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SATELLITE ATTITUDE STABILIZATION

WITH TWO CONTROL TORQUES

- A PRESTUDY -

TER M P ROJECT

FAL L 2003

ØYVIND H EGRENÆS

DEPARTMENT OF E NGINEERING C YBERNETICS

NORWEGIAN U NIVERSITY OF S CIENCE AND T ECHNOLOGY



Preface

The work presented in the following is the result of a term project, carried out in the 9thsemester of the Master of Science program at the Norwegian University of Science and Tech-

nology, NTNU, Department of Engineering Cybernetics. Even though the report is self-standing

by itself, it should be considered a prestudy of a potential procedure of achieving complete

three-axis stabilization of an underactuated satellite. At the time of writing the problem is still

unsolved in the literature.

I would like to thank my supervisor Professor Kristin Y. Pettersen and advisor Assistant Profes-

sor Jan Tommy Gravdahl for their support,valuable advice and interesting discussions during

this work.

Øyvind Hegrenæs

Trondheim 2003-11-28



ii



Abstract

The topic of this report is attitude stabilization of an underactuated rigid satellite. Underactu-ated mechanical systems are characterized by the fact that there are more degrees of freedom

than actuators. In our case, the satellite has three degrees of freedom, but only two available

actuators. The problem is interesting both from a practical and theoretical point of view and

has received much attention in the last decades.

The spacecraft is modelled as an ideal rigid body and various topics within spacecraft and

astrodynamics are included to provide a thorough insight in how the model is derived. To

represent its attitude, the relatively new (w, z)-parametrization is utilized. By means of its

properties, it is very interesting for the attitude control problem. It is a minimal and compact

parametrization with the singularity moved as far away from the origin as possible, and the

motion of the z-axis is decoupled from the rest of the system.

It is shown that the underactuated satellite model fails to satisfy Brockett’s necessary con-

dition, hence it can not be asymptotically stabilized by means of a continuous time-invariant

pure-state feedback. This is also shown to be applicable for the discontinuous case as well.

Solving the complete three-axis stabilization problem for an underactuated rigid spacecraft

is still an open problem in the literature. A novel approach is proposed, but unlike preceding

work, a strategy utilizing theory related to nonlinear systems in cascades is exploited. Together

with well known methods like Lyapunov design and backstepping, the proposed strategy seems

highly potential, and hopefully it will make a contribution in the problem of achieving com-

plete three-axis stabilization in sense of an underactuated satellite. The proposed approach aimtoward the rather ambitious goal of achieving global uniform asymptotic stability (GUAS) for

the closed-loop system.

Keywords: (w, z)-parametrization; spacecraft and astrodynamics; controllability; stabilizabil-

ity; cascaded systems; attitude control



iv



Contents

Preface i

Abstract iii

1 Introduction 1

1.1 A real-life scenario: An example . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Nonholonomic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Previous work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4 Outline of the report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Attitude parameterizations 5

2.1 The rotation matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.1 Kinematic differential equation . . . . . . . . . . . . . . . . . . . . . 6

2.1.2 Attitude deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Euler parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Rodrigues parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.1 The classical Rodrigues parameters . . . . . . . . . . . . . . . . . . . 7

2.3.2 The modified Rodrigues parameters . . . . . . . . . . . . . . . . . . . 8

2.4 The (w, z) parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4.1 Parametrization of the rotation matrix R ∈ SO(3) . . . . . . . . . . . 9

2.4.2 Kinematic differential equations . . . . . . . . . . . . . . . . . . . . . 12

2.4.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Spacecraft and astrodynamics 15

3.1 Attitude dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.1 Newton-Euler equations of motion for rigid bodies . . . . . . . . . . . 15

3.1.2 Disturbance torques . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1.3 Control torques and actuators . . . . . . . . . . . . . . . . . . . . . . 22

3.2 Celestial mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Model and control properties of an underactuated satellite 27

4.1 Satellite model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.3 Stabilizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31



vi CONTENTS

4.4 Investigation of the underactuated dynamics . . . . . . . . . . . . . . . . . . . 32

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5 A cascaded approach to stabilization 33

5.1 Stability in cascaded systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.2 The underactuated satellite - a cascaded approach . . . . . . . . . . . . . . . . 37

5.3 Control design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

6 Conclusions 43

6.1 Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Bibliography 45

A Theory 49

A.1 Mathematical preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49A.2 Lyapunov stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

A.3 Strict Lyapunov functions for time-varying systems . . . . . . . . . . . . . . . 50

A.4 Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

A.5 Stabilizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

B Newton-Euler equations of motion 55

B.1 Translational motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

B.2 Angular motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

B.3 Model summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

C Mathematica printouts 59



Chapter 1

Introduction

Control of underactuated mechanical systems has attracted many researchers in recent years.

This is due to their broad range of applications and the theoretical problems they have to of-

fer. Many real-life mechanical systems including aircrafts, helicopters, spacecrafts, underwater

vehicles, surface vessels, mobile robots, walking robots, and flexible systems are examples of

underactuated systems. Formally, underactuated mechanical systems are systems that have

fewer actuators than configuration variables (degrees of freedom). The restriction of the con-

trol makes the control design of these systems rather complicated.

The topic of this report is to establish and investigate a reasonable model of an underactu-

ated satellite, and then finally propose a control strategy with hopes of obtaining complete

attitude stabilization about the satellite’s three principal axes. The system to be studied hasthree degrees of freedom but only two available actuators, which makes it an underactuated

system. The motivation for studying this problem is both practical and theoretical. Usually an

actuator failure is handled by incorporating redundancy in the design. The disadvantage of this

approach is higher weight and a more complicated mechanical system. Especially for small

spacecrafts these factors may be crucial. An alternative is to use more complicated controllers

that manage to achieve the control objectives with only two actuators. From a theoretical point

of view the stabilization of an underactuated system is a challenging problem. Many interesting

control theoretical questions have to be answered and the problem by itself is highly nonlinear.

1.1 A real-life scenario: An example

At the time of writing a Norwegian micro-satellite named NSAT-1, initiated by the Norwegian

Defence Research Establishment (FFI), is being planned. The satellite is to be used for mar-

itime surveillance, and will give frequent coverage of Norwegian Ocean areas.

During a pass over Norwegian waters a passive receiver antenna onboard the satellite will

detect signals from active X-band radars from the vessels underneath. Based on the Angle of

Arrival (AOA) of the signals and pulse parameters, advanced onboard algorithms are used to

determine geographic position of the radar, and thereby the vessel holding it. This is known as

geolocation (Narheim et al., 2001). The author has been fortunate to have participated in the

study and specification phase for the particular satellite.



2 Introduction

From the mission described above it is obvious that the required accuracy must be high, and

the satellite platform should be able to obtain any desired attitude and remain in the same con-

figuration. This should also be the case when different torque perturbations or disturbances arepresent. Disturbances and perturbations are to be discussed subsequently. Since the receiver

antenna onboard needs to be oriented in a fixed direction, hence being able to detect radar sig-

nals in a particular observation area, complete 3-axis stabilization is desired.

To obtain the required performance for the system above, in terms of stability, it would be

desired to find a controller that globally uniformly asymptotically (GUAS) stabilizes the at-

titude configuration for the satellite. If possible, locally uniformly exponentially stability is

preferred, since this gives better convergence rate. If globally uniformly exponentially stability

was to be obtain, this gives strong robustness properties as well.

1.2 Nonholonomic systems

As mentioned earlier, control of underactuated mechanics systems has been a very active topic

of research during the last decades. The research on underactuated systems is a continuation

on the research on nonholonomic systems, as many underactuated systems are subject to non-

holonomic constraints. Nonholonomic constraints can be divided into first-order and second-

order nonholonomic constraints. First-order constraints are non-integrable constrains on the

form Φ(q, q ) = 0, where q and q are generalized coordinates and velocities. Second-order con-

straints are on the form Φ(q, q, q ) = 0, and constrain the acceleration of the system. A common

property of nonholonomic systems is that they can not be stabilized by a pure time-invariant

state feedback. This will later be shown to be the case for our system as well. An excellentintroduction to nonholonomic systems can be found in Kolmanovsky and McClamroch (1995).

The number of publications on nonholonomic and underactuated systems is extensive. In the

next section some of the most interesting publications concerning underactuated spacecrafts

are presented.

1.3 Previous work

There exist numerous research articles on the problem of attitude stabilization of spacecrafts.

Most of these deals with the case of complete control actuation using either reaction wheels,

thrusters or magnetic actuators. Some contributions from Scandinavian scientist are for in-stance Egeland and Godhavn (1994), Dalsmo and Egeland (1997), Skullestad and Gilbert

(2000) and Wisniewski and Blanke (1999). The following summary is taken from Fauske

(2002), and gives an overview of some of the most interesting publications concerning under-

actuated spacecrafts.

The angular velocity control of a rigid body with only one or two controls has been studied

extensively in the literature. The issue of feedback stabilization of the angular velocities has

been solved using various approaches. In Brockett (1985) it was shown by finding a Lyapunov

function, that the null solution of the angular velocity equations is asymptotically stabilizable

by two control torques aligned with two principal axes if the uncontrolled axis is not an axis

of symmetry. The angular velocity of a rigid body can in fact be asymptotically stabilized bysmooth feedback with a single control as long as the control is not aligned with a principal axis



1.3 Previous work 3

(Aeyels and Szafranski, 1988).

