## Analysis of a Linear System for Variable-thrust Control in the Terminal Phase of RendezvousA linear system for applying thrust to a ferry vehicle in the 3 terminal phase of rendezvous with a satellite is analyzed. This system requires that the ferry thrust vector per unit mass be variable and equal to a suitable linear combination of the measured position and velocity vectors of the ferry relative to the satellite. The variations of the ferry position, speed, acceleration, and mass ratio are examined for several combinations of the initial conditions and two basic control parameters analogous to the undamped natural frequency and the fraction of critical damping. Upon making a desirable selection of one control parameter and requiring minimum fuel expenditure for given terminal-phase initial conditions, a simplified analysis in one dimension practically fixes the choice of the remaining control parameter. The system can be implemented by an automatic controller or by a pilot. |

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500 statute miles acceleration components altitude of 500 apparent gravitational acceleration azimuth angle Cartesian coordinate system center of earth circular orbit continuously variable thrust control parameter analogous cosh fit critical damping dimensionless distance between satellite earth and axes effect of gravitation equation l8 equation of motion equations 22 expenditure of fuel feet per second ferry mass ferry's motion relative Figure fraction of critical fuel expenditure gravitation Gravitation neglected gravitational effect harmonic oscillator inertial frame initial conditions linear combination LINEAR SYSTEM mass variable NASA TN nonrotating orbital period phase of rendezvous position and velocity reference frame relative motion relative position Rendezvous Problems rocket motors satellite and ferry satellite orbit Satellite Rendezvous sinh fit system of type system with origin Terminal Guidance terminal phase terminal-phase rendezvous systems terminal-phase thrusting thrust acceleration thrust application Trajectory undamped natural frequency unit vectors universal gravitational constant velocity vectors x-j_