Introduction

In Chapters 5–8, various control laws were presented for attitude stabilization and maneuvering. The hardware used to implement the control laws were principally momentum exchange devices as well as magnetic and solar torque controllers. Such controllers work in a linear continuous mode. The torques that they can provide are in the range of 0.02–1 N-m for momentum exchange devices, 10−2–10−3 N-m for magnetic torque controllers, and 10−5–10−6 N-m for solar torque controllers.

This form of attitude control has two major disadvantages. First, the speed of attitude maneuvering is limited by the low-level maximal torques that can be delivered to the ACS. The second but no less important difficulty was encountered in orbit-maneuvering tasks. The high-level liquid thrusters (or solid propulsion motors) used for orbit changes induce parasitic torques due to physical irregularities of the propulsion system. The level of induced parasitic torques is of the order of several newton-meters. The only way to control the attitude of the spacecraft under such disturbance conditions is to use reaction thruster controllers (see also Section 8.8).

Reaction thrusters used in attitude control are activated in a pulsing mode only. There are no linear, continuous reaction thrust controllers. This fact somehow complicates the analytical treatment of attitude control systems using them as torque controllers. However, they can provide almost any torque level, as surveyed in Appendix C. Reaction torque levels ranging between 0.01 N-m and 30 N-m are very common in most spacecraft. For practical considerations, it is convenient to use thrusters of the same thrust level for all control tasks in the satellite, but if this is not feasible then thrusters with different thrust levels can be incorporated as part of a unified propulsion system.

This chapter deals with the analysis and design of reaction thruster attitude control. It also covers two principal difficulties caused by the pulsing mode of thruster firing: the limits on attitude accuracy that can be achieved with a given thruster, and the fuel penalty associated with sensor noise.

Introduction

From the moment that a satellite is launched into its initial orbit, commonly called a transfer orbit, multiple orbital changes must be performed. Changes of the transfer orbit are necessary in order to obtain the desired final orbit because a launch vehicle usually cannot put the satellite in its final orbit. Even if it were possible, a satellite launch that placed the s/c in its final orbit would not be optimal from the point of view of fuel consumption (see Duret and Frouard 1980).

Fuel consumption is a crucial factor in orbital maneuvers. Any orbital change is accompanied by a velocity change of the satellite, which necessitates a certain quantity of fuel consumption. As we shall see in this chapter, minimization of fuel consumption is essential because the weight of the useful payload that can be carried to the desired orbit depends on this minimization.

This chapter will also consider different kinds of orbit maneuvers and changes. For instance, orbits may be adjusted by single or multiple thrust impulses. It will be seen that with a single thrust impulse, very limited kinds of orbit changes can be achieved, whereas multiple thrust impulses can effect any desired orbit change. It is comparatively easy to analyze the change in orbit parameters due to an impulsive thrust; the analytical treatment usually ends with a closed-form solution. Unfortunately, an impulsive thrust is an idealization that cannot be met in practice.

Thus, another way of viewing orbital maneuvers concerns the duration of the thrust. During an orbit change, thrust is applied for a length of time (sometimes hours) that depends on the thrust magnitude (see Redding 1984). What we call an impulsive thrust depends very much on the thrust duration relative to the orbit's natural period. The principal drawback in applying a nonimpulsive thrust is that the application of a finite-time thrust is accompanied by a nonconstant thrust direction during burns. The nonconstant thrust direction results in a velocity loss because only a component of the thrust acceleration, rather than its entire value, acts along the mean thrust direction (see Robins 1966).

Introduction

As discussed in Chapter 4, a body spinning about its major or minor axis will keep the direction of its spinning axis fixed with respect to the inertial space. This direction, according to Euler's moment equations of angular motion, will change only if external moments are applied about its center of mass and perpendicularly to the spin axis.

Almost all spacecraft employ the spin effect during part or all of their lifetime in space. Many satellites are spin-stabilized during the orbital maneuvering stage (e.g., in the transfer from the initial orbit to the final mission orbit) so that parasitic torque disturbances, produced by the high thrust of the apogee boost motor, do not appreciably change the nominal direction of the additional vector velocity ΔV imparted to the satellite. The dynamic attributes of spinning bodies are used also to stabilize satellites' attitude within the final mission orbit. Spin stabilization was used in the first communications satellites in the early sixties, and in a large number of modern satellites (see e.g. Fagg and MacLauchlan 1981, Fox 1986).

Single-spin attitude stabilization is a very simple concept from the perspective of attitude control, but it has some crucial drawbacks with respect to communication efficiency. Dual-spin three-axis attitude stabilization, which is an extension of the single-spin stabilization principle, alleviates the communication deficiency. We shall find conditions under which passive nutation damping is feasible for dual-spin stabilized spacecraft.

Attitude Spin Stabilization during the ΔV Stage

As already mentioned, a common way to stabilize the attitude of a s/c during orbital maneuvering (the ΔV stage) is to spin-stabilize the axis along which the propulsion thruster is aligned. Unfortunately, the direction of the applied thrust does not pass exactly through the center of mass (cm) of the satellite, and a parasitic torque results that tends to change the nominal attitude of the spin axis. Since the thruster is fixed to the satellite's body, change of the attitude of the satellite will induce an error in the application of ΔV, with the result that the required new orbit parameters will not be achieved (see Section 3.4.3).

