Magnetic Attitude Control for Spacecraft with Flexible ... · Magnetic Attitude Control for...

Magnetic Attitude Control forSpacecraft with Flexible

Appendages

by

Julian Pierre Stellini

A thesis submitted in conformity with the requirementsfor the degree of Master of Applied Science

Graduate Department of Aerospace Science and EngineeringUniversity of Toronto

Copyright c© 2012 by Julian Pierre Stellini

Magnetic Attitude Control for Spacecraft with

Flexible Appendages


Master of Applied Science

Graduate Department of Aerospace Science and Engineering

University of Toronto

2012

Abstract

The design of an attitude control system for a flexible spacecraft using magnetic actuation

is considered. The nonlinear, linear, and modal equations of motion are developed for a

general flexible body. Magnetic control is shown to be instantaneously underactuated,

and is only controllable in the time-varying sense. A PD-like control scheme is proposed

to address the attitude control problem for the linear system. Control gain limitations

are shown to exist for the purely magnetic control. A hybrid control scheme is also

proposed that relaxes these restrictions by adding a minimum control effort from an

alternate three-axis actuation system. Floquet and passivity theory are used to obtain

gain selection criteria that ensure a stable closed-loop system, which would aid in the

design of a hybrid controller for a flexible spacecraft. The ability of the linearized system

to predict the stability of the corresponding nonlinear system is also investigated.

ii

Acknowledgements

First and foremost, I would like to thank my mother. She has supported me without

question throughout my academic career and my entire life. She helped me get through

the late nights and times of confusion and frustration. I could not have done anything

without her.

I would like to thank Dr. Christopher Damaren for all of his guidance and support.

He is an undeniable expert in his field and I am honoured to have been able to work

under his supervision. His enthusiasm in his work made my time with him exciting and

rewarding. I appreciate all of his help.

I would also like to thank Dr. James Forbes and Ludwik Sobiesiak for all of their

help. They kept myself and the entire lab sane and friendly. They were always there to

share ideas and help me overcome any hurdles during my time at UTIAS.

I would like to acknowledge UTIAS itself, for providing me with a stimulating and

educational environment to work in. I always had the resources I needed at my disposal,

and I appreciate the friendly and fun atmosphere that it provided.

Lastly, I would like to thank all of my friends for being there when I needed them

most. It is very easy to get so caught up in work and school that you lose focus of all

the other important things in life and I am grateful for all the needed distractions and

good times that they have given me.


June 2012

iii

Contents

Abstract ii

Acknowledgements iii

Table of Contents v

List of Tables vi

List of Figures viii

1 Introduction 1

1.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Background Concepts 8

2.1 Vectrix Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.1 Vector Dot and Cross Products . . . . . . . . . . . . . . . . . . . 9

2.1.2 Multiple Reference Frames and Rotation Matrices . . . . . . . . . 10

2.1.3 Angular Velocity and Vector Time Derivatives . . . . . . . . . . . 11

2.2 Attitude Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 Euler Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.2 Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Stability Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.1 Lyapunov Stability . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.2 Input-Output Stability . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 Floquet Stability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5 Lyapunov Stability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 18

iv

2.5.1 Direct Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.5.2 Indirect Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.6 Passivity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Spacecraft Mechanics 21

3.1 Reference Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Orbital Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.1 Orbit Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2.2 Orbital Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3 Spacecraft Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.4 Spacecraft Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.4.1 Rigid Spacecraft . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.4.2 Flexible Spacecraft . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4.3 Linearized Equations of Motion . . . . . . . . . . . . . . . . . . . 32

3.4.4 Modal Equations of Motion . . . . . . . . . . . . . . . . . . . . . 36

3.5 Archetypal Spacecraft/Orbit . . . . . . . . . . . . . . . . . . . . . . . . . 39

4 Magnetic Attitude Control 42

4.1 Magnetic Field Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2 Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.3 Hybrid PD-Like Control Law . . . . . . . . . . . . . . . . . . . . . . . . 48

5 Hybrid PD-Control Of Flexible Spacecraft 52

5.1 Floquet Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.2 Hybrid Controller Gain Selection Criteria . . . . . . . . . . . . . . . . . . 62

5.3 Stability of the Nonlinear System . . . . . . . . . . . . . . . . . . . . . . 71

6 Conclusions 75

6.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

References 79

Appendix A Geomagnetic Field Model Supplemental Calculations 82

Appendix B Hybrid Control Gain Selection For Rigid Spacecraft 84

Appendix C Floquet Stability Diagram for Rigid Spacecraft 88

v

List of Tables

3.1 Exact mode shape parameters for cantilevered beam. . . . . . . . . . . . 40

4.1 IGRF Coefficients for Epoch 2010. . . . . . . . . . . . . . . . . . . . . . . 44

5.1 Floquet stability prediction compared to nonlinear response. . . . . . . . 73

vi

List of Figures

1.1 A generic rigid spacecraft with two flexible beams attached. . . . . . . . 2

2.1 A generic reference frame and vector. . . . . . . . . . . . . . . . . . . . . 8

2.2 Block diagram of a system operator. . . . . . . . . . . . . . . . . . . . . 16

2.3 A generic feedback system. . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1 The inertial and body-fixed frames. . . . . . . . . . . . . . . . . . . . . . 22

3.2 Polar coordinate system for spacecraft orbital position. . . . . . . . . . . 23

3.3 The geometry of an ellipse. . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4 The orbital and inertial reference frames. . . . . . . . . . . . . . . . . . . 25

3.5 A generic rigid body. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.6 A generic unconstrained flexible body. . . . . . . . . . . . . . . . . . . . 31

3.7 A sample rigid spacecraft with two flexible beams attached. . . . . . . . . 40

4.1 An example of the geomagnetic field experienced in orbit. . . . . . . . . . 46

4.2 The cross-product relationship of the magnetic control torque. . . . . . . 47

4.3 A comparison of the geomagnetic field for polar and equatorial orbit. . . 48

4.4 A plot showing the existence of a gain limitation using PD-like control. . 50

5.1 Stability diagram obtained using Floquet theory. . . . . . . . . . . . . . . 57

5.2 Simulated spacecraft response demonstrating gain limitation. . . . . . . . 58

5.3 Interesting region of the Floquet stability diagram. . . . . . . . . . . . . 59

5.4 A plot showing the contribution of the flexible modes to the rigid ones. . 60

5.5 A plot illustrating the affect of flexibilty on the nonlinear system. . . . . 61

5.6 System response to purely magnetic control without elastic damping. . . 62

5.7 A plot showing a limitation of the Floquet analysis. . . . . . . . . . . . . 63

5.8 Block diagram for the flexible, hybrid-controlled system. . . . . . . . . . 66

5.9 Stability diagram obtained using passivity theory. . . . . . . . . . . . . . 67

vii

5.10 A plot illustrating the validity of the hybrid gain selection criteria. . . . . 68

5.11 A comparison of the attitude response for small/large initial conditions. . 72

5.12 An example of a stable and unstable nonlinear system response. . . . . . 74

B.1 Block diagram for a rigid, hybrid-controlled system. . . . . . . . . . . . . 86

C.1 Stability diagram for rigid spacecraft obtained using Floquet theory. . . . 88

C.2 Interesting region of the Floquet stability diagram for rigid spacecraft. . . 89

viii

Chapter 1

Introduction

The ability to control the attitude of spacecraft is a very important characteristic for any

space system to have. Spacecraft such as communications satellites rely on being able

to maintain line-of-sight with the earth as they traverse their orbits. Several methods

of providing active three-axis attitude control have been well documented and imple-

mented such as using reaction wheels, control-moment gyros, thrusters, etc. For small,

lightweight spacecraft that have very strict mass and power budgets with limited ability

to store fuel, the above techniques may be impractical to use.

Magnetic attitude control has recently been considered as a viable option for the

type of spacecraft described above. This method of control operates on the basis of

exploiting the interaction between the earth’s magnetic field and a set of three mutually

orthogonal electromagnetic actuators mounted on the spacecraft. These actuators could

be as simple as current loops aligned with the spacecraft body axes, which would create

three linearly independent dipole moments. This method of control is particularly suited

for low altitude orbits as the magnetic field is stronger in these regions and would allow for

a greater range of control torques [1]. The limitation of this technique is that the magnetic

control torque is only available perpendicular to the local geomagnetic field vector. This is

because the torque, τ−→(t), produced via an interaction between the actuator’s generated

dipole moment, m−→(t), and the magnetic field, b−→(t), is governed by the vector cross-

product

τ−→(t) = m−→(t)× b−→(t).

Thus, at most two axes are stabilizable at any given point in time, making the full three-

axis attitude stabilization problem unsolvable from a time-invariant point of view. This

1

Chapter 1. Introduction 2

fact is what poses the most difficulty when designing control laws for magnetic attitude

controllers. It is important to note however, that the geomagnetic field varies with time

along a spacecraft’s orbit, and [2] has shown that full control is possible if the variation

in the magnetic field is large enough (which is the case for near-polar orbits). Therefore,

magnetic attitude control is inherently a time-varying problem.

Lots of work has been done investigating various magnetic control schemes, and stabi-

lizing controllers exploiting the geomagnetic field have been successfully developed using

well-known methods such as optimal periodic control, linear-quadratic regulation (LQR)

techniques, and proportional/derivative (PD)-like control [3]. However, all of these mag-

netic control schemes are derived within the context of a rigid spacecraft. The topic

of attitude control systems for flexible spacecraft has been thoroughly studied as well,

and in this case controllers have also been successfully developed (i.e., see [4], [5] and

the references therein), but they involve types of actuation other than the geomagnetic

field and would therefore suffer the same disadvantages as described above if applied to

small/lightweight spacecraft.

There is a clear research gap: using magnetic actuation for the attitude control of

flexible spacecraft. Extending the magnetic control problem to the flexible case allows

for a broader class of spacecraft to be controlled by this method. For example, this

type of cotnrol would then be able to be used in controlling satellites with long, flexible

booms (on which items like scientific instruments, or solar panels may be mounted). The

general purpose of this thesis project is to commence bridging this gap by investigating

magnetic controller design methods for a specific subset of flexible spacecraft such as the

one mentioned above. In particular, the class of flexible spacecraft to be examined will

be limited to a rigid, central body with two flexible cantilever beams attached as shown

in Fig. 1.1.

F−→b

Rigid Body

Flexible Arm

ue1(y, t)

−ue2(y, t)

Flexible Arm

x

y

z

Figure 1.1: A generic rigid spacecraft with two flexible beams attached that will be used foranalysis and simulation purposes.


1.1 Literature Review

Before reviewing previous work done regarding this topic, it is important that the system

dynamics (including the interaction between the spacecraft and the geomagnetic field)

be controllable. An investigation on the controllability and accessibility conditions of a

nonlinear time-varying (NTV) spacecraft subject to magnetic attitude control was per-

formed in [6]. It is shown that the attitude dynamics are indeed controllable if the orbital

plane does not coincide with the geomagnetic equatorial plane and if the magnetic field is

periodic in time. See Chapter 4.2 for more specific information regarding controllability.

Since the literature regarding the attitude control of flexible spacecraft using magnetic

actuation is relatively scarce, the following review focuses on the techniques and control

schemes used for magnetic attitude control in the rigid spacecraft case. Looking at

how a rigid spacecraft may be controlled using these methods is an appropriate starting

point as they may possibly be extended to be applicable in the flexible case. A broad

exploration of the various approaches to magnetic attitude control for rigid spacecraft

can be found in [3], where linear methods (such as PD, H∞, and optimal control) and

nonlinear methods (such as magnetic predictive attitude control) are briefly compared

and contrasted.

Motivated by the Danish Ørsted satellite mission, [7] was able to develop a family of

controllers based on attitude and angular velocity feedback. The nonlinear, time-varying

system dynamics were considered in the controller design, and it was shown that the

control laws provided global three-axis attitude stabilization using magnetic torquers

only. The family of controllers were shown to be able to also provide joint control action

for satellite de-tumbling and nominal operation. It is important to note that there is

an error in this article, and its correction can be found in [8]. Also motivated by this

mission, [9] leveraged the periodic nature of the geomagnetic field for near-polar orbits

against the linear time-varying (LTV) dynamics to develop three types of controllers. Two

of them, finite and infinite horizon-based controllers, involves solving a periodic Riccati

equation, while the other was a constant-gain controller based on an approximation of

the monodromy matrix of the linearized system. A useful consequence of designing a

controller around a monodromy matrix is that a stability analysis using Floquet theory

can be easily applied.

The controllers proposed in [10] and [11] also exploit the periodic nature of the geo-

magnetic field. Asymptotic periodic LQR theory is used to design the controller in [10].


Integral action and saturation logic is included in the design in order to account for when

the magnetic actuators are providing the maximum magnetic moment that they are ca-

pable of. Simulation of the closed-loop LTV system exhibited robustness with respect to

modeling uncertainty and disturbance torques (such as residual dipoles). The controller

proposed in [11] uses periodic LQ optimal control techniques, and also relies on solving

a periodic Riccati equation. A unique feature of this controller is that it also includes

optimal estimation and compensation schemes for external disturbance rejection. The

disturbance estimation involves using a Kalman filter with output regulation strategies

similar to linear time-invariant (LTI) systems. In [12], H∞ control is used on LTV sys-

tems in order to design periodic controllers that perform well on the nonlinear system

and are also robust against large disturbance torques.

A PD-like control law is featured in the controllers designed in [13–16]. Static and

dynamic attitude and rate feedback is used to provide almost global attitude stabilization.

Reference [13] also looks at actuator saturation constraints and contingencies for when

one or more of the actuators becomes unavailable. Reference [14] proposes an adaptive

version of the PD-like control law, and uses a generalized averaging theory to prove

stability. In [15], it is demonstrated that stabilization without the rate feedback is also

possible with a minor adaptation to the control law. Note that the stability proofs

for all of these PD-like control laws rely on the generalized averaging theory described

in [17], which imposes mathematical limits on the gains in all of the controllers. These

mathematical limits can be shown to have a physical manifestation. This is a very

important result as it places restrictions on the performance of this simple and easy-to-

implement control strategy. Also, it is shown in [15] that the application of this relatively

simple control strategy requires that a matrix involving the geomagnetic field be positive

definite, restricting the use of this type of controller to non-equatorial orbits.

A family of optimal control programs called RIOTS is used by [18] to implement

a time-optimal open-loop controller. Using RIOTS allowed for designing a controller

around the NTV spacecraft dynamics with constrained inputs. A sub-optimal model

predictive feedback closed-loop control scheme is also proposed by [18], which attempts

to track a pre-calculated optimal trajectory. These controllers address the slow conver-

gence to equilibria inherent in other magnetic attitude control schemes, and can therefore

be used for time critical maneuvering. In [19], a discrete-time approach was taken. Three

optimal discrete-time controllers were proposed: a periodic optimal state feedback con-

troller, a predictive magnetic controller, and a fixed-structure projection-based controller.


Reference [20] proposes magnetic attitude control designs for the REIMEI microsatellite

similar to some of those already discussed. The novel contribution in [20] is an implemen-

tation of a residual magnetic moment observer and its feed-forward cancellation, which

suggests similar approaches may be taken for other disturbance torques. In [21] an LQR

control method (based on solving an algebraic Riccati equation) designed around the

linearized system is compared to another approach relying on the solution of a state-

dependent Riccati equation (using a state-dependent coefficient method) design around

the nonlinear system. It is shown that the latter method was found to be more robust

and stable as it includes the use of a variable-gain feedback.

Partial magnetic actuation has also been considered by some. These techniques in-

volve having the magnetic torquers as the main control source as well as other active

actuators, such as reaction wheels or thrusters, for support. Reference [22] shows that

using partial magnetic actuation can allow for the spacecraft dynamics to be treated as

an LTI system. The proposed control law is then a simple state feedback. Reference [22]

also addresses the inherent control allocation problem for optimum performance. Two

approaches were considered: a direct approach where the supporting actuators would

only provide control for the axis that the magnetic actuator can’t affect, and a quadratic

programming decision method that involves solving an optimization program. In [23] a

geometric approach is followed for the control allocation problem. A systematic way of

allocating the two control torques such that they do not overlap and potentially negate

each other is described. Another partial magnetic actuation strategy is proposed in [24],

where the PD-like control law of [16] is combined with a similar law for the other actua-

tion system. It is shown how the limitation on the attitude gain (proved in [16]) can be

alleviated with a minimum level of contribution from the supporting actuators.

The PD-like control strategy in [16] was shown to maintain its stability properties

when applied to a flexible spacecraft in [25]. However, this controller is still subject to the

gain limitation setbacks described above. It also does not use any information regarding

the flexible dynamics of the system, and its adequate performance suggests that there is

much room for improvement.

1.2 Purpose

The purpose of this thesis is to investigate controller design methods for the attitude

control of spacecraft with flexible appendages, using magnetic actuation. Certain con-


trollers designed for a rigid spacecraft will be extended to the flexible case. Conditions

for the stability of the resulting closed-loop systems will be investigated, thereby provid-

ing insight into the controller design. The analysis and design methods will be based on

the linearized dynamics, and the performance of the resulting controller on the nonlinear

system will be investigated.

1.3 Thesis Overview

Chapter 1 provides a brief introduction to the importance and relevance of exploiting

the earth’s magnetic field to control the attitude of spacecraft in general. The general

purpose and scope of the work done in this thesis is also explained. A review of the

relevant literature for this topic is provided as well.

