Stabilisation of the hyperbolic equilibrium of high area-to-mass spacecraft

In this paper we propose the exploitation of anti-heliotropic orbits, corresponding to the hyperbolic solution of the J2 and solar radiation pressure dynamical system, as gateway orbits between the low-eccentricity orbits where atmospheric drag does not affect the motion and the high eccentricity orbits which enter in drag regime. The eccentricity can be maintained in the neighborhood of the unstable point by means of a controller preserving the Hamiltonian structure of the system. In this way, any initial eccentricity close to the equilibrium conditions will lead to a bound trajectory around the controlled elliptic equilibrium. By selecting the time the controller is turned off, one of the two unstable manifolds leaving the equilibrium point can be followed, leading the orbit to become circular of to increase its eccentricity until natural decay occurs

Hamiltonian Neural Networks with Automatic Symmetry Detection

Recently, Hamiltonian neural networks (HNN) have been introduced to incorporate prior physical knowledge when learning the dynamical equations of Hamiltonian systems. Hereby, the symplectic system structure is preserved despite the data-driven modeling approach. However, preserving symmetries requires additional attention. In this research, we enhance HNN with a Lie algebra framework to detect and embed symmetries in the neural network. This approach allows to simultaneously learn the symmetry group action and the total energy of the system. As illustrating examples, a pendulum on a cart and a two-body problem from astrodynamics are considered

A Unification of Models of Tethered Satellites

In this paper, different conservative models of tethered satellites are related mathematically, and it is established in what limit they may provide useful insight into the underlying dynamics. An infinite dimensional model is linked to a finite dimensional model, the slack-spring model, through a conjecture on the singular perturbation of tether thickness. The slack-spring model is then naturally related to a billiard model in the limit of an inextensible spring. Next, the motion of a dumbbell model, which is lowest in the hierarchy of models, is identified within the motion of the billiard model through a theorem on the existence of invariant curves by exploiting Moser's twist map theorem. Finally, numerical computations provide insight into the dynamics of the billiard model

Analysis of Energy and QUadratic Invariant Preserving (EQUIP) methods

In this paper we are concerned with the analysis of a class of geometric integrators, at first devised in [14, 18], which can be regarded as an energy-conserving variant of Gauss collocation methods. With these latter they share the property of conserving quadratic first integrals but, in addition, they also conserve the Hamiltonian function itself. We here reformulate the methods in a more convenient way, and propose a more refined analysis than that given in [18] also providing, as a by-product, a practical procedure for their implementation. A thorough comparison with the original Gauss methods is carried out by means of a few numerical tests solving Hamiltonian and Poisson problems.Comment: 28 pages, 2 figures, 4 table

Dynamics and control of advanced space vehicles, volume 1

The following studies are reported: (1) Modal analyses of elastic continua for Liapunov stability analysis of flexible spacecraft; (2) development of general purpose simulation equations for arbitrary spacecraft; (3) evaluation of alternative mathematical models for elastic components of spacecraft; and (4) examination of the influence of vehicle flexibility on spacecraft attitude control system performance

Incorporation of Mission Design Constraints in Floquet Mode and Hamiltonian Structure-Preserving Orbital Maintenance Control Strategies for Libration Point Orbits

Libration point orbits are, in general, inherently unstable. Without the presence of corrective maneuvers a spacecraft will diverge from the vicinity of such trajectories. In this research effort, two orbital maintenance control strategies are studied: the impulsive Floquet Mode (FM) controller and the continuous Hamiltonian Structure-Preserving (HSP) controller. These two controllers are further developed to incorporate real-world mission design constraints. The FM controller is modified to accommodate feasible maneuver directions that are constrained to a plane or a line. This controller is shown to be applicable for orbital station-keeping of spin stabilized spacecraft that are only equipped with either tangential thrusters or axial thrusters. The HSP controller is extended for application to general three-dimensional hyperbolic libration point orbits, and then discretized to account for the minimum time required for orbit determination and/or scientific operations. Both controllers are applied to an unstable 1 halo orbit in the Sun-Earth/Moon system. The performances of these controllers are examined under the impacts of the spacecraft’s operation errors and mission design constraints. Simulation results suggest that the FM controller is capable of maintaining the motion of the spacecraft in the vicinity of the desired reference trajectory for the duration of the simulation, while satisfying all mission design constraints. The discrete-time MHSP controller proves to be able to improve the stability of the nominal trajectory by reducing the value of the unstable Poincare exponent of the reference orbit

