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Minimum-Time and Minimum-Fuel Low-Thrust Trajectory Design for Satellite Formation in Low-Earth Orbits
Abstract
Indirect formalism of optimal control theory is used to generate minimum-time and minimum-fuel trajectories for formation of two spacecraft (deputies) relative to a chief satellite. For minimum-fuel problems, a hyperbolic tangent smoothing method is used to facilitate numerical solution of the resulting boundary-value problems by constructing a one-parameter family of smooth control profiles that asymptotically approach the theoretically optimal, but non-smooth bang-bang thrust profile. Impact of the continuation parameter on the solution of minimum-fuel trajectories is analyzed. The fidelity of the dynamical model is improved beyond the two-body dynamics by including the perturbation due to the Earth’s second zonal harmonic, J 2 . In addition, a particular formation is investigated, where the deputies are constrained to lie diametrically opposite on a three-dimensional sphere centered at the chief.
... The satellite constellation problem is an interesting problem that was solved by the author using an indirect method in [163]. This problem considers the case of two "deputy" or "chaser" satellites being deployed by a launch vehicle into a low Earth orbit (LEO) . ...
... This constellation formation problem is simplified here to the case of one deputy assuming a leading orbit. However, two-deputy indirect solutions can be found in [163]. An interesting aspect of the problem is that it is solved in a Local-Vertical-Local-Horizontal (LVLH) frame of reference. ...
... Expressions for the J 2 acceleration in the ECI frame can be found in Curtis [32], and the standard Euler 3-1-3 rotation sequence can be used to map the perturbations in the LVLH frame. However, it was found in [163] that the J 2 perturbation had a negligible effect on the solution for this problem, so it is ignored. The problem data is given in Table 6.5. ...
The fields of trajectory optimization and optimal control are closely connected. Practical trajectory optimization techniques are built on the indirect and direct methods for solving optimal control problems. Indirect methods seek to satisfy the first-order necessary conditions of optimality, while direct methods discretize the problem and transcribe it into a large-scale nonlinear programming (NLP) problem. This study provides a basic overview of the direct and indirect methods. The indirect necessary conditions and Pontryagin's Minimum Principle are reviewed, and the approaches taken by direct methods are presented. The focus of this study then shifts to pseudospectral direct methods, which belong to the class of collocation methods. The theoretical groundwork of pseudospectral methods is laid, leading to a pseudospec-tral formulation and software for solving general optimal control problems. Several types of pseudospectral methods are presented, including the Legendre-Gauss and Chebyshev-Gauss methods. Special attention is given to the Legendre-Gauss-Radau (LGR) method, which is the primary transcription employed by the MATLAB-based Tiger Optimization Software (TOPS). TOPS is a general-purpose pseudospectral optimal control software developed by the author as part of their research in the Aero-Astro Computational and Experimental (ACE) Lab. A multi-interval pseudospectral method is presented, and the discrete form of the optimality conditions are derived. The study shifts focus from theory to application, and the practical aspects of pseudospectral optimal control methods are discussed. Another objective of this research is to compile the author's knowledge with regard to implementing pseudospectral techniques, thus enabling the reader to easily implement their own pseudospectral method. Several mesh refinement methods are compared and their merits are compared. In addition, several techniques that improve the efficiency of an NLP solver are presented, including efficient and exact derivative formulas and several scaling techniques. Finally, several example optimal control problems are solved using TOPS. The solutions obtained from TOPS are compared to the solutions obtained using indirect techniques to verify their accuracy and demonstrate the capabilities of TOPS.
