Gravity Gradient Torque Research Articles

Attitude stability of a spacecraft on a stationary orbit around an asteroid subjected to gravity gradient torque

Attitude stability of spacecraft subjected to the gravity gradient torque in a central gravity field has been one of the most fundamental problems in space engineering since the beginning of the space age. Over the last two decades, the interest in asteroid missions for scientific exploration and near-Earth object hazard mitigation is increasing. In this paper, the problem of attitude stability is generalized to a rigid spacecraft on a stationary orbit around a uniformly-rotating asteroid. This generalized problem is studied via the linearized equations of motion, in which the harmonic coefficients \(C_{20}\) and \(C_{22}\) of the gravity field of the asteroid are considered. The necessary conditions of stability of this conservative system are investigated in detail with respect to three important parameters of the asteroid, which include the harmonic coefficients \(C_{20}\) and \(C_{22}\), as well as the ratio of the mean radius to the radius of the stationary orbit. We find that, due to the significantly non-spherical shape and the rapid rotation of the asteroid, the attitude stability domain is modified significantly in comparison with the classical stability domain predicted by the Beletskii–DeBra–Delp method on a circular orbit in a central gravity field. Especially, when the spacecraft is located on the intermediate-moment principal axis of the asteroid, the stability domain can be totally different from the classical stability domain. Our results are useful for the design of attitude control system in the future asteroid missions.

Attitude Control and Momentum Management of Inertially Oriented Space Station

Manned Space Station features large size and suffers large environmental disturbance torque. To maintain space station attitude stable for long-time without frequent control moment gyros (CMGs) saturation interruption, an attitude and momentum management controller in inertial reference is designed. Firstly the space station attitude dynamics, angular momentum dynamics of CMGs, and aerodynamics are modeled, then the characteristics of gravity gradient torque for inertially oriented space station is analyzed. Subsequently the system dynamics is augmented with the filter states, and Linear Quadratic Regulator (LQR) method is used to design the feedback gain matrix. By rational setting of gain matrix of filter states, the space station can maintain inertially oriented stable without CMGs disaturation. Finally one middle-stage configuration of the Chinese Space Station is demonstrated as an example to validate the design, simulation results show that the angular momentum resulted from gravity gradient torque and aerodynamic torque does not accumulate respectively.

Analytical Solution of the Perturbed Oribt-Attitude Motion of a Charged Spacecraft in the Geomagnetic Field

In this work we investigate the orbit-attitude perturbations of a rigid spacecraft due to the effects of several forces and torques. The spacecraft is assumed to be of a cylindrical shape and equipped with a charged screen with charge density s. Clearly the main force affecting the motion of the spacecraft is the gravitational force of the Earth with uniform spherical mass. The effect of oblate Earth up to J2 is considered as perturbation on both the orbit and attitude of the spacecraft, where the attitude of the spacecraft is acted upon by what is called gravity gradient torque. Another source of perturbation on the attitude of the spacecraft comes from the motion of the charged spacecraft in the geomagnetic field. This motion generates a force known as the Lorentz force which is the source of the Lorentz force torque influencing the rotational motion of the spacecraft. In this work we give an analytical treatment of the orbital-rotational dynamics of the spacecraft. We first use the definitions of Delaunay and Andoyer variables in order to formulate the Hamiltonian of the orbit-attitude motion under the effects of forces and torques of interest. Since the Lorentz force is a non-conservative force, a potential like function is introduced and added to the Hamiltonian. We solve the canonical equations of the Hamiltonian system by successive transformations using a technique proposed by Lie and modified by Deprit and Kamel to solve the problem. In this technique we make two successive transformations to eliminate the short and long periodic terms from the Hamiltonian.

