This paper studies libration dynamics and stability of deorbiting nano-satellites by short and ba... more This paper studies libration dynamics and stability of deorbiting nano-satellites by short and bare electrodynamic tethers. A critical aspect of satellite deorbit by an electrodynamic tether is to maintain the tether aligned with the local vertical and stable while subjected to external perturbations. The dynamics of electrodynamic tether system in deorbit application is divided into the orbital motion of the center of system’s mass and the tether libration motion relative to that center. Major space environmental perturbations including the current-induced electrodynamic force, atmospheric drag, oblateness effect of the Earth, irregularity of geomagnetic field, variable plasma density, solar radiation pressure, and lunisolar gravitational attractions are considered in the dynamic analysis. Quantitative analyses are provided in order to characterize the order of the perturbative torques during the deorbit process. A single index is derived from the libration energy to stabilize the libration motion by regulating the current in the tether through simple on-off switching. Numerical results show that the libration dynamics of an electrodynamic tether has significant impacts on the deorbit process and the electrodynamic tether cannot effectively deorbit satellites without libration stability control. The proposed current regulation strategy is simple and very effective in stabilizing libration motion of an electrodynamic tether.
This paper studies the dynamics of nanosatellite deorbit by a bare electrodynamic tether. The orb... more This paper studies the dynamics of nanosatellite deorbit by a bare electrodynamic tether. The orbital dynamics of the tethered nanosatellite is modeled in Gaussian perturbation equations and the motion-induced voltage-current relationship along the electrodynamic tether is analyzed by using the 2000 International Geomagnetic Reference Field model including up to seventh-order terms and the International Reference Ionosphere 2007 model. The analysis reveals that the high-order magnetic model of Earth affects the dynamic characteristics of the tethered nanosatellite, especially in orbits with high inclination angles, by changing its orbit from circular to elliptical forms. This is beneficial for deorbiting the nanosatellite in near-polar orbits where the electrodynamic force is not as effective as in the equatorial orbit because the denser atmosphere at a lower perigee will provide a larger atmospheric drag. Moreover, the analysis shows that the electrodynamic force is always against the satellite motion in low Earth orbit even when the induced voltage/current across the tether reverses their polarities in near-polar orbits. Compared to the deorbit rate by the atmospheric drag only, the deorbit rate by an electrodynamic tether will be increased by several orders in magnitudes in both equatorial and polar orbits.
Electrodynamic tether systems orbiting the Earth are prone to libration instability because of pe... more Electrodynamic tether systems orbiting the Earth are prone to libration instability because of periodic changes in the geomagnetic field, plasma density, and lunisolar gravitational attractions in addition to nonperiodic changes resulting from the irregularity of the geomagnetic field, inhomogeneity of the Earth, and solar pressures. The long-term orbital and libration dynamics of a bare electrodynamic tether in deorbiting obsolete satellites is investigated by considering space environmental perturbations of current-induced electrodynamic force, atmospheric drag, Earth’s oblateness, irregularity of the geomagnetic field, variable space plasma density, solar radiation pressure, and lunisolar gravitational attractions. The electrodynamic tether is assumed to be rigid and the tethered spacecraft is modeled as a lumped mass. The study shows by numerical simulation that the out-of-plane libration is the primary source of libration instability in inclined orbits, which destabilizes the in-plane libration through nonlinear modal coupling. Accordingly, a simple stability criterion for current on/off switching control is derived from the libration energy of the tether to stabilize the out-of-plane libration by limiting the roll angle amplitude to a preset range. This in turn stabilizes the in-plane libration. The control requires only the feedback of the maximum roll angle with a minimum interval for current on/off switching imposed to avoid excessive current switching. The effectiveness of the control strategy has been demonstrated by analyzing the libration dynamics of electrodynamic tether with and without the current regulation in deorbiting satellites. Numerical results show that this approach is very effective in stabilizing both in-plane and out-of-plane libration of a tethered system subjected to periodic and nonperiodic perturbations.
