Trajectory Optimization Based On Tangent Thrust

The goal of the spacecraft trajectory optimization is to find a minimal energy orbit between arbitrarily two orbits. For the impulsive maneuvers, the analytical parameters of the terminal tangent transfer orbit can be obtained by the Lambert’s problem. However,the analytical solutions cannot be obtained under the continuous thrust maneuvers, and the nonlinear two-point boundary value problem is derived from the indirect method suffers from guessing the initial state and costate variables, which is still unanswered in the field of aerospace industry. In addition, the continuous tangential thrust orbit(the thrust vector is along the flight-path angle) has some advantages with less energy consumption and without changing direction. Then, the tangent orbit can be used as the initial design for the design of optimal continuous low-thrust orbit. Thus, this thesis focuses on the tangent orbit, the main contents of this paper are:(1) The two-impulse orbital rendezvous problem with a terminal tangent burn;(2) The continuous tangential thrust multiple-revolution transfer and rendezvous problems between coplanar orbits;(3) The continuous tangential thrust multiple-revolution transfer and rendezvous problems between non-coplanar orbits;(4) The optimal continuous low-thrust multiple-revolution transfer problems for the coplanar and non-coplanar orbits. The main contributions are as follows:The two-impulse orbital rendezvous problem with a terminal tangent burn between coplanar elliptical orbits is studied by considering a lower bound on perigee radius and an upper bound on apogee radius. This problem requires that two spacecraft arrive at the rendezvous point with the same arrival flight-path angle after the same flight time. The admissible range of the final true anomaly that meets the perigee and apogee constraints is obtained in closed form. The revolution number of the transfer orbit is expressed as a function of the true anomaly and the revolution numbers of the initial and final orbits.All the feasible solutions are obtained with a bound on the revolution number of the final orbit. Then, the minimum-fuel one is determined by comparing their costs. Finally,two numerical examples are provided to obtain all the feasible solutions for given initial impulse points and the optimal solution with the initial coasting arc. In addition, the minimum-fuel solution of the terminal tangent burn rendezvous is a little better than the cotangent rendezvous when the rendezvous time is long enough; however, the cotangent rendezvous may not exist when the rendezvous time is short, but there exists a solution using the terminal tangent burn.A radius shape function for the continuous tangential thrust coplanar orbits transfer and rendezvous problems is studied. It is proved that this shape is able to model oscillations of the radii between successive perigee and apogee, and satisfy the trajectory safety constraints. For the transfer problem, the shaping coefficients can be determined by the boundary conditions. For the rendezvous problem, the fourth shaping coefficient is obtained in its admissible, which is analytically derived from the trajectory constraints,then the admissible range of rendezvous time is determined by the range of the fourth shaping coefficient. For a given rendezvous time, the fourth shaping coefficient is solved by the secant method. The conditions for solution existence are also discussed for the short and long-enough rendezvous time, respectively. Finally, compared with the inverse polynomial shape, the shape-based method using the new radius shape function provides a lower-cost and lower maximum-thrust-acceleration suboptimal solution.For the continuous tangential thrust non-coplanar orbits transfer and rendezvous problems, an analytical expression for the elevation angle with respect to the inclination and right ascension of ascending node of the transfer orbit is derived from the geometry using the initial orbit plane as the reference plane of the spherical coordinate system.Moreover, we assume that the inclination and right ascension of ascending node shapes are both monotonically increasing functions with respect to the azimuthal angle, then the new elevation angle shape function is obtained. It is proved that the proposed elevationangle always be less than or equal to the angle between arbitrarily two orbit planes, that is, the trajectories are always between arbitrarily two orbit planes. In addition, in the case of large changes in inclination and right ascension of ascending node, especially for high inclinations, there exist solutions for the shape combination of the new elevation angle shape and the radius shape from the third chapter. Finally, compared with the Novak method, the shape-based method using the new shape combination provides a lower-cost and lower maximum-thrust-acceleration suboptimal solution.For the optimal continuous low-thrust orbit multiple-revolution transfer problem, the initial costate variables can be obtained by successively using the proposed shape-based method, first-order gradient and neighboring extremal algorithms. At first, the energy suboptimal solutions are obtained from the shape-based method in the third and fourth chapters. Secondly, we can take the suboptimal control variables as the initial iteration condition for the first-order gradient algorithm, then the rough initial costate variables are obtained, which can be used as the initial iteration condition for the neighboring extremal algorithm, then the high-accuracy initial costate variables are obtained. Numerical examples of coplanar and non-coplanar low-thrust orbits show the above three methods are complementarity, it is effectively to determine the value of the initial costate variables.Thus, the optimal trajectory is obtained.