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Minimum-fuel, multiple-impulse, time-fixed solutions are obtained for circle-to-circle rendezvous. The coplanar case and a restricted class of noncoplanar cases are analyzed. For several initial phase angles of the target relative to the vehicle, optimal solutions are obtained for a range of fixed transfer times and terminal radii. Primer vector theory is used to obtain the optimal number of impulses, their times and positions, and the presence of initial or final coasting arcs. For sufficiently large transfer times, the optimal solutions become the well-known time-open solutions, such as the Hohmann transfer for the coplanar case. The results obtained can be used to perform a time vs fuel trade-off for missions which have operational time constraints, such as space rescue operations and avoidance maneuvers.

Minimum-fuel, multiple-impulse orbital rendezvous is investigated for the case in which the transfer time is specified (time-fixed case). A method for obtaining optimal solutions which is applicable to rendezvous or orbit transfer between elliptical orbits of low eccentricity is presented. In this method, optimal solutions are constructed by satisfying the necessary conditions for the primer vector. It is assumed that the terminal orbits lie close enough to an intermediate circular reference orbit that the linearized equations of motion can be used to describe the transfer. The linear boundary-value problem for the rendezvous is then solved analytically. As an application of the method, optimal four-impulse, fixed-time rendezvous transfers between coplanar circular orbits are obtained for a range of transfer times. These linearized solutions offer physical insight into the problem and provide approximate initial conditions that can be used in an iterative numerical solution to the nonlinear optimal multiple-impulse problem.6 hy Nomenclature f = first derivative of ( ) with respect to time. Time derivatives of vectors are taken with respect to an inertial reference frame. = differentiation with respect to dimensionless time r = gravity gradient matrix = vector defined by Eq. (16) AIAA. t A boldface lower case letter denotes a column vector. The lightface letter denotes its scalar magnitude. H =(4X4) matrix having columns h/ I = identity matrix k = auxiliary variable defined by Eq. (26) m = auxiliary variable defined by Eq. (25) p = primer vector gy = auxiliary variable defined by Eq. (27) r = position vector 8R = nondimensional difference between final and initial circular orbit radii t = time t p = phasing time, used in definition of T Uy = thrust unit vector of jth impulse v = velocity vector (v = ir) AVy = vector velocity change due to jth thrust impulse AV = vector defined by Eq. (11) wy = vector defined by Eq. (9) W =(4X4) matrix having columns W/ x = state vector aj = dimensionless time interval defined by Eq. (18) Downloaded by UNIVERSITY OF CALIFORNIA -DAVIS on February 5, 2015 | http://arc.aiaa.org | OPTIMAL FOUR-IMPULSE FIXED-TIME RENDEZVOUS 929 5 = first variation of variable which it precedes A = a change in variable in precedes 0 = central angle X = radial component of primer vector M = circumferential component of primer vector v = out-of-plane component of primer vector T = dimensionless time, defined prior to Eq. (2) &ji = state transition matrix between times U and if co = mean motion of reference orbit Subscripts 0 = initial value F = final value 1,2,3... = numerical subscript denotes the number of the thrust impulse (first, second, third...) H = half-transfer time [Eq. (13)] j = the time of jth impulse r = radial component 0 = circumferential component

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