Chapter 2 summarizes the background mathematics and concepts relevant to the

work done in this thesis: using vectrix notation to represent vectors and vector operations;

spacecraft attitude representation using rotation matrices, Euler angles, and quaternions;

definitions of input-output and Lyapunov stability; and various stability theories such as

Lyapunov, Floquet, and passivity theory.

Chapter 3 provides a description of the relevant frames of reference, the orbital dy-

namics of a spacecraft orbiting the earth, and the mechanics (kinematics and dynamics)

of a general flexible and rigid spacecraft. The nonlinear and linear equations of motion

are derived. Modal analysis is performed on the linearized system and the equivalent

modal equations of motion are also developed.

Chapter 4 provides a more in-depth look at magnetic attitude control. The model used

to represent the earth’s magnetic field is included, as well as a summary of an investigation

into the controllability of spacecraft using this type of actuation. Approaches to magnetic

control relevant to this thesis are introduced and explored.

In Chapter 5, a hybrid PD-like control law is proposed and conditions under which

the closed-loop system is stable are investigated. A stability map for the controller is

obtained using Floquet theory. Passivity theory is used to obtain controller gain selection

criteria. The effect of the elastic properties of the flexible spacecraft on its stability are

also considered. The performance of the control law on the nonlinear system is explored

as well.

Finally, Chapter 6 summarizes the work done in the preceding chapters, and proposes

potential areas to explore in order to expand the results of this thesis. Supplementary


material required for a better understanding of some of the topics discussed in this thesis

are included in the appendices.

Chapter 2

Background Concepts

The derivation of the equations of motion, the controller development and the stability

analyses make use of certain mathematical concepts that are presented in the following

sections.

2.1 Vectrix Notation

Throughout this thesis, vectrix notation will be used to represent vectors. The develop-

ment of the equations of motion (see Chapter 3) relies heavily on this notation. Vectrix

notation is useful because it explicitly identifies the reference frame in which the vector

is being expressed. Consider a reference frame, denoted by Fa, and an arbitrary vector,

r−→, as seen in Fig. 2.1. A vectrix is defined to be F−→a =[

a−→1

a−→2

a−→3

]T

, which is a

r−→

a−→1

a−→2

a−→3

Fa

Figure 2.1: A generic reference frame and vector.

column containing the mutually orthogonal and right-handed basis vectors that form the

8

Chapter 2. Background Concepts 9

reference frame Fa. The vector r−→ can be then be expressed as

r−→ = r1 a−→1

+ r2 a−→2

+ r3 a−→3

=[

a−→1

a−→2

a−→3

]

r1

r2

r3

= F−→aTr

= rTF−→a,

where r =[

r1 r2 r3

]T

, r ∈ R3×1 is a column matrix containing the scalar components of

r−→ as expressed in Fa. Any equation or operation involving vectors can then be described

in terms of matrices, so long as all of the vectors are expressed in the same frame. More

detailed information regarding vectrix notation can be found in the appendices of [26].

2.1.1 Vector Dot and Cross Products

Given the vectrix notation described in Chapter 2.1, both the dot and cross products

between two vectors can be easily described in terms of matrix multiplication. Consider

two vectors, r−→ = F−→aTr and s

−→= F−→a

Ts (note how both are expressed in the same

reference frame). The dot product is then given by

r−→ · s−→

=[

r1 r2 r3

]

a−→1

a−→2

a−→3

·[

a−→1

a−→2

a−→3

]

s1

s2

s3

=[

r1 r2 r3

]

a−→1

· a−→1

a−→1

· a−→2

a−→1

· a−→3

a−→2

· a−→1

a−→2

· a−→2

a−→2

· a−→3

a−→3

· a−→1

a−→3

· a−→2

a−→3

· a−→3

s1

s2

s3

= rT1s

= rTs

= sTr ,


where 1 is used to denote the identity matrix. Similarily, the cross product is given by

r−→× s−→

=[

r1 r2 r3

]

a−→1

× a−→1

a−→1

× a−→2

a−→1

× a−→3

a−→2

× a−→1

a−→2

× a−→2

a−→2

× a−→3

a−→3

× a−→1

a−→3

× a−→2

a−→3

× a−→3

s1

s2

s3

=[

r1 r2 r3

]

0−→ a−→3

− a−→2

− a−→3

0−→ a−→1

a−→2

− a−→1

0−→

s1

s2

s3

=[

a−→1

a−→2

a−→3

]

0 −r3 r2

r3 0 −r1

−r2 r1 0

s1

s2

s3

= F−→aTr×s

= F−→aT(−s×r

),

where

r× =

0 −r3 r2

r3 0 −r1

−r2 r1 0

is a skew-symmetric matrix ((r×)T = −r×) that constructs the components of the cross

product.

2.1.2 Multiple Reference Frames and Rotation Matrices

Consider two reference frames denoted by F−→a and F−→b. The situation often arises when

a vector r−→ is given in one frame, yet its representation in the other frame is required.

Vectrix notation provides a simple way of resolving this issue. First, it is evident that

r−→ = F−→aTra = F−→b

Tr b and so

F−→bTr b = F−→a

Tra

F−→b · F−→bTr b = F−→b · F−→a

Tra

r b = Cbara,


where

Cba = F−→b · F−→aT

=

b−→1

· a−→1

b−→1

· a−→2

b−→1

· a−→3

b−→2

· a−→1

b−→2

· a−→2

b−→2

· a−→3

b−→3

· a−→1

b−→3

· a−→2

b−→3

· a−→3

.

This matrix is a direction cosine matrix describing the rotation between the two reference

frames. It belongs to SO(3), which is the group of 3 × 3 orthogonal rotation matrices

with determinant equal to unity. Thus, Cba takes a vector expressed in Fa and rotates it

into Fb.

2.1.3 Angular Velocity and Vector Time Derivatives

Suppose that Fb is rotating with respect to Fa, and let the angular velocity of Fb with

respect to Fa be denoted by ω−→ba. The magnitude of the angular velocity represents the

rate of rotation, and its unit vector represents the instantaneous axis of rotation.

For ω−→ba6= 0−→, the motion experienced in either frame is not the same. Let the vector

time derivative as seen in Fa be denoted by (·) and that in Fb by (). As defined, it is

clear that F−→a =

F−→b= 0−→. As for the derivative of Fb as seen in Fa, it can be shown using

vector calculus that [26]

F−→bT

= ω−→ba× F−→b

T.

Now consider a vector r−→; its time derivatives as seen in either frame is

r−→ = F−→aT

ra + F−→aTra = F−→a

Tra

r−→ =

F−→b

T

r b + F−→bT

r b= F−→bTr b,

noting that the vector time derivative of a column matrix is simply the regular time

derivative independent of frame, and is also denoted by (·). Using these relationships,

one can obtain the derivative of a vector in one frame in terms of the motion in the other


frame since

r−→ = F−→aTra = F−→b

Tr b + F−→bT

r b

= F−→bTr b + ω−→ba

× F−→bTr b

= F−→bT(r b + ω×

bar b).

2.2 Attitude Representation

Consider the rotation matrix Cba in Chapter 2.1.2. If F−→a and F−→b are chosen such that

they are inertial and body-fixed (i.e., as in Chapter 3.1), respectively, then Cba can be

considered as representing the orientation (or attitude) of the body with respect to inertial

space. The rotation matrix has nine entries, and so at first it would appear that there are

nine independent variables needed to describe the rotation. However, Euler’s Theorem

of Rotations says that any rotation can be described by defining an axis represented by

the unit vector a−→, about which the body is rotated, and an angle φ ∈ R, denoting the

degree of rotation [26]. These two variables can be combined into a single vector, ν−→,

where

ν−→ = φ a−→.

Thus, any rotation can be described by the vector ν−→, where its magnitude is the angle of

rotation, and its unit vector is the rotation axis. In three-dimensional space a minimum

representation of Cba would require only three parameters.

Many rotation matrix parameterizations exist, and choosing a suitable one depends

on the context of the problem. Some of the more common parameterization used in

spacecraft kinematics are Euler angles, Rodrigues parameters, and quaternions [26].

2.2.1 Euler Angles

An intuitive parameterization of the rotation matrix are Euler angles. First, it is noted

that SO(3) is closed under matrix multiplication, and that successive rotations can be

obtained simply by multiplying rotation matrices together. Furthermore, any rotation

can be decomposed into three successive principle rotations about a principal axes. A

principle rotation matrix, Ci (θ), denotes a rotation of θ radians about the i-th axis. Note


that

Ci (θ) =

1 0 0

0 cos(θ) sin(θ)

0 − sin(θ) cos(θ)

describes a rotation of θ radians about the i−→ axis, where i−→ = span

[

1 0 0]T

.

In this way an ijk-Euler angle, denoted by θ =[

θ1 θ2 θ3

]T

, represents the rotation

matrix

Cba (θ) = Ck (θ1)Cj (θ2)Ci (θ3) .

For example, letting 1−→, 2−→, 3−→ equal i−→, j−→

, k−→ respectively, a 321-Euler angle θ would

represent the following sequence of rotations:

1. A rotation of θ3 about the original (i.e., F−→a) 3−→ axis.

2. A rotation of θ2 about the intermediate 2−→ axis.

3. A rotation of θ1 about the transformed (i.e., F−→b) 1−→ axis.

The resulting rotation matrix would then look like

Cba (θ) = C1 (θ1)C2 (θ2)C3 (θ3)

=

c2c3 c2s3 −s2

s1s2c3 − c1s3 s1s2s3 + c1c3 s1c2

c1s2c3 + s1s3 c1s2s3 − s1c3 c1c2

,

where si = sin(θi), and ci = cos(θi). It is important to note that this particular Euler

angle representation has a singularity when θ2 =π2. When this occurs, θ1 and θ3 describe

the same degree of freedom, making it impossible for them to be uniquely determined.

Singularities like this occur with all Euler angle sequences, as well as with most other

parameterizations such as the Rodrigues parameters. Choice of Euler angle sequence

then becomes very important depending on the application, and care must be taken to

ensure that the system avoids those singularities.

2.2.2 Quaternions

A singularity-free parameterization of the rotation matrix can be obtained when using

quaternions (also called Euler parameters). This representation relies on the axis-angle


formulation (a, φ) of a rotation suggested by Euler’s Theorem. The variables ǫ = sin(φ2

)a

and η = cos(φ2

)form the Euler parameters, and when grouped together as q =

[

ǫT η

]T

they follow the mathematics of a unit quaternion. This parameterization uses four pa-

rameters, but they are not all independent as it is evident that ǫTǫ+ η2 = 1.

It can be shown that the rotation matrix based on a−→

and φ is given by [26]

Cba = cos(φ)1+ (1− cos(φ))aaT − sin(φ)a×. (2.1)

Using the definitions of ǫ and η, this equation then becomes

Cba = (η2 − ǫTǫ)1+ 2ǫǫT − 2ηǫ×

=

1− 2(ǫ22 + ǫ23) 2(ǫ1ǫ2 + ǫ3η) 2(ǫ1ǫ3 − ǫ2η)

2(ǫ2ǫ1 − ǫ3η) 1− 2(ǫ21 + ǫ23) 2(ǫ2ǫ3 + ǫ1η)

2(ǫ3ǫ1 + ǫ2η) 2(ǫ3ǫ2 − ǫ1η) 1− 2(ǫ21 + ǫ22)

. (2.2)

Although four parameters are used instead of the minimum (three), Eq. (2.2) shows

clearly that no representation singularity exists, and the parameters ǫ and η can uniquely

define any arbitrary orientation in R3.

2.3 Stability Definitions

The objective of most control problems is to provide inputs to a system in order to make

it “stable”. Any hope of realizing this objective requires a clear mathematical definition

of “stability”. Several concepts of stability exist; some that focus on how the system

responds to initial conditions, and others that deal with how the outputs of the system

are affected by the inputs. The following sections describe some of the main notions of

stability.

2.3.1 Lyapunov Stability

Lyapunov stability deals with how an unforced system responds to initial conditions.

It defines stability in terms of the behaviour of the system as it is perturbed from an

equilibrium. Consider the system

x = f(x, t), (2.3)


where x(t) ∈ Rn, t ∈ R

+, and f : Rn × R+ → R

n. The state x0 is an equilibrium of

Eq. (2.3) if f(x0, t) = 0. This equilibrium is said to be stable (or L-stable) if for any

ǫ > 0, there exists a δ > 0 such that

‖x(0)− x0‖ < δ ⇒ ‖x(t)− x0‖ < ǫ .

Furthermore, the equilibrium x0 is said to asymptotically stable if it is stable and there

exists a δ > 0 such that

‖x(0)− x0‖ < δ ⇒ x(t) → x0 as t→ ∞.

The equilibrium is globally asymptotically stable if it is stable and for any δ > 0, x(t) → x0

as t→ ∞. Finally, the equilibrium is considered unstable if it is not stable.

2.3.2 Input-Output Stability

Input-output (I/O) stability aims at describing how the inputs to a system affect the

outputs. There are many different classes of inputs that can potentially be applied to a

system, so for the purposes of this thesis admissible inputs will be required to belong to

the L2-space of functions defined by

L2 = u(t) | ‖u(t)‖2 <∞ ,

where ‖u(t)‖2 is the 2-norm of a vector function of time and is given as

‖u(t)‖2 =

√∫

∞

0

uT(t)u(t) dt.

Functions belonging to L2 can be thought of as having “finite energy”. In describing

I/O-stability it will be useful to define the truncation of a function as

uT (t) =

u(t) if t ≤ T

0 if t > T.


This gives rise to the idea of an extended L2-space denoted by

L2e = u(t) | uT (t) ∈ L2, 0 < T <∞ .

A “system” can mathematically be described as a mapping (or operator) G : L2e → L2e

which maps the input functions u(t) to its output functions y(t). This mapping operation

can be written as y = Gu, and described in block diagram form as seen in Fig. 2.2. The

u yG

Figure 2.2: Block diagram of a system operator.

system operator G is said to be input-output stable, or L2-stable, if

u ∈ L2 ⇒ y ∈ L2,

which can be interpreted as: if a finite energy input acts on the system, the corresponding

output also has finite energy. It is important to note that it can be shown that for any

function v ∈ L2, if v ∈ L2 then limt→∞v(t) = 0. This result links the concepts of

input-output stability with Lyapunov stability.

2.4 Floquet Stability Theory

Floquet theory provides a convenient way to determine the stability of a linear periodic

system of the form

x = A(t)x, (2.4)

where A(t + T ) = A(t). This section provides a summary of this theory as described

in [26] and [27].

Letting Φ(t, t0) represent the principal fundamental matrix solution of Eq. (2.4), it is

clear that

Φ(t, t0) = A(t)Φ(t, t0),

and it can be shown that the columns of Φ are linearly independent, thereby forming a


basis for the solution space of Eq. (2.4). More specifically, letting

Φ(t, 0) =[

ϕ1(t) ϕ2(t) · · · ϕn(t)]

,

where

ϕ1(t) = A(t)ϕ1(t), ϕ1(0) =[

1 0 · · · 0]T

ϕ2(t) = A(t)ϕ2(t), ϕ2(0) =[

0 1 · · · 0]T

......

ϕn(t) = A(t)ϕn(t), ϕn(0) =[

0 0 · · · 1]T

it is possible to form the solutions x(t) as a linear combination of the columns (ϕi) of

the principal matrix solution. This means that x(t) can be expressed as

x(t) = Φ(t, 0)x(0).

Since A(t) is T -periodic, Φ(t0 + T ) = Φ(t0). Therefore, after one orbit (from t = 0 to

t = T )

x(T ) = Φ(T, 0)x(0),

and after a second orbit (from t = T to t = 2T )the solution becomes

x(2T ) = Φ(2T, T )x(T )

= Φ(2T, T )Φ(T, 0)x(0)

= Φ(T, 0)Φ(T, 0)x(0)

= Φ2(T, 0)x(0),

since Φ(2T, T ) simply denotes the solution at t = 2T , given the initial condition at t = T ,

which is just Φ(T, 0) due to the periodicity of Φ. It is clear that x(kT ) = Φk(T, 0)x(0),

and so the fundamental matrix Φ(T, 0) embodies the stability properties of the entire

system. Floquet theory states that if

max (|λi Φ(T, 0)|) < 1,


where λi is an eigenvalue of Φ(T, 0), then Eq. (2.4) is asymptotically stable.

2.5 Lyapunov Stability Theory

Lyapunov stability theory deals with describing the stability of “unforced” systems of

the form

x = f(x, t). (2.5)

It also provides a means of inferring the stability of an equilibrium of a nonlinear system

by looking at the stability of the corresponding linear system. This section summarizes

the Lyapunov theory discussed in [26] and [28].

2.5.1 Direct Method

This method provides a means of determining the stability of a system by examining

the existence of a particular function V (x, t), called the Lyapunov function, that has

certain characteristics. This Lyapunov function is akin to a storage function. Consider

the system described by Eq. (2.5), with x(0) = x0 as an equilibrium (i.e., f(x0) = 0).

Also, it is assumed that x0 = 0 (which can be achieved via a coordinate transformation).

The equililibrium x0 of Eq. (2.5) is

• stable if there exists a C1 locally positive definite function V (x, t) such that V (x, t)

is locally negative semi-definite for all t ≥ 0;

• asymptotically stable if there exists a C1 locally positive definite function V (x, t)

such that V (x, t) is locally negative definite for all t ≥ 0;

• globally asymptotically stable if there exists a C1, positive definite, and radially

unbounded function V (x, t) such that V (x, t) is negative definite for all t ≥ 0.

