The Experts below are selected from a list of 279 Experts worldwide ranked by ideXlab platform
William M. Mceneaney - One of the best experts on this subject based on the ideXlab platform.
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Fundamental Solutions for Two-Point Boundary Value Problems in Orbital Mechanics
Applied Mathematics & Optimization, 2016Co-Authors: Seung Hak Han, William M. MceneaneyAbstract:We consider a two-point boundary value problem (TPBVP) in Orbital Mechanics involving a small body (e.g., a spacecraft or asteroid) and N larger bodies. The least action principle TPBVP formulation is converted into an initial value problem via the addition of an appropriate terminal cost to the action functional. The latter formulation is used to obtain a fundamental solution, which may be used to solve the TPBVP for a variety of boundary conditions within a certain class. In particular, the method of convex duality allows one to interpret the least action principle as a differential game, where an opposing player maximizes over an indexed set of quadratics to yield the gravitational potential. In the case where the time duration is less than a specific bound, there exists a unique critical point for the resulting differential game, which yields the fundamental solution given in terms of the solutions of associated Riccati equations.
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the principle of least action and fundamental solution of two point boundary value problems in Orbital Mechanics
European Control Conference, 2014Co-Authors: Seung Hak Han, William M. MceneaneyAbstract:The two-point boundary value problem (TPBVP) in Orbital Mechanics involving a small body (e.g., a spacecraft or asteroid) and N larger bodies is considered. The least action principle TPBVP formulation is converted into an initial value problem via the addition of an appropriate terminal cost to the action functional. The latter formulation is used to obtain a fundamental solution, which may be used to solve the TPBVP for a variety of boundary conditions within a certain class. In particular, the method of convex duality allows one to interpret the least action principle as a differential game, where an opposing player maximizes over an indexed set of quadratics to yield the gravitational potential. In the case where the time-duration is less than a specific bound, the fundamental solution is obtained as a set of solutions of Riccati equations associated with the resulting differential game.
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the principle of least action and two point boundary value problems in Orbital Mechanics
Advances in Computing and Communications, 2014Co-Authors: Seung Hak Han, William M. MceneaneyAbstract:We consider a two-point boundary value problem (TPBVP) in Orbital Mechanics involving a small body (e.g., a spacecraft or asteroid) and N larger bodies. The least action principle TPBVP formulation is converted into an initial value problem via the addition of an appropriate terminal cost to the action functional. The latter formulation is used to obtain a fundamental solution, which may be used to solve the TPBVP for a variety of boundary conditions within a certain class. In particular, the method of convex duality allows one to interpret the least action principle as a differential game, where an opposing player maximizes over an indexed set of quadratics to yield the gravitational potential. The fundamental solution is obtained as a set of solutions of associated Riccati equations.
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ACC - The principle of least action and two-point boundary value problems in Orbital Mechanics
2014 American Control Conference, 2014Co-Authors: Seung Hak Han, William M. MceneaneyAbstract:We consider a two-point boundary value problem (TPBVP) in Orbital Mechanics involving a small body (e.g., a spacecraft or asteroid) and N larger bodies. The least action principle TPBVP formulation is converted into an initial value problem via the addition of an appropriate terminal cost to the action functional. The latter formulation is used to obtain a fundamental solution, which may be used to solve the TPBVP for a variety of boundary conditions within a certain class. In particular, the method of convex duality allows one to interpret the least action principle as a differential game, where an opposing player maximizes over an indexed set of quadratics to yield the gravitational potential. The fundamental solution is obtained as a set of solutions of associated Riccati equations.
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ECC - The principle of least action and fundamental solution of two-point boundary value problems in Orbital Mechanics
2014 European Control Conference (ECC), 2014Co-Authors: Seung Hak Han, William M. MceneaneyAbstract:The two-point boundary value problem (TPBVP) in Orbital Mechanics involving a small body (e.g., a spacecraft or asteroid) and N larger bodies is considered. The least action principle TPBVP formulation is converted into an initial value problem via the addition of an appropriate terminal cost to the action functional. The latter formulation is used to obtain a fundamental solution, which may be used to solve the TPBVP for a variety of boundary conditions within a certain class. In particular, the method of convex duality allows one to interpret the least action principle as a differential game, where an opposing player maximizes over an indexed set of quadratics to yield the gravitational potential. In the case where the time-duration is less than a specific bound, the fundamental solution is obtained as a set of solutions of Riccati equations associated with the resulting differential game.