Stabilization of the angular velocities of a symmetric rigid body is addressed in Andriano(1993) and Outbib (1994). It is shown that that the angular velocities can be globally stabilized

by means of linear feedback when two control torques act on the body. In Reyhanoglu (1996)

it is shown that the angular velocity equation of a rigid body with two control torques cannot

be exponentially stabilized using a C1 feedback. Discontinuous feedback laws are proposed

that achieve asymptotic stability with exponential convergence rate.

In Mazenc and Astolfi (2000) the problem of semi-global stabilization of the angular velocity

of an underactuated rigid body in the presence of model errors is addressed and solved using

a smooth, time-varying, dynamic, output feedback control law. Robustness is also addressed

in Morin (1996), where homogeneity properties of the system are exploited, and in Astolfi and

Rapaport (1997). For more references on stabilization of the angular velocity of a rigid bodyrefer to Tsiotras and Doumtchenko (2000) and references therein.

The more difficult problem of feedback stabilization of both the the angular velocities and

attitude equations has also received much attention. One of the earliest investigations of the

attitude control problem was done in Crouch (1984), where necessary and sufficient conditions

for the controllability of a rigid body in the case of one, two and three independent control

torques was provided. In the case of momentum exchange devices it was shown that controlla-

bility is impossible with fewer than three devices.

In Byrnes and Isidori (1991) the longstanding problem concerning the existence of a time-

invariant smooth state feedback locally asymptotically stabilizing an underactuated rigid space-craft was settled in the negative. However stabilization about an attractor is possible, inducing

a closed-loop system with trajectories tending to a revolute motion about a principal axis. A

discontinuous control strategy was suggested in Krishnan et al. (1994). By switching between

various controllers a sequence of maneuvers were performed that stabilized the spacecraft to

any equilibrium attitude in finite time.

An article by Samson (1991) triggered the discovery that many systems that can not be stabi-

lized by continuous state-feedback can in fact be stabilized by smooth time-varying feedback.

A locally stabilizing smooth time-varying feedback was derived in Morin et al. (1995) by using

center manifold theory combined with averaging and Lyapunov techniques. However, due to

the smoothness of the control laws, the rate of convergence is only polynomial in the worstcase. Similar results where derived in Coron and Keraı (1996) for the general case of torques

that are not aligned with the principal axes of the satellite.

A stronger result was achieved in Morin and Samson (1997) where the attitude of the un-

deractuated rigid spacecraft was locally, exponential stabilized with respect to a given dilation.

The controller was periodic, time varying and non-differentiable at the origin and the construc-

tion relied on the properties of homogeneous systems.

For the axi-symmetric rigid body there exist a wide range of results. However, stabilization

is only possible for the restricted case of zero spin rate about the unactuated axis. Spin-

stabilization with two control torques is addressed in Tsiotras and Longuski (1994) based on a



4 Introduction

new formulation for the attitude dynamics. The new attitude formulation, described in Tsiotras

and Longuski (1995), was subsequently used in Tsiotras et al. (1995) and Tsiotras and Luo

(1996) to derive a time-invariant discontinuous control law that achieves arbitrary reorientationof the spacecraft. Due to the properties of the new parameterization, the control laws derived

were especially simple and elegant. In a more recent paper Tsiotras and Luo (2000), saturated

tracking and stabilization laws were developed for the underacutated axi-symmetric rigid body

under the assumption of zero spin rate. Another time-varying tracking law was developed in

Behal et al. (2002) using a Lyapunov-based approach and by exploiting several characteristics

of the quaternion attitude formulation. The spin rate was not required to be zero, however the

spacecraft could only be driven to an arbitrarily small neighborhood of the origin.

The topic of feasible trajectory generation for the underactuated spacecraft has not received

much attention. An exception is Tsiotras and Luo (2000) where feasible trajectories are gener-

ated for the axi-symmetric rigid body using the notion of differential flatness.

For an excellent overview of developments within actuator failures and control of underac-

tuated spacecraft the reader is referred to Tsiotras and Doumtchenko (2000) and references

therein. This papers also addresses problems that are still to be solved.

1.4 Outline of the report

The report is organized as follows:

• Chapter 2: Different parameterizations of the attitude and their properties are described,

with an emphasis on the relatively new (w, z) parametrization.

• Chapter 3: An introduction to spacecraft dynamics, actuators and astrodynamics is

given. The satellite model to be used later is derived based on the theory in this chapter.

• Chapter 4: A complete model of an underactuated satellite is presented and some im-

portant control properties are discussed.

• Chapter 5: Some theory regarding cascaded nonlinear systems, and how systems can

be transformed into this structure is given. A cascaded version of the satellite model

derived in Chapter 4 is suggested. A promising control strategy for complete attitude

stabilization is proposed, and some properties are discussed in detail.

• Chapter 6: Conclusions and recommendations for further work are given.

• Appendix A: Some mathematical preliminaries, including some important inequalities,

general nonlinear control theory, construction of strict Lyapunov functions, and control-

lability and stabilizability of nonlinear systems.

• Appendix B: Newton-Euler equation of motion for a rigid body.

• Appendix C: Different prints of Mathematica calculations for doing controllability and

stabilizability analysis.



Chapter 2

Attitude parameterizations

Rigid body dynamics is important for a wide range of control applications, and is essential in

robot control, ship control, control of aircraft and satellites, and vehicle control in automotive

systems. In the case of describing the dynamics of a rigid spacecraft, Euler’s equations of mo-

tion are commonly used to provide a complete and well defined framework. For the kinematics

the situation is different, due to the fact that the rotation matrix, which describes the relative

orientation between two reference frames, can be parameterized in more than one way. Which

parametrization to use is clearly dependent on the problem to be solved. This chapter gives

an overview of different attitude parameterizations and describes in detail the relatively new

(w, z) parametrization. The latter is to be used throughout this report.

A possible choice of attitude parametrization is the classical Euler angles, which are com-

monly used in naval and aerospace control problems. However, the use of Euler angles in the

equations of motion may introduce complicated nonlinear expression with inherent singulari-

ties. They are therefore not considered in this report.

2.1 The rotation matrix

The rotation matrix, also called the direction cosine matrix, has three interpretations:

• Describes the mutual orientation between two coordinate frames, where the column vec-

tor are cosines of the angles between the two frames.

• Transforms vectors represented in one reference frame to another.

• Rotates a vector within a reference frame.

The rotation matrix R from frame a to frame b is denoted Rab . A matrix R is a rotation matrix

if and only if it is an element of the set denoted by SO(3), that is,

SO(3) =R ∈ R

3×3 : RTR = I and detR = 1

, (2.1)

where I is the 3 × 3 identity matrix.

A useful parametrization of the rotation matrix is the angle-axis parametrization correspondingto a rotation θ ∈ R about a unit vector k ∈ R3. Rodrigues’ formula (Murray et al., 1994) gives



6 Attitude parameterizations

R(k, θ) = I + S(k)sin θ + S2(k)(1

−cos θ) (2.2)

S(·) is a skew-symmetric matrix operator and member of the set denoted by so(3), that is,

so(3) =S ∈ R

3×3 : ST = −S (2.3)

In coordinate vector notation we introduce the skew-symmetric form of the vector ω defined by

ω× = S(ω) =

0 −ωz ωy

ωz 0 −ωx

−ωy ωx 0

, ω =

ωx

ωy

ωz

(2.4)

2.1.1 Kinematic differential equation

From the properties of SO(3), it can be shown that the kinematic differential equation for the

rotation matrix can be given by the two alternative forms (Egeland and Gravdahl, 2002)

Rab = (ωa

ab)×Rab (2.5a)

Rab = Ra

b (ωbab)

×(2.5b)

where ωaab is the instantaneous angular velocity of frame b relative to frame a as seen from the

a frame. Similar, ωbab is the angular velocity of frame a relative to frame b as seen from the b

frame. Using (2.4) we can rewrite (2.5) as

Rab = S(ωa

ab)Rab = Ra

bS(ωbab) (2.6)

2.1.2 Attitude deviation

Let the frame a define a reference orientation and let frame b be a body fixed frame. Then the

rotation matrix R Rab will describe the orientation of the body. Suppose that the desired

orientation of the body is given by a rotation matrix Rd.

In the case of rotation matrices it does not make sense to subtract Rd from R as the result

would not be a valid rotation matrix. Instead the deviation between the desired and the actualorientation is described by the rotation matrix Ra ∈ SO(3) defined by

Ra RRTd (2.7)

It can be shown (Egeland and Gravdahl, 2002), by using composite rotations, that the kine-

matic differential equations for the attitude deviation can be calculated as

ωa = ωa − ωad (2.8a)

d

dtRa = (ωa)× Ra (2.8b)

Clearly, from (2.7) we se that when R ≡ Rd ⇒ Ra = I.



2.2 Euler parameters 7

2.2 Euler parameters

The Euler parameters, also called unit quaternions, are attractive due to their nonsingularparametrization and linear kinematic differential equations if the angular velocities are known.

The quaternion representation requires much less computations than for instance the Euler

angles representation, and is therefore useful in applications where computer resources are

limited.