In the conditions above, the time derivative of V are taken to be along the trajectories

of Eq. (2.5). That is,

V (x, t) =∂V

∂t+∂V

∂xTf(x, t).

It is important to note that this method only gives sufficient conditions for stability

meaning that even if a Lyapunov function doesn’t exist for the system, it may still be

stable.


2.5.2 Indirect Method

This method provides a way to determine (albeit in a limited manner) the stability of an

equilibrium of a nonlinear system by looking at the stability of its corresponding linear

system. Consider Eq. (2.5), also with x(0) = x0 as an equilibrium. Letting x = x0 + δx

and neglecting higher order terms in the corresponding Taylor expansion of Eq. (2.5),

the linearized system can be written as

δx = Aδx, (2.6)

where

A =∂f∂xT

∣∣∣∣x=x0

is the Jacobian of the nonlinear system about the equilibrium x0. Lyapunov’s Indirect

Method states that if the equilibrium δx = 0 of Eq. (2.6) is

• asymptotically stable then x0 is a locally asymptotically stable equilibrium of the

nonlinear system;

• unstable then x0 is an unstable equilibrium of the nonlinear system;

• stable then no conclusion can be inferred regarding the nonlinear system.

2.6 Passivity Theory

Passivity theory focuses on describing relationships between the inputs and outputs of

a given system. Thus, it provides a means of controller design ensuring input-output

stability. This section provides a summary of passivity theory as given in [28] and [29].

Consider a square system denoted by

y = Gu,

where u ∈ R, y ∈ R, and G : L2e → L2e. Note that, as in Chapter 2.3.2, G is considered

to be an operator that maps the system’s inputs, u, to its outputs, y. The operator G is

said to be passive if, for any u ∈ L2e, and for any T ≥ 0,

∫ T

0

uT (Gu) dt =∫ T

0

yTu dt ≥ 0.


Furthermore, the operator G is said to be strictly passive if, for any u ∈ L2e, and for any

T ≥ 0, there exists ǫ > 0 such that

∫ T

0

uT (Gu) dt =∫ T

0

yTu dt ≥ ǫ

∫ T

0

uTu dt.

The passivity theorem describes the input-output stability of the interconnection between

a passive and strictly passive system. Consider the feedback system given in Fig. 2.3.

The passivity theorem states that if G is passive and H is strictly passive, then the

+

−

u y1

y2

G

H

e

Figure 2.3: A generic, closed-loop feedback system.

feedback system is L2-stable. This implies that if the passivity theorem holds for this

system, then u ∈ L2 ⇒ y1 ∈ L2. See [29] for more detailed information regarding the

passivity theorem.

Chapter 3

Spacecraft Mechanics

Knowledge of how a system moves and responds under the influence of external forces

may give insight into controller design. This chapter aims to describe the mechanics of

a generic spacecraft system.

3.1 Reference Frames

There are two types of reference frames that are particularly useful in the context of

spacecraft attitude control: an inertial one and a body-fixed one. The inertial (non-

rotating) reference frame will be denoted by F−→i =[

i−→1i−→2

i−→3

]T

and is taken to be

the geocentric equatorial coordinate system. The origin of this frame is located at the

centre of the earth. The basis vector i−→1points in the direction of the vernal equinox

(also called Aries, or à), and i−→3points towards the geographical north pole. The

remaining basis vector lies in the equatorial plane with i−→1, such that it completes the

right-handed coordinate system. This particular inertial reference frame coincides with

the one that is typically used to describe the mechanics of a spacecraft orbiting the earth.

The second reference frame of interest is an orthogonal and right-handed one attached

to the spacecraft body itself, with the origin located at the body’s centre of mass. It is

denoted by F−→b =[

b−→1

b−→2

b−→3

]T

, and its relationship to the inertial frame is shown in

Fig. 3.1 (note that the vector r−→ in Fig. 3.1 describes the position of the centre of mass

of the spacecraft with respect to the centre of mass of the earth).

With these choices of reference frames the attitude of the spacecraft can then be rep-

resented by the rotation matrix Cbi, as described in Chapter 2.2. Also, it is important to

note that in subsequent chapters (particularly Chapters 3.3 and 3.4), the time derivative

21

Chapter 3. Spacecraft Mechanics 22

i−→1

i−→3

i−→2

à

b−→3

b−→2

b−→1

Earth

Spacecraft

Equator

North Pole

r−→

Figure 3.1: The inertial and body-fixed frames.

as seen in F−→i will be denoted by (·) and that as seen in F−→b by ().

3.2 Orbital Mechanics

Since the geomagnetic field vector experienced by a spacecraft depends on the spacecraft’s

location with respect to the centre of the earth (see Chapter 4.1), it is necessary, at least

for simulation purposes, to be able to calculate orbital positions as a function of time. To

this end, a brief summary of orbital dynamics will be presented (based on the material

in [30]).

The equations of motion describing the orbit shape can be found by examining the

two-body problem of classical mechanics. Taking into account that, of the two-bodies in

question (the earth and the spacecraft), the earth’s mass is considered to be much larger

than that of the spacecraft, the earth’s approximate position remains at one of the focal

points of the resulting orbit shape. It can also be shown that the angular momentum

per unit mass, h−→, of the system is constant. Since h−→ = r−→× r−→, the motion is confined

to a plane perpendicular to the angular momentum. Note that r−→ refers to the position

of the centre of mass of the spacecraft with respect to the centre of mass of the earth

(as in Fig. 3.1). With this in mind, the reference frames in Fig. 3.2 introduce a way of

describing r−→ in polar coordinates. The equations of motion then reduce to


Spacecraft

Earth

Orbit

θ

θ

1−→1

2−→1

1−→2

2−→2

r−→

Figure 3.2: Polar coordinate system for describing the position of the spacecraft with respectto the earth.

r =p

1 + e cos(θ), (3.1)

where r =∥∥∥ r−→

∥∥∥ and θ define the orbit position, p is the semilatus rectum, and e is

the eccentricity. The parameters p and e are measures of the angular momentum of

the system. Eq. (3.1) is just the polar equation of a conic section. In this thesis, only

Keplerian orbits will be considered so that 0 ≤ e < 1 and the resulting conic shape

describing the orbit is either an ellipse or a circle (e = 0). The geometry of an ellipse

is given in Fig. 3.3, where a and b are the semimajor and semiminor axes, respectively,

and F is the other focal point. It is clear from Fig. 3.3 that the minimum value of r, rp,

Spacecraft

EarthF

b

b

ae ae θ

a(1− e)

pr

a a

Figure 3.3: The geometry of an ellipse.

occurs at periapsis when θ = 0, and the maximum, ra, occurs at apoapsis when θ = π.


Therefore,

rp =p

1 + e

ra =p

1− e

and a relationship between the semimajor axis and the semilatus rectum can be estab-

lished by noting that

a =rp + ra

2

=p

1− e2. (3.2)

The significance of this relationship is mentioned in Chapter 3.2.2.

3.2.1 Orbit Orientation

The polar coordinate representation of the spacecraft’s orbit (Eq. 3.1) is useful because

it effectively reduces the three-dimensional problem to a planar one. However, the ob-

jective is to describe the position of the spacecraft with respect to the geocentric iner-

tial reference frame F−→i described in Chapter 3.1. Consider an orbital reference frame,

F−→o =[

o−→1

o−→2

o−→3

]T

, that is defined such that o−→1

points in the direction of periapsis,

o−→3

points in the direction of the angular momentum h−→, and whose origin coincides with

F−→i. Then it is clear that o−→1

and o−→2

are the same as 1−→1and 1−→2

in Fig. 3.2. Thus,

the orbit lies on the plane spanned by those two basis vectors and the polar coordinates

can be used. The orientation of the orbit with respect to the inertial frame can then

be described by a rotation matrix, Coi. Fig. 3.4 illustrates the orbital reference frame,

where the angles Ω, i, and ω specify its orientation with respect to F−→i, and n−→ (the line

of nodes) points towards the ascending node (the point where the spacecraft crosses the

equatorial plane from south to north). The parameter Ω ∈ [0, 2π) is called the right

ascension of the ascending node, and it is the angle between i−→1and n−→. The parameter

i ∈ [0, π] is called the inclination and it is the angle from the equatorial plane to the

orbital plane (which is equal to the angle between o−→3

and i−→3). The third parameter,

ω ∈ [0, 2π), is called the argument of periapsis and it is the angle between the line of

nodes ( n−→) and the periapsis direction ( o−→1

). These parameters are chosen such that they

form a 313-Euler angle, whose rotation matrix, Coi is easily obtained (see Chapter 2.2.1


i−→3

Spacecraft

Earth

i−→1

, à

n−→

i−→2

o−→1

Periapsis Directionr−→

h−→

, o−→3

Equator

i

θ

ω

Ω

Figure 3.4: The orbital and inertial reference frames.

for a description of Euler angles) and is given by

Coi = C3(ω)C1(i)C3(Ω)

=

cΩcω − sωcisΩ sΩcω + cisωcΩ sisω

−cΩsω − cωcisΩ −sΩsω + cωcicΩ sicω

sΩsi −sicΩ ci

. (3.3)

Once the orbital position of the spacecraft is known in F−→o, the inertial position can be

obtained by multiplying it by Cio = C−1oi = CT

oi (where the equalities follow from the

orthogonality of a rotation matrix).

3.2.2 Orbital Elements

It is clear that the parameters a and e fully specify the size and shape of the elliptical

orbit. The parameters Ω, i, and ω specify the orientation and location of the orbit with

respect to inertial space. One last parameter needs to be defined in order to temporally

locate an initial reference position of the spacecraft within the orbit. The time of periapsis

passage, t0 is typically used.

The six parameters a, e, t0, Ω, i, and ω are called the classical orbital elements and

together they are able to fully specify the position ( r−→) and velocity ( r−→) of the spacecraft


in F−→i at any point in time. From Fig. 3.4,

r−→ = F−→oTr =

r cos(θ)

r sin(θ)

0

.

Taking its derivative, it can be shown that

r−→ = F−→oTr =

√µ

p

− sin(θ)

cos(θ) + e

0

,

where µ comes from Newton’s Law of Gravitation and is equal to 6.67428 × 10−11 ·me

(where me is the mass of the earth in kilograms). Note that θ is actually a function of

time, implying that r−→ and r−→ are as well.

The angular coordinate θ(t), is also known as the true anomaly, and can be obtained

by first solving Kepler’s Equation at time t:

E − e sin(E) =

√µ

a3(t− t0),

for the eccentric anomaly E, and then solving the following relationship

tan

(θ

2

)

=

√

1 + e

1− etan

(E

2

)

.

These equations are based on Kepler’s Laws and their derivation can be found in [30].

Once θ is known, r can be obtained using Eq. (3.1), where p is given by Eq. (3.2), and

so r−→ and r−→ can be calculated in the orbital frame.

The position and velocity in the inertial frame are then obtained via the rotation

matrix Cio in Eq. (3.3) as

r−→(t) = F−→iTCior

v−→(t) = F−→i

TCior .


3.3 Spacecraft Kinematics

An important part of systems control is defining the system’s state variables; that is,

those variables whose motion needs to be controlled. For a spacecraft, the main objec-

tive tends to include being able to point it in a certain direction. In this case, the state

variables desiring control would be parameters describing its orientation (i.e., quater-

nions). Spacecraft kinematics deals with the rate of change of the spacecraft’s attitude

as it orbits around the earth. The orientation of the spacecraft will be characterized

using the quaternion parameterization of the rotation matrix Cbi as described in Chap-

ter 2.2.2. In the following discussions, all variables without a subscript will be taken

to be expressed in the body-fixed frame F−→b. The subscript ‘i’ will be used to denote

variables in the geocentric inertial frame F−→i.

To obtain the kinematic equations of motion, a relationship between the angular

velocity ω−→bi= F−→b

Tω and the inertial time derivative of the rotation matrix is needed.

The two frames can be related to each other by noting that

F−→iT = F−→b

TCbi.

Taking the inertial time derivative one obtains

F−→iT

= F−→bT

Cbi + F−→bTCbi

0−→ = ω−→× F−→bTCbi + F−→b

TCbi

0−→ = ωTF−→b × F−→bTCbi + F−→b

TCbi

0−→ = F−→bT

(

ω×Cbi + Cbi

)

,

which implies that

ω× = −CbiCbi. (3.4)

Substituting Eq. (2.1) into Eq. (3.4) one obtains

ω× = φa× − (1− cos(φ))(a×a

)×

+ sin(φ)a×,

and it follows that

ω = φa− (1− cos(φ))a×a+ sin(φ)a.


After careful algebraic manipulation of this expression (see [26]), it can be shown that

φ = aTω

a =1

2

(

a× − cot(φ

2)a×a×

)

ω.

Using the definition of ǫ and η in Chapter 2.2.2 with the above equation leads to the

following expression for the spacecraft kinematics:

[

ǫ

η

]

=1

2

[

ǫ× + η1

−ǫT

]

ω. (3.5)

3.4 Spacecraft Dynamics

Knowledge of how a system will respond to a given control input is an indispensable tool

to have in controller design. Spacecraft dynamics deals with the relationship between the

forces acting on a spacecraft and the resulting motion. Having a mathematical description

of the dynamics of a spacecraft system allows one to know apriori whether the controller

design will achieve the desired objective, or if those control inputs will cause the system’s

motion to go unstable.

For this thesis, the translational motion of the spacecraft in orbit will be ignored so

that the dynamics reduce to describing the rotational motion only. The next few sections

detail the development of the equations of motion of the spacecraft. The nonlinear equa-

tions will be obtained primarily via Newton’s Second Law. A Lagrangian approach will

be used for the linearized equations. Although this thesis focuses on magnetic attitude

control for flexible spacecraft, comparisons to the rigid case may be of value and so the

development of the equations for a rigid spacecraft are also included.

3.4.1 Rigid Spacecraft

Consider a generic rigid body as shown in Fig. 3.5. The rotational variation of Newton’s

Second Law states that

h−→ = τ−→,

where h−→ is the total angular momentum of the body about its centre of mass, and τ−→ is

the net torque also about the centre of mass. The total angular momentum is given by


F−→b

dm

ρ−→

b−→1

b−→2

b−→3

V

Figure 3.5: A generic rigid body.

h−→ =

∫

V

ρ−→

× ρ−→dm,

and its inertial time derivative is then

h−→ =

∫

V

ρ−→

× ρ−→

+ ρ−→

× ρ−→dm

=

∫

V

ρ−→

× ρ−→dm, (3.6)

since the cross product of a vector and itself is zero. The acceleration ρ−→

can be found

by noting that

ρ−→= 0−→ (by the rigid assumption) and so

ρ−→

=

ρ−→

+ω−→bi× ρ

−→

ρ−→

= ω−→bi× ρ

−→

ρ−→

= ω−→bi×

ρ−→

+

ω−→bi× ρ−→

+ ω−→bi×(

ω−→bi× ρ

−→

)

ρ−→

=


+ ω−→bi×(


)

.

Eq. (3.6) then becomes

h−→ =

∫

V

ρ−→

×(


)

+ ρ−→

×(

ω−→bi×(


))

dm

=

∫

V

− ρ−→

(

× ρ−→×

ω−→bi

)

dm+

∫

V

ρ−→

×(

ω−→bi×(


))

dm,


where the second term in the above equation can be written as

ρ−→

×(

ω−→bi×(


))

= − ρ−→

×(

ω−→bi×(

ρ−→

× ω−→bi

))

= ρ−→

×(

ρ−→

× ω−→bi

)

× ω−→

= −ω−→bi×(

ρ−→

×(

ρ−→

× ω−→bi

))

.

Expressing everything in F−→b, and noting that ω−→bi= F−→b

Tω is independent of the variable

of integration leads to the following equation:

h =

∫

V

(−ρ×ρ×

)dm ω + ω×

∫

V

(−ρ×ρ×

)dm ω.

It can be shown by direct expansion that the term∫

V(−ρ×ρ×) dm = I , where I is the

body’s mass moment of inertia. The dynamics of a rigid spacecraft, expressed in the

body frame, are then given by

I ω + ω×Iω = τ , (3.7)

which is just a statement of Euler’s Equation for rigid-body dynamics. Eqs. (3.5)

and (3.7) fully specify the motion of a rigid spacecraft.

3.4.2 Flexible Spacecraft

The equations of motion for a flexible spacecraft are obtained in a similary manner as

in Chapter 3.4.1. Consider a generic unconstrained flexible body where the undeformed

position of a mass element dm and its deformation are denoted in the body frame by ρ and

ue(ρ, t) respectively, as seen in Fig. 3.6. Assuming a Ritz expansion for the deformation

field, the deformation at a given point on the body can be described in F−→b as

ue(ρ, t) =N∑

i=1

ψi(ρ)qei(t), (3.8)

where qei are the generalized deformation degrees of freedom; and ψi are basis functions

that describe the deformation field while satisfying the following boundary conditions:

ψi(0) = 0

∇×ψi(0) = 0.


F−→b dm

ρ−→

u−→e

r−→

b−→3

b−→1

b−→2

Figure 3.6: A generic unconstrained flexible body.

The total acceleration of a differential mass element, r−→, as expressed in F−→b, can be

derived using a similar process as in Chapter 3.4.1:

r−→ = F−→bT (ρ+ ue)

r−→ = F−→bT(ω×ρ+ ue + ω

×ue

)

r−→ = F−→bT(ω×ω×ρ+ ω×ρ+ ue + 2ω×ue + ω

×ω×ue + ω×ue

).

