Hamiltonian Structure-preserving Research Articles

Hamiltonian Structure-Based Formation Flight Control Along Low-Energy Transfer Trajectory

This paper studies the three-dimensional formation-flying problem in multibody regime using the Hamiltonian structure-preserving (HSP) control strategy. A low-energy lunar halo transfer orbit is chosen as the reference trajectory, which can represent the dramatically varying dynamics in the vast Earth–Moon region, and demonstrates much more complex environments than the vicinity of libration points. This trajectory exhibits time-varying topology of four different types, with two new ones: saddle saddle center and complex saddle complex saddle center (to the authors’ best knowledge, never studied in the literature). Stability transitions occur with a tremendous variation of the eigenvalues, bringing new challenges to effective formation controller design. To obtain bounded motion, topology-type-dependent HSP controller is proposed using the local stable/unstable manifolds. Difficulty arises in the transition regions, where close-to-zero real (complex) eigenvalues appear and require huge control gains, which may be infeasible for onboard computers. Accordingly, the controller is improved by considering additionally the local center manifold, and the gains required are always lower than 25. The controller is simple to implement because only position information is required, and its validity and robustness in the presence of initial deviations and navigation errors are verified by Monte Carlo simulations.

Capturing a spacecraft around a flyby asteroid using Hamiltonian-structure-preserving control

The potentially hazardous asteroid is not only the target of planetary defense missions but also an ideal one for scientific exploration of near-Earth objects due to minimal launching budget during its close encounter with Earth. The flyby segment of asteroid shows to be hyperbolic in an Earth-centred inertial frame. To guide the spacecraft captured by the asteroid under the common Earth's gravity, the Hamiltonian-structure-preserving (HSP) controller, based only on the feedback of relative positions, is constructed to transform the topological structure of the equilibrium point from hyperbolic to elliptic. On account of the small mass of flyby asteroid, a hyperbolic two-body model is used to derive the form of controller. The controller is also proved to work effectively in the hyperbolic three-body model. Critical control gains for both transient and long-term stabilities are investigated, as well as the controlled frequencies caused by different gains. To avoid collision with an asteroid, the artificial potential function (APF) is employed to improve the HSP controller.

Application of Floquet theory to formation flying in elliptic orbits via a Hamiltonian structure-preserving control

In this study, an improved Hamiltonian structure-preserving controller is applied for formation flying in elliptic orbits. The Lyapunov-Floquet theory is applied to the Tschauner-Hempel (TH) equation in order to generate its transformed form with two zero eigenvalues and two pairs of imaginary eigenvalues. Then, an improved Hamiltonian structure-preserving (HSP) control is introduced to the transformed system to eliminate the instability of double zeros. All bounded trajectories can be generated by basic solutions, which effectively reveal the following performances of the controller, such as the relationship between the frequencies of basic motions and control gains are detected and the wide selection of gains for stability. Local and global optimizations on gains are discussed based on different performances of the controller. One focuses on the sensitivity of output with respect to input of controller and the other aims at a minimal fuel consumption for the maximum likelihood. The ergodic relationship between the fuel consumption and initial condition is established for the minimal consumption. To avoid in-orbit collision, a collision-free strategy based on the artificial potential function is proposed to enhance the proposed controller, which make sure a minimum safety distance for the formation.

Formation flying along unstable Libration Point Orbits using switching Hamiltonian structure-preserving control

Orbits for spacecraft formation flight along an unstable orbit are designed. The Hamiltonian structure-preserving (HSP) control is used to stabilize the motion of the spacecraft. Using a simple switching control strategy, the size of a circular orbit relative to a nominal trajectory can be systematically designed. To perform station-keeping of a spacecraft, the proposed HSP control is repeatedly applied. The nonlinear stability of the controller is also analyzed using Lagrange-Dirichlet criterion. To demonstrate the performance of the proposed switching HSP controller, a numerical simulation is performed for the formation flight of the spacecraft in Earth-Moon system's L2 halo orbit.

Dynamics of the spatial restricted three-body problem stabilized by Hamiltonian structure-preserving control

The local invariant (stable, unstable, and center) manifolds of a saddle $$\times $$ center $$\times $$ center equilibrium point can be used to construct a three-dimensional Hamiltonian structure-preserving (HSP) controller. The linear stability of the controlled Hamiltonian system is verified when one of the proposed criteria is satisfied. The nonlinear dynamics of a linearly stabilized Hamiltonian system is analyzed by normalizing the Hamiltonian high-order perturbation terms using the Lie series method. The analytical solutions of the invariant tori are then obtained in trigonometric series form. A theorem for the Nekhoroshev stability of the controlled Hamiltonian system is provided. When the constructed HSP controller is applied to a photogravitational restricted three-body problem with oblateness, a hyperbolic artificial equilibrium point at which a criterion is satisfied can be stabilized, and bounded Lissajous trajectories near the equilibrium point are obtained. The allocation law demonstrates that the attitude angles and lightness number of the solar sail can serve as control parameters to provide the required acceleration of the HSP controller.

