Fanghua Jiang

Practical Techniques for Low-Thrust Trajectory Optimization with Homotopic Approach

DOI: 10.2514/1.52476 This paper concerns the application of the homotopic approach, which solves the fuel-optimal problem of lowthrust trajectory by starting from the related and easier energy-optimal problem. To this end, some effective techniques are presented to reduce the computational time and increase the probability of finding the globally optimalsolution.First,theoptimalcontrolproblemismadehomogeneoustotheLagrangemultipliersbymultiplying the performance index by a positive unknown factor. Hence, normalization is applicable to restrict the unknown multipliersonaunithypersphere.Second,theswitchingfunction’s first-andsecond-orderderivativeswithrespectto time are derived to detect switching. The switching detection is embedded in the fourth-order Runge–Kutta algorithm with fixed step size to ensure integration accuracy for bang-bang control. Third, combined with the techniques of normalization and switching detection, the particle swarm optimization with well-chosen parameters considerably increases the probability of finding the approximate initial values of the globally optimal solution. Moreover, intermediate gravity assist, which brings complex inner constraints, is considered. To determine the approximate gravity assist date, analytical formulas are presented to evaluate the minimal maneuver impulse based on the results of Lambert problems. The first-order necessary conditions for gravity assist constraints are derived analytically.Theoptimalsolutioncanberapidlyobtainedbyapplyingthetechniquespresentedtosolvetheshooting function. The unknowns are far less than with direct methods, and the computational effort is also far lower. Two examples of fuel-optimal rendezvous problems from the Earth directly to Venus and from the Earth to Jupiter via Mars gravity assist are given to substantiate the perfect efficiency of these techniques.

Study on Relative Orbit Geometry of Spacecraft Formations in Elliptical Reference Orbits

This paper studies the relative orbit geometry of a leader–follower spacecraft formation flying in unperturbed elliptical reference orbits. The first-order relative position equations, derived using the reference orbital element approach under the condition that the follower and leader spacecraft have equal semimajor axes, are transformed fromtrigonometricformstoparametricandalgebraicforms.Theconditionsforandthenumberofself-intersections oftherelativeorbitprojectedontothethreecoordinateplanesoftheleaderlocal-vertical–local-horizontal frameare obtained. The relative orbit proves to be three-dimensional instead of planar in most cases, and may self-intersect spatially at most once. The collision between the follower and leader spacecraft corresponds to the case where the solution curve to the first-order relative motion equations passes through the origin. The conditions for collision are subsequently determined. For anondegenerate case(in which none of the relative motion in the radial, in-track, and cross-trackdirectionsvanish),threetypesofrelativeorbitarepossible.Mostfrequently,therelativeorbitisonaonesheet hyperboloid. Otherwise, when the relative orbit has a real or finite imaginary self-intersection, it rests on an elliptic cone. In the rest of the cases, including that with an imaginary self-intersection at infinity, the relative orbit is on an elliptic cylinder. The criteria for these three types are given, respectively, followed by examples.

Reachable Domain for Spacecraft with a Single Impulse

This paper analyzes the reachable domain for spacecraft with a single fixed-magnitude impulse, and overapproximation of the reachable domain is provided for cases involving spacial impulse. A low impulse is assumed, implying elliptical trajectories and a bounded reachable domain. An equation is developed for the trajectory, with no restriction on the eccentricity of the initial elliptical orbit. A method is given to determine both the envelope of the ellipsoids of revolution containing the trajectories and the envelope of the planes containing the trajectories, with the intersection of these two envelopes found to be the upper bound on the reachable domain. The reachable domain is given for three scenarios for different missions, with several examples given to demonstrate the efficiency of this method.