Using Newton’s second law, summing over the entire body, and premultiplying the ac-

celeration by a moment arm it is clear that

∫

V

ρ−→

× r−→ dm = τ−→,

where τ−→ is then the net torque. Expanding this expression and using the Ritz approx-

imation for ue, the net torque acting on the spacecraft is governed (as seen in the body

frame) by

τ = I ω +

[N∑

i=1

∫

V

−ρ×ψ×

i dm qei

]

ω + ω×Iω

+ ω×

[N∑

i=1

∫

V

−ρ×ψ×

i dm qei

]

ω + 2

[N∑

i=1

∫

V

−ρ×ψ×

i dm qei

]

ω

+

[N∑

i=1

∫

V

ρ×ψidm qei

]

, (3.9)


where once again I =∫

V−ρ×ρ×dm is the total mass moment of inertia of the body.

Similarily, if the shape functions are projected onto r−→ (which is equivalent to the force

per unit mass) one obtains ∫

V

ψT

i a dm = fe,

where a is the components of r−→ as expressed in F−→b, and fe =[

fe1 · · · feN

]T

is the force

per unit mass projected onto the shape functions. After expanding as above and adding

the contribution to fe due to the structural and damping forces one obtains

fei = ωT

[∫

V

ρ×ψ×

i dm

]

ω +

[∫

V

−ψT

i ρ×dm

]

ω +N∑

j=1

∫

V

ψT

i ψjdm qej

+ 2

[N∑

j=1

∫

V

−ψT

i ψ×

j dm qej

]

ω + ωT

[N∑

j=1

∫

V

ψ×

i ψ×

j dm qej

]

ω

+

[N∑

j=1

∫

V

−ψT

i ψ×

j dm qej

]

ω +N∑

j=1

(K ee)ij qej +N∑

j=1

(Dee)ij qej. (3.10)

The external forces fei are equal to zero for this particular system, since there is phys-

ically no actuator on the spacecraft to provide it. The stiffness term, K ee, comes from

including the resultant force on the system due to the deformation of the flexible body

(see Chapter 3.4.3). Chapter 3.4.4 shows how the damping term, Dee, is obtained. Com-

bined, Eqs. (3.9) and (3.10) represent the complete nonlinear attitude dynamics of the

flexible spacecraft, as expressed in the body frame.

3.4.3 Linearized Equations of Motion

Although the spacecraft dynamics are nonlinear, it is often the case where the system’s

state variables are small enough that the dynamics can be effectively approximated as

linear. A simple rate feedback such as B-control can be used to slow the spacecraft’s

motion down enough so it may be approximated by a linear system [28]. The complexity

of designing and proving the stability of a controller can be significantly reduced when

dealing with a linear system. Also, Lyapunov’s Indirect Method (see Chapter 2.5.2) states

that if the linearized system is asymptotically stable in the sense of Lyapunov, then so

is the nonlinear system (although not necessarily globally) [28]. Designing a controller

based on the linearized system can therefore be advantageous. Thus, a description of

the linearized spacecraft dynamics may be useful. A Lagrangian approach will be used


to obtain the linearized equations of motion. Without loss of generality, the nonlinear

system will be linearized about the operating point/equilibrium ǫ = 0, η = 1, ω = 0,

and qe = 0.

For the rigid spacecraft, it is noted that the source of nonlinearity is due to the second

term on the left-hand side of Eq. (3.7). Parameterizing the spacecraft attitude using an

Euler sequence (described in Chapter 2.2.1), the angular velocity, ω, is related to the

Euler angles, θ, via the relation

ω = S(θ)θ,

where the matrix S depends on the Euler sequence chosen; for a 321-Euler sequence it is

given by

S(θ) =

1 0 − sin(θ2)

0 cos(θ1) sin(θ1) cos(θ2)

0 − sin(θ1) cos(θ1) cos(θ2)

.

It is clear that for small angles (θi << 1) S ≈ 1, where 1 is the identity matrix, and so

ω ≈ θ. Substituting this into Eq. (3.7) and assuming that the angular rates are small as

well (θi << 1), the second-order terms become very small and can be neglected leading

to the heavily linearized equations of motion for the rigid spacecraft given by

I θ = τ . (3.11)

In terms of the kinematic equations of motion, for small angles θ ≈ 2ǫ.

To obtain the linear equations of motion for the flexible spacecraft Lagrange’s equa-

tions will be used. Recalling Fig. 3.6, the velocity of a differential mass element, r−→ =

F−→bTv, is given in the body frame by

v = −ρ×ω + ue + ω×ue.

Assuming that the elastic deflections and angular rates are small, it is defensable to

neglect the last term in the above equation.


The total kinetic energy, T , of the system is given by

T =1

2

∫

V

vTv dm

=1

2

∫

V

(−ρ×ω + ue

)T (−ρ×ω + ue

)dm

=1

2

∫

V

(−ρ×ω

)T (−ρ×ω

)+(−ρ×ω

)Tue + uT

e

(−ρ×ω

)+ uT

e ue dm

=1

2

∫

V

ωT(−ρ×ρ×

)ω

︸︷︷︸

a

+2ωTρ×ue︸︷︷︸

b

+ uT

e ue︸︷︷︸

c

dm.

Making the same small angle approximation as for the rigid case, ω ≈ θ, and using the

Ritz expansion for ue the terms a, b, and c in the kinetic energy integral can be written

as

a = ωT(−ρ×ρ×

)ω b = ωTρ×ue c = uT

e ue

= θT(−ρ×ρ×

)θ, = θT

(N∑

i

ρ×ψiqei

)

=

(N∑

i

ψT

i qei

)(N∑

j

ψj qej

)

= θT[

ρ×ψ1 · · · ρ×ψN

]

qe = qT

e

ψT

1ψ1 · · · ψT

1ψN

.... . .

...

ψT

Nψ1 · · · ψT

NψN

qe,

= qT

e

[

ρ×ψ1 · · · ρ×ψN

]T

θ,

where qe =[

qe1 · · · qeN

]T

, and N is the number of flexible degrees of freedom used in

the Ritz expansion. Letting

q =[

θT qT

e

]T

,

M rr = I =∫

V

−ρ×ρ× dm,

M re =

∫

V

[

ρ×ψ1 · · · ρ×ψN

]

dm, and

M ee =

∫

V

ψT

1ψ1 · · · ψT

1ψN

.... . .

...

ψT

Nψ1 · · · ψT

NψN

dm


the kinetic energy can be written as

T =1

2qTMq,

where

M = MT =

[

M rr M re

MT

re M ee

]

.

It is important to note that M > 0 (meaning that it is positive definite).

The potential energy, U , is just the strain energy of the elastic body and is given by

U =1

2

∫

V

εTEε dV,

where ε (ue) and E are the strain tensor and elastic modulus, respectively. The strain

tensor can be written as

ε(ue) = ε

(N∑

i=1

ψiqei

)

=N∑

i=1

ε (ψi) qei.

The potential energy then becomes

U =1

2

N∑

i=1

N∑

j=1

∫

V

ε (ψi)T Eε (ψi) dV qeiqej

=1

2qTKq ,

where

K = KT =

[

0 0

0 K ee

]

and

[K ee]ij =

∫

V

εT (ψei)Eε (ψej) dV.

Given the form of K , it is clear that K ≥ 0 (meaning it is positive semidefinite).

By examining the virtual work done by a force f−→e= F−→b

Tfe causing a virtual dis-

placement of the mass element dm, it can shown that the resulting generalized force fw


acting on the system is given by

fw =

[

g

fe

]

,

where g =∫

Vρ×fe dV = τ is the total torque acting on the spacecraft body, and

fei =∫

VψT

i fe dV is the total force projected onto the flexible degrees of freedom (which

is equal to zero for this particular system).

It is assumed that in any real system the flexible degrees of freedom have some

damping associated with them. This damping is added to the system in the form of a

generalized force f−→d= F−→b

Tfd, given by

fd = −Dq,

where

D = DT =

[

0 0

0 Dee

]

,

and Dee is obtained as in Chapter 3.4.4 so as to ensure that only the flexible modes are

being damped. Given the form of D, it is clear that D ≥ 0 as well.

Applying Lagrange’s equations, which are given by

d

dt

(∂L

∂qi

)

−∂L

∂qi= fi,

where L = T − U and f = fw + fd, the linearized equations of motion for the flexible

spacecraft can be written as

Mq + Dq + Kq = fw, (3.12)

where for this thesis fw =[

τT 0T

]T

. The linearized equations could also have been

obtained by letting ω = θ in Eqs. (3.9) and (3.10) and taking the Jacobian around the

operating point θ = θ = qe = qe = 0.

3.4.4 Modal Equations of Motion

Decomposing Eq. (3.12) into modal coordinates allows the dynamics of the system to

be evaluated in terms of its fundamental modes of motion. Most importantly, modal

analysis allows for the rigid and flexible motions to be investigated separately, making

it easier to assess the contribution of the flexibility of the spacecraft to the system’s


dynamics.

Consider the unforced version of Eq. (3.12), which is given by

Mq + Kq = 0. (3.13)

Note that the effect of damping can be considered as an external force acting on the

system. It is assumed that an arbitrary solution to the above equation can be written as

q(t) = qαeλαt,

where the subscript α refers to a particular degree of freedom. Substituting this into

Eq. (3.13) gives(λ2αM + K

)qα = 0, (3.14)

which is simply a statement of the eigenproblem for this particular system. For non-trivial

values of qα, the eigenproblem can be solved by finding the roots of the characteristic

equation

det(λ2αM + K

)= 0.

Premultiplying Eq. (3.14) by qHα (where (·)H refers to the conjugate transpose operation)

gives

λ2α = −qHα Kqα

qHα Mqα

.

Since M and K are both symmetric, the numerator and denominator of the above equation

are guaranteed to be real. Furthermore, since M > 0 and K ≥ 0 it follows that λ2α ≤ 0.

From the form of K (as given in Chapter 3.4.3), it is clear that there will be at least three

eigenvalues (λα) equal to zero, corresponding to the the three rigid degrees of freedom

describing a pure rotation of the entire spacecraft about each axis. The other eigenvalues

will be purely imaginary and can be written as

λα = ±jωα,

where ωα can be interpreted as the fundamental frequency of the α’th-mode of vibration.

It is easily shown that the resulting eigencolumns qα are mutually orthogonal with respect


to both the mass and stiffness matrices. If, for nonzero ωα, they are normalized such that

qT

αMqβ = δαβ,

where δαβ refers to the Kronecker delta, then it immediately follows that

qT

αKqβ = ω2αδαβ.

The eigenmatrix containing the eigencolumns for the flexible degrees of freedom can be

written as

Qe = row qαα=1,...,N .

The eigencolumns corresponding to the rigid modes (where ωα = 0) can also be normal-

ized and put into the following matrix form:

Qr =

[

1

0

]

.

Note that Qe ∈ R(3+N)×N , and Qr ∈ R

(3+N)×3. Letting Q =[

Qr Qe

]

, it is clear that

QTMQ =

[

I 0

0 1

]

QTKQ =

[

0 0

0 Ω2

]

,

where

Ω = diag ωα .

The benefit of this eigendecomposition is that it decouples the system dynamics as follows.

The eigencolumns span the entire solution space of the linearized system, and so the

solutions q(t) can be written as

q(t) = Qrηr(t) +N∑

α

qαηα(t)

= Qη(t),


where η(t) =[

ηr(t)T η1(t) · · · ηN(t)

]T

are the modal coordinates. Inserting this

expansion into Eq. (3.12) and premultiplying by QT leads to the following equation:

QTMQ η + QTDQη + QTKQη = QTfw

The mass and stiffness terms are decoupled, and it remains to construct the damping

matrix D such that it too is decoupled and acts only on the flexible degrees of freedom.

Since Eq. (3.12) represents a second order system, the damping coefficients, cα, for the

flexible degrees of freedom will be written as cα = 2ζωα. The damping matrix in modal

coordinates is then given by

Dmodal =

[

0 0

0 2ζΩ

]

.

The damping matrix in physical coordinates (D) is then obtained by

D =(QT)−1

DmodalQ−1.

Letting η =[

ηT

r ηT

e

]T

, where ηr and ηe =[

η1 · · · ηN

]T

denote the rigid and flexible

modes respectively, the above equations reduce to

I ηr = τ (3.15)

ηα + 2ζωαηα + ω2αηα = qT

eαfw α = 1, . . . , N, (3.16)

where fw =[

τT 0T

]T

, and qeα is the α’th column of Qe. Eqs. (3.15) and (3.16) represent

the modal equations of motion.

3.5 Archetypal Spacecraft/Orbit

For simulation purposes, a typical small satellite with two flexible appendages attached

has been identified with dimensions as shown in Fig. 3.7. The total moment of inertia,

I , of the spacecraft as expressed in the body frame is

I ≈

94.6 0 0

0 29.1 0

0 0 92.0

kg ·m2.


F−→b

Rigid Body

Flexible Arm

Flexible Arm

0.512 m

1.22 m

1.33 m2 m

0.2 m

0.2 m2 m

0.2 m

0.2 m

M = 106 kg

M = 20 kgM = 20 kg

z

y

x

Figure 3.7: A model rigid spacecraft with two flexible beams attached that will be used foranalysis and simulation purposes.

The appendages will be modeled as uniform cantilevered beams, with Young’s modulus

E = 4×105 Pa and damping ratio ζ = 0.1, and will be allowed to deflect about the body

‘x’ and ‘z’ axes only (i.e., no twisting about the ‘y’ axis). Furthermore, the deflections

about either axis are assumed to be decoupled and small enough such that the spacecraft

moment of inertia remains approximately constant. The shape functions, ψei, needed to

characterize the deformation field of the beam deflections have all been chosen to be of

the form

ψei(y) = cosh(kiy)− cos(kiy) + βi(sin(kiy)− sinh(kiy)),

which represent the exact mode shapes of a cantilevered beam [31]. The first three

modes shapes will be used for each deformation degree of freedom. Table 3.1 contains

the values of βi and ki for the first three modes. The resulting vector shape functions,

Table 3.1: Exact mode shape parameters for cantilevered beam [31].

i βi ki · L1 0.7341 1.87512 1.0185 4.69413 0.9992 7.8548Note: L is the length of the beam.

ψi (in Eq. (3.8)), are then given in a local frame by

ψ1=[

ψe1 0 0]T

ψ2=[

ψe2 0 0]T

ψ3=[

ψe3 0 0]T

ψ4=[

0 0 ψe1

]T

ψ5=[

0 0 ψe2

]T

ψ6=[

0 0 ψe3

]T

,


which describes the deformation field of the flexible arm on the right side of the rigid

body in Fig. 3.7, and

ψ7 =[

−ψe1 0 0]T

ψ8 =[

−ψe2 0 0]T

ψ9 =[

−ψe3 0 0]T

ψ10=[

0 0 ψe1

]T

ψ11=[

0 0 ψe2

]T

ψ12=[

0 0 ψe3

]T

,

which describes the deformation field of the left arm.

A sample orbit is also necessary for simulation purposes. The controllability require-

ments in Chapter 4.2 imply that magnetic attitude control is best suited for orbits that

are low-earth and near-polar. Therefore, the example orbit that will be used in all sim-

ulations (unless otherwise specified) is given by the following orbital elements:

a, e,Ω, i,Ω, t0 = Re+ 450 km, 0, 0, 87, 0, 0 s ,

where Re is the mean radius of the earth. These elements specify a circular orbit with

an inclination of 87 (note that a perfectly polar orbit would have i = 90, see Fig. 3.4).

Chapter 4

Magnetic Attitude Control

As mentioned in Chapter 1, the torque produced via an interaction between a dipole

moment and a magnetic field is given (in the body frame) by

τ = m×b, (4.1)

where τ is the control torque, m is the dipole moment produced by the spacecrafts mag-

netic actuators, and b is the local geomagnetic field vector. In the literature, particularly

for the PD-like control laws, the dipole moment is given by

m = b×ν,

where ν is a new control vector. The resulting expression for the control torque becomes

τ = −b×b×ν. (4.2)

The control torque is expressed in this manner since, for those PD-control laws, the

stability proofs require that

Γ = limT→∞

1

T

∫ T

0

−b×b× dt > 0.

In particular, in [13–16] the positive definiteness of Γ is critical in the Lyapunov stability

proofs. Along with [6], this condition is shown to hold for a dipole approximation of the

geomagnetic field as long as the orbital plane did not coincide with the earth’s equatorial

plane. Since the controller development in Chapter 5 is modelled after the analysis in the

42

Chapter 4. Magnetic Attitude Control 43

references above, Eq. (4.2) will be adopted. A secondary reason for expressing the control

torque as in Eq. (4.2) is that the new control vector ν can be thought of as the actual

resulting torque axis (as it takes into account the cross-product). That is, assuming the

desired torque axis lies on the plane perpendicular to the local geomagnetic field vector,

the axis defined by ν yields a dipole moment such that, after applying Eq. (4.2), τ is

parallel to ν.

