Abstract
We consider the optimal trajectories in the rendezvous problem calculated by calculus of variations techniques. We formulate the problem of translational motion of the spacecraft taking the asteroid rotation into consideration. The motion is supposed to occur outside of a sphere of influence of any planet, the gravity of the asteroid is defined like the one in the restricted two-body problem. We represent the equations of controlled motion in central gravity field and construct the calculus of variations problem, given the control constraints. The problem is solved applying Valentine’s method. We solve the Euler–Lagrange equations generated from the functional related to the fuel consumption and show the solutions in the special case of neglecting the asteroid gravity, both taking and not taking the Coriolis force into consideration. The optimal control in these solutions is occurred to be twofold: continuous (either smooth and piecewise smooth one) and piecewise continuous one, having one or more points of jump discontinuities. In case of these discontinuities the Weierstrass–Erdmann condition is applied. The optimality is validated by the Legendre condition. We note the way the control function and the trajectory change while varying some of the parameters of the problem: the initial speed, the flight duration, the control limit, the angular velocity of the asteroid. The descripted method allows one to evaluate some parameters of the flight which may be used in the rendezvous mission planning.
Highlights
Движение космического аппарата рассматривается в области, находящейся вне сфер действия планет Солнечной системы, гравитационное действие астероида определяется в рамках ограниченной задачи двух тел
We consider the optimal trajectories in the rendezvous problem calculated by calculus of variations techniques
We represent the equations of controlled motion in central gravity field and construct the calculus of variations problem, given the control constraints
Summary
Соответствующие интервалу времени t0,T : xi t0 xi0 ; xi T xi , i 1,6,. Где моменту времени t0 соответствуют начальные условия миссии сближения, моменту T – условия мягкой посадки на астероид. T0 где k – положительная величина, обратная скорости истечения газовой струи из реактивного сопла двигателя. Представим тот случай, когда для управления движением КА используются 6 одинаковых реактивных сопел, установленных по осям x1, y1, z1 , связанным с КА. Суммарный расход топлива за время движения определим в соответствии с уравнением И.В.
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