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Connecting Elements to Create a Track

Each element is specified in terms of local coordinates. In order to create a track, it is necessary to connect these elements together in a global setting. To do this, simple coordinate transformations are used.

Let $(x_{i},y_{i},z_{i})$ represent the local coordinates of the $i^{th}$ element, and let $(X_{i},Y_{i},Z_{i})$ represent the global coordinates of that element. Let $s$ and $\theta$ represent the arclength and azimuth angle at the current position, and let $s_{f}$ and $\theta_{f}$ represent the arclength and azimuth angle, respectively, at the end of that element. The position transformation from local to global coordinates requires a rotation about the local $z$ axis to line up the $x$-$y$ axes with the azimuth angle at the end of the previous element, as well as a translation to line up the local origin with the end of the previous element. This is accomplished as shown below, where the subscript $i$ is used to denote which element the quantities refer to.

$\displaystyle X_{i}(s)$ $\textstyle =$ $\displaystyle X_{i-1}(s_{f}) + x_{i}(s)\cos(\theta_{f_{i-1}}) - y_{i}(s)\sin(\theta_{f_{i-1}})$ (2.147)
$\displaystyle Y_{i}(s)$ $\textstyle =$ $\displaystyle Y_{i-1}(s_{f}) + x_{i}(s)\sin(\theta_{f_{i-1}}) + y_{i}(s)\cos(\theta_{f_{i-1}})$ (2.148)
$\displaystyle Z_{i}(s)$ $\textstyle =$ $\displaystyle Z_{i-1}(s_{f}) + z_{i}(s)$ (2.149)

The transformation is somewhat simpler for the forward and radial vectors than it is for the position, since they are not dependent on an origin. Only a rotation is required to line up these vectors properly. The forward vector transformation is shown here.

$\displaystyle f_{x_{i}}(s)$ $\textstyle =$ $\displaystyle f_{x_{i}}(s)\cos(\theta_{f_{i-1}}) - f_{y_{i}}(s)\sin(\theta_{f_{i-1}})$ (2.150)
$\displaystyle f_{y_{i}}(s)$ $\textstyle =$ $\displaystyle f_{x_{i}}(s)\sin(\theta_{f_{i-1}}) + f_{y_{i}}(s)\cos(\theta_{f_{i-1}})$ (2.151)
$\displaystyle f_{z_{i}}(s)$ $\textstyle =$ $\displaystyle f_{z_{i}}(s)$ (2.152)

The radial vector transformation is nearly identical.
$\displaystyle r_{x_{i}}(s)$ $\textstyle =$ $\displaystyle r_{x_{i}}(s)\cos(\theta_{f_{i-1}}) - r_{y_{i}}(s)\sin(\theta_{f_{i-1}})$ (2.153)
$\displaystyle r_{y_{i}}(s)$ $\textstyle =$ $\displaystyle r_{x_{i}}(s)\sin(\theta_{f_{i-1}}) + r_{y_{i}}(s)\cos(\theta_{f_{i-1}})$ (2.154)
$\displaystyle r_{z_{i}}(s)$ $\textstyle =$ $\displaystyle r_{z_{i}}(s)$ (2.155)

Refer to appendix B for a more general description of rotational transformations.

next up previous contents
Next: Dynamics Up: Geometry Previous: Clothoid 2:   Contents
Darla Weiss 2000-02-13