Hamiltonian Structure-preserving Control Research Articles

Energy-based trajectory tracking control of under-actuated unmanned surface vessels

In this paper, a novel energy-based trajectory tracking control strategy for under-actuated unmanned surface vessels (USVs) in the presence of unknown environmental disturbances is presented. The port-Hamiltonian framework is utilized to propose a passivity-based control model in body-fixed coordinates of the USVs. An adaptive disturbance estimation method is detailed and used to accurately estimate the environmental disturbances affecting USV motion. Furthermore, a passive and Hamiltonian structure-preserving controller is employed to achieve the desired trajectory of the USV system, and the stability of the desired target dynamic system is rigorously proven. The effectiveness of the proposed controller is demonstrated through simulations and experiments on a USV experimental platform, showcasing its capability of trajectory tracking performance and mitigating the effects of disturbances.

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.

Dynamics in the controlled center manifolds by Hamiltonian structure-preserving stabilization

A two-dimensional hyperbolic Hamiltonian system can be linearly stabilized by a Hamiltonian structure-preserving controller. A linear symplectic transformation and the Lie series method can successfully normalize the expanded Hamiltonian function around a controlled stable equilibrium point, then the dynamics in the controlled center manifolds of which can be described by a Poincaré section. With the implement of the inverse transformation of the Lie series, the analytical results of the controlled manifolds can be obtained. Applying normalization and analytical calculation to planar solar sail three-body problem, we can get the normal form of the corresponding Hamiltonian function and trajectory around the chosen equilibrium point by analytical results. Finally, typical KAM theory is used to analyze the nonlinear stability of the controlled equilibrium point, and the stable region of the control gains are given by numerical calculation.

Dynamics and Control for Nonideal Solar Sails Around Artificial Lagrangian Points

Due to their lack of fuel consumption, solar sails have potential applications for deep space exploration. However, due to the optical degradation and deformation of such a sail, a nonideal model is preferable for investigating their dynamics and control when moving around an artificial Lagrangian point. An improved imperfectly reflecting sail model is constructed to compare with the ideal one. With a Hamiltonian structure-preserving controller adopted, the trajectory around an artificial Lagrangian point is generated for ideal solar sails. The original trajectory changes and migrates when the imperfectly reflecting sail model is used. Two improvement strategies are proposed to adjust the changed trajectory. One strategy is to adjust the input parameters of the actuator to compensate for the change in acceleration. The other strategy is to design a new controller to enhance the change and migration of the trajectories. The effectiveness of the two strategies for coping with imperfectly reflecting solar sails is verified via numerical simulations.

Bounded Motions near Contact Binary Asteroids by Hamiltonian Structure-Preserving Control

This paper is devoted to stabilizing motions near a special type of equilibrium point of contact binary asteroids by Hamiltonian structure-preserving control. Aiming at noncollinear equilibrium points of the asteroid 1996 HW1, a Hamiltonian structure-preserving controller is built up using stable and unstable manifolds. Compared with traditional Hamiltonian structure-preserving controllers, it is capable of stabilizing the motion near a special kind of equilibrium point only using stable or unstable manifolds. The analytical solution of controlled motions is approximated by a nonlinear series using the method of variation of parameters. An optimization strategy is proposed including the numerical optimization of control gains by maximum-likelihood and the global optimization of initial deviation analytically. To achieve collision avoidance, the concept of an artificial potential function is introduced to the controller, which still has the capability to change equilibrium points’ topological type to elliptic without breaking the system’s Hamiltonian structure.

Cislunar Navigation Constellation by Displaced Solar Sails

Increasing lunar exploration activities are giving rise to higher demands for a navigation constellation system in cislunar space. A novel constellation of solar sails around the Sun-Earth Artificial Lagrangian Points (ALPs) is proposed for cislunar navigation in this paper, which benefits from the numberless and out-of-plane advantages of ALPs compared with the classical Lagrangian points. To relieve the technical pressure on sail equipment, a two-layer optimisation strategy including the navigation constellation architecture and trajectory design is developed to reduce the desired lightness number of the sail's motion. The constellation architecture is constructed in the shape of a regular tetrahedron, whose size and orientation are derived from the realisable lightness number at the ALPs. The powerful Hamiltonian structure-preserving controller and differential evolution algorithm are adopted to propagate the bounded quasi-periodic trajectory with minimum lightness number variation. With the premise of the sail's high feasibility in the mechanism, the numerical navigation simulations for a typical trans-lunar weak stability boundary trajectory indicate that the proposed navigation constellation has a low geometric dilution of precision factor and a good navigation performance.

