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88 Bernd Dachwaldcurrent target state x T (¯t i ), hence S : {(x SC , x T )} ↦→ {u}. If a propulsion system other thana solar sail is employed, the current propellant mass m P (¯t i ) might be considered as an additionalinput. The number of potential input sets, however, is still large because x SC and x Tmay be given in coordinates of any reference frame. At any time step ¯t i , each output neuronj ∈ {1, . . . , n o } gives a value Y j (¯t i ) ∈ (0, 1). The number of potential output sets is also largebecause there are many alternatives to define u, and to calculate u from Y . Good results havebeen obtained by letting the NC provide a three-dimensional output vector d ′′ ∈ (0, 1) 3 fromwhich a unit vector d is calculated. 2 d is interpreted as the desired thrust direction and is thereforecalled direction unit vector. For solar sailcraft, u = d, hence S : {(x SC , x T )} ↦→ {d}. Forelectric spacecraft, the output must include the engine throttle 0 ≤ χ ≤ 1, so that u = (d, χ),hence S : {(x SC , x T , m P )} ↦→ {d, χ}.Neurocontroller Fitness Assignment. In EAs, the optimality of a chromosome is rated by afitness function 3 J. The optimality of a trajectory might be defined with respect to various primaryobjectives (e.g., transfer time or propellant consumption). When an ENC is used for trajectoryoptimization, the accuracies of the trajectory with respect to the final constraints mustalso be considered as secondary optimization objectives because they are not enforced otherwise.If, for example, the transfer time for a rendezvous is to be minimized, the fitness functionmust include the transfer time T , the final distance to the target ∆r f = |r T (¯t f ) − r SC (¯t f )|, andthe final relative velocity to the target ∆v f = |ṙ T (¯t f ) − ṙ SC (¯t f )|, hence J = J(T, ∆r f , ∆v f ). If,for example, the propellant mass for a flyby problem is to be minimized, T and ∆v f are notrelevant but the consumed propellant ∆m P must be included in the fitness function, henceJ = J(∆m P , ∆r f ) in this case. Because the ENC unlikely generates a trajectory that satisfiesthe final constraints exactly (∆r f = 0 m, ∆v f = 0 m/s), a maximum allowed distance ∆r f,maxand a maximum allowed relative velocity ∆v f,max have to be defined. Because in the beginningof the search process most individuals do not meet the final constraints with the requiredaccuracy, a maximum transfer time T max must be defined for the numerical integration of thetrajectory. For a detailed description of the NC fitness assignment, the reader is referred to [4].Evolutionary Neurocontroller Design. This section details how an ENC may be applied forlow-thrust trajectory optimization. To find the optimal spacecraft trajectory, the ENC methodruns in two loops. Within the (inner) trajectory integration loop, a NC k steers the spacecraftaccording to its network function N wk that is completely defined by its parameter set w k . TheEA in the (outer) NC optimization loop holds a population P = {c 1 , . . . , c q } of NC parametersets including the additionally encoded initial conditions, and examines them for their suitabilityto generate an optimal trajectory. Within the trajectory integration loop, the NC takesthe current spacecraft state x SC (¯t i∈{0,...,τ−1} ) and that of the target x T (¯t i ) as input, and mapsthem onto some output. For electric spacecraft, the input includes the current propellant massm P (¯t i ) and the output includes the current throttle χ(¯t i ). The first three output values areinterpreted as the components of d ′′ (¯t i ), from which the direction unit vector d(¯t i ) is calculated.This way the spacecraft control u(¯t i ) is calculated from the NC output. Then, x SC (¯t i )and u(¯t i ) are inserted into the equations of motion and numerically integrated over one timestep to yield x SC (¯t i+1 ). The new state is fed back into the NC. The trajectory integration loopstops when the final constraints are met with sufficient accuracy or when a given time limit isreached (¯t i+1 = ¯t f,max ). Then, back in the NC optimization loop, the trajectory is rated by theEA’s fitness function J(c k ). The fitness of c k is crucial for its probability to reproduce and tocreate offspring. Under this selection pressure, the EA breeds more and more suitable steeringstrategies that generate better and better trajectories. Finally, the EA that is used within this2 via d ′ = 2d ′′ − (1, 1, 1) T ∈ (−1, 1) 3 and d = d ′ /|d ′ |3 This fitness function is also analogous to the cost function in optimal control theory. To emphasize this fact, it will be denotedalso by the symbol J.