Optimal Reconfigurations of Two-Craft Coulomb Formation in Circular Orbits

Optimal reconfigurations of a two-spacecraft Coulomb formation are determined by applying nonlinear optimal control techniques. The objective of these reconfigurations is to maneuver the two-craft formation between two charged equilibria configurations. The four optimality criteria considered are minimum time, minimum acceleration of the separation distance, minimum Coulomb and electric propulsion fuel usage, and minimum electrical power consumption. The reconfiguration between equilibra is first considered by varying the desired separation distance. In a radial relative equilibrium configuration, only the Coulomb force is required to control the in-plane motion and to steer the satellites from their initial to their final radial position. In this reconfiguration maneuver, the gravity gradient torque is exploited to stabilize the in-plane motion. For along-track and orbit normal equilibrium locations, the reconfiguration maneuver requires hybrid controls. Here the Coulomb force is varied to control the separation distance and inertial micro-thrusters are activated for control in the transverse directions. Second, a reconfiguration involving hybrid control is used to maneuver the crafts from any initial equilibrium position to a final one. The goal is to determine optimal maneuvers maximizing the use of Coulomb propulsion while minimizing the electric propulsion usage. The two-point boundary value problem optimization formulation is numerically solved via pseudo-spectral methods. Pontryagin’s Minimum Principle verifies the open loop solutions’ optimality.

Open-loop and closed-loop attitude dynamics and controls of miniature spacecraft using pseudowheels

This paper presents open-loop and closed-loop attitude control strategies of miniature spacecraft using pseudowheels, which are composed of multiple bimorphs and are mounted on the three principal axes of the spacecraft. Due to the conservation of angular momentum, the vibrations of the bimorph excited by applied voltages can rotate the spacecraft. By applying a sequence of rotations along the three axes, the attitude of the spacecraft can be changed. The modeling of the spacecraft–pseudowheel system includes the inverse piezoelectric effect and the gravity gradient torques. Since one cannot guarantee that a solution will be found for every rotation sequence, this paper proposes an approach to solve this problem. In addition, the voltages applied to the bimorphs are reduced to meet the specifications of the voltage amplifiers. Furthermore, a closed-loop attitude control strategy is presented to reduce the applied voltages and to compensate the attitude error caused by the gravity gradient torques.

Study of Stability of Rotational Motion of Spacecraft with Canonical Variables

This work aims to analyze the stability of the rotational motion of artificial satellites in circular orbit with the influence of gravity gradient torque, using the Andoyer variables. The used method in this paper to analyze stability is the Kovalev‐Savchenko theorem. This method requires the reduction of the Hamiltonian in its normal form up to fourth order by means of canonical transformations around equilibrium points. The coefficients of the normal Hamiltonian are indispensable in the study of nonlinear stability of its equilibrium points according to the three established conditions in the theorem. Some physical and orbital data of real satellites were used in the numerical simulations. In comparison with previous work, the results show a greater number of equilibrium points and an optimization in the algorithm to determine the normal form and stability analysis. The results of this paper can directly contribute in maintaining the attitude of artificial satellites.

CHAOTIC PITCH MOTION OF A MAGNETIC SPACECRAFT WITH VISCOUS DRAG IN AN ELLIPTICAL POLAR ORBIT

We study the pitch attitude dynamics of an asymmetric magnetic spacecraft in an almost circular orbit under the influence of a gravity gradient torque. The spacecraft is also subject to the influence of three perturbations: the small eccentricity of the elliptical orbit, a small magnetic torque due to the interaction with the Earth's magnetic field, and a small aerodynamic viscous drag generated by the action of the Earth's atmosphere. Under these perturbations, we show that the pitch motion exhibits heteroclinic chaotic behavior by means of the Melnikov method. This method gives us an analytical criterion for the existence of heteroclinic chaos in terms of the system parameters. This analytical criterion is confirmed numerically with good agreement. In spite of the chaos generated by the perturbations, we also find, by means of Poincaré surfaces of section that some periodic pitch motions persist in the perturbed system with the same period as the orbital motion of the spacecraft. Finally, we carry out a bifurcation analysis of these periodic motions by numerical continuation of them in terms of the perturbation parameters.

