Orbital Coordinate System Research Articles

The problem of satellite attitude stabilization in the geomagnetic field

A satellite moving along a circular Keplerian orbit in near-Earth space was explored, focusing on its attitude stabilization in the orbital coordinate system using the intrinsic magnetic and Lorentz force moments. Fluctuations in geomagnetic induction that occur as the satellite orbits cause the coefficients in the dynamical equations governing the satellite’s attitude motion to vary over time. The results show that, although the linearized system of differential equations of the satellite’s motion is non-stationary, it can be reduced to a stationary system of higher order, which holds even for high-precision multipole models of the geomagnetic field. Thus, a control law design was proposed to stabilize the satellite. The controllability of the system was analyzed, and an optimal stabilization algorithm based on the LQR method was developed. The effectiveness of the proposed approach was validated by computer modeling.

Powered–coasting–powered multi-phase mission reconfiguration method for launch vehicles under power system failures

For the final stage of a launch vehicle with powered–coasting–powered multi-phase flight, this paper investigates the joint reconfiguration method to determine subsequent flight trajectory and degraded orbit after power system failure in the first powered phase. While power system failures due to mass–flow–rate drop have been thoroughly studied in the existing literature, this paper focuses on the more challenging specific impulse drop failures, which result in loss of total launcher energy. This paper aims to design a fault-tolerant guidance method with onboard potential and to analyze the survivability and characteristics of the launcher in the event of such a failure. In the problem modeling, the strong nonlinearity of the orbital elements as terminal constraints is eliminated by analyzing the orbital properties in the orbital coordinate system to establish the primal trajectory optimization problem. For this multi-phase trajectory optimization problem, a joint optimization algorithm for the degradation orbit and the flight trajectory, based on convex optimization and Radau pseudospectral method, is designed to achieve autonomous mission reconfiguration. The algorithm introduces the relaxation–penalization idea, where the terminal constraints are relaxed and penalized in the performance index, allowing the algorithm to adaptively form an optimal degradation circular or elliptical orbit based on the remaining capacity of the launch vehicle. Simulation experiments demonstrate the effectiveness and superiority of the mission reconfiguration algorithm. Finally, the paper presents the first analysis of the optimal allocation of flight time between the first powered phase, the coasting phase, and the second powered phase in a propellant depletion scenario following the launcher power system failure.

On renormalization of the approximate solution of the orbital coordinate system equations of orientation

On renormalization of the approximate solution of the orbital coordinate system equations of orientation

A 3D Parametric Venusian Bow Shock Model with the Effects of Mach Number and Interplanetary Magnetic Field

Using global magnetohydrodynamics simulations, we have developed a three-dimensional parametric model for the Venusian bow shock based on a generalized conic section function defined by six parameters, with the effects of the solar wind magnetosonic Mach number (M MS) and the interplanetary magnetic field (IMF) involved. The parametric model’s results reveal the following findings: (1) The size of the Venusian bow shock is primarily determined by M MS. An increase in M MS results in the bow shock moving closer to Venus and a reduction in its flaring angle. (2) Both the subsolar standoff distance and the bow shock’s flaring angle increase with the strength of the IMF components that are perpendicular to the solar wind flow direction (B Y and B Z in the Venus-centered solar orbital coordinate system), whereas the parallel IMF component (B X ) has a limited impact on the subsolar standoff distance but affects the flaring angle. (3) The cross section of the bow shock is elongated in the direction perpendicular to the IMF on the Y–Z plane, and the elongation degree is enhanced with increasing intensities of B Y and B Z . (4) The quasi-parallel bow shock locates closer to the planet as compared to the quasi-perpendicular bow shock. These findings are in alignment with prior empirical and theoretical models. The influences of M MS and IMF on the bow shock’s position and geometry are attributed to the propagation of fast magnetosonic waves, showing the nature of the formation of a collisionless bow shock under the interaction of magnetized flow with an atmospheric object.

