The equations of motion of a secularly precessing elliptical orbit

The equations of motion of a secularly precessing ellipse are developed using time as the independent variable. The equations are useful when integrating numerically the perturbations about a reference trajectory which is subject to secular perturbations in the node, the argument of pericenter and the mean motion. Usually this is done in connection with Encke's method to ensure minimal rectification frequency. Similar equations are already available in the literature, but they are either given based on the true anomaly as the independent variable, or in mixed mode with respect to the time through the use of a supporting equation to track the anomaly. The equations developed here form a complete and independent set of six equations in the time. Reformulations both of Escobal's and Kyner and Bennett's equations are also provided which lead to a more concise form.Comment: Accepted in Monthly Notices of the Royal Astronomical Society. Paper presented at the "New Trends in Astrodynamics and Applications VI" conference, Courant Institute of Mathematical Sciences, New York University New York, NY, 6-8 June 201

Optimal options for rendezvous and impact missions to NEOs

In this paper some potentially interesting transfer options for missions to Near Earth Objects have been studied. Due to thehigh number of potential targets and to the large variety of possible missions that can be considered, especially if resorting to low-thrust propulsion, an extensive analysis of transfer options requires a preliminary approach oriented toward an effective global search, and an appropriately simplified trajectory transcription. Low-thrust options have been modeled through a novel shape-based approach and a global optimization method has been used to look for globally optimal transfers. Different targets have been identified and various mission scenarios have been considered: rendezvous, sample return missions both with and without Earth gravity assist and impact missions

Erratum to: Methods of harmonic synthesis for global geopotential models and their first-, second- and third-order gradients

n/

Proximal Splitting Meets Variance Reduction

Despite the rise to fame of incremental variance-reduced methods in recent years, their use in nonsmooth optimization is still limited to few simple cases. This is due to the fact that existing methods require to evaluate the proximity operator for the nonsmooth terms, which can be a costly operation for complex penalties. In this work we introduce two variance-reduced incremental methods based on SAGA and SVRG that can efficiently take into account complex penalties which can be expressed as a sum of proximal terms. This includes penalties such as total variation, group lasso with overlap and trend filtering, to name a few. Furthermore, we also develop sparse variants of the proposed algorithms which can take advantage of sparsity in the input data. Like other incremental methods, it only requires to evaluate the gradient of a single sample per iteration, and so is ideally suited for large scale applications. We provide a convergence rate analysis for the proposed methods and show that they converge with a fixed step-size, achieving in some cases the same asymptotic rate as their full gradient variants. Empirical benchmarks on 3 different datasets illustrate the practical advantages of the proposed methods

Thermo-mechanical-metallurgical model to predict geometrical distortions of rings during cooling phase after ring rolling operations

peer reviewedThe paper presents a validated numerical model able to predict geometrical distortions of rings during cooling phase after hot ring rolling operations. The model is capable to take into account the effects of all the phenomena resulting from the coupling of thermal, mechanical and metallurgical events. As simulation results strongly depend on the accuracy of input data, physical simulation experiments on real-material samples are developed and carried out to characterize material behaviour during phase transformation. The numerical model is then validated by an industrial case proving its effectiveness in predicting final ring distortions at room temperature

Analysis and Performance Evaluation of the ZEM/ZEV Guidance and its Sliding Robustification for Autonomous Rendezvous in Relative Motion

Devising closed-loop guidance algorithms for autonomous relative motion is an important problem within the field of orbital dynamics. In this paper, we study the guided relative motion of two spacecraft for which one of them is executing an autonomous rendezvous via the ZEM/ZEV feedback guidance and its robustified Optimal Sliding Guidance (OSG) counterpart. Starting from the classical Clohessy-Wiltshire (CW) model, we systematically analyze the ability of the ZEM/ZEV feedback guidance to generate closed loop trajectories that drive the deputy spacecraft to the chief satellite and evaluate its performance in terms of target accuracy and propellant consumption. It is shown that the guidance gains and the time of flight predicted by the theoretical solution generates a class of feedback trajectories that are accurate but suboptimal with respect to the open-loop fuel-optimal solution. Indeed, a parametric study shows that a different set of gains may generate relative guided trajectories that yields fuel consumption closer to the ideal optimal. The guidance algorithms are also demonstrated to be accurate in guiding the relative motion of the deputy toward a chief spacecraft in highly elliptical orbit where the Linearized Equations of Relative Motions (LERM) are employed to compute the Zero-Effort-Miss (ZEM) and Zero-Effort-Velocity (ZEV) necessary to compute the acceleration command as prescribed by the theory