4.1 Magnetic Field Model

Since magnetic attitude control relies on the earth’s magnetic field, it is important to

have a suitable magnetic field model, especially for simulation purposes. Although several

models exist, the spherical harmonic model outlined in [1] will be used. This model

assumes that the dominant portion of the earth’s magnetic field, which is denoted by b−→

,

can be described by the gradient of a scalar potential function V . That is, the field can

be described by the relation

b−→

= −∇−→V, (4.3)

where V can be represented by a series of spherical harmonics:

V (r, θ, φ) = a

k∑

n=1

(a

r

)n+1n∑

m=0

(gmn cosmφ+ hmn sinmφ)Pmn (θ) ,

where a is the equatorial radius of the Earth, gmn and hmn are the International Geo-

magnetic Reference Field (IGRF) Gaussian coefficients, Pmn are the Schmidt normalized

Legendre functions, k is the highest spherical harmonic to be considered, and r, θ, and

φ are the geocentric distance, coelevation, and East longitude, which define any point in

space. Note that the above equation only applies if r > a. The gaussian coefficients for

the year 2010 will be used and are given in Table 4.1 (for k = 1, . . . , 5). It is important

to note that the local magnetic field is dependent on the orbital position of the space-

craft, which is a function of time. Therefore, the magnetic field vector experienced by

the spacecraft is implicitly a function of time as well. From Eq. (4.3), the magnetic field


Table 4.1: IGRF Coefficients for Epoch 2010 [32].

n m g (nT) h (nT)1 0 -29496.5 n/a1 1 -1585.9 4945.12 0 -2396.6 n/a2 1 3026.0 -2707.72 2 1668.6 -575.43 0 1339.7 n/a3 1 -2326.3 -160.53 2 1231.7 251.73 3 634.2 -536.84 0 912.6 n/a4 1 809.0 286.44 2 166.6 -211.24 3 -357.1 164.44 4 89.7 -309.25 0 -231.1 n/a5 1 357.2 44.75 2 200.3 188.95 3 -141.2 -118.15 4 -163.1 0.15 5 -7.7 100.9

vector can be expressed as

Br =−∂V

∂r=

k∑

n=1

(a

r

)n+2

(n+ 1)n∑

m=0

(gn,m cos(mφ) + hn,m sin(mφ))P n,m(θ)

Bθ =−1

r

∂V

∂θ= −

k∑

n=1

(a

r

)n+2n∑

m=0

(gn,m cos(mφ) + hn,m sin(mφ))∂P n,m(θ)

∂θ

Bφ =−1

r sin(θ)

∂V

∂φ=

−1

sin(θ)

k∑

n=1

(a

r

)n+2n∑

m=0

m (−gn,m cos(mφ) + hn,m sin(mφ))P n,m(θ),

(4.4)

where gn,m, hn,m, and P n,m(θ) are related to gmn , hmn , and P

mn (θ), respectively (see Ap-

pendix A for their calculation). The reference frame in which Eq. (4.4) holds can be


related to the geocentric inertial frame F−→i (from Chapter 3.1 by noting that

b−→

= F−→iT

(Br cos(δ) +Bθ sin(δ)) cos(α)−Bφ sin(α)

(Br cos(δ) + Bθ sin(δ)) sin(α) + Bφ cos(α)

(Br sin(δ)− Bθ cos(δ))

, (4.5)

where α is the right ascension, and δ = π2− θ is the declination of the spacecraft position

in F−→i. Letting r−→ = F−→iTr = F−→i

T

[

r1 r2 r3

]T

be the position of the spacecraft with

respect to the centre of the earth (as in Fig. 3.1), the right ascension and declination are

given by

α = tan−1

(r2

r1

)

δ = tan−1

(

r3√

r21 + r22

)

.

It is important to note that east longitude φ is related to the right ascension α by

φ = α−

(

αG0 +dαG

dtt

)

,

where αG(t) is the right ascension of the Greenwich meridian and αG0 = αG(0). This

term takes into account the rotation of the earth. Also, the coelevation θ is related to

the declination δ by

θ =π

2− δ.

An important aspect of this model is how the magnetic field observed by a space-

craft as it orbits the earth is approximately periodic. Fig. 4.1 shows the magnetic field

vector expressed in F−→i over a few orbits. The 450 km orbit was taken to have i = 87,

with all other orbital elements equal to zero (see Chapter 3.2.2 for a description of the

orbital elements). The almost periodic nature of the magnetic field means that periodic

controllers (like the ones in [9]) can be developed. It also means that stability theories

that rely on the periodic nature of the system dynamics (such as the Floquet theory in

Chapter 2.4) can be used.


0 0.5 1 1.5 2 2.5 3−5

−4

−3

−2

−1

0

1

2

3

4x 10

−5

# of Orbits

Mag

netic

Fie

ld (

T)

b

1

b2

b3

Figure 4.1: An example of the geomagnetic field experienced by a spacecraft in orbit.

4.2 Controllability

When first attempting to solve the control problem for a given system, it is of interest

to know whether or not the system can actually be controlled (i.e., whether or not there

exists a control vector that is able to take the system from one arbitrary state to a

particular desired state). In the case of magnetic attitude control, the control torque

is goverened by the vector cross-product τ−→ = m−→ × b−→, as mentioned in Chapter 4.

Because of the inherent nature of the cross-product the resulting torque must lie on a

plane perpendicular to the local geomagnetic field vector as seen in Fig. 4.2. This plane

is spanned by two degrees of freedom only, and since the orientation of a spacecraft has

three degrees of freedom in Euclidean space, it is evident that at any point in time, the

spacecraft mechanics are instantaneously underactuated and therefore, uncontrollable.

However, since the spacecraft is orbiting around the earth, the local geomagnetic field

vector that acts on the spacecraft changes with time. It was shown in [6] that under

certain conditions, the mechanics of a spacecraft using magnetic actuation can indeed

be controllable in a time-varying sense. This necessarily makes the control problem a

time-varying one.

The analysis in [6] first develops conditions for accessibility and controllability for

general time-varying systems. Those conditions were then considered for the case of the


b−→

m−→

τ−→

= m−→

× b−→

control torque axis limited to this plane

Figure 4.2: The cross-product relationship of the magnetic control torque.

time-varying magentic attitude control problem. It was shown that if bi(t)×bi(t) 6= 0

for any t ∈ R, then the attitude dynamics described by Eqs. (3.5), (3.7), and (4.1) are

strongly accessible. It is also proven that this condition on the magnetic field implies

that

Γdef= lim

T→∞

1

T

∫ T

0

−b×

i b×

i dt > 0,

which, according to [16], is a sufficient condition for stabilizibility. Note that the positive

definiteness of Γ gives motivation to the form of the control torque given in Eq. (4.2). It

is then shown that if, in addition to the conditions above, the magnetic field is periodic

in nature, then the dynamics are controllable. The article goes on to show that the time

variation of a constant dipole approximation of the geomagnetic field along an elliptical

orbit satisfies these conditions as long as the orbital plane is not equatorial.

The magnitude of control torque available depends on how the strength of the dipole

the magnetic actuators can produce, as well as the strength of the geomagnetic field.

Fig. 4.3 compares the geomagnetic field vector along a polar (i = 90) and equatorial

(i = 0) orbit in the inertial frame. It can be seen that closer-to-polar orbits experience

a higher strength geomagnetic field. Also, from Eq. (4.3) it is clear that for the dipole

approximation of the geomagnetic field, the field strength is proportional 1r3

where r is

the geocentric distance of the spacecraft in orbit. Thus, the magnetic field is stronger

closer to the earth. These observations lend further credence to the fact that this type

of control is better suited for near-polar, low-earth orbits.


0 0.5 1 1.5 2 2.5 3−1

0

1

2

3x 10

−5

Mag

netic

Fie

ld (

T)

Equatorial Orbit

b

1

b2

b3

0 0.5 1 1.5 2 2.5 3−5

0

5x 10

−5

# of Orbits

Mag

netic

Fie

ld (

T)

Polar Orbit

b

1

b2

b3

Figure 4.3: A comparison of the geomagnetic field for polar and equatorial orbit.

4.3 Hybrid PD-Like Control Law

The controller development for the flexible system (described in Chapter 3) in this thesis

will be based on proportional-derivative (PD) control due to its simplicity, ease of imple-

mentation, and effectiveness. As mentioned in the literature review, several controllers

based on PD-control have been proposed for a rigid spacecraft. Recall that the equations

of motion for the rigid system (see Chapters 3.3 and 3.4.1) are given (in the body frame)

by

[

ǫ

η

]

=1

2

[

ǫ× + η1

−ǫT

]

ω

I ω + ω×Iω = τ . (4.6)


For this rigid system, [13] proposes the following PD-like control law in order to stabilize

the equilibrium ω = 0,[

ǫT η

]T

=[

0 0 0 1]T

:

τ = −b×b×ν

ν = −I−1(ε2kpǫ+ εkdω

). (4.7)

It is important to note that Eq. (4.7) relies on knowledge of the spacecraft inertia matrix

I , potentially making it not robust to model uncertainty. This control law utilizes both

angular velocity and attitude feedback. The stability proof in [13] places an upper bound,

ε∗, on the controller parameter ε, which limits the effectiveness of this type of control.

This bound is also shown to exist for the similar PD-like controllers proposed in [14–16].

In [13], the closed-loop equations of motion are re-cast (after a coordinate transformation)

into the form

x = εf(x, ε),

and the generalized averaging theory described in [17] is applied, which states that for

ε < ε∗ the trajectories of a non-autonomous system remain close to the trajectories of

the averaged system. The non-autonomous part of Eq. (4.7) is the term −b×b×, and it

is replaced by its average (similar to Γ in Chapter 4.2). The averaged equations then

describe an LTI system, and stability for the averaged system is proven via Lyapunov’s

Direct Method. The existence of ε∗ can be reasoned as ensuring that the response of the

system (determined by x) is much slower than the excitation (given by f), making the

response predominantly determined by the average of the excitation. A more physical

approach can also be taken to explain the existence of ε∗. Consider the energy E of the

linearized version of the above control system, given by

E =1

2θTI θ +

1

2θTKθ,

where K = −ε2kpI−1b×b×I−1. Its time rate of change is given by

E =1

2θTKθ − θTDθ,

where D = −εkdI−1b×b×I−1 ≥ 0. It is clear that since the first term is indefinite, and

the second term is negative semi-definite, the controller parameters must be carefully

chosen such that the rate of change of the energy is not increasing in order to have any


chance for stability. This requirement manifests itself as a bound on ε. It should be

noted that without the attitude feedback, K = 0 and E ≤ 0 is satisfied for any choice of

ε. Thus, it is evident that ε∗ places restrictions on the ratio of proportional to derivative

control. Fig. 4.4 demonstrates the existence of ε∗ for the controller given by Eq. (4.7),

on the nonlinear rigid system described in Chapter 3. The plots on the left show the

trajectories of the system when ε = 0.001, and those on the right show when ε = 0.003,

with initial conditions of ω =[

0.05 0.05 0.05]T

and all else zero.

0 1 2 3−1

0

1

2ε = 0.001

Qua

tern

ions

ǫ1ǫ2ǫ3η

0 1 2 3−0.05

0

0.05

0.1

0.15

# of Orbits

Ang

ular

Vel

ocity

(ra

d/s)

ω1

ω2

ω3

0 1 2 3−1

0

1

2ε = 0.003

ǫ1ǫ2ǫ3η

0 1 2 3−0.05

0

0.05

0.1

0.15

# of Orbits

ω1

ω2

ω3

Figure 4.4: Plots of the trajectories of a system using the control law in Eq. (4.7), demonstrat-ing the existence of ε∗.

In [24], it is shown that the limitation on the control gains under purely magnetic

actuation (for the system shown above) can be removed with the addition of a mini-

mum level of alternate, independent three-axis actuation (i.e., using thrusters or reaction

wheels). In order to stabilize the same equilbirium as above, [24] proposes the following

control law:

τ = −‖bi‖−2 b×b×ν + u,


where

ν = −I−1(2ε2kpǫ+ εkdω

)

is the control effort given by the magnetic actuators and

u = −γI−1(2ε2kpǫ+ εkdω

)

is that given by the alternate three-axis actuation system. The law for the control

vector u includes the parameter ε so that the new control parameter γ can be seen as a

dimensionless scaling that is a kind of measure of the amount of torque provided by the

three-axis actuation over that provided by the magnetic actuators. It is shown in [24]

that if γ >kpλ2

max

k2d

, with ε, kp, and kd all greater than zero, and where λmax is the

largest principal moment of inertia, then the system is asymptotically stable. Thus, with

minimum control effort provided by the alternate three-axis actuator, the system can be

guaranteed to be stable, even for ε > ε∗. The controller design in the following chapter

will attempt to extend the work done in [24] to the flexible spacecraft case.

Chapter 5

Hybrid PD-Control Of Flexible

Spacecraft

Without loss of generality, the control problem is defined to be the stabilization of the

equilibrium ω = 0,[

ǫT η

]T

=[

0 0 0 1]T

. The controller design in this section is

based on the PD-like control methods discussed in Chapter 4.3. It is important to note

that [25] proposes a PD-like control law in order to stabilize the flexible system using

magnetic actuation. The control law takes the form

τ = −‖bi‖−2 b×b×ν,

where

ν = −(ε2kpI−1ǫ+ εkdIω

).

As with the other laws that contained attitude feedback, limitations are placed on the

size of the control gains. Also, since the moment of inertia appears in the control law, it

lacks robustness against model uncertainty. A more robust version that eliminates the

controller’s dependency on the spacecraft inertia can be found (for the rigid case) in [15].

It will serve as the starting point of the controller design for the flexible case.

The nonlinear equations of motion for the flexible system (see Chapter 3) are very

complex to work with, and so the controller development will focus on the linearized

system. After stabilizing the linear system, Lyapunov’s Indirect Method can then be

applied to draw conclusions for the nonlinear case. Also, working with the linearized sys-

tem allows for the application of Floquet analysis, which is a useful tool for investigating

the control gain stability criteria.

52

Chapter 5. Hybrid PD-Control Of Flexible Spacecraft 53

Recall the linearized equations of motion for the flexible spacecraft, which are given

by

Mq + Dq + Kq = fw, (5.1)

where fw =[

τT 0T

]T

(see Chapter 3.4.3 for the definitions of the other variables in

Eq. (5.1)). It is assumed that the spacecraft orbit is such that

Γi = limT→∞

1

T

∫ T

0

Γi dt > 0,

where

Γidef= −‖bi‖

−2 b×

i b×

i ≥ 0.

It is shown in [25] that for sufficiently small angular velocities, if the above holds true

then it also holds when bi is replaced with b, the magnetic field vector in the body frame,

meaning that

Γ = limT→∞

1

T

∫ T

0

Γ dt > 0,

where

Γdef= −‖bi‖

−2 b×b× ≥ 0.

Note that this requirement was shown in [6] to be true for the dipole approximation of

the geomagnetic sphere, as long as the orbital plane was not equatorial. Also, b = Cbibi

which for small angles Cbi = 1 − θ×. With the linear assumption, the −θ× term leads

to products of small angles and rates, meaning that b can, and will, be approximated by

bi.

Consider the purely magnetic control law given by

τ = Γν

ν = −(2ε2kpǫ+ εkdω

),

which, after linearizing (i.e., letting θ ≈ 2ǫ, θ ≈ ω) becomes

τ = −Γ(

ε2kpθ + εkdθ)

= −Γc(ε2kpq + εkdq

), (5.2)

where c =[

1 0T

]

, and q =[

θT qT

e

]T

. Substituting the above into Eq. (5.1), the


closed-loop equations of motion become

M clq + Dclq + K clq = 0, (5.3)

where

M cl = M > 0, Dcl =

[

εkdΓ 0

0T Dee

]

≥ 0, and K cl =

[

ε2kpΓ 0

0T K ee

]

≥ 0.

Since b varies with time along an orbit, so does Γ. This means that Eq. (5.3) describes a

linear, time-varying system. In order to apply the generalized averaging theory described

in [17] (as is done in [13]), the following coordinate transformation is introduced:

x1 = q, x2 =1

εq.

Eq. (5.3) can then be written in canonical form as

[

x1

x2

]

= ε

[

0 1

−M−1cl K cl −M−1

cl Dcl

][

x1

x2

]

︸︷︷︸

f

x = εf (x, t, ε) , (5.4)

where

K cl =

[

kpΓ 0

0 1ε2

K ee

]

, and Dcl =

[

kdΓ 0

0 1εDee

]

.

Due to the periodic nature of the geomagnetic field, Eq. (5.4) is continuous and bounded

and matches all the criteria to apply the averaging theory in [17]. Therefore, there exists

an ε∗ such that, for 0 < ε < ε∗, the trajectories of Eq. (5.4) are sufficiently close to

the trajectories of its autonomous version, which is obtained by replacing Γ by Γ. The

averaged version of Eq. (5.3) is given by

M clq + Dclq + K clq = 0, (5.5)

where

Dcl =

[

εkdΓ 0

0 Dee

]

> 0, and K cl =

[

ε2kpΓ 0

0 K ee

]

> 0.


The positive semidefiniteness of Dcl and K cl where due to the fact that Γ ≥ 0. However,

since Γ > 0, Dcl and K cl are positive definite as well.

The stability of the averaged system can be shown via Lyapunov’s Direct Method.

Consider the candidate Lyapunov function given by

V =1

2qTM clq + qTK clq > 0.

The function V is clearly positive definite and furthermore, it is radially unbounded. Its

derivative is given by

V = −qTDclq ≤ 0.

Although V is negative semidefinite with respect to the system state[

qT qT

]T

, it is

evident that q = 0 ⇒ q = 0 ⇒ K clq = 0 ⇒ q = 0 and so LaSalle’s theorem can be

applied. Therefore, the averaged, autonomous, closed-loop system described by Eq. (5.5)

is asymptotically stable. This implies that for ε < ε∗, the non-autonomous, closed-loop

system given by Eq. (5.3) is also asymptotically stable about the equilibrium q = q = 0,

since its trajectories remain close to those of the autonomous system.

