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

(Bertil Jonell) wrote in message ...
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-

Try googling "Pontryagin constant thrust orbit transfer" and see what you can find?

Dave

(Bertil Jonell) wrote in message ...
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-

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..

(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?

Bertil Jonell wrote:
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-

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

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