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Midland Helicopters . HeliProz . ZoomsHobbies

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Main Discussion > Linear Servo travel analysis
 
 
AugustorrAdvertiser - Location: San Diego, CA -
It's interesting to see all the talk about linear servos. But some time ago I did the required calculations and realized that linear servos are just a mute point with today’s radios.

First of all in some cases such as say the Freya's elevator, you have a circular servo arm movement controlling a circular "X arm" controlling the swashplate. In this case 10 degrees anywhere in the travel will be 10 degrees in the elevator swashplate rotation. You can't get more linear than that and that's why they say for FAI the mechanical mixing is the best.

Nevertheless with today's radios whenever you need linear from a circular to straight movement you can make it almost perfectly linear with the mixing options/expos in your radio. As a matter of fact just applying expo makes it so close to linear that's just a waste of time to try to make it better.

To demonstrate the point and for those of you who like the reasoning behind an opinion here's an example with the calculations.


Above is an example of a typical non-linear servo installation.
Let’s study the case of a servo where as usual you hopefully made the pushrod 90 degrees with the servo arm at the center of the travel.

What we are interested is to find a relationship between servo angular travel and effective linear travel. To do that lets name the following quantities:


Ed = Effective distance
Edi = Effective distance after servo moved i degrees
T = Total effective travel = Ed - Edi

In this example we have four known quantities:

Rs = Servo arm Radius = Known quantity
Lp = Length of pushrod = Known quantity
s = angle of movement at servo arm = Known quantity
b = 90 degrees in this example

Lets now calculate the rest:

Looking at the figure to the left:
From Pythagoras we know that:
Ed^2 = Lp^2 + Rs^2
so
Ed = sqrt( Lp^2 + Rs^2 )
By the way "^2" is used here to mean "squared" or “raised to the power of two”.

Therefore "Ed" is now a known quantity. (Known so far: RS, Lp, s, Ed, b )

b = 90 degrees
Being the "b" angle 90 degrees we can assume that the tangent of "c" is Rs/Lp
Tan(c) = Rs/Lp
then
c = arcTan ( Rs / Lp )
now "c" is also a know quantity (Known so far: RS, Lp, s, Ed, b, c )


From trigonometry we know that the sum of all angles of any triangle totals 180 degrees.(1)
a + b + c = 180
a = 180 - b - c = 180 - 90 - c = 90 - c
Since we know "c" then a = 90 - c
So "a" is now a know quantity.(Known so far: RS, Lp, s, Ed, b, c, a )

Going to the figure on the right:

s = servo arm travel angle
The new angle "ai" is the sum of the old angle "a" plus the servo arm angel "s"
ai = a + s
Since a is known and s is also know then a1 is now known too.(Known so far: RS, Lp, s, Ed, b, c, a, ai )

From trigonometry using the “Law of the Sines” we know that the ratio of the side divided by the opposing angle is the same for every side/opposing angle pair so;
Edi / Sin(bi) = Lp / Sin(ai) = Rs / Sin(ci)

Using the bi angle and the ai angle pairs and since we are looking for "Edi" we have;
Edi = ( Lp / Sin(ai) ) * Sin(bi)

Lp = Known
ai = Known
bi = ?

To get the bi angle we need the ci angle so lets find the ci angle;
Lp / Sin(ai) = Rs / Sin(ci)

ci = arcSin( (Rs * Sin(ai) ) / Lp)
Since Rs, ai and Lp ar know then ci is also known now.(Known so far: RS, Lp, s, Ed, b, c, a, ai, ci)

From (1) again;
ai + bi + ci = 180
bi = 180 - ai - ci
Since we know a1 and c1 now bi is also known. (Known so far: RS, Lp, s, Ed, b, c, a, ai, ci, bi)

Now that we have ai and bi determined and since Lp is also known we can now calculate Edi using the previous formula:
Edi = ( Lp / Sin(ai) ) * Sin(bi)
(Known: RS, Lp, s, Ed, b, c, a, ai, ci, bi, Edi)

Now to calculate the actual effective travel "T" we only need to subtract Ed minus Edi to get "T"
T = Ed – Edi

Following is a list of the results and a chart:


Above is a list of the calculated values using a servo radius of 1 inch and a pushrod length of 10 inches.


The blue one is the actual travel without any compensation. Notice that the actual linear travel per degree close to the center of travel is higher than at the ends.
The ideal travel (in green) is the one that gives a linear travel across the whole travel.
If we take the difference between the actual and the ideal and subtract if from the ideal we can see what a correcting curve should look like in Red. Guess what it looks like? Yes, it looks like an expo travel compensation so there you have your linear traveling servo. Actually you always had it in your radio. If you want to get picky you can use the formulas above and using your actual measured lengths calculate the exact expo required for your application.