As is done in [24], an independent control vector u will be added representing the

torque caused by an alternate three-axis actuator such as reaction wheels or thrusters.

The purpose of adding another control actuator is to alleviate the gain restriction of

purely magnetic attitude control. The control vector fw (from Eq. (5.1)) can then be

written as

fw =

[

τ + u

0

]

,

where τ is the magnetic control given by Eq. (5.2), and

u = −γ(2ε2kpǫ+ εkdω

).

In this control law, ε is included so that the additional control parameter γ acts as a

dimensionless scaling of the amount of non-magnetic control. This further emphasizes

the fact that u is meant to augment the magnetic control; it is not the primary method

of actuation. After linearizing and manipulation, the final hybrid control law is given by

fw = −B(ε2kpq + εkdq

), (5.6)


where

B =

[

(γ1+ Γ) 0

0 0

]

.

It should be noted that for purely non-magnetic control (i.e., τ = 0), the closed-loop

equations of motion would be autonomous, and the stiffness and damping matrices would

be positive definite. Asymptotic stability could be shown by choosing a Lyapunov func-

tion identical to V above. The following sections investigate the gain criteria achieving

stability with a minimum contribution from u (measured by the control parameter γ).

5.1 Floquet Stability Analysis

The Floquet theory described in Chapter 2.4 was applied to the baseline sample spacecraft

outlined in Chapter 3.5. Since Floquet theory is a linear analysis, the linearized equations

of motion were used. The dynamics are summarized by

Mq + Dq + Kq = fw,

where fw represents the hybrid control law given as

fw = −B(ε2kpq + εkdq

).

From the equations above, the system matrix A(t) corresponding to the one in Eq. 2.4

in Chapter 2.4 can be written as

A(t) =

[

0 1

−M−1 (K + ε2kpB) −M−1 (D + εkdB)

]

,

with the system state variable being x =[

qT qT

]T

. The Floquet stability analysis was

used to investigate the values of ε and γ that would yield a stable system. For each

ε and γ pair, the theory was applied and the stability of the system was investigated.

Fig. 5.1 represents a stability matrix illustrating the results of the analysis. The control

parameter ε was varied from 0.0001 to 0.02, and γ was varied from 0 to 0.0035 (both

using a uniformally spaced mesh). Note that kp = 25000 kg·m2

s2and kd = 25000 kg·m2

sfor

all graphs unless otherwise specified. In the diagram, a “∗” signifies that the system was

found to be asymptotically stable whereas a “” denotes an unstable system.


0 0.5 1 1.5 2 2.5 3 3.5

x 10−3

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

γ

ε

Figure 5.1: Stability diagram obtained using Floquet theory for the baseline sample spacecraft(∗ = L-stable, = unstable).

Fig. 5.1 clearly confirms the existence of an ε∗ since when looking at the vertical

axis (γ = 0), it is evident that above a certain value of ε the system goes unstable and

remains unstable. Based on the diagram ε∗ ≈ 0.0006853 but if a finer mesh is adopted,

Floquet theory predicted that ε∗ ≈ 0.0002. In general, Fig. 5.1 shows that as ε increases,

less control effort from the non-magnetic actuation system is needed. Typically with the

PD control of LTI systems, larger gains equates to better performance. Fig. 5.2 shows a

comparison of the simulated response of the spacecraft under purely magnetic actuation

for ε = 0.0001 and ε = 0.0002, with an initial condition of θ =[

0.001 0.001 0.001]T

rads

(all else zero). It is clear that for ε = 0.0002 the spacecraft motion eventually goes

unstable, and the system’s states start to grow without bound. Note that for each of the

simulations the stability of the flexible degrees of freedom (qe) matched that of the rigid

degrees of freedom and are omitted from the plot.

An interesting aspect of Fig. 5.1 is the region roughly spanned by 0 < γ < 0.0002

and 0.012 < ε < 0.017. This region is plotted in Fig. 5.3. It can be seen that there is

a curious spur in the stability map where, as γ is increased, the system’s stability goes


0 5 10 15−1

−0.5

0

0.5ǫ = 0.0001

Eul

er A

ngle

s (r

ad)

θ1

θ2

θ3

0 5 10 15−1

−0.5

0

0.5

1x 10

−3

# of Orbits

Ang

ular

Vel

ocity

(ra

d/s)

θ1

θ2

θ3

0 5 10 15−4000

−2000

0

2000ǫ = 0.0002

Eul

er A

ngle

s (r

ad)

θ1

θ2

θ3

0 5 10 15−2

0

2

4

# of Orbits

Ang

ular

Vel

ocity

(ra

d/s)

θ1

θ2

θ3

Figure 5.2: Simulated spacecraft motion for ε = 0.0001 and ε = 0.0002, demonstrating thegain limitations when using purely magnetic control.

from unstable to stable to unstable to stable. Causes for this curiosity are not quite

clear. Simulating the system at the boundaries of the spur confirmed the results of the

Floquet analysis, suggesting that it is not merely an artifact of the stability theory. A

similar spur was found to exist in [33], where the stability properties (using the hybrid

control law in [24]) of a spacecraft with different inertia properties was investigated.

The stability maps for the equivalent rigid spacecraft are given in Appendix C. In

comparing the plots it is evident that they are the same; the stability predictions for

the flexible case were identical to those for the rigid case. This would suggest that the

flexibility of the spacecraft (as modeled) does not affect its stability properties, at least in

the linear analysis performed here. Even when simulating both rigid and flexible space-

craft under the same conditions, a miniscule difference can be seen in their trajectories.

Fig. 5.4 shows a plot of the contribution of the flexible modes to the rigid ones (given

by Qreη) for purely magnetic control with ε = 0.0001. The modal equations of motion


0 1 2

x 10−4

0.012

0.0125

0.013

0.0135

0.014

0.0145

0.015

0.0155

0.016

0.0165

0.017

γ

ε

Figure 5.3: Interesting region of the Floquet stability diagram (∗ = L-stable, = unstable).

given in Chapter 3.4.4 were used. As can be seen, the flexible modes add very little mo-

tion to the rigid ones. This was also the case for other values of the control parameters

and initial conditions. The result is not too surprising given that the flexibility of the

spacecraft’s appendages is asymptotically stable in itself due to the presence of damping.

It would appear that the only way to increase the contribution of the flexible modes

to a point where it impacts the rigid motion would be if the controller input was such

that a resonant frequency of any of the flexible modes was excited. However, even for ε

and γ leading to unstable dynamics, the frequency content of the control effort (approxi-

mated by taking a Fourier transform of the control signal) was still much lower than the

lowest fundamental frequency of the flexible body, suggesting that exciting resonance in

the system is a very unlikely event. The additional nonlinearities due to the presence

of flexibility in the appendages may affect the rigid motion for larger angles and rates,

where the linear approximation is invalid. In this case, the system must be described

using the nonlinear equations of motion, and it is possible that those additional nonlin-

earities become more prevalent. Fig. 5.5 shows evidence of this possibility. The attitude

response of both the rigid and flexible nonlinear systems is shown for two different initial


0 5 10 15−8

−7

−6

−5

−4

−3

−2

−1

0

1x 10

−6

# of Orbits

Qre

η

Figure 5.4: A plot showing the contribution of the flexible modes to the rigid ones.

conditions: for ω01 =[

0.0001 0.0001 0.0001]T

rads

and for ω02 =[

1 1 1]T

rads. The

control gains used were ε = 0.0001 and γ = 0.0035. For ω01, where the dynamics can

even be well-approximated by the linearized equations of motion, the attitude response

is almost exactly the same in both the rigid and flexible case. For ω02, the response of

the flexible spacecraft is seen to be much different than the rigid version, suggesting that

it is indeed the nonlinearities imposed by the flexibility of the system that is causing the

discrepancy. Thus, the linear Floquet analysis is limited in the sense that its stability

predictions may be invalid for larger angles and rates where the dynamics are no longer

linear. Chapter 5.3 investigates this issue in further detail.

The Floquet stability analysis was performed for various values of the appendages’

Young’s modulus E. Changing the stiffness did not alter the stability predictions. This

may be due to the very small impact the flexible modes have on the rigid ones. The

damping ratio was also varied and the stability predictions still remained the same. An

important observation is that even with zero damping (ζ = 0), the flexible modes (and

hence motion of the appendages) may still damp out even using purely magnetic control.

Recalling the modal equations of motion (see Chapter 3.4.4) and noting once again that


0 5 10−0.04

−0.02

0

0.02ω01

Qua

tern

ions

ǫ1ǫ2ǫ3

0 5 10−1

−0.5

0

0.5

1ω02

ǫ1ǫ2ǫ3

0 5 10−0.04

−0.02

0

0.02

Qua

tern

ions

# of Orbits

ǫ1ǫ2ǫ3

0 5 10−1

−0.5

0

0.5

1

# of Orbits

ǫ1ǫ2ǫ3

Figure 5.5: A plot illustrating the affect of flexibilty on the nonlinear system for larger anglesand rates. The top two plots are for the rigid spacecraft, and the bottom two are for the flexibleversion.

the eigenmatrix Q can be written as

Q =

[

1 Qre

0 Qee

]

,

and that q = Qη, the existence of Qre implies that the flexible modes contribute to the

rigid ones so that any flexible motion will cause the controller to provide a control torque

until the flexible modes are attenuated. The linear system was simulated using purely

magnetic control with ε = 0.0001 and ζ = 0, and the system response (including the tip

deflection ue(L) for one arm only) is shown in Fig. 5.6. As can be seen, even without the

help of physical damping the purely magnetic control law causes the tip deflection to go

to zero.

Another limitation of the Floquet analysis is that it says nothing about controller

performance. Fig. 5.7 shows the system response with the control gains set to ε =

0.0006853 and γ = 0.003191. As can be seen, although the dynamics will eventually


0 5 10 15−1

−0.5

0

0.5

Eul

er A

ngle

s (r

ad)

θ1

θ2

θ3

0 5 10 15−1

−0.5

0

0.5

1x 10

−3

Ang

ular

Vel

ocity

(ra

d/s)

θ1

θ2

θ3

0 5 10 15−1

−0.5

0

0.5

1x 10

−6

# of Orbits

Tip

Def

lect

ion

(m)

u

ex

uez

Figure 5.6: A plot illustrating that the proposed purely magnetic control scheme attenuatesflexible motion even without elastic damping (note: tip deflection plot is zoomed in for clarity).

stabilize, it takes a very long time.

5.2 Hybrid Controller Gain Selection Criteria

This section attempts to extend the analysis in [24] to the flexible spacecraft case. The

idea is to make use of the passivity theorem (see Chapter 2.6) in order to establish

conditions on the control parameter γ (in Eq. (5.6)) that will make the closed-loop linear

system asymptotically stable despite the gain restrictions described in Chapter 5. In

contrast to using Floquet theory to investigate gain selection, which heavily relies on the

simulation of the equations of motion, this approach is more analytical in nature.


0 20 40 60 80 100−0.05

0

0.05

Eul

er A

ngle

s (r

ad)

θ1

θ2

θ3

0 20 40 60 80 100−1

−0.5

0

0.5

1x 10

−4

Ang

ular

Vel

ocity

(ra

d/s)

# of Orbtis

θ1

θ2

θ3

Figure 5.7: A plot showing how the Floquet theory predicts stability but not controller per-formance.

For the rigid case, a completely analytic result can be obtained, imposing a condition

on the choice of γ in order to achieve stability for any ε. It is shown in Appendix B that

stability for the closed-loop rigid system using the hybrid control scheme is achieved as

long as γ > kpk2d

λmax, where λmax is the spacecraft’s largest principle inertia.

For the flexible case, a similar procedure (as in [24] and Appendix B) is performed in

order to establish gain selection criteria. Consider, once again, the closed-loop equations

of motion for the flexible linear system given by Eqs. (3.12) and (5.6), which can be

written collectively as

Mq + Dq + Kq = −B(ε2kpq + εkdq

).

With the intention of decoupling the dynamics as much as possible, and of obtaining

simpler equations to work with the above equations are cast into modal coordinates

η = Q−1q (as shown in Chapter 3.4.4), yielding the transformed equations which are

written as

Mmodη + Dmodη + Kmodη = −QTBQ(ε2kpη + εkdη

), (5.7)


where

Mmod =

[

I 0

0 1

]

, Dmod =

[

0 0

0 2ζΩ

]

, and Kmod =

[

0 0

0 Ω2

]

.

It is important to note that Dmod and Kmod are diagonal matrices and that Mmod can be

made diagonal (i.e., making I diagonal) by either choosing the body frame to coincide

with the spacecraft’s principal axes or by normalizing Q such that I is diagonalized. Also,

the control matrix B can be written as

B =

[

(µγ1+ Γ) 0

0 µM ee

]

+

[

(1− µ)γ1 0

0 −µM ee

]

,

where 0 < µ < 1. The closed-loop system can be decomposed into a traditional plant

and controller type structure (as shown in Figs. 2.2 and 2.3). The plant’s output, y(t),

is defined as

y(t) = εkdη(t) + ε2kpη(t),

and its control input, u(t), as

−u(t) = QT

[

(µγ1+ Γ) 0

0 µM ee

]

Q

︸︷︷︸

H(t)

y(t).

Defining the control input in this way removes the time-varying part of the closed-loop

system from the plant dynamics. Furthermore, the mapping between controller input y

and controller output u, denoted by the operator H, is then strictly passive since

∫ T

0

yT (Hy) dt =∫ T

0

yTΘ(t)y(t) dt

≥ σmin

∫ T

0

yTy dt,

where

Θ(t) = QT

[

(µγ1+ Γ(t)) 0

0 µM ee

]

Q > 0,

and σmin is the smallest eigenvalue of Θ. Note that Θ is positive definite for any t > 0

since (µγ1+ Γ) and M ee are both positive definite. Thus, the operator H simply corre-

sponds to time-varying negative feedback using a positive definite gain matrix, which is


strictly passive (as shown). The passivity theorem then requires that the plant, described

by the operator G, be passive in order to guarantee input-output stability. With these

definitions for input and output, Eq. (5.7) can be written as

Mmodη(t) + Dmodη(t) + Kmodη(t) = u(t)−Ξy(t),

where

Ξ = QT

[

(1− µ)γ1 0

0 −µM ee

]

Q.

Since the above system is LTI, it can be described in the frequency domain by a transfer

matrix. To this end, a Laplace transform (with the transform variable denoted by s) is

performed, giving

(s2Mmod + sDmod + Kmod

)η(s) +Ξy(s) = u(s).

Noting that y(s) = (εkds+ ε2kp)η(s), the resulting system can be written as

(s2Mmod + sDmod + Kmod

εkds+ ε2kp

)

y(s) +Ξy(s) = u(s)

y(s) =[(

s2Mmod + sDmod + Kmod

εkds+ ε2kp

)

+Ξ

]−1

︸︷︷︸

G(s)

u(s)

y(s) = G(s)u(s).

The plant is therefore described by the transfer matrix G(s) as defined above. Fig. 5.8

shows the block diagram representation of the closed-loop system. It is of the same form

as the generic system in Fig. 2.3, for which the passivity theorem was derived. It is

important to note that each element of G(s) is a real and rational function of s. It is

known that an LTI operator (such as G) is passive if its corresponding transfer matrix

G(s) is positive real (PR). Finding conditions for which the plant is PR would then mean

that the requirements for L2-stability are satisfied via the passivity theorem. According

to [34], G(s) is PR if and only if

(a) no element of G(s) has a pole in Res ≥ 0;

(b) GH(jω) + G(jω) ≥ 0 for all real ω, with jω not a pole of any element of G;


+

−

0 y

−u

H(t)

G(s)

QT

[

(µγ1+ Γ(t)) 0

0 µMee

]

Q

Figure 5.8: Block diagram for the flexible, hybrid-controlled system.

(c) for any jω0 that is a pole for some element of G, it is at most a simple pole, and the

residue matrix, given by K 0 = lims→jω0

(s− jω0)G(s) for finite ω0 and K∞ = limω0→∞

G(jω0)jω0

if ω0 is infinite, is non-negative definite Hermitian,

where (·)H refers to the conjugate transpose operation for a complex-valued matrix. Note

that in G, there are damping terms for both the rigid and flexible components; the rigid

damping coming from the rate feedback of the control law and the flexible damping

inherent in the physical spacecraft structure itself. The presence of this damping is

enough to ensure that there are no purely imaginary poles, and so condition (c) is satisfied

automatically. Condition (b) can be satisfied using adequate choices for the control

parameters. The control parameters can be estimated numerically by fixing ε, kp, and

kd, and finding the minimum γ such that GH(jω) + G(jω) ≥ 0 for a large range of

ω. The other parameters may be changed and the procedure repeated. Fig. 5.9 shows

the minimum value of γ satisfying (b) over a range of ε (with kp = 25000 kg·m2

s2and

kd = 25000 kg·m2

s) for the sample system given in Chapter 3.5. For each pair of ε and γ

in Fig. 5.9, condition (a) was verified numerically using the pole() function in MATLAB.

Therefore, the graph obtained shows values for the control parameters ensuring G is PR,

implying that the system given by Eq. (5.3) is input-output stable.