The Euler parameters are defined in terms of the angle-axis parameters θ and k, briefly dis-

cussed in regards to equation (2.2). The mapping is defined as

η = cos θ

2, = k sin

θ

2 (2.9)

which gives the corresponding rotation matrixR(η, ) = I + 2ηS() + 2S2() (2.10)

It is found in Egeland and Gravdahl (2002) that the derivatives of the Euler parameters can be

given as functions of the angular velocity, which gives the kinematic differential equations

η = −1

2Tω (2.11a)

= 1

2 [ηI + S()]ω (2.11b)

In component form (2.11) can be written as

η = −12

(ε1ω1 + ε2ω2 + ε3ω3) (2.12a)

ε1 = 1

2(ηω1 − ε3ω2 + ε2ω3) (2.12b)

ε2 = 1

2(ε3ω1 + ηω2 − ε1ω3) (2.12c)

ε3 = 1

2(−ε2ω1 + ε1ω2 + ηω3) (2.12d)

2.3 Rodrigues parameters

The classical and modified Rodrigues parameters can be interpreted as the coordinates resultingfrom a stereographic projection of the four-dimensional Euler parameter hypersphere onto a

three-dimensional hyperplane (Schaub et al., 1995). The difference between them is how the

projection point and mapping hyperplane is chosen.

2.3.1 The classical Rodrigues parameters

The classical Rodrigues parameters can be derived from the Euler parameters with the trans-

formation

q =

η (2.13)

Combining (2.13) and (2.9) yields

q = k tan θ2

(2.14)




Clearly, the classical Rodrigues parameters have a singular condition for θ = ±π, where |q| →∞. The kinematic differential equation is derived from (2.11)

q = d

dt

η =

η− η

η2 (2.15)

which gives the quadratic nonlinear differential equation for the kinematics, that is

q = 1

2

I + S(q) + qqT

(2.16)

In component form (2.16) can be written as

q 1 = 1

2 (1 + q 21)ω1 + (q 1q 2 − q 3)ω2 + (q 1q 3 + q 2)ω3

(2.17a)

q 2 = 12(q 1q 2 + q 3)ω1 + (1 + q 22)ω2 + (q 2q 3 − q 1)ω3 (2.17b)

q 3 = 1

2

(q 3q 1 − q 2)ω1 + (q 3q 2 + q 1)ω2 + (1 + q 23)ω3

(2.17c)

Unlike the Euler parameters, the Rodrigues parameters are numerically unique. They uniquely

define a rotation on the open range of (−π, π). As is evident in equation (2.13), reversing the

sign of the Euler parameters has no effect on the q.

2.3.2 The modified Rodrigues parameters

The modified Rodrigues parameters can be derived from the Euler parameters with the trans-

formationσ =

1 + η (2.18)

Combining (2.13) and (2.18) yields

σ = k tan θ

4. (2.19)

Clearly the the modified Rodrigues parameters have a singular condition for θ = ±2π, which

allow twice the principal rotation angle compared to the classical Rodrigues parameters. From

(2.13) and in (2.11) we get the differential kinematic equations

σ = 1

4 (1 − σT

σ)I + 2S(σ) + 2σσTω (2.20)

which can be written in component form as

σ1 = 1

4(1 + σ2

1 − σ22 − σ2

3)ω1 + 1

2(σ1σ2 − σ3)ω2 +

1

2(σ1σ3 + σ2)ω3 (2.21a)

σ2 = 1

2(σ2σ1 + σ3)ω1 +

1

4(1 − σ2

1 + σ22 − σ2

3)ω2 + 1

2(σ2σ3 − σ1)ω3 (2.21b)

σ3 = 1

2(σ3σ1 − σ2)ω1 +

1

2(σ3σ2 + σ1)ω2 +

1

4(1 − σ2

1 − σ22 + σ2

3)ω3 (2.21c)

The equations display a similar degree of nonlinearity as do the corresponding equations in

terms of the classical Rodrigues parameters. However, unlike the classical Rodrigues parame-ters, the modified Rodrigues parameters are not unique. This can be seen in equation (2.18).



2.4 The (w, z) parametrization 9

(a) (b)

Figure 2.1: Rotation sequence of the (w, z) parametrization. Left hand side shows the initial

rotation about the body i3-axis. Right hand side shows the resulting coordinate system after

the second rotation.

2.4 The (w, z ) parametrization

The three-dimensional (w, z) parametrization is a relatively new formulation for describing

the relative orientation of two reference frames using two perpendicular rotations. Although it

uses three parameters to describe the motion, two of the parameters can be combined to a single

complex variable. The complex variable is used to designate the second of the two rotations

and it is derived using stereographic projection (Conway, 1978). The remaining parameter

represents the initial rotation. As will be shown, the two rotations can be completely decoupled,which has important implications and advantages. The parametrization is probably unfamiliar

to the reader and is therefore derived in detail. The material in this section is based on the work

by Tsiotras and Longuski (1995, 1996).

2.4.1 Parametrization of the rotation matrixR ∈ SO(3)

As mentioned above, we wish to derive a parametrization of the rotation matrix R in section

2.1 using two successive rotations. The resulting rotation matrix can thus be decomposed as

R(w, z) Rbi(w, z) = Rb

o(w)Roi (z) (2.22)

This can be interpreted as follows. Consider a inertial frame i, defined by three orthogonal unit

vectors i1, i2 and i3. Another frame b is fixed to the body and is defined by the three orthogonal

unit vectors b1, b2 and b3. The rotation matrix R in (2.22) is then the rotation matrix from the

the body frame to the inertial frame. The intermediate frame o, defined by the three orthogonal

unit vectors o1, o2 and o3 is the result of the first rotation.

The first rotation is shown in figure 2.1 a) and represents a simple rotation about the inertial i3-axis. This resulting rotation matrix can be written as

Roi (z) = cos z sin z 0

− sin z cos z 00 0 1

R1(z) (2.23)




The next step is to find an expression for the second rotation matrix Rbo(w) R2(w). Recall

that a rotation matrix can be computed using an angle-axis description, which was shown in

section 2.1 by using Rodrigues’ fomula (2.2). The formula is repeated here for clarity;

R(k, θ) = I + S(k)sin θ + S2(k)(1 − cos θ) (2.24)

Assume that o3 = a b1 + b b2 + c b3, which is the o3-axis in the intermediate frame, transformed

to the body frame. In sense of the rotation matrix Rbo(k, θ) this can be written asa

bc

= Rbo(k, θ)o3 (2.25)

The resulting vector [a,b,c]T is clearly the third column in the rotation matrix R2. Also, it can

be shown that (−a, −b, c) are the directed cosines of the b3-axis in the o frame, that is

b3 = −ao1 − bo2 + co3 (2.26)

The angle between b3 and o3 is simply found from the vector dot product as

θ = cos−1 o3 · b3

|o3|| b3|= cos−1 c (2.27)

The axis of rotation can be found from (note that k has the same coordinates in both frames)

k =

o3

× b3

o3 × b3 (2.28)

Using o3 = [0, 0, 1]T, when refereed to the intermediate reference frame, and (2.26), the axis

of rotation can be calculated as

k = bo1 − ao2√

a2 + b2 (2.29)

Now the angle-axis description of the rotation matrix can be used. Insertion of (2.29) and (2.27)

into (2.24) gives the rotation matrix from the intermediate frame to the body frame, that is

Rob(k, θ) =

c + b2

1+c − ab1+c −a

− ab1+c c + a2

1+c

−b

a b c

(2.30)

By taking the transpose of (2.30) we get the following expression for the rotation matrix R2(w)

Rbo(k, θ) =

c + b2

1+c − ab1+c a

− ab1+c c + a2

1+c b

−a −b c

R2(w) (2.31)

Expanding (2.22) with (2.23) and (2.31) gives the complicated matrix

R(w, z) = c cos z+ab sin z+(b2+c2)cos z

1+cc sinz−ab cos z+(b2+c2)sin z

1+c a

−c sinz+(c2+a2)sin z+ab cos

1+c z c cos z+(c2+a2)cos z−ab sin z

1+c b−b sin z − a cos z −b cos z − a sin z c (2.32)




Figure 2.2: Stereographic projection of a point (a,b,c) on a unit sphere on to a complex plane.