Augusto

Fury, Logo 20 & 10, Vigor, Raptor, Sceadu.
02-20-2002 01:38 AM
 
 
capebob
Veteran
Location: Cape Coral, FL

Augusto,

WOW!

I suspect that you're absolutely right. However, according to the latest issue of Rotory Magazine in the case of CCPM one would still need a linear servo to prevent cyclic interaction at extreme collective positions. Or they could be wrong.

Those of us old enough to remember the Heathkit radios of the early 70s probably remember the servos that came with the old "blue box". You had to build it yourself, it was very large by today's standard, it wasn't very strong, it was very slow, and centering changed as voltage dropped. That was the down side. The up side was that it did have linear outputs. Also, it didn't have a variable feedback resistor, it had a variable capaciter feedback which was one of the reasons it was so large.

Bob
02-20-2002 02:05 AM
HOMEPAGE  
 
 
rickc
Elite Veteran
Location: Creve Coeur IL. (Peoria Area)

>>in the case of CCPM one would still need a linear servo to prevent cyclic interaction at extreme collective positions.<<

Isn't this the reasoning behind the 140 degree CCPM, to get away from the cyclic interactions at extreme collective positions?

Rick
02-20-2002 02:18 AM
HOMEPAGE  
 
 
Alex Rodriguez
Senior Heliman
Location: Ohio

This kids is why you should never tell your math teachers that "I will never need to know this in the future"
02-20-2002 02:19 AM
 
 
Angelos
Key Veteran
Location: nr Oxford, OX11, UK

Calculations are totaly correct, but I think the expo is applied to the input from the sticks, not the outputs of the radio.

Thus if the output is a result of mixing like CCPM, it will probably not result to linear movements. Am I right?

However, this thread gave me an idea... what about making a miniature pass-through on-board box that will correct the expo! I could probably make one using a single microcontroller (for all 3 servos) and I don't expect it to weight more that 10-20gramms and be larger than a servo.
02-20-2002 02:24 AM
 
 
capebob
Veteran
Location: Cape Coral, FL

Rick,

I'm not sure. I know that 140 CCPM reduces one cause of interaction but don't know if it's the interaction caused by rotary servos.

Bob
02-20-2002 02:28 AM
HOMEPAGE  
 
 
gyroman
Senior Heliman
Location: Pasadena, CA

And I was proud because I figured out the CCPM Swashplate. I feel so stupid.
02-20-2002 03:09 AM
HOMEPAGE  
 
 
SemiArticulate
Veteran
Location: On Location

Those are the slider crank equations and that is the output for a slider crank.

Your vector loop should be:
R2-R3-R4-R1=0

In your case, R2 = Rs (incorrect), R3=Lp, R4=0, R1=Ed (incorrect)

But Rs is the position vector with the varying magnitude and/or angle and is not R2. I think you are using Ed and Edi for Rs. R1 would equal Rs in your example. There is no offset so R4 = 0. Thats the terminology anyway.

But that is beside the point. You are trying to linearize the servo. That is ok but that is not what the system is doing. Your angular output on the servo is used as angular input throughout the heli. The swash does not act as a slider crank on the cyclic. If it did, the output is still angular at the systems output. The only argument of a slider crank would be on the collective which is offset by fact that most systems pivot this on a fixed point also. Ie. the raptor and freya. Trying to linearize the servo via a slider crank would do more to make the output nonlinear than leaving it as is. Maybe I misunderstand and that is what you are demonstrating. But that example is not on our helis. Like the Raptor and Freya, every angular motion leads to another angular motion. The only effect of expo in those systems is to dull the motion about center stick. You need to look at the input (servos) and the output (pitch changes). Thats where you see if the system is linear or not. All the junk that happens in between is irrelevant. That "junk" is the system.
02-20-2002 03:27 AM
 
 
Maxx
Key Veteran
Location: Shreveport Louisana

I believe the point of the article was that you can have a truly linear 120 degree system using 3 linear digital servos and use no wheels, bellcranks or going into brain strain! While Agusto is correct as far as the math goes the point is this: Yes; while you COULD make the servo wheel/bellcrank movement approximate the output of a true linear servo, when you start talking about setting three servos EXACTLY the same...not to mention the need for the bellcranks to be set to the 10th degree cubed to match the servo rotation(rotarional distance). Any divergence between the servo output and the bellcrank rotation would require different mixing at different points in the rotation to compensate. Now factor in the swashplate bias to counteract the CW rotation of the Main Rotor and the equation gets polluted...getting one servo to achive linear motion from rotory output takes time, effort, and Coors Light...getting three rotory servos to work like three REAL linear servos would take an act of God...If a servo was available to drive a swashplate straight up and down (with the slight rotational divergance of swashplate tilt), it would keep the headaches to a bare minimum...but until we see a True Linear output servo on the market we can make do with what we have... Chris
02-20-2002 03:33 AM
 