Two-Point Boundary Value Problem Solutions to Spacecraft Formation Flying

The two-point boundary value problem of a leader-follower spacecraft formation flying in unperturbed elliptical reference orbits is studied. The initial and final relative positions and times and the orbit of the leader are known, and the orbit of the follower must be determined. This problem will be called the relative Lamberts problem. First, we will show this relative Lamberts problem can be solved like the classical Lamberts problem. Then, a set of approximate analytic solutions are obtained through linearizing the Lagranges time equation. Meanwhile, a simplified Newton-Raphson algorithm is applied to obtain numerical solutions, and the relevant quantities of the leader are used as initial conditions. Here, our special efforts are dedicated to the study of the periodic relative orbits of the follower. In particular, a set of first-order analytic solutions to the relative Lamberts problem are derived from the periodic solutions of Lawdens equations, and a constraint on the leaders true anomalies (implicitly in time) and relative positions is obtained. From that constraint on the radial/in-track plane of the leader local-vertical-local- horizontal frame, we found that, for specified initial and final times, the locus of final positions of the follower with fixed initial position is a straight line, and so is the locus of initial positions with fixed final position. Furthermore, for fixed initial and final positions, the transfer times with either specified initial time or final time can be expressed as the real roots of a cubic equation, for which there are at most three solutions. Several examples will be given to support these conclusions.

Asteroid body-fixed hovering using nonideal solar sails

Asteroid body-fixed hovering problem using nonideal solar sail models in a compact form with controllable sail area is investigated in this paper. The nonlinear dynamic equations for the hovering problem are constructed for a spherically symmetric asteroid. The feasible region for the body-fixed hovering is solved from the above equations by using a shooting method. The effect of the sail models, including the ideal, optical, parametric and solar photon thrust, on the feasible region is studied through numerical simulations. The influence of the asteroid spinning rate and the sail area-to-mass ratio on the feasible region is discussed in a parametric way. The required sail orientations and their corresponding variable lightness numbers are given for different hovering radii to identify the feasibility of the body-fixed hovering. An attractive mission scenario is introduced to enhance the advantage of the solar sail hovering mission.

Autonomous Navigation of Mars Probes by Combining Optical Data of Viewing Martian Moons and SST Data

Autonomous navigation has become a key technology for deep space exploration missions. Phobos and Deimos, the two natural moons of Mars, are important optical navigation information sources available for Mars missions. However, during the phase of the probe orbiting close to Mars, the ephemeris bias and the difference between the barycentre and the centre of brightness of a Martian moon will result in low navigation accuracy. On the other hand, Satellite-to-Satellite Tracking (SST) can achieve convenient and high accuracy observation for autonomous navigation. However, this cannot apply for a Mars mission during the Mars orbit phase only by SST data because of a rank defect problem of the Jacobian matrix. To improve the autonomous navigation accuracy of Mars probes, this paper presents a new autonomous navigation method that combines SST radio data provided by two probes and optical measurement by viewing the natural Martian moons. Two sequential orbit determination algorithms, an Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF) are compared. Simulation results show this method can obtain high autonomous navigation accuracy during the probes Mars Orbit phase.

Systematic low-thrust trajectory optimization for a multi-rendezvous mission using adjoint scaling

A deep-space exploration mission with low-thrust propulsion to rendezvous with multiple asteroids is investigated. Indirect methods, based on the optimal control theory, are implemented to optimize the fuel consumption. The application of indirect methods for optimizing low-thrust trajectories between two asteroids is briefly given. An effective method is proposed to provide initial guesses for transfers between close near-circular near-coplanar orbits. The conditions for optimality of a multi-asteroid rendezvous mission are determined. The intuitive method of splitting the trajectories into several legs that are solved sequentially is applied first. Then the results are patched together by a scaling method to provide a tentative guess for optimizing the whole trajectory. Numerical examples of optimizing three probe exploration sequences that contain a dozen asteroids each demonstrate the validity and efficiency of these methods.

Autonomous Navigation of Mars Probes by Single X-ray Pulsar Measurement and Optical Data of Viewing Martian Moons

In order to achieve high accuracy of autonomous navigation for Mars probes, an integrated navigation method using X-ray pulsar measurement and optical data of viewing Martian moons is proposed. For single X-ray pulsar measurement on board a Mars probe, navigation accuracy is low due to its poor observability. On the other hand, Phobos and Deimos, two natural moons of Mars, are important optical navigation information sources available for Mars missions. However, the Martian moons ephemeris bias and the differences between barycentre and centre of brightness of Martian moons will result in low navigation accuracy. The method of integrated navigation using X-ray pulsar measurement and optical data of viewing Martian moons can overcome the defect and achieve accurate navigation. Two sequential orbit determination algorithms, Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF), are compared. The simulation results show this method can obtain high autonomous navigation accuracy during the phase of a probe orbiting Mars.