Rigid Body Dynamics for Space Applications

By Vladimir Aslanov

Rigid Body Dynamics for Space Applications explores the modern problems of spaceflight mechanics, such as attitude dynamics of re-entry and space debris in Earth's atmosphere; dynamics and control of coaxial satellite gyrostats; deployment, dynamics, and control of a tether-assisted return mission of a re-entry capsule; and removal of large space debris by a tether tow.

Most space systems can be considered as a system of rigid bodies, with additional elastic and viscoelastic elements and fuel residuals in some cases. This guide shows the nature of the phenomena and explains the behavior of space objects. Researchers working on spacecraft attitude dynamics or space debris removal as well as those in the fields of mechanics, aerospace engineering, and aerospace science will benefit from this book.

• Provides a complete treatise of modeling attitude for a range of novel and modern attitude control problems of spaceflight mechanics
• Features chapters on the application of rigid body dynamics to atmospheric re-entries, tethered assisted re-entry, and tethered space debris removal
• Shows relatively simple ways of constructing mathematical models and analytical solutions describing the behavior of very complex material systems
• Uses modern methods of regular and chaotic dynamics to obtain results

• Language: English
• Publisher: Elsevier Science
• Release date: Apr 22, 2017
• ISBN: 9780081018743

Vladimir Aslanov

Professor Aslanov has been Department Head/Professor at Samara State Aerospace University since 1989. Since 2019, he has also served as the head of the ‘Space Flight Mechanics Laboratory’ at the Moscow Aviation Institute. His scientific interests include classical mechanics, nonlinear oscillations and chaotic dynamics, mechanics of space flight, dynamics of gyrostats, and dynamics of tethered satellite systems and spacecraft stability. He has published multiple books, including Dynamics of Tethered Satellite Systems and Rigid Body Dynamics for Space Dynamics, both with Elsevier.

Book preview

Rigid Body Dynamics for Space Applications

First Edition

Vladimir S. Aslanov

Table of Contents

Cover image

Title page

Copyright

Dedication

Preface

Chapter 1: Mathematical Mechanical Preliminaries

Abstract

1.1 Mathematics

1.2 Rigid Body Kinematic

1.3 Rigid Body Dynamics

1.4 Chaotic Motion

Chapter 2: Reentry Attitude Dynamics

Abstract

2.1 Introduction

2.2 Aerodynamics of Reentry Vehicles

2.3 The Equations of Motion

2.4 Analytical Solutions of the Undisturbed Equation for Sinusoidal Aerodynamic Moment

2.5 Analytical Solutions of the Undisturbed Equation for Biharmonical Aerodynamic Moment

2.6 Quasistatic Solutions for the Disturbed Equation of Motion

2.7 Adiabatic Invariants and the Approximate Solution for the Disturbed Motion

2.8 Bifurcation and Ways of Its Elimination at the Descent of Spacecraft in the Rarefied Atmosphere

2.9 Chaotic Attitude Motion of Reentry Vehicle With an Internal Moving Mass

2.10 Chaotic Behavior of Bodies in a Resistant Medium

2.11 Chaotic Motion of a Reentry Capsule During Descent into the Atmosphere

Chapter 3: Dynamics and Control of Coaxial Satellite Gyrostats

Abstract

3.1 Introduction

3.2 Attitude Motion Equations

3.3 Integrable Cases in the Dynamics of Coaxial Gyrostats

3.4 The Exact Analytical Solutions

3.5 Dynamics and Chaos Control of the Gyrostats

3.6 Dynamics and Control of Dual-Spin Gyrostat Spacecraft With Changing Structure

3.7 Adiabatic Invariants in the Dynamics of Axial Gyrostats

Chapter 4: Deployment, Dynamics, and Control of a Tether-Assisted Return Mission of a Reentry Capsule