A novel methodology is proposed for designing low-thrust trajectories to quasi-periodic, near rectilinear Halo orbits that leverages ephemeris-driven, "invariant manifold analogues" as long-duration asymptotic terminal coast arcs. The proposed methodology generates end-to-end, eclipse-conscious, fuel-optimal transfers in an ephemeris model using an indirect formulation of optimal control theory. The end-to-end trajectories are achieved by patching Earth-escape spirals to a judiciously chosen set of states on pre-computed manifolds. The results elucidate the efficacy of employing such a hybrid optimization algorithm for solving end-to-end analogous fuel-optimal problems using indirect methods and leveraging a composite smooth control construct. Multiple representative cargo re-supply trajectories are generated for the Lunar Orbital Platform-Gateway (LOP-G). A novel process is introduced to incorporate eclipse-induced coast arcs and their impact within optimization. The results quantify accurate Δ costs required for achieving efficient eclipse-conscious transfers for several launch opportunities in 2025 and are anticipated to find applications for analogous uncrewed missions.
Finding a solution to a low-thrust minimum-fuel transfer trajectory optimization problem in the circular restricted three-body problem (CRTBP) is already known to be extremely difficult due to the strong nonlinearity and the discontinuous bang-bang control structure. What is less commonly known is that such a problem can have many local solutions that satisfy all the first-order necessary conditions of optimality. In this paper, the existence of multiple solutions in the CRTBP is fully demonstrated, and a practical technique that involves multiple homotopy procedures is proposed, which can robustly search as many as desired local solutions to determine the best solution with the optimal performance index. Due to the common failure of the general homotopy method in tracking a homotopy path, the parameter bounding fixed-point (PBFB) homotopy is coupled to construct the double-homotopy, which can overcome the difficulties by tracking the discontinuous homotopy path. Furthermore, the PBFB homotopy is capable to achieve multiple solutions that lie on separate homotopy path branches. The minimum-time problem is first solved by the double-homotopy on the thrust magnitude to infer a reasonable transfer time. Then the auxiliary problem is solved by the double-homotopy and the PBFP homotopy, and multiple local solutions with smooth control profiles are obtained. By tracking homotopy paths starting from these solution points, multiple solutions to the minimum-fuel problem are obtained, and the best solution is eventually identified. Numerical demonstration of the minimum-fuel transfers from a GEO to a halo orbit is presented to substantiate the effectiveness of the technique.
The minimum-time transfer of a satellite from a low and eccentric initial orbit toward a high geostationary orbit is considered. This study is preliminary to the analysis of similar transfer cases with more complicated performance indexes (maximization of payload, for instance). The orbital inclination of the spacecraft is taken into account (3D model), and the thrust available is assumed to be very small (e.g. 0.3 Newton for an initial mass of 1500 kg). For this reason, many revolutions are required to achieve the transfer and the problem becomes very oscillatory. In order to solve it numerically, an optimal control model is investigated and a homotopic procedure is introduced, namely continuation on the maximum modulus of the thrust: the solution for a given thrust is used to initiate the solution for a lower thrust. Continuous dependence of the value function on the essential bound of the control is first studied. Then, in the framework of parametric optimal control, the question of differentiability of the transfer time with respect to the thrust is addressed: under specific assumptions, the derivative of the value function is given in closed form as a first step toward a better understanding of the relation between the minimum transfer time and the maximum thrust. Numerical results obtained by coupling the continuation technique with a single–shooting procedure are detailed.
This paper presents an efficient indirect optimization method to solve time-and fuel-optimal asteroid landing trajectory design problems. The gravitational field of the target asteroid is approximated with two methods: 1) a simple two-body (point-mass) model, and 2) a high-fidelity polyhedral model. A homotopy approach, at the level of the gravity-model, is considered to solve the resulting boundary-value problems. Moreover, in fuel-optimal trajectories, the difficulties associated with control discontinuities (i.e., the so-called bang-off-bang profile of thrust) are overcome by employing a recently introduced hyperbolic tangent smoothing method. We generate time-and fuel-optimal trajectories for spacecraft landing on asteroid (101955) Bennu with low-thrust propulsion. The trajectories are validated through comparison with the trajectories obtained by other methods. The results indicate that the two-body gravity model can be used for generating high-quality solutions to "warm-start" numerical methods when a high-fidelity gravity model is used.
