Libration Angle Research Articles

Review of Deployment Controllers for Space Tethered System

This review article investigates the performance of newly designed controllers for Tethered Space Systems (TSS) deployment, comparing them to past and current research. Specifically, the study delves into the PD type controller, known for its simplicity and early development. While PD controllers adequately manage non-coplanar scenarios with external perturbations, they are limited to linearized cases. In contrast, advanced controllers like Sliding Mode Control (SMC) effectively handle TSS’s highly nonlinear dynamics due to their robust nature. The study introduces a sigma function, derived from Djebli’s literature, to regulate tether deployment rate and length across all controllers. Simulation results demonstrate the feasibility of controlling out-of-plane libration angles solely through the tether tension. Among the controllers tested, the advanced sigma-SMC exhibits superior accuracy during deployment, while the modified SMC deploys the tether fastest albeit with significant steady-state errors and deflection angles. Numerical results show that the original SMC controller performs well as the most fuel-efficient option. This comparison focuses solely on tether deployment with J2 and gravity-gradient perturbations, offering a foundation for further exploration into TSS missions, including factors like aerodynamic drag, solar radiation, third body perturbations, tether flexibility, and retrieval phase control, tailored to specific mission requirements.

Modeling and control of perturbation torques and mass distribution impact on a tethered system for a 12U CubeSat in sun-synchronous orbit

Tethered CubeSat systems are characterized by momentum exchange and gravity gradient stabilization, and encounter stability challenges, especially with shorter tethers in highly inclined and eccentric orbits. This study investigates the libration dynamics of a 12U CubeSat in its stowed configuration, which separates into two satellites connected via tether, throughout its deployment, station-keeping, and retrieval phases. Two deployed configurations, symmetrical 6U-6U, and asymmetrical 8U-4U connected by a non-conductive tether with a maximum length of 100m are analyzed. The study accounts for perturbations including Earth’s oblateness, atmospheric drag, solar radiation pressure, and lunisolar gravitation, modeling the tether as a rigid, extendable rod and the satellites as lumped masses. The translational and rotational dynamics are decoupled, assuming the system’s center of mass follows an unperturbed Keplerian Sun-synchronous orbit at altitudes between 400 and 600km. A tension control strategy based on Lyapunov’s direct method and supplemented by external actuation torques is explored. Results show high orbit eccentricity significantly affects the maximum in-plane libration angle. The 6U-6U system experiences smaller disturbance torques but greater tether tension variations than the 8U-4U system. Both configurations exhibit larger in-plane oscillations compared to near-zero out-of-plane oscillations, nonetheless, eclipse passages exacerbate the out-of-plane libration of the 8U-4U system. Relative stability is maintained during deployment, however, retrieval is chaotic, with notable oscillations in libration angles and tether tension. The tension control strategy effectively dampens oscillations during retrieval but loses effectiveness as tether tension approaches zero at retrieval’s conclusion. External actuation torques, ranging from 1-2mNm for deployment to 15mNm for retrieval, complement tension control.

Autonomous assembly of multiple spacecraft by tether-aided rendezvous and docking: From theory to experiment

Utilizing the tether-aided rendezvous and docking to realize an on-orbit autonomous assembly of multiple spacecraft is a novel and prospective space mission. The paper presents the dynamics and control of this assembly mission mode in a spinning scenario numerically and experimentally. The system equations in general form are first established. Based on several feasible assumptions, the coupling effects among the chaser spacecraft are eliminated, a set of simplified governing equations described by the tether length and the libration angle are then obtained. By employing the trapezoidal tether retrieval law, an assembly strategy composed of the tangential and normal thrust controllers is proposed. The operation constraints of the tether tension and the libration angle are explicitly considered. Numerical simulation results verify the feasibility and effectiveness of the designed control strategy. In addition, the corresponding tether-aided assembly process is further validated by the air-bearing experiment based on the upgraded system SOOHLS.

