Hexi Baoyin

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.

ORBITAL DYNAMICS IN THE VICINITY OF ASTEROID 216 KLEOPATRA

This paper studies the orbital dynamics around the main-belt asteroid Kleopatra through a detailed look at the motion around the equilibria using observational data. The shapes of zero-velocity surfaces are represented at assigned values of the Jacobi integral to explore the connections between forbidden regions in three-dimensional space and the final fates of the nearby trajectories. All four equilibria are found to be nonlinearly unstable, and the degree and mode of instability are further clarified by decomposing the motion into respective local manifolds. This study suggests that there are many sensitive and uncertain trajectories near the equilibria due to the multiplicity of unstable branching trajectories around the center manifolds. Six continuous major families of periodic orbits are obtained close to the equilibria, which are proved to be unstable by Floquet theory.

Solar sail equilibria in the elliptical restricted three-body problem

The existence and dynamical properties of artificial equilibria for solar sails in the elliptical restricted threebody problem is investigated.We show that planar two-dimensional equilibrium curves exist, embedded in threedimensional space, in a nonuniformly rotating, pulsating coordinate system. However, due to the stretching of the system plane coordinates and unstretching of the out-of-plane coordinate, the equilibrium surfaces do not exist in the three-dimensional elliptical restricted three-body system. Control in the neighborhood of an equilibrium point is investigated through a pole assignment scheme. This permits practical out-of-plane equilibria in elliptical three-body systems with small eccentricity.

Equilibria, periodic orbits around equilibria, and heteroclinic connections in the gravity field of a rotating homogeneous cube

This paper investigates the dynamics of a particle orbiting around a rotating homogeneous cube, and shows fruitful results that have implications for examining the dynamics of orbits around non-spherical celestial bodies. This study can be considered as an extension of previous research work on the dynamics of orbits around simple shaped bodies, including a straight segment, a circular ring, an annulus disk, and simple planar plates with backgrounds in celestial mechanics. In the synodic reference frame, the model of a rotating cube is established, the equilibria are calculated, and their linear stabilities are determined. Periodic orbits around the equilibria are computed using the traditional differential correction method, and their stabilities are determined by the eigenvalues of the monodromy matrix. The existence of homoclinic and heteroclinic orbits connecting periodic orbits around the equilibria is examined and proved numerically in order to understand the global orbit structure of the system. This study contributes to the investigation of irregular shaped celestial bodies that can be divided into a set of cubes.

Trajectories to and from the lagrange points and the primary body surfaces

This paper investigates ballistic trajectories to and from the vicinity of the Lagrange points L1 and L2 and the surfaces of the primaries in the circular restricted three-body problem. The study focuses on trajectories from the Lagrange points and their Lyapunov orbits that can access the entire surfaces of the primary bodies for the sun-Earth and Earth-moon systems. By a symmetry property, trajectories leaving the surface of the primary bodies and reaching the Lagrange points and their Lyapunov orbits can also be found. The lowest velocity increment providing such whole-surface coverage from the Lagrange points and the lowest-energy Lyapunov orbit providing whole-surface coverage are found. Applications of such trajectories are to be found in sample return missions and future crewed missions that use the Lagrange points as exploration staging posts.

Topological classifications and bifurcations of periodic orbits in the potential field of highly irregular-shaped celestial bodies

This paper studies the distribution of characteristic multipliers, the structure of submanifolds, the phase diagram, bifurcations, and chaotic motions in the potential field of rotating highly irregular-shaped celestial bodies (hereafter called irregular bodies). The topological structure of the submanifolds for the orbits in the potential field of an irregular body is shown to be classified into 34 different cases, including six ordinary cases, three collisional cases, three degenerate real saddle cases, seven periodic cases, seven period-doubling cases, one periodic and collisional case, one periodic and degenerate real saddle case, one period-doubling and collisional case, one period-doubling and degenerate real saddle case, and four periodic and period-doubling cases. The different distribution of the characteristic multipliers has been shown to fix the structure of the submanifolds, the type of orbits, the dynamical behaviour and the phase diagram of the motion. Classifications and properties for each case are presented. Moreover, tangent bifurcations, period-doubling bifurcations, Neimark–Sacker bifurcations, and the real saddle bifurcations of periodic orbits in the potential field of an irregular body are discovered. Submanifolds appear to be Mobius strips and Klein bottles when the period-doubling bifurcation occurs.

