If one mounts a camera to a sturdy tripod and opens the shutter for several minutes, as would be required to capture even a fairly bright object, the developed picture will be trailed, with every star creating a short or long streak depending on the length of the exposure and the part of the sky being imaged. Because of the Earth's motion, the positions of the stars shift during any long exposure, thereby limiting the practical exposure time for a fixed camera. This limit works out to around 30 seconds for a normal wide-angle lens. Of course, very long exposures exploiting this effect have their own artistic appeal, but accurate images of a particular object will not be possible.
To counteract the Earth's rotation and allow longer exposures, some kind of
drive system is needed. If an equatorially-driven telescope is available,
it can provide the tracking with the camera mounted ``piggy-back'' on the
side. For beginning astrophotographers seeking a more economical approach,
the "barn door" mount is a simple way to track a camera for wide-angle
pictures.
The simplest barn door drive consists of two pieces of wood which are hinged
at one end. One piece is mounted to a firm support like a tripod and has a
threaded rod going up through the end opposite the hinge (see Figure 1).
This drive screw can be turned by hand at periodic intervals, or driven by
a motor, in order to gradually push the boards apart. If the camera is
attached to the top board and the hinge axis is aligned with the north
celestial pole (very near to the star Polaris), a crude tracking system is
obtained. This is sometimes referred to as a tangent arm drive,
because the drive screw is always perpendicular to the bottom board. If the
screw is turned at a constant rate, the tracking speed is initially correct
but error accumulates rapidly.
A more popular variation is the isoceles or single-arm drive. Here the drive screw is pivoted at both boards so that an isoceles triangle is always formed as the boards open up. In published designs, the screw is almost always driven at a constant rate using a stepper motor and a oscillator/divider circuit built from standard CMOS logic. Some simple math shows how the angular velocity of the mount is related to the motion of the drive screw:
L/2
sin(theta/2) = ---
x
L/2
theta = 2*arcsin ---
x
dtheta dL/dt
omega = ------ = --------------------
dt x*sqrt(1 - L^2/4x^2)
In other words, the angular velocity of the mount, omega, will not be constant when the drive screw is turned at a constant rate which yields a constant speed of board separation, dL/dt, at the drive screw. The approximation is much better than in the tangent arm case, but significant errors will accumulate after ten or twenty minutes.
Suppose we pick L = L(t) = 2x*sin(at), where a is some constant. Now,
2x*sin(at)
theta = 2*arcsin ---------- = 2at
2x
and the mount is turning at a constant angular rate--just what we need. Note that dL/dt = 2ax*cos(at), indicating that perfect tracking is achieved when we let the drive rate slow down with time. Mechanically, the smart barn door uses the single-arm (isoceles) design, as this is fairly easy to construct and yet gives a considerable weight savings over the double-arm designs. For the control circuit, an 87C51FA microcontroller is used--the EPROM version of Intel's popular 8051 family. This choice was based on the parts on hand in the lab; many other 8051 derivatives could be used with the same or very similar assembly code and only minor circuit changes. Four of the port 3 pins are used to drive NPN darlington pair transistors of sufficient power-handling ability to energize the four stepper motor windings. (If a motor type other than four phase unipolar is used, a different drive scheme will be required). Three of the port 1 pins control status LEDs, and four more serve as switch inputs.
[ View the Schematic diagram ]
The drive screw speed, and hence the stepping rate of the motor, must be varied according to a sine function, and this poses an implementation problem. Any sort of extended-precision or fractional arithmetic on the 8051 would be difficult to implement on its 8-bit architecture. A data table of timer values for each step was considered, but this would use too much memory because of the tens of thousands of motor steps needed over the course of a few hours. Since a performance simulation in MATLAB was desired anyway, a unique solution was developed: The simulation was used to generate the needed tracking corrections for a chordal approximation to the ideal sine curve drive rate. In this approach, the stepper motor is started at some initial rate, then the time between steps is increased or decreased only as needed to keep the tracking error within specified bounds. As the simulation runs, it also stores the speed corrections to a file. Each correction consists of two numbers: a step number (counting from zero with the two boards fully closed) and the new stepping interval (taken in milliseconds). Here are the first few data points from one simulation run:
.word 1474
.word 254
.word 1616
.word 253
.word 2998
.word 254
.word 3153
.word 253
[ View the MATLAB simulation code ]
[ View the complete 8051 assembly code ]
As long as the software can keep track of the number of steps executed since the closed position, it can figure out what the proper stepping interval should be. Note the since corrections are relatively infrequent, especially at the beginning of tracking, the required data table size in the microcontroller is drastically reduced. Best of all, the simulation results from which these numbers were derived guarantee an arbitrary level of accuracy. In reality, of course, mechanical imprecision and less-than-ideal polar alignment of the mount will cause errors which completely swamp out the computed error based on geometry and timing alone. The best that can be said is that the smart barn door drive has no inherent errors designed-in.
Another interesting plot is the graph of stepping interval versus time. Although one might initially expect the stepping interval to increase monotonically, constraining these values to one millisecond units requires that the stepping interval be alternated back and forth to keep within the specified error bounds. This is analogous to pulse-width modulation, and the gradually changing "duty cycle" of the trace is easily seen on the graph.
The user interface of the tracker is simple and follows a cassette tape player model. Four switches are used in the usual orientation for rewind (a fast reverse drive to prepare for another exposure), stop, play (normal tracking operation), and fast forward. The barn door mount is smart enough to automatically slow down and stop when it is in fast reverse mode and the end of travel is reached with the boards touching.
Should the mount lose synchronization, i.e. if the software thinks it is at step zero when the two boards are not touching, tracking error will result. Errors may also be caused by a difference between the actual board lengths and the lengths set in the MATLAB simulation.
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Last updated 9 November 2000 by Mark Haun