Model and Dynamic Analysis of a Three-Body Tethered Satellite System in Three Dimensions

The three-body tethered satellite system is a new potential technology for the purposes of orbital transportation. In contrast to conventional orbit transfer methods, this system is expected to transport space supplies to a predetermined orbital altitude without consuming fuel; however, the unwanted libration resulting from the Coriolis force acting on moving subsatellites may induce tumbling within the system. In order to analyze the strongly coupled characteristics of the libration motion and the variable-length tethers, a six-DOF dynamic model of the system based on Newton’s law is established. By utilizing the dynamic equations and stability criterion for linear systems, three equilibrium configurations of the satellite system with two constant-length tethers are given. The coupling characteristics of the libration angles are analyzed based on mechanical features and simulations. The results demonstrate that part of the libration energy can transfer from out-of-plane motion to in-plane motion during out-of-plane libration, but not vice versa or when in-plane libration occurs alone. Furthermore, the dynamic characteristic of this system with a predesigned deployment strategy is surveyed. This investigation reveals that such a strategy can rapidly suppress the out-of-plane libration motion while maintaining purely sinusoidal oscillations in the in-plane motion.

Data-driven optimal controller design for sub-satellite deployment of tethered satellite system

<abstract><p>In the paper, we presented a data-driven optimal control method for the sub-satellite deployment of tethered satellite systems during the release phase, with the objective of reducing the libration angle fluctuation during system work. First, the dynamic equation of the tethered satellite system was established and processed dimensionless. Considering the presence of noise when sensors were put on the satellite to measure the libration angle, the corresponding state equation was derived. Next, we estimated the system state using an unscented Kalman filter and established a performance index and the Hamilton function. Based on the index, we designed an optimal control strategy and provided sufficient conditions for the asymptotic stability of the closed-loop system. We used critic-actor neural networks based on measurement data to implement the data-driven control algorithm to approximate the performance index function and the control policy, respectively. Finally, taking a tethered satellite system as an example, a simulation showed that the unscented Kalman filter can effectively estimate the system state and improve the impact of noise on the system, and the proposed optimal control strategy ensured that the tethered satellite system is stable, which shows the applicability and effectiveness of the control strategy in reducing the tether vibration of the disturbed system.</p></abstract>

Influence of orientational disorder in the adsorbent on the structure and dynamics of the adsorbate: MD simulations of SO2 in ZSM-22

Structural and dynamical behavior of SO2 molecules within ZSM-22 is studied using molecular dynamics simulations, to understand the influence of orientational disorder (OD) and inter – crystalline spacing in ZSM-22 as a function of adsorbate loading. Addition of inter-crystalline space provides connectivity of isolated pores in ZSM-22 and is shown to suppress both translational and rotational motion of SO2. We infer that geometry and dimensionality of the connecting space are important factors in determining the effects of pore connectivity on the adsorbed species behavior. As a function of OD, decrease in self-diffusion coefficient of SO2 in ZSM-22 is observed. An increase in rotational correlation time and a decrease in libration angle with OD is observed, ascribed to the restriction imposed on the orientational freedom of the adsorbate by an increase in OD. The behavior of SO2 results from an interplay of guest-host interactions and the dimensionality and geometry of confinement.

Covariance analysis of periodic and quasi-periodic orbits around Phobos with applications to the Martian Moons eXploration mission

Understanding the internal composition of a celestial body is fundamental for formulating theories regarding its origin. Deep knowledge of the distribution of mass under the body’s crust can be achieved by analyzing its moments of inertia and gravity field. In this regard, the two moons of the Martian system have not yet been closely studied and continue to pose questions regarding their origin to the space community; thus, they deserve further characterization. The Martian Moons eXploration mission will be the first of its kind to sample and study Phobos over a prolonged period. This study aims to demonstrate that the adoption of periodic and quasi-periodic retrograde trajectories would be beneficial for the scientific value of the mission. Here, a covariance analysis was implemented to compare the estimation of high-order gravitational field coefficients from different orbital geometries and for different sets of processed observables. It was shown that the adoption of low-altitude non-planar quasi-satellite orbits would help to refine the knowledge of the moon’s libration angle and gravitational field.