This paper studies the deployment and retrieval of a tethered satellite system that is composed b... more This paper studies the deployment and retrieval of a tethered satellite system that is composed by two satellites connected with a flexible tether. A hybrid hinged-rod is developed based on conventional hinged-rod and bead models. The tether is discretized into rigid rods connected by springs and dampers at hinges and the end-satellites are modeled as rigid spheres. Newton-Euler method is applied to establish equations of motion for each rigid body. By adopting an instantaneity assumption and transfer mechanism between beads and rods, the proposed hybrid hinged-rod model is able to model the tether deployment and retrieval processes while avoiding the difficulty encountered due to dimension variation of rod element in deployment and retrieval. Two tethered satellite systems moving on a circular orbit with different tether lengths are investigated. Simulation results demonstrated the effectiveness and robustness of the newly developed hybrid hinged-rod model by comparing the differences between the results of the bead model and the current model with different rod element lengths.
This paper studies the optimal control problem of a nano-satellite deorbiting by a short electrod... more This paper studies the optimal control problem of a nano-satellite deorbiting by a short electrodynamic tether. The optimal control theory is introduced by forming the control problem as a cost index minimisation subjected to several constraints. A direct method based on Hermite-Simpson discretisation is adopted to solve the constraint cost minimisation problem, resulting in an optimal trajectory including the time history of the states and control input, which achieves best deorbiting efficiency and libration stability simultaneously under the given mission requirements. In order to reduce the computation efforts, the continuous deorbiting process of an electrodynamic tether is discretised into a sequential time intervals, where during each interval the slowly varying orbital parameters of the electrodynamic tether are assumed constant. Thus, the whole optimal trajectory is obtained by combining the solutions to the optimal control problems in the intervals. Numerical simulations are performed to test the performance of the optimal trajectory by applying the control input profile to an electrodynamic tether under complex environment perturbations.
This paper studies the long term dynamics and optimal control of a nano-satellite deorbit by a sh... more This paper studies the long term dynamics and optimal control of a nano-satellite deorbit by a short electrodynamic tether. The long term deorbit process is discretized into intervals and within each interval a two-phase optimal control law is proposed to achieve libration stability and fast deorbit simultaneously. The first-phase formulates an open-loop fast-deorbit control trajectory by a simplified model that assumes the slow-varying orbital elements of electrodynamic tethered system as constant and ignores perturbation forces other than the electrodynamic force. The second phase tracks the optimal trajectory derived in the first phase by a finite receding horizon control method while considering a full dynamic model of electrodynamic tether system. Both optimal control problems are solved by direct collocation method base on the Hermite–Simpson discretization schemes with coincident nodes. The resulting piecewise nonlinear programing problems in the sequential intervals reduces the problem size and improve the computational efficiency, which enable an on-orbit control application. Numerical results for deorbit control of a short electrodynamic tethered nano-satellite system in both equatorial and highly inclined orbits demonstrate the efficiency of the proposed control method. An optimal balance between the libration stability and a fast deorbit of satellite with minimum control efforts is achieved.
This paper proposes a piecewise two-phased optimal control scheme for fast nanosatellite deorbit ... more This paper proposes a piecewise two-phased optimal control scheme for fast nanosatellite deorbit by a short electrodynamic tether. The first phase concerns the open-loop control trajectory optimization, where the optimal control problem is formulated only for the tether libration motion by assuming the slow-varying orbital elements of the electrodynamic tether system as constant within a discretized interval. The second phase deals with the closed-loop optimal control for tracking the derived optimal reference trajectory subject to multiple major orbital perturbations. The finite receding horizon control method is used in the optimal trajectory tracking. Both optimal control problems are solved by a direct collocation method based on the Hermite–Simpson method using discretization schemes with coincident nodes. The resulting nonlinear programming problem significantly reduces the problem size and improves the computational efficiency. Numerical results for fast nanosatellite deorbit by an electrodynamic tether in both equatorial and highly inclined orbits show the proposed method achieves high control accuracy and efficiency.
