# Targeted Sub-Orbital Optimization Problem

I was messing around with some orbital dynamics for a simulator I am working on, and I came across a snag that I have not been able to figure out. Specifically, I am trying to minimize the required deltaV for an impactor probe. Say for instance, you had a probe at some initial position (r0 vector) and initial velocity (v0 vector), and you wanted it to hit a specific point on the moon (rf vector) with the least amount of dV expended. How would one go about solving a problem like this, and what formulas would I need to use?

Thank you all!

-Frank

• You seem to be asking two different questions here, one about transforming orbits, and one about crashing stuff into a specific point on a moon. It is better to ask one question per post. I'm also not entirely sure what you're trying to do with the orbit projection, but crashing stuff should be straightforward. If you rewrite your question just to ask about crashing, it might be easier to answer. Mar 30, 2022 at 8:33
• I just changed it, The second part was just talking about my current attempt, but I can see how that would be confusing. The goal of this project is for an impactor probe model, so thats really all I am looking for. Thank you for the advice Mar 30, 2022 at 13:16
• Ah thank you, Just changed it Mar 30, 2022 at 13:23

This is a nonlinear optimization problem. There are many ways to find the optimal solution. We have multiple questions with answers on similar problems. Look for questions/answers about porkchop plots (or pork chop plots).

There always exists at least two or more conic sections that take a vehicle from point $$A$$ in inertial space at time $$a$$ to point $$B$$ in inertial space at time $$b$$ that is greater than time $$a$$. Finding these solutions is the essence of Lambert's problem.

What you'll need to do is determine the cost of transferring the vehicle to the target point given different choices of times $$a$$ and $$b$$. You'll need to be able to determine the vehicle state $$A$$ at time $$a$$ and the target state $$B$$ at time $$b$$. Given that information, you can solve Lambert's problem. Given a cost function (e.g., $$\Delta \text v$$) you will be able to hone in on an optimal solution.

There is one gotcha here: The algorithms typically used to solve Lambert's problem have singularities at a 180° transfer. It is those 180° transfers that are oftentimes the optimal solution. You'll have to do something special with regard to transfers at or near 180°.

A shortcut to a solution that doesn't involve pork chop plots: If your vehicle will at some point in time be diametrically opposed (180° transfer) to where the target will be, look for and solve for the $$\Delta \text v$$ for that possibility. This is a non-Lambertian transfer, but it might well be optimal.

• Some places (eg. this related answer) refer to Gauss' Problem, rather than Lambert's problem. Are they the same thing? And if so, have people really been arguing over the correct name for 200 years? Mar 30, 2022 at 14:56
• @StarfishPrime afaik they are the same Mar 30, 2022 at 15:17
• @StarfishPrime The answerer in that related answer explicitly says that he doesn't know the correct name, and that Bates et al. (which is a good basic reference) calls it Gauss's problem while others call it Lambert's problem. I've always seen it called Lambert's problem in a professional setting. Mar 30, 2022 at 17:09