Out-of-plane chaotic motion and suppression for tethered tug-debris systems with thrust perturbation

This paper investigates the characterization and suppression of chaotic motion in an out-of-plane tethered tug-debris system that is towed by a constant thrust with time-varying micro-perturbations at a fixed non-zero in-plane libration angle. These perturbations arise due to imperfections in the thruster’s mechanical system on the tug. Static equilibrium solutions and homoclinic orbits of the corresponding Hamiltonian system are derived, and a subcritical fork bifurcation is identified and presented. By the Melnikov method, it is found that the micro-fluctuation in the thrust could induce chaos in the system near unstable saddle points. Then, a chaotic parameter region is produced to identify chaos efficiently. The pure out-of-plane chaotic motion would appear, provided system parameters are in the region. A sliding mode controller is designed to suppress the chaotic oscillation by deploying and retrieving the tether. Finally, the impact of the system parameters on the chaotic region is estimated numerically. Simulation results demonstrate that chaotic motions exist and can be identified efficiently by the chaotic parameter region and that the control method is effective in suppressing the chaotic motion.

Existence proof of librational invariant tori in an averaged model of HD60532 planetary system

We investigate the long-term dynamics of HD60532, an extrasolar system hosting two giant planets orbiting in a 3:1 mean motion resonance. We consider an average approximation at order one in the masses which results (after the reduction in the constants of motion) in a resonant Hamiltonian with two libration angles. In this framework, the usual algorithms constructing the Kolmogorov normal form approach do not easily apply and we need to perform some untrivial preliminary operations, in order to adapt the method to this kind of problems. First, we perform an average over the fast angle of libration which provides an integrable approximation of the Hamiltonian. Then, we introduce action-angle variables that are adapted to such an integrable approximation. This sequence of preliminary operations brings the Hamiltonian in a suitable form to successfully start the Kolmogorov normalization scheme. The convergence of the KAM algorithm is proved by applying a technique based on a computer-assisted proof. This allows us to reconstruct the quasi-periodic motion of the system, with initial conditions that are compatible with the observations.

Effects of Dynamical Degrees of Freedom on Magnetic Compass Sensitivity: A Comparison of Plant and Avian Cryptochromes.

The magnetic compass of migratory birds is thought to rely on a radical pair reaction inside the blue-light photoreceptor protein cryptochrome. The sensitivity of such a sensor to weak external magnetic fields is determined by a variety of magnetic interactions, including electron-nuclear hyperfine interactions. Here, we investigate the implications of thermal motion, focusing on fluctuations in the dihedral and librational angles of flavin adenine dinucleotide (FAD) and tryptophan (Trp) radicals in cryptochrome 4a from European robin (Erithacus rubecula, ErCry4a) and pigeon (Columba livia, ClCry4a) and cryptochrome 1 from the plant Arabidopsis thaliana (AtCry1). Molecular dynamics simulations and density functional theory-derived hyperfine interactions are used to calculate the quantum yield of radical pair recombination dependent on the direction of the geomagnetic field. This quantity and various dynamical parameters are compared for [FAD•- Trp•+] in ErCry4a, ClCry4a, and AtCry1, with TrpC or TrpD being the third and fourth components of the tryptophan triad/tetrad in the respective proteins. We find that (i) differences in the average dihedral angles in the radical pairs are small, (ii) the librational motions of TrpC•+ in the avian cryptochromes are appreciably smaller than in AtCry1, (iii) the rapid vibrational motions of the radicals leading to strong fluctuations in the hyperfine couplings affect the spin dynamics depending on the usage of instantaneous or time-averaged interactions. Future investigations of radical pair compass sensitivity should therefore not be based on single snapshots of the protein structure but should include the ensemble properties of the hyperfine interactions.