해밀턴 구조와 외란 관측기 기법을 이용한 라그랑주 점 궤도 주변에서의 경계 상대 운동 및 궤도유지

학위논문 (박사) -- 서울대학교 대학원 : 공과대학 기계항공공학부, 2020. 8. 김유단.In this dissertation, a novel strategy for station-keeping and formation flight of spacecraft in the vicinity of unstable libration point orbits is presented, and its performance and stability are analyzed. The presented control strategy leverages the Hamiltonian nature of the equations of motion, rather than simply applying the control theory from the perspective of ``signal processing". A filtered extended high-gain observer, a kind of disturbance observer, is designed to mitigate the performance degradation of the control strategy due to model uncertainties and external disturbances. Canonical coordinates are adopted to design a controller that exploits the mathematical structure of Hamiltonian system inherent in orbital mechanics, and then the equations of motion of spacecraft are represented in the form of Hamilton's equation with generalized coordinates and momenta. The baseline controller, utilizing the canonical form of the Hamiltonian system, is divided into two parts: i) a Hamiltonian structure-preserving control, and ii) an energy dissipation control. Hamiltonian structure-preserving control can be designed in accordance with the Lagrange-Dirichlet criterion, i.e., a sufficient condition for the nonlinear stability of Hamiltonian system. Because the Hamiltonian structure-preserving control makes the system marginally stable instead of asymptotically stable, the resultant motion of the Hamiltonian structure-preserving control yields a bounded trajectory. Through the frequency analysis of bounded relative motion, a circular motion can be achieved for particular initial conditions. By appropriately switching the gain of the Hamiltonian structure-preserving control, the radius of bounded motion can be adjusted systematically, which is envisioned that this approach can be applied to spacecraft formation flight. Furthermore, the energy dissipation control can be activated to make the spacecraft's bounded relative motion converge to the nominal orbit. On the other hand, a filtered extended high-gain observer is designed for the robust station-keeping and formation flight even under highly uncertain deep-space environment. The filtered extended high-gain observer estimates the velocity state of the spacecraft and disturbance acting on the spacecraft by measuring only the position of the spacecraft. The filtered extended high-gain observer includes an integral state feedback to attenuate navigation error amplification due to the high gain of the observer. The global convergence of the observer is shown, and it is also shown that the tracking error is ultimately bounded to the nominal libration point orbit by applying the Hamiltonian structure-based controller. Numerical simulations demonstrate the performance of the designed control strategy. Halo orbit around the L2 point of the Earth-Moon system is considered as an illustrative example, and various perturbations are taken into account.