Cubesat solar sail 3-axis stabilization using panel translation and magnetic torquing

A Cubesat mission with a deployable solar sail of 5 meter by 5 meter in a sun-synchronous low earth orbit is analyzed to demonstrate solar sailing using active attitude stabilization of the sail panel. The sail panel is kept parallel to the orbital plane to minimize aerodynamic drag and optimize the orbit inclination change caused by the solar pressure force normal to the sail surface. A practical control system is proposed, using a combination of small 2-dimensional translation of the sail panel and 3-axis magnetic torquing which is proved to have sufficient control authority over the gravity gradient and aerodynamic disturbance torques. Miniaturized CMOS cameras are used as sun and nadir vector attitude sensors and a robust Kalman filter is used to accurately estimate the inertially referenced body rates from only the sun vector measurements. It is shown through realistic simulation tests that the proposed control system, although inactive during eclipse, will be able to stabilize the sail panel to within ± 2 ° in all attitude angles during the sunlit part of the orbit, when solar sailing is possible.

Fault Tolerant Attitude Synchronization Control during Formation Flying

It is necessary for precise satellite formations to keep each satellite in accurate relative attitude synchronization and to maintain relative angular velocity synchronization. An adaptive fuzzy terminal sliding mode control is proposed for a formation flying tracking system which includes acquisition, tracking, and pointing. The relative attitude of the leader and follower spacecraft under gravity gradient torque and external periodic disturbances is controlled by four reaction wheels. First, a terminal sliding mode controller is used to obtain stable and fast tracking. Then, the adaptive fuzzy logic system is used to approximate and learn the system uncertainty with the online fault and time-varied external disturbances. The system is proved to be stable, and the parameters are proved to be bounded under the control scheme, using a Lyapunov methodology. Finally, simulation results (under actuator faults like wheel failure and wheel degradation) show that the proposed control method has the advantages of...

Quaternion based model for momentum biased nadir pointing spacecraft

In this paper, we propose a quaternion based spacecraft model that describes the nadir pointing spacecraft with momentum wheel under gravity gradient torque disturbance. This state space model uses only three components of the quaternion. From this nonlinear model, we derive a linearized state space model for the spacecraft system. We show that unlike all existing quaternion models, this linearized state space model is fully controllable. Therefore, all modern control system design methods can be directly applied to the attitude control system design for momentum biased nadir pointing spacecraft.

Using a MEMS pendulum to measure the gravity gradient in orbit: a new concept for a miniaturized Earth sensor

We present the fabrication and test results of a novel inertial sensor for use onboard satellites, to obtain the Earth vector. Current state-of-the-art Earth sensors determine the Earth vector by imaging the Earth’s horizon in the IR. This requires multiple optical heads on different faces of the satellite, with associated mounting and thermal considerations. The MEMS-based approach reported here is based on measuring the gravity gradient vector by measuring the gravity gradient torque on a 4 cm long Si-pendulum. This approach eliminates the need for multiple external access ports, allowing a compact sensor to be situated anywhere inside the spacecraft.

Attitude stability of artificial satellites subject to gravity gradient torque

The stability of the rotational motion of artificial satellites is analyzed considering perturbations due to the gravity gradient torque, using a canonical formulation, and Andoyer’s variables to describe the rotational motion. The stability criteria employed requires the reduction of the Hamiltonian to a normal form around the stable equilibrium points. These points are determined through a numerical study of the Hamilton’s equations of motion and linear study of their stability. Subsequently a canonical linear transformation is used to diagonalize the matrix associated to the linear part of the system resulting in a normalized quadratic Hamiltonian. A semi-analytic process of normalization based on Lie–Hori algorithm is applied to obtain the Hamiltonian normalized up to the fourth order. Lyapunov stability of the equilibrium point is performed using Kovalev and Savchenko’s theorem. This semi-analytical approach was applied considering some data sets of hypothetical satellites, and only a few cases of stable motion were observed. This work can directly be useful for the satellite maintenance under the attitude stability requirements scenario.