 
SemiArticulate
Veteran
Location: On Location

"equation gets polluted"
Never

You missed my point. YOU DO NOT WANT LINEAR SERVOS. They will give you the headaches you think they would fix. Stop looking at the swash. Look at the input and the output. Forget the swash. The swash is twisting your brain. The swash is nothing.
02-20-2002 03:39 AM
 
 
Maxx
Key Veteran
Location: Shreveport Louisana

So fly your heli without the swash and report back...
the swash is the end device...the litmus test...it don't matter HOW you got linear motion...the swash will know. That's what the exercise is about...output is cool...drivetrain is neato...output is the question in need of an answer... Maybe Burt Rutan said it best (you remember him; Voyager flew around the World un-refueled in the mid-'80's when NASA said the technology didn't exist at the time to accomplish the feat) : sometimes engineers make problems for themselves by not looking at the problem form the easiest, cheapest,most logical way instead of from the "solve the equation" mindset. No offense to inganeeres...we'll never get linear servos without them . I wish I had taken some inganeering courses at La. Tech...them alphabet soup thingeys done confused me... Chris (don't confuse me with the facts)
02-20-2002 04:10 AM
 
 
SemiArticulate
Veteran
Location: On Location

Thats about right, the topic involves some small amount of engineering. You get an engineer to answer (how i don't know). The answer does not agree with yours. Therefore it must be some kind of bizaar thinking of the engineer that must be the problem. I don't know why people bring these things up if they are not looking for the answers. I am telling you the easiest method. Its the same @!#$ method that your heli is using now. I'm sorry that does not justify spending more money on worthless crap. How about this: Purple anodized aluminum will make your heli more linear. There now you can go spend some money. Its about the same thing.
02-20-2002 04:32 AM
 
 
Maxx
Key Veteran
Location: Shreveport Louisana

SemiArtirculate: No offfence dude, the point I was trying to make is ..you; the engineer, may be able to Prove Engineering Principle, in this case we make an servo do a linear motion dance from a rotary motion input! But you have John Q Heli flying these things...will he (or she) understand or have the equipment to match your settings in the shop/field enviroment? The Linear servo question is not (IMO) can it be done...it's can it be used by the afore mentioned John Q. Pilot. BTW; there is no small amount of Engineering...,Everything is Start to Finish. Don't knock the PurpleS*#t either..some of it ain't bad (from the Engineeering Point O' Veiw!) Chris -Shade Tree Engineer
02-20-2002 05:06 AM
 
 
SemiArticulate
Veteran
Location: On Location

I and I think ColeopTer are trying to tell John Q public that life will be easier for them if they forget about "linear" servos. I am trying to say in several of my posts that the use of a "linear" servo will do more to throw things out of whack. I was not debating Coleopter in my first post, I was trying to clarify what his point was. You'll need to tell me if that statement is simple enough for John Q Public to understand. If they add this pretend linear servo, they will make getting linear output much harder. They will make setup much harder. I don't know how else to say it.
02-20-2002 05:25 AM
 
 
Rotomotion
Heliman
Location: Charleston, SC

good article

Thanks for the article!! Very informative. That makes sense and was explained so that a dummy (me) could understand it (Expo/Linearity for Dummies).

Seems like the swash is independent of servo type (linear or expo adjusted normal) because everything in the head has a circular movement, doesn't that cancel out any differential in the movement transfers? I guess the circular movement of the components might have a different radius in some cases....

...but that's all really outside the point of our article anyways!!
02-20-2002 08:22 AM
HOMEPAGE  
 
 
Doug
Elite Veteran
Location: Naples Florida....

Where else could we get such an intelligent discussion without any crap, I'm proud to be a member
Thanks Mark
02-20-2002 01:10 PM
 
 
ed vega
Key Veteran
Location: nyc, queens

coleopter , nice work, can you factor in the

"1 1/2 spline differential method "? ..

remember reading articles on how to increase or extend the throw with curve in a attempt to acheive a linear throw output ..

anyone remember this mechanical form of exponential ?
02-20-2002 01:25 PM
 
 
Augusto
rrAdvertiser
Location: San Diego, CA

The funny part of all this is that at the end of the day no matter how much we linearize the responses or fix all the interactions the whole thing is driven by a thumb or finger-thumb that's either good or bad and that'll ultimately determine how that helicopter will fly.

But going back to the original intent of the post, the important point is that there is an input which is the stick's position and an output which is the swashplate movement and a transfer function between them that's the result of both the electronics and the mechanics working together.

