 2 is the region in which the curvature varies from $1/r$ back to zero. The azimuth angle change experienced by the car as it traverses clothoid 2 is $\gamma_{2}$. This angle must be greater than zero to guarantee curvature continuity. ( $\gamma_{2} > 0$)

height $(h)$

The elevation change of the curve from beginning to end.

To determine the equations of a curve, the $x$-$y$ projection is first created from $r$, $\gamma_{1}$, $\delta$, and $\gamma_{2}$. The arclength of the $x$-$y$ projection and the height, $h$, are then used to calculate the vertical angle, defined to be constant throughout the curve. The height change throughout the curve is therefore linear with respect to arclength along the curve.

In order to describe the clothoid regions of the curve, the scaling parameters, $a_{1}$ and $a_{2}$, as well as the range of the variable $t$ for each clothoid, need to be determined.^2.3 Both the entrance and exit clothoid have zero curvature at the end of the curve element. The variable $t$ will be defined to be $t_{0}$ at that end. The following can be found from equation A.6.

$$\begin{eqnarray*} k_{t=t_{0}} & = & \frac{\pi t_{0}}{a} \\ t_{0} & = & \frac{k_{t=t_{0}} a}{\pi} = 0 \end{eqnarray*}$$

The azimuth angle change of the curve through a clothoid region is given by $\gamma$. This can be used to determine the final value of the variable $t$. See equation A.7.

$$\begin{eqnarray*} \gamma & = & \frac{\pi t_{f}^{2}}{2} \\ t_{f} & = & \sqrt{\frac{2 \gamma}{\pi}} \end{eqnarray*}$$

The clothoid scaling parameter can now be found from the value of $t_{f}$ and equation A.6. The junction of the clothoid and the circular region occurs when $t=t_{f}$. The curvature is $1/r$ at this point.

$$\begin{eqnarray*} a & = & \frac{\pi t_{f}}{k_{t=t_{f}}} \\ & = & \pi r t_{f} \end{eqnarray*}$$

The arclength of a clothoid is given by equation A.5.

$$s = at$$

The length of the projected curve in the $x$-$y$ plane is found from the equations for the arclength of the clothoids and of the circle.

$$l_{proj} = a_{1}t_{1_{f}} + r \delta + a_{2}t_{2_{f}}$$

Figure 2.3: Curve Vertical Angle.
This figure illustrates the relationship between the total length of the curve, the length of the projection in the $x$-$y$ plane, and the elevation change. Note that $\phi$ is a constant angle throughout the curve, as explained in section 2.1.

The vertical angle, $\phi$, is depicted in figure 2.3. The hypotenuse in the figure represents the total length of the curve. The lower leg, labeled $l_{proj}$, represents the length of the projection of the curve in the $x$-$y$ plane. The angle $\phi$ can therefore be found from the following.

$$\phi = \arctan \left( \frac{h}{l_{proj}} \right)$$

The initial and final vertical angles are both $\phi$, as the vertical slope does not change throughout a curve element.

$$\phi_{0} = \phi_{f} = \phi$$ (2.18)

The total arclength of the curve is found from the Pythagorean Theorem, since it can be viewed as the hypotenuse of a right triangle. The legs are the height and the length of the projection of the curve in the $x$-$y$ plane.

$$l = \sqrt{l_{proj}^{2} + h^{2}}$$ (2.19)

The azimuth angle change a car undergoes is the rotation its forward vector experiences. In a curve element, the azimuth angle change through clothoid 1 is $\gamma_{1}$. Through the circular region the car rotates $\delta$ and through clothoid 2 the rotation is $\gamma_{2}$. The total curve angle is given by $\theta_{f}$, which is the azimuth angle at the end of the curve.

$$\theta_{f} = \gamma_{1} + \delta + \gamma_{2}$$ (2.20)

Recall that the required geometric functions of an element are the position vector, $\ensuremath \mathbf{\widehat{p}}$, the forward and radial vectors, $\ensuremath \mathbf{\widehat{f}}$ and $\ensuremath \mathbf{\widehat{r}}$, and the curvature, $k$. In order to calculate these functions for a curve, the projected arclength in the $x$-$y$ plane, $s_{proj}$, will be used instead of the arclength along the curve, $s$. The conversion is made with the following, which can be inferred from figure 2.3.

$$s_{proj} = s \cos\phi$$

The equations of the curve are derived assuming a left-hand turn. If it is a right-hand turn, the curve will be a mirror image of the one calculated. Since the initial forward direction is in the $x$-$z$ plane, all that must be done is to negate the values of $y$, thereby producing the mirror image.

Figure 2.4 illustrates the construction of a curve, and labels many of the coordinates that will be discussed.

Figure 2.4: Curve Construction.
This figure illustrates how a curve is created. First clothoid 1 is created, beginning at the origin with $t=0$. The end of clothoid 1 occurs when $t=t_{1_{f}}$, at $(x_{1_{f}},y_{1_{f}})$. The center of the circular region, $(x_{c},y_{c})$, can then be found. The circular region is added, beginning where clothoid 1 ended. The beginning of clothoid 2, $(x_{2_{0}},y_{2_{0}})$, is then found. Since clothoid 2 starts with non-zero curvature, $t=t_{2_{f}}$ at the beginning of the clothoid and $t=0$ at the end. The clothoid is inially created such that it ends horizontally, and is then rotated $\theta_{f}$ radians about the $z$ axis to line up the azimuth angle at the beginning of clothoid 2 with the azimuth angle at the end of the circular region.

^2.3 See appendix A for a complete description of clothoids.