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Space Forum / Space Flight / November 2003Optimum constant-thrust transfers?
There has been a lot of work done on Hohman and Bielliptic transfers, but has there been anything on constant-thrust transfers? Of course, the "optimum" in this case won't be lowest deltaV, but shortest time for a given acceleration? Does anyone have any pointers to online papers? -bertil-  Signature"It can be shown that for any nutty theory, beyond-the-fringe political view or strange religion there exists a proponent on the Net. The proof is left as an exercise for your kill-file." > There has been a lot of work done on Hohman and Bielliptic transfers, >but has there been anything on constant-thrust transfers? There was considerable theoretical work on it in the late 50s and early 60s, when people thought that low-thrust propulsion would be in use soon. You need to look at some pretty old and dusty sources to find it, though. As far as I know, there's no single source that includes all of the significant results. Analytical approaches only go so far with low-thrust stuff, however, because it's just plain difficult to deal with mathematically. For real problems, you end up burning lots of computer time. Good methods for *that* are still experimental, an active research topic today. >Of course, the >"optimum" in this case won't be lowest deltaV, but shortest time for a >given acceleration? Could be either, depending on the conditions of the problem. They are sometimes, but not always, synonymous. SignatureMOST launched 30 June; first light, 29 July; 5arcsec | Henry Spencer pointing, 10 Sept; first science, early Oct; all well. | henry@spsystems.net > There has been a lot of work done on Hohman and Bielliptic transfers, > but has there been anything on constant-thrust transfers? Of course, the [quoted text clipped - 4 lines] > > -bertil- Try googling "Pontryagin constant thrust orbit transfer" and see what you can find? Dave > There has been a lot of work done on Hohman and Bielliptic transfers, > but has there been anything on constant-thrust transfers? Of course, the [quoted text clipped - 4 lines] > > -bertil- Quite a lot of work has been done on this. Look up solar sails or ion engins, you'll be sure to find a few.. > There has been a lot of work done on Hohman and Bielliptic transfers, >but has there been anything on constant-thrust transfersp? Of course, the >"optimum" in this case won't be lowest deltaV, but shortest time for a >given acceleration? If you are talking about a transfer between two coplanar, circular orbits, the optimum trajectory as acceleration asymptotes to zero is a constant spiral under circumferential thrust, the delta-V requirement turns out to be the difference between the circular orbit velocities of the initial and destination orbits, and the time is whatever time is required to deliver that delta-V with your (very small) acceleration. For non-coplanar but still circular orbits, something called Edelbaum's Approximation applies, and the delta-V becomes DV = ( Vo^2 + Vf^2 - 2 Vo Vf cos (pi/2 Di) )^0.5 Vo = circular velocity of initial orbit Vf = circular velocity of final orbit Di = inclination change pi = 3.14159... These approximations are good enough for all but the most detailed level of mission planning, so long as your maximum acceleration is less than one hundredth or so the local acceleration due to gravity. If either the initial or the destination orbit is noncircular, or if third-body effects are involved, the situation gets rather complex and is not amenable to analytic solution. > Does anyone have any pointers to online papers? None that I know of. Most of the original work was done in the '60s, and so mostly isn't available online. Most of what work has been done recently, is either in-depth analysis of convoluted special cases that assumes you've read all the papers from the '60s, or is the internal, very proprietary work of comsat operators who have started doing low-thrust partial orbit transfers using the ion or plasma thrusters their birds now carry for stationkeeping purposes. Signature*John Schilling * "Anything worth doing, * *Member:AIAA,NRA,ACLU,SAS,LP * is worth doing for money" * *Chief Scientist & General Partner * -13th Rule of Acquisition * *White Elephant Research, LLC * "There is no substitute * *schillin@spock.usc.edu * for success" * *661-951-9107 or 661-275-6795 * -58th Rule of Acquisition * >If either the initial or the destination orbit is noncircular, or if >third-body effects are involved, the situation gets rather complex >and is not amenable to analytic