Control of Spacecraft and Aircraft

by Arthur E. Bryson Jr.

Overview

Here a leading researcher provides a comprehensive treatment of the design of automatic control logic for spacecraft and aircraft. In this book Arthur Bryson describes the linear-quadratic-regulator (LQR) method of feedback control synthesis, which coordinates multiple controls, producing graceful maneuvers comparable to those of an expert pilot.


The first half of the work is about attitude control of rigid and flexible spacecraft using momentum wheels, spin, fixed thrusters, and gimbaled engines. Guidance for nearly circular orbits is discussed. The second half is about aircraft attitude and flight path control. This section discusses autopilot designs for cruise, climb-descent, coordinated turns, and automatic landing. One chapter deals with controlling helicopters near hover, and another offers an introduction to the stabilization of aeroelastic instabilities. Throughout the book there is a strong emphasis on the mathematical modeling necessary for designing a good feedback control system. The appendixes summarize analysis of linear dynamic systems, synthesis of analog and digital feedback control, simulation, and modeling of flexible vehicles.

Product Details

ISBN-13:	9780691087825
Publisher:	Princeton University Press
Publication date:	06/05/1994
Pages:	408
Product dimensions:	6.00(w) x 9.25(h) x (d)

About the Author

Arthur E. Bryson, Jr., is Paul Pigott Professor of Engineering at Stanford University. Coauthor, with Y. C. Ho, of Applied Optimal Control (Blaisdell/ Hemisphere), he is a member of the National Academy of Engineering and the National Academy of Sciences.

Read an Excerpt

Control of Spacecraft and Aircraft

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By Arthur E. Bryson Jr.

PRINCETON UNIVERSITY PRESS

Copyright © 1994 Princeton University Press
All rights reserved.
ISBN: 978-0-691-08782-5

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CHAPTER 1

Natural Motions of Rigid Spacecraft

1.1 Translational Motions in Space

The natural motions of the center of mass of a body in space are described by Newton's equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.2)

where

m = mass of the spacecraft,

([??], [??])= (velocity, position) of the center of mass with respect to inertial space,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = time rate of change with respect to inertial space,

[??] = sum of external forces.

In the absence of external forces, the velocity stays constant, and the position changes linearly with time.

For motions taking place in times very much less than an orbit period (either around the Sun or the Earth), neglecting gravity yields a useful approximation for the translational motions.

1.2 Translational Motions in Circular Orbit

In circular orbit, centrifugal force balances gravitational force. For small deviations from circular orbit, the equations of motion of the center of mass are conveniently written in locally-horizontal-vertical (LHV) coordinates that rotate with the orbital angular velocity, n (see Fig. 1.1):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.4)

where

δ[??] = velocity deviation,

δ[??] = position deviation,

δ[??] = force deviation,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = denotes time derivative of components with respect to LHV axes,

[??]L = angular velocity of LHV axes with respect to inertial axes.

Following NASA standard notation, δx is in-track position deviation (positive in direction of orbital velocity), δy is cross-track position deviation, and δz is vertical deviation (positive down). Thus,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.6)

[??]L = -n[??], (1.7)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] orbital angular velocity and ([??], [??], [??]) are unit vectors along the (x, y, z) axes. The only force is the inverse-square gravitational force

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.8)

where g = gravitational force per unit mass at radial distance, R, from the attracting center. Thus the deviation in force is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.9)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.10)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.11)

Substituting (5)–(7) and (9–11) into (1) and (2), we obtain the equations of motion for small deviations from circular orbit. They decouple into a set governing cross-track motions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.12)

and a set governing in-track/radial motions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.13)

where (Tx, Ty, Tz) = thrust components.

The characteristic equation of the system (12) is

s2 + n2 = 0, (1.14)

so there is one purely oscillatory mode at frequency n. The natural motion is of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.15)

where c and β are arbitrary constants, and the complex eigenvector corresponding to s = nj is (n, –j)T. The real part of this eigenvector is the coefficient of cos (nt + β), while the imaginary part is the coefficient of [– sin (nt + β)]. The motion may be interpreted as a slight change in orbit plane, so that the spacecraft crosses the reference orbital plane twice per revolution, and thus appears to oscillate right-left with orbital frequency n.