On control of spacecraft orientation to the ground data acquisition station

The article dwells on the spacecraft attitude control to point the onboard antenna to the ground data acquisition station during the communication session. Antenna is fixed relative to the spacecraft body. Pur-pose of the antenna is to receive the flight task aboard the spacecraft and to downlink the telemetry infor-mation. When orbiting, the spacecraft position relative to the ground data acquisition station changes contin-uously. It is due to the diurnal rotation of the Earth, spacecraft orbital motion and angular motion of the spacecraft relative to the center of mass under the impact of the disturbing and control moments. To tilt the spacecraft uses reaction wheels, installed in axes of coordinate system coupled with spacecraft center of mass. Electromagnets are used to unload the reaction wheels. The reaction wheels control law is suggested, which tilts the spacecraft to point the antenna to the ground data acquisition station. Mathematical model of the spacecraft dynamics relative to center of mass is given, using the suggested reaction wheels control law. The following external disturbing moments, acting on the spacecraft in flight, are taken into consideration: gravitational, magnetic, aerodynamic moments and solar radiation moment of forces. Dipole model of the magnetic field of the Earth is used to calculate the magnetic moments. Software was developed and space-craft dynamics was simulated on the personal computer with the specified initial data. Simulation initial con-ditions correspond to the attitude control mode of the spacecraft relative to the orbital coordinate system with the specified accuracy. Simulation results verify the applicability of the suggested reaction wheel control law. Key words: electrical axis of the antenna, mathematical model, coordinate system, transformation matrix, vector.

Research for Optimal Programs for Controlling the Relative Motion of a Spacecraft with Limited Thrust

The problem of optimal control of the relative motion of a spacecraft with a finite thrust engine in arbitrary circumcircular orbits using the Pontryagin’s maximum principle is considered. Motion is studied in an orbital cylindrical coordinate system, using variables written in the form of secular and periodic components of relative motion in the orbital plane. The main attention is paid to the analysis of the optimal control structure with a free and transversal orientation of the thrust vector in the presence of passive areas on the trajectory. As a criterion for choosing the optimal control, the motor operating time of the corrective motors was considered. Characteristic control structures for various areas of initial driving conditions have been determined, and estimates of the marginal costs of motor time have been obtained.

Interplanetary Magnetic Field Effect on the Location of the Martian Bow Shock: MAVEN Observations

The Martian bow shock (BS) is generated with the mass-loading and magnetic pileup processes when the solar wind interacts with the Martian ionosphere. In this vein, the interplanetary magnetic field (IMF) frozen in the solar wind can affect the location of the Martian BS, which is less reported. Based on the Mars Atmosphere and Volatile EvolutioN (MAVEN) mission, we manually identify 10,283 BS crossings during a period of the gradually declining solar cycle phase (2014 October–2020 December) and investigate the effects of the intensity and orientation of the IMF on the Martian BS. In the Mars Solar Orbital coordinate system, our results show the following: (1) The Martian BS, including the subsolar and flank regions, linearly moves away from Mars when the IMF intensity increases, which confirms the theoretical and the MHD simulation results. (2) Under the radial IMF condition, we first demonstrate that the subsolar and flank regions of the Martian BS are situated closer to Mars compared to other IMF situations. This might be caused by the weaker magnetic pileup process and the “low-pressure magnetosheath” model under the radial IMF condition. (3) Moreover, the cross section of the Martian BS is elongated in the north–south direction when the Y component of the IMF is dominant, which is on account of the fast magnetosonic speed effect and verifies the elongation phenomenon of the terrestrial BS. The IMF intensity and orientation effects cannot be ignored and should be considered in future models of the Martian BS.

Robust control of a hub-spoke tethered formation system of microsatellites using Hamilton-Jacobi inequality

The problem of controlling a rotating hub-spoke tethered formation system in low Earth orbit is considered, in which microsatellites are located radially around the central spacecraft (hub) and connected to it by tethers (spokes). To analyze the dynamics of the tethered system, a mathematical model is developed in the orbital coordinate system by Lagrange method, in which the central spacecraft is regarded as a rigid body. In the proposed control scheme, the spin motion of the central body is regulated by the control moment, and tether deployment control law is proposed by robust approach, which is carried out by regulating the tether tensions and low thrusts acting on the microsatellites. The robustness and stability of the system are investigated using Lyapunov theory and Hamilton-Jacobi inequality, which is used to determine the robustness index of the control system. The results of numerical calculations are presented, which confirm that the proposed control scheme is effective when taking into account periodic gravitational perturbations, external perturbations and perturbations associated with uncertainty in the initial states of the system and with the rotation of the central body.