The foregoing matrix is redundant, in the sense that the elements a,b,c are not independent, but

satisfy the constrainta2 + b2 + c2 = 1 (2.33)

We can therefor eliminate one of the elements in (2.32) to obtain a simplified form. One way

of doing this is to use a stereographic projection. From the constraint in (2.33) we introduce

the set S 2 =

(x1, x2, x3) ∈ R3 : x21 + x2

2 + x23 = 1

∈ R3, which is the unit sphere. For

(a,b,c) ∈ S 2, we get the mapping σ : S 2\{(0, 0, −1)} → C, (a,b,c) → w. The stereographic

projection is shown in Figure 2.2 and is defined as

w w1 + iw2 = b − ia

1 + c (2.34)

It can be verified that the inverse map σ−1 : C → S 2\{(0, 0, −1)}, w → (a,b,c) is given by

a = i(w − w)

1 + |w|2 , b = w + w

1 + |w|2 , c = 1 − |w|21 + |w|2 (2.35)

where w is the complex conjugate of w and |w|2 = ww = w21 + w2

2. The basis of the projection

is the point (0, 0, −1), which is the south pole S of the unit sphere. Note that when c = −1the projection has a singularity and w → ∞. The singularity corresponds to an upside-down

orientation of the body. Using (2.35) we can express R2(w) in terms of (2.34) as follows

R2(w) =

1

1 + w21 + w22 1 + w2

1 − w22 2w1w2 −2w2

2w1w2 1 − w

2

1 + w

2

2 2w12w2 −2w1 1 − w2

1 − w22 (2.36)

When using complex notation, a more compact matrix can be given as

R2(w) = 1

1 + |w|2

1 + Re(w2) Im(w2) −2Im(w)Im(w2) 1 − Re(w2) 2Re(w)2Im(w) −2Re(w) 1 − |w|2

(2.37)

The total rotation matrix R(w, z) is then given in terms of w and z as follows

β · (1 + w2

1 − w22)cz − 2w1w2sz (1 + w2

1 − w22)sz + 2w1w2cz −2w2

2w1w2cz − (1 − w21 + w

22)sz 2w1w2sz + (1 − w

21 + w

22)cz 2w1

2w2cz + 2w1sz 2w2sz − 2w1cz 1 − w21 − w2

2

(2.38)




where cz and sz denotes cos(z) and sin(z), respectively, and β = 1/(1 + |w|2).

The foregoing matrix can be written more compactly as

R(w, z) = 1

1 + |w|2

Re(1 + w2)eiz Im(1 + w2)eiz −2Im(w)Im(1 − w2)e−iz Re(1 − w2)e−iz 2Re(w)

2Im(weiz) −2Re(weiz) 1 − |w|2

(2.39)

2.4.2 Kinematic differential equations

From (2.6) we can derive the kinematic differential equations for the attitude motion of the rigid

body. The differential equation for (2.22) becomes R(w, z) = S(ω)R(w, z), where ω = ωbbi

is the angular velocity of the inertial frame relative to the body frame, as seen from the body

frame. It can be shown however, that ωb

bi = −ωb

ib (Kane et al., 1983). This relation makes

it possible to derive the kinematics using the more intuitive angular velocity ωbib, which is the

angular velocity of the body frame relative to the inertial frame, as seen from the body frame.

For the reminder we let ω = ωbib [ω1, ω2, ω3]T, and the redefined differential equation for

(2.22) becomes R(w, z) = −S(ω)R(w, z), where S(ω) was defined in (2.4).

It can easily be verified that the third column of R(w, z) must satisfya

bc

= −S(ω)

abc

(2.40)

Recall from (2.34) that w is defined as

w = b − ia

1 + c (2.41)

Differentiation of (2.41) gives

w =b − ia − wc

1 + c (2.42)

Using the relations in (2.35) and (2.40) gives the differential equation for w ∈ C

w = −iω3w + ω

2 +

ω

2w2 (2.43)

where

ω = ω1 + iω2, ω = ω1 − iω2, i =√ −1 (2.44)

An alternative formulation, which will be used throughout, is given as

w1 = ω3w2 + ω2w1w2 + ω1

2 (1 + w2

1 − w22) (2.45a)

w2 = −ω3w1 + ω1w1w2 + ω2

2 (1 + w2

2 − w21) (2.45b)

To find the differential equation for z we start with the scalar form of the differential equation

for a rotation matrix,tr[ R(w, z)] = tr[−S(ω)R(w, z)] (2.46)




where tr(·) denotes the trace of the matrix. Taking the trace of R(w, z) gives

tr[ R(w, z)] = ddt (tr[R(w, z)]) = ddt 2cos z + 21 + w21 + w2

2− 1

= − 2z sin z

1 + w21 + w2

2

− 4(1 + cos z)(w1 w1 + w2 w2)

(1 + w21 + w2

2)2 (2.47)

Combining (2.45a) and (2.45b) gives the relation

2 w1 w1 + w2 w2

(1 + w21 + w2

2)2 = ω1w1 + ω2w2 (2.48)

Substituted (2.48) into (2.47) gives the expression

tr[ ˙R(w, z)] =

2

1 + w21 + w2

2 z sin z + (1 + cos z)(ω1w1 + ω2w2) (2.49)

Expanding the right hand side of (2.46) we obtain

tr[−S(ω)R(w, z)] = −2

1 + w21 + w2

2

[(1 + cos z)(ω1w1 + ω2w2) + (ω3 − ω1w2 + ω2w1)sin z]

(2.50)

Equating (2.50) with (2.49), we obtain the following differential equation for the angle z

z = ω3 − ω1w2 + ω2w1 (2.51)

or equivalently,

z = ω3 + i

2(ωw

−ωw) (2.52)

To summarize the discussion above, the differential kinematic equations for the (w, z) parametriza-

tion arew1 = ω3w2 + ω2w1w2 +

ω1

2 (1 + w2

1 − w22) (2.53a)

w2 = −ω3w1 + ω1w1w2 + ω2

2 (1 + w2

2 − w21) (2.53b)

z = ω3 − ω1w2 + ω2w1 (2.53c)

Alternatively they can be written more compactly as

w = −iω3w +

ω

2 +

ω

2 w

2

, (2.54a)

z = ω3 + i

2(ωw − ωw) (2.54b)

Remark 2.4.1. It is straight forward to verify that the kinematic differential equations for the

(w, z) parametrization can be written as

d

dt|w|2 = (1 + |w|2)Re(ωw) (2.55a)

z = ω3 + Im(ωw) (2.55b)

Remark 2.4.2. Equation (2.34) is only one of the possible definitions of the parameter w.

Other combinations will provide different kinematic equations for w and z. This is shown inTable 2.1.




w Kinematics ω

a+ib1+c

w =−

i ω3w−

ω

2

+ ω

2

w2z = ω3 + 12 (ωw + ωw) ω1 + iω2

b−ia1+c

w = −iω3w + ω2 + ω

2 w2

z = ω3 + i2 (ωw − ωw)

ω1 + iω2

b+ic1+a

w = −i

ω1w − ω2 + ω

2 w2

z = ω1 + 12 (ωw + ωw)

ω2 + iω3

c−ib1+a

w = −iω1w + ω2 + ω

2 w2

z = ω1 + i2 (ωw − ωw)

ω2 + iω3

c+ia1+b

w = −i

ω2w − ω

2 + ω2 w2

z = ω2 +

1

2 (ωw + ωw)

ω3 + iω1

a−ic1+b

w = −iω2w + ω2 + ω

2 w2

z = ω2 + i2 (ωw − ωw)

ω3 + iω1

Table 2.1: Stereographic coordinate w and corresponding kinematics

2.4.3 Properties

The (w, z) parametrization has some unique properties that makes it useful in attitude control

problems.

• The kinematic equations are compact and have a clear physical interpretation. It can be

realized using two rotations about perpendicular axes.

• The z parameter does not appear in (2.53a) and (2.53b). This means that in some appli-

cations the control problem can be decomposed into one of controlling only w and one

of controlling z.

• A three dimensional parametrization will always involve singularities. A singularity

appears in the (w, z) parametrization when the body is upside down and consequently

w → ∞. The equilibrium (w1, w2, z) = (0, 0, 0) is nevertheless as far away from the

singularity as possible.

• The (w, z) parametrization can easily be connected to other well known parametrizations

like the Eulerian angles, Euler-Rodrigues parameters and Angle-Axis parametrization.

2.5 Discussion

The previous sections have shown that there are many attitude parameterizations to choose

from. However, the (w, z) parametrization will be used in the rest of this report to describe the

attitude dynamics of an underactuated spacecraft. First of all this parametrization is minimal

and it is easy to avoid the singularity as long as we ensure that |w1| and |w2| does not approach

∞. Secondly, if the actuator failure is about the z-axis, the dynamics of the unactuated axis

can be decoupled from the rest of the system.



Chapter 3

Spacecraft and astrodynamics

The study of the dynamics of objects in interplanetary or interstellar space is called astrody-

namics and has two major divisions. Celestial mechanics or orbit dynamics is concerned with

the motion of the center of mass of objects in space, whereas attitude dynamics is concerned

with the motion about the center of mass. Since the attitude dynamics is the main focus of this

report, only a brief review will be given on the celestial mechanics. For the interested reader,

Sellers (2000) is highly recommended as an introduction to astrodynamics. For more in-depth

information see Hughes (1986) and Wertz (1978, 1999). The work in this chapter is based on

these references.

3.1 Attitude dynamics

The first part of this chapter considers Newton-Euler equations of motion for rigid bodies, as

well as giving an overview of disturbance and control torques. Some of the equations found in

this chapter, together with the differential kinematic equations that were found in the foregoing

chapter, make up the system to be investigated in later chapters. Unless otherwise is stated, the

principal axes in the equations will coincide with the body reference system.

3.1.1 Newton-Euler equations of motion for rigid bodies

The angular motion of a spacecraft can be modelled as an ideal rigid body. However, most

spacecrafts have flexible parts like for instance antennas and solar panels. Also, internal effects

like fuel sloshing and thermal deformations are not accounted for using a rigid body model.

Nevertheless, for many problems the rigid body model is a good approximation, especially for

small spacecrafts.

