Introduction P pointing control and rapid maneuvering capabilities have long been part of many space missions; examples include military observation platforms, communications satellites, and exploratory space missions. Rapid retargeting may be an intrinsic part of the mission profile, as in military and exploratory applications, or it may be required to correct, periodically, the guidance and navigation sensors of a spacecraft. Consequently, research in attitude maneuvers and time-optimal controls has been a consistently strong field of study. Of particular interest is the time-optimal attitude maneuver; this problem combines aspects of many different fields, such as mathematics, optimal control theory, spacecraft dynamics, elasticity, and structural mechanics. In the study of attitude maneuvers and control, researchers have investigated the response of satellite control systems to disturbances caused by the influence of solar pressure, gravity gradients, the effectiveness of active control devices, and the best placement of actuators and sensors on a given structure. Numerous studies have been conducted for specific satellite configurations to determine a time-optimal attitude control for a predetermined maneuver. Further, the mathematical procedures required to investigate this problem have inspired a number of papers on techniques to address the difficulties which arise from time-optimal control. This survey reviews the literature of the last thirty years, including many of the more recent advances in the field, covers the aforementioned topics as well as specific attitude maneuvers, and traces the development of time-optimal and near time-optimal control algorithms. We begin by considering the formulation of the time-optimal control problem and the difficulties faced in obtaining the solution. The equations for the rotational motion of any rigid body (Ruler's equations) are quite elegant and compact; they are also coupled and highly nonlinear. In addition, the high angular velocities of time-optimal maneuvers will cause gyroscopic stiffness through the nonlinear terms. Ruler's equations are expressed as

Survey of time-optimal attitude maneuvers

E demand on precise orientation of a satellite, relative to an inertial or an orbiting frame of reference, has resulted in a whole new area of scientific endeavor referred to as attitude dynamics In the early stages of space exploration when spacecraft tended to be small, mechanically simple, and essentially inflexible, the elastic deformations were relatively insignificant Numerous investigations involving active and passive stabilization procedures and accounting for internal as well as external forces have been carried out assuming satellites to be rigid* However, in a modern space vehicle carrying lightweight deployable members, which are inherently flexible, this is no longer true Motivations for this trend are many 1) Every satellite goes through "a brief interval of vigorous acceleration and vibration during boost, followed by a prolonged functioning in a quiescent mode of operation characterized by extremely small loads and acceleration" The universal solution to this dilemma has been to design large members (solar panels, booms, antennas, etc) as flexible bodies that can be stored in a compact fashion (and hence behave like rigid bodies) during the launch but emerge like a butterfly from its cocoon after the boost terminates 2) In response to demanding mission requirements, satellite configurations have become increasingly complex to maintain a delicate balance between diverse constraints of space, weight, performance, control, and their optimum realization For example, ever-increasing demand on power for operation of the on board instrumentation, scientific experiments, communications systems, etc, has been reflected in the size of the solar panels It has increased to a point where their flexibility can no longer be neglected For example, the proposed Canadian Communications Technology Satellite (CTS), to be launched in 1975, is designed to carry two solar panels, 375 ft x 24 ft each, to generate 12 kw ~ 8 3) Use of larger members may be essential in some missions For example, Radio Astronomy Explorer (RAE) Satellite' used four 750 ft antennas for detecting low-frequency signals 4) Certain multipurpose missions, particularly those requiring simultaneous relative measurements from points appreciably apart, may find elastically supported or the recently proposed cable connected multibody systems attractive" 5) Modular construction leading to a gigantic space station through elastic linkages can no longer be considered a fantasy In this connection, studies of far-reaching significance by Roberson, Wittenburg, and others are of particular relevance"" 6) For longer life span, passive or at least semi-passive attitude control procedures are ideal They normally involve use of long booms, large radiation reflectors, or aerodynamic flaps" The structural flexibility may interact with attitude control systems of spacecraft in a variety of ways A NASA special report focuses attention on the significance of this problem through a discussion of the anomalous behavior of several spacecraft Besides citing useful references, it represents, with several others,' excellent review of the work in this area and puts current and anticipated problems in proper perspective Furthermore, there are a few recommended procedures for analysis, simulation, and design A paper by Noll et al effectively summarizes the NASA document Experience to date suggests that in most cases problems arise not because of a lack of available analytical/numerical design procedures but because of failure on our part to recognize and appreciate the mechanism of the attitude control/structural flexibility interactions The review papers just mentioned serve as an ideal introduction to the subject They effectively summarize dynamical problems through illustration of several past and projected configurations where flexibility played or is likely to play an important role The problem of flexibility is neither of a recent origin nor is it restricted to spacecraft Missile, aircraft, and surface vehicle designers have long been dealing with it As pointed out by Ashley, several of the techniques are directly applicable here In many ways, however, spacecraft differ from other dynamical systems and require development of modified approaches specifically suited to the emerging field of astroelasticity However,

