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42 V.M. Becerra, D.R. Myatt, S.J. Nasuto, J.M. Bishop, and D. Izzo2.3 Gravity assist thrust constraintTwo constraints are added in order to maximise the probability of gravity assists being feasible.The first such constraint is the gravity assist thrust constraint, which limits the maximumabsolute difference between incoming and outgoing velocities during a gravity assist to somethreshold, T v . This threshold is set separately for each gravity assist. The following is thenperformed for every arrival time at a planet:1. Calculate the bounds on incoming velocity, v i min and vi max.2. Invalidate any outgoing trajectories that do not have outgoing velocities in the range[v i min − T v − L v , v i max + T v + L v ], where L v is an appropriate tolerance based on theLipschitzian constant of the current phase plot.3. Calculate the modified bounds on outgoing velocity, v f min and vf max.4. Invalidate any incoming trajectories with velocities outside the range [v f min −T v−L v , v f max+T v + L v ].2.4 Gravity assist angular constraintThe gravity assist angular constraint removes infeasible swingbys from the search space on thebasis of them being associated with a hyperbolic periapse under the minimum safe distancefor the given gravity assist body. This is determined over every arrival date at a planet asfollows, assuming i valid incoming trajectories and j valid outgoing trajectories:1. For all i incoming trajectories2. For all j incoming trajectories3. If the swingby is valid for the current incomingand outgoing trajectory, mark both incoming andoutgoing trajectory as valid.4. End5. End6. Invalidate all trajectories not marked as validThe swingby angle is decreased by an appropriate Lipschitzian tolerance θ L , in order to compensatefor the effects of the grid sampling of the search space.3. Time and space complexityThis section determines the time and space complexity of the GASP algorithm. It will beshown that GASP scales quadratically in space and quartically in time with respect to thenumber of gravity assist manoeuvres considered. For simplicity, the following analysis assumedthat the initial launch window and all phase times are the same.3.1 Space ComplexityConsider a launch window discretised into k bins and a mission phase time also discretisedinto k bins. For the first phase k 2 Lambert problems must be sampled. The next phase willneed to sample (k + k)k = 2k 2 , as the number of possible times that the planet may be arrivedat is doubled (minimum launch date, minimum phase time to maximum launch date, maximumphase time). The third phase will require 3k 2 Lambert function evaluations, and the n thphase nk 2 . This gives the seriesO(n) = k 2 + 2k 2 + 3k 2 + . . . + nk 2 (3)