Zhen Yang - One of the best experts on this subject based on the ideXlab platform.
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A review of uncertainty propagation in Orbital Mechanics
Progress in Aerospace Sciences, 2017Co-Authors: Ya-zhong Luo, Zhen YangAbstract:Abstract Orbital uncertainty propagation plays an important role in space situational awareness related missions such as tracking and data association, conjunction assessment, sensor resource management and anomaly detection. Linear models and Monte Carlo simulation were primarily used to propagate uncertainties. However, due to the nonlinear nature of Orbital dynamics, problems such as low precision and intensive computation have greatly hampered the application of these methods. Aiming at solving these problems, many nonlinear uncertainty propagators have been proposed in the past two decades. To motivate this research area and facilitate the development of Orbital uncertainty propagation, this paper summarizes the existing linear and nonlinear uncertainty propagators and their associated applications in the field of Orbital Mechanics. Frameworks of methods for Orbital uncertainty propagation, the advantages and drawbacks of different methods, as well as potential directions for future efforts are also discussed.
Seung Hak Han - One of the best experts on this subject based on the ideXlab platform.
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Fundamental Solutions for Two-Point Boundary Value Problems in Orbital Mechanics
Applied Mathematics & Optimization, 2016Co-Authors: Seung Hak Han, William M. MceneaneyAbstract:We consider a two-point boundary value problem (TPBVP) in Orbital Mechanics involving a small body (e.g., a spacecraft or asteroid) and N larger bodies. The least action principle TPBVP formulation is converted into an initial value problem via the addition of an appropriate terminal cost to the action functional. The latter formulation is used to obtain a fundamental solution, which may be used to solve the TPBVP for a variety of boundary conditions within a certain class. In particular, the method of convex duality allows one to interpret the least action principle as a differential game, where an opposing player maximizes over an indexed set of quadratics to yield the gravitational potential. In the case where the time duration is less than a specific bound, there exists a unique critical point for the resulting differential game, which yields the fundamental solution given in terms of the solutions of associated Riccati equations.
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the principle of least action and fundamental solution of two point boundary value problems in Orbital Mechanics
European Control Conference, 2014Co-Authors: Seung Hak Han, William M. MceneaneyAbstract:The two-point boundary value problem (TPBVP) in Orbital Mechanics involving a small body (e.g., a spacecraft or asteroid) and N larger bodies is considered. The least action principle TPBVP formulation is converted into an initial value problem via the addition of an appropriate terminal cost to the action functional. The latter formulation is used to obtain a fundamental solution, which may be used to solve the TPBVP for a variety of boundary conditions within a certain class. In particular, the method of convex duality allows one to interpret the least action principle as a differential game, where an opposing player maximizes over an indexed set of quadratics to yield the gravitational potential. In the case where the time-duration is less than a specific bound, the fundamental solution is obtained as a set of solutions of Riccati equations associated with the resulting differential game.
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the principle of least action and two point boundary value problems in Orbital Mechanics
Advances in Computing and Communications, 2014Co-Authors: Seung Hak Han, William M. MceneaneyAbstract:We consider a two-point boundary value problem (TPBVP) in Orbital Mechanics involving a small body (e.g., a spacecraft or asteroid) and N larger bodies. The least action principle TPBVP formulation is converted into an initial value problem via the addition of an appropriate terminal cost to the action functional. The latter formulation is used to obtain a fundamental solution, which may be used to solve the TPBVP for a variety of boundary conditions within a certain class. In particular, the method of convex duality allows one to interpret the least action principle as a differential game, where an opposing player maximizes over an indexed set of quadratics to yield the gravitational potential. The fundamental solution is obtained as a set of solutions of associated Riccati equations.