Since only rotational motion will be considered throughout, the translational case will not be

taken into account at this stage. However, a detailed derivation of the equations of motion for

both cases is given in Appendix B. For the rotational case, when referred to the center of mass

and the body reference system, the well known equations for a rigid body can be written as

Mωbib + S(ωb

ib)Mωbib = τ =

i

τ i (3.1)

whereM is the inertia matrix for the rigid body, referred to the center of mass, τ = [τ x, τ y, τ z]T

is the total torque acting on the body, and ωbib = [ω1, ω2, ω3]T is the angular velocity as ex-



16 Spacecraft and astrodynamics

plained in chapter 2. The torques τ i, acting on the individual mass elements in the body, are

due to both forces between individual mass elements and externally applied forces. Usually

the internal torques sum to zero and the resultant torque is simply the torques due to externalforces. The external torques τ e can be divided into two groups, called disturbance torques and

control torques. The first case is caused by environmental effects such as aerodynamic drag

and gravity gradient torque, while the latter is deliberately applied torques from devices such

as thrusters or magnetic coils. Both cases will be discussed in the following.

Assuming a diagonal inertia matrix, M = diag{m11, m22, m33}, the dynamics in equation

(3.1) can easily be found to be given in component form as

m11 ω1 + (m33 − m22)ω2ω3 = τ x (3.2a)

m22 ω2 + (m11 − m33)ω3ω1 = τ y (3.2b)

m33 ω3 + (m22 − m11)ω1ω2 = τ z (3.2c)

Remark 3.1.1. By defining the angular momentum h Mωbib, and assuming only external

torques, equation (3.1) can be rewritten as

dh

dt = τ e − S(ωb

ib)h (3.3)

From this equation it can easily be seen that the angular momentum h, and hence ωbib, is not

constant in the body frame, even when the external torque τ e is equal to zero. The resulting

motion is called nutation. Rotational motion without nutation only occurs when ωbib h, that

is, only if the rotation is about a principle axis of the rigid body.

Remark 3.1.2. A spacecraft equipped with reaction or momentum wheels is not a rigid bodyin the sense that they cause a redistribution of the angular momentum between the wheels and

the spacecraft body. The wheels do not change the total angular momentum of the spacecraft,

hence they can not be external torques.

However, in the case of using reduction or momentum wheels in the spacecraft body, the equa-

tions above can still be used with one minor modification. To encounter for the angular mo-

mentum of the wheels, we redefine the total angular momentum for the spacecraft, including

wheels, as

hb = Mωbib + hw (3.4)

where the inertia matrix M includes the mass of the wheels and the vector hw = [h1, h2, h3]T

is the net angular momentum due to the rotation of the wheels relative to the body. Using asimilar procedure as when deriving (3.1) we get the following equation of motion

Mωbib + S(ωb

ib)(Mωbib + hw) = τ − dhw

dt (3.5)

The quantity dhw/dt is the net torque applied to the wheels from the spacecraft body, so by

Newton’s 3rd law of motion, -dhw/dt [τ wx, τ wy , τ wz ]T is the torque applied to the spacecraft

body by the wheels. Writing (3.5) in component form in the body system, referred to the center

of mass, yields

m11 ω1 + (m33 − m22)ω2ω3 + h3ω2 − h2ω3 = τ x + τ wx (3.6a)

m22 ω2 + (m11

−m33)ω3ω1 + h1ω3

−h3ω1 = τ y + τ wy (3.6b)

m33 ω3 + (m22 − m11)ω1ω2 + h2ω1 − h1ω2 = τ z + τ wz (3.6c)



3.1 Attitude dynamics 17

Remark 3.1.3. A rigid body with one or more spinning wheels is commonly called a gyrostat .

An alternative representation of the multi-spin system described in (3.5) is derived in Hughes

(1986), in the case of using only one wheel. In Tsiotras et al. (2001) the representation is

expanded to include any number of wheels, which makes it quite practical to use. The rotational

equations of motion for a N -wheel gyrostat can be written as

hb = τ − J−1(hb −Aha)

× hb (3.7a)

ha = τ a (3.7b)

or equally

Jωbib = τ −Aha − S(ωb

ib)(Jωbib + Aha) (3.8a)

ha = τ a (3.8b)

where ha is the N × 1 vector of the axial angular momenta of the wheels, τ is the 3 × 1 vectorof the total torque acting on the body, not including wheel torques, τ a is the N ×1 vector of the

internal axial torques applied by the platform to the wheels, and A is the 3 × N matrix whose

columns contain the axial unit vectors of the N momentum exchange wheels. The vector hb is

the total angular momentum for the spacecraft in the body frame, given by

hb = Jωbib + Aha (3.9)

J is the inertialike matrix defined as

J M−AMsAT (3.10)

where M is the moment of inertia matrix for the spacecraft, including wheels, and the matrix

Ms = diag{Ms1,Ms1,...,MsN } contains the axial moments of inertia of the wheels on thediagonal. The axial angular momenta of the wheels can be written in terms of the body angular

velocity and the wheels’ axial angular velocities relative to the body, ωs, as

ha = MsATωb

ib + Msωs (3.11)

Note that ωs = [ωs1,ωs2, ...,ωsN ]T is a N × 1 vector, and that these relative angular ve-

locities are those that would for instance be measured by tachometers fixed to the platform.

We denote the resultant axial angular velocity of the wheels, relative to the inertial frame as

ωc = [ωc1,ωc2, ...,ωcN ]T. Using this notation we can write (3.11) as

ha = Msωc (3.12)

where ωc

= ωs + ATωb

ib. Note that because typically the wheels spin at a much higher speed

than the spacecraft itself, ωs ωbib and we have that ωc ≈ ωs.

3.1.2 Disturbance torques

As mentioned above, the environmental disturbances contribute as torques on the spacecraft

body, making them noneligible when doing attitude prediction in real-life. If they were to be

added to the equations of motion they would have to be modelled as a function of time as well

as the orientation of the spacecraft. Worth noticing is that the external torques will change the

angular momentum of the spacecraft, while the internal torques only will affect the distribution

of momentum between the moving parts. Some disturbance torques are given in Table 3.1,

indicating at what height above the surface they are most likely to dominate. In the following

an overview of the most important disturbance torques is given. This could contribute as agood starting point, if advanced models or robustness issues are to be considered.




Table 3.1: Environmental disturbance torques

External torque Region of space where

source dominant∗

Aerodynamic < 500km†

Gravity gradient 500km to 35000km

Magnetic 500km to 35000km

Solar pressure > 700km†

Thrust misalignment all heights

Internal torque

source

Mechanical and electrical devices

Fuel sloshing

General mass movement

Flexible appendages

∗The specific altitude at which the various torques

dominate are highly spacecraft dependent†Value depends upon the level of solar activity

Aerodynamic torque

The interaction of the upper atmosphere with a spacecraft’s surface produces a torque about the

center of mass. The effect is clearly dependent on the area and shape of the exposed surface. In

general the impact of the atmospheric molecules can be modelled as an elastic impact without

reflection. For low orbit spacecrafts the air density is high enough to influence the attitude

dynamics of the body. Calculating the aerodynamic torques and forces can be done in several

ways, and more or less all approaches lead to rather complicated expressions. Use of empiric

data is also common.

If the spacecraft surface comprises a collection of small incremental areas dA, each with out-

wards unit normal n, then the force on a surface element is given by

df aero = −1

2ρv2C D(n · v)vdA (3.13)

where v is the translational velocity of the surface element relative to the incident stream, and vis the unit vector in the same direction. The coefficients ρ and C D are the atmospheric density

and the drag coefficient, respectively. Performing a summation over all such areas gives the

simplified expression for the aerodynamic torque, that is

τ aero =

rs×df aero (3.14)

where rs is the position vector from the center of mass of the body to the surface element

dA. Usually this integral is not amenable to simple solutions for a surface associated with acomplex structure. A commonly used alternative is to represent the spacecraft as a collection




of simple geometrical elements. The torque about the center of mass of the spacecraft is then

the vector sum of the individual torques for each of these geometrical simplifications, that is

τ aero =i

ri × Faero,i (3.15)

The vector ri is in this case the vector distance from the center of mass of the spacecraft to

the center of pressure of the the specific geometric shape and Faero,i is the force acting on the

component. Aerodynamic forces for some simple geometric figures are listed in Table 3.2.

Geometric figures Faero,i

Sphere of radius R −12ρv2C DπR2v

Plane with surface area A andnormal unit vector n −1

2ρv2

C DA(n · v)v

Table 3.2: Aerodynamic force for some simple geometric figures

Remark 3.1.4. The integral in (3.14) is taken over the spacecraft surface for which n · v > 0,

that is, the surface exposed to the incoming flow of atmospheric particles.

Gravity gradient torque

As a result of the nonuniform gravitational field surrounding the earth, any nonsymmetrical

object in orbit is subject to a gravitational torque. It is important to emphasize that this can

only occur as long as there are variations in the specific gravitational force over the spacecraft.

By first defining an orbiting frame o as in Figure 3.1, the expression for the gravity gradi-

ent torque in (3.16) can be derived. The unit vector o3 that appear in the equation is called the

local vertical and by definition it is always pointing toward nadir (center of the earth).