solution. There are *some* analytical results for non-circular orbits, but they don't add up to a useful complete picture. For example, there is an analytical result for the low-thrust delta-V for escape from an elliptical orbit using tangential thrust (along the velocity vector), and another for the optimal thrust direction for escape from an elliptical orbit, but they don't match up -- the optimal thrust direction is *not* tangential. (This is actually true even for circular orbits, but there the differences in both optimal direction and resulting delta-V are almost negligible. For elliptical orbits, the differences are too large to ignore.) SignatureMOST launched 30 June; first light, 29 July; 5arcsec | Henry Spencer pointing, 10 Sept; first science, early Oct; all well. | henry@spsystems.net Henry> the optimal thrust direction is *not* tangential. Ah hah! Can you suggest, or give a reference that would suggest, a better parameterized model for pointing during launch? So far I've thought of: - point along velocity vector (what I do now). - point into the wind (air is rotating around earth center, so not the same as pointing along velocity). - point parallel to earth surface (orthogonal to altitude vector) I suspect there are better pointing profiles, but it's tough to experiment without a model to constrain the optimization space. At the same time, I'm at a loss to explain why pointing not along the velocity vector would be anything but less efficient, since E = F*d Any force orthogonal to motion would appear to add no energy to the vehicle. Is there some sort of rotating-frame thing going on here? >Henry> the optimal thrust direction is *not* tangential. > >Can you suggest, or give a reference that would suggest, a better >parameterized model for pointing during launch? *Launch* is a somewhat different story than low-thrust orbit maneuvering. Generally, while within the atmosphere, it is obligatory to point into the wind (maintaining angle of attack at 0) to avoid excessive aerodynamic loads on the vehicle. Usually this is done with a precalculated pitch program, but sometimes refinements like active sensing and pointing are added to reduce wind-gust loads. (The Saturn V was precalculated, the shuttle does some active pointing.) After max Q has passed, sometimes it can be worth incurring a bit of loading by cranking in a little bit of pitch-up, so as to get some body lift. After exiting the atmosphere, it is common to use closed-loop optimizing guidance algorithms which don't lend themselves to simple description. (The closed-loop guidance on the Saturn V did amazing things after the Apollo 6 double engine failure. It did reach orbit, but the guidance data was a sight to behold.) That said, a first approximation is to drive the pitch angle (above the local horizontal) theta to satisfy tan(theta) = A + B*t where t is time, A is an initial pitch (usually somewhat above the flight path) and B is usually negative (so pitch declines with time and is zero or slightly negative at insertion). Finally, just before insertion it is usual to freeze the pitch angle and limit optimizing guidance to controlling cutoff time. Trying to actively chase the last little errors in position and velocity can lead to wild gyrations as the error magnitudes shrink rapidly and the error directions become almost random. >...I'm at a loss to explain why pointing not along the >velocity vector would be anything but less efficient, since > E = F*d >Any force orthogonal to motion would appear to add no energy to the >vehicle... True, but there are two other issues. One is that although it does not add energy, the thrust component perpendicular to the velocity rotates the velocity vector, which can be desirable if you are in the vicinity of some large hard object (e.g. the Earth) that you don't want to smack into while maneuvering. This is an important issue for launch. The other is that your goal is not to optimize the instantaneous rate of energy addition, which is F dot v , but to optimize the total energy added, which is integral(F dot v dt) . And because the problem is nonlinear, these two strategies are *not* equivalent for low-thrust orbit maneuvering: energy added yesterday changes the orbit and thus changes v today, so it can be better to accept a lower rate of energy addition yesterday if it will give better conditions today. SignatureMOST launched 30 June; first light, 29 July; 5arcsec | Henry Spencer pointing, 10 Sept; first science, early Oct; all well. | henry@spsystems.net > The other is that your goal is not to optimize the instantaneous rate of > energy addition, which