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The Trajectories

In this section we will calculate the trajectory of the slider and camera as a function of time. Three positions (x, y, z) and three attitudes (roll, pitch, and yaw) are required for a complete trajectory description. The origin of the luge is at the center of mass of the luge. The x axis of the luge is tangent to the path the luge takes. The luge y axis is normal to the x axis and tangent to the track. The luge z axis is orthogonal to the luge x and y axes. The roll, pitch, and yaw angles describe the relationship between the luge and the global (luge world) coordinate systems. A climbing right hand turn gives positive attitudes. A level luge traveling along the global x axis has an attitude of zero. The camera and slider share attitudes but are offset along the luge z axis.

We will first calculate the trajectory at track level. Using this trajectory, we will then calculate the position of the camera by applying the appropriate translations and rotations. The transformation from global to sled coordinates occurs via a series of Euler angle transformation in yaw, pitch, and roll, respectively. The sliders position is then determined by interpolating between the two.

Our luge run is defined in spatial coordinates but we need temporal trajectory functions for the animation. We will first calculate the path length of the luge at track level by numerically integrating the position function (in x). The Pythagorean theorem is applied repeatedly to points along the path and the individual lengths are summed.

[Graphics:../Images/index_gr_44.gif]
[Graphics:../Images/index_gr_45.gif]

A typical velocity during a luge run is 60 miles per hour which is 88 feet per second.

[Graphics:../Images/index_gr_46.gif]

At this velocity, the time to traverse one cycle of the luge track (and one cycle of the animation) can be calculated.

[Graphics:../Images/index_gr_47.gif]

We will calculate the x position of the slider at 72 position increments of deltap and time increments of deltat.

[Graphics:../Images/index_gr_48.gif]
[Graphics:../Images/index_gr_49.gif]
[Graphics:../Images/index_gr_50.gif]

The Pythagorean theorem is used to calculate the x position for a constant velocity.

[Graphics:../Images/index_gr_51.gif]

The x position is transformed from index based to time based using an interpolation function. The y and z position functions are also calculated.

[Graphics:../Images/index_gr_52.gif]

Here is a plot of the track level position functions (x, y, z) in black, blue, and red respectively.

[Graphics:../Images/index_gr_53.gif]

[Graphics:../Images/index_gr_54.gif]

The attitude of the sled is determined by its contact with the track. The sled has length in the fore/aft and side to side directions. Therefore, the angle of the sled is calculated as the difference in the sled's position rather than the instantaneous slope of the track. The luge's fore/aft (x) and side to side (y) widths are given below.

[Graphics:../Images/index_gr_55.gif]

The attitudes are calculated using vector geometry. For the yaw and pitch angles, a vector which runs the length of the luge x axis is created at track level. This vector is projected to the horizontal. The angle the projected vector makes with the luge world x axis gives the yaw angle. The pitch angle is the angle between the luge x axis with the projected vector. For the roll angle, a vector which runs the width of the luge y axis is created. Roll is given by the angle this vector makes with the horizontal.

[Graphics:../Images/index_gr_56.gif]
[Graphics:../Images/index_gr_57.gif]
[Graphics:../Images/index_gr_58.gif]

The attitudes as a function of x are converted to functions of time by substitution of the x position.

[Graphics:../Images/index_gr_59.gif]

Here is a plot of yaw, pitch, and roll in black, blue, and, red, respectively, in units of degrees.

[Graphics:../Images/index_gr_60.gif]

[Graphics:../Images/index_gr_61.gif]

The camera position, cpos, will be located at the midpoint of the sled at 1.25 feet off the track surface. A vertical vector of this length is transformed through roll, pitch, and yaw rotation matrices (from Geometry`Rotations`) to obtain the correction dct which is added to the surface position.

[Graphics:../Images/index_gr_62.gif]
[Graphics:../Images/index_gr_63.gif]

The slider's position, spos, is midway between the track position and the camera position.

[Graphics:../Images/index_gr_64.gif]

Here is a plot of the track, camera position, and the slider in freeze frame.

[Graphics:../Images/index_gr_65.gif]
[Graphics:../Images/index_gr_66.gif]

[Graphics:../Images/index_gr_67.gif]


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