Theory and Design
Linear Velocity (m/s)
Angular Velocity (rad/s)
Figure 3.5 (a) The linear velocity during a sweep. (b) The angular velocity during a sweep. These velocities do not contain the spikes that appear during an entire scan.
ssuming the spikes in velocity between sweeps can be ignored, the maximum velocities during a typical sweep may be analysed. Figure 3.5a shows the linear velocity of the scan- ner during a single sweep. The sweep starts off slowly and rises to a maximum velocity of approximately 0.52 m/s before dropping away again. The angular velocity shows a dif- ferent picture (Figure 3.5b) and remains low throughout the sweep. Intuitively this seems
reasonable since the operator keeps the scanner at a fairly constant angle to the surface being scanned. In a broad sense, the surface is likely to be flat during a sweep and there-
fore the angle of the scanner remains approximately constant.
s the angular velocity is
low (reaching a maximum of 0.59 rad/s) it is not analysed further. analysed to determine the maximum tracking speed.
The linear velocity is
The maximum sweep velocity in m/s can be transformed into a velocity on the image plane in pixels/frame. These units will be the native units used by the firmware. For linear translations, parallax causes objects closer to a camera to appear to move faster than objects further away from the camera. The minimum working distance is 0.5 m so the maximum image velocity is calculated using this figure. It is a function of the velocity of the scanner, the FOV of the camera, the frame rate of the camera, and the distance between the markers and the camera. The velocity v0 in pixels/frame is given by
v , f T r (3.19) zk
where v is the velocity in m/s, z is the distance between the camera origin (the pinhole) and t h e m a r k e r , k i s t h e l e n g t h o f a p i x e l a t t h e i m a g e s e n s o r i n m e t r e s , T r i s t h e f r a m e p e r i o d and f is the focal length of the camera lens. This equation is derived using the perspective ,
transform in Equation 3.4 and is illustrated in Figure 3.6. For example, using a focal length of 6 mm, a distance of 0.5 m between the camera and marker, a frame period of 1/60 s, and