Prediction of distortion during cooling of steel rolled rings using thermal-mechanical-metallurgical finite element model

peer reviewedThis work takes place in the framework of a CRAFT European project gathering three universities, three companies who produce rings through the ring rolling process and a manufacturer of temperature and dimension measurement devices. The final goal of the project is to develop and set up a system, integrated in the industrial process, capable of predicting the geometrical characteristics of final pieces just after the ring rolling stage and to allow the rolling process to avoid dimensional defects through online adaption. In fact, ring rolling production does not imply only the rolling process, but also the cooling and quench stages of steel rings. During all these phases, the dimensions of the pieces change dramatically. In particular, due to the lack of symmetry in the cooling conditions, ring distortions include contraction and rotation of the ring section. The modeling of the cooling phase requires taking into account a large number of phenomena resulting from the coupling of thermal, mechanical and metallurgical effects. A numerical model has been implemented in the non-linear finite element code LAGAMINE, developed by the University of Liège. Such a model can help to better understand the evolution of the geometry during the cooling phase and also the effects of each physical and microstructural parameter implemented in the model on the ring final shape. Effectively, several parameters can affect the ring distortions and the model should take them into account; in particular, the mechanical and thermal behavior of each phase present in the material (metastable austenite, ferrite, pearlite, bainite and martensite). Phase transformation modeling implies the integration of a wide data base of material properties (thermo-physical and mechanical properties of the phases, TTT and CCT diagrams, enthalpy and strain of phase transformation, strain of transformation plasticity…) but only a few of these data are available in literature. Some of them have been found for the reference material (42CrMo4 steel), but additional laboratory experiments have been performed at the Universities of Padua and Liège in order to characterize thermal, mechanical and plastic behaviour of phases. Finally, this paper presents the model validation on an industrial case (measurements of temperature and dimensions of rings have been provided by the manufacturer). Then, some applications are presented, demonstrating the importance of some factors such as some material properties, the shape of the rings, the type of cooling (and the cooling rate) or the symmetry of the cooling scheme on final ring distortion

Dynamical orbital effects of General Relativity on the satellite-to-satellite range and range-rate in the GRACE mission: a sensitivity analysis

We numerically investigate the impact of GTR on the orbital part of the satellite-to-satellite range \rho and range-rate \dot\rho of the twin GRACE A/B spacecrafts through their dynamical equations of motion integrated in an Earth-centered frame over a time span \Delta t=1 d. Instead, the GTR effects connected with the propagation of the electromagnetic waves linking the spacecrafts are neglected. The present-day accuracies in measuring the GRACE biased range and range-rate are \sigma_\rho\sim 1-10 \mum, \sigma_\dot\rho\sim 0.1-1 \mum s^-1; studies for a follow-on of such a mission points toward a range-rate accuracy of the order of \sigma_\dot\rho\sim 1 nm s^-1 or better. The GTR range and range-rate effects turn out to be \Delta\rho=80 \mum and \Delta\dot\rho=0.012 \mum s^-1 (Lense-Thirring), and \Delta\rho=6000 \mum and \Delta\dot\rho=10 \mum s^-1 (Schwarzschild). We also compute the dynamical range and range-rate perturbations caused by the first six zonal harmonic coefficients J_L, L=2,3,4,5,6,7 of the classical multipolar expansion of the terrestrial gravitational potential in order to evaluate their aliasing impact on the relativistic effects. Conversely, we also quantitatively, and preliminarily, assess the possible a-priori \virg{imprinting} of GTR itself, not solved-for in all the GRACE-based Earth's gravity models produced so far, on the estimated values of the low degree zonals of the geopotential. The present sensitivity analysis can also be extended, in principle, to different orbital configurations in order to design a suitable dedicated mission able to accurately measure the relativistic effects considered.Comment: LaTex, 24 pages, 5 figures, 9 tables. Accepted for publication in Advances in Space Research (ASR

Orbital effects of spatial variations of fundamental coupling constants

We deal with the effects induced on the orbit of a test particle revolving around a central body by putative spatial variations of fundamental coupling constants $\zeta$. In particular, we assume a dipole gradient for \zeta(\bds r)/\bar{\zeta} along a generic direction \bds{\hat{k}} in space. We analytically work out the long-term variations of all the six standard Keplerian orbital elements parameterizing the orbit of a test particle in a gravitationally bound two-body system. It turns out that, apart from the semi-major axis $a$, the eccentricity $e$, the inclination $I$, the longitude of the ascending node $\Omega$, the longitude of pericenter $\pi$ and the mean anomaly $\mathcal{M}$ undergo non-zero long-term changes. By using the usual decomposition along the radial ($R$), transverse ($T$) and normal ($N$) directions, we also analytically work out the long-term changes $\Delta R,\Delta T,\Delta N$ and $\Delta v_R,\Delta v_T,\Delta v_N$ experienced by the position and the velocity vectors \bds r and \bds v of the test particle. It turns out that, apart from $\Delta N$, all the other five shifts do not vanish over one full orbital revolution. In the calculation we do not use \textit{a-priori} simplifying assumptions concerning $e$ and $I$. Thus, our results are valid for a generic orbital geometry; moreover, they hold for any gradient direction (abridged).Comment: Latex2e, 20 pages, 1 figure, 7 tables. Version accepted by Monthly Notices of the Royal Astronomical Society (MNRAS). Error in the caption of Table 5 corrected. References update