Abstract

4.1 Introduction

4.2 Mathematical Model of a Satellite With a Tethered Payload

4.3 Analytical Solution in the Case of a Slow Changing of the Parameters

4.4 Oscillations of the Satellite With a Vertical Elastic Tether

4.5 Oscillations in the Case of an Elliptic Orbit

4.6 Swing Principle for Deployment of a Tether-Assisted Return Mission of a Reentry Capsule

4.7 Tether-Assisted Return Mission From an Elliptical Orbit Taking Into Account Atmospheric Stage of Reentry

Chapter 5: Removal of Large Space Debris by a Tether Tow

Abstract

5.1 Introduction

5.2 Dynamics of Orbital Debris Connected to Spacecraft by a Tether in a Free Space

5.3 Dynamics of Large Orbital Debris Removal Using Tethered Space Tug in the Earth's Gravitational Field

5.4 Behavior of Tethered Debris With Flexible Appendages

5.5 Dynamics, Analytical Solutions and Choice of Parameters for Towed Space Debris With Flexible Appendages

5.6 The Motion of Tethered Tug-Debris System With Fuel Residuals

5.7 Dynamics of Towed Large Space Debris Taking Into Account Atmospheric Disturbance

5.8 Chaos Behavior of Space Debris During Tethered Tow

Chapter 6: Original Tasks of Space Mechanics

Abstract

6.1 Introduction

6.2 Gravitational Stabilization of the Satellite With a Moving Mass

6.3 The Dynamics of the Spacecraft of Variable Composition

6.4 Restoration of Attitude Motion of Satellite Using Small Numbers of Telemetry Measurements

Index

Copyright

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Dedication

To my parents, and wife Lyudmila

–Vladimir S. Aslanov

Preface

Vladimir S. Aslanov

Soviet sputnik ushered in the space era in Oct. 1957. This launch occurred almost 70 years ago and nonetheless interest in the creation and use of space technics continues unabated. New challenges formulate new ideas, which be implemented in the new space programs. Currently, global unexpected, including, and environmental concerns associated with big population of nonfunctional and abandoned satellites, spent upper stages and fragments. Each new space program is unique and requires new technologies and careful research based on mathematical modeling. Results of the mathematical simulation and the analytical solutions allow a better understanding of the phenomenon and processes of spacecraft functioning, and choose the conceptual design of future aerospace systems. Most of the space systems can be considered as a system of rigid bodies, and in some cases, with additional elastic and viscoelastic elements, and with fuel residuals.

The purpose of the book is to show the nature of the phenomena and to explain features of the behavior of space objects, as a system of rigid bodies, based on the knowledge of classical mechanics, regular and chaotic dynamics. The author tried to show relatively simple ways of constructing mathematical models and analytical solutions describing the behavior of very complex mechanical systems. The book contains many analytical and approximate analytical solutions that help to understand the nature of the studied phenomena. It is based on the recent papers of the author in international journals, which have been reviewed by leading scientists of the world, thus the results can be trusted. This book covers modern problems of spaceflight mechanics, such as attitude dynamics of reentry capsule in Earth's atmosphere, dynamics and control of coaxial satellite gyrostats, dynamics and control of a tether-assisted return mission, removal of large space debris by a tether tow.

The author hopes that this book will be helpful for a wide range of scientists, engineers, graduate students, university teachers, and students in the fields of mechanics, and aerospace science. Graduate students and researchers find in the book the new results of studies in a wide range of aerospace applications, and they can also use it as tool for obtaining new knowledge. Aerospace engineers can get engineering approaches to the development of new space systems. University teachers can use the text for preparation of new sections in the course of the mechanics of space flight and students will have updated courses of lectures.

