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Third-body plant matrix validation with respect to the numerical reference for relative motion perturbed by only Sun and Moon third-body effect.
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In this paper we made an analysis of self-similar solutions for the mean curvature flow (MCF) by surfaces of revolution and ruled surfaces in $\mathbb{R}^{3}$. We prove that self-similar solutions of the MCF by non-cylindrival surfaces and conical surfaces in $\mathbb{R}^{3}$ are trivial. Moreover, we characterize the self-similar solutions of the...
In this paper we made an analysis of self-similar solutions for the mean curvature flow (MCF) by surfaces of revolution and ruled surfaces in R 3. We prove that self-similar solutions of the MCF by non-cylindrival surfaces and conical surfaces in R 3 are trivial. Moreover, we characterize the self-similar solutions of the MCF by surfaces of revolut...
In this paper we made an analysis of self-similar solutions for the mean curvature flow (MCF) by surfaces of revolution and ruled surfaces in R 3. We prove that self-similar solutions of the MCF by non-cylindrival surfaces and conical surfaces in R 3 are trivial. Moreover, we characterize the self-similar solutions of the MCF by surfaces of revolut...
In this paper we made an analysis of self-similar solutions for the mean curvature flow (MCF) by surfaces of revolution and ruled surfaces in R 3. We prove that self-similar solutions of the MCF by non-cylindrival surfaces and conical surfaces in R 3 are trivial. Moreover, we characterize the self-similar solutions of the MCF by surfaces of revolut...
Citations
... The formulations for the three different plant matrices are taken respectively from [12], [10], and [3]. The density term ρ in the drag plant matrix can be computed onboard with any atmospheric model of choice, naturally, a better model improves the accuracy and optimality of the algorithm. ...
This paper proposes the MPC strategy designed to cope with the autonomous control of the two formation-flying VULCAIN CubeSats, designed to orbit on VLEO. VULCAIN is an Earth Observation scientific mission to image in VIS-IR the Earth volcanoes by means of two 12U CubeSats in train formation. The embarked scientific cameras need for instruments co-pointing to ensure that their footprints overlap. Therefore, an onboard optimal station keeping plan shall be generated on board to cope with the perturbed dynamics according to scientific acquisitions time windows, along which active control is forbidden. The proposed controller compensates the relative orbital deviations from the reference slots exploiting the system linearized and convexified dynamics expressed in quasi-nonsingular Relative Orbital Elements (ROE), allowing the use of fast and efficient solvers for the online optimization. Due to the MPC strong dependence on the state estimation quality, ROE are computed onboard with the desired accuracy from GNSS measurements integrated with state information coming from the inter-satellite link via appropriate filtering techniques exploiting the knowledge of the absolute and relative orbital dynamics. Numerical simulations run with the controller in closed loop with a high-fidelity orbital propagator including major perturbances in VLEO and considering realistic measurement errors for GNSS receivers. Results highlight good accuracy of the designed GNC scheme in maintaining the reference slot with feasible control profiles for the formation, while prescribing control actions only when no other operation is foreseen. Light computational times confirm the applicability of this solution for on board implementation.
... In the Keplerian two-body problem, the general linearized relative motion of a deputy satellite relative to the chief for arbitrary eccentricities is described in terms of ROE as reported by Guffanti et al. (2017): ...
... In particular, first-order secular effects of the second-order zonal geopotential harmonic J2 on the orbit geometry are included in the ROE propagation. The formulation that will be used is the one introduced by Guffanti et al. (2017) and reported in the following: In a drag-free environment, a smart design of J2invariant orbits allows to maintain the formation geometry with collision-free motion for hundreds of orbits with no additional station keeping (Morgan et al., 2012). That is why the majority of current research is focused on formation reconfiguration between invariant orbits, rather than formation maintenance for extended periods of time. ...
... Since the error caused by perturbations vary slowly with time, it is simpler and higher-fidelity to use ROEs to describe relative motion. To date, these research results can be divided into two main directions [24,25]. The first one was developed by Alfriend and Gim [26]. ...
... Generally speaking, there are two types of perturbations that cause changes in the orbital elements, one is conservative, and the other is non-conservative [24]. In the former case, the differential perturbation effect of the Chief and the Deputy is determined by the difference in orbital geometry, which can be recorded as . ...
An accurate description of the relative motion of satellites is critical for space missions in the complex geostationary earth orbit space environment. Here, a new relative motion model is established considering major perturbations. To avoid the singularity of the classical orbital elements, a set of quasi‐non‐singular orbital elements are used to represent the relative motion states, which is further extended by considering the SRP coefficient difference. On the basis of existing classical model, an analytical model is derived by taking the Taylor expansion of the time derivative of the perturbation functions to the first order in the neighbourhood of the reference satellite, which contains the earth's oblateness perturbation, the third‐body perturbations and the solar radiation pressure perturbation. This method avoids solving the complex differential equations of relative motion directly and provides a general framework for including arbitrary perturbations into the model. The new model is verified in uncontrolled natural relative motion scenarios and position‐keeping control scenario, respectively. The results show that the proposed model has high accuracy and strong competitiveness.