본 논문에서는 불안정한 동적특성을 갖는 라그랑주 점 궤도 주변에서 위성의 궤도유지 및 편대비행을 위한 제어기와 관측기를 설계하였으며, 설계된 제어기와 관측기의 안정성 그리고 전체 시스템의 안정성을 분석하였다. 설계한 기준 제어 전략은 신호처리 관점의 제어이론을 기반으로 하지 않고, 라그랑주 점 궤도의 자연적인 수학적 구조를 활용하였다. 모델 불확실성과 외부 외란으로 인한 기준 제어 전략의 성능저하를 완화하기 위해 외란관측기의 일종인 확장 고이득 관측기를 설계하였다. 본 논문에서는 궤도역학에 내재되어 있는 해밀턴 시스템의 구조를 활용하는 제어기를 설계하기 위해 정준좌표를 도입하였으며, 좌표변환을 통해 위성의 운동방정식을 해밀턴 시스템의 정준형식으로 나타내었다. 해밀턴 시스템의 정준형식으로 표현된 운동방정식을 이용해 설계한 기준 제어기는 해밀턴-구조 보존제어와 에너지 소산제어로 분리 설계된다. Lagrange-Dirichlet 기준은 정준형식으로 나타낸 해밀턴 시스템의 비선형 안정성을 판별하는 충분조건으로, 해밀턴-구조 보존제어 설계의 기준이 된다. 기준 라그랑주 점 궤도 주위에서 해밀턴-구조 보존 제어를 적용한 결과, 위성은 기준궤도로 수렴하지 않고 기준궤도와 유한한 거리를 유지하는 경계운동을 하였다. 경계운동의 주파수 분석을 통하여 특정한 초기조건 하에서는 원형 경계운동이 가능하였으며, 더 나아가 해밀턴-구조 보존제어의 제어이득 값을 적절히 설정함으로 원형 경계운동의 크기를 체계적으로 조절할 수 있고 이를 위성 편대비행에 응용할 수 있음을 보였다. 추가적으로 에너지 소산제어 입력을 설계하여 위성이 기준 라그랑주 점 궤도로 점근 수렴하는 운동도 가능함을 수학적으로 증명하였다. 한편, 심우주상의 예측하기 어려운 섭동력 및 불확실성 하에서도 강건한 궤도유지와 편대비행을 수행하기 위해 확장 고이득 관측기를 설계하였다. 확장 고이득 관측기는 위성의 위치 정보만을 이용하여 위성의 속도와 위성에 작용하는 외란을 동시에 추정하며, 추정된 상태변수를 이용하여 기준이 되는 피드백 제어입력을 생성한다. 추정된 외란은 피드포워드 형태의 제어입력으로 구성되어 제어기의 성능을 강건하게 만든다. 심우주 공간상의 위성의 궤도결정 결과로 얻어지는 위치정보는 상대적으로 큰 오차를 갖는데, 확장 고이득 관측기는 위치 오차를 증폭시킨다는 단점이 있다. 본 연구에서는 이러한 단점을 완화하고자 적분 관측기 형태로 개선된 필터링된 확장 고이득 관측기를 설계하고 수렴성을 분석하였다. 그리고 필터링된 확장 고이득 관측기와 시스템의 해밀턴 구조를 활용하는 제어기를 적용한 전체 시스템의 안정성을 분석하였다. 불안정한 라그랑주 점 궤도 주변에서 위성의 궤도유지와 편대비행을 위해 설계된 제어기법의 성능을 확인하고자 수치 시뮬레이션을 수행하였다. 수치 시뮬레이션을 위해 지구-달 시스템의 L2 주변 헤일로 궤도를 기준궤도로 설정하였으며, 심우주 공간에서의 다양한 섭동력 및 모델 불확실성을 고려하였다. 궤도결정 오차로 인한 위성의 위치 및 속도 불확실성이 존재 하더라도 제안한 제어기법을 통해 위성이 궤도유지와 편대비행을 만족스럽게 수행함을 보였다.1 Introduction 1 1.1 Background and Motivation 1 1.2 Literature Review 3 1.2.1 Spacecraft Station-Keeping in the Vicinity of the Libration Point Orbits 3 1.2.2 Spacecraft Formation Flight in the Vicinity of the Libration Point Orbits 5 1.3 Contributions 7 1.4 Dissertation Outline 10 2 Background 13 2.1 Circular Restricted Three-Body Problem 14 2.1.1 Equilibrium Solutions and Periodic Orbits 16 2.1.2 Stability of Periodic Orbits 20 2.2 Hamiltonian Mechanics 21 2.2.1 Hamiltonian Approach to CR3BP 21 2.2.2 Hamiltonian Approach to LPO Tracking Problem 22 3 Hamiltonian Structure-Based Control 25 3.1 Classical Linear Hamiltonian Structure-Preserving Control 27 3.2 Switching Hamiltonian Structure-Preserving Control 29 3.2.1 Orbital Properties of Spacecraft 33 3.2.2 Switching Point 1: From a Circular Orbit to an Elliptical Orbit 34 3.2.3 Switching Point 2: From an Elliptical Orbit to a Circular Orbit 37 3.3 Hamiltonian Structure-Based Control 39 3.3.1 Potential Shaping Control 39 3.3.2 Energy Dissipation Control 45 4 Filtered Extended High-Gain Observer and Closed-Loop Stability 49 4.1 Filtered Extended High-Gain Observer and Its Convergence 51 4.2 Closed-Loop Stability Analysis 56 5 Numerical Simulations 67 5.1 Disturbance Model 67 5.2 Navigation Error Model 68 5.3 Simulation Results 69 5.3.1 Simulation 1 71 5.3.2 Simulation 2 77 5.3.3 Simulation 3 81 5.3.4 Simulation 4 93 5.3.5 Simulation 5 98 6 Conclusion 101 6.1 Concluding Remarks 101 6.2 Further Work 103 Bibliography 105 국문초록 127Docto