This is an analysis of the variables which affect the ability to control a multi-rotor as the size is scaled up.

This will show why at some point, variable pitch must be introduced.

I hope to follow this with another article which includes Phase Margin and Gain Margin, but right now I'm trying to refresh my memory from 30 years ago on Laplace transforms and Bode Plots.   Ugh!

I hope this makes some sense.

MultiRotorControllability.pdf

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Comment by Rustom Jehangir on April 22, 2016 at 10:45pm

Randy - Nice analysis. I wrote an a paper a few years ago about a control system for large multicopter. One of the most interesting issues is the scalability of the differential torque yaw method used on multicopters and how its effect diminishes to almost nothing with scale.

We ended up tilting the props with actuators to get sufficient yaw authority.

Here's a link to the paper if you are interested.

-Rusty


Developer
Comment by Andrew Tridgell on April 23, 2016 at 3:43am

Thanks for the analysis. I'm trying to see how to tally this with real-world experience. My "quad" is 13kg (a big quadplane) and has arm spacing of 1m. My 250 racing quad is 4x smaller in arm spacing, and 10x smaller in weight.  Going by the table on page 9 would I expect 1024x smaller roll response? That doesn't seem to tally with what I see in practice.

What have I missed?

Comment by Rob_Lefebvre on April 23, 2016 at 4:47am

Interesting work Randy.  As you may know, I'm a proponent of helicopters.  Anything we can do to put the brakes on the out of control multirotor bandwagon is good.  :)   It is my belief, and I think that anybody who seriously considers the situation would agree, that once you get to the point of needing variable pitch propellers on a quadcopter, there is simply no point to it's existence.  Multirotors have the advantage of extreme mechanical simplicity.  4 sticks.  4 direct drive brushless motors.  4 fixed pitch props.  And a microcontroller.  Beautiful.  They are not more efficient than a helicopter.  They really don't do anything better than a helicopter other than being simple (and being new).  Once you take away the simplicity there is no good reason to build them.  Adding 4 servos, 4 variable pitch mechanisms, and possibly 4 powertrains for central power distribution, they have lost all their simplicity.

Now, a few comments, I only skimmed over it. First, the R22 is a 2 seater. Secondly, you aren't actually coming to a conclusion about where to draw the line in multirotor scale?  You also do not address the scale factor of external disturbance force which may actually help larger machines.  ie: A 40 km/h wind gust should have less effect on a large multirotor than a small one, requiring less control response.  I think.  I'm not sure how to model this.

Comment by Vladimir Kvitnevskiy on April 23, 2016 at 7:17am

Neat analysis. I like that someone went out of their way to document this stuff.

The last sentence, "This is primarily due to the exponential increase in the polar inertia of the rotor blade", leaves a bad taste. P^5 is not exponential, 5^P is. If you want to write a technical paper, don't use "exponential" to mean "fast-growing".

Comment by Michael Torrie on April 23, 2016 at 7:38am

Kind of interesting to see how everything old becomes new again.

Early designers grappled with the problems of full-scale multi-rotor craft, which led to the invention of the helicopter.  The moment one adds variable pitch and swash plates to multi-rotor craft one has re-invented the helicopter.  Perhaps the future is in multi-rotor helicopters (which is where this seems to be going)?  Do they provide more stability and are easier to control?  If a computer is already managing the flight characteristics of a multi-rotor craft, could not it just as easily stabilize a single-rotor helicopter?  From a full-scale perspective, I wonder if a single rotor with swash plate is more reliable than multiple rotors with swash plates.

Comment by Rob_Lefebvre on April 23, 2016 at 9:18am

Vladimir, exponential is commonly used to describe any function involving an exponent.

Comment by Rob_Lefebvre on April 23, 2016 at 9:21am

If a computer is already managing the flight characteristics of a multi-rotor craft, could not it just as easily stabilize a single-rotor helicopter?

Yes, yes it could.  That is why ArduCopter can control a Helicopter.  And it does a good job.

I don't think one could draw any hard and fast rule about reliability of a swashplate vs. multiple variable pitch props.  It depends too heavily on the particular application.

Comment by Marc Dornan on April 23, 2016 at 9:36am
Rob. You are correct about helicopter advantages. One thing that is hard to overlook is safety. One large blade is far more dangerous and hazardous than four smaller ones.
Comment by Rob_Lefebvre on April 23, 2016 at 9:42am

It's not quite so simple.  Depends on the blade design in each case.  Quadcopter blades can be rigid and razor sharp, capable of cutting aluminum (as I have personally seen), or they can be blunt and flexible.  Helicopters blades can be turning fast, or slow, and they can be strong carbon fiber, or balsa that simply explodes on impact.

In larger system like we are talking about in this blog post, it just doesn't matter.  You are arguing about which machine will kill you deader.

Comment by Gary McCray on April 23, 2016 at 10:33am

Hi Randy,

An excellent mathematical analysis of results already observed in the real world.

For a while there was great emphasis on simply making quadcopters bigger and indeed, there are props available that are even larger than 28 inches in diameter and motors designed to power them.

Several companies came out with large quadcopters because of both their additional payload capacity and the increased efficiency gained from going with the larger props.

But as your model suggests they also ended up with decreased responsiveness and reduced controllability.

As a result multicopters with greater number of props Hexes and octos or Y6 or X8 designs with top and bottom props became the accepted means for carrying larger payloads.

Of course both of these give up the additional advantage of the efficiency of the larger prop diameter and in the case of the Y6 and X8 design reduce efficiency even further by the interaction of the counter-rotating props in the same thrust stream.

Although there are still some quadcopters in use with props in the 24" to 28" range, that seems to be about the practical upper limit and I suspect that under unfavorable wind / gust conditions or at performance limits their controllability can degrade very quickly.

Essentially for big copters, helis simply have the edge, period.

Hexes and Octos do have the singular advantage of providing recovery with one or more motors out, but that is compromised by also having more elements to fail.

Nothing can beat that big single rotor for efficiency and the pitch control mechanisms are evolved sufficiently at this point to be reliable and long lasting.

Supposedly Brad Hughey has designed an automatically pitch compensating propeller / rotor which will alleviate the controllability problem for multi rotors and there was also some work on this from one of the major Universities, but we will just have to wait and see.

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