Chapter 1. Introduction

Transcription

1 Chapter 1 Introduction Vehicle dynamics and control is a rich subject involving a variety of topics from mechanics and control theory. In a first course in dynamics, students learn that the motion of a rigid body can be divided into two types of motion: translational and rotational. For example, the motion of a thrown ball can be studied as the combined motion of the mass center of the ball in a parabolic trajectory and the spinning motion of the ball rotating about its mass center. Thus a first approximation at describing the motion of a ball might be to model the ball as a point mass and ignore the rotational motion. While this assumption may give a reasonable approximation of the motion, in actuality the rotational motion and translational motion are coupled and must be studied together to obtain an accurate description of the ball s trajectory. The motion of the mass center of a curve ball is an excellent example of how the rotational and translational motions are coupled. 1 The ball does not follow the parabolic trajectory predicted by analysis of the translational motion, because of the unbalanced forces and moments acting on the spinning ball. The study of vehicle dynamics is similar to the study of baseball dynamics. One first develops an understanding of the translational motion of the mass center using particle dynamics techniques, and then rotational motion is studied. Thus, for example, the usual study of spacecraft dynamics begins with a course in orbital dynamics (translational motion), usually in the junior or senior year, followed by a course in attitude dynamics (rotational motion) in the senior year or the first year of graduate study. Similarly, aircraft dynamics is usually studied as aircraft performance (translational motion), followed by stability and control (rotational motion). In this treatment I assume that the student has successfully completed courses in the translational and rotational motion of aircraft, spacecraft, or marine vessels at the undergraduate level. However, I do not assume any specific details of that coverage; thus this course is relatively self-contained and should be accessible to any beginning graduate student in aerospace or ocean engineering. Another important decomposition of dynamics problems is into kinematics and kinetics. For translational motion, kinematics is the study of the change in position 1-1

2 1-2 CHAPTER 1. INTRODUCTION for a given velocity, whereas kinetics is the study of how forces cause changes in momentum and hence velocity. For rotational motion, kinematics is the study of the change in orientation for a given angular velocity, and kinetics is the study of how moments cause changes in the angular momentum and hence angular velocity. Translational kinematics is relatively easy to learn, since it only involves the motion of a point in three-dimensional space. Rotational kinematics, however, is usually more difficult to master, since it involves the orientation of a reference frame in threedimensional space. Kinetics { }} { Force affects Velocity affects Position Moment affects Angular Velocity affects Orientation } {{ } Kinematics In this chapter, I begin by describing how translational and rotational dynamics and control arise in the operation of aerospace and other vehicles. This discussion is followed by a description of the fundamental modelling and analysis techniques that are in widespread use. Finally, I give an overview of the textbook. 1.1 Translational Dynamics and Control in Vehicle Operations In the sense used in this course, a vehicle is a mechanical system for transporting objects in space. Land vehicles move across the surface of the Earth, and there is a rich body of literature associated with their motion (see, for example, Ref. 2). Marine vehicles move across water surfaces or under water; the former are called surface vessels, and the latter are called submersibles (see Fossen 3 ). Aircraft have arguably been the most important class of vehicles in the development of modern three-dimensional dynamics and control (see, e.g., Stengel 4 ). More recently, the operation of spacecraft has led to many important results in the field (see, e.g., Wie 5 ). Additional references are cited as needed. Translational motion of aircraft is typically studied in a course on Aircraft Performance in the second year of undergraduate studies. Anderson 6 is a standard text. This topic begins by formulating equations of motion for the airplane modelled as a point mass, where the forces acting on the aircraft include aerodynamic forces (lift L and drag D), gravitational force (weight W), and propulsion force (thrust T). For example, in the simple case of straight and level flight of an aircraft with an aligned propulsion system, the equations of motion are written as T = D and L = W

