Enter your RPM and drive bolt thread details below to calculate the distance needed between your hinge and drive bolt.Stars are always in a fixed position in the sky and appear to move across the night sky as the earth rotates. The rate at which the earth rotates relative to the stars in known as sidereal time and a sidereal day is the time it takes for the earth to perform a full rotation. Essentially, it is the time it takes for a any star in the sky to return to that same position.A sidereal day on Earth is 23 hours 56 minutes and 4.091 seconds long or 1436 minutes (about 4 minutes shorter than a standard solar day) and we can use this time to calculate the rate at which the barn door tracker needs to turn.If we think of the top board of the barn door tracker drawing an imaginary line as it moves, it will eventually draw a full circle if it is able to move 360° on it’s hinge. This means that any point on the board can be through of as the radius of an imaginary circle so we need to work out what size circle will match the sidereal rate of the Earth and this is dependent on the rate at which the top board will move which is determined by the size of the thread on the drive bolt and the speed at which it is turned.
One rotation of Earth is 2Π radians so the sidereal rate of the Earth is 2Π / 1436 which is approximately 0.004375 radians per minute. Using this value we come to the final formula of:Radius (in inches) = RPM / ((2Π/ 1436) * TPI) or approximately RPM / (0.004375 * TPI)where RPM is the revolutions per minute of the drive bolt and TPI is it’s threads per inch. The drive bolt should be centered at this distance from the hinge.In part 1 of this series, I described how to construct an Arduino based motor controller. This time around it is time to look at the mathematics behind the movement of the mount. As noted previously, the mount will be driven by a threaded rod. As the motor rotates the rod in a nut attached to the camera board, it generates linear motion, however, the board needs to open up with constant angular motion. For simplicity it is intended to construct a type 1 barndoor mount with an isosceles drive rod, as illustrated in the following diagram In the above diagram the threaded rod has length R between the two boards, forming an angle θ.
It can readily be seen from the diagram that the isosceles triangle formed by the two boards and the rod can be split into a pair of identical right angle triangles. Basic trigonometry tells us that the sine of the principal angle in a right angle triangle is equal to the ratio between the opposite and hypotenuse: In our diagram above, the principal angle is θ/2, the length of the opposite is R/2 and the hypotenuse is L. Plugging those symbols into the first formula we get: In order to drive the threaded rod, the value we want to calculate is R, so we need to re-arrange the formula to get R on the left hand side: The Arduino isn’t directly controlling the length of the rod though, rather it is controlling its rotation. The length of the rod is the ratio between the number of rotations and the number of threads per centimetre. This gives us a second formula for R Lets substitute this new formula for R, into the previous formula: A few moments ago we mentioned that the quantity we actually control is the rotation of the rod, so the formula must be re-arranged to get the number of rotations on the right hand side:
With this formula, we know how many rotations are needed to achieve a given angle between the boards, but what exactly is the angle we need ? The Earth doesn’t take exactly 24 hours to rotate a full 360 degrees, in fact it is about 23 hours, 56 minutes, 4.0916 seconds. This value is known as the sidereal time or rate. With this information we can now define a formula to derive the value for θ at any given time t, since starting operation of the mount from a closed position: Since θ is the value we need, lets re-arrange that formula to get θ on the right hand side This formula for θ can now be substituted into the earlier formula for calculating the rotations of the rod: The final term can be slightly simplified by removing a factor of 2 This formula operates in degrees, but when doing calculations in software it is desirable to measure angles in radians. There is a simple formula for converting from degrees to radians: With this information it is now possible to update the formula for calculating rotations to operate in radians by substituting in the conversion formula:
A little while ago it was said that the Arduino is controlling the number of rotations of rod. This isn’t strictly true, it is in fact controlling the number of steps of the motor. A motor will have a certain number of steps it can do in one rotation, which gives a step size in degrees. The formula for calculating rotations can thus be adapted to calculate the number of steps This is our last formula and it has 4 variables to be filled in with accurate values This formula could be executed directly on the Arduino board, but a sine calculation is somewhat heavy for the microcontroller. Realistically the mount isn’t going to be doing exposures of longer than 2-3 hours. It is fairly trivial to thus this formula and calculate the required number of rotations for each minute of each hours and produce a 180 entry table. The Arduino then merely has to keep track of time and do a trivial table lookup to determine the rotation rate. An algorithm for doing this will be the subject of a later blog post.
Now read: part 3, drive control softwareStep 2: Some calculationsShow All Items A mean sidereal day is 23 hours 56 minutes 4.0916 seconds (23.9344696 hours), this is the rate at which the stars appear to revolve around our planet termed diurnal motion and is the rate of travel required in the barn door mechanism. So, 360°/23.9344696 = 15.041068635170423830908707498578° per hr = 0.25068447725284039718181179164296° per min to match the diurnal rate. The M6 drive rod has a pitch rate of 1mm in 1 min, so we need to calculate the length needed to achieve that diurnal rate, ie 0.25068447725284039718181179164296° per min. 1/(tan 0.25068447725284039718181179164296°)=228.55589mm Nice to know: M8 x 1.25 rod would need a rod to hinge distance of 285.69486mm M5 x 0.8 rod would need a rod to hinge distance of 182.8447mm « PreviousNext »View All Steps Download A barn door tracker, also known as a Haig or Scotch mount, is a device used to cancel out the diurnal motion of the Earth for the observation or photography of astronomical objects.
It is a simple alternative to attaching a camera to a motorized equatorial mount. Astronaut Don Pettit operates a barn door tracker located in the Destiny lab of the International Space Station. He made the mount from spare parts he had accumulated from aboard the station. The barn door tracker was created by George Haig. His plans were first published in Sky & Telescope magazine in April 1975. Modified versions of the tracker were published in the magazine's February 1988 and June 2007 editions. In late 2002 and early 2003, NASA astronaut Don Pettit, part of International Space Station Expedition 6, constructed a barn door tracker using spare parts he had accumulated from around the space station,[2] permitting sharper high resolution images of city lights at night from the ISS. A simple single-arm barn door tracker can be made by attaching two pieces of wood together with a hinge. A camera is mounted on the top board, usually with some sort of ball joint to allow the camera to be pointed in any direction.
The hinge is aligned with a celestial pole and the boards are then driven apart (or together) at a constant rate, usually by turning a threaded rod or bolt. This is called a tangent drive. This type of mount is good for approximately 5–10 minutes before tracking errors become evident when using a 50 mm lens. This is due to the tangent error. That length of time can be increased to about 20 minutes when using an isosceles mount. A curved drive bolt in lieu of either a straight tangent or isosceles mount will greatly extend the useful tracking time. These designs were further improved upon by Dave Trott, whose designs were published in the February 1988 issue of Sky & Telescope. By using a second arm to drive the camera platform - thus adding complexity to the fabrication - tracking accuracy was greatly increased, and can lead to exposure times of up to one hour. The most accurate of these designs is the Type-4. A modified double arm design minimizes tangent error by raising the point of rotation of the arm on which the camera is mounted.