Density of high area-to-mass objects in Geostationary and Medium Earth orbits through semi-analytical equations and differential algebra

This paper introduces and combines two novel techniques. Firstly, we introduce an efficient numerical method for the propagation of entire sets of initial conditions in the phase space and their associated phase space densities based on Differential Algebra (DA) techniques. Secondly, this DA density propagator is applied to a DA-enabled implementation of Semi-Analytical (SA) averaged dynamics, combining for the first time the power of the SA and DA techniques. While the DA-based method for the propagation of densities introduced in this paper is independent of the dynamical system under consideration, the particular combination of DA techniques with SA equations yields a fast and accurate method to propagate large clouds of initial conditions and their associated probability density functions very efficiently for long time. This enables the study of the long-term behavior of particles subjected to the given dynamics. To demonstrate the effectiveness of the proposed method, the evolution of a cloud of high area-to-mass objects in Medium Earth Orbit is reproduced considering the effects of solar radiation pressure, the Earth’s oblateness and luni-solar perturbations. The computational efficiency is demonstrated by propagating 10; 000 random samples taking snapshots of their state and density at evenly spaced intervals throughout the integration. The total time required for a propagation for 16 years in the dynamics is on the order of tens of seconds on a common desktop PC

Dealing with uncertainties in angles-only initial orbit determination

A method to deal with uncertainties in initial orbit determination (IOD) is presented. This is based on the use of Taylor differential algebra (DA) to nonlinearly map uncertainties from the observation space to the state space. When a minimum set of observations is available, DA is used to expand the solution of the IOD problem in Taylor series with respect to measurement errors. When more observations are available, high order inversion tools are exploited to obtain full state pseudo-observations at a common epoch. The mean and covariance of these pseudo-observations are nonlinearly computed by evaluating the expectation of high order Taylor polynomials. Finally, a linear scheme is employed to update the current knowledge of the orbit. Angles-only observations are considered and simplified Keplerian dynamics adopted to ease the explanation. Three test cases of orbit determination of artificial satellites in different orbital regimes are presented to discuss the feature and performances of the proposed methodology

Time-Varying Lyapunov Control Laws with Enhanced Estimation of Distribution Algorithm for Low-Thrust Trajectory Design

Enhancements in evolutionary optimization techniques are rapidly growing in many aspects of engineering, specifically in astrodynamics and space trajectory optimization and design. In this chapter, the problem of optimal design of space trajectories is tackled via an enhanced optimization algorithm within the framework of Estimation of Distribution Algorithms (EDAs), incorporated with Lyapunov and Q-law feedback control methods. First, both a simple Lyapunov function and a Q-law are formulated in Classical Orbital Elements (COEs) to provide a closed-loop low-thrust trajectory profile. The weighting coefficients of these controllers are approximated with various degrees of Hermite interpolation splines. Following this model, the unknown time series of weighting coefficients are converted to unknown interpolation points. Considering the interpolation points as the decision variables, a black-box optimization problem is formed with transfer time and fuel mass as the objective functions. An enhanced EDA is proposed and utilized to find the optimal variation of weighting coefficients for minimum-time and minimum-fuel transfer trajectories. The proposed approach is applied in some trajectory optimization problems of Earth-orbiting satellites. Results show the efficiency and the effectiveness of the proposed approach in finding optimal transfer trajectories. A comparison between the Q-law and simple Lyapunov controller is done to show the potential of the potential of the EEDA in enabling the simple Lyapunov controller to recover the finer nuances explicitly given within the analytical expressions in the Q-law

A high order method for orbital conjunctions analysis: Monte Carlo collision probability computation

Three methods for the computation of the probability of collision between two space objects are presented. These methods are based on the high order Taylor expansion of the time of closest approach (TCA) and distance of closest approach (DCA) of the two orbiting objects with respect to their initial conditions. The identification of close approaches is first addressed using the nominal objects states. When a close approach is identified, the dependence of the TCA and DCA on the uncertainties in the initial states is efficiently computed with differential algebra (DA) techniques. In the first method the collision probability is estimated via fast DA-based Monte Carlo simulation, in which, for each pair of virtual objects, the DCA is obtained via the fast evaluation of its Taylor expansion. The second and the third methods are the DA version of Line Sampling and Subset Simulation algorithms, respectively. These are introduced to further improve the efficiency and accuracy of Monte Carlo collision probability computation, in particular for cases of very low collision probabilities. The performances of the methods are assessed on orbital conjunctions occurring in different orbital regimes and dynamical models. The probabilities obtained and the associated computational times are compared against standard (i.e. not DA-based) version of the algorithms and analytical methods. The dependence of the collision probability on the initial orbital state covariance is investigated as wel

Nonlinear control of leader-follower formation flying

This paper considers the problem of relative motion control involved in a leader-follower formation keeping mission. More specifically, center of mass dynamics of two Earth orbiting satellite is modeled, including the nonlinearity due to Earth oblateness. Next, the differential algebra is exploited to compute an high order Taylor expansion of the State-Dependent Riccati Equation (SDRE) solution. This new approach reduces the computational cost of the online Algebraic Riccati Equation solution required by SDRE algorithm; in fact, the differential algebraic formulation gives a polynomial representation which can be directly evaluated for SDRE solutions or exploited to define an initial first guess for iterative SDRE algorithms

Diffraction limited optics for single atom manipulation

We present an optical system designed to capture and observe a single neutral atom in an optical dipole trap, created by focussing a laser beam using a large numerical aperture N.A.=0.5 aspheric lens. We experimentally evaluate the performance of the optical system and show that it is diffraction limited over a broad spectral range (~ 200 nm) with a large transverse field (+/- 25 microns). The optical tweezer created at the focal point of the lens is able to trap single atoms of 87Rb and to detect them individually with a large collection efficiency. We measure the oscillation frequency of the atom in the dipole trap, and use this value as an independent determination of the waist of the optical tweezer. Finally, we produce with the same lens two dipole traps separated by 2.2 microns and show that the imaging system can resolve the two atoms.Comment: 8 pages, 9 figures; typos corrected and references adde

Probabilistic data association: the orbit set

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