3 1.2. ROTATIONAL DYNAMICS AND CONTROL IN AIRCRAFT OPERATIONS1-3 where T is the propulsion system thrust, D is the aerodynamic drag, L is the aerodynamic lift, and W is the aircraft weight. Combining these equations with the definitions of lift and drag coefficients allows the aircraft performance analyst to characterize the thrust and power required to achieve straight and level flight for various flight speeds and standard altitudes. Although rotational motion is not included in this analysis, angle of attack is an important parameter in the study of these problems, and ultimately angle of attack is controlled by rotation of the aircraft. Translational motion of spacecraft is typically studied in a course on Astrodynamics in the third or fourth year of undergraduate studies. The inexpensive paperback by Bate et al. 7 is a standard text. This topic begins with the formulation of equations of motion for a point mass spacecraft, where the forces acting on the spacecraft are solely due to the inverse-square gravitational law. The resulting nonlinear equations of motion admit a well-known solution in terms of conic sections. The Astrodynamics course includes a variety of techniques used to study problems such as ground coverage, the time-of-flight problem, and the Hohmann transfer, and leads to more advanced trajectories, involving other gravitational and thrust forces, such as planetary flyby maneuvers and low-thrust orbit transfers. Rotational motion is not included in these analyses, but since the direction of the thrust vector is often important, ultimately spacecraft attitude is an important element of trajectory analysis. Translational motion of marine vessels is almost always coupled with the rotational motion; hence a first course in Ship Dynamics includes both translational and rotational dynamics. Fossen 3 is a standard text. 1.2 Rotational Dynamics and Control in Aircraft Operations From takeoff to landing, aircraft execute many types of rotational maneuvers. Takeoff begins with translational motion down the runway, until the aircraft reaches a speed where its lift exceeds its weight. The actual liftoff is usually effected by rotating the aircraft about its pitch axis. The term pitch refers to the angular motion about an axis that can generally be thought of as passing from the center of mass (c.m.) of the aircraft to the right wingtip. The roll axis passes from the c.m. to the nose of the aircraft, and the yaw axis passes from the c.m. out the bottom of the vehicle. We will give a more careful definition at the appropriate point, but for now, let us note that the roll-pitch-yaw description of rotational motion is equally applicable to other vehicles. Aircraft execute rotational maneuvers about the roll, pitch, and yaw axes throughout their missions; however, a typical aircraft may spend a significant fraction of its flight time in straight and level flight. An aircraft usually measures its rotational motion using rate gyroscopes and accelerometers, and controls its rotational motion using aerodynamic actuators such as rudders and ailerons.

4 1-4 CHAPTER 1. INTRODUCTION 1.3 Rotational Dynamics and Control in Spacecraft Operations Essentially all space vehicles include one or more subsystems intended to interact with or observe other objects. Typically there is one primary subsystem that is known as the payload. For example, the primary mirror on the Hubble Space Telescope is one of many instruments that are used to observe astronomical objects. The communications system on an Intelsat satellite is its payload, and the infrared sensor on board a Defense Support System (DSP) satellite is its payload. In each case, the payload must be pointed at its intended subject with some accuracy specified by the customer who purchased the spacecraft. This accuracy is typically specified as an angular quantity; e.g., 1, 10 arcseconds, or 1 milliradian. The attitude control system designer must design the attitude determination and control subsystem (ADCS) so that it can meet the specified accuracy requirements. It costs more than $10,000 to put a kilogram of mass into low-earth orbit (LEO), and even more to put it into geostationary orbit (GEO). 8 A typical spacecraft masses about 500 kg, and costs tens to several hundred millions of dollars to design, manufacture, test, and prepare for launch. All of this money is spent to purchase the mission capability of the spacecraft. The attitude control system, propulsion system, launch vehicle, and so forth, are only there so that the mission may be performed effectively. If the mission can be accomplished without an ADCS, then that mass can be used to increase the size of the payload, decrease the cost of launch, or in some other way improve the performance or reduce the cost. The bottom line is that the payload and its operation are the raison d être for the spacecraft. This focus justifies our spending a little time on describing how the ADCS fits into the operations of the payload. There are many spacecraft payloads, but most fit into one of just two categories: communications and remote sensing. On a communications satellite, the payload comprises the radio transceivers, multiplexers, and antennas that provide the communications capability. Historically, most communications satellites have been in the geostationary belt, and have been either dual-spin or three-axis stabilized. More recently, a host of LEO commsats has been put into orbit, including the 66-satellite Iridium constellation, the 36-satellite OrbComm constellation, and the planned 48- satellite GlobalStar constellation. The Iridia use hydrazine propulsion for three-axis stabilization, whereas the OrbComms are gravity-gradient stabilized. The GlobalStar spacecraft are three-axis stabilized, using momentum wheels, magnetic torquers, and thrusters. In commsats, the mission of the ADCS is to keep the spacecraft pointed accurately at the appropriate ground station. The more accurate the ADCS, the more tightly focused the radio beam can be, and the smaller the power requirements will be. However, ADCS accuracy carries a large price tag itself, so that design trades are necessary. There are two basic types of remote sensing satellites: Earth-observing and space-

