Drone Physics
iahmed.me
iahmed.me
Having familiarity with all 3, as soon as I read this sentence I knew it was all down hill for me.
The readers have a right to complain about LLM slop content. Especially when publishers use credential stuffing to sound authoritative with nonsense. =3
There is a good SE answer about deriving net torques given each props' thrust force in less PhD language than this article: https://drones.stackexchange.com/a/416/11402
If anyone is interested in tackling UAV simulation, make sure to check out this prop/motor performance database, helped me a lot building a virtual drone with realistic properties (mainly motor thrust & torque): https://database.tytorobotics.com/tests
Of course there are 6, 8, and larger numbers of rotors used in actual drones. The advantage of more rotors is that redundancy to failure can be built in, and that rotor tip speed for a given lift can be somewhat reduced at the cost of efficiency.
Tricopters are possible as tilt-rotors, where you have two motors spinning opposite directions on mirrored sides, and the third motor is on the midline in front or back and able to tilt left and right. This allows the vehicle to control yaw despite having an inherent yaw imbalance in hover. I suppose you could do any odd number of rotors this way, and more rotors would mean less inherent yaw imbalance if there's only one extra motor spinning a given direction.
Two is possible if the payload hangs below the props, and the props are able to independently tilt.
One requires collective pitch and still at least a small tail rotor to cancel out the yaw, as a helicopter, but that is far more complex and fragile. But it is the most efficient, as fewer larger props are always more efficient than more smaller ones.
So four just turns out to be minimum to have no moving parts besides the props themselves, and still have full control authority, and the control logic is more straightforward than tilt rotors and such.
I think this is incorrect, it's the somewhat unintuitive rocket pendulum fallacy.
There's a few demonstrated here (26-32 seconds in): https://www.youtube.com/watch?v=vBsyFj8bEJk
And here's one that uses the ailerons instead of being able to tilt the motors: https://www.youtube.com/watch?v=L2fgL97rgK0
I could've sworn Peter Sripol built one while on Flite Test too, but I can't find it atm.
Looks like the rocket pendulum fallacy is about expecting meaningful passive stability from the location of the center of thrust vs center of mass, but even 2-rotors need to be able to tilt independently (or deflect thrust) for active control.
Theoretically this could still work even if the center of mass was above center of thrust, but the tilting/vectoring responsiveness would need to be very high. These RC models at least move so slowly that the air resistance of swinging back and fourth really does help dampen oscillations passively, but they all still have active flight controllers that are trying to keep angular velocity at zero without control input.
> the air resistance of swinging back and fourth really does help dampen oscillations passively
But they don't swing back and forth like a pendulum (unless your PIDs are off) because of the center of mass, that is the entire issue with the fallacy.
Tandem rotor helicopters exist.
I was arguing against the COG of the payload, not of the existence of bicopters entirely.
As other replies note, 4 is the simplest arrangement mechanically and control-wise, as the control math is quite simple (just rotor speeds/torque) and the only moving parts are the fixed-pitch rotors.
The minimum, as seen in real (and model) helicopters, is either two (approximately) constant-speed rotors with swashplate control, or one (approximately) constant-speed rotor with swashplate control and one tail rotor, either with (approximately) constant speed and variable pitch, or with variable speed. At the scale of real helicopters, two rotors may often be more powerful and efficient (e.g. CH-47, V-22) but the size and weight of the gearbox needed to transmit so much power is a significant contribution to the weight and cost of the helicopter, and thus having a single main gearbox is much lighter and cheaper. The notable difficulties of shaft drive between multi-rotor helicopters, particularly with distributed engines (see a number of V-22 issues) strongly discourages helicopters with more than 2 rotors.
Which I think is a good thing :)
If you are just concerned with moving the center of mass to different locations in space, or have it travel with a specific velocity, you're fine.
This sort of limitation is common vehicles of all sorts; having 4 independent DOF is above average!
What I mean by not required is, I've written drone firmware and didn't directly use this; the core can be done with a PID for rate controls (Compare measured rate along each axis with commanded; nudge motor power proportional to the diff), and commanding attitudes can be done with fundamental quaternion operations, as a slower outer loop.
I would skip the Tait-Bryan stuff in the article, in favor of pure quaternions. Actually, I'm kind of floored the word "quaternion" doesn't appear in the article.
Then of course you can add on GPS for absolute position and route planning in 3 dimensions, and a ground height sensor for auto-landing, then you can add distance sensors on the sides for obstacle avoidance... it's all incredibly intuitive from a game programmer perspective. Then you can add in some signal filtering to mask out the range of vibrations from the motors and props being imperfectly balanced.
The hard part seems to be smooth rapid vertical descent. It's impossible to predict how the prop wash will interact with the wind and push the drone around as it descends into its own turbulence. I was tracking betaflight development for a while and was wondering if we'd ever see some kind of prop-wash calibration. Is there some adjustment of PID gains while descending through prop wash that could improve stability?
The phenomenon is called Vortex Ring State* and it's not a problem that can be solved with a better calibration/control. A basic description of the problem is the prop moves into it's own prop-wash replacing happy lift with sad turbulence. The solutions are to:
1. Don't descend vertically, always have a reasonable degree of lateral motion. 2. Descend vertically slowly, how slow is vehicle specific 3. Angle the propellers so that their thrust angle is off vertical. 4. Descend with no power at all, thus avoiding the creation of prop-wash entirely, halting the descent will require the use of 1-3.
If you do happen upon a solution that can be applied to traditional helicopters there's probably a good deal of money in it for you.
Ducted fans might help a little, since they can send the turbulence further away faster, but then you have the additional mass of the ducts, and the inefficiency of accelerating the air to higher speeds, since larger slower props are more efficient (but less responsive), to the point that this is possible: https://www.youtube.com/watch?v=emK-qIbuJ-k
(I wonder if the drone people from Ukraine already have this? I suspect it'd make drones even harder to defend against with anti aircraft system designed and optimised to shoot down fighter/bomber planes instead of swarms of inexpensive drones.)
In summary: I take F=ma and extend it for rotational motion. (1) Calculating linear motion when the vehicle containing sensors is rotating. (2) Calculating rotation of the vehicle itself due to thrust/yaw force acting about its center of mass.
I'll echo what the other commenter said: this is no way PhD math. It may appear so - but I'm only being verbose with simpler concepts like cross products and rotation matrices.