What you're looking for when you want a linear response is that given a certain portion of stick movement lets say 25% of the total travel your swashplate moves also 25% of the total travel. The method used is not important whether it's expo or spline or some mechanical contraption, the result has to be a one-to-one relation between the stick movement and the swashplate movement.

When you're using a CCPM swashplate there is a certain software mix in the radio that will translate the stick movements to a combination of the three servos movements that gives as a result either elevator or aileron control. This software algorithm uses the stick position as read by an analog to digital converter in number form and processes it mathematically to give a set of three servo positions that mimic the stick movement. Lets call this the CCPM algorithm.

The EXPO algorithm is one that takes the digitized stick position and passes it through an exponential transfer function that will modify it to make it fit the expo function as required.

The CCPM function receives the digitized position not directly from the converter but already pre-processed by the EXPO function so the number it receives has already been modified and adjusted by the expo to fit a "linearizing mix". That's why using EXPO in CCPM linearizes the response as much as in the mechanical mixed case.

Regarding dissimilar servo responses. If you measure the three servos and select the one with the lowest travel as the reference, you can make the remaining two servos have an identical response by working on that particular servo ATV travel. You just diminish the travel % of the particular servo until the mechanical travel matches the one with the lowest travel. This ATV travel override is independent of both the EXPO and the CCPM algorithm. Once all the ATV’s are adjusted you will have three servos that will travel exactly the same for a given stick throw.

If you do all these things you can rest assured your response will be practically perfect.

Another issue that's confused very often is the one about total servo travel limiting the CCPM algorithm. If you select a "PITCH" percentage too high what will happen is that when at full corner stick if you raise and lower the collective you'll notice that the servos run out of travel and your some servos will no longer be able to keep moving and some will still move making the swashplate change its angle at the top. This is not "Servo interaction" this is just bad programming. You need to select the maximum pitch percentage that allows you to keep the servos from running out of travel when applying full pitch but not more.

Another common mistake confused with "Servo interaction" is the one that happens when moving the stick up and down very fast. In this case you will see the center swashplate ball go up and down momentarily and you might think you have an interaction problem. What you're seeing is the result of the two front servos advancing a smaller total travel faster than the back servo who needs to travel much more (and therefore longer) to catch-up with the front ones. The correct way to check this is to move the stick slowly and observing the swashplate's center ball for any vertical movement. If it doesn't move then you don't have interaction. This is the reason why it's a good idea to use fast digital servos.

Ed,

You're right; the spline method is an interpolation method.
In other words, out of several points you measure you can calculate the remaining ones using a numerical method. It could be used for this.

In the spline method, the slope of the curve at the measuring point is used for both the left and the right side of the curve that you're going to create and the curves themselves are piecewise cubic curves that's why the complete name is cubic spline interpolation.

The reason why it's called differential is because the slope you're looking for is actually the second derivative. So the second differential is used to make each section of the curve smoothly transition into the next.

The result is that you can calculate pretty accurately the required curve by moving the stick into several positions and measuring the servo output. Using the measurements you do a cubic spline interpolation and you get all the percentages for lets say a 13 point mixing curve that will be pretty much ideal. That would require you to replace the expo by a custom mix. I would use this method in a case where an expo curve wouldn't result in a linear enough response due to the geometry of the mechanics.

Avant Aurora Ultimate
02-21-2002 12:00 AM
 
 
xcelllogo
Senior Heliman
Location: San Diego

OK Augusto you need to go work for N.A.S.A and then maybe our space program would be going to light speed by now!!! That is cool I never looked at it like that.. But you told me that you don't run expo? Does this mean you are going to start?


Adam
02-21-2002 12:37 AM
HOMEPAGE  
 
 
SemiArticulate
Veteran
Location: On Location

heli72, yes thats pretty much it.

But after reading the thread again. Coleopter and I are not talking about the same thing. I thought he was trying to show how linearizing did not work as input. Because providing a linear input does not cover up nonlinearity in this system. It adds to it.

You need to look at the input (the rotation of the servo) vs output (cyclic and collective pitch) to determine the linearity of the system. Cutting the system in half and only looking at the swash will only give you half the story. If that. Not to mention the fact that the swash rotates about the ball for cyclic and thus IS NOT A SLIDER CRANK. The equations don't apply. You could argue that on ECCPM the collective is acting on a slider crank but then the servo arm is translated back to rotational movement after the swash. So again, you can't just look at the swash. So if I were to follow Coleopter's logic I would need to conclude that ECCPM is inferior to standard mix. Which I don't believe although I think ECCPM is over hyped and not any better than MCCPM. But if you want to have a mathmatical model then you need an accurate model. One that actually synthesizes the linkages.
02-21-2002 12:58 AM
 
 
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