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I'm trying to simulate a gravity turn using a gimbaled thruster. I'm using the dynamical system described in a previous post of mine to execute a gravity turn by changing the angle $\phi_T$ between the thrust vector and the velocity vector by some number of degrees once the rocket reaches a certain height $h_{turn}$ and leaving the gimbal angle constant until burnout. After experimentation with a couple of rocket designs, I've found that I've needed quite large angles to make it happen (almost 45 degrees), and I don't think this is really feasible in reality.

So, I'm curious how the pitch-over program is actually executed in reality. For example, take the space shuttles. How large of a gimbal angle was needed to execute the gravity turn? Was it constant or did the angle change over time (like with an automatic control loop)? For how much time were the gimbals angled to produce the turn?

Is there a database showing the gimbal controls over time for different spacecraft to execute a gravity turn?

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Gimbaling the engines off the line intersecting the center of mass produces torque, which yields a rotation rate; if the vehicle is stable then a very small deflection should eventually bring you to the desired pitch angle.

The basic logic is an automatic control loop using something like a PID controller, usually with several additional constraints -- like not exceeding a certain angle of attack limit. The actual gimbal angle would be changing constantly throughout the flight, since it has to compensate for all the real-world things happening to the vehicle like gusts of wind, uneven thrust, propellant slosh, and so on.

Nice features of a well-tuned PID controller are that it produces a larger control signal (i.e. larger gimbal angle) when it's further off target (the P-for-proportional part), increases its effort if the proportional control isn't keeping up for one reason or another (the I-for-integral part), and avoids overcompensation and unwanted oscillation around the target (the D-for-derivative part).

I believe the shuttle's update loop executed every 40 milliseconds.

The Saturn V's guidance system adjusted the gain of the proportional and derivative portions of its controller at scheduled times during flight to adjust for the vehicle getting lighter and into less dense air. (I don't know if this implies that its control was PD rather than PID). This document on the Saturn V guidance system might be an informative read.

I'm not sure where your sim is going wrong; I think turning gimbaled main engines to 45 degrees and holding them there will flip the rocket pretty quickly. (Are you modeling the distance between the engines and the center of mass correctly and getting reasonable torque values therefrom?)

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  • $\begingroup$ As a matter of fact, i did not take into consideration the distance from the engines to the center of mass. I simply prescribed the angle once the vehicle reached a given height. I guess I'll have to go back and figure out how to account for that distance. Any advice on that front? $\endgroup$ – Paul Sep 22 '17 at 20:00
  • $\begingroup$ Ah, so you're getting the change in thrust direction, but not the torque? And assuming that the rocket turns to match its velocity vector? $\endgroup$ – Russell Borogove Sep 22 '17 at 20:02
  • $\begingroup$ For a rough approximation you can treat the rocket as one cylinder per stage, with the CoG of each stage at the center of the cylinder, with the overall vehicle CoG being the mass weighted sum of the stage CoGs -- taking into account fuel consumption for the currently running stage. For my own launch sim I explicitly did not model gimbal angle, instead assuming that the rocket magically points to the commanded direction. $\endgroup$ – Russell Borogove Sep 22 '17 at 20:05
  • $\begingroup$ To answer your questions, yes to both. I simply have a term $\frac{Tcos(\phi_T)}{m}$ and $\frac{Tsin(\phi_T)}{m}$ in the $\dot{v}$ and $\dot{\gamma}$ equations, respectively, where $\phi_T$ is a nonzero constant after reaching a target height. $\endgroup$ – Paul Sep 22 '17 at 23:51
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Here are some shuttle ascent guidance basics so you can interpret my answer.

During first stage, the shuttle flew a predesigned (therefore open-loop) pitch-yaw-roll profile based, among other things, on the measured winds of the day. (Note that only the attitude targets were open-loop. The shuttle used a closed-loop control system to fly to those targets.)

After solid rocket booster separation, a closed-loop guidance scheme called Powered Explicit Guidance was used, which steered the vehicle to its Main Engine Cutoff (MECO) targets. During this phase, the commanded attitude was continuously calculated.

So, given that, even if the shuttle was flying a constant attitude, the engines would be continually gimbaling to keep the thrust vector pointed through the center of mass as the propellant was used up.

Also note that during first stage, attitude control was primarily controlled via the SRB thrust vector control, due to the enormous SRB thrust and long moment arms from the nozzles to the c.g.

I have some notes that show the center SSME pitch gimbal started out at about 4.25 degrees at liftoff, ramped up slowly to slightly over 5 degrees at SRB sep, then moved sharply to a little over 1 degree at sep. During second stage, the gimbal moved up slowly and wound up around 4.25 degrees again at MECO.

Please note that these gimbal angles are measured from the null mounting position. As you can see in this side view, there was a built-in pitch angle in the engine installed position so that the null mounting position pointed roughly through the average c.g.

enter image description here

The installed null position for the left and right main engines is 10° up from the X axis and 3.5° outboard from an engine centerline parallel to the X axis. The center engine's installed null position is 16° above the X axis centerline for pitch and on the X axis centerline for yaw.

So the maximum angular deflection from the centerline of the External Tank that I know of would be about 5 degrees plus the 16 degree null mounting angle, for approximately 21 degrees. Given the rather strange shuttle side-mounted engine configuration, I'd have to say that your 45 degree gimbal angle is excessive, especially if you have a more symmetrical vehicle.

Sources

DOLILU overview presentation

Shuttle Crew Operations Manual 2-13.47

personal notes

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  • $\begingroup$ Off topic, but do you happen to know why the engines were canted outward like that? Avoiding plume interaction or something? $\endgroup$ – Russell Borogove Sep 23 '17 at 0:12
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    $\begingroup$ to keep each of the engines thrust passing through the CoM, so it wouldn't go into a cartwheel when an engine was shutdown (STS51) $\endgroup$ – JCRM Sep 23 '17 at 0:18
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    $\begingroup$ "so that the null mounting position pointed roughly through the average c.g." $\endgroup$ – Organic Marble Sep 23 '17 at 1:20
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    $\begingroup$ Derp, of course. Gotcha. $\endgroup$ – Russell Borogove Sep 23 '17 at 2:17

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