Optimal Finite-Thrust Rendezvous Trajectories Found via Particle Swarm Algorithm

Abstract

The particle swarm optimization technique is a population-based stochastic method developed in recent years and successfully applied in several fields of research. The particle swarm optimization methodology aims at taking advantage of the mechanism of information sharing that affects the overall behavior of a swarm, with the intent of determining the optimal values of the unknown parameters of the problem under consideration. This research applies the technique to determining optimal continuous-thrust rendezvous trajectories in a rotating Euler-Hill frame. Five distinct applications, both in two dimensions and in three dimensions, are considered. Hamiltonian methods are employed to translate the related optimal control problems into parameter optimization problems. The transversality condition, which is an analytical condition that arises from the calculus of variations, is proven to be ignorable for these problems, and this property greatly simplifies the solution process. For each of the five applications considered in the paper, despite its simplicity, the swarming algorithm successfully finds the optimal control law corresponding to the minimum-time trajectory with great accuracy. Nomenclature a = thrust acceleration of P with components a x ; a y ; a z in the r T ;θ T ;ĥ T frame, DU∕TU 2 or km∕s 2 a k = lower bound for the kth component of the position vector χ b k = upper bound for the kth component of the position vector χ a P = magnitude of the thrust acceleration of P, DU∕TU 2 or km∕s 2 c = effective exhaust velocity of the propulsive system, DU∕TU or km∕s d k = upper bound for the kth component of the velocity vector w H = Hamiltonian function h i = specific angular momentum of spacecraft i (i P or T) J = objective function J j opt = minimum value of the objective function found by the particle swarm optimization algorithm up to iteration j J = objective function considered by the particle swarm optimization algorithm (in the presence of equality constraints) l r = rth equality constraint function (r 1; : : : ; m) m = number of equality constraints m 0 = initial mass of P N = number of particles N IT = number of iterations n = number of unknown parameters n 0 = initial thrust acceleration, DU∕TU 2 or km∕s 2 P = pursuing spacecraft p = parameter vector r i = inertial position vector of spacecraft i (i P or T) R T = radius of the circular orbit of the target vehicle, DU or km r T , θ T ,ĥ T = orbital reference frame T = target vehiclẽ T = propulsive thrust t = time, TU or s t f = final time, TU or s t 0 = initial time, TU or s u = control vector v i = inertial velocity vector of spacecraft i (i P or T) wi = velocity vector of particle i with components fw k ig x = state vector α = in-plane thrust angle, deg or rad β= out-of-plane thrust angle, deg or rad η r = weighting coefficient for penalty term r Θ = state transition matrix associated with the linear equations of motion λ = adjoint variable conjugate to the dynamics equations with components fλ i g μ = gravitational parameter of the attracting body, ...

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