The book consists of six chapters. It begins from the necessary fundamentals. Chapter 1 covers basic aspects of mathematics and mechanics, including elliptic functions, rigid body kinematics, Serret–Andoyer canonical variables, and Poincare and Melnikov’s methods. Chapter 2 explores uncontrolled descent of the reentry capsule into an atmosphere by the averaging method and methods of chaotic dynamics. Chapter 3 deals with attitude motion of free dual-spin satellite gyrostats. Exact analytical solutions of the undisturbed motion are presented for all possible ratios of inertia moments of the gyrostats. Chapter 4 is devoted to a tether-assisted reentry capsule return mission. Chapter 5 considers a problem of removal of large space debris by a space transportation system, which is composed of a space tug connected by a tether with the space debris. Chapter 6 contains several separate issues of space flight mechanics, which are of great practical interest, but were not included in previous chapters: the problem of the gravitational stabilization of the satellite by a controlled motion of a point mass on board, dynamics of a space vehicle during retrorocket engine operating, and restoration of attitude motion of satellite using small numbers of telemetry measurements.

I would like to acknowledge brilliant Russian scientist in the field of Aeronautics and Astronautics Professor Vasiliy Yaroshevskiy for special attention and support in the beginning of my academic career, and my first research supervisor Professor Vitali Belokonov. I would like to thank all of my friends and colleagues who helped me make my researches, in particular Dr. Viktor Boyko, Professor Ivan Timbay, Dr. Anton Doroshin, Dr. Alexander Ledkov, and Dr. Vadim Yudintsev. Especially, I would like to express my appreciation to Dr. Alexander Ledkov and Dr. Vadim Yudintsev for their help in the work on this manuscript. I also thank Elsevier for their support and publication of this book, and Samara National Research University in the person of Rector Evgeniy Shakhmatov and President Viktor Soifer for productive environment and the opportunity to work with interesting people.

This book contains results of researches that has been supported in part by the Russian Science Foundation (project no. 16-19-10158, Chapter 5), the Russian Foundation for Basic Research (project no. 15-01-01456-A), and the Ministry Education and Science of Russia (Contract No. 9.540.2014/K).

Chapter 1

Mathematical Mechanical Preliminaries

Abstract

This chapter presents the basics of mathematics and mechanics used in the book. In the first section, the elliptic integrals and elliptical functions are considered. In mechanics, these mathematical objects describe nonlinear oscillations, unlike the trigonometric functions, which describe linear oscillations. In the second section, rigid body kinematics is considered. The section deals with two methods to describe attitude motion of rigid body, which are used in the book: two types of Euler angles (ZYZ and XYZ rotation sequences) and directional cosine matrix. Kinematic equations are written for each type of coordinates. The kinematic equations are the relation between angular velocity of the rigid body and the derivatives of the coordinates described in orientation of the rigid body. In the third section, dynamic equations of the rigid body are presented including equations in Serret-Andoyer canonical variables, which allow reducing torque-free rotational dynamics of a rigid body to two-dimensional Hamiltonian system. In the last section, a chaotic motion is considered. The Poincaré method is described, which is used to investigate the motion of complex systems. In addition, the Melnikov method is described, which is used in the book to prove the existence of chaos in considered mechanical systems.

Keywords

Elliptic integrals; Elliptical functions; Rigid body kinematic; Rigid body dynamics; Chaos; Poincaré section; Melnikov function

1.1 Mathematics

This section contains basic information on elliptic integrals and elliptical functions with examples of their use. Detailed information about elliptic integrals, elliptical functions, and their applications can be found in Refs. [1–6].

1.1.1 Elliptic Integrals

An elliptic integral is an integral that can be written in the form:

   (1.1)

where R(x,y) is a rational function and P(x) is a polynomial of the third or fourth degree in x.

Let us consider the following integral:

   (1.2)

which is called incomplete integral of the first kind. The incomplete elliptic integral is a function of angle φ .

, the incomplete integral of the first kind is said to be complete elliptic integral of the first kind and is denoted as K(k):

   (1.3)

Definite integral

   (1.4)

   (1.5)

Incomplete elliptic integral of the third kind is

   (1.6)

Complete elliptic integral of the third kind is

   (1.7)

where n is called the characteristic.

