> This article describes drone physics. Audience should have familiarity with introductory linear algebra, introductory calculus, and introductory classical mechanics.
Having familiarity with all 3, as soon as I read this sentence I knew it was all down hill for me.
In general, no... most drones do not support negative altitude in their guidance systems.
The readers have a right to complain about LLM slop content. Especially when publishers use credential stuffing to sound authoritative with nonsense. =3
If you're into simulators then you may try applying forces directly rather than integrating ODE. If sim's has tiny numerical instabilities or integrator schedule is wrong the whole thing gonna fall spectacularly. The interest is with running large-scale simulations where along with ODE you need to handle collisions and deformations. The best one currently is MuJoCo.
I studied this subject when I was building FPV sim from scratch. I noticed that yaw action is often misunderstood, falsely believing that it's caused by a change in angular momentum (think reaction wheels).
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
4, 6, and 8 are all fairly standard, as quadcopter, hexacopter, and octacopter, because you have a pair of motors rotating in each direction, which allows you to slow down all the motors spinning in one direction and speed up the others (as long as each set spinning the same direction is on opposite sides of the center of gravity), to maintain constant lift while yawing left or right.
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 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.
Sorry, I wasn't arguing against bicopters being possible, obviously something like the osprey exists.
> 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.
He specifically says if the motors independently tilt. This is active control. The Rocket pendulum fallacy is only valid for systems requiring passive stability.
If you look at larger multi-copters, 6 and 8 are quite common since they allow for some redundancy, allowing for safe landing and/or continued flight following the failure of one of the rotors.
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.
Not the OP, but due to neighboring rotor interactions an even number (or a single helicopter like lift rotor and a tail rotor ignoring airfoil lift possibilities) rotor makes design easier and improves performance significantly. Think of the rotors like gears meshed together and realize that an even number spins easily, while an odd number is locked.
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.
I was looking for an explanation of how you deal with the fact that quad rotors are under-actuated, and therefore you lack the complete control authority to do any arbitrary motion in space. So path planning has to take this into account, but how is this described?
Pitch and roll are coupled to fwd/back and Left/right respectively. So, you can make it move however you want, but its attitude can't be set independently of its motion. That's a simplification. One consequence is, for example, if you have a fixed camera mounted fwd etc, the angle it looks will correlate with the speed you have it move.
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!
Yes and no. The math here is useful, albeit not required. It is not specialized PhD material; it's linear algebra. (And not the abstract quantum mechanics/chem kind!)
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.
I always found it interesting that basic quadcopters work on nothing but a gyroscope which measures angular velocity in 3 dimensions, and optionally an accelerometer for auto-levelling, with 3 dimensions of acceleration.
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 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.
You could, but once you need to turn around and thrust upwards to avoid hitting the ground, you'll need to run the motors HARD, which will make a lot of turbulence right below you, which you then fall into. So you're falling into wind that is already blowing towards the ground, so you need to push it down even faster, and so on.
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, for autonomous (or semi autonomous) flight, there's a way to make the drone flip and "power dive" then fly a carefully configured/programmed half-loop at the bottom? Seems there might be scope that way to have an autonomous "reduce altitude as rapidly as possible to 5m" function on a button that'd fly a full power j shape and compensate for the semicircular radius of the "deceleration without flying through your own prop wash" part of the maneuver? Similar (but more complex) to the "flip" and auto land functions on "toy" drones?
(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.)
My intuition is that you don't want to touch your gains/tuning for external disturbance. The tuning should be the most aggressive that maintains stability margins. The external input doesn't matter because you're looking at open loop stability margins (See: https://en.wikipedia.org/wiki/Nyquist_stability_criterion ). All this assumes linear systems and such but as as general principle.
The main anti prop wash thing in BF seems to be using a good frame design so that you don't get frame resonances below ~100+ Hz and that allows you to raise the lowpass gyro filter threshold high enough to let the PID loop handle propwash.
You'd be right to be surprised by lack of mention of quaternions. I (blog author) was not too familiar with them when I did the work this post is based on. With the benefit of hindsight, I may yet revise the post.
Hello, blog author here. I agree the math appears overkill. I wrote this as a learning aid for myself with the benefit of hindsight of having worked with some drone sims. I wanted to dispel any doubt in my mind that I could derive drone physics from basic principles, instead of copy-pasting state equations. I went further and tried to motivate transport theorem, rotational analog of F=ma etc from scratch.
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.
Ok man, for me it's PhD math. Waaaay above my high school level math.
Perhaps you will write a version for layman like me, without advanced math and rather with more verbal explanations and animations.
> This article describes drone physics. Audience should have familiarity with introductory linear algebra, introductory calculus, and introductory classical mechanics.
Having familiarity with all 3, as soon as I read this sentence I knew it was all down hill for me.
Going down hill is easy, right?
In general, no... most drones do not support negative altitude in their guidance systems.
The readers have a right to complain about LLM slop content. Especially when publishers use credential stuffing to sound authoritative with nonsense. =3
If you're into simulators then you may try applying forces directly rather than integrating ODE. If sim's has tiny numerical instabilities or integrator schedule is wrong the whole thing gonna fall spectacularly. The interest is with running large-scale simulations where along with ODE you need to handle collisions and deformations. The best one currently is MuJoCo.