τ g = 3µ

R3c

[o3 × (Mo3)] (3.16)

The parameters in (3.16) are summarized in the following table

Symbol Explanation

µ Gravitational coefficient, µ = 3.986 · 1014 Nm2/kgRc Distance to center of the earth (m)

M Spacecraft inertia matrix

o3 Unit vector toward nadir

Remark 3.1.5. The expression in (3.16) is rather simplified due to the four assumptions

a) Only one celestial primary is considered. In most cases the primary will be the earth.

b) The primary possesses a spherically symmetrical mass distribution.

c) The spacecraft is small compared to its distance from the mass center of the primary.

d) The spacecraft consists of a single body.

In most spacecraft situations these are realistic assumptions.




Figure 3.1: Orbit frame

Remark 3.1.6. There is no gravitational torque about the local vertical, that is

τ g · o3 = 0

As can be seen from (3.16), the vector equation is given in the local orbiting frame o. However,

in most cases it would be useful to represent the torque in the body fixed reference frame. By

letting the rotation matrix Rbo represent the rotation matrix from the body frame to the orbit

frame, we get the following expression for the gravitational torque, as referred to the body

frame

τ bg = 3µ

R3c

rb3 ×

Mrb3

(3.17)

where rb3 = [r13, r23, r33]T is the third column in the rotation matrix Rbo. Independent of the

attitude parametrization we use to represent the rotation matrix, and assuming a diagonal inertiamatrix M = diag{m11, m22, m33}, the gravitational torque in (3.17) simplifies to

τ bg = 3µ

R3c

(m33 − m22)r23r33(m11 − m33)r33r13(m22 − m11)r13r23

(3.18)

Solar pressure torque

Solar radiation pressure produces a force on a surface, which depends upon its distance to the

sun. Since light carries momentum, it represents an exchange of momentum with the surface

when it is reflected. For most applications, the forces may be modelled adequately by assum-ing that the incident radiation is either absorbed, reflected specularly, reflected diffusely, or in

some combination of these.

If the spacecraft surface comprises a collection of small incremental areas dA, each with out-

wards unit normal n, and s is the unit vector from the spacecraft to the sun, then the force on a

surface element due to solar radiation is given by

df solar = −P cos θ

(1 − f s)s + 2(f s cos θ +

1

3f d)n

dA (3.19)

where P is the mean momentum flux (∼ 4.67 · 10−6

N m2

at the earth), and f s and f d are thecoefficients of specular and diffuse reflection, respectively. The angle of incidence radiation is




given as θ = cos−1(s · n). Performing a summation over all such areas gives the expression

for the solar pressure torque, that is

τ solar = rs×df solar (3.20)

where rs is the position vector from the center of mass of the body to the surface element

dA. This integration is in general difficult to solve for a surface associated with a complex

structure. However, as for the aerodynamic torque, a commonly used alternative is to represent

the spacecraft as a collection of simple geometrical elements. The total torque about the center

of mass of the spacecraft is then the vector sum of the individual torques for each of these

geometrical simplifications.

τ solar =

iri × Fsolar,i (3.21)

The vector ri is in this case the vector distance from the center of mass of the spacecraft to

the center of pressure of the the specific geometric shape and Fsolar,i is the force acting on the

component. Solar radiation forces for some simple geometric figures are listed in Table 3.3.

Geometric figures Fsolar,i

Sphere of radius R −P (4πR2)14 + f d

9

s

Plane with surface area A

and normal unit vector n :

θ = cos−1(s · n)

−P A cos θ

(1 − f s)s + 2

f s cos θ + f d3

n

Table 3.3: Solar radiation force for some simple geometric figures

Remark 3.1.7. The integral in (3.20) is taken over the spacecraft surface for which s · n > 0,

that is, the surface exposed to the incoming radiation.

Magnetic torque

The residual magnetic field generated by a spacecraft interacts with the local field from the

earth and thereby exerts a couple on the body. The effect of the magnetic torque is altitude

dependent and strongest at low altitudes. The instantaneous magnetic disturbance torque τ mag

due to the spacecraft effective magnetic moment m is given by

τ mag = m ×B (3.22)

where B is the geocentric magnetic flux density and m is the sum of the individual magnetic

moments caused by permanent and induced magnetism and spacecraft generated current loops.

Internal torques

Internal torques are defined as torques exerted on the spacecraft body by such internal moving

parts as reaction and momentum wheels, fuels or liquids inside partially filled containers orflexible constructions. The reaction and momentum wheels are usually not to be considered




as disturbances, but are included here for definition. In a highly hypothetical scenario, the in-

ternal torques would also exist if the spacecraft was to be removed entirely from all external

influences in space.

As mentioned earlier the external torques will change the angular momentum of the space-

craft, while the internal torques only will affect the distribution of momentum between the

moving parts. However, even though the angular momentum remains constant in the absence

of external torques, the kinetic energy for the body could change and in most cases would the

redistribution of the angular momentum between the moving parts lead to a change in the dy-

namic characteristics.

In general the internal disturbance torques are undesired, hence must be encountered for using

external torques. Also, when designing a spacecraft the effects of internal disturbances could

be tried encountered for in the sense of placing the different devices in a clever manner.

3.1.3 Control torques and actuators

Deliberately applied torques can be generated using various approaches, and commonly as a

combination of several different actuators. The types of actuators that can be used to control

the orientation of a spacecraft can usually be divided into three categories. These are thrusters,

momentum exchange devices and magnetic actuators. The actuator can also be categorized to

be either active or passive. Some control torques, and their properties, are given in Table 3.4.

A short explanation of some of the different designs is also given subsequently.

An important part of the design of a spacecraft is to decide the size and properties of the actu-ators since it is crucial to have enough control to overcome the disturbances discussed above,

as well as getting the spacecraft into its desired configuration. However, this topic will not be

discussed any further in this report, and the reader is referred to Wertz (1999) for details.

In later chapters the system will only have two available torques, hence making the system

underactuated. More details about this matter will be given shortly.

Gravity gradient

As mentioned earlier any nonsymmetrical object in orbit is subject to a gravitational torque.

Although this effect often is considered as a disturbance, it can also be utilized as a passivecontrol torque. This is commonly done using a gravitational boom. However, in the sense of

stabilizing a spacecraft, the body will only be in a stable equilibrium if its axes of minimum

inertia is aligned with the local vertical. Due to low accuracy and the need for damping makes

the use of other control torques necessary as well. In the Danish satellite Ørsted (Wisniewski

and Blanke, 1999) the gravity gradient was utilized together with magnetic actuators to archive

complete three-axis stabilization.

Solar radiation

In the previous section on disturbance torques, it was explained how the solar radiation causes

a passive torque on an exposed spacecraft. This can be utilized with controllable panels or solarsails. The torques achieved are nevertheless low.




Type Advantages Disadvantages

External torque Control momentum build-up

Thrusters Insensitive to altitude Requires fuel

Suit any orbit On-Off operation only

Create torque about any axis Minimum impulse

Exhaust plume contaminants

Gravity No fuel or energy needed No torque about the local vertical

gradient Low accuracy

Low torque, altitude sensitive

Libration mode needs damping

Magnetic No fuel required No torque about local field direction

Control torque magnitude Altitude and latitude sensitive

Can cause magnetic interference

Solar radiation No fuel required Needs controllable panels

Very low torque

Internal torque No fuel required Uncontrollable momentum build-up

Can store momentum

Control torque magnitude

Reaction wheels Continuous Nonlinearity at zero speed

Fine-pointing capability

Momentum wheels Provide momentum biasControl moment Suitable for three-axis control Complicated

gyroscope Provide momentum bias Potential reliability problems

Table 3.4: Control torques

Thrusters

Thrusters or gas jets produce torque by expelling mass, and are potentially the largest source of

force and torque on a spacecraft. They are highly active sources, and being external they will

affect the total momentum. They can be used both for attitude and position control. In fact,they are the only actuators that can increase the altitude of a spacecraft in orbit. When used

for attitude control a pair of thrusters on opposite sides of the spacecraft is activated to create

a couple. The main advantage of using thrusters is that they can produce an accurate and well

defined torque on demand, as well as being independent of altitude. The main disadvantage is

that a spacecraft can only carry a limited amount of propellant.

Reaction wheels

Torquers associated with momentum storage such as reaction wheels are essentially active

internal torquers, suitable for attitude control but not for controlling the angular momentum.

By definition, a reaction wheel is a flying wheel with a body fixed axis designed to operate atzero bias. A flying wheel is any rotation wheel or disk used to store or transfer momentum.




When the spacecraft is exposed to a perturbation or it is accelerated, so are the wheels mounted

inside, and the result is generated torques from the wheels in the opposite direction, that is

Mrw ωrw = −Mωbib (3.23)

As seen from (3.23) the wheels have to be accelerated in order to create a torque. Neglecting

friction effects, the torque applied to a set of reaction wheels can be written as (Kaplan, 1976)

τ bw = dh

dt + ωb

ib × h (3.24)

where h = Mrwωrw [h1, h2, h3]T is the total angular momentum of the wheels, and τ bwdenote the torque applied to the wheels by the spacecraft body. By Newton’s 3rd law, the

torque employed to the spacecraft body from the wheels is therefor given by τ wb =

−τ bw. By

defining -dh/dt [τ wx, τ wy , τ wz ]T we get the following equation

τ wb =

τ wx − h3ω2 + h2ω3

τ wy − h1ω3 + h3ω1

τ wz − h2ω1 + h1ω2

(3.25)

This is consistent with the results derived earlier, as equation (3.6) can be obtained by substi-

tuting the torque in (3.25), in addition to some external torque, for the torque in (3.1).

