### 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

• Rusty,

A while back you posted a link regarding reduced yaw control authority on large multi-copter.

I am still working on my large quad-rotor, and my current plan is to run rotors at fixed speed, and vary their pitch.

Based on the paper you linked me to, I am also incorporating an 8 degree tilt on each of the rotor arms (around the arm axis) to generate yaw moments.  I chose 8 degrees based on an arbitrary 1% loss of lift

arc-cos(0.99)=8 degrees.  This gives me 14% of my thrust as yaw (sin(8)=0.14.  But I'm still relying on the difference in thrust between opposite rotating rotors to generate the yaw torque.  This approach is different than your approach which used actuators to tilt the rotors.

Do you have any info to help me quantify what magnitude yaw torque I will need to keep the aircraft stable?  Where do yawing disturbances come from in open-air flight?  With a quad, I'm not envisioning a need to have super response to yaw control inputs, but do want to keep the yaw axis stable with regard to random disturbances from air currents, etc.

Here is a sample calc:

My vehicle is 340 Lbs gross with payload and fuel.  Each rotor thrust is 85/.99=85.85 Lb.

Yaw force is 0.14*85.85=12 Lbs. (per rotor).

So if I increase the CW rotors thrust by 1% I get 12.12 Lbs yaw force.

Decrease CCW rotors by 1%: 11.88 Lbs.

Difference = 0.24 Lbs x 2 pairs of rotors = 0.48 Lbs acting at a 4.5 ft lever arm (length of support arm) ~

=2.15 Ft*Lb yaw torque.  (~2.92 N*m)

I've estimated vehicle rotational inertia as 50 kg*m^2, so angular accel would be A=T/I = 2.92/50

=0.058 rad/sec^2  = 3.33 degrees / sec^2.

At this accel rate, it would take about 7.3 seconds to change yaw orientation by 45 degrees (d=(1/2)at^2).

(3.7 sec accel, 3.7 sec decel)

That seems a bit sluggish to me.  Again, not anticipating a need for super fast yaw response, but not sure if this is enough (yaw authority).

Another thing I'm considering, is how, in order to yaw, I need to split the lift between the CW and CCW rotors.

I have a nominal total lift (power) factor based on max lift ~ 125% max gross weight.  I know this is not as much headroom as toy quads, but I'm not looking for super agility.  My thoughts are more about how that total lift gets distributed between the 4 rotors, and what is the expected magnitude of the dynamic lift component for each rotor.  I can imagine a scenario where the sum of the 4 control components (altitude, pitch, roll, yaw) might give one of the rotors a pitch request signal which is out of its design performance envelope (say nearing stall).

I'm thinking I either need a limit for each rotor...(but then this will affect stability as the complementary rotor pairs would no longer be balanced)....or I need a "load shedding" function which will override signals from the various components on a prioritize basis - first yaw, then altitude, then pitch and roll. (the goal to always remain upright).

I'm interested in any thoughts or experience on this subject area.

• Developer

@Randy, the weights and rotors are:

quadplane: has been flown as both a octocopter and as a quadcopter. 13kg TOW. Uses 18x5.5 2-blade props

Logs available if you want them for estimation of control rates.

btw, in both cases the eCalc forumulas are pretty good, and in general the community finds eCalc to work within about 20%. So a good starting point for testing your maths may be to test it against eCalc.

• @Hector

Thanks for your complete response.  I am thrilled that I've gotten such informed response from so many folks.

I acknowledge that my model was simplistic and only addressed some very basic relationships, and yes as John said "there is a lot going on here".  I tried to acknowledge I was making some knowingly invalid assumptions in the analysis,

• @Andrew - I'm curious, do you have the weights and rotor diameters of your two quads?  It would be a good start for me to collect some empirical data. My theoretical formulas show that power ~ wt^1.5 power and power ~ 1/sqrt(rotor diam).

That means that power required for hover should change with P^4 for a quad where every component is scaled up proportionally. I'm curious where the real number for your two quads is.

• @Randy

Please, accept my apologizes. I did not mean to be rude or non-polite. I would like to point out why your result could not be considered correct or conclusive (do not take it personal please, some of them are just mathematical facts).

- First a minor issue. It is convenient (even for a informal tech note) to add references if other people have done employed the same model/assumptions as you. Some of your claims when you are constructing your model are not straightforward. I am very suspicious about your claim of a "magic factor" to the power of five.

- Another minor issue. Number or tag your equations, otherwise it is difficult for other people to communicate/discuss with you about your note (like me now :P).