The characteristic equation of the system (13) is

s2(s2 + n2) = 0, (1.16)

so there is one purely oscillatory mode at frequency n, and two stationary modes. The natural motions are of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.17)

where c1, β, c2, and c3 are arbitrary constants. The first two column vectors in (17) are the real and imaginary parts of the eigenvector corresponding to s = nj; the third and fourth column vectors are the principal and secondary eigenvectors corresponding to s = 0 (see Appendix A). The motion of the perturbed spacecraft in each of these three modes, as observed from an unperturbed spacecraft in the same circular orbit, is shown in Fig. 1.2.

The first mode (c1 ≠ 0) corresponds to a slightly elliptic orbit so that the spacecraft goes above the circular orbit for half a period and slows down, then goes below and speeds up for the other half. The second mode (c2 ≠ 0) corresponds to the spacecraft being in the same circular orbit but slightly ahead of (or behind) the reference point. The third mode (c3 ≠ 0) corresponds to the spacecraft being in a lower (or higher) circular orbit that has a faster (or slower) orbital velocity.

1.3 Rotational Motions in Space

The rotational (attitude) motions of a rigid spacecraft in space are described by Euler's equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.18)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = moment of momentum of spacecraft about the c.m.,

[??] = moment of inertia dyadic with respect to the c.m. of the S/C,

[??] = angular velocity of S/C with respect to inertial space,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = time rate of change with respect to body-fixed axes,

[??] = resultant external torque.

Since the moment of inertia dyadic, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], is constant in body-fixed axes, Equation (18) is usually used in body-axis components. If the body-axis components of the angular velocity are [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and we use principal axes so that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where ([??], [??], [??] are unit vectors along the (x, y, z) principal body axes, then (1) becomes

Ix[??] – (Iy – Iz)qr = Qx, (1.19)

Iy[??] – (Iz – Ix)rp = Qy, (1.20)

Iz[??] – (Ix – Iy)pq = Qz, (1.21)

where (Qx, Qy, Qz) are the body-axis components of the torque.

1.3.1 Kinematic Equations

Angular position may be described by three Euler angles (or by four Euler parameters (quaternions), or nine direction cosines (cf. Kane)). The NASA standard Euler angles are shown in Fig. 1.3; the first rotation, ψ, is about the body z-axis; the second rotation, θ, is about the new position of the body y-axis; the third rotation, φ, is about the new position of the body x-axis. Fig. 1.4 shows a two-gimbal interpretation of these Euler angles. If the body-axis components of the angular velocity are (p, q, r), then the Euler angle rates are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.22)

1.3.2 Small Attitude Changes of a Nonspinning Spacecraft

For small attitude changes of a nonspinning spacecraft with respect to inertial space, that is, magnitudes of φ, θ, ψ small compared to unity, the nonlinear equations of motion (19)–(22) are well approximated by the following linearized equations of motion:

Ix[??] [congruent to] Qx, (1.23)

Iy[??] [congruent to] Qy, (1.24)

Iz[??] [congruent to] Qz, (1.25)

[??] [congruent to] p, (1.26)

[??] [congruent to] q, (1.27)

[??] [congruent to] r. (1.28)

Thus angular motion about each of the three principal axes is uncoupled from motion about the other two principal axes. Eliminating (p, q, r) from (23)–(28), we obtain three independent second-order systems:

Ix[??] [congruent to] Qx, (1.29)

Iy[??] [congruent to] Qy, (1.30)

Iz[??] [congruent to] Qz. (1.31)

1.4 Rotational Motions in Circular Orbit

1.4.1 Gravity Torque

The very small torque acting on a rigid body in an inverse-square gravitational field may be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.32)

where

g = gravitational force per unit mass at radial distance R,

[??] = unit vector in the radial direction,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = moment-of-inertia dyadic of the rigid body.

This torque is zero whenever a principal axis is parallel to [??], since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [??] × (λ[??]) = 0.

If the body principal axes (x, y, z) are aligned as shown in Fig. 1.5, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.33)

where (Ix, Iy, Iz) are the principal moments of inertia and ([??], [??], [??]) are unit vectors pointing along the body (x, y, z) axes.

For small angular departures from this nominal position

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.34)

1.4.2 Relationship between Euler Angle Rates and Angular Velocity

Let (p, q, r) be body-axis components of the angular velocity of the body with respect to inertial axes, and let (φ, θ, ψ) be Euler angles of the body axes with respect to locally horizontal axes, (LHA). Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.35)

where n = angular velocity of LHA with respect to inertial axes and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.36)

A useful mnemonic for these rotations (thanks to Professor Holt Ashley) is – s( ) is above the row with a 1 in it, cyclically.