Application of idealized constant-specific-impulse, constant-thrust electric thruster performance models or curve-fitted polynomials is quite common for spacecraft trajectory design. However, incorporation of realistic performance models of multi-mode electric thrusters leads to notable challenges, and at the same time, offers unprecedented system-level optimization opportunities. In this paper, a framework is developed and demonstrated for co-optimization of 1) spacecraft trajectory, 2) operation modes of multi-mode propulsion systems, and 3) solar array size. The selection of the most optimal operation modes is in accordance to Pontryagin's minimum principle for solving a payload-mass-maximization problem. The novelty of the work lies in solving a mixed-integer trajectory optimization problem featuring user-defined constraints on the maximum number of operating modes along the trajectory. Utility of the framework is demonstrated through optimization of a multi-year trajectory from Earth to comet 67P/Churyumov–Gerasimenko using an SPT-140 Hall thruster with 21 operating modes; the results are interesting and of significant practical utility.
Indirect optimization methods convert optimal control problems (OCPs) into two-or multi-point boundary-value problems. A highly desirable feature of indirect methods, specifically for space applications, is that high-resolution trajectories can be generated, which satisfy the first-order necessary conditions of optimality. A recently developed Composite Smoothing Control (CSC) framework is utilized to formulate and solve the problem of simultaneous trajectory optimization and propulsion subsystem design of spacecraft. A reasonable parameterized breakdown of the spacecraft mass is adopted, which captures the impact of power produced by the solar arrays and its contribution to the total spacecraft mass. Thus, the implicit trade-offs can be considered in the indirect optimization approach. The function space co-optimization problem of spacecraft power subsystem parameters along with the main trajectory is solved with the objective to maximize the payload delivered. The proposed framework amounts to an invariant embedding that reduces the original, difficult-to-solve, multi-point boundary-value problem into a two-point boundary-value problem with continuous, differentiable control inputs. Utility of the proposed construct is demonstrated through a low-thrust, multi-revolution, multi-year rendezvous maneuver to asteroid Dionysus with a variable-specific-impulse, variable-thrust modeled engine. This is the first time that indirect optimization methods have tackled such a complex co-optimization problem using the CSC framework.
Efficient performance of a number of engineering systems is achieved through different modes of operation - yielding systems described as “hybrid”, containing both real-valued and discrete decision variables. Prominent examples of such systems, in space applications, could be spacecraft equipped with 1) a variable-Isp, variable-thrust engine or 2) multiple engines each capable of switching on/off independently. To alleviate the challenges that arise when an indirect optimization method is used, a new framework — Composite Smooth Control (CSC) — is proposed that seeks smoothness over the entire spectrum of distinct control inputs. A salient aftermath of the application of the CSC framework is that the original multi-point boundary-value problem can be treated as a two-point boundary-value problem with smooth, differentiable control inputs; the latter is notably easier to solve, yet can be made to accurately approximate the former hybrid problem. The utility of the CSC framework is demonstrated through a multi-year, multi-revolution heliocentric fuel-optimal trajectory for a spacecraft equipped with a variable-Isp, variable-thrust engine.
Equipping a spacecraft with multiple solar-powered electric engines (of the same or different types) compounds the task of optimal trajectory design due to presence of both real-valued inputs (power input to each engine in addition to the direction of thrust vector) and discrete variables (number of active engines). Each engine can be switched on/off independently and “optimal” operating power of each engine depends on the available solar power, which depends on the distance from the Sun. Application of the Composite Smooth Control (CSC) framework to a heliocentric fuel-optimal trajectory optimization from the Earth to the comet 67P/Churyumov-Gerasimenko is demonstrated, which presents a new approach to deal with multiple-engine problems. Operation of engine clusters with 4, 6, 10 and even 20 engines of the same type can be optimized. Moreover, engine clusters with different/mixed electric engines are considered with either 2, 3 or 4 different types of engines. Remarkably, the CSC framework allows us 1) to reduce the original multi-point boundary-value problem to a two-point boundary-value problem (TPBVP), and 2) to solve the resulting TPBVPs using a single-shooting solution scheme and with a random initialization of the missing costates. While the approach we present is a continuous neighbor of the discontinuous extremals, we show that the discontinuous necessary conditions are satisfied in the asymptotic limit. We believe this is the first indirect method to accommodate a multi-mode control of this level of complexity with realistic engine performance curves. The results are interesting and promising for dealing with a large family of such challenging multi-mode optimal control problems.
