Two-point Boundary Value Problem For Spacecraft Orbital Rendezvous And Orbital Transfer

The two-point boundary-value problem for spacecraft is a fundamental problem ofastrodynamics. The problem requires the determination of a conic section between giveninitial and final points, which satisfies some special conditions. For diferent missions,these conditions are mainly divided into three cases:(1) the transfer time is given;(2)the initial (departure) or final (arrival) flight-direction angle is given;(3) the energy con-sumption is minimum. The first condition can be directly used in the orbit rendezvousproblem, which can be described by the absolute motion (Lambert’s problem) and therelative motion. The second condition is the specified initial/final flight-direction angleproblem, which can be used when a specified arrival angle is needed for atmosphericreentry, and the tangent orbit problem. The third condition can be used in the optimaltime-free two-impulse (or multiple-impulse) transfer problem. For these three problem-s, the main contents of this dissertation include multiple-revolution Lambert’s problem,optimal time-free two-impulse transfer, orbital rendezvous with constant thrust, and thetangent orbit problem. The main contributions are as follows:For a constrained multiple-revolution Lambert’s problem, which requires a lowerbound for the perigee altitude and an upper bound for the apogee altitude, all the fea-sible solutions and the optimal two-impulse solution are solved. The characteristics arefirstly analyzed for short-path orbit and for long-path orbit, respectively. Then a solutionprocedure for the semimajor-axis range is proposed that takes these two constraints intoaccount. Based on the semimajor-axis range, the solutions with a feasible number of rev-olutions can be easily selected. Furthermore, the transfer orbit solution can be obtainedfor each feasible number of revolutions. For the minimum-fuel two-impulse problemin the constrained multiple-revolution Lambert’s solutions, the optimal time-free two-impulse transfer problem implies finding the real roots of an eighth-order polynomial.Then, by comparing the optimal time-free solution with the feasible solutions, the opti-mal semimajor axis for the fixed-time two-impulse rendezvous problem is identified.For the constant-thrust case, two rendezvous methods are proposed: one is basedon the linear relative motion equations; the other is based on Lambert’s solutions. For both methods, the process of rendezvous is composed of three time intervals, i.e., the firstengine maneuver, a coasting subarc, and the second engine maneuver. For the methodbased on the linear relative motion equations, an analytical propagation of the relative s-tate is obtained under a constant external acceleration. Then the required relative velocityin the first maneuver and the ignition time of the second maneuver are solved analytical-ly and numerically, respectively. The first maneuver aligns the thrust direction with therelative velocity-to-be-gained vector. In the second maneuver, the thrust direction is keptinvariable in a solved direction. For the method based on Lambert’s solutions, a newmethod is proposed to verify the optimality of the velocity-to-be-gained guidance, whichis adopted in both the first and the second engine maneuvers.For the two-body tangent orbit technique, an analytical study is given by providingsolution-existence conditions. The flight-direction angle is used to describe and solve thisproblem. Closed-form solutions are obtained for three classic problems: specified arrivalflight-direction angle, specified departure flight-direction angle, and cotangent transfers.Not all of the problems admit solutions; thus, closed-form conditions for solution ex-istence are provided by imposing a positive semilatus rectum constraint and a negativetransfer-orbit energy (elliptic orbit transfer) constraint. The final solution-existence con-dition is then provided in terms of the true anomaly range for initial or final orbit. Thesingularity problem of180deg orbit transfer is also analyzed.A tangent orbital rendezvous problem for two cooperative spacecraft with an as-signed rendezvous point is studied. Two spacecraft are required to arrive at this pointwith the same velocity direction after the same transfer time. The discussion for the ex-istence of solution is based on the intersection range of initial flight-path angle for twotransfer orbits. For the multiple-revolution case, there are many solutions for this prob-lem, which can be obtained by a numerical iterative algorithm, e.g., the secant method.Although the tangent rendezvous orbit is not the minimal energy one, the maneuver ofnulling the relative velocity would be dramatically simplified.The noncoplanar tangent orbit technique in3D is proposed and solved. Based ona new “definition” of orbit tangency in3D, solutions are obtained for three noncoplanarorbit transfer problems: tangent to initial orbit, tangent to final orbit, and cotangent trans-fer. The flight-direction angle of the transfer orbit is obtained, and then the tangent to initial/final orbit problem in3D can be solved analytically. Moreover, a simple expres-sion for the transfer angle with respect to the initial true anomaly is derived, which canbe used to solve the cotangent transfer problem. The initial true anomaly is solved bythe secant method for this problem. There may exist several solutions for the cotangenttransfer problem. Finally, the conditions for solution existence are also discussed.