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It's done using convex optimization https://www.nap.edu/read/23659/chapter/10#39 https://www.nap.edu/read/23659/chapter/10#39
by sonium 7y ago
It's done using convex optimization
https://www.nap.edu/read/23659/chapter/10#39 https://www.nap.edu/read/23659/chapter/10#39
- amelius 7y agoHow is "booster landing" mapped to a convex optimization problem?
- theoh 7y agoLars Blackmore, who works for SpaceX, has a page about the techniques that are used here: http://www.larsblackmore.com/losslessconvexification.htm http://www.larsblackmore.com/losslessconvexification.htm See also: https://www.aa.washington.edu/facultyfinder/behcet-acikmese https://www.aa.washington.edu/facultyfinder/behcet-acikmese
- dom96 7y agoCan anyone give a ELI5 (for developers) of how this technique works?
- theoh 7y agoThe control parameter space for the rocket is non-convex because, for one thing, it can't be throttled down close to zero thrust. You can think of the parameter space as having a hole in it. An optimal control algorithm might want to make very delicate low-thrust corrections, but that's not possible. This technique is based on "the idea of relaxing the nonconvex control constraints to a convex set in such a way that the optimal solution to the relaxed problem is guaranteed to be the optimal solution to the original problem." By "relaxing", what they mean is that the new model (the convex set) actually contains some parameter values that aren't achievable, but OTOH it is geometrically susceptible to being analyzed by an efficient optimization technique. So it's a clever replacement of a hard problem with an easier one. The complex bit, which I don't understand, is how they show that this replacement will always give the same value as if they had solved the (real) hard problem, i.e. that the solution will never actually use the "illegal" parameter values. Maybe someone else can give more insight.
- CarVac 7y agoI read a paper on this—it's convex because any optimal trajectory will have the engines at full throttle whenever they're on, so they just fill up the parameter space, confident that illegal values will never be required.
- chasd00 7y agocan you link to the paper please?
- theoh 7y agoI don't know which paper they mean, but here's a relevant paragraph from one of the papers available from Blackmore's page: "In this paper, we unify the convex optimization approaches of [1], [2], [17] and extend them to handle thrust pointing constraints. While convexifying the problem with nonconvex thrust pointing constraints, we develop a geometrical insight into the problem that establishes a connection with “normal systems” [18]. A normal linear system is defined in the context of optimal control theory where the system is said to be normal with respect a set of feasible controls if it maximizes the Hamiltonian at a unique point of the set of feasible controls. In the case when the set of feasible controls is convex, a system being normal implies that the Hamiltonian is maximized at an extreme point of the set [18]. Our convexification result has a similar geometric interpretation since it establishes lossless convexification by ensuring that the Hamiltonian is maximized at the extreme points of a projection of the relaxed set of feasible controls. This set is then shown to be contained in the original nonconvex set of feasible controls, thereby estab- lishing that we can obtain optimal solutions of the original nonconvex problem via solving its convex relaxation."