As is well known in celestial mechanics, coordinate choices have significant consequences in the analytical and
computational approaches to solve the most fundamental initial value problem. The present study focuses on the
impact of various coordinate representations of the dynamics on the solution of the ensuing state/costate two-point
boundary-value problems that arise when solving the indirect optimal control necessary conditions. Minimum-fuel
trajectory designs are considered for 1) a geocentric spiral from geostationary transfer orbit to geostationary Earth
orbit, and 2) a heliocentric transfer from Earth to a highly eccentric and inclined orbit of asteroid Dionysus. This study
unifies and extends the available literature by considering the relative merits of eight different coordinate choices to
establish the state/costate differential equations. Two different sets of orbit elements are considered: equinoctial
elements and a six-element set consisting of the angular momentum vector and the eccentricity vector. In addition to
the Cartesian and spherical coordinates, four hybrid coordinate sets associated with an osculating triad defined by the
instantaneous position and velocity vectors that consist of a mixture of slow and fast variables are introduced and
studied. Reliability and efficiency of convergence to the known optimal solution are studied statistically for all eight
sets; the results are interesting and of significant practical utility.
Application of optimal control principles on a number of engineering systems reveal bang-bang and/or bang-off-bang structures in some or all of the control inputs. These abrupt changes introduce undesired non-smoothness into the equations of motion, and their ensuing numerical propagation, which requires special treatments. In order to alleviate these induced difficulties, a generic smoothing technique is proposed that is straightforward to implement while providing additional flexibility in the rate of change of the control. The proposed technique does not affect the standard derivation of the so-called indirect methods (i.e., optimality conditions, form of the cost functional, and the structure of the switching function), which makes it ideal for programming purposes. In essence, the smoothing is applied directly at the control level. In many cases, modest smoothing can be introduced with full visibility of the frequently near-negligible loss of performance relative to smoothing the discontinuous controls. The utility of the proposed method is illustrated via two different problems: 1) a minimum-fuel multiple-revolution low-thrust interplanetary trajectory design problem, and 2) a rest-to-rest minimum-time attitude reorientation problem. Numerical results indicate that the domain of convergence of the TPBVP enlarges significantly where random initialization of the unknown co-states is sufficient to get a feasible solution. This solution can be used in a numerical continuation method (and over the smoothing parameter) to guide the solution towards the solution of the original problem.
Description
This book provides a comprehensive treatment of dynamics of space systems starting with the basic fundamentals. This single source contains topics ranging from basic kinematics and dynamics to more advanced celestial mechanics; yet all material is presented in a consistent manner. The reader is guided through the various derivations and proofs in a tutorial way. The use of "cookbook" formulas is avoided. Instead, the reader is led to understand the underlying principle of the involved equations and shown how to apply them to various dynamical systems.The book is divided into two parts. Part I covers analytical treatment of topics such as basic dynamic principles up to advanced energy concept. Special attention is paid to the use of rotating reference frames that often occur in aerospace systems. Part II covers basic celestial mechanics treating the two-body problem, restricted three-body problem, gravity field modeling, perturbation methods, spacecraft formation flying, and orbit transfers.A Matlab® kinematics toolbox provides routines which are developed in the rigid body kinematics chapter. A solutions manual is also available for professors. Matlab® is a registered trademark of The MathWorks, Inc.