Overview

Space technology is relatively young compared to other modern technologies, such as aircraft technology. However, in only forty years this novel domain has achieved a tremendous level of complexity and sophistication. The reason for this is simply explained: most satellites, once in space, must rely heavily on the quality of their onboard instrumentation and on the design ingenuity of the scientists and engineers who produced them. Recent achievements of repairing satellites while in orbit testify to the complexity involved in space technology. The desire of humans to conquer space within the solar system will surely encourage new technological achievements that are not yet imagined.

The technical fields in which satellites are used are numerous – telecommunications, scientific research, meteorology, and others. According to the specific task for which they are designed, satellites are very different from one another. They may be in orbits as low as 200 km or as high as 40,000 km above the earth; other spacecraft leave the earth toward planets in the solar system. Satellites may be very heavy: an inhabited space station, for example, could weigh several tons or more. There also exist very light satellites, weighing 20 kg or less. Small satellites may be relatively cheap, of the order of a million dollars apiece. Despite their differences, satellites possess fundamental features that are common to all. The physical laws that govern their motion in space and their dynamics are the same for all spacecraft. Hence, the fundamental technologies that evolved from these laws are common to all.

A satellite's life begins with the specific booster transferring it to some initial orbit, called a transfer orbit, in which the satellite is already circling the earth. For a satellite that will stay near earth, the next stage will be to “ameliorate” the orbit; this means that the satellite must be maneuvered to reach the precise orbit for which the satellite was designed to fulfill its mission. Next, the satellite's software must check for the proper functioning of its instrumentation and its performance in space, as well as calibrate some of the instruments before they can be used to control the satellite. The final stage is the one for which the satellite was designed and manufactured.

Introduction

A typical spacecraft structure consists of two principal parts. The first one is the body of the spacecraft, which contains all the payload instrumentation and control hardware pertaining to the Attitude and Orbit Control System (AOCS). Its structure must be very rigid in order to withstand mechanical loads during the launch stage, and also to assure correct positioning of the control torquers and attitude sensors for achieving the necessary pointing accuracy in the s/c mission stage. The necessity to save on weight leads to mechanical design tradeoffs between weight and rigidity, which results in a body that is only quasirigid and in which structural vibration modes should be anticipated.

The second part of the spacecraft structure consists of large flexible appendages: parabolic antennae, large synthetic-aperture radar, and very large flexible solar arrays built from light materials in order to reduce their weight. Spacecraft structures are now. becoming extremely complicated because of these appendages, which also induce structural oscillation under the excitation of external torques and forces. Finally, we must likewise consider vibrations due to the liquid contained in fuel tanks. The vibrational dynamics evolving from flexible appendages and liquid sloshing interferes strongly with the attitude control dynamics, and puts severe limitations on the achievable qualities of the attitude control system (ACS).

This chapter is primarily concerned with this second class of structural vibrations. First, we must write simplified structural dynamics models for solar panels and liquid sloshing before coupling them to the rigid-body attitude dynamics of spacecraft. An analytical model for the rigid body and the flexible structural modes will allow us to evaluate the limitations on the bandwidth of the ACS.

Modeling Solar Panels

Although the techniques of structural modeling described in this chapter are applicable to any nonrigid appendage to the solid body, we will concentrate on the structural dynamics of solar panels, which are common to almost all types of satellites.

Introduction

The present and remaining chapters deal with attitude control of spacecraft; this section serves as an introduction to all of them. The expression attitude control has the general meaning of controlling the attitude of the satellite. In practice, there exist a multitude of variations to this simple and apparently straightforward expression. The following are some examples of primary control tasks for which the attitude control system is responsible.

1. In orbital maneuvering and adjustments, the attitude of the satellite must be pointed and held in the desired ΔV direction.

2. A spin-stabilized satellite may be designed to keep the spin axis of its body pointed at some particular direction in space.

3. A nadir-pointing three-axis-stabilized satellite must keep its three Euler angles close to null relative to the orbit reference frame; this is true of most communications satellites.

4. In earth-surveying satellites, the attitude control system is designed to allow the operative payload to track defined targets on the earth's surface.

5. A scientific satellite observing the sky must maneuver its optical instruments toward different star targets on the celestial sphere in some prescribed pattern of angular motion.

The few examples listed and the many others not mentioned suggest a multitude of different tasks and missions to be performed by the attitude control system. However, we shall see that some features are common to all such systems.

An important distinction for attitude control concepts is between passive and active attitude control. Passive attitude control is attractive because the hardware required is less complicated and relatively inexpensive. Natural physical properties of the satellite and its environment are used to control the s/c attitude. However, the achievable accuracies with passive attitude control are generally much lower than those that are possible with active attitude control, which uses sophisticated (and much more expensive) control instrumentation.