... (1) yields [21] ( ) = ...
... Same maneuver direction and timing under uncertainty the proposed controller uses the desired formation and the initial relative position as input as shown in Fig.5 and determines the timings and direction of the maneuver to correct the disturbed relative orbit after orbit cycle in Eqs. (21) and (22) ...
Laser ablation is a vital technology for contactless active debris removal, where a service satellite with a laser system irradiates laser pulses to a target satellite to generate the ablation force for deorbiting. The deorbiting force decelerates the target, and the service satellite needs to maintain its relative position and keep irradiating. In other words, both the service satellite and the target are supposed to be deorbited simultaneously, where both satellites have accelerations. The difficulty of simultaneous deorbiting stems from the relative motion between the service satellite and the target in powered flight because conventional formation flying missions assume that only a service satellite maneuvers. This paper derives the relative equations of motion between the service satellite and the target in powered flight. A control law for the simultaneous deorbit is proposed, which determines the timing and direction of the laser ablation and the electrical thrust so that the formation periodically returns to the desired formation. Numerical simulations are performed for two test cases to verify the control law under the uncertainties of thrust magnitude and orbital perturbation.
... Using VoP and isolating the effect of non-integrable dynamics on the IC state has various advantages. Firstly, it fosters the formulation of highly efficient analytical closed-form dynamics models [18][19][20][21]. Secondly, it permits to directly account/compensate for the effects of non-integrable dynamics in guidance and control strategies [11,22]. ...
... As described in the introduction, one approach uses the OE as IC state, while the other approach uses the ROE. From an operational stand-point, for spacecraft in closed proximity, modeling directly the relative motion is advantageous both from a dynamics modeling standpoint [18][19][20][21] and navigation accuracy standpoint [40]. In particular, about the latter, spacecraft absolute motion navigation solutions are less accurate than relative motion ones, usually by at least one order of magnitude. ...
... Both a two and five orbits transfers are considered, in which the 2 -norm of the control input is minimized, and PS is guaranteed in RN plane for at least two orbits ( ), of at least 3 or 5m ( ), at 3-confidence ( ). The uncertainty stems from differential GPS navigation [40], cold-gas impulsive actuation [42] and process-noise modeling the discrepancy between the full-force ground-truth [37,40] and the on-board dynamics model [19][20][21]. In particular, looking at Table 2 on the right, the relation between maneuver magnitude and associated actuation execution uncertainty is proportional, which motivates the uncertainty model formalized in Section II. ...
This paper develops a novel dynamics, guidance, and control framework that enables a multi-agent system to enforce optimally and efficiently motion safety guarantees even in case of sudden loss of control capabilities by any agent, i.e., guarantees of passive safety (PS). The main application are distributed space systems (DSS) employing miniaturized low-size-weight-and-power and commercial-off-the-shelf technology, which reduce mission financial costs at the expense of reliability. The framework explores the method of variation of parameters, which models the effects of non-integrable dynamics on the integration constants of an integrable portion of the governing dynamics itself, to reduce the number of constraints required to guarantee PS by one polynomial degree in the number of discrete time samples, while compensating for non-integrable dynamics and uncertainty effects. This enables efficient enforcement of PS within a multi-agent optimal control problem solvable using direct methods, as well as the formulation of novel closed-form solutions of DSS PS in closed eccentric orbits. Experimental results on the upcoming VIrtual Super Optics Reconfigurable Swarm (VISORS) flight mission, as well as complementary challenging formation-flying test cases in eccentric orbits, demonstrate the proposed advantage in terms of achieved safety guarantees, control accuracy and gained fuel and computational efficiency.
... Some work has also been done to analyze the relative motion problem in other coordinate systems, notably curvilinear coordinates and orbit element differences (Schaub, 2004). Additionally, some authors have considered the effects of various perturbations common in planetary orbits, including atmospheric drag (Carter and Humi, 2002), zonal gravity (Schweighart and Sedwick, 2002;Biria and Russell, 2018), and solar radiation pressure (Guffanti et al., 2017;Parsay and Schaub, 2016). These works are especially valuable for multi-spacecraft formations in low-Earth orbit (LEO) to medium-Earth orbit (MEO). ...
This paper explores methods for approximating and analyzing the dynamics of highly perturbed spacecraft formations with an emphasis on computationally efficient approaches. This facilitates on-board computation or rapid preliminary mission design analysis. Perturbed formation dynamics are often approximated as linear time-varying (LTV) systems, for which Floquet theory can be used to analyze the degree of system instability. Furthermore, the angular momentum of the relative orbital state can be computed with the approximate dynamics to provide additional insight. A general methodology is developed first and then applied to the problem of unstable formation dynamics in asteroid orbits. Here the dominant perturbative effects due to low-order gravitational harmonics and solar radiation pressure are modeled. Numerical simulations validate the approach and illustrate the approximation accuracy achieved.