This paper investigates the solution of the bang-bang optimal control problem that applied to formation-flying satellites reconfiguration. Based on the Tschauner-Hempel equations, first-order necessary optimality conditions are derived from Pontryagin's maximum principle for the two thruster configurations. Those necessary optimality conditions are numerically satisfied by solving the associated shooting functions. To settle the initial guess sensitivity and control discontinuity, a homotopic approach algorithm is proposed. Analytical expressions of the initial costate are derived from the simplified energy-optimal reconfiguration problem, thus the homotopic approach algorithm can start automatically for the general reconfiguration problems. It is proven that the six identical thrusters configuration would consume more fuel than the single thruster configuration. Finally, the performance of the algorithm is illustrated by simulating a circular reference orbit reconfiguration problem.
Space agencies are now realizing that much of what has previously been achieved using hugely complex and costly single platform projects-large unmanned and manned satellites (including the present International Space Station)-can be replaced by a number of smaller satellites networked together. The key challenge of this approach, namely ensuring the proper formation flying of multiple craft, is the topic of this second volume in Elsevier's Astrodynamics Series, Spacecraft Formation Flying: Dynamics, control and navigation. In this unique text, authors Alfriend et al. provide a coherent discussion of spacecraft relative motion, both in the unperturbed and perturbed settings, explain the main control approaches for regulating relative satellite dynamics, using both impulsive and continuous maneuvers, and present the main constituents required for relative navigation. The early chapters provide a foundation upon which later discussions are built, making this a complete, standalone offering.
Fuel optimal maneuvers of spacecraft relative to a body in circular orbit are investigated using a point mass model in which the magnitude of the thrust vector is bounded. All nonsingular optimal maneuvers consist of intervals of full thrust and coast and are found to contain at most seven such intervals in one period. Only four boundary conditions where singular solutions occur are possible. Computer simulation of optimal flight path shapes and switching functions are found for various boundary conditions. Emphasis is placed on the problem of soft rendezvous with a body in circular orbit.
Formation flying is defined as a set of more than one spacecraft whose states are coupled through a common control law. This paper provides a comprehensive survey of spacecraft formation flying control (FFC), which encompasses design techniques and stability results for these coupled-state control laws. We divide the FFC literature into five FFC architectures: (i) multiple-input multiple-output, in which the formation is treated as a single multiple-input, multiple-output plant, (ii) leader/follower, in which individual spacecraft controllers are connected hierarchically, (iii) virtual structure, in which spacecraft are treated as rigid bodies embedded in an overall virtual rigid body, (iv) cyclic, in which individual spacecraft controllers are connected non-hierarchically, and (v) behavioral, in which multiple controllers for achieving different (and possibly competing) objectives are combined. This survey significantly extends an overview of the FFC literature provided by Lawton, which discussed the L/F, virtual structure and behavioral architectures. We also include a brief history of the formation flying literature, and discuss connections between spacecraft FFC and other multi-vehicle control problems in the robotics and automated highway system literatures.
Formation flying is defined as a set of more than one spacecraft whose states are coupled through a common control law. This paper provides a comprehensive survey of spacecraft formation flying control (FFC), which encompasses design techniques and stability results for these coupled-state control laws. We divide the FFC literature into five FFC architectures: (i) Multiple-Input Multiple-Output, in which the formation is treated as a single multiple-input, multiple-output plant, (ii) Leader/Follower, in which individual spacecraft controllers are connected hierarchically, (iii) Virtual Structure, in which spacecraft are treated as rigid bodies embedded in an overall virtual rigid body, (iv) Cyclic, in which individual spacecraft controllers are connected non-hierarchically, and (v) Behavioral, in which multiple controllers for achieving different (and possibly competing) objectives are combined. This survey significantly extends an overview of the FFC literature provided by Lawton, which discussed the L/F, Virtual Structure and Behavioral architectures. We also include a brief history of the formation flying literature, and discuss connections between spacecraft FFC and other multi-vehicle control problems in the robotics and Automated Highway System literatures.
A survey of spacecraft formation flying guidance and control (part 1): guidance
- D Scharf
- F Hadaegh
- S Ploen
D. Scharf, F. Hadaegh, and S. Ploen, "A survey of spacecraft formation
flying guidance and control (part 1): guidance," in Proceedings of the
2003 American Control Conference, 2003., vol. 2, 2003, pp. 1733-1739.