Replies to This Discussion

Permalink Reply by Dave_fr on August 25, 2009 at 12:19pm

Hi wayne.

Writing this post I realized that the formula is in actual fact RoT = G*tan(Bank)/Airspeed.

No vector here whatsoever. I just used a basic vector calculation to obtain the result, so basic that I managed to make a mistake :-) .
G is the earth gravity (9,81m/s²) it's a constant.
Essentially, in a turn the lift is rotated inside the turn by the value of the bank. By projecting the rotated lift on the horizontal axis you come to the conclusion that the plane sustains a force that pulls the plane toward the center of the turn that is equal to m*G*tan(Bank) ( m:mass ).
Now in the plane's referencial, this force is balanced in strengh by the centrifugal force in other words m*g*tan(Bank)= CentrifugalForce. Centrifugal force = m*Airspeed*RoT ( learnt formula )
We then have m*g*tan(Bank) = m*Airspeed*RoT => RoT = G*tan(Bank)/Airspeed

If you want I can draw you up a draft to explain the formula more in detail.

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DeveloperPermalink Reply by Ryan Beall on August 25, 2009 at 12:33pm

I guess I see more what you are talking about.. Your approach will most likely not help the problem you are trying to solve. You will have more bandwidth on your heading controller that is true. However the reason the aircraft skews way past the waypoint and takes forever to get back on is because that's the dynamics you are accecpting in the waypoint switching. Each aircraft has a min radius for a turn and you want to set up the waypoint switch early such that it leads the turns, turns, and lags the turn to get back on...ie: for a left turn you must turn right first then make a min radius left turn and then another right turn to get back on track. All three turns utilizing min radius turns...this is the most dynamicly sound way to make turns that don't skew way off track like is currently programmed in the nav code.

This is some code I wrote and tested in matlab. The circles represent the min radius I was talking about...they are tangent to eachother they just don't map well in non spherical 2d space. I hope this makes sense. the skewed line with an x is the aircraft and its heading

http://diydrones.com/photo/track_smoothing-1?context=user

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DeveloperPermalink Reply by Ryan Beall on August 25, 2009 at 12:50pm

Another assumption that would kill this implemenation is the over simplification of the ROT equation. This would work for very small AOA's but nothing outside of 15 deg probably even less than 10 deg. Here is why

ROT = g*sqrt(n^2-1)/Vinf

where g=gravity, n=load factor, Vinf=airspeed
I guess you could back out n or assume its 1 or...I don't know but in a 60 deg AOB turn its 2 for a steady level turn. You can see how quickly this will walk your equation away. The main thing is....Its a simplification and I think the truth data recieved from it would only be good for steady level at low bandwidth. So you would probably be in the same boat. I just don't think assuming n =1 in turns is a simplification you can overlook.

a better assumption would be to assume the turn is steady level and it would look likewise

n = 1/(cos(phi)); phi = roll angle
ROT = same as above

Hope this helps. Great idea and very easy to implement!!

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Permalink Reply by Dave_fr on August 25, 2009 at 1:01pm

Do you do full size flying Ryan ?
If yes have you ever flown a trajectory only using a slow gps as heading reference?
I'm just introducing a very reactive way to get a good real time estimate of the heading over ground. If that was useless we wouldn't have gyro stabilised direction indicator on most of full size aeroplane.
Believe me, when flying IFR it helps a little...
I just think that you're confused with where I want to go.
I was just talking about high bank turns as a positive consequence. Having a real time heading allows it. Having 1 heading a second only allows a very slow establishment to the new desired heading.

Best regards,

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ModeratorPermalink Reply by wayne garris on August 25, 2009 at 1:17pm

Okay thank for the clarification G not g( local ) . I see how this could be useful . but a few questions , are you assuming a level(no alt loss) coordinated turn?what would happen if the plane was banked w/o turning ?

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Permalink Reply by Dave_fr on August 25, 2009 at 2:14pm

Interesting question.

It should work as long as the ailerons are not fighting the rudder in other words, if there is a fma copilot always keeping the wings leveled and the rudder kicking the nose one way to the next waypoint, yes it would be a problem. The reason being that because in this precise case it is the fuselage and the engine thrust that turns the plane, no direct relation between the bank and RoT can be made. However a coefficient could possibly be applied but I don't think that it would work as such.
Now with aileron control I can't see any problem neither with the 2 axis (rudder only). Even if with the 2 axis (rudder only) planes, the turn is uncoordinated only momentaneously it is still the wing Lift that changes the direction of the plane the dissymetrical airflow pushes left and right as rudder is applied but averages out in the end.

Now, if there is altitude loss it means that Lift hasn't been increased as it should ( pull on the stick ). It is actually the assumption that I made when I said that the RoT=G*sin(Bank)/Airspeed. The difference betwen the formula is just a tan() instead of a sin() where the difference become only noticeable after 30° bank.
Anyway the lift is always somehow compensated for (increase in speed) otherwise it would mean that the plane keeps going faster and faster (down force not compensated for = constant downward acceleration).

Best regards,

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DeveloperPermalink Reply by Ryan Beall on August 25, 2009 at 3:58pm

Yeah, I fly for the Navy. As far as the heading controller at 1Hz not being adequate. I have built my own fully capable imu auto pilot from the ground up and have used both COG from GPS as well as the IMU estimated (smoothed) psi for heading control. Surpirsingly the COG from GPS will hold the heading just fine(moderate wind).
When i'm flying i can't read the RMI at 1Hz down to the 6th decimal .....just saying I'm not that good but my auto pilot is and its perfectly adequate from what I've seen in the feild.....It is the nav logic setting up the turns that helps a ton. However I don't know how the code is written for the track holding....You may just have to increase the cross track error gain... Speaking in the terminology that I use on my auto pilot?? Do you know the equivalent in the code?

I compared your equation to mine and they are the same. I stand corrected....They both assume steady level coor turns.

http://diydrones.com/photo/straightlinehold-1?context=user
This is a matlab plot I used to check my code. chaning the crosstrack gain causes the heading controller to converge quicker on the track....just a thought?

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Permalink Reply by Dave_fr on August 25, 2009 at 4:19pm

Well, it looks like we are talking about the same thing now.
As far as the code is concerned I'm just taking Chris and Jordi's. No cross track for the moment just permanent gotos (as far as I've seen). As far as the coordinated turn assumption is concerned I've just answered Wayne.

Best regards,

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DeveloperPermalink Reply by Ryan Beall on August 25, 2009 at 4:36pm

I am assuming they use some cross track just from some of the google earth plots I have seen. Doesn't seem like they are doing direct to navigation. I could be wrong....I'm intrigued...I'll see if I can find out
-Ryan

DeveloperPermalink Reply by Ryan Beall on August 25, 2009 at 4:43pm

Yeah just glanced at the Ardupilot code for the first time.. I'm impressed with how organized it is. Yeah there is no cross track vectoring. So yeah much improvement in the logic here would greatly improve your ground nav!! It is very simple if you have any questions Very simple!

I bet if you just simply added this (which is what I'm running) you would be completely satisfied with its heading performance!

Never the less I like your filter idea!

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