Practical Techniques for Low-Thrust Trajectory Optimization with Homotopic Approach

DOI: 10.2514/1.52476 This paper concerns the application of the homotopic approach, which solves the fuel-optimal problem of lowthrust trajectory by starting from the related and easier energy-optimal problem. To this end, some effective techniques are presented to reduce the computational time and increase the probability of finding the globally optimalsolution.First,theoptimalcontrolproblemismadehomogeneoustotheLagrangemultipliersbymultiplying the performance index by a positive unknown factor. Hence, normalization is applicable to restrict the unknown multipliersonaunithypersphere.Second,theswitchingfunction’s first-andsecond-orderderivativeswithrespectto time are derived to detect switching. The switching detection is embedded in the fourth-order Runge–Kutta algorithm with fixed step size to ensure integration accuracy for bang-bang control. Third, combined with the techniques of normalization and switching detection, the particle swarm optimization with well-chosen parameters considerably increases the probability of finding the approximate initial values of the globally optimal solution. Moreover, intermediate gravity assist, which brings complex inner constraints, is considered. To determine the approximate gravity assist date, analytical formulas are presented to evaluate the minimal maneuver impulse based on the results of Lambert problems. The first-order necessary conditions for gravity assist constraints are derived analytically.Theoptimalsolutioncanberapidlyobtainedbyapplyingthetechniquespresentedtosolvetheshooting function. The unknowns are far less than with direct methods, and the computational effort is also far lower. Two examples of fuel-optimal rendezvous problems from the Earth directly to Venus and from the Earth to Jupiter via Mars gravity assist are given to substantiate the perfect efficiency of these techniques.