Study Of Indirect And Pseudospectral Methods For Low Thrust Trajectory Optimization In Deep Space Exploration

Low thrust propulsion is one of the key techniques for deep space exploration, andsuch propulsion systems are really efficient due to their high specific impulse. However,the thrust is very ‘low’ and applied on the spacecraft continuously&lastingly, whichrenders the dynamic equations nonintegrable and thus makes design of low thrusttrajectories totally different from that of the impulsive trajectory. It is now an urgentneed to construct effective and reliable ways to design low thrust trajectory in aerospaceengineering. With such practical demands in mind, this thesis studies indirect andpseudospectral methods systematically for their applications to low thrust trajectoryoptimization.Based upon optimal control theory, a systematic way for designing low thrusttrajectory via multiple gravity assists (GAs) is presented. Different from the commonshaped-based approach and parameter optimization, gravity assists are taken as interiorpoint constraints here and the first order optimal necessary conditions are derived usingoptimal control principle. The trajectory optimization with multiple GAs is formulatedas a multiple point boundary value problem (MPBVP) and the associated shootingequations are then constructed. The ways to generate reasonable initial guesses for theshooting variables are detailed and the MPBVP is finally solved by the simple shootingmethod. In comparison with parameter optimization, the present method’s efficiencyand optimality is satisfactory.The thesis systematically studies the pseudospectral method for its applications tolow thrust trajectory optimization. A pseudospectral scheme for low thrust trajectoryoptimization is first formulated. Its equivalence with the indirect method is verified andthe associated covector mapping is established. The continuous trajectory optimizationis transformed into a discrete nonlinear programming problem (NLP) and theenergy-optimal solutions are obtained with a NLP solver. Furthermore, the thesisanalyzes various difficulties confronted by the pseudospectral method when solvingfuel-optimal bang-bang controls and a pseudospectral-homotopic hybrid method is thenproposed to circumvent these difficulties. With the homotopic algorithm, the hybridscheme first transforms the fuel-optimal problem to an energy-optimal one which can be solved efficiently by the pseudospectral method. And the energy-optimal results arethen continued to the desirable fuel-optimal ones by the homotopic algorithm. Thishybrid method combines beneficial features of the pseudospectral method andhomotopic approach, yielding bang-bang controls more accurately and efficiently thanthe former, and also resolving the latter’s initiation difficulty via the former.A multi-phase pseudospectral method is constructed for low thrust trajectoryoptimization involving time varying interior point constrains (IPCs), and the associatedcovector mapping is formulated and proved. Through introducing multi-phasepseudospectral grids, the continuous trajectory with IPCs is discretized into a NLP,whose first-order necessary conditions (i.e., Karush–Kuhn–Tucker or KKT conditions)are then derived. The (interior) KKT conditions are shown equivalent to the (interior)optimal necessary conditions in the indirect method, thus extending the previouscovector mapping to the cases involving time varying IPCs. This extended covectormapping connects the multi-phase pseudospectral method and the MPBVP. It canestimate the costates, as well as the Lagrange multipliers associated with the IPCs andthe interior time. Thus, it is not only a theoretical improvement of the pseudospectralmethod, but also the theoretical basis for extending the pseudospectral-homotopichybrid method to trajectory optimization involving time varying IPCs.