Solar Sail Orbits at Artificial Sun-Earth Libration Points

In this Note a new family of solar sail orbits will be investigated in the sun-Earth circular restricted three-body problem. It will be shown that periodic orbits can be developed that are displaced above or below the plane of the restricted three-body system. Whereas traditional halo orbits are centered on the classical libration points, these new orbits are associated with artificial libration points. The orbits are retrograde, circular orbits with a period half that of the orbit period of the two primary masses of the problem. Numerical analysis of stability and controllability of the orbits shows that the orbits are unstable but completely controllable with both lightness number (sail areal density) and sail attitude.

Passive Stability Design for Solar Sail on Displaced Orbits

This paper investigates the passive stability design of a sailcraft on a displaced orbit with the orbit and attitude considered simultaneously. The objective is to stabilize the orbit-attitude coupled system passively by designing the sailcraft parameters. The design results show that the system can bemarginally stable with passive control. First, the criteria for the passive stability of the sailcraft are given, and the coupled equations ofmotion are obtained. Then, the whole design problem is converted into an optimization problem by converting the design parameters into the optimization parameters, the static requirements of the stability criteria into constraints, and the dynamical requirements into the optimization function. Finally, an example of the optimal design is given. The simulation results under different initial conditions show that the sailcraft is marginally stable on the orbit.

Solar Sail Formation Flying Around Displaced Solar Orbits

R ECENTLY, attention has been focused on solar sail missions, such as the new artificial Lagrange points created by solar sails to be used to provide early warning of solar plasma storms, before they reach Earth [1,2]. There are several prior references with regard to such orbits in the literature. As early as 1929, Oberth mentioned in his study that solar radiation pressure would displace a reflector in an Earth polar orbit in the anti-sun direction, so that the orbit plane did not contain the center-of-mass of the Earth [3]. Later, in 1977, Austin et al. [4] noted that propulsive thrust can be used to displace the orbit of an artificial body, but only small displacements were considered for spacecraft proximity operations, and no analysis of the problem was provided. Similarly, Nock suggested a displaced orbit above Saturn’s rings for in situ observation, however, again no analysis was given [3]. In 1981, Forward [5] considered a displaced solar sail north or south of the geostationary ring. However, because he did not use an active control, subsequent analysis has criticized thiswork and claimed that such orbits were impossible. More recently, McInnes and Simmons have done work in which large families of displaced orbits were found by considering the dynamics of a solar sail in a rotating frame [6], and the dynamics, stability, and control of different families of displaced orbits were investigated in detail [7,8]. Based on McInnes’ and Simmons’ work [6], Molostov and Shvartsburg considered a more realistic solar sail model with nonperfect reflectivity and discussed the effect of finite absorption of the sail on the displaced orbits [9,10]. However, studies of the relative motion of solar sails are rare in the literature. The original idea of formation flying around a displaced orbit considered in this note comes from the concept of combining a displaced orbit with formation flying to achieve greater resolution than a single sail for science missions. This note outlines the characteristics of the relative motion around a displaced solar orbit and proposes some possible control strategies. Because the relative distance between the sails is very small compared with the distance from the sun to the sails, the relative equation of motion is linearized in the vicinity of a displaced solar orbit. Based on the linearized equation, two types of formations, seminatural and controlled formations, are discussed. The seminatural formations are performed with only sail attitude variations, but configurations of the relative orbits strongly depend on the orbit of the leader sail. Therefore, more complex controllers are adopted to build more sophisticated formations to meet special demands on the relative orbit configurations.

Collision and annihilation of relative equilibrium points around asteroids with a changing parameter

In this work, we investigate the bifurcations of relative equilibria in the gravitational potential of asteroids. A theorem concerning a conserved quantity, which is about the eigenvalues and number of relative equilibria, is presented and proved. The conserved quantity can restrict the number of non-degenerate equilibria in the gravitational potential of an asteroid. It is concluded that the number of non-degenerate equilibria in the gravitational field of an asteroid varies in pairs and is an odd number. In addition, the conserved quantity can also restrict the kinds of bifurcations of relative equilibria in the gravitational potential of an asteroid when the parameter varies. Furthermore, studies have shown that there exist transcritical bifurcations, quasi-transcritical bifurcations, saddle-node bifurcations, saddle-saddle bifurcations, binary saddle-node bifurcations, supercritical pitchfork bifurcations, and subcritical pitchfork bifurcations for the relative equilibria in the gravitational potential of asteroids. It is found that for the asteroid 216 Kleopatra, when the rotation period varies as a parameter, the number of relative equilibria changes from 7 to 5 to 3 to 1, and the bifurcations for the relative equilibria are saddle-node bifurcations and saddle-saddle bifurcations.