On The Problem Of Fuel-optimal Pinpoint Soft Landing

It’s an expensive but the only way to send payloads into space by launch vehicles.To make exploring space more affordable it’s necessary to make launch vehicles reusable.By recovering the stages which were regularly discarded it would be a sharp decline in the expenditure.However recovering launch vehicles with high speed from high orbit is an extremely difficult task which a lot of advance techniques are involved in.Guidance navigation and control aka GNC is an important item of the techniques.And the guidance algorithm has to be effective to compute an optimal thrust and an ideal landing traj ectory.The landing error is directly related to it.During the landing phase the rocket perform powered descent and land to a drone ship or a prescribed position precisely.The fuel,which is reserved for powered landing,is additional payload during launching phase.Thus a fuel-optimal pinpoint soft landing problem is proposed in the purpose of consuming fuel as less as possible during the landing phase.For this problem here are the steps taken in the thesis to deal with it.An optimal control problem is presented.By some basis assumptions the dynamics and constraints on thrust,position and velocity are described in mathematics equations or inequalities.A study on the property for the optimal control problem is carried on meanwhile a solving algorithm is proposed in the thesis.At first a simple problem which there’s only vertical channel landing is researched.A conclusion that there are exactly four patterns of optimal thrust is derived by Pontryagin’s Maximum Principle.More than that an analytic solution can be solved under the assumption that the minimum thrust level is zero.Similarly the property of the fuel-optimal precision landing problem can be derived.It can be proved by Pontryagin’s Maximum Principle that there are exactly five patterns of the magnitude of the optimal thrust.And then a convex programming is utilized to get the numeric solution of the optimal thrust.A second order cone programming aka SOCP is presented after introducing slack variable,changing variables,approximating constraints and discretization.SOCP,a particular type of convex programming,can be solved by Inner Point Method aka IMP in polynomial time.At last a few simulations in Matlab are presented to illustrate the feasibility of the algorithm.And the difference between two landing trajectory under two different initial conditions are presented as well.