Lyapunov stability can also be shown, assuming that G is PR. A minimal realization

of G can be written as G(s) = C (s1− A)−1 B, for some matrices A,B,C. Letting

x =[

ηT ηT

]T

, a state-space representation of the block diagram in Fig. 5.8 can be


3.75 3.8 3.85 3.9 3.95 4 4.05 4.1 4.15 4.2

x 10−3

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

γ

ε

Figure 5.9: Stability diagram obtained using passivity theory.

given as

x = Ax + Bν

y = Cx

ν = −Hy.

By the Kalman-Yakubovich lemma, the positive realness of G(s) implies that there exists

some P = PT > 0 and Q = QT > 0 such that ATP+ PA = −Q and PB = CT. Letting

V =1

2xTPx,

it is easy to show that

V = −1

2xTQx − yT

Hy.

The positive definiteness of V and the negative definiteness of its derivative imply that

it is a valid Lyapunov function, meaning that the closed-loop system is asymptotically

stable (via Lyapunov’s Direct Method). Since the values for ε and γ in Fig. 5.9 render

G(s) PR, they also ensure asymptotic stability, and so the figure can be thought of as a

stability matrix diagram comparable to the figures in Chapter 5.1. The trend shown in


Fig. 5.9 shows a very conservative estimate for the minimum γ needed to ensure stability

at a particular ε, when compared to Fig. 5.1. It is clear that this passivity analysis yields

sufficient, though not neccesary, conditions on γ, since choosing γ = 0.0038 would ensure

stability for any ε. An example validating this analysis can be seen in Fig. 5.10, where

the spacecraft’s unstable motion under purely magnetic control (with ε = 0.003) is shown

to become stable when choosing γ = 0.0038 as predicted by Fig. 5.9. Note that Floquet

theory estimated that stability is obtained for γ ≈ 0.0012.

0 1 2−5

0

5

10x 10

5 ǫ = 0.003, γ = 0

Eul

er A

ngle

s (r

ad)

θ1

θ2

θ3

0 1 2−0.2

−0.1

0

0.1

0.2ǫ = 0.003, γ = 0.0038

θ1

θ2

θ3

0 1 2−2000

−1000

0

1000

2000

Ang

ular

Vel

ocity

(ra

d/s)

# of Orbits

θ1

θ2

θ3

0 1 2−5

0

5

10x 10

−4

# of Orbits

θ1

θ2

θ3

Figure 5.10: A plot illustrating the validity of the hybrid gain selection criteria. Purelymagnetic control on the left, hybrid control on the right.

This passivity analysis was performed for various values of the damping ratio and

the Young’s modulus as well. Larger damping ratios gave less conservative results. For

example, given ε = 0.003, with ζ = 0.1 stability was achieved if γ ≈ 0.00381, but with

ζ = 0.5 all that was needed was that γ = 0.00376. When the Young’s modulus was

increased, a single value for γ was obtained for all ε. With E = 4 × 1011 Pa, the single

value for γ obtained was γ = 0.00791. It is clear that changing the material properties

of the flexible appendages does have a small affect on the passivity analysis. The more

“rigid” the appendages were made to be, the closer the analysis results were to the rigid


case as is expected. It is important to note that a stabilizing γ could not be found

for some values of the material properties. A reason for the analysis breaking down is

discussed below, where it shown that for weak coupling between the flexible and rigid

modes there are simultaneous bounds on γ and ε.

A more analytical result may be obtained if it is assumed that the coupling between

the flexible and rigid modes is weak. Noting once again that the eigenmatrix Q can be

written as

Q =

[

1 Qre

0 Qee

]

,

the weak-coupling assumption amounts to saying that the Qre term is negligible. The

transfer matrix G(s) is then completely decoupled, and the rigid and flexible dynamics

can be investigated independently. With this assumption,

G(s) ≈

[

1

εkds+ ε2kp

(

s2

[

I 0

0 1

]

+ s

[

0 0

0 2ζΩ

]

+

[

0 0

0 Ω2

])

+

[

(1− µ)γ1 0

0 −µ1

]]−1

,

where, in each matrix the top-left quadrant represents the rigid dynamics, and the

bottom-right represents the flexible dynamics. Once again, it is necessary to find condi-

tions on the parameters in G such that it is positive real (which would then imply that

it is also passive). Since it is diagonal, this requirement implies that each of its elements,

Gi(s), must be positive real. As shown in [35], Gi(s) is strictly positive real (SPR, which

is a stronger condition implying positive realness) if

(a) it is analytic in Res ≥ 0;

(b) ReGi(jω) > 0, for ω ∈ (−∞,∞);

(c) limω→∞

ω2ReGi(jω) > 0.

For the rigid subsystem, Appendix B shows that all three conditions are met for each

of its elements if

kp > 0, kd > 0, ε > 0, µ < 1, and γ >kp

k2dλmax, (5.8)

where λmax is the largest principal moment of inertia. It is important to note that this

bound on γ differs from that in [24] by a factor of λmax since the moment of inertia was

not included in the control law.


For the flexible subsystem, each element of the transfer matrix becomes

Gi(s) =

[s2 + 2ζωis+ ω2

i

εkds+ ε2kp− µ

]−1

.

Letting µ→ 0 and s = jω it can be shown that

Gi(jω) =[ε2kp (ω

2i − ω2) + 2εkdζωiω

2] + j [εkdω (ω2i − ω2)− 2ε2kpζωiω]

(ω2i − ω2)

2+ (2ζωiω)

2,

from which it is easy to show that

limω→∞

ω2ReGi(jω) = 2εkdζωi − ε2kp.

Condition (a) is automatically satisfied if γ > 0, kp > 0, kd > 0, ǫ > 0, and µ < 1. From

the above equation, it is clear that condition (c) is satisfied if ε < 2ζωikdkp. To satisfy

condition (b), ε must be chosen such that 2kdζωiω2 + εkpω

2i > εkpω

2. If the bound on ε

from condition (c) holds, then the above inequality is also satisfied since

ε < 2ζωikd

kp⇒ εkpω

2 < 2ζkdωiω2 < 2ζkdωiω

2 + εkpω2i .

To summarize, after imposing the weak-coupling assumption, each element of G (and

hence G itself) is SPR if

kp > 0, kd > 0, µ < 1, γ >kp

k2dλmax, and 0 < ε < 2ζωmin

kd

kp,

where ωmin is the lowest natural frequency of the flexible system. Therefore, if these

bounds on the control parameters hold, then the closed loop system is input-output

stable. Asymptotic stability can also be shown in a similar way as was done for the

coupled case.

With kp = 25000 kg·m2

s2and kd = 25000 kg·m2

s, the gain selection criteria given by the

above conditions are that 0 < ε < 0.406 and γ > 0.00378 in order to guarantee stability.

When compared to the stability map (Fig. 5.1) predicted using Floquet analysis, this

selection criteria provides a conservative method of choosing the control gains. When

compared to the stability diagram obtained using passivity theory (see Fig. 5.9) for

the case where the weak-coupling assumption was not imposed, the stability criteria


agrees and gives a slightly less conservative estimate. This agreement with the other

stability analyses suggests that using the weak-coupling assumption to obtain analytically

obtainable stability criteria is a valid and useful approach. The value in using this

analytical approach is that it provides a very quick and relatively easy way to place

conditions on the control gains ensuring stability, which is vital in control design.

Since there is an explicit bound on the gains it is easy to see what the effects of elastic

damping and material stiffness would be on the stability of the system. The bound on

γ is independent of both of those properties, and so would remain the same. The bound

on ε depends on both the damping ratio and the minimum natural frequency of the

appendages. It is clear that increasing the damping or stiffness properties of the flexible

structure will raise the upper bound on ε.

5.3 Stability of the Nonlinear System

The stability of the hybrid controller developed in the preceding sections was proven

for the linearized system. This section will explore how well the stability predictions

(obtained using the linear Floquet analysis) apply to the nonlinear equations of mo-

tion. Recall that the closed-loop, flexible nonlinear dynamics are described by Eqs. (3.9)

and (3.10), where fe = 0, and τ (which in this case represents the hybrid control law) is

given by

τ = −(γ1+ Γ)(2ε2kpǫ+ εkdω

).

In general, the linearization of a nonlinear system accurately describes the dynamics

only when the state variables and initial conditions differ from the equilibrium point

by small values. For initial conditions outside of the linear approximation range, the

linearized system does not accurately describe the dynamics. Fig. 5.11 shows a compar-

ison of the state response of both the linear and nonlinear system for initial conditions

that are within, and outside, of the linear approximation range. The control parameters

used were ε = 0.0001 and γ = 0.0038 with kp and kd the same as in Chapter 5.1. The

initial conditions were ω01 =[

0.0001 0.0001 0.0001]T

rads, which is within the linear

approximation range, and ω01 =[

0.1 0.1 0.1]T

rads, which is outside it. All other state

variables were initially zero. As can be seen, for ω01 the linear system accurately describes

the nonlinear dynamics but for ω02, they are very different. Although not plotted, the

flexible degrees of freedom (qe) followed the same trend. This is a limitation of designing


a control law for a nonlinear system around its linearization. When evaluating if the

stability analyses based on the linear system apply in the nonlinear case, it is important

to keep this limitation in mind.

0 5 10−0.04

−0.02

0

0.02

Eul

er A

ngle

s (θ

/2 r

ad)

θ1

θ2

θ3

0 5 10−1

0

1

2x 10

−4

Ang

ular

Vel

ocity

(ra

d/s)

θ1

θ2

θ3

0 5 10−0.04

−0.02

0

0.02

Qua

tern

ions

# of Orbits

ǫ1ǫ2ǫ3

0 5 10−1

0

1

2x 10

−4

Ang

ular

Vel

ocity

(ra

d/s)

# of Orbits

ω1

ω2

ω3

0 5 10−40

−20

0

20

Eul

er A

ngle

s (θ

/2 r

ad)

θ1

θ2

θ3

0 5 10−0.1

0

0.1

Ang

ular

Vel

ocity

(ra

d/s)

θ1

θ2

θ3

0 5 10−1

0

1

Qua

tern

ions

# of Orbits

ǫ1ǫ2ǫ3

0 5 10−0.2

0

0.2

Ang

ular

Vel

ocity

(ra

d/s)

# of Orbits

ω1

ω2

ω3

Figure 5.11: A plot comparing the system response for initial conditions in, and outside, thelinear approximation range. The top four plots have an initial angular velocity of ω01, and thebottom four have an intial angular velocity of ω02.


The closed-loop linear and nonlinear system dynamics were simulated for various

values of ε and γ along the stability boundaries seen in Figs. 5.1 and 5.3. The L-

stability of the dynamics was deduced by inspection. Table 5.1 shows a comparison

between the predicted stability (using Floquet theory), the linear system response, and

the nonlinear system response for both initial conditions (ω01 and ω02). It is important

to note that the linear dynamics are not accurate for ω02, but are included anyways.

Floquet theory predicts the stability (of the linear system) for any initial condition,

even those such that the dynamics are no longer linear. Thus, as seen in Table 5.1, the

simulated linear response confirms the Floquet prediction even for ω02, which is outside

the linear approximation range. An example of a stable and unstable response for the

Table 5.1: Comparison between the Floquet stability prediction, and the simulated lin-ear/nonlinear response.

ε γFloquet ω01 ω02

Prediction Linear Nonlinear Linear∗ Nonlinear0.0001 0 stable stable stable stable stable

0.0006853 0 unstable unstable unstable unstable unstable0.0006853 0.003088 unstable unstable unstable unstable unstable0.0006853 0.003191 stable stable stable stable stable0.001856 0.001441 unstable unstable unstable unstable unstable0.002441 0.001647 stable stable stable stable stable0.006538 0.0003088 unstable unstable unstable unstable unstable0.006538 0.0004118 stable stable stable stable stable0.01217 0.000145 unstable unstable unstable unstable unstable0.0155 0.000065 unstable unstable unstable unstable unstable0.0155 0.000075 stable stable stable stable stable0.0155 0.00009 unstable unstable unstable unstable unstable0.01533 0.00011 stable stable stable stable stable0.01824 0.003191 stable stable stable stable stable

Note: the term “stable” as used here means “asymptotically stable”.∗The linear system response is not accurate for this initial condition.

nonlinear system can be seen in Fig 5.12, where the conditions in rows 2 (unstable) and 8

(stable) of Table 5.1 were simulated with ω02 as an initial angular velocity. Note that the

tip deflection for only one appendage is shown. Table 5.1 also shows that the stability

of the nonlinear dynamics matched the Floquet prediction, suggesting that the linear

stability analysis might be able to predict the stability of the nonlinear system, even for

larger angles and rates. This, of course, is merely a suggestion and does not constitute a


0 1 2 3−1

0

1ǫ = 0.0006538, γ = 0.0004118

Qua

tern

ions

ǫ1ǫ2ǫ3

0 5 10−2

0

2ǫ = 0.0006853, γ = 0

ǫ1ǫ2ǫ3

0 1 2 3−0.1

0

0.1

Ang

ular

Vel

ocity

(ra

d/s)

ω1

ω2

ω3

0 5 10−0.2

0

0.2

ω1

ω2

ω3

0 1 2 3−5

0

5x 10

−4

# of Orbits

Tip

Def

lect

ion

(m)

u

ex

uez

0 5 10−5

0

5x 10

−4

# of Orbits

u

ex

uez

Figure 5.12: An example of a stable and unstable nonlinear system response (note: tip deflec-tion plots are zoomed in for clarity).

rigorous stability analysis for the nonlinear equations of motion. However, it does show

that it is likely that the controller designed around the linear system can be used for the

nonlinear case.

As the Floquet analysis seems to be able to predict the stability of the nonlinear sys-

tem well, the gain selection criteria discussed in Chapter 5.2 would be expected to apply

as well. The criteria provided a very conservative estimate with respect to the Floquet

analysis, and so it should also provide likewise conservative controller gain estimate for

the nonlinear system.

Chapter 6

Conclusions

The purpose of this thesis was to investigate using magnetic actuators for the attitude

control of a flexible spacecraft. The PD-like magnetic control techniques previously used

for rigid spacecraft were extended to the flexible case. The gain limitations inherent

with this type of control are shown to be alleviated with the addition of a minimum

amount of control effort from an alternate, independent, three-axis actuation system

(such as reaction wheels or thrusters). The hybrid control law in [24] designed for a rigid

spacecraft was adopted and modified for the flexible case.

The relevant background concepts needed to understand and investigate the control

problem were presented. A brief summary of the notation used throughout the thesis

was given, as it was critical in the development of the equations of motion. Spacecraft

“attitude” was defined using a rotation matrix, and two parameterizations (Euler angles

for the linear dynamics, quaternions for the nonlinear dynamics) were presented. The two

main definitions of stability (input-ouput and Lyapunov) were also stated, and several

stability theories used in the closed-loop system’s stability analyses were summarized:

Floquet, Lyapunov, and passivity theory.

The mechanics of a flexible spacecraft in an orbit around the earth was then devel-

oped. A geocentric inertial reference frame and a frame affixed to the spacecraft body

were defined in order to describe the mechanics. Since magnetic attitude control requires

knowledge of the geomagnetic field at a particular point along the spacecraft’s orbit, the

orbit definition and mechanics in terms of orbital elements was presented. Quaternions

were used to represent the spacecraft kinematics, which describes the evolution of the

spacecraft’s orientation over time. Vectrix notation was used to develop the dynamical

equations of motion for both a rigid and flexible spacecraft. The flexibility of the space-

75

Chapter 6. Conclusions 76

craft was incorporated using the Rayleigh-Ritz method, where the elastic deflections of

the body are represented by a series of shape functions describing the deformation field

coupled with generalized coordinates for the deformation degrees of freedom. The lin-

earized equations of motion were also derived, as the controller development and stability

analyses focused on the linear dynamics. The linearized system was further broken down

into modal coordinates, so that the rigid and flexible modes could be analyzed indepen-

dently. An example spacecraft consisting of a rigid hub with two flexible appendages

attached was presented as an archetype.

A model for the earth’s magnetic field was presented. It assumes that the magnetic

field can be represented by the gradient of a scalar potential function consisting of a

series of spherical harmonics. Controllability of the spacecraft magnetic attitude control

problem was discussed. It was shown that the spacecraft is instantaneously underac-

tuated since the interaction between the geomagnetic field and a dipole created by the

magnetic actuators is governed by a vector cross product. However, if the magnetic field

experienced by the spacecraft varies enough along the orbit, the attitude dynamics can be

controllable in a time-varying sense. The constant dipole approximation of the geomag-

netic field (obtained by taking the first harmonic only of the spherical harmonic model)

is able to satisfy the controllability requirements, and is the model that was adopted for

this thesis. The requirements also suggest that magnetic attitude control is best suited

for spacecraft in low-earth and polar orbits. After establishing controllability, a more

indepth summary of the PD-like control law designed for rigid spacecraft was explored

since the controller design for flexible spacecraft was based on it. Limitations on the

gains for this type of control were shown to exist, and a hybrid control scheme (for the

rigid case) that relaxes these limitations was presented.

These rigid controllers were then extended to the flexible case. A PD-like control law

was proposed for the linear system. The lack of any spacecraft properties (i.e., moment

of inertia) in the controller made it more robust than the one for the rigid case (on which

it was based). Stability of the closed-loop system was explored and it was shown that

the gain limitations still exist. A hybrid controller was then proposed that augments

the purely magnetic control with an alternate, independent, three-axis actuation system.