I studied this subject when I was building FPV sim from scratch. I noticed that yaw action is often misunderstood, falsely believing that it's caused by a change in angular momentum (think reaction wheels).
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
Why does four seem optimal? Why not three or five? or eight?
4, 6, and 8 are all fairly standard, as quadcopter, hexacopter, and octacopter, because you have a pair of motors rotating in each direction, which allows you to slow down all the motors spinning in one direction and speed up the others (as long as each set spinning the same direction is on opposite sides of the center of gravity), to maintain constant lift while yawing left or right.
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.
> Two is possible if the payload hangs below the props
I think this is incorrect, it's the somewhat unintuitive rocket pendulum fallacy.
Here is a video of one: https://www.youtube.com/watch?v=qO747JB4Nr8
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.
Sorry, I wasn't arguing against bicopters being possible, obviously something like the osprey exists.
> 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.
He specifically says if the motors independently tilt. This is active control. The Rocket pendulum fallacy is only valid for systems requiring passive stability.
Tandem rotor helicopters exist.
Rockets also have active control via gimballing motors. Bicopters tilt their motors to control pitch and yaw, roll is still governed by motor speed.
I was arguing against the COG of the payload, not of the existence of bicopters entirely.
Too late to edit, but I meant cyclic pitch, not collective pitch.
If you look at larger multi-copters, 6 and 8 are quite common since they allow for some redundancy, allowing for safe landing and/or continued flight following the failure of one of the rotors.
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.
Not the OP, but due to neighboring rotor interactions an even number (or a single helicopter like lift rotor and a tail rotor ignoring airfoil lift possibilities) rotor makes design easier and improves performance significantly. Think of the rotors like gears meshed together and realize that an even number spins easily, while an odd number is locked.
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.
This feels very https://ciechanow.ski/
Which I think is a good thing :)
It's my blog. Bartosz Ciechanowski's work was my inspiration!
I was looking for an explanation of how you deal with the fact that quad rotors are under-actuated, and therefore you lack the complete control authority to do any arbitrary motion in space. So path planning has to take this into account, but how is this described?
Pitch and roll are coupled to fwd/back and Left/right respectively. So, you can make it move however you want, but its attitude can't be set independently of its motion. That's a simplification. One consequence is, for example, if you have a fixed camera mounted fwd etc, the angle it looks will correlate with the speed you have it move.
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!
I wrote something similar (though admittedly more in depth) some time ago: https://www.cggonzalez.com/blog/index.html
Previously: https://news.ycombinator.com/item?id=48781539
I would have liked to understand this, but the math behind it is overkill. One may need a PhD in aerospace to digest it.
Yes and no. The math here is useful, albeit not required. It is not specialized PhD material; it's linear algebra. (And not the abstract quantum mechanics/chem kind!)
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.
I always found it interesting that basic quadcopters work on nothing but a gyroscope which measures angular velocity in 3 dimensions, and optionally an accelerometer for auto-levelling, with 3 dimensions of acceleration.
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 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.
* https://en.wikipedia.org/wiki/Vortex_ring_state
Why not just flip over and accelerate? I bet some in the drone racing crowd do this.
You could, but once you need to turn around and thrust upwards to avoid hitting the ground, you'll need to run the motors HARD, which will make a lot of turbulence right below you, which you then fall into. So you're falling into wind that is already blowing towards the ground, so you need to push it down even faster, and so on.
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, for autonomous (or semi autonomous) flight, there's a way to make the drone flip and "power dive" then fly a carefully configured/programmed half-loop at the bottom? Seems there might be scope that way to have an autonomous "reduce altitude as rapidly as possible to 5m" function on a button that'd fly a full power j shape and compensate for the semicircular radius of the "deceleration without flying through your own prop wash" part of the maneuver? Similar (but more complex) to the "flip" and auto land functions on "toy" drones?
(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.)
My intuition is that you don't want to touch your gains/tuning for external disturbance. The tuning should be the most aggressive that maintains stability margins. The external input doesn't matter because you're looking at open loop stability margins (See: https://en.wikipedia.org/wiki/Nyquist_stability_criterion ). All this assumes linear systems and such but as as general principle.
The main anti prop wash thing in BF seems to be using a good frame design so that you don't get frame resonances below ~100+ Hz and that allows you to raise the lowpass gyro filter threshold high enough to let the PID loop handle propwash.
One can predict forces involved in landing for specific hardware using machine learning, e.g. https://arxiv.org/abs/1811.08027.
You'd be right to be surprised by lack of mention of quaternions. I (blog author) was not too familiar with them when I did the work this post is based on. With the benefit of hindsight, I may yet revise the post.
Hello, blog author here. I agree the math appears overkill. I wrote this as a learning aid for myself with the benefit of hindsight of having worked with some drone sims. I wanted to dispel any doubt in my mind that I could derive drone physics from basic principles, instead of copy-pasting state equations. I went further and tried to motivate transport theorem, rotational analog of F=ma etc from scratch.
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.
Ok man, for me it's PhD math. Waaaay above my high school level math. Perhaps you will write a version for layman like me, without advanced math and rather with more verbal explanations and animations.
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