Electrodynamic tethers have the ability to remove space debris from Earth’s orbit without the use... more Electrodynamic tethers have the ability to remove space debris from Earth’s orbit without the use of propellant. Unfortunately, the periodic variations of electrodynamic force will lead to the tumbling of tethers, and attitude control is needed to achieve a successful deorbit of space debris at the cost of deorbit efficiency. This paper develops an optimal current switching control scheme to enable a fast and stable spacecraft deorbit simultaneously by electrodynamic tethers. In addition, the computational effort of the proposed optimal control is significantly reduced by using a piecewise treatment that discretizes the deorbit process into consecutive time intervals. Within each interval, the system dynamic model is simplified based on different timescales of state variables and the current on–off switching is optimized by solving a constrained minimization problem of a control index representing the deorbit efficiency. Direct Hermite–Simpson discretization is adopted to convert the optimal control problem into a standard nonlinear programming problem. The validity and efficiency of the proposed control strategy is shown by numerical simulations. The deorbit rate increases significantly with the proposed optimal current switching control compared with the existing simple current on–off switch, whereas the computational efforts are reduced.
This paper studies libration dynamics and stability of deorbiting nano-satellites by short and ba... more This paper studies libration dynamics and stability of deorbiting nano-satellites by short and bare electrodynamic tethers. A critical aspect of satellite deorbit by an electrodynamic tether is to maintain the tether aligned with the local vertical and stable while subjected to external perturbations. The dynamics of electrodynamic tether system in deorbit application is divided into the orbital motion of the center of system’s mass and the tether libration motion relative to that center. Major space environmental perturbations including the current-induced electrodynamic force, atmospheric drag, oblateness effect of the Earth, irregularity of geomagnetic field, variable plasma density, solar radiation pressure, and lunisolar gravitational attractions are considered in the dynamic analysis. Quantitative analyses are provided in order to characterize the order of the perturbative torques during the deorbit process. A single index is derived from the libration energy to stabilize the libration motion by regulating the current in the tether through simple on-off switching. Numerical results show that the libration dynamics of an electrodynamic tether has significant impacts on the deorbit process and the electrodynamic tether cannot effectively deorbit satellites without libration stability control. The proposed current regulation strategy is simple and very effective in stabilizing libration motion of an electrodynamic tether.
This paper studies the dynamics of nanosatellite deorbit by a bare electrodynamic tether. The orb... more This paper studies the dynamics of nanosatellite deorbit by a bare electrodynamic tether. The orbital dynamics of the tethered nanosatellite is modeled in Gaussian perturbation equations and the motion-induced voltage-current relationship along the electrodynamic tether is analyzed by using the 2000 International Geomagnetic Reference Field model including up to seventh-order terms and the International Reference Ionosphere 2007 model. The analysis reveals that the high-order magnetic model of Earth affects the dynamic characteristics of the tethered nanosatellite, especially in orbits with high inclination angles, by changing its orbit from circular to elliptical forms. This is beneficial for deorbiting the nanosatellite in near-polar orbits where the electrodynamic force is not as effective as in the equatorial orbit because the denser atmosphere at a lower perigee will provide a larger atmospheric drag. Moreover, the analysis shows that the electrodynamic force is always against the satellite motion in low Earth orbit even when the induced voltage/current across the tether reverses their polarities in near-polar orbits. Compared to the deorbit rate by the atmospheric drag only, the deorbit rate by an electrodynamic tether will be increased by several orders in magnitudes in both equatorial and polar orbits.
