For the last two weeks myself and Agis ( @azisi ) have been trying to specify the required values for the third version of the SatNOGS tracker in order to design it. First of all we started by measuring the current version characteristics. The theoretical value of angular velocity is ω=150steps/s * 1.8deg/step / 60(ratio) = 4.5deg/s
and the one of torque, given that the steppers we are using have τ = 59Ncm, is τ = 354Kgcm (gear ratio 60:1).
However, practically we found out that torque was approximately 38kgcm measured at 24V, 0.5A and the drivers’ Vref set at 1V. (At this point it is worth noting that the chinese pololus had different torque in each direction, and we had spent many hours before we understood why this was happening)
Some of the reasons we are not approaching those theoretical values are
- The combination of stepper motors and stepper drivers don’t work
- Lot of friction between moving parts, the lack of bearings.
- Oscillation is created because of non smooth movement together with
elasticity of the box and the axis.
- Worm gear is not optimal designed to have the maximum efficiency
given the materials it is made of.
So we made a research on the net to find other commercial and DIY (only the documented) solutions. Because this forum doesn’t support tables, here is the link to an ethercalc with our results
After this, and since all the information we could find about antenna mounting and tracking stations were scattered around the web and many of them were not so scientific, we made our analysis about torque, nominal and stall, angular velocity for ALT and AZ and accuracy.
The greatest force the tracker needs to withstand is the force created by strong wind. The worst case is when one antenna is elevated at 90 degs, facing the direction of the wind. We based our calculations on an article found online after comparing it to others. We “translated” the second table in metric (because we don’t understand imperial and because we needed same units system in our calculations)
and we applied the worst case model (EIA-222-F) in 3 different antennas: in the biggest one of our designs, and in two others, for which we obtained data from yaesu G800 rotator manual at page 3. We assumed that antennas are mounted in 1m away from the azimuth axis. For our antenna with 2m length (actual, not wavelength), made by 2cm square tube, the generated torque was ≈600Kgcm. For the 144MHz 10-elements Yagi from the article is ≈6000Kgcmand for the third 430MHz, 12-elements Yagi is ≈1800Kg*cm
Moment of inertia
Now for the moment of inertia: (for all installation methods we assumed that antennas are counterbalanced in the elevation axis) the worst case scenario here is to use two 3kg (our designs are less than 1kg) back mounted yagis with 3kg counterbalances both mounted in 0.75m away from azimuth axis. The torque you need in order to accelerate this system from ω=0deg/s angular velocity to ω=5deg/s (the math about angular velocity is below) in one second is about 60kgcm.
Note: we suppose that the mass of antennas is near to the altitude axis, so the torque of this axis that is needed to accelerate is approximately 0.
M1: torque of Azimuth axis
L: length of center of mass of antennas from azimuth axis (0.75m)
m: mass of antennas and of counterweight (3kg + 3kg = 6kg)
I: moment inertia
a: angular acceleration of azimuth axis 5deg/s^2
I = I1 + I2 = mL^2 + mL^2 = 2mL^2 = 6.75 kgm^2
M1 = I*a = 6.75kgm^2 * 0.087rad/s^2 = 0.58 Nm = 5.8 kgm = 58 kgcm
(How well do you remember trigonometry?)For the angular velocity max needed in altitude axis the things are straightforward. The closer is the satellite the larger the velocity. According to the wikipedia article about LEO, the lowest height limit is 160 km and the speed unit to orbit earth in this altitude is 7,8 km/s. As a result, maximum velocity in ALT axis is 2,8 deg/s. In ALT AZ rotator design there is a well known limitation: the closer something passes near zenith the biggest gets the velocity of the AZ axis. Therefore, we have analysed this problem to figure out the optimal velocity and how high we are allowed to track a target in relation to AZ velocity. The picture below illustrates a ground station B which tracks a satellite Γ in X degrees altitude. The satellite velocity at this point is vertical to the screen (page) plane.
The equations that lead to maximum altitude at which we can track in relation to AZ angular velocity are
ω : angular velocity of AZ DOF in rad/s
H = ΑΕ + ΕΓ : Minimum Height of LEO, 160 km
R = ΑΕ : Radius of Earth, 6500 km
u : linear velocity of satellite that rotates in 160km height is 7.8 km/s
ΒΔ = u / ω : ΒΔ in km
α = atan(ΒΔ / R)
δ = π - α
γ = asin( sqrt(R^2+ΒΔ^2) * sin(δ) / (H+R) )
ά = π - δ - γ
ΓΔ = (H+R) * sin(ά) / sin(δ)
χ = atan(ΓΔ / ΒΔ)
Below you can see the plot of the equations mentioned above, where horizontal axis represents angular velocity (ω) in deg/s and vertical axis shows the max track altitude (χ) for lower bound of LEO.
After studying this diagram, we came up to the conclusion that an angular velocity of 5 deg/s is adequate. For this decision, we took into consideration the main lobe of antenna (Δ3db) which in most situations is about 20 deg.
Together with the above mentioned specifications, we would also like for the 3rd version of SatNOGS rotator to be:
- inexpensive (less than €300, if possible)
- lightweight and portable
- rigid and durable
- easy to build and fix (try to use easily available materials)
- electromagnetically shielded, so that noise in reception is reduced
- accurate (backslash reduction and use of encoders at the axis)