3 ms·

Drone Physics

138 points by wrxd 3mo ago

huqedato 3mo ago
I would have liked to understand this, but the math behind it is overkill. One may need a PhD in aerospace to digest it.
- the__alchemist 3mo ago
  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.
  - hnuser123456 3mo ago
    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?
    - MadnessASAP 3mo ago
      > 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 https://en.wikipedia.org/wiki/Vortex_ring_state
      - chasd00 3mo ago
        Why not just flip over and accelerate? I bet some in the drone racing crowd do this.
        hnuser123456 3mo ago
        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 https://www.youtube.com/watch?v=emK-qIbuJ-k
  - hazrmard 3mo ago
    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.
- deleted 3mo ago
  [deleted]
- hazrmard 3mo ago
  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.
  - huqedato 3mo ago
    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.
too_root 3mo ago
This feels very https://ciechanow.ski/ https://ciechanow.ski/ Which I think is a good thing :)
- hazrmard 3mo ago
  It's my blog. Bartosz Ciechanowski's work was my inspiration!
cwiz 3mo ago
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.
ivanjermakov 3mo ago
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 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 https://database.tytorobotics.com/tests
- fragmede 3mo ago
  Why does four seem optimal? Why not three or five? or eight?
  - kurthr 3mo ago
    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.
  - deleted 3mo ago
    [deleted]
  - hnuser123456 3mo ago
    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.
oren-speiser 3mo ago
[flagged]
katzgrau 3mo ago
> 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.
- dmd 3mo ago
  Going down hill is easy, right?
  - Joel_Mckay 3mo ago
    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
tolerance 3mo ago
Previously: https://news.ycombinator.com/item?id=48781539 https://news.ycombinator.com/item?id=48781539
cgg1 3mo ago
I wrote something similar (though admittedly more in depth) some time ago: https://www.cggonzalez.com/blog/index.html https://www.cggonzalez.com/blog/index.html
Isamu 3mo ago
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?
- the__alchemist 3mo ago
  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!
jdw64 3mo ago
[dead]

138 points by wrxd 3mo ago

I would have liked to understand this, but the math behind it is overkill. One may need a PhD in aerospace to digest it.

the__alchemist 3mo ago
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.
- hnuser123456 3mo ago
  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?
  - MadnessASAP 3mo ago
    > 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 https://en.wikipedia.org/wiki/Vortex_ring_state
    - chasd00 3mo ago
      Why not just flip over and accelerate? I bet some in the drone racing crowd do this.
      - hnuser123456 3mo ago
        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 https://www.youtube.com/watch?v=emK-qIbuJ-k
- hazrmard 3mo ago
  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.
deleted 3mo ago
[deleted]
hazrmard 3mo ago
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.
- huqedato 3mo ago
  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.

too_root 3mo ago

This feels very https://ciechanow.ski/ https://ciechanow.ski/ Which I think is a good thing :)

hazrmard 3mo ago
It's my blog. Bartosz Ciechanowski's work was my inspiration!

cwiz 3mo ago

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.

ivanjermakov 3mo ago

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 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 https://database.tytorobotics.com/tests

fragmede 3mo ago
Why does four seem optimal? Why not three or five? or eight?
- kurthr 3mo ago
  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.
- deleted 3mo ago
  [deleted]
- hnuser123456 3mo ago
  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.

oren-speiser 3mo ago

[flagged]

katzgrau 3mo ago

> 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.

dmd 3mo ago
Going down hill is easy, right?
- Joel_Mckay 3mo ago
  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

tolerance 3mo ago

Previously: https://news.ycombinator.com/item?id=48781539 https://news.ycombinator.com/item?id=48781539

cgg1 3mo ago

I wrote something similar (though admittedly more in depth) some time ago: https://www.cggonzalez.com/blog/index.html https://www.cggonzalez.com/blog/index.html

Isamu 3mo ago

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?

the__alchemist 3mo ago
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!

jdw64 3mo ago

[dead]