A new approach for computing time-optimal open-loop control inputs for uncertain flexible spacecraft is developed. In particular, the single-axis, rest-to-rest maneuvering problem of flexible spacecraft in the presence of uncertainty in model parameters is investigated. Robust time-optimal control inputs are obtained by solving a parameter optimization problem subject to robustness constraints. A simple dynamical system with a rigidbody mode and one flexible mode is used to illustrate the concept.

Robust Time-Optimal Control of Uncertain Flexible Spacecraft

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

A control scheme is described for slewing e exible spacecraft with both suppression of dee ection during the slew and elimination of residual oscillations. The method minimizes the maneuver time subject to constraints on residual vibration magnitude, sensitivity to modeling errors, rest-to-rest slew distance, and the transient dee ection amplitude. Furthermore, a solution is sought that provides inherent fuel efe ciency. The feasibility of the approach is demonstrated with linear and nonlinear computer simulations.

Slewing Flexible Spacecraft with Deflection-Limiting Input shaping

Active Control Technology for Large Space Structures

The near-minimum time single-axis slewing of a flexible spacecraft with simultaneous suppression of vibration of elastic modes is considered. The hyperbolic tangent (tahh) function is used as a smooth approximation to the discontinuous sgn function occurring in the rigid body "bang-bang" control. Variable structure control concepts are used to identify the necessary characteristics of the control switching line. Simulations of the rest-to-rest and tracking maneuvers indicate that the elastic energy can be reduced by several orders of magnitude with only a modest increase in the maneuver time. Introduction M ANY future large space structures, due to mass con- traints, will be flexible. For the purpose of analysis these systems can be modeled with a large number of modes of vibration. For certain applications, it will be desirable that such a spacecraft be able to slew as rapidly as possible, within the operating limits of the control actuators. The problem of control design for rotational maneuver and vibration suppres- sion of flexible spacecraft has been addressed extensively.1'10 Optimal control theory can be applied to enforce quiescent terminal conditions on the flexible modes, usually by applying a quadratic cost function that weights the control rates and the states. The order of the system model grows rapidly as the number of flexible modes to be controlled is increased, mak- ing it impractical to attempt to control more than a few flexible modes. Furthermore, there is no rigorous means by which control magnitude constraints may be enforced. Al- though a feedback method such as the linear quadratic regula- tor (LQR) with terminal constraints9"11 is attractive because it can enforce quiescent conditions oh the elastic modes, its computational complexity is burdensome. The optimal control solution to the minimum time, single- axis, rotational maneuver problem for a rigid body gives a control scheme characterized by saturation of the control throughout the maneuver with at most one control switch that is instantaneous (bang-bang). Although not exactly achiev- able in physical systems, and even though the trajectories are extremely sensitive to variations in spacecraft parameters, this control law has wide application for rigid systems using on-off thrusters. When rigorously applied to flexible systems, how- ever, the result is typically multiple control switches and excessive excitation of the flexible modes of vibration. Previ- ous investigations of the near-time optimal maneuver for flexible systems7'8 are open-loop designs requiring extensive computational effort. An alternative is to compute the re- quired parameters for a number of initial conditions off-line and use table lookup and interpolation for a particular ma- neuver. This may involve excessive storage. We start with the assumption that the minimum time solution for a rigid space- craft can serve as an initial approximation to the near-mini- mum time solution for a flexible structure. In other words, we

R. M. Byers

Papers

Near-Minimum Time, Closed-Loop Slewing of Flexible Spacecraft