Length-rate control for libration reduction during retraction of tethered satellite systems

Retrieval of tethered satellite systems is an unstable process. Several complex control methods have been proposed to stabilize this process. Recently, a simple, heuristic bang–bang control algorithm was found to be effective at significantly reducing libration angles by introducing short periods of extension during the retraction phase. In this paper, an analytical method is derived for the determination of effective retraction/extension speeds for bang–bang length rate control. The performance of the derived control profiles is validated by comparison to optimized two-switch bang–bang profiles. Additionally, particle swarm optimization is leveraged to optimize switch times and tether length rates for control profiles with increased numbers of switches. It is verified that the analytical methodology for control speed selection can be employed to define both extension/retraction and retraction-only controls. It is determined that speeds of retraction and extension contribute greatly to the libration motion of the tethers, and that proper speed selection is especially critical when the tethered satellite system is near its equilibrium state. Accurate switch timing is also found to be particularly important, as slight variations can significantly affect the tether libration response. While an increased number of switches is found to improve the tether response and to reduce the retraction time, the achievable benefits may not be sufficient to justify the increased complexity.

The irradiance instrument subsystem (IRIS) on the airborne-lunar spectral irradiance (air-LUSI) instrument

The objective of the airborne lunar spectral irradiance (air-LUSI) project is to make low uncertainty, SI-traceable measurements of the LUSI in the visible to near-infrared region from an aircraft above most of the optically absorbing components of the atmosphere. The measurements are made from a NASA ER-2 aircraft, which can fly at altitudes of approximately 20 km above sea level. Air-LUSI measurements, corrected for residual atmospheric attenuation, are designed to provide a matrix of low uncertainty top-of-the-atmosphere lunar irradiances at known lunar phase and libration angles to be compared and combined with other lunar irradiance data sets to constrain the uncertainties in models of lunar irradiance and reflectance. The measurements are also expected to provide insight into the differences between models and satellite sensor measurements of lunar irradiance. This paper describes the development and characterization of the air-LUSI subsystem for acquiring lunar measurements, called the irradiance instrument subsystem, prior to flight.

Structure, hydrogen bond dynamics and phase transition in a model ionic liquid electrolyte.

We show that solid-state NMR spectroscopy is a suitable method for characterizing the structure, hydrogen bond dynamics and phase transition behavior in protic ionic liquids (PILs). Deuteron line shape and spin relaxation time analysis provide a description of the structural and dynamical heterogeneity in the solid state of the model PIL triethyl ammonium bis(trifluoromethanesulfonyl)amide [TEA][NTf2]. Therein, we observed two deuteron quadrupole coupling constant for the ND bond of the TEA cation, indicating differently strong hydrogen bonds to the nitrogen and oxygen atoms of the NTf2 anion, as we could confirm by DFT calculations. The transition processes in the dynamically heterogeneous phase are characterized by two standard molar enthalpies and thus different stages of melting. We provide geometry, rates and energetics of the cation in the solid and liquid states of the PIL. Comparison with PILs having stronger interacting anions shows higher enthalpy change between the solid and liquid states, lower activation barriers of tumbling motion and higher amplitude of librational motion for the TEA cation in the presence of the weakly interacting anion NTf2. We provide reasonable relations between microscopic and macroscopic properties, as is relevant for any kind of application.

Passivity-Based Model Predictive Control for Tethered Despin of Massive Space Objects by Small Space Tug

This paper studies the stable control of tethered despin of a massive rotating space object in a central gravity field by a small space tug. A control scheme under the passivity-based Model Predictive Control framework is designed to ensure the constraints of state and input (bounded libration angle, positive tether tension and bounded thrust) are satisfied. Furthermore, an additional passivity constraint is introduced into the passivity-based Model Predictive Control to guarantee the asymptotic stability of the closed-loop control system. The attainable equilibrium configuration of the tethered system is first analyzed. Then, a framework of the storage energy function is constructed by the potential energy shaping methodology to establish the passivity mapping of the tethered system from input to output. Finally, the strictly asymptotic stability of despinning control of the tethered system under the proposed control law is theoretically proved by the Invariance theorem and Lyapunov stability theory. The effectiveness of the proposed control scheme is verified by numerical simulation.