Formation of a rotating ring-shaped three-body tethered nanosatellite system with limited control

The problem of forming a rotating ring-shaped tethered system consisting of three nanosatellites is considered. To analyze the dynamics of the tether system, a mathematical model is developed in the orbital coordinate system using the Lagrange method. Using the sliding mode control method, two control programs for the deployment of tethers are proposed, in which tether tensions and thrust forces created by low-thrust engines are used as controls. In the first control program, the control actions are directly limited by the permissible limits of tether tensions forces and thrust forces, and when designing the second control program, an auxiliary dynamic system is added into the control system, which introduces control corrections that take into account the saturation effect. The stability of motion of the tethered formation system for both control programs is investigated using the Lyapunov theory. The results of numerical simulations confirmed the possibility of using the proposed control programs for the formation of a rotating triangular tethered system in the form of a regular triangle in the presence of disturbances and with account of the given constraints.

A mission planning method for sunlight avoidance period of GEO optical remote sensing satellite

GEO optical remote sensing satellites need to avoid sunlight entering the camera optical system at midnight hours. For the existing evasion methods such as thermal doors and posture planning, nearly 15% of the satellite in-orbit life cannot be reasonably utilized due to the passive evasion methods. To solve this problem, a method based on mission planning is proposed. When evasion begins, the satellite changes the orbital coordinate system into the inertial coordinate system and maintains the safe attitude in inertial space. If space missions are planned such as camera calibrations and celestial bodies observations, the algorithm can determine the compliance of the missions with attitude constraints and energy constraints, and the eligible missions are executed. When evasion ends, the satellite maneuvers toward the Earth proceeding with Earth observation missions. The mathematical simulation indicates the method is feasible, ensuring the safety of the optical camera and executing the space missions meanwhile. The mission planning method extends relatively the service life of the satellite and makes the expansion of the space observation field possible.

Genetic Algorithm of Energy Consumption Optimization for Reorientation of the Spacecraft Orbital Plane

The paper is dedicated to the problem of finding optimal spacecraft trajectories. The equations of spacecraft motion are written in quaternion form. The spacecraft moves on its orbit under acceleration from the limited in magnitude jet thrust. It is necessary to minimize the energy costs for the process of reorientation of the spacecraft orbital plane. The equations of spacecraft motion are written in orbital coordinate system. It is assumed that spacecraft orbit is circular and control has constant value on each part of active spacecraft motion. In this case the lengths of the sections of the spacecraft motion are unknown. We need to find the length of each section, their quantity and value of control on each section. The equations of the problem were written in dimensionless form. It simplifies the numerical investigation of the obtained problem. There is a characteristic dimensionless parameter in the phase equations of the problem. This parameter is a combination of dimension variables describing the spacecraft and its orbit. Usually the problems of spaceflight mechanic are solved with the maximum principle. And we have to solve boundary value problems with some kind of shooting method (Newton’s method, gradient descent method etc.) Each shooting method requires initial values of conjugate variables, but we have no analytical formulas to find them. In this paper spacecraft flight trajectories were found with new genetic algorithm. Each gene contains additional parameter which equals to " True" , if the gene forms the control and equals to "False" otherwise. It helps us determine the quantity of spacecraft active motion parts. The input of proposed algorithm does not contain information about conjugate variables. It is well-known that the differential equations of the problem have a partial solution when the spacecraft orbit is circular and control is constant. The genetic algorithm involves this partial solution and its speed is increased. Numerical examples were constructed for two cases: when the difference between angular variables for start and final orientations of the spacecraft orbital plane equals to a few (or tens of) degrees. Final orientation of the spacecraft plane of orbit coincides with GLONASS orbital plane. The graphs of components of the quaternion of orientation of the orbital coordinate system, the longitude of the ascending node, the orbit inclination and optimal control are drawn. Tables were constructed showing the dependence of the value of the quality functional and the time spent on the reorientation of the orbital plane on the maximum length of the active section of motion.

