### Multi-Rotor Scalability

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

• For a large quad the safety factor goes away very fast when you realize that if you loose a motor it falls out of the sky at full speed. These quads are 10-20kg and that falling from 100m is extremely dangerous. With a heli, at least there is the possibility of autorotation if there is an issue with the power system. With that said, a servo failure is probably catastrophic.

At the end of the day, any flying object at a high altitude is not safe and care should be taken when using these devices.

• @rustom - thanks for link to paper.  I will read it. Interesting about yaw.  Obviously my analysis only looked at pitch and roll.  It never occurred to me that yaw would be a problem, but obviously knowing the lift:drag ratio of rotor is in the 50 to 100 range, it would follow that yaw actuating torques are much smaller.

@Rob - Yes, my bad - the R22 is a 2 seater.

@Andrew - I made some assumptions in my analysis that might not be obvious.  For example, I didn't have a model for how the power available to the craft would need to scale up.  And hence, all of my analysis assumed a fixed input of a given motor current which produced a fixed motor torque.  Obviously as you scale up a vehicle, the power (current) and the motor torque coefficient would increase as a result of the increased scale.  However, I do not believe they would come close to compensating for the 5 power rotor inertia factor.  However, it could feasibly reduce that 5th power factor to a 3rd or 4th power of the scaling factor.  Which might be more in line with observed reality.

@Rob - Yes, I know you are a helicopter proponent, and your points about a "computer" (AKA flight controller) stabilizing a full scale heli are the obvious solution.  Yet, there is still something compelling about the challenge to make a large quad.  I don't think the arguments of simplicity, safety, or reliability can be easily determined unless you have a specific design to analyze.  I'm not saying that the heli doesn't beat the variable pitch quad in these area, but that they are highly dependent on the design.

@all - I'm just getting into the control algorithms for different flight modes, and am wondering if any of the control modes employ a model based control with trajectory planning which plans out a control response that is known to be within the limits of the physical model constraints.  A form of this is the Smith-Predictor, which runs a software model in parallel with the real world system, and uses the difference in response of the two systems to generate closed loop corrective actions.

Anyway, this whole exercise was just to shed some light on the subject, and get some idea about how control is affected by scale, and which parameters are most important.  It wasn't intended to give an absolute answer of the max weight of a quad, but rather to show how the uphill battle get very steep indeed as size increases.

I plan to do a similar exercise with a variable pitch quad model which I'm thinking will give a good result, so long as the collective servos are quick.  That's another days work....

• MR60

This study is interesting but does not reflect what is observed on reality because you forgot in your study that the energy (watt hours) does not remain constant when scaling up. This is why we do not observe a factor of 1024 in response reduction between a 250 and a one meter class quadcopter!

• Brilliant, this morning I was thinking about a project and I was exactly wondering how does the controllability scales, thanks a lot!

Any experience from real life? How many kilos can a quad weigh before becoming too slow to be reliably controlled?

• 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.

• 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.

• 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.
• 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.

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

• 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.