is F dot v , but to optimize the total energy [quoted text clipped - 3 lines] > v today, so it can be better to accept a lower rate of energy addition > yesterday if it will give better conditions today. I might add that the topic of optimizing low-thrust orbitla transfers is one of the hottest topics (well, the only hot topic) in the orbit planning business these days. The solutions and strategies are very closely guarded industrial secrets. Brett > The other is that your goal is not to optimize the instantaneous rate of > energy addition, which is F dot v , but to optimize the total energy [quoted text clipped - 3 lines] > v today, so it can be better to accept a lower rate of energy addition > yesterday if it will give better conditions today. ..in addition, while both analytical and numerical methods are known to solve differential equations (the first, local one), solving integro- differential equations (the second one) is a black art, and likely to remain so. Jan > That said, a first approximation is to drive the pitch angle (above the > local horizontal) theta to satisfy tan(theta) = A + B*t where t is time, > A is an initial pitch (usually somewhat above the flight path) and B is > usually negative (so pitch declines with time and is zero or slightly > negative at insertion). Thank you. I'll give this a shot and see what I get. One thing that has struck me is that my stage-two burn times (this is a TSTO) are nearly always around 400 odd seconds, over very wide variations in delta-V between first and second stage. I think the SSME burn is about this long too. There may be a reason for this consistency: It takes 271 seconds to fall straight down to dirt from a dead stop at 360 km. It takes 383 seconds to fall straight down from 360 km in half gravity. If you launch from a dead stop and end up in orbit, the (gravitational-centripetal) acceleration will decrease from 9.8 m/s/s to 0. A few km/s of head start from a first stage doesn't change this much because v^2/r is still pretty small. And although this acceleration to ground doesn't time average to 9.8/2 m/s/s, it's going to be somewhat near that value because that's the median. The first stage head start will reduce the time average acceleration a bit, and increasing acceleration as propellant burns off will decrease it. And finally, you can't burn much of your second-stage delta-V resisting gravity because you need most of it to get orbital velocity. But any thrust pointed radially away from earth does reduce the time average acceleration a bit. Finally, the number of seconds of second-stage burn is related to the square root of the time average acceleration, so over time averages of 4.0 to 4.5 m/s^2, the burn time varies from 424 to 400 seconds. Anyway, the reason this is at all interesting is that if the burn time is fixed at around 410 seconds, then for a fixed engine size you have a fixed tank size and the only way to increase the stage's mass ratio is to decrease the size of the payload. Extra fuel doesn't do much because there is no time to burn it. > One is that although it does not add energy, the thrust component > perpendicular to the velocity rotates the velocity vector, which can be > desirable if you are in the vicinity of some large hard object (e.g. the > Earth) that you don't want to smack into while maneuvering. This is an > important issue for launch. Right. My intuition says that it might be worthwhile, then, to point a little above the velocity vector at some point during the flight, in order to give the engines a bit longer to burn, in order to increase the mass ratio and thus delta-V without decreasing the payload mass. Which is just what you suggest, above. I tried launches where the first engine cut is at a low perigee, and then an apogee burn circularizes things. For low earth orbits, this doesn't seem to do very much since the apogee burn has such a small delta-V. I think the small decrease in average toward-ground acceleration is mostly cancelled by perigee being closer so there is that much less to fall. >One thing that has struck me is that my stage-two burn times (this is a >TSTO) are nearly always around 400 odd seconds, over very wide variations >in delta-V between first and second stage. I think the SSME burn is about >this long too. There may be a reason for this consistency... Hmm, yes, an interesting observation, and at first glance your explanation for it seems reasonable. It's always possible to *reduce* the burn time, of course, but that will mean heavier engines and there isn't any obvious advantage to make up for this. >Anyway, the reason this is