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86 Bernd DachwaldFor spacecraft with high thrust, optimal interplanetary trajectories can be found relativelyeasily because few thrust phases are necessary during its "free fall" within the gravitationalfield of the solar system. These can be approximated by singular events that change thespacecraft’s velocity instantaneously while its position remains fixed. Low-thrust propulsionsystems, in contrast, are required to operate for a significant part of the transfer to generatethe necessary ∆V . A low-thrust trajectory is obtained from the numerical integration of thespacecraft’s equations of motion. Besides the inalterable external forces, the spacecraft trajectoryx SC [t] = (r SC [t], ṙ SC [t]) is determined entirely by the thrust vector history F [t] (’[t]’ denotesthe time history of the preceding variable, x SC is called the spacecraft state). The thrust vectorF (t) of low-thrust propulsion systems is a continuous function of time. It is manipulatedthrough the n u -dimensional spacecraft control function u(t) that is also a continuous functionof time. Therefore, the trajectory optimization problem is to find, in infinite-dimensionalfunction space, the optimal spacecraft control function u ⋆ (t) that yields the optimal trajectoryx ⋆ SC[t]. This renders low-thrust trajectory optimization a very difficult problem that cannot be solved except for very simple cases. What can be solved, at least numerically, however,is a discrete approximation of the problem. Dividing the allowed transfer time interval[t 0 , t f,max ] into τ finite elements, the discrete trajectory optimization problem is to find the optimalspacecraft control history u ⋆ [¯t] (the symbol ¯t denotes a point in discrete time) that yieldsthe optimal trajectory x ⋆ SC[t]. Through discretization, the problem of finding the optimal controlfunction u ⋆ (t) in infinite-dimensional function space is reduced to the problem of findingthe optimal control history u ⋆ [¯t] in a finite but usually still very high-dimensional parameterspace. For optimality, some cost function J must be minimized. If the used propellant mass∆m P = m P (¯t 0 ) − m P (¯t f ) is to be minimized, J = ∆m P is an appropriate cost function, if thetransfer time T = ¯t f − ¯t 0 is to be minimized, J = T is an appropriate cost function.2. Evolutionary Neurocontrol as a Smart Global Low-ThrustTrajectory Optimization MethodTraditionally, low-thrust trajectories are optimized by the application of numerical optimalcontrol methods that are based on the calculus of variations. All these methods can generallybe classified as local trajectory optimization methods (LTOMs), where the term optimizationdoes not mean to find the best solution but rather to find a solution. The convergence behaviorof LTOMs is very sensitive to the initial guess, which has to be provided prior to optimizationby an expert in astrodynamics and optimal control theory. Because the optimization processrequires nearly permanent expert attendance, the search for a good trajectory can becomevery time-consuming and expensive. Even if the optimizer finally converges to an optimaltrajectory, this trajectory is typically close to the initial guess and that is rarely close to the (unknown)global optimum. Another drawback of LTOMs is the fact that the initial conditions(launch date ¯t 0 , initial propellant mass m P (¯t 0 ), initial velocity vector ṙ SC (¯t 0 ), etc.) – althoughthey are crucial for mission performance – are generally chosen according to the expert’s judgmentand are therefore not explicitly part of the optimization problem.To evade the drawbacks of LTOMs, a smart global trajectory optimization method (GTOM)was developed by the author [4]. This method – termed evolutionary neurocontrol (ENC)– fuses artificial neural networks (ANNs) and evolutionary algorithms (EAs) into so-calledevolutionary neurocontrollers (ENCs). The implementation of ENC for low-thrust trajectoryoptimization was termed InTrance, which stands for Intelligent Trajectory optimization usingneurocontroller evolution. To find a near-globally optimal trajectory, InTrance requiresonly the target body and intervals for the initial conditions as input. It does not require aninitial guess or the attendance of a trajectory optimization expert. During the optimizationprocess, InTrance searches not only the optimal spacecraft control but also the optimal initialconditions within the specified intervals.