- I am very suspicious about that you can assume that the roll angle can be decoupled from the rest of the system so easily. Under what requirements is this true? what are your assumptions?

- You assume all the factors there are constant and you only have the roll angle as the only variable. Furthermore you assume that you have a linear system. If you check the literature, this is a very strong assumption by you that in reality is not true.

- And the most important point. It is NOT a surprise (it is a very trivial result in fact) that a sinusoidal input in a second order system has as an output another sinusoidal with an phase offset of 3pi/2 (basic calculus), so what? This is not a red flag for anything. If in your model I add an extra order (for example, as another user said, by introducing the electronics in the equation) then I will have a 0 offset, so will I have a "green flag" again? The stability of a first/second/third.... order (linear) system DOES NOT DEPEND on the order itself! In fact, this 3pi/2 in your equation only plays a role for the initial conditions of your solution! It can be removed by just choosing appropriated initial conditions.

- Since you are dealing with a linear system, the result about w_f^3 is either a surprise. It is a basic result from the harmonic oscillator (second order linear system). What would you expect?

- The most important point when you have a "big scale" rotorcraft are the aerodynamics (which are completely missing in your note), the pitch/roll rate coefficients (which depends among other things on the angle of attack, which also depends on other variables of the system, so everything is highly coupled) are the most important factors in order to determine the stability of such a system. This coefficients depends mostly on the shape of your rotorcraft.

Please do not take this as I wanted to "mine" your work. I believe it is very nice that people want to spend their time and effort in such topics. However, it is also important to note when one is pushing not in the right direction. I hope my comments will be useful for your next iteration of your work.

p.d. Exponential means exp(x), and not x^a, The latter means polynomial.

• T3

What would be a more interesting study is to take a single quadrotor with a variety of battery sizes and test duration of flight (start with hover then maybe a mission) with various payloads to find the sweet spot for highest flight duration given a weight. Then do that for a few other frames and motor/ESC combos and see how those compare.

I always get people coming to me and asking me about putting this and that on a quad at work and while I try to sway them from a quad and to use a helicopter, or heaven forbid a balloon, that would be a better suited for their application, they always go back to wanting a quad because it's a more sexy solution. No convincing will sway those types so it would at least be useful to have a study that tries to show at what point does adding more battery capacity to extend flight become a detriment to flight time. Maneuverability doesn't seem to matter so much to most people I've talked with as they're mainly looking at carrying a sensor but that could be another aspect one studies afterwards. Just a thought.

• @Hector

The -3pi/2 indicates that the pitch angle response to a sine wave input from the pilot will lag behind by 3pi/2 phase angle (in radians) or by 270 degrees.  This is a big clue that the system will have some serious controlability challenges with it being a 3rd order system.  If my model had been more complex, that phase angle would be frequency dependant because the rotor speed system actually resembles a first order low pass filter instead of a pure integrator.

And, I do not have any idea about how you derived such a formula   I thought I included a fairly comprehensive road map for how I derived the formula.  If you don't understand it I would certainly entertain a request to clarify, but to declare it "completely useless" from a position of lack of understanding might be considered "less than polite".

@John, You are right there is A LOT going on.  My model was extremely simple.  As I keep learning I'm continually fascinated by the complexity of the myriad interactions occurring in mult-rotors, and especially in traditional heli's.  There are certainly no shortage of questions to be answered.  And because of that, I'm working to build a prototype of a rotor system to verify some of my computations.  I'm sure I'll learn a lot from that.

Thanks all.

• Developer

There is A LOT going on when you move a copter around. Body mass, momentum, drag, wind resistance. Same for propellers mass, momentum/inertia, efficiency, drag, lift etc. that will also vary depending on rpm. And on top of this you have ESC and motor efficiency, sync and reaction times etc.. And none of these scale in a linear fashion when you start to increase the size.

This is part of why you usually end up with a more practical approach using 'known variations' for certain popular sizes, based on experiments and experience. This is also where diy communities shine, since you get distributed 'intelligent/selective' brute force attacks on the problem.

• I do not have any idea about how you derived such a formula and whether is correct or not. But for sure this 3pi/2 inside the sin function is completly useless xD.
• Developer

@Randy, I don't think the power system is enough to explain it. My big quad hovers at 120A of 6S, which is 2.9kW. The 250 quad hovers at 10A on 4S, which is 160W, which is a ratio of 18 to 1. That leaves a factor of 56x to explain to get it to the 1024 times for 4:1 scale from your calculations. I don't think the roll rate is 56x higher in the 250 as compared to the big quad.

I think there is something else going on.