Electrodynamic tether systems orbiting the Earth are prone to libration instability because of pe... more Electrodynamic tether systems orbiting the Earth are prone to libration instability because of periodic changes in the geomagnetic field, plasma density, and lunisolar gravitational attractions in addition to nonperiodic changes resulting from the irregularity of the geomagnetic field, inhomogeneity of the Earth, and solar pressures. The long-term orbital and libration dynamics of a bare electrodynamic tether in deorbiting obsolete satellites is investigated by considering space environmental perturbations of current-induced electrodynamic force, atmospheric drag, Earth’s oblateness, irregularity of the geomagnetic field, variable space plasma density, solar radiation pressure, and lunisolar gravitational attractions. The electrodynamic tether is assumed to be rigid and the tethered spacecraft is modeled as a lumped mass. The study shows by numerical simulation that the out-of-plane libration is the primary source of libration instability in inclined orbits, which destabilizes the in-plane libration through nonlinear modal coupling. Accordingly, a simple stability criterion for current on/off switching control is derived from the libration energy of the tether to stabilize the out-of-plane libration by limiting the roll angle amplitude to a preset range. This in turn stabilizes the in-plane libration. The control requires only the feedback of the maximum roll angle with a minimum interval for current on/off switching imposed to avoid excessive current switching. The effectiveness of the control strategy has been demonstrated by analyzing the libration dynamics of electrodynamic tether with and without the current regulation in deorbiting satellites. Numerical results show that this approach is very effective in stabilizing both in-plane and out-of-plane libration of a tethered system subjected to periodic and nonperiodic perturbations.
This paper studies the deployment and retrieval of a tethered satellite system that is composed b... more This paper studies the deployment and retrieval of a tethered satellite system that is composed by two satellites connected with a flexible tether. A hybrid hinged-rod is developed based on conventional hinged-rod and bead models. The tether is discretized into rigid rods connected by springs and dampers at hinges and the end-satellites are modeled as rigid spheres. Newton-Euler method is applied to establish equations of motion for each rigid body. By adopting an instantaneity assumption and transfer mechanism between beads and rods, the proposed hybrid hinged-rod model is able to model the tether deployment and retrieval processes while avoiding the difficulty encountered due to dimension variation of rod element in deployment and retrieval. Two tethered satellite systems moving on a circular orbit with different tether lengths are investigated. Simulation results demonstrated the effectiveness and robustness of the newly developed hybrid hinged-rod model by comparing the differences between the results of the bead model and the current model with different rod element lengths.
This paper studies the optimal control problem of a nano-satellite deorbiting by a short electrod... more This paper studies the optimal control problem of a nano-satellite deorbiting by a short electrodynamic tether. The optimal control theory is introduced by forming the control problem as a cost index minimisation subjected to several constraints. A direct method based on Hermite-Simpson discretisation is adopted to solve the constraint cost minimisation problem, resulting in an optimal trajectory including the time history of the states and control input, which achieves best deorbiting efficiency and libration stability simultaneously under the given mission requirements. In order to reduce the computation efforts, the continuous deorbiting process of an electrodynamic tether is discretised into a sequential time intervals, where during each interval the slowly varying orbital parameters of the electrodynamic tether are assumed constant. Thus, the whole optimal trajectory is obtained by combining the solutions to the optimal control problems in the intervals. Numerical simulations are performed to test the performance of the optimal trajectory by applying the control input profile to an electrodynamic tether under complex environment perturbations.
This paper studies the long term dynamics and optimal control of a nano-satellite deorbit by a sh... more This paper studies the long term dynamics and optimal control of a nano-satellite deorbit by a short electrodynamic tether. The long term deorbit process is discretized into intervals and within each interval a two-phase optimal control law is proposed to achieve libration stability and fast deorbit simultaneously. The first-phase formulates an open-loop fast-deorbit control trajectory by a simplified model that assumes the slow-varying orbital elements of electrodynamic tethered system as constant and ignores perturbation forces other than the electrodynamic force. The second phase tracks the optimal trajectory derived in the first phase by a finite receding horizon control method while considering a full dynamic model of electrodynamic tether system. Both optimal control problems are solved by direct collocation method base on the Hermite–Simpson discretization schemes with coincident nodes. The resulting piecewise nonlinear programing problems in the sequential intervals reduces the problem size and improve the computational efficiency, which enable an on-orbit control application. Numerical results for deorbit control of a short electrodynamic tethered nano-satellite system in both equatorial and highly inclined orbits demonstrate the efficiency of the proposed control method. An optimal balance between the libration stability and a fast deorbit of satellite with minimum control efforts is achieved.
