The Application Research Of Indirect Method In Continuous Thrust Fuel-Optimal Cooperative Rendezvous

The trajectory optimization of spacecraft is one of the key technologies in the aerospace field,and it is also a research hotspot and difficulty.The research content covers multiple disciplines and fields,including spacecraft orbital mechanics,nonlinear system theory,optimal control theory,numerical calculation and analysis.In the earlier research,most of the spacecraft engines were based on impulse thrust.With the complexity of space missions,continuous thrust has received extensive attention and application because of its characteristics of large specific gravity,high efficiency and long working hours.For such propulsion systems,the spacecraft trajectory optimization problem based on finite continuous thrust was investigated in this paper,taking spacecraft cooperative rendezvous as background.The fuel-optimal cooperative rendezvous problem was transformed into the two-point boundary value problem(TPBVP)by using Pontryagin's maximum principle.The fuel-optimal control law of the spacecraft was derived based on the first-order optimality necessary conditions.In order to avoid the non-smooth integral problem caused by the discontinuous fuel-optimal bang-bang control law,this paper introduced the configuration variables to artificially configure the spacecraft's switching sequence to replace the optimal control of the engine switch in fuel-optimal control law.By discretizing the discontinuous optimal control equations,the differential equations were integrated in stages,and the optiral control problem was transformed into a high-dimensional parameter optimization problem.This method belongs to the hybrid method.In the process of optimization simulation with the hybrid method,for the simple space maneuvering task,the spacecraft control strategies were preset as four burn arcs and four coast arcs.Then for the same space maneuvering task,the spacecraft engine thrust was reduced,the spacecraft flight time increased consequently,and more maneuvers were required to complete the space mission.The optimization simulation showed that four burn arcs and four coast arcs were difficult to meet the rendezvous requirements,leading to the waste of fuel or the spacecraft being unable to achieve rendezvous.For this reason,this paper increased the spacecraft control strategies to eight burn arcs and eight coast arcs.However,with the increase of the number of preset burn and coast arcs artificially,more and more parameters were involved,and the calculation efficiency and convergence were decreased.Therefore,the hybrid method cannot increase the preset control strategese infinitely,and it is not suitable for solving complex space rendezvous tasks that the spacecraft thrust is low and the engine requires multiple switching to complete.For such tasks,considering that in the indirect method,the spacecraft engine switch is completely determined by the switching function.There is no need for artificial preset control strategy and no intermediate variables are introduced,leading to few parameters to be sought.Moreover,it has higher precision because the optimal control satisfies the first-order optimality necessary conditions.Therefore,the indirect method has a natural theoretical advantage over the hybrid method.However,because its feasibility in solving fuel-optimal cooperative rendezvous was not high,the paper intended to improve the feasibility of the indirect method through a series of improved techniques,and to take advantages of the indirect method in solving low thrust complex space maneuvering task.Aiming at the discontinuous fuel-optimal bang-bang control,this paper adopted homotopy technique to carry on the smoothing processing.It started from the energy-optimal problem with continuous optimal control law and smoothly transitioned to the fuel-optimal problem with discontinuous optimal control law in the form of exponential decreasing.It avoided oscillation of differential equations and overcame the difficulties of non-smooth integral caused by the discontinuity of the right side of fuel-optimal control differential equations.In the indirect method,the convergence domain of the shooting function is extremely narrow,and it is difficult to obtain the guess value in the convergence domain.The shooting process is extremely sensitive to the initial value guess of the costate variables,and the precision requirements for initial values is extremely high.Besides,in view of the cooperative rendezvous problem,there are still the following difficulties:Both two spacecraft are active spacecraft and both will maneuver,causing the number of parameters doubled.The terminal rendezvous time and terminal rendezvous orbit is unknown.The above two points increased the complexity of the system significantly and further increased the optimizing difficulty.Therefore,in order to improve the probability of converging to the globally optimal solution of the indirect method,higher requirements must be put forward for the design and implementation of the optimization process.First of all,in the face of costate variables with lack of physical meaning and unknown scope,the initial value of the costate variables were normalized to improve the efficiency of optimization.Then,the quantum particle swarm optimization(QPSO)algorithm,which is suitable for multi-parameter optimization and has strong global convergence,was used to preliminarily s'earch for the initial costate variables of the high-dimensional energy-optimal problem.Finally,the results obtained were further corrected by the sequence quadratic programming(SQP)algorithm to improve the accuracy of the initial value of energy-optimal problem and reduce the impact of the sensitivity of the shooting function.Based on these modified initial costates,the smooth energy-optimal results were transitioned to the desirable nonsmooth fuel-optimal results by the homotopy method.The optimization and simulation results showed that the accuracy of the initial value of the energy-optimal problem directly affected the merits of the fuel-optimal solution.Through the correction to the initial value of energy-optimal by modified algorithm,the convergence accuracy and stability of the fuel-optimal problem were improved obviously.Comparing the indirect method with the hybrid method,the simulation results showed that the applicability of the indirect was more extensive.In the face of the problem of coplanar rendezvous with large thrust,the results obtained by the hybrid method and the indirect method were basically the same.Howerver,as the thrust of spacecraft decreased and the number of maneuvers required for rendezvous increased,the advantages of the indirect method over the hybrid method were beginning to emerge.