at all interesting is that if the burn time >is fixed at around 410 seconds, then for a fixed engine size you have >a fixed tank size and the only way to increase the stage's mass ratio >is to decrease the size of the payload. Or to refine things like structure. >I tried launches where the first engine cut is at a low perigee, and then >an apogee burn circularizes things. For low earth orbits, this doesn't >seem to do very much since the apogee burn has such a small delta-V... Yes, a Hohmann insertion doesn't buy much over a direct insertion for seriously low orbits. (Note that the shuttle was the first US manned spacecraft to use Hohmann insertion.) Above a few hundred kilometers, though, the advantage starts to increase rapidly. SignatureMOST launched 30 June; first light, 29 July; 5arcsec | Henry Spencer pointing, 10 Sept; first science, early Oct; all well. | henry@spsystems.net >>One thing that has struck me is that my stage-two burn times >>(this is a TSTO) are nearly always around 400 odd seconds, over [quoted text clipped - 4 lines] > Hmm, yes, an interesting observation, and at first glance your > explanation for it seems reasonable. The explanation is much simpler. The expression for burn time, t, is: t = integral(dmp / mdot) where t = burn time mp = mass of propellant mdot = propellant mass flow rate If mdot is constant (the constant thrust case) the equation becomes t = mp / mdot Since mp is typically a very high fraction of stage mass and the initial T/W ratio is typically about 1 the expression becomes t ~ T / g / mdot ~ Isp in seconds So burn time is to a large extent depends on propellant choice. This is also the reason that RP-1 fuelled SSTOs have lower gravity losses than LH2 fuelled SSTOs. Jim Davis Jim> Since ... the initial T/W ratio is typically about 1 Isn't there a fair bit of variation in initial T/W? Does the Space Shuttle even get to T/W = 1 at SRB seperation? I would imagine that anything resembling an SSTO would have much higher T/W at takeoff (but of course, those would have to throttle as well, voiding my assumption). My explanation suggests that upper stages should take 410 seconds +/- maybe 10% or so, which is a lot less variation than I would expect in initial T/W ratios. And although I'm sure you wouldn't really consider it a data point, my gun-launched rocket simulations with 2000 - 2500 m/s muzzle velocity burn about 400 seconds, and begin with T/W ~= 0.6. >The explanation is much simpler. The expression for burn time, t, is: >t = integral(dmp / mdot) >If mdot is constant (the constant thrust case) the equation becomes >t = mp / mdot >Since mp is typically a very high fraction of stage mass and the >initial T/W ratio is typically about 1... Hold it. Where did that "typically about 1" come from? That's not a law of nature; there's no inherent reason why upper-stage initial T/W has to be in that neighborhood. It's that assumption which produces your "t ~ Isp" result. But this doesn't explain anything without an explanation of that assumption. The *reason* why upper-stage initial T/W is typically around 1 is that that's the value required to keep the burn time down to where you don't fall out of the sky before the burn is finished. Roughly how low does that burn time have to be? See Iain's explanation. So no, this isn't a simpler explanation -- it just brings us full circle, back to Iain's "how long does it take to fall out of the sky?" calculation. SignatureMOST launched 30 June; first light, 29 July; 5arcsec | Henry Spencer pointing, 10 Sept; first science, early Oct; all well. | henry@spsystems.net > There has been a lot of work done on Hohman and Bielliptic transfers, > but has there been anything on constant-thrust transfers? Of course, the [quoted text clipped - 4 lines] > > -bertil- Not online, but the latest AIAA Journal of Guidance Control and Dynamics, Nov-Dec 2003, Vol 26 Number 6, has a paper: "Minimum-Time Orbital Phasing Manuevers," C.D. Hall, V.Collazo-Perez. This discusses constant thrust coplanar transfer over less than a Hohmann 180 deg phase angle. Describes 4 types of thrust profiles - Matt >> There has been a lot of work done on Hohman and Bielliptic transfers, >> but has there been anything on constant-thrust transfers? Of course, the [quoted text clipped - 4 lines] >> >> -bertil- Ok, I've saved the thread. Thanks! everyone who answered. -bertil- Signature"It can be shown that for any nutty theory, beyond-the-fringe political view or strange religion there exists a proponent on the Net. The proof is left as an exercise for your kill-file." |
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