The elliptic integrals satisfy the following relations:

   (1.8)

   (1.9)

   (1.10)

   (1.11)

, complete elliptic integrals can be expanded into series as follows:

   (1.12)

   (1.13)

.

1.1.2 Elliptic Functions

The inverse functions of incomplete elliptic integral of the first kind form the elliptical functions. The amplitude function is defined as (see Eq. 1.2)

   (1.14)

The elliptical sine function sn(u,k) is given by

   (1.15)

The elliptical cosine function cn(u,k) is given by

   (1.16)

Elliptic functions sn(u,k) and cn(u,k) have period 4K(k) for the argument u. Delta amplitude function dn(u,k) is given by the expression:

   (1.17)

This function has period 2K(k).

The elliptic functions satisfy the following relations:

   (1.18)

   (1.19)

The derivatives of the elliptic functions are given by the following expressions:

   (1.20)

   (1.21)

   (1.22)

   (1.23)

, Eq. (1.2) has the form:

   (1.24)

that means

   (1.25)

Solving Eq. (1.25) for sinφ, we get

   (1.26)

Therefore, taking into account Eq. (1.18), we obtain

   (1.27)

Taking into account Eq. , we get

   (1.28)

,

   (1.29)

So we get

   (1.30)

and

Thus, when k=0, elliptical functions degenerate to trigonometric functions.

Elliptic integrals and elliptical functions are used in mechanics and engineering. For example, they used to describe nonlinear oscillations of mechanical systems. Let us consider the motion of a physical pendulum as an example of the application of elliptical functions [6].

A physical pendulum is the generalized case of the simple pendulum. It consists of rigid body of mass m that oscillates about a pivot point. The moment of inertia of the body around the pivot point is J. The in-plane dynamics of the physical pendulum is given by the following initial value problem:

   (1.31)

. This equation has the first integral, which can be obtained using the law of conservation of energy for the system considered:

   (1.32)

Let us rewrite the first integral (1.32) as follows:

   (1.33)

. Let us assume that the pendulum oscillates through a maximum angle of β from its vertical equilibrium position. In this case,

   (1.34)

The first integral (1.32) can be rewritten as

   (1.35)

Let us make the following substitutions:

   (1.36)

First, integral (1.35) now can be rewritten as

   (1.37)

And now we can integrate

   (1.38)

The integral at the left side is an incomplete integral of the first kind K(ψ,k1), so we can write down the solution as

   (1.39)

or according to Eq. (1.36)

   (1.40)

Function φ has period

   (1.41)

Expanding K(k1) in series in the powers of k1, we obtain

   (1.42)

so for small β, we get well-known approximation of the pendulum period

   (1.43)

or more precise estimation of T

   (1.44)

1.2 Rigid Body Kinematic

The orientation of the rigid body can be described by several methods: using orthogonal matrices, Euler angles, and quaternions. First two methods, used in this book, are addressed in the next subsections. Each method has strengths and weaknesses compared with other methods.

1.2.1 Orthogonal Matrices

Any vector R can be represented as a linear combination of the orthogonal basis vectors e1¹,  e2¹,  e3¹ (Fig. 1.1):

Fig. 1.1 Coordinates of the vector R in the basis e ¹ .

   (1.45)

where R(1) is a column vector of the coordinates of the vector R .

Any unit vector of the basis can be represented as a linear combination of another basis vectors. Let us consider two bases e1¹,e2¹,e3¹ and e1²,e2²,e3². All basis vectors of the second basis can be represented as a linear combination of the first basis (Fig. 1.2):

Fig. 1.2 Coordinates of the basis vector e 1 ² in the basis e ¹ .

