Rocket pitch and yaw control question : is is decoupled from roll control?

[DanyloMalyuta]

Hello,

I'm designing a controller for rocket Pitch and Yaw control (aim is for rocket to remain upright, so zero pitch and yaw angles). I'm having trouble developing the dynamic equations and can't find existing material corresponding to my needs elsewhere. I've got the following non-linear dynamic equations:

I'm trying to get a transfer function so I need to linearize these. This is where my problem comes in.

The actuator that I will implement will be unable to roll-stabilize the rocket so the rocket is free to spin, and I want pitch and yaw angles to remain 0==> ψ=0 and θ=0. The problem is then that if I linearize around an equilibrium position where roll angle φ=0 and roll rate P=0, then the linear set of equation that I get is:

These contain no information about roll! Can you confirm that effectively pitch and yaw control is decoupled roll control in rockets?

Best,

Danylo.

 

[imcjoek]

Disclaimer: I am strictly terrestrial. Don't know much bout exploding trash cans.

That said, taking a step back from differential equations...
The only way for the fins to induce a rolling moment would be if they are actuated in opposite directions. If they're adjusting for pitch/yaw, they are are not doing this. Minus tolerances.
Ditto gimbaled engines.

 

[DanyloMalyuta]

I agree. However, imagine this situation. The rocket has a certain spin when a set of fins adjusting pitch are actuated. At time t, the corrective force created by these fins points in one direction. But, since the rocket is rolling (let's assume its roll rate is much faster than the "speed of adjustment" in pitch), then at time t+Δt the corrective force finds itself pointing in an entirely different direction - if the roll is super fast, perhaps in a direction that is almost opposite to what it was at time t. So now you see my problem - the set of linear differential equations I need to create my transfer function need to have some info about the rocket's current roll angle and rate, otherwise the controller actuating the fins will have no idea about roll and hence, as in the situation described in the first few sentences of this post, the rocket will quickly go haywire.

In doing some reading, it seems that ever author is allowing him/herself to decouple pitch and yaw from roll, and from each other even, just as you described because they assume that the rocket is roll-stabilized so the situation of "super fast roll rate" will never arrive. In my situation, it is a model rocket and I have no interest in adding additional mass for an internal inertia wheel for roll control. I will work more on my problem and post back results. Meanwhile, if anyone has suggestions for attitude control of roll uncontrolled rockets reading (I've found none), please drop in a word.

Cheers,

Danylo.

 

[Compositepro]

Spin stabilized rockets do not have active pitch and yaw control. To have active pitch and yaw control you must measure pitch and yaw, and not just in the rocket's frame of reference but in an inertial frame of reference. How else would the rocket know where it is or where it is going? If you know the pitch and yaw in an inertial frame of reference then you also know the roll. It is the derivative.

It sounds like you are trying to control the rocket like a model airplane with visual feedback. Then you have to design the rocket like a model airplane, using large fins (wings) and trim tabs to minimize roll. It has to be stable enough to fly using only visual feedback. Generally, rockets are too fast for manual control to work.

 

[DanyloMalyuta]

Not at all. The model rocket is conventional (not resembling an RC airplane) and will have automatic control via an on-board micro-controller. My issue is with linearizing the non-linear set of equations when the rocket is free-spinning (free-rolling...). Since it is free to roll, there is no "equilibrium" state so I cannot very well say let's linearize around φ=0 and be done with it, since φ will NOT be zero and neither will the rolling rate Q, since neither has an "equilibrium".

 

[Compositepro]

How are you measuring pitch and yaw?

 

[DanyloMalyuta]

I'll have an accelerometer (BMA180 probably) and an IMU. I'm a mechanical engineering student; I have no experience in sensors and I have some experience in control. I'm doing the GNC algorithm for the rocket. If you are experienced, I would really appreciate help regarding how pitch and yaw stabilization is done professionally. I'd also like to know whether it is even possible to do a pitch/yaw but not roll-stabilized rocket, or does that create huge complications? Today I successfully wrote the in-plane 2D rocket flight simulation and I imagine that that would suffice for a roll-stabilized case (as pitch and yaw are decoupled and for my rocket the moments of inertia about pitch and yaw axes is the same).

 

[IRstuff]

I would have thought that gyros would have been more appropriate for this type of measurement. A 3-axis gyro package wouldn't be particularly expensive compared to 3-axis accels.

TTFN
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Of course I can. I can do anything. I can do absolutely anything. I'm an expert!

 

[DanyloMalyuta]

Yes, a gyro would be the right choice!

 

[GregLocock]

I think (but am not sure) the answer to your original question is hidden above, if the rocket is pitching and you apply a yaw correction then it will roll as well, due to precession. So you must have oversimplified the linear equations too much, or it may be that your pitch and yaw velocities are small enough that you can ignore it.

