Reaching a target orbit with control of nothing but thrust direction

[Zander]

hey this is my first time posting in this area. I have an interesting problem. I need a step by step guide for how to reach a target orbit. The catch is, nothing can be changed except the direction of thrusting.

I have the information for ownships mass, thrust, position, altitude, speed, and anything else needed. The body is earth.

I have all the orbital elements for the target orbit.

apogee perigee period inclination descending node and eccentricity

What I need to know is, how can I calculate which direction to thrust in to reach the target orbit.

Im planning to write the calculations in C# and run it several times a second updating ownships properties each cycle And use this to produce a second by second thrusting direction command to feed to the gimbals.

[JtPB]

the intensity of the thrust can be changed? (including zero thrust)

[Zander]

the intensity of the thrust can be changed? (including zero thrust)

No the engine is not throttleable and the engine cannot be turned on again. So once it turns off the rocket has to be in the target orbit.

So the only thing that can change is the direction of thrusting(the gimbals of the engine)

[Brotoro]

Do you want to start from the surface of the Earth, or from some other orbit?

I wouldn't expect continuous thrusting to be an optimal way of doing this…although it would certainly be doable with a low thrust engine if you just wanted to transfer between orbits.

[Zander]

Do you want to start from the surface of the Earth, or from some other orbit?

I wouldn't expect continuous thrusting to be an optimal way of doing this…although it would certainly be doable with a low thrust engine if you just wanted to transfer between orbits.

It starts at around 100km in a suborbital trajectory that is going in the general direction of the target orbit which will be circular with an apoapsis and periapsis of around 190km. And yes it is a low TWR engine that will burn continuously at 100% power with no throttling ability.

[K^2]

If TWR is sufficiently low, burning continuously prograde puts your ship on a gentle spiral, which carries you from one circular orbit to another. No need to invent anything there. If, however, there is eccentricity change involved, you'll need a more clever control algorithm. But it is doable. So have fun with it.

[PUNiSH3R]

I actually just finished reading an article on this; its by Francesco Topputo and Edward Belbruno, the title is "Optimization of Low-Energy Transfers" and it has been published as Chapter 16 of Springer's "Modeling and Optimization in Spacecraft Engineering". And it does in fact describe an optimal control algorithm for a continuous low-thrust transfer. An Earth-Moon transfer and a Lagrangian transfer are examined.

Keep in mind these types of techniques require utilization of a restricted three-body or four-body problem - planar circular restriction to be exact - solved via numerical methods, so bust out the PETSc and start grinding...

[Zander]

Really?? Im quite surprised at these responses. Its as if nobody has every tried this before.. The continuous burn to orbit is like the most common way in real life to launch satelites and rockets.. I was expecting someone to already have an equation at hand.

All right I guess I will have to do some research.