5 1.4. ROTATIONAL DYNAMICS AND CONTROL IN MARINE VESSEL OPERATIONS1-5 observing. An Earth-observing spacecraft could be nadir-pointing, it could be scanning the land or sea in its instantaneous access area, or it could selectively point to and track specific ground targets. In the first case, a passive gravity-gradient stability approach might suffice, whereas in the second and third cases, an active ADCS would likely be required, using some combination of momentum wheels, magnetic torquer rods, or thrusters. A space-observing system could simply point away from the Earth, with an additional requirement to avoid pointing at the sun. This anti-earth pointing is essentially the ADCS requirement for the CATSAT mission being built by the University of New Hampshire. The CATSAT ADCS uses momentum wheels and magnetic torquer rods. More complicated space-observing systems require both large-angle slewing capability and highly accurate pointing control. The Hubble Space Telescope is a well-known example. It uses momentum wheels for attitude control, and performs large-angle maneuvers at about the same rate as the minute-hand of a clock. The HST does not use thrusters because the plume would contaminate the sensitive instruments. One problem with momentum wheels, reaction wheels, and control moment gyros is momentum buildup: external torques such as the gravity gradient torque and solar radiation pressure will eventually cause the wheel to reach its maximum speed, or the CMG to reach its maximum gimbal angle. Before this saturation occurs, the spacecraft must perform an operation called momentum unload, or momentum dump: external torques are applied, using thrusters of magnetic torquer rods, that cause the ADCS to decrease the wheels speeds or the CMGs gimbal angles. Depending on the spacecraft, this type of maneuver may be performed as often as once per orbit. Another ADCS operation involves keeping the spacecraft s solar panels pointing at the sun. For example, when the HST is pointing at a particular target, it still has a degree of freedom allowing it to rotate about the telescope axis. This rotation can be used to orient the solar panel axis so that it is perpendicular to the direction to the sun. Then the panels are rotated about the panel axis so that they are perpendicular to the sun direction. This maneuver is known as yaw steering. 1.4 Rotational Dynamics and Control in Marine Vessel Operations Controlled rotational motion of marine vessels is generally associated with steering to control the translational motion. Marine vessels experience significant rotational motion due to environmental torques, and much of the literature on this topic is associated with understanding and predicting these effects. Submersible vehicles are Cooperative Astrophysics and Technology SATellite. See Refs. 9 and 10, and the website