Non-certainty-equivalent adaptive satellite attitude control using solar radiation pressure

The article presents a new adaptive controller for attitude control of satellites using solar radiation pressure (SRP). The satellite has two identical solar flaps for pitch angle control. The model includes gravity gradient torque and the torque due to SRP. Unlike adaptive control design published in the literature for this model, here an adaptive law for large pitch angle trajectory control based on the non-certainty-equivalence principle is derived. The control system has a modular structure consisting of an estimator and a control module designed based on a backstepping method. Using Lyapunov analysis, it is shown that the closed-loop system is stable and the pitch angle trajectory asymptotically follows the reference trajectory. The control system designed here has some special features. In the closed-loop system, the trajectory is ultimately confined to a manifold in the space of state variables and parameter errors. Interestingly, once the estimated parameter values coincide with their true values during the adaptation process, these remain frozen and the estimation error remains zero thereafter, and the system recovers the performance of the deterministic controller. Simulation results are presented for a set of uncertain parameters of the model. These results show that in the closed-loop system, precise large pitch angle control is accomplished despite large uncertainties in the system parameters.

Solar Sail Dynamics and Coning Control in Circular Orbits

Attitude dynamics of a solar sail are described relative to an orbiting local-vertical/local-horizontal frame under the influence of aerodynamic, solar, and gravity-gradient torques. The complete set of equilibria of the sail normal vector in local-vertical/local-horizontal is described as a function of sailcraft spin rate and orbit and sailcraft design parameters. These equilibria enable orbit-rate coning of the sail normal that is useful for changing orbit parameters using solar thrust. Lyapunov stability properties of the equilibria are analyzed, providing conditions for stable equilibria. Feedback control is considered for stabilization and additional damping, enabling any of the equilibria to be asymptotically stabilized. This approach enables long-term thrust-vector control in the presence of large environmental torques for significant orbit-parameter variation over time. An example of a 3 kg microsailcraft is used to illustrate the equilibrium and stability conditions and the performance of the attitude control.

Unscented KALMAN Filtering for Spacecraft Attitude and Rate Determination Using Magnetometer

An Unscented Kalman Filter (UKF) for estimation of the attitude and rate of a spacecraft using only magnetometer vector measurement is developed. The attitude dynamics used in the estimation is the nonlinear Euler’s rotational equation which is augmented with the quaternion kinematics to construct a process model. The filter is designed for small satellite in low Earth orbit, so the disturbance torques include gravity-gradient torque, magnetic disturbance torque, and aerodynamic drag torque. The magnetometer measurements are simulated based on time-varying position of the spacecraft. The filter has been tested not only in the standby mode but also in the detumbling mode. Two types of actuators have been modeled and applied in the simulation. The PD controller is used for the two types of actuators (reaction wheels and thrusters) to detumble the spacecraft. The estimation error converged to within 5 deg for attitude and 0.1 deg/s for rate respectively when the two types of actuators were used. A joint state parameter estimation has been tested and the effect of the process noise covariance on the parameter estimation has been indicated. Also, Monte-Carlo simulations have been performed to test the capability of the filter to converge with the initial conditions sampled from a uniform distribution. Finally, the UKF performance has been compared to that of the EKF and it demonstrates that UKF slightly outperforms EKF. The developed algorithm can be applied to any type of small satellites that are actuated by magnetic torquers, reaction wheels or thrusters with a capability of magnetometer vector measurements for attitude and rate estimation.

Satellite attitude dynamics and estimation with the implicit midpoint method

We describe the application of the implicit midpoint integrator to the problem of attitude dynamics for low-altitude satellites without the use of quaternions. Initially, we consider the satellite to rotate without external torques applied to it. We compare the numerical solution with the exact solution in terms of Jacobi’s elliptic functions. Then, we include the gravity-gradient torque, where the implicit midpoint integrator proves to be a fast, simple and accurate method. Higher-order versions of the implicit midpoint scheme are compared to Gauss–Legendre Runge–Kutta methods in terms of accuracy and processing time. Finally, we investigate the performance of a parameter-adaptive Kalman filter based on the implicit midpoint integrator for the determination of the principal moments of inertia through observations.

Chaos and its control in the pitch motion of an asymmetric magnetic spacecraft in polar elliptic orbit

We study the pitch attitude dynamics of an asymmetric magnetic spacecraft in a polar almost circular orbit under the influence of a gravity gradient torque. The spacecraft is perturbed by the small eccentricity of the elliptic orbit and by a small magnetic torque generated by the interaction between the Earth’s magnetic field and the magnetic moment of the spacecraft. Under both perturbations, we show that the pitch motion exhibits heteroclinic chaotic behavior by means of the Melnikov method. Numerical methods applied to simulations of the pitch motion also confirm the chaotic character of the spacecraft attitude dynamics. Finally, a linear time-delay feedback method for controlling chaos is applied to the governing equations of the spacecraft pitch motion in order to remove the chaotic character of initially irregular attitude motions and transform them into periodic ones.