... The SRP is modeled in the Cartesian Earth-Centered-Inertial (ECI) frame as a differential perturbing acceleration purely dependent on the ballistic coefficient difference between the spacecraft and subsequently applied to the ROE state through linear maps. Following the research track started by D'Amico in his thesis [22], Guffanti et al. [31] presented a linear model for the SRP valid in arbitrary eccentric orbit around Earth including both effects due to spacecraft ballistic coefficient difference and orbital position differences. ...
... The methodology used to derive the linear dynamics models for the two different state representations has been developed by the authors in [28,31]. First, the time derivative of the relative state is expressed as ...
... In Eq. (4) the ballistic coefficient difference is assumed constant. This method for deriving the plant is general and has been used in [28,31] to compute plants of various perturbations. The feasibility of solving Eq. (4) in closed form to formalize an STM depends on the nature and structure of the plant matrix. ...
Many scientific applications require the implementation of spacecraft formation-flying around Earth and other celestial bodies. The control of these formations calls for simple relative dynamics models able to accurately and efficiently incorporate secular and long-periodic effects of relevant perturbations. This paper presents novel linear analytical models of the effect of solar radiation pressure (SRP) on the relative motion of two spacecraft in arbitrary eccentric orbit for two state definitions based on relative orbital elements. The new models are valid both around Earth and around other celestial bodies such as near-Earth asteroids. The linear models are derived by augmenting the state with force model parameters and Taylor expanding the differential equations of relative motion, including the secular and long-periodic effects of SRP. New state transition matrices including J2 and SRP are formalized and validated through comparison with a high-fidelity orbit propagator around Earth and around a near-Earth asteroid. In addition, the dominant trends caused by SRP on the relative orbital elements are captured in closed-form. The new models have accuracy in the range of meter to few tens of meters in all orbit scenarios considered and can be leveraged for formation guidance, control, and design.
... These methods demonstrate that an empirical formulation is able to capture the mean effects of differential drag perturbations. Spiridonova [23], and Guffanti et al. [24] further extended the developments of [21,22] to model solar radiation pressure and third-body perturbations due to the sun and moon for the relative motion at near-circular orbits. ...
... The ROE as defined by D'Amico [2] are nonlinear combinations of the Keplerian orbital elements and IC of the homogeneous nonlinear equations of relative motion [16]. The effect of control and perturbations on the ROE is modeled by means of the Gauss Variational Equations [16][17] [18], i.e., the ICV equation. Optimally control the satellite relative motion in ROE space permits to intrinsically leverage the integrable-Keplerian dynamics, and moreover, to include the effect of perturbations at the first-order and compensate them geometrically [5] [16]. ...
This paper explores the advantages of a state
parameterization in integration constants to optimally path plan
multi-dimensional systems governed by ordinary differential
equations. The idea is inherited from astrodynamics, where the
integration constants of the unperturbed equations of orbital
motion – orbital elements – are used to path plan optimally
artificial satellites in space. This paper shows that the advantages
gained from this state parameterization can be extended to
other applications. In order to do so, firstly, it presents a general
methodology to model the governing equations – in general nonintegrable
and nonlinear – using the integration constants of a
reduced integrable part. Subsequently, it formalizes the optimal
path planning problem in integration constant space, where a
geometric interpretation of the minimum input cost path is
provided together with an algorithmic procedure to identify it.
It is shown that working in integration constant space simplifies
the path planning. In particular, the integrable dynamics can
be exploited intrinsically, while the part of the dynamics which
is not integrable can be isolated and compensated.
Virtual Super-resolution Optics with Reconfigurable Swarms (VISORS) is a distributed telescope mission for high-resolution imaging of the Sun using two 6U CubeSats flying in formation in a Sun-synchronous low-Earth orbit. An optics spacecraft carries a photon sieve acting as a high-resolution lens in the extreme ultraviolet spectrum, while the image passing through the sieve is focused on a detector spacecraft. This paper presents the newly conceived design of the on-board guidance, navigation and control (GNC) system, which is highly autonomous , robust, passively safe, and validated under realistic mission simulations. The primary objective of the GNC system is to establish a passively safe and high-precision formation alignment at 40-meter separation, with sub-centimeter relative navigation and position control accuracy, over repeated observations of 10-second duration. Science mission success rates are assessed via Monte-Carlo analyses under realistically modelled uncertainties stemming from sensing errors, maneuver errors, unmodelled dynamics, and erroneous knowledge of internal spacecraft components. Precise real-time relative navigation is achieved by carrier phase differential GPS with integer ambiguity resolution. Precise control over short baselines is achieved via closed-loop optimization-based stochastic model predictive control with centimeter-level accuracy. Control at far range and during approach is achieved by closed-form impulsive control with meter-level accuracy. Fault detection with isolation and recovery is constantly assessed by the GNC system, while passive safety is enforced throughout the mission to mitigate collision risks even under critical subsystem failure. Beyond VISORS, this work also realizes the crucial insight that the described GNC architecture is generalizable to other distributed space missions where accuracy and fault-tolerant safety are key requirements, such as rendezvous, proximity operations, and swarming missions.