Another important distinction is between attitude-maneuvering and nadir-pointing (earth-pointing) stabilized satellites.

The goal of this book is to provide the reader with the basic engineering notions of controlling a satellite. In the author's experience, one of the most important facts to be taught from the beginning is practical engineering reality. Theoretical, “nice” control solutions are seriously hampered when practical problems (e.g., sensor noise amplification, unexpected time delays, control saturation effects, structural modes, etc.) emerge at a later stage of the design process. The control algorithms must then be redesigned, with the inevitable loss of time and delay of the entire program. Early anticipation of these effects shortens the design process considerably. Hence it is of utmost importance to analyze different concepts for engineering solutions of spacecraft control tasks in the preliminary design stages, so that the correct one will be selected at the outset. This is why several approaches may be suggested for a given control task.

Part of the material in this textbook has been used as background for a single-semester course on “Spacecraft Dynamics and Control” – offered since 1986 at the Tel Aviv University and also more recently at the Israel Institute of Technology, the Technion, Haifa. All the material in this book is appropriate for a course of up to two semesters in length. The book is intended for introductory graduate-level or advanced undergraduate courses, and also for the practicing engineer. A prerequisite is a first course in automatic control, continuous and sampled, and a first course in mechanics. This, in turn, assumes knowledge of linear algebra, linear systems, Laplace transforms, and dynamics.

A sequential reading of the book is advised, although the chapters are for the most part self-contained. A preliminary overview is recommended in order to acquire a feeling for the book's contents; this will help enormously in the second, and deeper, reading.

Modern spacecraft control concepts are based on a vast choice of physical phenomena: single- and dual-spin stabilization; gravity gradient attitude control; three-axis stabilization; momentum-bias stabilization; and solar, magnetic, or reaction torque stabilization. It is important to master the essential qualities of each before choosing one as an engineering solution.

Introduction

Hardware items that are mandatory for realizing almost any spacecraft attitude and orbit control system can be divided into two classes: instrumentation for measuring the attitude of the satellite; and instrumentation for providing forces and torques. The latter category will be treated in Appendix C. Appendix B deals with attitude sensors, but is not an extensive treatment of the subject. There are excellent textbooks and technical papers providing complete treatment of the hardware from both analytical and practical points of view; Wertz (1978) is especially recommended. However, for completeness and for the reader's convenience, a short exposition of the basic principles of satellite attitude hardware will be presented here, together with examples of existing space-proven commercial instruments.

Attitude measurement hardware is used to determine the attitude of the satellite with respect to a defined reference frame. The final product may be, for instance, the Euler angles of the satellite in the orbit reference frame, or (in a different context) the sun vector components in the body axis frame. Attitude determination hardware includes:

1. earth sensors (in particular, infrared earth sensors);

2. sun sensors;

3. star sensors;

4. rate and rate integrating sensors, based on gyroscopic, laser, or other solid-state principles; and

5. magnetometers.

The quality of the instruments is responsible for the accuracy that can be achieved in the attitude control system. For instance, there are sun sensors that can measure the direction of the sun with an accuracy of 0.015°, whereas others have an accuracy of only 0.5° however, analytical processing of the two sensors° output is basically the same. The commonly used attitude reference sources are the earth, the sun, and the stars.

The earth is used in two different aspects, optical and magnetic. The more important one is the optical aspect. Unlike the sun (which appears as a small illuminated disk) or the stars (which can be treated as illuminated points), the earth – as seen from nearby space – has a complex appearance that must be adequately modeled for accurate attitude determination.

Introduction

The purpose of Appendix C is to introduce the reader to the basic features of the control hardware used to provide translational and angular accelerations to spacecraft.

The level of control forces that can be obtained depends on the source characteristics. For example, ion thrusters can produce forces of the order of tens of milli-newtons; liquid propellant thrusters used in spacecraft control provide forces in the range of hundreds of newtons; and solid propellant motor-produced forces are in the range of hundreds of thousands of newtons.

As far as attitude control is concerned, we have seen that torques can be produced with the aid of momentum exchange devices, magnetic torqrods, or solar torque controllers. The torque levels we can obtain with these devices are generally low: in the range of 0.01 to 1 N-m for momentum exchange devices (control moment gyros excluded), of the order of a few centinewton-meters with magnetic torqrods, and tens of micronewton-meters with solar torque controllers. With reaction propulsion means we can achieve torques of about 20–30 N-m, but also much lower torques of the order of 0.1 N-m if needed. Reaction propulsion is also used for momentum dumping – unloading the parasitic angular momentum accumulated in the spacecraft.

The sources of force and torques used for control can be classified as follows:

1. propulsion systems, which can provide translatory and angular accelerations (forces and torques) to the satellite;

2. solar radiation pressure, which can produce forces and torques;

3. momentum exchange devices, which can provide torques and angular momentum; and

4. magnetic torqrods, which can provide only torques.

Two special controllers that are not discussed in this book are (i) atomic energy-based thrusters for translation accelerations, and (ii) control moment gyros, which are used in large inhabited space structures to provide torques in the range of hundreds of newton-meters.