Optimal Bounded Low-Thrust Rendezvous with Fixed Terminal-Approach Direction

Abstract

Thebounded,low-thrust,e xed-time, fuel-optimalconstrainedterminalapproachdirection rendezvoususing the relative linearized equationsof motion is investigated. A minimum fuel, two-stage solution isdeveloped. In thee rst stage, optimal transfer from initial conditions to an intermediate point in the e nal line approach is implemented. In the second stage, an optimal thrust program guaranteeing directional approach is applied. Optimal rendezvous with e xed terminal approach is achieved by minimizing overall fuel consumption as a function of the intermediate point relative location and velocity in the e nal line approach. Solutions are generated and analyzed with e nal time as a parameter. Numerical results for a representative case are presented. tem, which is characterized by variableexhaustvelocity and limited power, produces low thrust with high specie c impulse, greatly re- ducing the initial spacecraft mass. This mass reduction makes pos- sible a new generation of small and sophisticated spacecraft for a variety of missions, from satellite station keeping to interplanetary missions, as well as maneuvers in the vicinity of space stations and even small comets or asteroids. For efe ciency, an optimal trajectory is desired. In most cases, the objective is to guide the spacecraft from an initial state to a e nal state (position and velocity ), with minimum fuel expenditure. Op- timal trajectory analyses of such systems were discussed in several comprehensive works, such as that by Marec. 2 Advanced analy- sis should consider practical issues, such as bounds on the thrust levels. Optimum unbounded thrust rendezvous programs for power- limited propulsion systems were examined by Lembeck and Prussing 3 using equations linearized about a nominal circular orbit and by Carter 4 using equations linearized about general Keplerian orbits. Exact, analytical expressions were obtained for the required control accelerations. Pardis and Carter 5 and Kechichian 6 derived bounded, low-thrust rendezvous trajectories with power-limited propulsion systems. Carter and Pardis 7 extended the work for the casewherethecontrollerhasbothupper andlowerbounds.Toavoid saturation, they proposed the use of multiple engines, where each one has continuous thrust bounded by upper and lower limits. In all of these works no constraints were imposed on the chaser' s e nal approach to the target. However, the e nal approach of a real rendezvous is constrained. Proximity maneuvering is started at a certain distance from the target and is to be performed along the target-docking axis. Therefore, these path constraints at the e nal approachhavetobetakenintoaccounttoperformterminalapproach along a prespecie ed inertial or target-relative direction. In Ref. 8, a minimum fuel, two-stage solution was developed for the power-limited unbounded thrust rendezvous with a e xed e nal direction approach. In the e rst stage, optimal transfer from initial conditions to an intermediate point in the e nal line approach was implemented.Inthesecondstage,anoptimalthrustprogramguaran- teeing directional approach was applied. Optimal rendezvous with e xed terminal approach was achieved by minimizing overall fuel consumption as a function of the intermediate point relative loca- tion and velocity in the e nal line approach. This two-stage solution