Generalization of Lagrange's implicit function theorem to N-dimensions

Feagin, T.; Gottlieb, Robert G.

doi:10.1007/bf01228035

Cited by 8 publications

(2 citation statements)

References 3 publications

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“…The reversion of series for a function of one or many variables (Feagin and Gottlieb 1971), can also be written using tensors of increasing order (Turner 2003). In the present case, following the development in Turner (2003), a reversion of series is applied on (22), comprising linear and quadratic terms only.…”

Section: Inverse Map Using Series Reversionmentioning

confidence: 99%

Second-order state transition for relative motion near perturbed, elliptic orbits

Sengupta

Vadali

Alfriend

2006

Celestial Mech Dyn Astr

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This paper develops a tensor and its inverse, for the analytical propagation of the position and velocity of a satellite, with respect to another, in an eccentric orbit. The tensor is useful for relative motion analysis where the separation distance between the two satellites is large. The use of nonsingular elements in the formulation ensures uniform validity even when the reference orbit is circular. Furthermore, when coupled with state transition matrices from existing works that account for perturbations due to Earth oblateness effects, its use can very accurately propagate relative states when oblateness effects and second-order nonlinearities from the differential gravitational field are of the same order of magnitude. The effectiveness of the tensor is illustrated with various examples.

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Section: Inverse Map Using Series Reversionmentioning

confidence: 99%

Second-order state transition for relative motion near perturbed, elliptic orbits

Sengupta

Vadali

Alfriend

2006

Celestial Mech Dyn Astr

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“…However, both the above solutions of second-order equations are approximate such that the precision of initial relative velocity can not be adequately guaranteed. Feagin and Gottlieb (1971) and Turner (2003) used series reversion method to solve the two-point boundary value problem. Lizia et al (2008) gave a high order expansion of the solution to the two-point boundary value problem using differential algebraic (DA) techniques.…”

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confidence: 99%

A second-order solution to the two-point boundary value problem for rendezvous in eccentric orbits

Zhang

Zhou

2010

Celest Mech Dyn Astr

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A new second-order solution to the two-point boundary value problem for relative motion about orbital rendezvous in one orbit period is proposed. First, nonlinear differential equations to describe the relative motion between a chaser and a target are presented considering the second-order terms in the gravity. Then, by regarding the second-order terms as external accelerations, we establish second-order state transition equations. Moreover, the J2 perturbations effects can also be considered in the state transition equations. Last, the initial relative velocity to fulfill a rendezvous is determined by solving the state transition equations. Numerical simulations show that the new second-order state transition equations are accurate. The second-order solution to the two-point boundary value problem on eccentric orbits is valid even if the relative range is farther than 500 km.

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Formule D’Inversion de Lagrange et son Application a la Theorie des Perturbations

Inoue¹

1979

Dynamics of the Solar System

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Generalization of Lagrange's implicit function theorem to N-dimensions

Cited by 8 publications

References 3 publications

Second-order state transition for relative motion near perturbed, elliptic orbits

Second-order state transition for relative motion near perturbed, elliptic orbits

A second-order solution to the two-point boundary value problem for rendezvous in eccentric orbits

Formule D’Inversion de Lagrange et son Application a la Theorie des Perturbations

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