High-Precision Angle Prediction of Sun for Space Remote Sensing Instrument

In order to grasp the timing of sun calibration in advance, this paper introduces a high-precision method to predict the solar angle by using the current broadcast time and orbital instantaneous root of the satellite platform. By calculating the sun’s apparent right ascension and apparent declination, the conversion matrix from the geocentric inertial coordinate system to the orbital coordinate system, and the satellite attitude correction matrix, the sun vector in the satellite body coordinate system is obtained. This method is used to predict the sun angle of a sun synchronous orbit in the satellite coordinate system, and the prediction results are compared with the STK simulation results. The results show that the sun angle prediction error of this method is less than ±0.003°. It can meet the requirements of on-orbit solar calibration. The main error sources in the prediction method are analysed.

Control of a service satellite during its mission on space debris removal from orbits with high inclination by implementation of an ion beam method

The largest population of space debris is observed in the low-Earth-orbit region with altitudes of 600–1000 km and inclinations of 80–100°. As follows from the predictive calculations, space debris removal from this region exactly is of the first priority. For this reason, our article discusses the problems of controlling the motion of service spacecraft – space debris object cluster when the space debris object is removed from a low orbit with high inclination. In the conceptual design of the spacecraft presented hereinafter, the electric propulsion system contains two electric propulsion thrusters of stationary plasma thruster type, each of which is mounted with the possibility of independent rotation respect to two mutually perpendicular directions. An ion beam injector forms an ion beam with 65 mN of thrust. Spacecraft motion control, including the thrust vector control, is provided by coordinated rotation of thrusters. For such concept, a strategy and an algorithm for controlling the motion of the spacecraft center of mass are proposed, which are based on the control of the spacecraft lateral and longitudinal motion that is understood as the control of the relative distance between the spacecraft and the space debris object. The strategy of spacecraft lateral motion control assumes that due to the thrusters' rotation the spacecraft should be displaced in such a way that the vector of the relative range between the spacecraft and the space debris object coincides with the direction of the transversal of the orbital coordinate system. For the practical implementation of the control, an algorithm has been developed for controlling the thrusters' rotation to create the required values of the spacecraft acceleration projections. The mathematical model includes the equations of motion of the spacecraft center of mass taking into account the aerodynamic deceleration. The solar panels' orientation to the Sun is provided by the spacecraft roll rotation relative to its longitudinal axis and turning the solar panels relative to their axis of rotation. The change in the ion beam momentum transfer is considered using a simple simulation model. Mathematical modeling was performed for the following parameters of the initial orbit: altitude: 800 km; eccentricity: 0.0005; and inclination: 900. For the proposed control algorithm, a range of possible thrust of a single thruster was 51–60 mN. The simulation results show that the proposed control can be used to remove space debris object from low orbits with high inclination, taking into account the roll spacecraft rotation to keep the solar panels’ orientation to the Sun.

Algorithm for solving optimal open-loop terminal control problem for spacecraft rendezvous with account of constraints on the state

The paper proposes an algorithm for solving the optimal open-loop terminal control problem of two spacecraft rendezvous with constraints on their states. A system of nonlinear differential equations that describes the dynamics of the active (maneuvering) spacecraft relative to the passive spacecraft (station) in the central gravitational field of the Earth in the orbital coordinate system of coordinates related to the passive spacecraft center-of-mass is considered as an initial model. The obtained nonlinear model of the active spacecraft dynamics is linearized relative to the specified reference state trajectory of the passive spacecraft, and then it is discretized and reduced to linear recurrence relations. Mathematical formalization of the spacecraft rendezvous problem under consideration is carried out at a specified final moment of time for the obtained discrete-time controlled dynamical system. The quality of solving the problem is estimated by a convex functional taking into account the geometric constraints on the active spacecraft states and the associated control actions in the form of convex polyhedral-compacts in the appropriate finite dimensional vector space. We propose a solution of the problem of optimal terminal control over the approach of the active spacecraft relative to the passive spacecraft in the form of a constructive algorithm on the basis of the general recursive algebraic method for constructing the availability domains of linear discrete controlled dynamic systems, taking into account specified conditions and constraints, as well as using the methods of direct and inverse constructions. In the final part of the paper, the computer modeling results are presented and conclusions about the effectiveness of the proposed algorithm are made.