Attitude Dynamics of Satellites with Flexible Appendages- A Brief Review

The dynamic equations of motion of a multibody flexible spacecraft with topological tree configuration are derived by the direct path method. The terminal bodies are treated as flexible and the interconnected bodies as rigid. Bodies are connected by gimbal hinges supplied with torsional springs and dampers. The three gimbal rotational axes may be either completely free or locked sequentially. Three approaches are suggested in this analysis, namely; the Lagrangian-Newtonian approach, the complete Newtonian approach, and the complete Lagrangian approach. The direct paths, the direct position vectors, and the incidence matrix are the key features of this analysis. They efficiently govern the contribution from the single body to the overall dynamic equations of the system. From the physical point of view, the complete Newtonian approach is the most straightforward. Torque equations are obtained from directly taking the moment about the hinge points rather than going through the lengthier differentiation of the angular momentum. This approach also gives better physical understanding to the interaction among motions of the individual bodies. From the mathematical point of view, the complete Lagrangian approach is the fastest. The lengthy elimination of the interacting forces and torques does not appear in this approach.

Direct Path Method for Flexible Multibody Spacecraft Dynamics

The problem of modal analysis of an elastic appendage on a rotating base is examined to establish the relative advantages of various mathematical models of elastic structures and to extract general inferences concerning the magnitude and character of the influence of spin on the natural frequencies and mode shapes of rotating structures. In realization of the first objective, it is concluded that except for a small class of very special cases the elastic continuum model is devoid of useful results, while for constant nominal spin rate the distributed-mass finite-element model is quite generally tractable, since in the latter case the governing equations are always linear, constant-coefficient, ordinary differential equations. Although with both of these alternatives the details of the formulation generally obscure the essence of the problem and permit very little engineering insight to be gained without extensive computation, this difficulty is not encountered when dealing with simple concentrated mass models.

Mathematical modeling of spinning elastic bodies for modal analysis.

A general and rigorous method for the stability analysis of a spinning body which is part rigid and part elastic is presented. The motion of the body is described by a "hybrid" system of equations, namely a system consisting of both ordinary and partial differential equations. The stability analysis is based on the Liapunov direct method and it works directly with the hybrid set of equations in contrast with the common practice of replacing the partial differential equations with ordinary differential equations by means of modal truncation or spatial discretization. This new approach permits a more rigorous analysis which not only produces sharper stability criteria but also avoids questions as to the effect of the truncation and discretization processes on the results. The general formulation can be used to test the stability of orbiting satellites with various types of elastic members and should also be applicable to the stability analysis of hybrid systems encountered in other areas.

Stability of a spinning body containing elastic parts via Liapunov's direct method

A new method for the stability analysis of force-free dynamical systems described by simultaneous sets of ordinary and partial differential equations of motion is presented. Such systems, referred to as hybrid, arise naturally in connection with the motion of spinning flexible bodies. The method is based on Liapunov's second method and works directly with the hybrid system of equations. It involves the construction of a Liapunov functional that takes into account automatically motion integrals resulting from the absence of external forces on the system. The general theory is particularly suitable for the stability analysis of torque-free spinning satellites containing distributed elastic members. As an illustration, the case of spin stabilization of a satellite with flexible appendages is solved.

Leonard Meirovitch

Papers

Stability of a spinning body containing elastic parts via Liapunov's direct method

A Method for the Liapunov Stability Analysis of Force-Free Dynamical Systems

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