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ACC - The principle of least action and two-point boundary value problems in Orbital Mechanics
2014 American Control Conference, 2014Co-Authors: Seung Hak Han, William M. MceneaneyAbstract:We consider a two-point boundary value problem (TPBVP) in Orbital Mechanics involving a small body (e.g., a spacecraft or asteroid) and N larger bodies. The least action principle TPBVP formulation is converted into an initial value problem via the addition of an appropriate terminal cost to the action functional. The latter formulation is used to obtain a fundamental solution, which may be used to solve the TPBVP for a variety of boundary conditions within a certain class. In particular, the method of convex duality allows one to interpret the least action principle as a differential game, where an opposing player maximizes over an indexed set of quadratics to yield the gravitational potential. The fundamental solution is obtained as a set of solutions of associated Riccati equations.
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ECC - The principle of least action and fundamental solution of two-point boundary value problems in Orbital Mechanics
2014 European Control Conference (ECC), 2014Co-Authors: Seung Hak Han, William M. MceneaneyAbstract:The two-point boundary value problem (TPBVP) in Orbital Mechanics involving a small body (e.g., a spacecraft or asteroid) and N larger bodies is considered. The least action principle TPBVP formulation is converted into an initial value problem via the addition of an appropriate terminal cost to the action functional. The latter formulation is used to obtain a fundamental solution, which may be used to solve the TPBVP for a variety of boundary conditions within a certain class. In particular, the method of convex duality allows one to interpret the least action principle as a differential game, where an opposing player maximizes over an indexed set of quadratics to yield the gravitational potential. In the case where the time-duration is less than a specific bound, the fundamental solution is obtained as a set of solutions of Riccati equations associated with the resulting differential game.
Ya-zhong Luo - One of the best experts on this subject based on the ideXlab platform.
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A review of uncertainty propagation in Orbital Mechanics
Progress in Aerospace Sciences, 2017Co-Authors: Ya-zhong Luo, Zhen YangAbstract:Abstract Orbital uncertainty propagation plays an important role in space situational awareness related missions such as tracking and data association, conjunction assessment, sensor resource management and anomaly detection. Linear models and Monte Carlo simulation were primarily used to propagate uncertainties. However, due to the nonlinear nature of Orbital dynamics, problems such as low precision and intensive computation have greatly hampered the application of these methods. Aiming at solving these problems, many nonlinear uncertainty propagators have been proposed in the past two decades. To motivate this research area and facilitate the development of Orbital uncertainty propagation, this paper summarizes the existing linear and nonlinear uncertainty propagators and their associated applications in the field of Orbital Mechanics. Frameworks of methods for Orbital uncertainty propagation, the advantages and drawbacks of different methods, as well as potential directions for future efforts are also discussed.
Michael Efroimsky - One of the best experts on this subject based on the ideXlab platform.
