Three-Dimensional Guidance Law for Landing on a Celestial Object

Abstract

Introduction A S spacevehicletechnologydevelops,the rendezvousof a space vehicle with a space station or a celestial object becomes possible and imminent. Many rendezvousguidance strategies for landing on an asteroid or the docking of two vehicles have been studied recently. The rendezvousproblems of two space vehicles in two dimensionswere presented in Refs. 1 and 2. Under similar dynamical equations, the rendezvous of a space vehicle with an asteroid was discussed in Refs. 3 and 4. However, the existing studies tended to neglect the combined impact of atmosphere and gravity. The Viking Lander descent to Mars through an unknown atmosphere density proŽ le, winds, and terrain characteristics was described in Ref. 5. Autonomous spacecraft navigation and control for a comet landing with model error and the effects of environmental disturbances was analyzed by Monte Carlo simulation in Ref. 6. The rendezvouswith a celestial object usually consists of three successive phases: 1) cruiseor transfer to the vicinityof the celestialbody, 2) approach, and 3) maneuvers near the celestial body. This Note is concerned with maneuver near the celestial body by using the generalized three-dimensionalmodel with the effects of both gravity and atmospheric drag. Moreover, in this Note, the unmodeled perturbing forces such as solar radiation pressure and nonspherical gravitational effects are treated as disturbances. These perturbing forces usually cause periodic variations8 and will be formulated as trigonometric functions. The main goal of this Note is, from a theoretic point of view, to proposea new landingguidancelaw for ensuringthe vehiclelanding on the celestial body in a Ž nite time and satisfying the landing constraint in the terminalphasewhen the spacevehicleapproachesa celestial body. That is, v ! 0 as r ! ra , where ra denotes the celestial radius.By properly selecting the desired trajectory, the landingprocess is transformed into a tracking problem. The variable structure control (VSC) technique is then applied to the design of a tracking control law. By the construction of a time-varying boundary layer, the tracking performance is shown to be achieved at an exponential convergence rate. Moreover, the modiŽ ed guidance law attained is continuous and alleviates the classical chattering drawback of the VSC control scheme. An illustrative example is also presented to demonstrate the use of the results.