This paper proposes a piecewise two-phased optimal control scheme for fast nanosatellite deorbit ... more This paper proposes a piecewise two-phased optimal control scheme for fast nanosatellite deorbit by a short electrodynamic tether. The first phase concerns the open-loop control trajectory optimization, where the optimal control problem is formulated only for the tether libration motion by assuming the slow-varying orbital elements of the electrodynamic tether system as constant within a discretized interval. The second phase deals with the closed-loop optimal control for tracking the derived optimal reference trajectory subject to multiple major orbital perturbations. The finite receding horizon control method is used in the optimal trajectory tracking. Both optimal control problems are solved by a direct collocation method based on the Hermite–Simpson method using discretization schemes with coincident nodes. The resulting nonlinear programming problem significantly reduces the problem size and improves the computational efficiency. Numerical results for fast nanosatellite deorbit by an electrodynamic tether in both equatorial and highly inclined orbits show the proposed method achieves high control accuracy and efficiency.
Electrodynamic tethers have the ability to remove space debris from Earth’s orbit without the use... more Electrodynamic tethers have the ability to remove space debris from Earth’s orbit without the use of propellant. Unfortunately, the periodic variations of electrodynamic force will lead to the tumbling of tethers, and attitude control is needed to achieve a successful deorbit of space debris at the cost of deorbit efficiency. This paper develops an optimal current switching control scheme to enable a fast and stable spacecraft deorbit simultaneously by electrodynamic tethers. In addition, the computational effort of the proposed optimal control is significantly reduced by using a piecewise treatment that discretizes the deorbit process into consecutive time intervals. Within each interval, the system dynamic model is simplified based on different timescales of state variables and the current on–off switching is optimized by solving a constrained minimization problem of a control index representing the deorbit efficiency. Direct Hermite–Simpson discretization is adopted to convert the optimal control problem into a standard nonlinear programming problem. The validity and efficiency of the proposed control strategy is shown by numerical simulations. The deorbit rate increases significantly with the proposed optimal current switching control compared with the existing simple current on–off switch, whereas the computational efforts are reduced.
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Papers by Rui Zhong
the tethered nanosatellite is modeled in Gaussian perturbation equations and the motion-induced voltage-current
relationship along the electrodynamic tether is analyzed by using the 2000 International Geomagnetic Reference Field
model including up to seventh-order terms and the International Reference Ionosphere 2007 model. The analysis
reveals that the high-order magnetic model of Earth affects the dynamic characteristics of the tethered nanosatellite,
especially in orbits with high inclination angles, by changing its orbit from circular to elliptical forms. This is beneficial
for deorbiting the nanosatellite in near-polar orbits where the electrodynamic force is not as effective as in the
equatorial orbit because the denser atmosphere at a lower perigee will provide a larger atmospheric drag. Moreover,
the analysis shows that the electrodynamic force is always against the satellite motion in low Earth orbit even when the
induced voltage/current across the tether reverses their polarities in near-polar orbits. Compared to the deorbit rate
by the atmospheric drag only, the deorbit rate by an electrodynamic tether will be increased by several orders in
magnitudes in both equatorial and polar orbits.
