Structured-Chromosome Evolutionary Algorithms for Variable-Size Autonomous Interplanetary Trajectory Planning Optimization

Abstract

In interplanetary trajectory optimization, events such as planetary gravitational-assist maneuvers (swingbys) and deep-space maneuvers can be added/removed from the trajectory plan to reduce the cost or the flight time. This renders the number of design variables in the optimization problem variable. Global optimization methods that optimize this type of multimodal objective function can only handle problems with a fixed number of design variables. This paper presents the structured-chromosome evolutionary algorithm framework that is developed to handle variable-size design space optimization problems. In this framework, a solution (chromosome) is represented by a hierarchical data structure where the genes in the chromosome are classified as dependent and nondependent genes. This structure provides the capability to apply genetic operations between solutions of different lengths, and thus to automatically determine the number of swingbys, the planets to swingby, launch and arrival dates, and the number of deep-space maneuvers, as well as their locations, magnitudes, and directions, in an optimal sense. This new method is applied to several interplanetary trajectory design problems. Results show that solutions obtained using this tool match known solutions for some complex problems. A comparison between genetic algorithms and differential evolution in the structured-chromosome framework is presented. Nomenclature a = first selected chromosome a = thrust acceleration vector b = second selected chromosome c = third selected chromosome F = fitness (cost function) F 0 = modified fitness (cost function) f = flight direction G = derating function h = pericenter altitude, km h = normalized swingby pericenter altitude iter = iteration m = number of swingby maneuvers NR = number of runs n = number of deep-space maneuvers in a single leg R = mean radius, km r = heliocentric position vector in inertial frame, km r = spacecraft position vector r = spacecraft acceleration vector SC = success counter SR = success rate sol = solution T = time of flight, days t = Julian date tmp = temporary chromosome u = solution feature v ∞ = hyperbolic velocity vector relative to the planet, km∕s w = differential weight ΔV x = x component of impulsive maneuver velocity ΔV y = y component of impulsive maneuver velocity ΔV z = z component of impulsive maneuver velocity Δv = impulsive maneuver velocity vector, km∕s Δv T = total mission cost, km∕s ε = epoch of a deep-space maneuver as a fraction of transfer time η = swingby plane rotation angle, rad μ = gravitational constant, km 3 ∕s 2 Subscripts a = arrival d = departure l = leg number p = swingby's planet ps = powered swingby req = required s∕c = spacecraft T = total Superscripts − = incoming = outgoing

An Overview of Evolutionary Algorithms toward Spacecraft Attitude Control

“…A more recently derived rule of thumb is that for "small" parameter sets, the population size is effective if scaled with the number of parameters with 10 m, and for larger spaces the population size scales with ln m ðÞ [42], where the definition of large is different for each author. For a simple parameter set GA can be quite effective for optimization problems in many space based applications [23,24,28,[43][44][45][46][47] which stem from aeronautical control [20,22], and ground-based robotic systems [48][49][50].…”

“…The most prominent evolutionary algorithm is the genetic algorithm officially introduced by John Holland in his 1975 book titled "Adaptation in Natural and Artificial Systems" [27] and its primary variants involving the concepts of chromosomes, elitism, parallel populations [28][29][30], and adaptation [31][32][33] which are derived from the concept of Darwinian evolution of animal species across many generations, also known as natural selection. Genetic Algorithms will be discussed more thoroughly in Section 2.…”

High-Lift Devices Topology Optimisation using Structured-Chromosome Genetic Algorithm

“…A variety of strategies for facing variable-size global optimisation is described in the literature [9], [10]. The hidden gene adaptation of GA for the optimisation of interplanetary trajectories is introduced in [10].…”

“…A more complex, but efficient and flexible adaptation of GA is proposed in [9]. In this case, a hierarchical multi-level chromosome structure is adopted in place of the standard string one.…”

Autonomous Generation of Observation Schedules for Tracking Satellites with Structured-Chromosome GA Optimisation

“…A more complex, but efficient adaptation of GA is proposed in [4], [5]. In these cases, a hierarchical multi-level chromosome structure is adopted in place of the standard string one.…”