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gauge freedom in Orbital Mechanics
Annals of the New York Academy of Sciences, 2005Co-Authors: Michael EfroimskyAbstract:: Both Orbital and attitude dynamics employ the method of variation of parameters. In a non-perturbed setting, the coordinates (or the Euler angles) are expressed as functions of the time and six adjustable constants called elements. Under disturbance, each such expression becomes ansatz, the “constants” being endowed with time dependence. The perturbed velocity (linear or angular) consists of a partial time derivative and a convective term containing time derivatives of the “constants.” It can be shown that this construction leaves one with a freedom to impose three arbitrary conditions on the “constants” and/or their derivatives. Out of convenience, the Lagrange constraint is often imposed. It nullifies the convective term and thereby guarantees that under perturbation the functional dependence of the velocity upon the time and “constants” stays the same as in the undisturbed case. “Constants” obeying this condition are called osculating elements. The “constants” chosen to be canonical, are called Delaunay elements, in the Orbital case, or Andoyer elements, in the spin case. (Because some of the Andoyer elements are time dependent, even in the free-spin case, the role of “constants” is played by the initial values of these elements.) The Andoyer and Delaunay sets of elements share a feature not readily apparent: in certain cases the standard equations render these elements non-osculating. In Orbital Mechanics, elements calculated via the standard planetary equations turn out to be non-osculating when perturbations depend on velocities. To keep elements osculating under such perturbations, the equations must be amended with additional terms that are not parts of the disturbing function (Efroimsky and Goldreich 2003, 2004). For the Kepler elements, this merely complicates the equations. In the case of Delaunay parameterization, these extra terms not only complicate the equations, but also destroy their canonicity. So under velocity-dependent disturbances, osculation and canonicity are incompatible. Similarly, in spin dynamics the Andoyer elements turn out to be non-osculating under angular-velocity-dependent perturbation (a switch to a noninertial frame being one such case). Amendment of the dynamical equations only with extra terms in the Hamiltonian makes the equations render nonosculating Andoyer elements. To make them osculating, more terms must enter the equations (and the equations will no longer be canonical). It is often convenient to deliberately deviate from osculation by substituting the Lagrange constraint with an arbitrary condition that gives birth to a family of nonosculating elements. The freedom in choosing this condition is analogous to the gauge freedom. Calculations in nonosculating variables are mathematically valid and sometimes highly advantageous, but their physical interpretation is nontrivial. For example, nonosculating Orbital elements parameterize instantaneous conics not tangent to the orbit, so the nonosculating inclination will be different from the real inclination of the physical orbit. We present examples of situations in which ignorance of the gauge freedom (and of the unwanted loss of osculation) leads to oversights.
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Gauge Freedom in Orbital Mechanics
Annals of the New York Academy of Sciences, 2005Co-Authors: Michael EfroimskyAbstract:In Orbital and attitude dynamics the coordinates and the Euler angles are expressed as functions of the time and six constants called elements. Under disturbance, the constants are endowed with time dependence. The Lagrange constraint is then imposed to guarantee that the functional dependence of the perturbed velocity on the time and constants stays the same as in the undisturbed case. Constants obeying this condition are called osculating elements. The constants chosen to be canonical are called Delaunay elements, in the Orbital case, or Andoyer elements, in the spin case. (As some Andoyer elements are time dependent even in the free-spin case, the role of constants is played by their initial values.) The Andoyer and Delaunay sets of elements share a feature not readily apparent: in certain cases the standard equations render them non-osculating. In Orbital Mechanics, elements furnished by the standard planetary equations are non-osculating when perturbations depend on velocities. To preserve osculation, the equations must be amended with extra terms that are not parts of the disturbing function. In the case of Delaunay parameterisation, these terms destroy canonicity. So under velocity-dependent disturbances, osculation and canonicity are incompatible. (Efroimsky and Goldreich 2003, 2004) Similarly, the Andoyer elements turn out to be non-osculating under angular-velocity-dependent perturbation. Amendment of only the Hamiltonian makes the equations render nonosculating elements. To make them osculating, more terms must enter the equations (and the equations will no longer be canonical). In practical calculations, is often convenient to deliberately deviate from osculation by substituting the Lagrange constraint with a condition that gives birth to a family of nonosculating elements.
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The method of variation of constants and multiple time scales in Orbital Mechanics
Chaos (Woodbury N.Y.), 2003Co-Authors: William I. Newman, Michael EfroimskyAbstract:The method of variation of constants is an important tool used to solve systems of ordinary differential equations, and was invented by Euler and Lagrange to solve a problem in Orbital Mechanics. This methodology assumes that certain “constants” associated with a homogeneous problem will vary in time in response to an external force. It also introduces one or more constraint equations. We show that these constraints can be generalized in analogy to gauge theories in physics, and that different constraints can offer conceptual advances and methodological benefits to the solution of the underlying problem. Examples are given from linear ordinary differential equation theory and from Orbital Mechanics. However, a slow driving force in the presence of multiple time scales contained in the underlying (homogeneous) problem nevertheless requires special care, and this has strong implications to the analytic and numerical solutions of problems ranging from celestial Mechanics to molecular dynamics.