From (35) and (36) it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.37)

If (ψ, θ, φ) are small in magnitude compared to 1 radian, then (37) may be approximated by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.38)

The average value, over an orbit period, of the magnitude of the nonlinear terms ([??]θ, [??]φ, [??]φ) in (38) is small compared to the average magnitude of the linear terms if φ are θ are oscillatory about zero or if the magnitudes of [??] and [??] are small compared to the orbit rate n. In such cases, (38) may be approximated as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.39)

1.4.3 Moment of Momentum

The angular velocities of a rigid spacecraft may be determined from Euler's equations for moment of momentum:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.40)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = moment of momentum,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = angular velocity body with respect to inertial space,

[??] = external torques on body.

1.4.4 Equations of Motion, Earth-Pointing Satellite

Assuming that the body axes are principal axes, the moment of momentum of the spacecraft is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.41)

From (32), (34), (39), and (40), the equations of motion for small perturbations from locally horizontal axes decouple into two sets, one for pitch and one for roll/yaw. The pitch set is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.42)

[??] – n [congruent to] q. (1.43)

The roll/yaw set is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.44)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.45)

[??] – nψ [congruent to] p, (1.46)

[??] – nφ [congruent to] r, (1.47)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] orbital angular velocity and (Qx, Qy, Qz) are external torques. The gyroscopic coupling terms, n(Iy – Iz)r in (44) and n(Iy – Ix)p in (45), arise from the rotation of the locally horizontal axes at orbit rate n.

1.4.5 Pitch Librations

The characteristic equation of the pitch system (Equations (42) and (43)) in circular orbit is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.48)

If Ix > IZ, the system has undamped oscillations (librations) at a frequency

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.49)

which is called the pitch libration frequency. If Ix< IZ, the system is unstable.

1.4.6 Roll/Yaw Librations

Eliminating p and r from Equations (44)–(47), the Laplace transform of the roll/yaw equations may be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.50)

where

a [??] (Iy – Iz)/Ix, (1.51)

b [??] (Iy – Ix)/Iz. (1.52)

The characteristic equation of (50) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.53)

The magnitudes of a and b are less than or equal to unity since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and similarly

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For b = constant, (53) may be written in Evan's root locus form as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.54)

where s is in units of n. Fig. 1.6 shows the root loci vs. a for fixed values of b, and Fig. 1.7 summarizes the results. The spacecraft roll/yaw motions are oscillatory for a > 0, b > 0, and for a small region when a< 0, b< 0; elsewhere the motions are unstable.

(Continues...)

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Excerpted from Control of Spacecraft and Aircraft by Arthur E. Bryson Jr.. Copyright © 1994 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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Table of Contents

List of Figures

List of Tables

Preface and Acknowledgments

Ch. 1 Natural Motions of Rigid Spacecraft 1

Ch. 2 Spacecraft Sensors and Attitude Determination 16

Ch. 3 Attitude Control with Thrusters 28

Ch. 4 Attitude Control with Reaction Wheels 37

Ch. 5 Attitude Stabilization with Spin 50

Ch. 6 Attitude Control with a Gimbaled Momentum Wheel 74

Ch. 7 Attitude Control during Thrust Maneuvers 93

Ch. 8 Control of Translational Motions 102

Ch. 9 Flexibility and Fuel Slosh 114

Ch. 10 Natural Motions of Rigid Aircraft 142

Ch. 11 Aircraft Sensors 164

Ch. 12 Control of Longitudinal Motions of Aircraft 179

Ch. 13 Control of Lateral Motions of Aircraft 232

Ch. 14 Control of Helicopters near Hover 271

Ch. 15 Aeroelastic Systems 284

Appendix A - Linear System Representations 301

Appendix B - Steady-State Control 311

Appendix C - Synthesis of Analog Control Logic 317

Appendix D - Synthesis of Digital Control Logic 328

Appendix E - Simulation 343

Appendix F - Modeling Flexible Systems 350

References 369

Index 377

Editorial Reviews

"The text reads very smoothly, and I found it very well edited. . . . I very much enjoyed reading it and find it a welcome addition to the available dynamics and control texts."

American Scientist

"The text reads very smoothly, and I found it very well edited. . . . I very much enjoyed reading it and find it a welcome addition to the available dynamics and control texts."—American Scientist

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