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I want to launch my ship to the purple ellipse orbit. But because of non-ideal world I have non-ideal ship orientation and non-ideal thrust impulse. So, my initial speed is a bit bigger and it's direction has a tiny error. But it leads to huge error in the final destination (point of intersection of ellipses with circle).

The purple ellipse is ideal orbit what I want to reach. The cyan ellipse is my real orbit.

How to calculate correction impulse to turn my real orbit into ideal before spacecraft left the cian circle (which is my destination orbit that I should arrive in the given position where purple ellipse intersects with cyan circle)?

I can measure my real velocity and position and calculate the needed velocity and position of ideal orbit at any moment. But I don't know how to turn one to another.

Real (cyan ellipse with big excentricity) and ideal (purple ellipse) orbits

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    $\begingroup$ The honest, best advice i can give you is... Play a few hundred hours of KSP. It gives one an intuitive grasp of orbits that months of formal studying just cannot match. $\endgroup$ Nov 5 at 10:57
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    $\begingroup$ Have you read about the Hohmann transfer ? $\endgroup$
    – Uwe
    Nov 5 at 14:26
  • $\begingroup$ Hohmann transfer can't be used here because it transfers you between pericenter/apocenter. I need to correct my orbit before I leave the circle. Should I use the Lambert's solver again but from current position and with time left? $\endgroup$
    – Robotex
    Nov 5 at 15:01
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    $\begingroup$ I’m voting to close this question because it is far too broad. What you are asking about is the subject of multiple graduate level text books and is the subject of multiple peer reviewed journals such as the Journal of Spacecraft and Rockets, the Journal of Guidance, Control, and Dynamics, and many more. $\endgroup$ Nov 7 at 5:17
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    $\begingroup$ looks good enough to me; voting to reopen so that the OP can post an answer to their own question. $\endgroup$
    – uhoh
    Nov 17 at 23:37
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If you want to do it in one impulse, you have to do it at the two intersections (assuming this is 2D).

To find those intersections, you can make use of the fact that the distance to the central mass is equal for the two orbits at those points.

Radius, in terms of argument of periapsis ($\theta$), periapsis distance ($r_P$) and apoapsis distance ($r_A$)

$$r(\theta) =\frac{2r_Ar_P}{(r_A - r_P)\cos(\theta) + r_A + r_P}$$

If we say the relative argument of periapsis between them (angle between the apsis lines) is $\omega$, you can solve for:

$$r_{ideal}(\theta) = r_{real}(\theta + \omega)$$

At those distances, you can calculate the velocities, and the radial and tangential components of both orbits:

$$v_(r) = \sqrt{\mu\left(\frac{2}{r} - \frac{2}{r_A + r_P}\right)}$$

$$v_{tangential}(r) = \frac{\mu r_P\left(\frac{2}{r_P} - \frac{2}{r_A + r_P}\right)}{r}$$

$$v_{radial}(r) = \sqrt{v_(r)^2 - v_{tangential}(r)^2}$$

For a final burn of:

$$\Delta v = \sqrt{\left(v_{tangential_{real}} - v_{tangential_{ideal}}\right)^2 + \left(v_{radial_{real}} - v_{radial_{ideal}}\right)^2}$$


Multiple impulses

The above equations should be sufficient to calculate these aproaches as well.

Some easy to calculate routes:

strategy 1

  1. Remove relative angle, by doing an impulse at $pi + \omega/2$
  2. Lower periapsis.
  3. Lower apoapsis

Or, even easier to calculate as it has only periapsis and apoapsis burns:

strategy 2

  1. Circularise at apoapsis
  2. Lower periapsis
  3. Lower apoapsis

Advanced: general bi-tangential transfers

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  • $\begingroup$ It's not the appropriate solution for me. I need to make all orbit corrections before the spacecraft go outside the cyan circle. My target is intersection of cyan circle and purple ellipse because I'm trying to make rendezvouz and this orbit was calculated to be on needed position at needed time. So, I need to detect my position-velocity errors and correct it quckly before it became too big. Is it enough to correct just velocity errors and ignore position errors? How spacecraft that transit from Earth to Mars correct their orbits errors during transit? $\endgroup$
    – Robotex
    Nov 5 at 11:24
  • $\begingroup$ @Robotex The first half of the answer is about making the correction at the cyan circle. $\endgroup$ Nov 5 at 11:27
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    $\begingroup$ And rendezvouz wasn't mentioned anywhere in your question, and is a different and much worse problem entirely. $\endgroup$ Nov 5 at 11:31
  • $\begingroup$ because question is not about rendezvouz, it's about correction the orbit disturbed by some factors. I've already found the ideal orbit for rendezvouz transfer (the purple) and I need to fix my real orbit errors to return to ideal if disturbed $\endgroup$
    – Robotex
    Nov 5 at 11:46
  • $\begingroup$ You can see that two orbits are very close before the pericenter and then position error becomes too big. So, I need to fix it before it becomes too big and needs difficult maneuvres. Is it enough to correct just velocity errors and ignore position errors at each moment of transit? $\endgroup$
    – Robotex
    Nov 5 at 11:50
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Finally, I found nice solution:

To correct my orbit and turn it into ideal non-perturbed orbit I need to know my real radius-vector $\vec{R_{real}}$ and velocity $\vec{V_{real}}$

Also, I need to know my ideal radius-vector $\vec{R_{ideal}}$ and ideal velocity $\vec{V_{ideal}}$ for current moment. It can be calculated from the keplerian params of needed orbit.

After that I can find the velocity that I should to have at the current moment for orbit correction:

$\vec{V_{needed}}$ = $\vec{V_{ideal}}$ + ($\vec{R_{ideal}}$ - $\vec{R_{real}}$)

And then I can calculate the direction and amount of impulse for orbit correction:

$\vec{P_{correction}}$ = $\vec{V_{needed}}$ - $\vec{V_{real}}$

This vector gives direction in which I should to accelerate and time of acceleration impulse can be given by

$t$ = $\frac{\vec{P_{correction}}*m_{spacecraft}}{F_{thrust}}$

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