Remark 3.1.8. When the wheels reach their maximum speed, the storage of momentum will

be at its maximum as well. Therefor, it will be necessary to restore the nominal values, using

external torques. This process is known as momentum dumping.

Momentum wheels

Momentum wheels are very similar to reaction wheels, but in contrast to the reaction wheels

they are designed to operate at biased , or nonzero, momentum. As for the reaction wheels they

need to be used in conjunction with other external actuators.

Magnetic actuators

An active magnetic actuator takes advantage of the natural torque caused by the magnetic field

surrounding the earth. The magnetic disturbance that was described earlier is exploited by in-

stalling magnetic coils or torquers inside the spacecraft. The principle can best be explainedwith the well known compass needle that attempts to align itself with the local field.

Magnetic actuators offer a cheap, reliable and robust way to control a spacecraft’s attitude.

Unfortunately they are only effective for low earth orbit (LEO) spacecrafts and requires a com-

plex model of the geomagnetic field surrounding the earth.

Electromagnets may be used to provide an external torque, which can be modelled as

τ mag = m ×B = niA (c×B) (3.26)

where m is the magnetic dipole moment generated by the coils in the magnet. The otherparameters are listed in the following table.



3.2 Celestial mechanics 25

Symbol Explanation

B Local geomagnetic field vector

c Unit vector in the direction of the coil’s axisi Control current in the coil

n Number of coil windings

A Cross-sectional area of the coil

3.2 Celestial mechanics

As mentioned at the very beginning of this chapter, the study of astrodynamics can be divided

into celestial mechanics and attitude dynamics. Sofar we have only considered the latter, in

the sense of giving an overview of different topics considering the motion about the center of mass. Only a brief discussion on celestial mechanics will be given at this point. Even though

the effects of the celestial mechanics are assumed negligible throughout this report, the motion

of the center of mass of objects in space is highly relevant, and for a real-life system to be

revealing, these effects should be taken into account when designing the control systems.

In general, the theory of celestial mechanics underlies all the dynamical aspects of the orbital

motion of a spacecraft. Different approaches exist to provide the necessary equations needed

to calculate orbital elements from position and velocity, and to predict the future position and

velocity of the spacecraft. In the case of circular orbits and spherical earth it is relatively easy

to determine these relations by the use of gradients, combined with rotation matrices. In a

more general case the use of classical orbit elements, Keplerian orbit elements, is adequate.Combined with perturbation theory this provides an excellent reference. Since the equations

concerning perturbations are rather extensive, they are not considered here. For further details

about this subject, the reader should refer to any textbook about spacecraft geodesy.



Chapter 4

Model and control properties of an

underactuated satellite

Sofar the main focus has been on different approaches to model the kinematic and dynamic

differential equations for describing the configuration of a rigid body. The purpose of this

chapter is to choose an adequate model based on these equations, and describe some of its most

important properties. In our special case, the model is considered to be a satellite. Furthermore,

the satellite is assumed to be underactuated, which can be immediately design related or due

to for instance actuator failures. Throughout the rest of this report, the third principal axis in

the satellite body system contributes as the underactuated part in the sense that no torques can

be applied directly about this axis. As will be shown, many challenging control problems arise

when doing attitude stabilization due to this property.

4.1 Satellite model

To simplify the analysis it is important to choose a reasonable model that is not too complicated.

Therefor, because of its useful properties, the (w, z) parametrization presented in chapter 2 is

chosen to characterize the kinematics, while the dynamic equations are based on the Newton-

Euler representation in chapter 3. In the following the model to be used is derived.

A preliminary and rather complex model can be written as1

ω = J−1τ − J−1Aha − J−1S(ω)(Jω + Aha) (4.1a)

ha = τ a (4.1b)

z = ω3 − ω1w2 + ω2w1 (4.1c)

w1 = ω3w2 + ω2w1w2 + ω1

2 (1 + w2

1 − w22) (4.1d)

w2 = −ω3w1 + ω1w1w2 + ω2

2 (1 + w2

2 − w21) (4.1e)

1Let Q be an m × n matrix of rank k . If k = m = n, then Q is nonsingular and has a unique inverse, Q−1.

The inverse is both left and right inverse, that is

QQ−1

= Q−1

Q = I



28 Model and control properties of an underactuated satellite

where ωbib = [ω1, ω2, ω3]T is written as ω for simplicity, and τ =

τ control +

τ disturbance.

The model arises immediately from (3.8) and (2.53), and both control and disturbance torques

were discussed in chapter 3. As mentioned earlier, it is important to emphasize that torquesrelated to the wheels should not be included in τ since these are already accounted for.

An interesting model to control and simulate would be a satellite with wheels and thrusters,

and also including the gravity gradient as a disturbance. The thrusters could then be used to

implement the torques for large and fast (slew) maneuvers during attitude initialization and tar-

get acquisition phases, while providing momentum management when necessary. The wheels

could be used to supply reference torques in the sense of overcoming the disturbances, and also

for storing and releasing kinetic energy. A potential model could have been obtained for this

scenario from (4.1), and then assuming no torques except from those mentioned above.

However, even though this model is somewhat simplified compared to a real-life scenario,it is still considered too rigorous for our purpose. For that reason, different assumptions will

be regarded subsequently to obtain even further simplifications. We also want the model to be

associated with an underactuated satellite as mentioned in the introduction. Taking the com-

plexity of the present model in (4.1) into account, a simplified model can be derived as

ω1 = (m22 − m33)ω2ω3

m11+

τ xm11

(4.2a)

ω2 = (m33 − m11)ω3ω1

m22+

τ ym22

(4.2b)

ω3 =

(m11

−m22)ω1ω2

m33 +

τ z

m33(4.2c)

w1 = ω3w2 + ω2w1w2 + ω1

2 (1 + w2

1 − w22) (4.2d)

w2 = −ω3w1 + ω1w1w2 + ω2

2 (1 + w2

2 − w21) (4.2e)

z = ω3 − ω1w2 + ω2w1 (4.2f)

Consider now the case when there are no available torques about the third principal axis in

the body system, or equally, the ω3 state in the model can not be altered directly through the

control input. A trivial redefinition of the control inputs finally yields the model

ω1 = τ 1

ω2 = τ 2

ω3 = ε ω1ω2

w1 = ω3w2 + ω2w1w2 + ω1

2 (1 + w2

1 − w22)

w2 = −ω3w1 + ω1w1w2 + ω2

2 (1 + w2

2 − w21)

z = ω3 − ω1w2 + ω2w1

Table 4.1: Simplified model of an underactuated satellite



4.1 Satellite model 29

The assumptions needed for deriving this simplified model from (4.1) are stated as follows

a) All values associated with reaction or momentum wheels are set equal to zero,

hence (4.1a) and (4.1b) are reduced to (3.1).

b) The inertia matrix for the satellite is diagonal, M = diag{m11, m22, m33}.

c) There are no torques about the third principal axis in the body system.

d) The available control torques are provided by pairs of thrusters, and they can

implement angular velocity about the actuated axes directly.

e) The control torques τ 1 and τ 2 in the model arise from the transformation

τ 1 = m22 − m33

m11ω2ω3 +

1m11

τ x

τ 2 = m33 − m11

m22ω1ω3 +

1

m22τ y

f) The satellite is nonsymmetric about its underactuated axis, ε = m11−m22

m33= 0.

Definition 4.1 (Axi-symmetric rigid body). When ε = 0 ⇒ ω3 = 0, and the rigid body isaxi-symmetric.

Remark 4.1.1. The variable ε gives a measure of the body asymmetry about the underactuated

axis. In case the spacecraft is nearly symmetric about this axis, |ε| 1. Without additional

assumptions, the model is uncontrollable at the origin when ε = 0. In fact, it is not even

accessible in this case. However, a real spacecraft can never be completely axi-symmetric,

implying that even if ε is very small there will be a slow rotation about the symmetric axis.

Several underactuated vehicles can be described by the general model (Pettersen, 1996)

Mν + C(ν )ν + D(ν )ν + g(η) =τ

0

(4.5a)

η = J(η)ν (4.5b)

where η ∈ Rn1 , ν ∈ Rn2 , n1 ≥ n2, τ ∈ Rm, m < n2. The matrix M is nonsingular

and M = 0 in the body reference system. The matrix J has full rank, i.e. det(J) = 0. The

vector ν denotes linear and angular velocities, and η denotes the position and orientation of

the vehicle. Gravitational and buoyant forces and torques are denoted by g(η), M is the inertia

matrix, C(ν ) is the Coriolis and centripetal matrix and D(ν ) is the damping matrix. Some

examples of vehicles described by (4.5) are underactuated surface vessels, underwater vehicles

and spacecrafts.