Symbolic-Analytic Methods for Studying Equilibrium Orientations of a Satellite on a Circular Orbit

This paper proposes two simple methods for determining equilibrium orientations of a satellite moving in a central Newtonian force field along a circular orbit under the action of the gravitational torque. The first method uses linear algebra approaches, while the second one employs computer algebra algorithms. The equilibrium orientations of the satellite in the orbital coordinate system for given values of principal central moments of inertia are determined by the roots of a system of nonlinear algebraic equations. To determine the equilibrium solutions, the system of algebraic equations is decomposed using linear algebra methods and algorithms for Grobner basis construction. The equilibria of the satellite are determined by analyzing the real roots of the algebraic equations from the Grobner bases constructed. Using the proposed approach, it is shown that the satellite with unequal principal central moments of inertia has 24 equilibrium orientations on a circular orbit.

Monoaxial Electrodynamic Stabilization of an Artificial Earth Satellite in the Orbital Coordinate System via Control With Distributed Delay

An artificial Earth satellite (AES) with three different principal central moments of inertia is under consideration. The AES moves along a Keplerian circular equatorial near-Earth orbit. The AES is equipped with electrodynamic attitude control system that simultaneously generates Lorentz and magnetic control torques. The possibility of using an electrodynamic attitude control system for monoaxial attitude stabilization of AES in the orbital coordinate system is analyzed. The development of the concept of electrodynamic attitude control, including the use of a restoring torque with a distributed delay (integral term), is proposed. The conditions are found under which the electromagnetic attitude control system with distributed delay solves the problem of AES monoaxial stabilization in the presence of the disturbing gravitational torque. In a nonlinear formulation, sufficient conditions for the asymptotic stability of the AES equilibrium position are obtained. A theorem on the asymptotic stability of the AES programmed attitude motion is proved. The effectiveness of the constructed attitude control with a distributed delay is confirmed by numerical modeling.

Averaging Method in the Problem of the Lorentz Stabilization of the Indirect Equilibrium Position of a Satellite in the Orbital Coordinate System

A dynamically symmetric satellite in a circular orbit of small inclination is considered. The problem of the Lorentzian stabilization of the satellite in the orbital coordinate system in the indirect equilibrium position is solved under the conditions of the perturbing effect of the gravitational torque. To solve this problem, which is characterized by incomplete control, a method of averaging differential equations is developed. Using the original construction of the unsteady Lyapunov function, we obtain sufficient conditions for the asymptotic stability of the programmed satellite motion mode in the form of constructive inequalities with respect to the control parameters.

Averaging technique in the problem of satellite attitude stabilization in indirect position in the orbital reference frame with the use of Lorentz torque

A dynamically symmetric satellite in a circular orbit of small inclination is considered. The problem of the Lorentzian stabilization of the satellite in the orbital coordinate system in the indirect equilibrium position is solved under the conditions of the perturbing effect of the gravitational torque. To solve this problem, which is characterized by incomplete control, a method of averaging differential equations is developed. Using the original construction of the unsteady Lyapunov function, we obtain sufficient conditions for the asymptotic stability of the programmed satellite motion mode in the form of constructive inequalities with respect to the control parameters.

Influence of external force fields on the evolution of space debris clouds

In this paper, the evolution of a cloud of interacting particles in their orbital motion around the Earth is considered.In numerical simulations in a non-inertial orbital coordinate system, there is a large difference in the orders of the quantities used, which prevents calculations. To prevent this from happening, the equations of motion of particles are written in a special form.The proposed research algorithm allows one to obtain the relative configurations of the clouds of particles with allowance for various perturbations and with different required computational accuracy.

Mathematical Simulation of Satellite Motion with an Aerodynamic Attitude Control System Influenced by Active Damping Torques

The dynamics of a satellite moving in a central Newtonian force field in a circular orbit under the influence of aerodynamic and active damping torques depending on projections of the satellite’s angular velocity is studied. A method for determining all equilibrium positions (equilibrium orientations) of the satellite in the orbital coordinate system given the values of aerodynamic torque, damping coefficients, and the principal central moments of inertia is proposed. In the case when the axes of the coordinate system attached to the satellite coincide with the axes of the orbital coordinate system, necessary and sufficient conditions for the asymptotic stability of the corresponding zero equilibrium position are obtained using the Routh–Hurwitz criterion. The domains with satisfied asymptotic stability conditions for the zero equilibrium position are analyzed depending on various dimensionless parameters of the problem. The damping of spatial oscillations of the satellite is numerically studied for various values of aerodynamic torque and damping coefficients.