Cheers

Greg Locock


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[DanyloMalyuta]

@GregLocock I see exactly what you are trying to say. My problem is though that the linearized equations must be obtained by Taylor first-order expansion around an "equilibrium point". For pitch and yaw, this equilibrium point is 0 pitch angle (θ), 0 pitch rate (Q), 0 yaw angle (ψ), 0 yaw rate (R). But, since the rocket has no roll control, I cannot say that the rocket has an equilibrium at 0 roll angle, 0 roll rate. For all I know, the roll angle one second after lift-off may be 270° offset from the at-launch roll angle (which we may define as 0). Another second later, this roll angle may become 570°! You see, I can't say anything about roll angle or rate. The linearized equations in my original post are effectively linearized around 0 roll angle and 0 roll rate and, as you can see, have become decoupled ==> this is why roll angle is no longer present in them as, for small roll angles and roll rates (as would be assured by a roll controller), pitch and yaw may be considered independently from each other and from roll. But since my rocket has no roll controller, the linearized equations are simply not true and I'm looking for whether someone knows how to handle this kind of situation without recurring to non-linear control. I'm pretty sure it's possible without going into non-linear control because, as you can see in this video the Apollo-era Little Joe rocket handled the no-roll-control situation rather well (and I'm sure they didn't have non-linear controllers back in the day??).

 

[btrueblood]

There is a fair bit of information on the AIM 9 sidewinder missile that is in the public domain. The fins on these missiles incorporate a device called a "rolleron", which minimizes the roll rate in an automatic fashion. Google is your friend.

Most, if not all, later boosters incorporated roll control via either thrust vectoring from a pair of gimballed engines, or by attitude control thrusters, or by aero control surfaces, or some combination of the three.

 

[DanyloMalyuta]

@btrueblood Thanks for the info, will look into it. Roll control for boosters is crucial as I imagine that the structure of these cannot tolerate the high aerodynamic loads induced by a roll rate at supersonic velocities (you can see this in the Little Joe video in my last post, the rocket ends up disintegrating). But for model rockets, these spin all the time with no problem and I'm not interested in adding mass to control the roll as it would not cause structural failure. I'm looking for a way to control pitch and yaw without needing to also control roll just to be able to make the linearization of my first post.

 

[btrueblood]

1. I'm not sure the video you posted shows a rocket disintegrating from excessive rotation (either inertia or aero forces), but instead one which was given a ground abort command due to loss of control, i.e. the pitch/yaw gimbal control could no longer maintain the planned flight path due to the very type of instability you described earlier (gimbal actuation speed could no longer keep up with the rotation speed of the missile). This is even more likely given that the escape rockets with their fairly frail structure didn't come apart, but instead had enough time to accomplish their task and safely extract the capsule. Seems unlikely for that to happen if structural failure occurred, but likely to happen if the event was preprogrammed, including a delay after escape rocket firing before actuating the casing decompression on the main solid booster.

2. In a model rocket, roll instability is compounded by the higher likelihood of aero surface misalignment causing a roll rate proportional to forward speed, and due to a low roll-direction mass moment of inertia, both in relative terms to full-scale orbital boosters.

Given 1 and 2, it is even more likely that an attempt to control yaw and pitch via ailerons, on a typical model rocket, is doomed from the start given the assumption that theta-dot remains below a threshold that causes the instability.

Do look at the "AIM 9" and/or "rolleron" entry in wikipedia, and/or search google images for the term rolleron. They are a pretty low-mass, self-actuated solution to the problem.

 

[DanyloMalyuta]

@btrueblood Thanks a lot for the information and explanation. Could you please clarify what you said here:

an attempt to control yaw and pitch via ailerons, on a typical model rocket, is doomed from the start given the assumption that theta-dot remains below a threshold that causes the instability.

I don't quite understand what you mean after the "given the assumption that theta-dot..." as it seems to me like that very assumption is what would ensure that the control system works (given that we mean theta-dot is the roll rate).

 

[DanyloMalyuta]

Hello,

So I've re-written the dynamic equations and simulated them in MATLAB. I attached a PDF of the equations as well as the 2 Matlab files you need to run the simulation. They seem to work well, but disclaimer : this is work in progress!

Now comes the linearization step, which I will report on once I do it. In the PDF of the equations, the notation is:
x,y,z = coordinates of rocket CG with x the roll axis, y the pitch axis and z the yaw axis. If the rocket is drawn upright on paper then x is up, y is right and z is into the paper, with the origin at the c.m. (Center of Mass).
ψ,θ,φ = yaw, pitch, roll Euler angles executed in that order.
V = rocket velocity
U = air velocity
W = velocity of rocket with respect to the air, which in the body axes computes as vec(U_B)-vec(V_B)
Anything with subscript "B" ==> means "in the body axes frame of reference".

I plan to write a paper with the step by step development once I am done. I will publish the paper here.

 

[DanyloMalyuta]

Oops, seems I can only upload 1 file at a time, so here goes the second MATLAB file.

 

[DanyloMalyuta]

... And here is the PDF of the non-linear dynamic equations.

 

[btrueblood]

Danyl,

Read the whole sentence, which starts with "Given 1 and 2", 2 being a statement that (for two independent reasons) in a model rocket, theta-dot is extremely unlikely to remain a small value once the fire gets lit.

 

[DanyloMalyuta]

@btrueblood ==> yup, reared the sentences and understood your point. I'll try to make a controller that will attempt to keep theta-dot small and report back my results.