The study shows by numerical simulation that the out-of-plane libration is the primary source of libration instability in inclined orbits, which destabilizes the in-plane libration through nonlinear modal coupling. Accordingly, a simple stability criterion for current on/off switching control is derived from the libration energy of the tether to stabilize the out-of-plane libration by limiting the roll angle amplitude to a preset
range. This in turn stabilizes the in-plane libration. The control requires only the feedback of the maximum roll angle with a minimum interval for current on/off switching imposed to avoid excessive current switching. The effectiveness of the control strategy has been demonstrated by analyzing the libration dynamics of electrodynamic tether with and without the current regulation in deorbiting satellites. Numerical results
show that this approach is very effective in stabilizing both in-plane and out-of-plane libration of a tethered system subjected to periodic and nonperiodic perturbations.
assumption and transfer mechanism between beads and rods, the proposed hybrid hinged-rod model is able to model the tether deployment and retrieval processes while avoiding the difficulty encountered due to dimension variation of rod element in deployment and retrieval. Two tethered satellite systems moving on a circular orbit with different tether lengths are investigated. Simulation results demonstrated the effectiveness and robustness of the newly developed hybrid hinged-rod model by comparing the differences between the results of the bead model and the current model with different rod element lengths.
discretisation is adopted to solve the constraint cost minimisation problem, resulting in an optimal trajectory including the time history of the states and control input, which achieves best deorbiting efficiency and libration stability
simultaneously under the given mission requirements. In order to reduce the computation efforts, the continuous deorbiting process of an electrodynamic tether is discretised into a sequential time intervals, where during each interval
the slowly varying orbital parameters of the electrodynamic tether are assumed constant. Thus, the whole optimal trajectory is obtained by combining the solutions to the optimal control problems in the intervals. Numerical simulations are performed to test the performance of the optimal trajectory by applying the control input profile to an electrodynamic tether under complex environment perturbations.
that assumes the slow-varying orbital elements of electrodynamic tethered system as constant and ignores perturbation forces other than the electrodynamic force. The second phase tracks the optimal trajectory derived in the first phase by a finite receding horizon control method while considering a full dynamic model of electrodynamic tether system. Both optimal control problems are solved by direct
collocation method base on the Hermite–Simpson discretization schemes with coincident nodes. The resulting piecewise nonlinear programing problems in the sequential intervals reduces the problem size and improve the computational efficiency, which enable an on-orbit
control application. Numerical results for deorbit control of a short electrodynamic tethered nano-satellite system in both equatorial and highly inclined orbits demonstrate the efficiency of the proposed control method. An optimal balance between the libration stability and a fast deorbit of satellite with minimum control efforts is achieved.
Unfortunately, the periodic variations of electrodynamic force will lead to the tumbling of tethers, and attitude control
is needed to achieve a successful deorbit of space debris at the cost of deorbit efficiency. This paper develops an optimal
current switching control scheme to enable a fast and stable spacecraft deorbit simultaneously by electrodynamic
tethers. In addition, the computational effort of the proposed optimal control is significantly reduced by using a
piecewise treatment that discretizes the deorbit process into consecutive time intervals. Within each interval, the
system dynamic model is simplified based on different timescales of state variables and the current on–off switching is
optimized by solving a constrained minimization problem of a control index representing the deorbit efficiency. Direct
Hermite–Simpson discretization is adopted to convert the optimal control problem into a standard nonlinear
programming problem. The validity and efficiency of the proposed control strategy is shown by numerical simulations.
The deorbit rate increases significantly with the proposed optimal current switching control compared with
the existing simple current on–off switch, whereas the computational efforts are reduced.
the tethered nanosatellite is modeled in Gaussian perturbation equations and the motion-induced voltage-current
relationship along the electrodynamic tether is analyzed by using the 2000 International Geomagnetic Reference Field
model including up to seventh-order terms and the International Reference Ionosphere 2007 model. The analysis
reveals that the high-order magnetic model of Earth affects the dynamic characteristics of the tethered nanosatellite,
especially in orbits with high inclination angles, by changing its orbit from circular to elliptical forms. This is beneficial
for deorbiting the nanosatellite in near-polar orbits where the electrodynamic force is not as effective as in the
equatorial orbit because the denser atmosphere at a lower perigee will provide a larger atmospheric drag. Moreover,
the analysis shows that the electrodynamic force is always against the satellite motion in low Earth orbit even when the
induced voltage/current across the tether reverses their polarities in near-polar orbits. Compared to the deorbit rate
by the atmospheric drag only, the deorbit rate by an electrodynamic tether will be increased by several orders in
magnitudes in both equatorial and polar orbits.