6 1-6 BIBLIOGRAPHY often spacecraft-like in the sense that rotational motion is used for the pointing of observational instruments on board the vehicle. 1.5 Overview of the Textbook Most textbooks on related subjects begin with some treatment of kinematics and then proceed to a study of a variety of dynamics problems, with some control problems perhaps included. Our approach is similar, but our aim is to spend more time up front, both in motivation of the topics, and in developing an understanding of how to describe and visualize rotational motion. To this beginning, this introductory chapter on Mission Analysis should help readers to develop an appreciation for how dynamics and control problems fit into the overall mission of various vehicles. For a more traditional course, this chapter could be read quickly or even skipped entirely. Chapter 2 introduces the notation and equations describing translational motion of a particle, introducing linear momentum, Newtons laws, and coordinate frames. Simple examples are developed using topics from aircraft performance, orbital mechanics, and launch vehicle trajectories. Chapter 3 introduces attitude kinematics, developing the classical topics in some detail, and introducing some new topics that may be used in a second reading. Chapter?? covers the important topic of attitude determination. This topic is not usually covered in an introductory course, but I believe that mastery of this subject enhances the student s appreciation for the remaining material. In Chapter 4 we develop the standard equations of motion and relevant results for rigid body dynamics. These topics lead directly to Chapter 5 introduces the fundamentals of linearization, linear system analysis, and linear control theory. In Chapter 6 we introduce and develop equations of motion for a variety of applications in aircraft, spacecraft, marine vessels, and missiles. 1.6 References and Further Reading Elementary textbooks that provide useful background for this course include Anderson s Introduction to Flight, 11 Hale s Introduction to Space Flight. 12 There is no elementary textbook on marine vessel dynamics, but Fossen s Guidance and Control of Ocean Vehicles 3 provides the required elementary introduction. Dankowicz 13 is a useful reference for general topics in characterizing and visualizing rigid body motion. More advanced references are included in the appropriate chapters. Bibliography [1] Neville de Mestre. The Mathematics of Projectiles in Sport, volume 6 of Australian Mathematical Society Lecture Series. Cambridge University Press, Cam-

7 1.7. EXERCISES 1-7 bridge, [2] Gaetano Cocco. Motorcycle Design and Technology: How and Why. Giorgio Nada Editore Publications, Vimodrone, Italy, [3] Thor I. Fossen. Guidance and Control of Ocean Vehicles. John Wiley & Sons, Chichester, [4] Robert F. Stengel. Flight Dynamics. Princeton University Press, Princeton, [5] Bong Wie. Space Vehicle Dynamics and Control. AIAA, Reston, Virginia, [6] John D. Anderson, Jr. Aircraft Performance and Design. McGraw-Hill, New York, [7] Roger R. Bate, Donald D. Mueller, and Jerry E. White. Fundamentals of Astrodynamics. Dover, New York, [8] John R. London III. LEO on the Cheap Methods for Achieving Drastic Reductions in Space Launch Costs. Air University Press, Maxwell Air Force Base, Alabama, [9] Chris Whitford and David Forrest. The CATSAT attitude control system. In Proceedings of the 12th Annual Conference on Small Satellites, number SSC98- IX-4, Logan, Utah, September [10] Ken Levenson and Kermit Reister. A high capability, low cost university satellite for astrophysical research. In 8th Annual Conference on Small Satellites, Logan, Utah, [11] John D. Anderson, Jr. Introduction to Flight. McGraw-Hill, New York, fifth edition, [12] Francis J. Hale. Introduction to Space Flight. Prentice-Hall, Englewood Cliffs, NJ, [13] Harry Dankowicz. Multibody Mechanics and Visualization. Springer-Verlag, London, Exercises 1. Select an aircraft in each of the following categories: civil aviation, military fighter, military cargo, military autonomous, commercial airliner. For each aircraft, find estimates for the following quantities: dry mass, maximum mass, dimensions, moments of inertia, typical maximum speed.

8 1-8 BIBLIOGRAPHY 2. Select a spacecraft in each of the following categories: low-earth orbit (LEO) science, LEO manned, geostationary (GEO) communications, interplanetary exploration. For each spacecraft, find estimates for the following quantities: dry mass, maximum mass, dimensions, moments of inertia, typical maximum speed. 3. Select a marine vessel in each of the following categories: private yacht, aircraft carrier, military submarine, submersible autonomous, commercial cruise ship. For each vessel, find estimates for the following quantities: dry mass, maximum mass, dimensions, moments of inertia, typical maximum speed.