Approximately optimal discrete law of spacecraft desecent control with asymmetry in Mars atmosphere

DOI: 10.34759/vst-2022-2-179-188

Аuthors

Bakry I.

Samara National Research University named after Academician S.P. Korolev, 34, Moskovskoye shosse, Samara, 443086, Russia

e-mail: ibrahimbakry0@gmail.com

Abstract

The spacecraft orientation stability these days is of utter importance for both public and private space agencies and companies. The growing interest to the Red Planet increases the number of space missions, which include orbital apparatuses, landers or Mars rovers. Since 1960s up to now, more than forty nine missions were sent to Mars from different countries. The majority of them end in failure, either fly far away from the Mars orbit (did not enter an orbit), crash upon its surface, do not reach the target, or connection is being lost prior to the target reaching. This indirectly indicates errors at the stages of navigation, control, stabilization or design.

The following missions are the example of failed missions to Mars, which are either lost or crashed due to failures in the navigation system, or incorrect orientation. They are 1M, 2M, 2MV, 3MV and 3MS (1960-1971), Mars-1 (1962), Mars-2 lander (1971), Mars-6 and Mars-7 landers (1973), Phobos-1 (1988), Mars Observer (1992), Mars-96 (1996), Mars Polar Lander (1999), Deep Space-2 (1999), Beagle-2 (2003), Yinghuo-1 (2011), Schiaparelli EDM lander (2016).

The presented article considers a dynamic model describing the spacecraft perturbed motion as a rigid body with significant aerodynamic and mass asymmetries relative to the spacecraft center of mass in the rarefied atmosphere of Mars.

The purpose of this work consists in obtaining an approximate discrete optimized control law of a spacecraft attitude employing dynamic programming and averaging methods. The system of quasi-linear equation was considered and averaged to obtain a simpler system of equations, which can be modeled applying the dynamic programming method.

Optimal control laws were determined based on the quadratic optimization criterion by Bellman principle, and, besides, the system of discrete equations, employing analytical Z-transform, reverse Z-transform and numerical discrete Euler method, was developed and solved. Reliability of the obtained analytical control laws is being confirmed by the results of numerical integration by the numerical Euler Method.

Euler method integration was being performed employing fixed and variable integration steps. The results obtained with a variable step appeared to be more exact than those obtained with the fixed step with the Z-transform method. The conversion behavior of both the angle of attack and the angular velocity at comparing them with the found solutions while similar studies for a significant aerodynamic and inertial asymmetry relative to the center of mass come closer to the results of this study.

The numerical results of this work confirm that the obtained approximate discrete expressions for control optimization ensure the in angular velocity and spatial angle of attack reduction to the required small values in a time commensurable with the time from the free movement start of the spacecraft uncontrolled descent to the braking parachute system initializing.

By applying these laws to a lander with asymmetries in both vehicle aerodynamics and mass, the values of angular velocity and the angle of attack will converge to zeros enforcing the stabilization.

The practical significance of the obtained discrete laws of the two-channel control is being confirmed by application of the small jet engines running in discrete mode.

Keywords:

optimal orientation control with Bellman method, discrete orientation control, aerodynamic and mass asymmetry, motion in Marts atmosphere

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