The study shows by numerical simulation that the out-of-plane libration is the primary source of libration instability in inclined orbits, which destabilizes the in-plane libration through nonlinear modal coupling. Accordingly, a simple stability criterion for current on/off switching control is derived from the libration energy of the tether to stabilize the out-of-plane libration by limiting the roll angle amplitude to a preset
range. This in turn stabilizes the in-plane libration. The control requires only the feedback of the maximum roll angle with a minimum interval for current on/off switching imposed to avoid excessive current switching. The effectiveness of the control strategy has been demonstrated by analyzing the libration dynamics of electrodynamic tether with and without the current regulation in deorbiting satellites. Numerical results
show that this approach is very effective in stabilizing both in-plane and out-of-plane libration of a tethered system subjected to periodic and nonperiodic perturbations.
assumption and transfer mechanism between beads and rods, the proposed hybrid hinged-rod model is able to model the tether deployment and retrieval processes while avoiding the difficulty encountered due to dimension variation of rod element in deployment and retrieval. Two tethered satellite systems moving on a circular orbit with different tether lengths are investigated. Simulation results demonstrated the effectiveness and robustness of the newly developed hybrid hinged-rod model by comparing the differences between the results of the bead model and the current model with different rod element lengths.
discretisation is adopted to solve the constraint cost minimisation problem, resulting in an optimal trajectory including the time history of the states and control input, which achieves best deorbiting efficiency and libration stability
simultaneously under the given mission requirements. In order to reduce the computation efforts, the continuous deorbiting process of an electrodynamic tether is discretised into a sequential time intervals, where during each interval
the slowly varying orbital parameters of the electrodynamic tether are assumed constant. Thus, the whole optimal trajectory is obtained by combining the solutions to the optimal control problems in the intervals. Numerical simulations are performed to test the performance of the optimal trajectory by applying the control input profile to an electrodynamic tether under complex environment perturbations.
that assumes the slow-varying orbital elements of electrodynamic tethered system as constant and ignores perturbation forces other than the electrodynamic force. The second phase tracks the optimal trajectory derived in the first phase by a finite receding horizon control method while considering a full dynamic model of electrodynamic tether system. Both optimal control problems are solved by direct
collocation method base on the Hermite–Simpson discretization schemes with coincident nodes. The resulting piecewise nonlinear programing problems in the sequential intervals reduces the problem size and improve the computational efficiency, which enable an on-orbit
control application. Numerical results for deorbit control of a short electrodynamic tethered nano-satellite system in both equatorial and highly inclined orbits demonstrate the efficiency of the proposed control method. An optimal balance between the libration stability and a fast deorbit of satellite with minimum control efforts is achieved.
Unfortunately, the periodic variations of electrodynamic force will lead to the tumbling of tethers, and attitude control
is needed to achieve a successful deorbit of space debris at the cost of deorbit efficiency. This paper develops an optimal
current switching control scheme to enable a fast and stable spacecraft deorbit simultaneously by electrodynamic
tethers. In addition, the computational effort of the proposed optimal control is significantly reduced by using a
piecewise treatment that discretizes the deorbit process into consecutive time intervals. Within each interval, the
system dynamic model is simplified based on different timescales of state variables and the current on–off switching is
optimized by solving a constrained minimization problem of a control index representing the deorbit efficiency. Direct
Hermite–Simpson discretization is adopted to convert the optimal control problem into a standard nonlinear
programming problem. The validity and efficiency of the proposed control strategy is shown by numerical simulations.
The deorbit rate increases significantly with the proposed optimal current switching control compared with
the existing simple current on–off switch, whereas the computational efforts are reduced.