Entry Trajectory Tracking Law via Feedback Linearization

Abstract

Asasteptowardextendingthetwo-dimensional(longitudinal)entrypredictive/trackingguidanceschemeusedby the U.S. Space Shuttle Orbitertothreedimensions, a control law fortrackinga three-dimensional entrytrajectory isdesigned. Thetrackinglawcommandstheanglesofattack and of bank thatarerequired tofollow aground track specie ed as a function of energy. Feedback linearization is used to design the tracking law. Some extensions to the existing theory are required to accommodate features of the entry tracking problem. Downrange and crossrange angles serve as output variables for the feedback linearization and lead to state and control transformations that convert the entry dynamics to an equivalent linear system in an approximate sense, which is dee ned. A feedback tracking law is then designed, taking advantage of the linear structure of the system dynamics in the transformed variables. This tracking law is shown to achieve bounded tracking of the output variables. Simulation results indicate the effectiveness of the tracking law in compensating for initial offsets from a reference trajectory. URRENT efforts to develop reusable launch vehicles (RLVs), a space station crew return vehicle, and a military spaceplane provide motivation for investigating potential improvements to the entry guidance capability for a lifting unpowered e ight vehicle.The state of the art is represented by the U.S. Space Shuttle Orbiters' entry guidance, as described by Harpold and Graves. 1 The Shuttle's longitudinal guidance combines predictive guidance, i.e., model- basedreferencetrajectoryplanningwithin-e ightupdating,withref- erence trajectory tracking. Mease and Kremer 2 revisited the Shuttle tracking law derivation in the framework of feedback linearization andshowedthattheShuttlelawisa linearized version ofthepropor- tional integral derivative (PID) type law that the feedback lineariza- tion approach can produce. By not linearizing the tracking law, the nonlinearities in the dynamics can be compensated for, provided they can be adequately modeled. Roenneke and Well 3 simulated a variety of low-lift re-entry eights and showed that the longitudinal PIDtracking lawwith feedback linearization yields uniformly good tracking performance and effectively compensates for 20% errors in air density and large initial position errors. A related nonlinear tracking law has been developed and applied to RLV guidance by Lu. 4 The entry guidance requirements for future unpowered entry ve- hicleswilllikelybe,astheyarefortheShuttle,tosteerthevehicleon afeasibletrajectory,atrajectorywithintheentrycorridor,dee nedby heating, acceleration, dynamic pressure, and controllability limits, thatachievesthe specieedtargetconditionwithin thespecie ederror margin. Additionalrequirements thatdrive our study are eying over specie edwaypointsandincreasedcrossrangecapability.Theformer may be required toavoid e ying over populated areas; the latter may be required for abort scenarios and to reduce the waiting time for returnfromorbit.TheShuttleentryguidancehandlescrossrangetar- getingbybankreversallogic.Crossrangetargetingtakessecondpri- ority todownrange targeting,and some crossrange capability issac- rie ced asa result. The additional requirements suggest that it would be desirable to extend the Shuttle longitudinal predictive/tracking guidance to longitudinal and lateral predictive/tracking guidance. The entry requirements can be met by a guidance scheme com- prising a rapid trajectory planner that generates a feasible ground