Optimal Rendezvous Trajectories as a Function of Thrust-to-Weight Ratio

Abstract

An optimal rendezvous trajectory analysis of long-range guidance (or phasing and transfer to a target orbit) related with design of low-thrust propulsion system is considered. Typical rendezvous missions to low Earth orbit space stations for an active spacecraft with different thrust-to-weight ratios (or thrust acceleration) are studied. The method of pseudoimpulse sets is used for the computations. This approach combines large-scale linear programming algorithms with the well-known discretization of the trajectories on small segments and uses discrete pseudo-impulse sets which are considered independently for each segment. Such method is very suitable for spacecraft trajectory optimization with arbitrary thrust from high to low. Existence of low-thrust optimal solutions is briefly discussed. An analysis of thrust-to-weight ratio factor for phasing orbit is presented. As for the high thrust case, for low-thrust trajectories there also are optimal phase solutions, i.e. there is a range of initial phase angles for which required characteristic velocity is almost equal to optimal orbit transfer between initial and target orbit. Possible rendezvous trajectory types in a wide ranges of thrust acceleration values from high to low (including a continuous multirevolution burn) and phase angles are described. Nomenclature a, a = thrust acceleration vector and its magnitude A = matrix of inequality constraints A e = matrix of equality constraints e, e r , e n , e b = thrust direction unit vector and its components in local vertical/local horizontal coordinate system -radial, along-track, cross-track i = segment number j = pseudo-impulse number J = performance index k = quantity of pseudo-impulses at each segment m = number of boundary conditions n = quantity of segments q = weight coefficient vector ρ = vector of relative coordinates N R = specified number of revolutions for rendezvous mission P = boundary condition vector 2 r ω . = mean radius of circular reference orbit t = time Δt AN = relative time from ascending node V = vector of relative velocity T = orbit period X = vector of decision variables Δt i = duration of i-th segment ΔV i (j) = characteristic velocity of j-th pseudo-impulse at i-th segment ΔV x = characteristic velocity μ = gravitational parameter for the Earth ϑ = pitch angle , Fig. 4 ϕ = phase angle, Fig. 1 ψ = yaw angle, Fig. 4Ф, Ф ρρ , Ф ρV , Ф Vρ , Ф VV = transition matrix and its sub-matrices ω = mean angular velocity of orbit motion

Trajectory Optimization for Spacecraft Proximity Operations with Constraints

“…In a similar way, a three-dimensional case for the possible thrust directions can be considered (see details in Refs. [17][18][19][20]. …”

“…The adjacent segments among these should be joined in burns. More detailed description of the methods with application examples of spacecraft trajectory optimization is given in [17][18][19][20].…”

“…The method use discretization of the spacecraft trajectory on segments and sets of pseudoimpulses for each segment [17][18][19][20]. Suppose that the final time of the trajectory t f is specified.…”

“…the thrust direction and magnitude) by sets of pseudoimpulses with an inequality constraint for each segment. In a sense, the paper is a continuation of the author's previous works [17][18][19][20] in which the key idea was used for several optimization problems: maintenance of American Institute of Aeronautics and Astronautics 2 a 24-hour elliptical orbit, coplanar and non-coplanar orbit transfers, various rendezvous trajectories for near-circular orbits (with high-, medium-, and low-thrust), a three-dimensional launch trajectory from the Moon's surface to a circumlunar orbit with constraints, lunar landing trajectories with interior-point constraints.…”