Optimum constant-thrust transfers?

(Bertil Jonell) writes:

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.


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In article ,
John Schilling wrote:
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.)
--
MOST launched 30 June; first light, 29 July; 5arcsec | Henry Spencer
pointing, 10 Sept; first science, early Oct; all well. |

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?

In article ,
Iain McClatchie wrote:
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.
--
MOST launched 30 June; first light, 29 July; 5arcsec | Henry Spencer
pointing, 10 Sept; first science, early Oct; all well. |

Henry Spencer wrote:

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.

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
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.

...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.

In article ,
Iain McClatchie wrote:
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.
--
MOST launched 30 June; first light, 29 July; 5arcsec | Henry Spencer
pointing, 10 Sept; first science, early Oct; all well. |

Henry Spencer wrote:

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.

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