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In this paper, the development of relative guidance and control algorithms for proximity operations to satellite in elliptical orbit are presented. The paper aims to discuss these issues.

The process of autonomous proximity is divided into three phases: proximity manoeuvre, flyaround manoeuvre, and hovering manoeuvre. The characteristics of the three phases are analyzed. Different guidance algorithms are based on using the analytical closed-form solution of the Tschauner-Hempel (TH) equations that is completely explicit in time. Lastly, the linear quadratic regulators control algorithm based on the linearized TH equations is developed to minimize the initial state errors in the last phase.

This paper defines three phases in the satellite proximity operations and develops the guidance and control algorithms. Then, the relative guidance and control algorithms are illustrated through different numerical examples. And the results demonstrate the effectiveness and simplicity of using a TH model in autonomous proximity.

The findings indicate that a TH model is clearly effective at estimating the relative position and velocity and controlling the relative trajectory. In addition, this model is not restricted to a circular orbit, but it can be used as well for an elliptical orbit. Furthermore, by using this model, simple guidance and control algorithms are developed to approach, flyaround and hover from a target satellite.

Based on the guidance algorithms, the manoeuvre-flight period can be set in accordance with the mission requirement. Flyaround with different types of trajectory and a feedback control scheme to achieve stable hovering state are studied. Consequently, this proposed guidance algorithms can effectively implement guidance and control for satellite proximity operations.

The purpose of this paper is to study the effect of J₃ perturbation of the Earth’s oblateness on satellite orbit compared with J₂ perturbation.

Based on the parametric variation method in the time domain, considering more accurate Earth potential function by considering J₃-perturbation effect, the perturbation equations about satellite’s six orbital elements (including semi-major axis, orbit inclination, right ascension of the ascending node, true anomaly, eccentricity and argument of perigee) has been deduced theoretically. The disturbance effects of J₂ and J₃ perturbations on the satellite orbit with different orbit inclinations have been studied numerically.

With the inclination increasing, the maximum of the semi-major axis increases weakly. The difference of inclination disturbed by the J₂ and J₃ perturbation is relative to orbit inclinations. J₃ perturbation has weak effect on the right ascension and argument of perigee. The critical angle of the right ascension and argument of perigee which decides the precession direction is 90° and 63.43°, respectively. The disturbance effects of J₂ and J₃ perturbations on the argument of perigee, right ascension and eccentricity are weakened when the eccentricity increases, simultaneously, the difference of J₂ and J₃ perturbations on argument of perigee, right ascension and argument of perigee decreases with eccentricity increasing, respectively.

In the future, satellites need to orbit the Earth much more precisely for a long period. The J₃ perturbation effect and the weight compared to J₂ perturbation in LEO can provide a theoretical reference for researchers who want to improve the control accuracy of satellite. On the other hand, the theoretical analysis and simulation results can help people to design the satellite orbit to avoid or diminish the disturbance effect of the Earth’s oblateness.

The J₃ perturbation equations of satellite orbit elements are deduced theoretically by using parametric variation method in this paper. Additionally, the comparison studies of J₂ perturbation and J₃ perturbation of the Earth’s oblateness on the satellite orbit with different initial conditions are presented.