Floquet analysis was used to determine how much effort the additional actuation system

needed to provide in order to ensure stability. It was seen that as the control gains for

the purely magnetic control increased, less control effort was needed from the additional

actuation system. The Floquet analysis also provided a stability map that could be used


for hybrid controller design. Floquet analysis was also performed on the rigid version

of the archetypal spacecraft and it was seen that the stability predictions were identical

to the flexible case. This suggests it might be possible that controllers designed for the

rigid system can be adopted for the flexible case “as is”. The modal equations of motion

were used to show that the flexible modes had little effect on the rigid ones using the

hybrid control scheme. Even when both the damping ratio and stiffness (represented by

the Young’s modulus) were varied, the Floquet predictions did not change.

Passivity theory was used in order to obtain gain selection criteria for the hybrid

controller. The closed-loop system was decomposed into a linear time-invariant plant

with a time-varying strictly passive feedback. Conditions on the control gains were

then obtained (numerically) that made the plant passive and hence, the closed-system

input-output stable. Asymptotic stability was shown to follow from the input-output

result. Making the assumption that the flexible modes contribute little to the rigid ones

allowed for bounds on the control gains to be obtained analytically. The Floquet analysis

confirmed the validity of these bounds, and so the analytical gain criteria and passivity

analysis proved to be a relatively effective tool for the practical design of a controller

using the hybrid control scheme.

The hybrid controller proposed in this thesis was based on the linear system dynamics.

The stability of the nonlinear closed-loop system was investigated via simulation. The

stability predictions based on the Floquet and passivity analyses were tested. It was

found that the Floquet predictions for the linear system also predicted the stability of

the nonlinear system, even for angles and rates outside the linear approximation range. It

was concluded that the hybrid PD-like controller design for the actual nonlinear system

could be based on the linearized version, meaning that the gain selection criteria and

stability maps obtained via the linear analyses would likely apply to the nonlinear case.

6.1 Future Work

This thesis aims at providing a design method for a particular class of control. The main

drawback of using the purely magnetic control is the gain limitations embodied in the

existence of ε∗. This gain limitation was estimated using numerical analyses; finding

a way to calculate it analytically would be an important addition to the work done in

this thesis. The analyses performed focused on determining the gain values that would

lead to a stable system without regard for controller performance. Investigating the gain


criteria for optimal controller performance would be another great contribution. The

stability analyses all focused on the linear system dynamics, and so the controller had to

be design around the linear system as well. The nonlinear system stability was inferred

in a non-rigorous way. Finding a technique that analyzes the stability of the nonlinear

system would make the results of this thesis applicable to a broader range of motion.

Future work could also include examining the situation where the sensors and actua-

tors are not collocated, as they are assumed to be in this thesis. Non-collocated sensors

and actuators would potentially introduce destabilizing spillover from flexible modes that

have not been accounted for in the spacecraft model. Also, there are other ways of de-

scribing the flexibility of a body, and an investigation of these other methods might lead

to more insight regarding the flexible dynamics. Another limitation of the work done in

this thesis is that the spacecraft was axisymmetric, and so the deflections caused by rigid

motion would contain symmetry as well. Extending the controller design for spacecraft

that are not symmetric would be interesting.

This thesis deals primarily with PD-like control. Future work could include extending

other types of control methods (such as the ones briefly mentioned in Chapter 1.1) to

the flexible case, and comparing them among each other to see if any particular control

technique allows for easier design.

References

[1] J. Wertz, Spacecraft Attitude Determination and Control. D. Reidel Publishing Co.,Dordrecht, The Netherlands, 1978.

[2] S. Bhat and A. Dham, “Controllability of spacecraft attitude under magnetic actu-ation,” in IEEE Conference on Decision and Control, 2003.

[3] E. Silani and M. Lovera, “Magnetic spacecraft attitude control: a survey and somenew results,” Control Engineering Practice, vol. 13, pp. 357–371, 2005.

[4] R. Ryan, “Simulation of actively controlled spacecraft with flexible appendages,”Journal of Guidance, Control, and Dynamics, vol. 13, no. 4, pp. 691–702, 1990.

[5] H. Bang, C. Ha, and J. Kim, “Flexible spacecraft attitude maneuver by applicationof sliding mode control,” Acta Astronautica, vol. 57, pp. 841–850, 2005.

[6] S. Bhat, “Controllability of nonlinear time-varying systems: Applications to space-craft attitude control using magnetic actuation,” IEEE Transactions on Automatic

Control, vol. 50, no. 11, pp. 1725–1735, 2005.

[7] R. Wisniewski and M. Blanke, “Fully magnetic attitude control for spacecraft subjectto gravity gradient,” Automatica, vol. 35, pp. 1201–1214, 1999.

[8] C. Damaren, “Comments on: Fully magnetic attitude control for spacecraft subjectto gravity gradient,” Automatica, vol. 38, p. 2189, 2002.

[9] R. Wisniewski, “Linear time-varying approach to satellite attitude control using onlyelectromagnetic actuation,” Journal of Guidance, Control, and Dynamics, vol. 23,no. 4, pp. 640–647, 2000.

[10] M. Psiaki, “Magnetic torquer attitude control via asymptotic periodic linearquadratic regulation,” Journal of Guidance, Control, and Dynamics, vol. 24, no. 2,pp. 386–394, 2001.

[11] M. Lovera, E. D. Marchi, and S. Bittanti, “Periodic attitude control techniques forsmall satellites with magnetic actuators,” IEEE Transactions on Control Systems

Technology, vol. 10, no. 1, pp. 90–95, 2002.

[12] J. Kulkarni and M. Campbell, “An approach to magnetic torque attitude control ofsatellites via ‘H∞’ control for LTV systems,” in 43rd IEEE Conference on Decision

and Control, 2004.

79

References 80

[13] M. Lovera and A. Astolfi, “Global magnetic attitude control of inertially pointingspacecraft,” Journal of Guidance, Control, and Dynamics, vol. 28, no. 5, pp. 1065–1067, 2005.

[14] M. Lovera and A. Astolfi, “Global magnetic attitude control of spacecraft in the pres-ence of gravity gradient,” IEEE Transactions on Aerospace and Electronic Systems,vol. 42, no. 3, pp. 796–805, 2006.

[15] A. Astolfi and M. Lovera, “Global spacecraft attitude control using magnetic actu-ators,” in Proceedings of the American Control Conference, pp. 1331–1335, 2002.

[16] M. Lovera and A. Astolfi, “Spacecraft attitude control using magnetic actuators,”Automatica, vol. 40, pp. 1405–1414, 2004.

[17] H. K. Khalil, Nonlinear Systems. Prentice Hall, Upper Saddle River, NJ, 1996.

[18] J. Liang, R. Fullmer, and Y. Chen, “Time-optimal magnetic attitude control forsmall spacecraft,” in 43rd Conference on Decision and Control, pp. 255–260, 2004.

[19] M. Lovera and A. Varga, “Optimal discrete-time magnetic attitude control of satel-lites,” in Proceedings of the 16th IFAC World Congress, 2005.

[20] S. Sakai, Y. Fukushima, and H. Saito, “Studies on magnetic attitude control systemfor the reimei microsatellite,” in AIAA, 2006.

[21] J. Oh, S. Park, and K. Choi, “Magnetic torque attitude control of a satellite usingthe state-dependent riccati equation technique,” in AIAA Guidance, Navigation,

and Control Conference and Exhibit, 2006.

[22] T. Pulecchi and M. Lovera, “Attitude control of spacecraft with partially magneticactuation,” in IFAC Symposium on Automatic Control in Aerospace, vol. 17, 2007.

[23] J. Forbes and C. Damaren, “Geometric approach to spacecraft attitude control usingmagnetic and mechanical actuation,” Journal of Guidance, Control, and Dynamics,vol. 33, no. 2, pp. 590–595, 2010.

[24] C. J. Damaren, “Hybrid magnetic attitude control gain selection,” Proc. IMechE,

Part G, J. of Aerospace Engineering, vol. 223, pp. 1041–1047, 2009.

[25] F. Schiavo, M. Lovera, and A. Astolfi, “Magnetic attitude control of spacecraft withflexible appendages,” in 43rd IEEE Conference on Decision and Control, pp. 1545–1550, 2006.

[26] P. Hughes, Spacecraft Attitude Dynamics. Dover Publications Inc., Mineola, NY,2004.

[27] C. Chicone, Ordinary Differential Equations with Applications. Springer-Velag NewYork Inc., 1999.

References 81

[28] C. J. Damaren, “AER1503H Spacecraft Dynamics and Control 2 Course Notes,”tech. rep., University of Toronto Institute for Aerospace Studies, 2010.

[29] C. Desoer and M. Vidyasagar, Feedback Systems: Input-Output Properties. AcademicPress Inc., 1975.

[30] C. J. Damaren, “AER506H Spacecraft Dynamics and Control 1 Course Notes,” tech.rep., University of Toronto Institute for Aerospace Studies, 2009.

[31] J. DeLaurier, “AER503H Aeroelasticity Course Notes,” tech. rep., University ofToronto Institute for Aerospace Studies, 2007.

[32] C. C. Finlay, S. Maus, C. D. Beggan, and T. N. Bondar, “International geomagneticreference field: Eleventh generation,” Geophysical Journal International, vol. 183,pp. 1216–1230, October 2010.

[33] S. Chee, “Stability of a hybrid magnetic attitude control system,” Master’s thesis,University of Toronto Institute for Aerospace Studies, 2009.

[34] B. Anderson and S. Vongpanitlerd, Network Analysis And Synthesis. Dover Publi-cations Inc., Mineola, NY, 2006.

[35] P. Ioannou and G. Tao, “Frequency domain conditions for strictly positive realfunctions,” IEEE Transactions on Automatic Control, vol. 32, pp. 53–54, 1987.

[36] B. Wie, Space Vehicle Dynamics and Control. Washington, DC: AIAA, 1998.

Appendix A

Geomagnetic Field Model

Supplemental Calculations

The transformed Gaussian coefficients gn,m and hn,m are defined as

gn,m ≡ Sn,mgmn

hn,m ≡ Sn,mhmn ,

where

Sn,m ≡

[(2− δ0m)(n−m)

(n+m)

]1/2(2n− 1)!!

(n−m)!.

The parameter δ0m = 1 if m = 0, and is 0 otherwise. The Gauss functions P n,m can be

obtained recursively by noting that

P 0,0 = 1

P n,n = sin(θ)P n−1,n−1

P n,m = cos(θ)P n−1,m −Kn,mP n−2,m,

where

Kn,m ≡

(n−1)2−m2

(2n−1)(2n−3)n > 1

0 n = 1.

82

Appendix A. Geomagnetic Field Model Supplemental Calculations 83

The partial derivatives of the Gauss functions can also be obtained recursively:

∂P 0,0(θ)

∂θ= 0

∂P n,n(θ)

∂θ= sin(θ)

∂P n−1,n−1(θ)

∂θ+ cos(θ)P n−1,n−1 n ≥ 1

∂P n,m(θ)

∂θ= cos(θ)

∂P n−1,m(θ)

∂θ− sin(θ)P n−1,m −Kn,m∂P

n−2,m(θ)

∂θ.

Appendix B

Hybrid Control Gain Selection For

Rigid Spacecraft

This section outlines the proof of the stability condition on γ in the rigid spacecraft

version of the hybrid magnetic control scheme presented in Chapter 5. It closely follows

the analysis in [24] which is the basis of the proposed control law.

Consider a spacecraft system controlled using both magnetic actuation as well as

another three-axis control actuator (i.e., thrusters). This system can be described by the

equations (see Chapters 3 and 4.3)

˙q =1

2

[

q41+ q×

−qT

]

ω

I ω + ω×Iω = u + τ , (B.1)

where q =[

qT q4

]T

are the quaternions representing the spacecraft attitude, ω is the

spacecraft angular velocity, u is the control due to the alternate three-axis actuation

system and τ is due to the magnetic control all in the body frame. Consider the control

law given by

u = −γ[ǫkdω + 2ǫ2kpq

](B.2)

and

τ = −b×m

m = ‖bi‖−2 b×ν

ν = −[ǫkdω + 2ǫ2kpq

], (B.3)

84

Appendix B. Hybrid Control Gain Selection For Rigid Spacecraft 85

where b and bi is the geomagnetic field vector in the body and geocentric inertial frame,

respectively. Note that this control law is the same as the one in [24] except that the

moment of inertia has been changed to the identity matrix, making the controller more

robust to model uncertainty.

For the case when m = 0 (no magnetic control), and ǫ, γ, kp, kd all greater than zero,

the equilibrium ω = q = 0 of Eqs. (B.1) and (B.2) can be shown to be globally asymp-

totically stable. The proof relies on using Lyapunov’s Direct Method (see Chapter 2.5.1)

with the Lyapunov function (similar to the one proposed in [36]) given by

V =1

2ωTIω + 2γǫ2kp

[qTq + (q4 − 1)2

],

and noting that V is radially unbounded, positive definite, and its derivative is given by

V = −γǫkdωTω,

which is negative semi-definite. The global asymptotic stability result is achieved after

applying the Krasovskii-LaSalle theorem.

Consider the case when u = 0 (purely magnetic control), kp > 0, kd > 0. It is assumed

that the spacecraft orbit is such that

Γ = limT→∞

1

T

∫ T

0

Γ dt > 0,

where

Γ = −‖bi‖−2 b×

i b×

i ≥ 0.

It is shown in [15] that there exists an ǫ∗ such that for 0 < ǫ < ǫ∗, the equilibrium

ω = q = 0 of Eqs. (B.1) and (B.3) is asymptotically stable.

The linearization of Eq. (B.1) can be obtained by assuming small angles and rates

(θ, θ) such that θ ≈ 2q, θ ≈ ω, and b ≈ bi. The closed-loop linearized system is then

given by

I θ + (γ1+ Γ)(

ǫkdθ + ǫ2kpθ)

= 0. (B.4)

Introduce the eigen-decomposition of the inertia matrix, so that I = EΛET, where E−1 =

ET is the orthognal matrix composed of the eigenvectors of I , and Λ = diagλ1, λ2, λ3 is

the diagonal matrix of the principal moments of inertia (eigenvalues). Letting θ = Eψ,

and left-multiplying Eq. (B.4) by ET, the linear equations of motion can be written as

Λψ + ET(γ1+ Γ)E(ǫkdψ + ǫ2kpψ) = 0.


Letting Ξ = ETΓE, and y =(

ǫkdθ + ǫ2kpθ)

, the linear system can further be reduced

to

Λψ + (1− µ)γy = u, (B.5)

where 0 < µ < 1 and −u = (Ξ+ µγ1) y. Taking the Laplace transform (with s as the

transform variable) of Eq. (B.5) yields

y(s) =[

s2

ǫkds+ ǫ2kpΓ+ (1− µ)γ1

]−1

︸︷︷︸

G(s)

u(s),

where G(s) is the system transfer function and is diagonal. In Fig.B.1, Eq. (B.5) is

represented in block diagram form. As is shown in [24], the operator H is strictly

+

−

0 y

−u

G(s)

H

Ξ + µγ1

Figure B.1: Block diagram of Eq. (B.5).

passive, so if γ is chosen such that G (the operator corresponding to G(s)) is passive,

then input-output stability is guaranteed by the passivity theorem (see Chapter 2.6). To

this end, it is realized that the LTI operator G is passive if its corresponding transfer

matrix (G(s)) is positive real. Since it is diagonal, this requirement implies that each of

its elements, Gi(s), must be positive real. As shown in [35], Gi(s) is strictly positive real

(which is a stronger condition implying positive realness) if

(a) it is analytic in Res ≥ 0;

(b) ReGi(jω) > 0, for ω ∈ (−∞,∞);

(c) limω→∞

ω2ReGi(jω) > 0.

Condition (a) is automatically satisfied given that γ > 0, kp > 0, kd > 0, ǫ > 0, and

µ < 1. Also, manipulating the equation for G(s) yields

ReGi(jω) =ǫ2[ǫ2k2pγ(1− µ) + ω2(k2dγ(1− µ)− kpλi

]

[(1− µ)γǫ2kp − ω2λi]2 + [(1− µ)γǫkdω]

2


and

limω→∞

ω2ReGi(jω) =ǫ2 [k2dγ(1− µ)− kpλi]

λ2i.

From these equations it is clear that a sufficient condition for (b) and (c) is that, letting

µ→ 0,

k2dγ − kpλi > 0,

which is satisfied for i = 1, 2, 3 if

γ >kp

k2dλmax. (B.6)

Therefore, if Eq. (B.6) holds, then G(s) is strictly positive real and the linearized system

is input-output stable via the passivity theorem. The rest of the analysis in [24] takes

this result and shows that it also implies asymptotically stability. It is important to note

that the result of Eq. (B.6) differs from that in [24] by a factor of λmax.

Appendix C

Floquet Stability Diagram for Rigid

Spacecraft

This section contains stability diagrams illustrating the results of the Floquet stability

analysis on the rigid version of the baseline spacecraft given in Chapter 3.5. The diagrams

are comparable to Figs. 5.1 and 5.3, which show the results of the same analysis but for

the flexible case.

0 0.5 1 1.5 2 2.5 3 3.5

x 10−3

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

γ

ε

Figure C.1: Stability diagram obtained using Floquet theory for the rigid version of the baselinesample spacecraft (∗ = L-stable, = unstable).

88

Appendix C. Floquet Stability Diagram for Rigid Spacecraft 89

0 1 2

x 10−4

0.012

0.0125

0.013

0.0135

0.014

0.0145

0.015

0.0155

0.016

0.0165

0.017

γ

ε

Figure C.2: Interesting region of the Floquet stability diagram for rigid spacecraft (∗ =L-stable, = unstable).

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