Analysis of gravity-gradient-perturbed rotational dynamics at the collinear lagrange points

This paper studies the dynamics and stability of a rigid spacecraft subjected to gravity gradient torques exerted by the Sun and the Earth in the circular restricted three-body problem. We focus on the dynamics in a close vicinity to the Lagrangian collinear equilibrium points, and show that the linear stability domain predicted by the Beletskii-DeBra-Delp method in the two-body problem is modified due to the presence of an additional gravitating primary. The nonlinear differential equations are derived using a Hamiltonian formalism and are subsequently investigated using Poincare maps. The effect of the gravity gradient torque is accentuated using difference Poincare maps. The Melnikov integral method is utilized for studying the chaotic behavior of the gravity-gradient-perturbed system.

Robust Attitude Tracking Control of Spacecraft in the Presence of Disturbances

I T IS well known that spacecraft usually operate in the presence of various disturbances and that their control input usually is limited because of actuator saturation. Therefore, two problems, disturbance rejection and saturation constraint accommodation, should be addressed in attitude control. Both problems have attracted considerable research interest in the existing literature. However, only limited results explicitly dealt with them simultaneously [1–5]. Di Gennaro [1] designed a dynamic controller that can globally asymptotically stabilize the spacecraft in the presence of saturation constraint and gravity gradient torque. However, the rejection of other disturbances is not guaranteed. Boskovic et al. [2,3] designed sliding-mode controllers, which are discontinuous and can cause chattering [6]. To avoid chattering, the boundary layer method was used in [2,3]. However, this method can only guarantee bounded attitude and angular velocity errors. To achieve global asymptotic stability with a smooth controller, Boskovic et al. [4] designed a continuous adaptive tracking controller with a condition necessary for guaranteeing the convergence of the attitude error to zero. However, it is difficult to know in what cases that condition can be satisfied. Moreover, according to [7], there do exist some disturbances in the presence of which that condition cannot be satisfied. Wallsgrove and Akella [5] designed a smooth controller using a hyperbolic tangent function. Although the angular velocity error converges to zero, the convergence of the attitude error to zero is not proved in [5]. Moreover, this controller gives rise to numerical problems in simulations of long duration. This paper proposes a controller without the aforementioned drawbacks. II. Spacecraft Attitude Dynamics

Analytical attitude propagation with non-singular variables and gravity gradient torque for spin stabilized satellite

An analytical approach for the spin stabilized satellite attitude propagation is presented using the non-singular canonical variables to describe the rotational motion. Two sets of variables were introduced for Fukushima in 1994 by a canonical transformation and they are useful when the angle between z-satellite axis of a coordinate system fixed in artificial satellite and the rotational angular momentum vector is zero or when the angle between Z-equatorial axis and rotation angular momentum vector is zero. Analytical solutions for rotational motion equations and torque-free motion are discussed in terms of the elliptic functions and by the application of some simplification to get an approximated solution. These solutions are compared with a numerical solution and the results show a good agreement for many rotation periods. When the mean Hamiltonian associated with the gravity gradient torque is included, an analytical solution is obtained by the application of the successive approximations’ method for the satellite in an elliptical orbit. These solutions show that the magnitude of the rotation angular moment is not affected by the gravity gradient torque but this torque causes linear and periodic variations in the angular variables, long and short periodic variations in Z-equatorial component of the rotation angular moment and short periodic variations in x-satellite component of the rotation angular moment. The goal of this analysis is to emphasize the geometrical and physical meaning of the non-singular variables and to validate the approximated analytical solution for the rotational motion without elliptic functions for a non-symmetrical satellite. The analysis can be applied for spin stabilized satellite and in this case the general solution and the approximated solution are coincidence. Then the results can be used in analysis of the space mission of the Brazilian Satellites.