   (1.46)

   (1.47)

   (1.48)

where aij are dot products of the unit vectors

   (1.49)

   (1.50)

   (1.51)

Now it is possible to represent transformation of the coordinates of the vector R from the basis e² to the basis e¹ and vice versa. Let us assume that the coordinates of the vector R in the basis eand the coordinates of the vector in e. Now we can write

   (1.52)

   (1.53)

   (1.54)

or in the matrix form

   (1.55)

where

   (1.56)

is a directional cosine matrix, which transforms coordinates of the vector from basis e² to basis e, which is equal to transpose matrix AT due to orthogonality of the matrix A:

   (1.57)

   (1.58)

where I is an 3×3 identity matrix.

Matrix, which transforms coordinates from the basis e² resulting from rotation of the basis e¹ around its x axis (e1¹) has the form

   (1.59)

The matrix that transforms coordinates from the basis e² rotated around the y axis of the basis e¹ is

   (1.60)

and the matrix that transforms coordinates from the basis e² rotated around the z axis of the basis e¹ is

   (1.61)

Using these matrices, one can describe the complex orientation of the basis e² relative to the basis e¹.

The orthogonal basis e² can be linked with the rigid body so the directional cosine matrix A can be used to describe the orientation of that rigid body.

1.2.2 Euler Angles

Any orientation of rigid body can be achieved by composing three elementary rotations about frame axes. The first method is using Euler angles introduced by Leonhard Euler. There are several sets of Euler angles, depending on the axes about which the rotations are performed. Euler angles are a minimal representation of the orientation of a rigid body unlike to the directional cosine matrix.

Let us consider orientation of the body-fixed coordinate system Ox2y2z2 relative to the fixed frame Ox1y1z1. Orientation of the Ox2y2z2 frame can be constructed as a sequence of three rotations. It is supposed that Ox2y2z2 frame is initially aligned with the fixed frame Ox1y1z1. The first rotation is carried out about Oz1 axis to the angle ψ, so we get the coordinate system Ox′y′z′ (Fig. 1.3). At the second step, we rotate Ox′y′z′ frame around Ox′ axis to the angle θ. At the third step, we rotate Ox′y″z2 frame around z2 axis and get the final orientation of the frame Ox2y2z2. This rotation sequence is called Euler angles (313 or ZYZ sequence).

Fig. 1.3 Euler angles.

The transformation matrix, which transform coordinates from the basis Ox2y2z2 to the basis Ox1y1z1, can be represented in the Euler angles as

   (1.62)

or in the full form

   (1.63)

where

.

, a coordinate singularity exists. In this case, infinitely many pairs of ψ and φ represent the same orientation of the basis e² relative to the basis e¹ (so we get the loss of a degree of freedom)

   (1.64)

Other sequences of three rotations can be used to represent orientation of the basis e² relative to the basis e¹. For example, Tait-Bryan angles represent rotations about three distinct axes (123) instead of Euler angles (313). Fig. 1.4 demonstrates this formalism.

Fig. 1.4 Tait-Bryan angles.

The first rotation is carried out about Ox1 axis to the angle α1, so we get the frame Ox1y′z′. At the second step, we rotate Ox1y′z′ frame around Oy′ axis to the angle α2. At the third step, we rotate Ox′y′z2 frame around z2 axis and get the final orientation of the frame Ox2y2z2.

   (1.65)

or in the full form

   (1.66)

.

there are infinitely many pairs of α1 and α3 represent on and the same orientation of the basis e² relative to the basis e¹:

   (1.67)

1.2.3 Kinematic Equations

Kinematic equations establish a link between the angular velocity of the rigid body in the body frame:

   (1.68)

and the derivatives of the coordinates that describe the orientation of the body.

For the elements of the orthogonal matrix, the kinematic relation has the following form [7]:

   (1.69)

where A is a skew-symmetrical matrix of the angular velocity components in the body frame:

   (1.70)

Angular velocity in the body frame also can be represented as the sum of the three derivatives of the Euler angles (Fig. 1.3):

   (1.71)

So we can express the coordinates of the angular velocity vector as

   (1.72)

   (1.73)

   (1.74)

The derivatives of the Euler angles are

   (1.75)

   (1.76)

   (1.77)

Kinematic equations for Tait-Bryan angles have the forms [7,8]:

   (1.78)

   (1.79)

   (1.80)

The derivatives of the for Tait-Bryan angles are

   (1.81)

   (1.82)

   (1.83)

Additional information about other forms of kinematic equations can be found in Refs. [7,8].