Prescribed performance based dual-loop control strategy for configuration keeping of partial space elevator in cargo transportation

The partial space elevator is a space transportation system that conveys cargo by a climber moving along the tether that connects the main satellite and the end body. This work proposed a novel prescribed performance based dual-loop control strategy for configuration keeping of the partial space elevator in cargo transportation. The control loop I controls the libration angle of the climber by a newly proposed prescribed performance control law to track the desired state with bounded performance. The Lyapunov stability of the control law is proved analytically. Control loop II stabilizes the overall configuration of the PSE by applying thrust and deploying/retrieving the tether simultaneously at the end body. To enforce the constraint that the total tether length is unchanged at the end of transportation, an event-triggered loaning control algorithm is proposed to further improve the Control loop II. Numerical simulations validate the prescribed performance based dual-loop control strategy. The results show that the stable configuration of the PSE is kept effectively with no libration motion in the period of cargo transportation.

Tethered Satellite-Controlled Re-Entry Dynamics From the International Space Station

Payloads left in space at the end of life create debris. A large amount of space debris surrounds our planet and within a few years, experts argue, it will no longer be possible to send payloads safely into space. Our study strives to demonstrate the ability to bring a payload back to Earth without the use of an active propulsion system in close proximity of the International Space Station (ISS). The use of a classical chemical propulsion system near sensitive and inhabited space areas is risky and causes contamination due to the fuel ejection. Consequently, the design of a passive but controlled vehicle, that satisfies safety and free-pollution requirements, needs a new propulsive technology. A possible solution is using a Tether Subsystem, mounted onboard a re-entry capsule, employed to execute the first phase of the release/deployment maneuver. The tether deployment trajectory must be controlled in order to provide a large libration angle of about 40° and a radial velocity near zero at the end of tether deployment. The control algorithm adopted is based on reliable and easy to measure dynamics parameters: the deployed length and length rate are the inputs of the control loop that forces the tethered capsule to follow a predetermine reference trajectory during deployment. Relevant details of the IR sensors (i.e., photocells) that are planned for measuring the input parameters are also presented. The aim of our study is to propose a safe and pollution-free solution for separating small satellites from the ISS, preventing hazards, and minimizing external contamination.

Analysis of analytical and numerical dynamic lunar ephemerides

This work considers the analysis of libration dynamics of the planets’ moons. To study such processes, it is necessary to use methods for investigating complex systems, as an influence from neighboring celestial bodies on spin-orbital motion of a moon should be taken into account. This fully applies to the Moon’s rotation about its axis. The construction of modern theories of the physical libration of the Moon (PhLM) is based on gravimetric, seismic, satellite observations, and lunar laser ranging, which in its turn has allowed to increase the accuracy by several orders of magnitude compared to the PhLM theories of the end of the 20th century. To satisfy the modern requirements for accuracy of the physical libration theory, it is necessary to apply a precise ephemeris of the Moon’s motion. In this paper, the DE421 modern numerical ephemeris is analyzed. The features of its construction and practical application are studied. The accurate values of ephemeris parameters are produced and an algorithm for determining the Moon’s center of mass coordinates and establishing the angles of physical libration on the basis of DE421 is developed. As a result, a comparative analysis of the physical libration angles with the values produced by the analytic theory of physical libration is conducted. The calculations are made over a time period of 800 years. The comparative analysis shows that significant differences in the ephemerides of 15 and 80 arc seconds in longitude and latitude respectively are caused by the planets’ gravity field, and those impacts should be considered when working with the analytic ephemeris.