Low-thrust displaced orbits by weak Hamiltonian-Structure-Preserving control

For the requirements from some missions, the spacecraft is placed near the unstable region around the hyperbolic equilibrium. To stabilize the motion, a novel concept of the weak Hamiltonian-Structure-Preserving (HSP) control parameterized by the Coriolis acceleration is introduced. Based on the dynamical model of low-thrust displaced orbits, this paper searches for the mechanism in orbital stability change by the weak HSP control and provides specific strategies on how the control is implemented. Since the weak control has no effect on the topology of the original system, two invariant equilibria, a hyperbolic one and an elliptic one are solved. Dynamical system techniques are employed to investigate the controlled motions near the two equilibria, illustrating the Lyapunov orbit in different Coriolis acceleration cases and presenting two basic periodic modes measuring the bounded trajectories in the interior region. Controlled trajectories in the neck region are analyzed for their characteristics, classification and transition to make preparation for the stability change. One of the important contributions of this paper is to numerically demonstrate that the weak HSP control preserves the homoclinic orbits to the hyperbolic equilibrium as well as to the Lyapunov orbit, and another is to achieve the orbital transfer within, or even beyond KAM tori based on two methods in determining whether the controlled trajectory integrated from an initial point can exhibit transit.

Dynamics and control of high area-to-mass ratio spacecraft and its application to geomagnetic exploration

This paper studies the dynamics and control of a spacecraft, whose area-to-mass ratio is increased by deploying a reflective orientable surface such as a solar sail or a solar panel. The dynamical system describing the motion of a non-zero attitude angle high area-to-mass ratio spacecraft under the effects of the Earth's oblateness and solar radiation pressure admits the existence of equilibrium points, whose number and the eccentricity values depend on the semi-major axis, the area-to-mass ratio and the attitude angle of the spacecraft together. When two out of three parameters are fixed, five different dynamical topologies successively occur through varying the third parameter. Two of these five topologies are critical cases characterized by the appearance of the bifurcation phenomena. A conventional Hamiltonian structure-preserving (HSP) controller and an improved HSP controller are both constructed to stabilize the hyperbolic equilibrium point. Through the use of a conventional HSP controller, a bounded trajectory around the hyperbolic equilibrium point is obtained, while an improved HSP controller allows the spacecraft to easily transfer to the hyperbolic equilibrium point and to follow varying equilibrium points. A bifurcation control using topologies and changes of behavior areas can also stabilize a spacecraft near a hyperbolic equilibrium point. Natural trajectories around stable equilibrium point and these stabilized trajectories around hyperbolic equilibrium point can all be applied to geomagnetic exploration.

Trajectory optimization for solar sail in cislunar navigation constellation with minimal lightness number

In view of the limitations of the existing libration-point satellite navigation systems in cislunar space, this paper replaces satellites with solar sails to construct a cislunar navigation constellation and is devoted to the trajectory optimization of solar sails to minimize the lightness number control. The Artificial Lagrangian Points (ALPs) yielded by solar sail in the Sun–Earth+Moon system benefit from the advantages of numberless equilibria and out-of-plane displacement, when compared with the classical Lagrangian points. Limited to the manufacturing of sail film in practice, the candidate constellation architecture in the shape of a cube is constructed based on the optimization of the average lightness number required at ALPs. Considering the lunar gravity, the Hamiltonian-structure-preserving (HSP) controller achieved by changing the sail's attitude and lightness number is developed to stabilize the sails' trajectories near the ALPs. Moreover, an optimal quasi-periodic trajectory with minimum lightness number control is searched for through differential evolution algorithm evolving the controller gains and initial states of orbits. There are three important contributions of the trajectory optimization for a sail in the cislunar navigation constellation: firstly, the large amounts of ALPs break the restrictions on the number and plane of the five classical Lagrangian equilibrium solutions to enlarge the selection of constellations; secondly, the station keeping tool HSP controller powerfully ensures the boundedness of the ALP's trajectory; thirdly, using the optimization algorithm to generate ALP orbits effectively avoids the time consumption of differential correction, which is more convenient and general for the natural trajectory design of ALPs.

Formation flying on quasi-halo orbits in restricted Sun–Earth/Moon system

Due to the strong nonlinear perturbations near the libration points, the continuous low-thrust technique has many potential applications in stationkeeping relative motions. The Hamiltonian structure-preserving (HSP) control is employed in this paper to stabilize formation flying on quasi-periodic orbits near LL1 of the restricted Sun–Earth–Moon–spacecraft system. In the bi-circular model (BCM), a multiple shooting corrector is developed to refine quasi-periodic orbits as the chief spacecraft's reference trajectories. The linearized variation equation in BCM is used to design the stationkeeping control. A HSP controller is constructed to change the topology of the equilibrium from hyperbolic to elliptic using only relative position feedbacks consisting of stable, unstable and center manifolds. The critical control gains for transient and long-term stabilities are presented to guide the selection of control gains.