Reachable domain for interception at hyperbolic speeds

Vinh, Nguyen X.; Gilbert, Éric; Howe, R. M.; Sheu, Donglong; Lu, Ping

doi:10.1016/0094-5765(94)00132-6

Cited by 36 publications

(18 citation statements)

References 2 publications

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“…The problem of determining attainable regions for a single-impulse transfer from a given initial Keplerian elliptical orbit has been considered by Kirpichnikov. 20 Vinh et al 12 introduced the concept of the reachable domain, which is the set of points attainable at a given time by a free-ying interceptor using a fuel potential represented by a total characteristic velocity. They also considered possible applications of this concept.…”

Section: ) Linearizedmentioning

confidence: 99%

Direct High-Speed Interception: Analytic Solutions, Qualitative Analysis, and Applications

Ulybyshev

2001

Journal of Spacecraft and Rockets

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Trajectories for high-speed interception in a thin spherical shell of an inverse-square central gravity eld are considered. The solutions are obtained for the problem of exoatmospheric ight, open time intercept of a nonmaneuvering target in an orbit during free-ight phase. The xed-fuel interceptor is assumed to have the capability of generating xed magnitude thrust or a speci c impulsive velocity change. A closed-form solution for the control law is derived. The thrust direction is a linear function of the relative states between the interceptor and target and a nonlinear function of the transfer time. This transfer time is obtained as an analytic solution to a quadratic equation. First-and second-order methods are developed. A qualitative analysis and a descriptive geometric interpretation of a space interception are considered. The results show that the optimal guidance law for the boost phase of an interceptor is the well-known constant bearing course relative to the target. Descriptive existence conditions of trajectories for a xed-fuel interceptor are derived and a computation method to determine the reachable domain, that is, the boundary set of initial positions of the interceptor and target, is developed. Numerical examples of an asteroid interception and satellite interception are presented. Nomenclature a = initial thrust acceleration; Eq. (1) a i = polynomial approximation of ±r for free-ight phase; Eq. (B3) b i = polynomial approximation of ±r for boost phase; Eq. (C2) C 1 ; C 2 = constants; Eq. (42) D = discriminant e = unit vector of thrust direction H = Hamiltonian function P m = speci c fuel usage rate Q v ; Q ½ = perturbed terms; Eqs. (30) and (31) r = radius r = radius vector r m = mean radius of thin spherical shell S v ; S ½ = perturbed terms; Eqs. (30) and (31) t = time t 0 f = minimal range time in an inverse-square gravity eld V = velocity vector, km/s W ½ ; W v = scaled distance and velocity changes for homogeneous central gravity eld; Eq. (A7) 0 = zero vector ®;¯;°; º = constant vectors 1i = inclination angle interceptor orbital plane; see Fig. 9 1t f = t f ¡ t k 1V = speci c impulsive velocity change, km/s ±r = r ¡ r m ±t f = relative displacement of minimal range time, .t f ¡ t 0 f /=t 0 f ±½ min = ½ min =½ 0 µ = angle between vectors V and 1 = adjoint vector; Eq. (11) ¹ = gravitational constant of central body ½ = relative distance between interceptor and target ½ = relative position vector ½ min = minimal range in an inverse-square gravity eld . Member AIAA. ' = interceptor or target position angle; see Fig. 9 ! = mean motion of circular orbit with radius of r m ; Eq. (5) Subscripts F = nal G = gravity term I = interceptor K = end of boost phase T = target V = velocity 0 = initial ½ = range Superscript T = transposition

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Section: ) Linearizedmentioning

confidence: 99%

Direct High-Speed Interception: Analytic Solutions, Qualitative Analysis, and Applications

Ulybyshev

2001

Journal of Spacecraft and Rockets

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“…2). Thus, p can be expressed under S xyz as p = cos θ sin θ cos i sin θ sin i T (1) where θ ∈ [0, 2π] is the angle between vectors r 0 and p, and i is the angle between the initial orbit plane and plane M , as shown in Fig. 2.…”

Section: Coordinate Systemsmentioning

confidence: 99%

“…Vector t is defined as the tangent directions of L. Given that i c and i ρ are two asymptotes of L, the polar angle of t, denoted by ϕ t , satisfies ϕ + π < ϕ t 2π, if ϕ 2 < ϕ π 2π < ϕ t ϕ + π, if π < ϕ < ϕ 1 (A.1)…”

Section: A Proof Of Propositionmentioning

confidence: 99%

“…Early studies on RD conducted by Vihn et al [1] used RD to evaluate the area accessible for a satellitebased interceptor. In recent years, RD has attracted increasing attention with regard to dealing with orbital maneuvers [2][3][4][5][6][7][8][9].…”

Section: Introductionmentioning

confidence: 99%

“…In recent years, RD has attracted increasing attention with regard to dealing with orbital maneuvers [2][3][4][5][6][7][8][9]. In existing RD studies [1][2][3][4][5][6][7][8][9], an orbital maneuver is regarded as a single velocity increment Δv. Due to the constraint in fuel consumption, Δv is constrained in magnitude (no larger than an available characteristic velocity change, Δv max ) but free in direction, thereby indicating that Δv is isotropically wenchangxuan@gmail.com distributed within a sphere with the radii of Δv max .…”

Section: Introductionmentioning

confidence: 99%

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Reachable domain for spacecraft with ellipsoidal Delta-V distribution

Wen

Peng

2018

Astrodyn

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Conventional reachable domain (RD) problem with an admissible velocity increment, Δv, in an isotropic distribution, was extended to the general case with Δv in an anisotropic ellipsoidal distribution. Such an extension enables RD to describe the effect of initial velocity uncertainty because a Gaussian form of velocity uncertainty can be regarded as possible velocity deviations that are confined within an error ellipsoid. To specify RD in space, the boundary surface of RD, also known as the envelope, should be determined. In this study, the envelope is divided into two parts: inner and outer envelopes. Thus, the problem of solving the RD envelope is formulated into an optimization problem. The inner and outer reachable boundaries that are closest to and farthest away from the center of the Earth, respectively, were found in each direction. An optimal control policy is then formulated by using the necessary condition for an optimum; that is, the first-order derivative of the performance function with respect to the control variable becomes zero. Mathematical properties regarding the optimal control policy is discussed. Finally, an algorithm to solve the RD envelope is proposed. In general, the proposed algorithm does not require any iteration, and therefore benefits from quick computation. Numerical examples, including two coplanar cases and two 3D cases, are provided, which demonstrate that the proposed algorithm works efficiently. KEYWORDS reachable domain maneuverability orbit uncertaintyResearch Article

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