An elastic model of Phobos’ libration

Context. Study the rotation of a celestial body is an efficient way to infer its interior structure, and then may give information of its origin and evolution. In this study, based on the latest shape model of Phobos from Mars Express (MEX) mission, the polyhedron approximation approach was used to simulate the gravity field of Phobos. Then, the gravity information was combined with the newest geophysical parameters such as GM and k2 to construct the numerical model of Phobos’ rotation. And with an appropriate angles transformation, we got the librational series respect to Martian mean equator of date. Aims. The purpose of this paper is to develop a numerical model of Phobos’ rotational motion that includes the elastic properties of Phobos. The frequencies analysis of the librational angles calculated from the numerical integration results emphasize the relationship between geophysical properties and dynamics of Phobos. This work will also be useful for a future space mission dedicated to Phobos. Methods. Based on the latest shape model of Phobos from MEX mission, we firstly modeled the gravity field of Phobos, then the gravity coefficients were combined with some of the newest geophysical parameters to simulate the rotational motion of Phobos. To investigate how the elastic properties of Phobos affect its librational motion, we adopted various k2 into our numerical integration. Then the analysis was performed by iterating a frequency analysis and linear least-squares fit of Phobos’ physical librations. From this analysis, we identified the influence of k2 on the largest librational amplitude and its phase. Results. We showed the first ten periods of the librational angles and found that they agree well with the previous numerical results which Phobos was treated as a perfectly rigid body. We also found that the maximum amplitudes of the three parameters of libration are also close to the results from a rigid model, which is mainly due to the inclination of Phobos and moments of inertia. The other amplitudes are slightly different, since the physics contained in our model is different to that of a previous study, specifically, the different low-degree gravity coefficients and ephemeris. The libration in longitude τ has the same quadratic term with previous numerical study, which is consistent with the secular acceleration of Phobos falling onto Mars. We investigated the influence of the tidal Love number k2 on Phobos’ rotation and found a detectable amplitude changes (0.0005°) expected in the future space mission on τ, which provided a potential possibility to constrain the k2 of Phobos by observing its rotation. We also studied the influence of Phobos’ orbit accuracy on its libration and suggested a simultaneous integration of orbit and rotation in future work.

OSSOS. XVIII. Constraining Migration Models with the 2:1 Resonance Using the Outer Solar System Origins Survey

Abstract Resonant dynamics plays a significant role in the past evolution and current state of our outer solar system. The population ratios and spatial distribution of Neptune’s resonant populations are direct clues to understanding the history of our planetary system. The orbital structure of the objects in Neptune’s 2:1 mean-motion resonance (“twotinos”) has the potential to be a tracer of planetary migration processes. Different migration processes produce distinct architectures, recognizable by well-characterized surveys. However, previous characterized surveys only discovered a few twotinos, making it impossible to model the intrinsic twotino population. With a well-designed cadence and nearly 100% tracking success, the Outer Solar System Origins Survey (OSSOS) discovered 838 trans-Neptunian objects, of which 34 are securely twotinos with well-constrained libration angles and amplitudes. We use the OSSOS twotinos and the survey characterization parameters via the OSSOS survey simulator to inspect the intrinsic population and orbital distributions of twotinos. The estimated twotino population, with H r < 8.66 (diameter ∼100 km) at 95% confidence, is consistent with the previous low-precision estimate. We also constrain the width of the inclination distribution to a relatively narrow value of and find that the eccentricity distribution is consistent with a Gaussian centered on e c = 0.275 with a width e w = 0.06. We find a single-slope exponential luminosity function with α = 0.6 for the twotinos. Finally, for the first time, we meaningfully constrain the fraction of symmetric twotinos and the ratio of the leading asymmetric islands; both fractions are in the range of 0.2–0.6. These measurements rule out certain theoretical models of Neptune’s migration history.

Reorientational Dynamics of Amyloid-β from NMR Spin Relaxation and Molecular Simulation.

Amyloid-β (Aβ) aggregation is a hallmark of Alzheimer’s disease. As an intrinsically disordered protein, Aβ undergoes extensive dynamics on multiple length and time scales. Access to a comprehensive picture of the reorientational dynamics in Aβ requires therefore the combination of complementary techniques. Here, we integrate 15N spin relaxation rates at three magnetic fields with microseconds-long molecular dynamics simulation, ensemble-based hydrodynamic calculations, and previously published nanosecond fluorescence correlation spectroscopy to investigate the reorientational dynamics of Aβ1–40 (Aβ40) at single-residue resolution. The integrative analysis shows that librational and dihedral angle fluctuations occurring at fast and intermediate time scales are not sufficient to decorrelate orientational memory in Aβ40. Instead, slow segmental motions occurring at ∼5 ns are detected throughout the Aβ40 sequence and reach up to ∼10 ns for selected residues. We propose that the modulation of time scales of reorientational dynamics with respect to intra- and intermolecular diffusion plays an important role in disease-related Aβ aggregation.