Solar Radiation Pressure Hamiltonian Feedback Control for Unstable Libration-Point Orbits

This work investigates a Hamiltonian structure-preserving control that uses the acceleration of solar radiation pressure for the stabilization of unstable periodic orbits in the circular restricted three-body problem. This control aims to stabilize the libration-point orbits in the sense of Lyapunov by achieving simple stability. It also preserves the Hamiltonian nature of the controlled system. The Hamiltonian structure-preserving control is then extended to a general case in which complex and conjugate eigenvalues occur at high-amplitude orbits. High-amplitude orbits are currently of interest to the European Space Agency for future libration-point orbit missions because they require a lower insertion compared to low-amplitude orbits. Based on the design of the feedback control, the purpose of this work is to verify when the use of solar radiation pressure is feasible and to determine the structural requirements and the spacecraft’s pointing accuracy.

Cluster flight control for fractionated spacecraft on an elliptic orbit

This paper deals with the stabilization of cluster flight on an elliptic reference orbit by the Hamiltonian structure-preserving control using the relative position measurement only. The linearized Melton’s relative equation is utilized to derive the controller and then the full nonlinear relative dynamics are employed to numerically evaluate the controller’s performance. In this paper, the hyperbolic and elliptic eigenvalues and their manifolds are treated without distinction notations. This new treatment not only contributes to solving the difficulty in feedback of the unfixed-dimensional manifolds, but also allows more opportunities to set the controlled frequencies of foundational motions or to optimize control gains. Any initial condition can be stabilized on a Kolmogorov–Arnold–Moser torus near a controlled elliptic equilibrium. The motions are stabilized around the natural relative trajectories rather than track a reference relative configuration. In addition, the bounded quasi-periodic trajectories generated by the controller have advantages in rapid reconfiguration and unpredictable evolution.

The Hamiltonian Structure-Preserving Control and Some Applications to Nonlinear Astrodynamics

A systematic research on the structure-preserving controller is investigated in this paper, including its applications to the second-order, first-order, time-periodic, or degenerated astrodynamics, respectively. The general form of the controller is deduced for the typical Hamiltonian system in full feedback and position-only feedback modes, which is successful in changing the hyperbolic equilibrium to an elliptic one. With the poles assigned at any different positions on imaginary axis, the controlled Hamiltonian system is Lyapunov stable. The Floquet multiplier is employed to measure the stability of time-dependent Hamiltonian system, because the equilibrium of periodic system may be unstable even though the equilibrium is always elliptic. One type of periodic orbits is achieved by the resonant conditions of control gains, and another type is making judicious choice in the foundational motions with different frequencies. The control gains are selected from the viewpoint of both the local and global optimizations on fuel cost. This controller is applied to some astrodynamics to achieve some interesting conclusions, including stable lissajous orbits in solar sail’s three-body problem and degenerated two-body problem, quasiperiodic formation flying on aJ2-perturbed mean circular orbit, and controlled frozen orbits for a spacecraft with a high area-to-mass ratio.

Application of Hamiltonian structure-preserving control to formation flying on a J 2-perturbed mean circular orbit

The bounded quasi-periodic relative trajectories are investigated in this paper for on-orbit surveillance, inspection or repair, which requires rapid changes in formation configuration for full three-dimensional imaging and unpredictable evolutions of relative trajectories for non-allied spacecraft. A linearized differential equation for modeling J 2 perturbed relative dynamics is derived without any simplified treatment of full short-period effects. The equation serves as a nominal reference model for stationkeeping controller to generate the quasi-periodic trajectories near the equilibrium, i.e., the location of the chief. The developed model exhibits good numerical accuracy and is applicable to an elliptic orbit with small eccentricity inheriting from the osculating conversion of orbital elements. A Hamiltonian structure-preserving controller is derived for the three-dimensional time-periodic system that models the J 2-perturbed relative dynamics on a mean circular orbit. The equilibrium of the system has time-varying topological types and no fixed-dimensional unstable/stable/center manifolds, which are quite different from the two-dimensional time-independent system with a permanent pair of hyperbolic eigenvalues and fixed-dimensions of unstable/stable/ center manifolds. The unstable and stable manifolds are employed to change the hyperbolic equilibrium to elliptic one with the poles assigned on the imaginary axis. The detailed investigations are conducted on the critical controller gain for Floquet stability and the optimal gain for the fuel cost, respectively. Any initial relative position and velocity leads to a bounded trajectory around the controlled elliptic equilibrium. The numerical simulation indicates that the controller effectively stabilizes motions relative to the perturbed elliptic orbit with small eccentricity and unperturbed elliptic orbit with arbitrary eccentricity. The developed controller stabilizes the quasi-periodic relative trajectories involved in six foundational motions with different frequencies generated by the eigenvectors of the Floquet multipliers, rather than to track a reference relative configuration. Only the relative positions are employed for the feedback without the information from the direct measurement or the filter estimation of relative velocity. So the current controller has potential applications in formation flying for its less computation overload for on-board computer, less constraint on the measurements, and easily-achievable quasi-periodic relative trajectories.