We consider the problem of cooperation among a collection of vehicles performing a shared task using intervehicle communication to coordinate their actions. Tools from algebraic graph theory prove useful in modeling the communication network and relating its topology to formation stability. We prove a Nyquist criterion that uses the eigenvalues of the graph Laplacian matrix to determine the effect of the communication topology on formation stability. We also propose a method for decentralized information exchange between vehicles. This approach realizes a dynamical system that supplies each vehicle with a common reference to be used for cooperative motion. We prove a separation principle that decomposes formation stability into two components: Stability of this is achieved information flow for the given graph and stability of an individual vehicle for the given controller. The information flow can thus be rendered highly robust to changes in the graph, enabling tight formation control despite limitations in intervehicle communication capability.

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Information flow and cooperative control of vehicle formations

A tutorial account of variable structure control with sliding mode is presented. The purpose is to introduce in a concise manner the fundamental theory, main results, and practical applications of this powerful control system design approach. This approach is particularly attractive for the control of nonlinear systems. Prominent characteristics such as invariance, robustness, order reduction, and control chattering are discussed in detail. Methods for coping with chattering are presented. Both linear and nonlinear systems are considered. Future research areas are suggested and an extensive list of references is included. >

Variable structure control: a survey

This paper assumes a requirement for an unmanned multiunit satellite to be assembled in orbit. The requirement to be met is to bring the satellites together so tha t they do not collide but actually rendezvous. The equations of motion of the rendezvous satellite in a relative coordinate system are derived and used to compute a final injection velocity which would effect collision after a time r. The velocity is corrected periodically by a command guidance system and just before impact retrothrust is applied. A terminal infrared homing sj^stem is required to actually accomplish physical contact and joining of the satellites. The first satellite placed in orbit is the "control satellite" and controls all the satellites to be assembled and contains the ccmputer, command guidance equipment, precision orientation equipment, and other features necessary to effect rendezvous. The succeeding satellites contain a propulsion system, a rough at t i tude control system, and a command receiver plus whatever scientific equipment they carry to perform their basic mission. This paper presents the following:

Terminal Guidance System for Satellite Rendezvous

Analytical Mechanics of Space Systems, Third Edition provides a comprehensive treatment of dynamics of space systems, starting with the fundamentals and covering topics from basic kinematics and dynamics to more advanced celestial mechanics. The reader is guided through the various derivations and proofs in a tutorial way and is led to understand the principles underlying the equations at issue, and shown how to apply them to various dynamical systems. Part I covers analytical treatment of topics such as basic dynamic principles up to advanced energy concepts. 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. MATLAB[registered], Mathematica[registered], Python and C-Code toolboxes are provided for the rigid body kinematics routines discussed in chapter 3, and the basic orbital 2-body orbital mechanics routines discussed in chapter 9. The third edition streamlines the presentation of material by adding additional examples, homework problems, and illustrations. It includes expanded discussion on: Numerically integrating MRPs and using heading measurements and evaluating a three-dimensional orientation; Numerically integrating the complex VSCMG differential equations of motion; The Lyapunov function and stability definitions; implementing a rate-based attitude servo control solution, and integrating an integral feedback component with a reaction wheel based attitude control, featuring new examples; developing acceleration-based VSCMG steering laws for three-axis attitude control developments; and, new Appendix I describes how to implement Kalman-Filter estimating MRP coordinates in a non-singular fashion.

Analytical Mechanics of Space Systems

This paper provides a survey of modern nonlinear filtering methods for attitude estimation. Early applications relied mostly on the extended Kalman filter for attitude estimation. Since these applications, several new approaches have been developed that have proven to be superior to the extended Kalman filter. Several of these approaches maintain the basic structure of the extended Kalman filter, but employ various modifications in order to provide better convergence or improve other performance characteristics. Examples of such approaches include: filter QUEST, extended QUEST and the backwards-smoothing extended Kalman filter. Filters that propagate and update a discrete set of sigma points rather than using linearized equations for the mean and covariance are also reviewed. A twostep approach is discussed with a first-step state that linearizes the measurement model and an iterative second step to recover the desired attitude states. These approaches are all based on the Gaussian assumption that the probability density function is adequately specified by its mean and covariance. Other approaches that do not require this assumption are reviewed, Associate Professor, Department of Mechanical & Aerospace Engineering. Email: johnc@eng.buffalo.edu. Associate Fellow AIAA. Aerospace Engineer, Guidance, Navigation and Control Systems Engineering Branch. Email: Landis.Markley@nasa.gov. Fellow AIAA. Postdoctoral Research Fellow, Department of Mechanical & Aerospace Engineering. Email: cheng3@eng.buffalo.edu. Member AIAA.