1.3 Rigid Body Dynamics

1.3.1 Kinetic Energy of Rigid Body

Kinetic energy of the rotational motion of a rigid body can be described by using angular velocity of the body ω in the body frame:

   (1.84)

where J is the inertia tensor of the rigid body. In the body, principal frame J is a diagonal matrix:

   (1.85)

G is the angular momentum vector of the body in the body frame:

   (1.86)

For the angular velocity parametrized by derivatives of the Euler angles, the expressions (1.84) can be rewritten as

   (1.87)

1.3.2 Canonical Variables

Euler angles ψ,θ,φ with the momentums

   (1.88)

form canonical set of variables of rigid body. Angular velocities p,q,r can be expressed in the momentums using

   (1.89)

   (1.90)

   (1.91)

Taking into account Eqs. (1.78)–(1.80),

   (1.92)

   (1.93)

   (1.94)

The momentums pψ, pθ, and pφ can be expressed in terms of scalar product of the angular momentum vector and basis vectors of the body frame and inertial frame:

   (1.95)

   (1.96)

   (1.97)

1.3.3 Serret-Andoyer Canonical Variables

Dynamics of an attitude motion of a rigid body can be described using Serret-Andoyer canonical variables. Serret-Andoyer canonical variables allow reducing the torque-free rotational dynamics to one-and-a-half degrees of freedom [9–11]. Let us consider two frames of reference: an inertial frame Ox1y1z1 and a body-fixed frame Ox2y2z2. We suppose that the axes of the body-fixed frame aligned with the principal axes of inertia of the body. Orientation of the frame Ox2y2z2 relative to the frame Ox1y1z1 can be described by the angles h, g, and l. Angle h describes orientation of the line OB in the inertial reference plane Ox1y1. Line OB is the intersection of the plane Ox1y1 and the invariable plane OBA that is perpendicular to the angular momentum vector G. Angle g describes orientation of the line OA in the OBA plane. Angle l describes orientation of the Ox2 axis relative to the line OA, which is the intersection of the plain OBA and the plane of the rigid body Ox2y2 (Fig. 1.5).

Fig. 1.5 Serret-Andoyer variables [ 11].

There are three generalized momentums that correspond to the coordinates h, g, and l [9]:

   (1.98)

Kinetic energy of the body can be written using Serret-Andoyer variables:

   (1.99)

The angular velocity components of the body can be expressed using Serret-Andoyer variables as follows [11]:

   (1.100)

1.3.4 Dynamic Equations

Dynamic equations of the rigid body can be written using the relation between the torque acting on the body and the derivative of the angular momentum:

   (1.101)

In the body frame, where J for a rigid body is a constant matrix, Eq. (1.101) has simple form:

   (1.102)

or in the scalar form:

   (1.103)

   (1.104)

   (1.105)

where Mx, My, and Mz are the projections of the external torque M on the body frame axes.

The dynamic equations (1.103)–(1.105) can be rewritten in terms of Euler angles:

   (1.106)

   (1.107)

   (1.108)

Canonical equations for torque-free motion in Serret-Andoyer variables have the form [9]:

   (1.109)

   (1.110)

   (1.111)

   (1.112)

   (1.113)

We see that while T is constant equations, Eqs. (1.111), (1.113) are separable and they can be solved in a closed form [9].

1.4 Chaotic Motion

Chaotic dynamic studies dynamical systems that are highly sensitive to initial conditions. Chaotic systems are characterized by two main properties: instability and topologically mixing. Instability mixing means that any smallest change in the initial conditions can lead to arbitrarily large changes in motion. Topologically mixing means that any region of the