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Survey of nonlinear attitude estimation methods

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. Intended for graduate students, professors and academic researchers in the fields of aerospace and mechanical engineering, mathematics, astronomy and astrophysics, "Spacecraft Formation Flying" is a technical yet accessible, forward-thinking guide to this critical area of astrodynamics. This is the first book dedicated to spacecraft formation flying, written by leading researchers and professors in the field. This title develops the theory from an astrodynamical viewpoint, emphasizing modeling, control and navigation of formation flying satellites on Earth orbits. It provides examples that are used to illustrate the main developments, with a sample simulation of a formation flying mission included to illustrate high fidelity modeling, control and relative navigation.

Spacecraft Formation Flying: Dynamics, Control and Navigation

The spacecraft large-angle maneuver problem is treated using the principles of variable-structure control theory. A control law that constrains the state to follow a specified path (the so-called sliding mode) in the state-space is designed on the basis of a simplified model of the spacecraft dynamics. The sliding mode is obtained by solving an optimal control problem posed in a reduced space, the solution being the angular velocities of the spacecraft as functions of the attitude variables (Euler parameters or quaternions in the present context). It is shown that the motion along the sliding mode is insensitive to parameter variations and unmodeled effects.

Variable-Structure Control of Spacecraft Large-Angle Maneuvers

Two nonlinear feedback control laws are presented for reestablishing a desired J2 invariant relative orbit. Since it is convenient to describe the relative orbit of a deputy with respect to a chief satellite in terms of mean orbit element differences, and because the conditions for a relative orbit being J2 invariant are expressed in terms of mean orbit elements, the first control law feeds back errors in terms of mean orbit elements. Dealing with mean orbit elements has the advantage that short period oscillations are not perceived as tracking errors; rather, only the long term tracking errors are compensated for. The second control law feeds back traditional Cartesian position and velocity tracking errors. For both of the control laws, the desired orbit is computed using mean orbit elements. A numerical study compares and contrasts the two feedback laws.

Spacecraft Formation Flying Control Using Mean Orbit Elements

The design of nonrobust and robust time-optimal controllers for linear systems in the frequency domain is presented. The bang-bang profile is represented as the superposition of time-delayed step inputs or the output of a time-delay filter subject to a step input. A parameter optimization problem is formulated to minimize the final time of the maneuver with the constraint that the time-delay filter cancels all of the poles of the system. The issue of robustness to errors in the model is addressed by placing multiple zeros of the time-delay filter at the estimated locations of the poles of the system. The design technique is illustrated on representative models of large space structures, for rest-to-rest, time-optimal, and robust time-optimal maneuvers. Spin-up maneuvers are shown to be special cases of the general formulation. IME-OPTIMAL control of flexible spacecraft is a topic of current interest.1 Many computational approaches and analyses have been presented in the recent literature to deal with the effects of flexibility. Most of these works deal with planar (singleaxis) rest-to-rest maneuvers under two categories: near-minimumtime control and exact-minimum-time control. The first category of methods is based on smooth approximations to minimum-time control for an equivalent rigid body. This class of methods has been shown to be well suited when applied moments or torques are produced by either throttlable thrusters or reaction wheels.2'3 Higher modes of the system are not excited due to the smoothness of the control profile. The second category of methods deals with on-off thrusters directly. Rajan4 formulates the problem including

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Robust time-optimal control - Frequency domain approach

Variable speed control moment gyroscopes (CMG) are single-gimbal gyroscopes where the fly wheel speed is allowed to be variable. The equations of motion of a generic rigid body with several such variable speed CMGs attached are presented. The formulation is such that it can easily accommodate the classical cases of having either control moment gyros or reaction wheels to control the spacecraft attitude. A globally asymptotically stabilizing nonlinear feedback control law is presented. For a redundant control system, a weighted minimum norm inverse is used to determine the control vector. This approach allows the variable speed control moment gyroscopes to behave either more like classical reaction wheels or more like control moment gyroscopes, depending on the local optimal steering logic. Where classical control moment gyroscope control laws have to deal with singular gimbal angle configurations, the variable speed control moment gyroscopes are shown not to encounter any singularities for many representative examples considered. Both a gimbal angle velocity and an acceleration-based steering law are presented. Further, the use of the variable speed CMG null motion is discussed to reconfigure the gimbal angles to preferred sets. Having a variable reaction wheel speed allows for a more general redistribution of the internal momentum vector

Srinivas R. Vadali

Papers

Spacecraft Formation Flying: Dynamics, Control and Navigation

Variable-Structure Control of Spacecraft Large-Angle Maneuvers

Spacecraft Formation Flying Control Using Mean Orbit Elements

Robust time-optimal control - Frequency domain approach

Feedback Control Law for Variable Speed Control Moment Gyros