The spacecraft model in Table 4.1 can be written in the form (4.5) by using

J(η) = 1

2(1 + w

2

1 − w

2

2) w1w2 w2w1w2

12(1 + w2

2 − w21) −w1

−w2 w1 1

, η = [w1, w2, z]T (4.6)

C(ν ) = S(ω)M, D(ν ) = 0, g(η) = 0, ν = ω = [ω1, ω2, ω3]T, τ = [τ 1, τ 2]T (4.7)

Remark 4.1.2. The fundamental difference between the underactuated spacecraft model and

the general model (4.5), is the lack of a damping term, i.e D(ν ) = 0.

Remark 4.1.3. A spacecraft with a gravity gradient will experience gravitational torques,

hence g(η) = 0.

4.2 Controllability

Assuming that we have a smooth nonlinear system x = f (x,u), it can generally be converted

into an affine system quite easily, that is, into a system of the form

x = f (x) +mi=1

q i(x)ui (4.8)

From (4.6)-(4.7), and by letting

f (η,ν ) =

−M−1C(ν )ν J(η)ν

(4.9)

q 1 = [1, 0, 0, 0, 0, 0]T, q 2 = [0, 1, 0, 0, 0, 0]T (4.10)

the system in (4.5), or equally, the system in Table 4.1, may be written asν

η

= f (η,ν ) + q 1τ 1 + q 2τ 2 (4.11)

Sussmann and Jurdjevic (1972) demonstrated that a necessary and sufficient condition for ac-

cessibility of these systems is that the Lie algebra generated by the system have full rank, the

so-called Lie algebra rank condition (LARC).

Lemma 4.1. The nonsymmetric underactuated satellite model (4.11) is locally strongly acces-

sible from every (η,ν ) ∈ R6 when = 0.

Proof. The accessibility distributionC {q 1, q 2, [f, q 1], [f, q 2], [q 2, [f, q 1]], [f, [[q 1, f ], q 2]]} ∈R6×6 spans a six-dimensional space for every (η,ν ) ∈ R6, since the determinant of C can be

found to be −14(1+ w2

1+w22)22 = 0, and consequently rankC = 6 for every (η,ν ) ∈ R6.

Remark 4.2.1. For a general nonlinear system, if in addition to LARC, f in (4.8) is identically

zero, there is a theorem saying that the system is controllable.

It should also be noted that if a linearized system is controllable ⇒ nonlinear system is acces-

sible. However, if a linearized system is uncontrollable nonlinear system is unaccessible.

Proposition 4.1. The nonsymmetric underactuated satellite model (4.11) is small time locallycontrollable (STLC) from any equilibrium.



4.3 Stabilizability 31

Proof. From the proof of Lemma 4.1 it is seen that the system satisfies the LARC conditions,

and that any brackets with degree greater than 4 can be expressed as a linear combination of

lower order brackets. We further more note from the definitions in Appendix A.4 that thedegree of ”bad” brackets must be odd. The bad bracket of degree 1 is f , which vanishes at

any equilibrium. The ”bad” brackets of degree 3 are [g1, [f, g1]] and [g2, [f, g2]] which are

identically zero vector fields. Consequently, the nonsymmetric underactuated satellite model

(4.11) satisfies the Sussmann sufficient condition for STLC (Theorem A.3) with θ = 1, at every

equilibrium.

4.3 Stabilizability

In the following some control properties for the underactuated satellite model are discussed.

Proposition 4.2. There exists no continuous nor discontinuous time-invariant pure-state feed-back that renders the underactuated satellite system in Table 4.1 asymptotically stable about

the origin.

Proof. (Based on Proposition 2.3 in Pettersen (1996)) Consider the mapping f (η,ν , τ ) : R3 ×R3 ×R2 → R6 defined by

f (η,ν , τ ) =

−M−1C(ν )ν + M−1

τ

0

J(η)ν

(4.12)

It can easily be verified that this mapping is equivalent to (4.11). To prove Proposition 4.2 we

must show that f

(η,ν ,τ ) is not locally surjective. Consider a point ε inR6

of the form

ε =

M−1

α1

α2

β

03×1

(4.13)

where (α1, α2, β ) ∈ R, arbitrary and non-zero. Points on the form ε exist in any neighborhood

of 0 in R6. For f (η,ν , τ ) to be surjective then for any ε ∈ R6 there exists an δ ∈ R6 ×R2 for

which f (δ) = ε. As J(η) has full rank, the equation f (η,ν , τ ) = ε implies that ν = ω = 0.

Thus it implies

τ a

τ b0 =

α1

α2β (4.14)

Clearly, (4.14) has no solution since β = 0. From Theorem A.4 it then follows that there exists

no continuous pure-state feedback that renders the origin asymptotically stable.

By Theorem A.5, and the fact that the satellite model (4.11) cannot be asymptotically sta-

bilized by a pure-state continuous feedback law, it can neither be asymptotically stabilized by

means of a discontinuous pure-state feedback law.

Remark 4.3.1. In Pettersen (1996) a more general result was derived. It was in fact shown that

there exists no continuous nor discontinuous pure-state feedback law that makes the origin of

(4.5) asymptotically stable if gu

(η) has a zero element. The vector gu

(η) is the elements of g(η) corresponding to the underactuated dynamics.




Proposition 4.3. The nonsymmetric underactuated satellite model (4.11) is locally asymptoti-

cally stabilizable in small time by means of an almost smooth periodic time-varying feedback

law.

Proof. By Corollary A.2, as dim[(η,ν )] > 4 and the vector fields f, q 1, q 2 in (4.11) are ana-

lytic, this follows from the proof of Proposition 4.1.

4.4 Investigation of the underactuated dynamics

Some important insight can be gained about the system in Table 4.1 by considering the case

when (w1, w2) ≈ (0, 0), that is, the system simplifies to

ω1 = τ a (4.15a)

ω2 = τ b (4.15b)

ω3 = ε ω1ω2 (4.15c)

w1 = 1

2ω1 (4.15d)

w2 = 1

2ω2 (4.15e)

z = ω3 (4.15f)

First of all we see from (4.15f ) and (4.15c) that we have no direct control of z and ω3. How-

ever, z can be manipulated indirectly through the term ε ω1ω2 in (4.15c). If the spacecraft is

axi-symmetric, i.e. ε = 0, we have no control at all.

Assume that it is possible to manipulate the angular velocities directly. Consider the timevarying, periodic controllers

ω1 = −ω3 sin t, ω2 = sin t (4.16)

Insertion of the controllers into (4.15c) gives

ω3 = −ε ω3 sin2 t (4.17)

which has the average value

˙ω3 = − 1

T

T 0

ε ω3 sin2 tdt = −1

2ε ω3 (4.18)

with T = π. This means that it in average is possible to control ω3 by using time-varying peri-

odic controllers. This is the basic and intuitive idea behind many controllers for underactuatedand nonholonomic systems. Several other control strategies exist, but in order to circumvent

Brockett’s necessary condition (Brockett, 1985) they must be time-varying or discontinuous.

See for instance Kolmanovsky and McClamroch (1995) for a comparison of different control

strategies for nonholonomic systems.

4.5 Summary

In this chapter the nonsymmetric underactuated satellite model in Table 4.1 was derived. Sev-

eral properties of the system were then established, hereby the fact that any three-axis stabi-

lizing controller for the system has to be time-varying (periodic). This fact will be utilized inlater chapters when proposing and discussing a potential control strategy.



Chapter 5

A cascaded approach to stabilization

As mentioned in the introduction, the topic of this report is to choose a practical model, inves-

tigate some of it’s properties, and then suggest a control strategy to obtain attitude stabilization

about the satellite’s three principal axes. The system to be studied has three degrees of freedom

but only two available actuators, which makes it an underactuated system. From a theoretical

point of view the stabilization of an underactuated system is a challenging problem. Many in-

teresting control theoretical questions have to be answered and the problem is highly nonlinear.

The angular velocity stabilization problem for the case of one or two actuator failures is well

understood and can be considered solved, although some work still remains for momentum

exchange actuators. On the other hand, despite extensive attempts, the complete attitude sta-

bilization problem is not completely understood in the underactuated case. Consequently, no

globally stabilizing control law has so far been reported in the literature, as far as the author

knows.

In the following a novel strategy will be proposed, the final goal being complete three-axis

stabilization. Unlike most previous work however, a cascaded approach will be exploited.

The first part of this chapter considers some theory of stability for cascaded systems. Then,

in the following section, the theory is utilized when discussing the potential control scheme. A

globally uniformly asymptotically stabilized (GUAS) closed-loop system is desired.

5.1 Stability in cascaded systems

Consider the nonlinear time-varying system

x1 = f 1(t,x1,x2) (5.1a)

x2 = f 2(t,x2,u) (5.1b)

where x1 ∈ Rn, x2 ∈ Rm and u ∈ Rl. The function f 1(t,x1,x2) is both measurable and

continuously uniformly differentiable in (x1,x2) in t. The structure of the system is given in

Figure 5.1. As mentioned the goal is to get a completely stabilized system, hence we need to

find a control input u = u(t,x1,x2) or u = u(t,x2) such that the cascade interconnection

becomes globally uniformly asymptotically stable (GUAS). Several structural properties of thesystem have to be satisfied however, in order to obtain this objective. In the following, sufficient



34 A cascaded approach to stabilization

Figure 5.1: A cascaded system

conditions to guarantee that a GUAS nonlinear time-varying system

Σ1 : x1 = f 1(t,x1, 0) (5.2)

remains GUAS when it is perturbed by the output of another GUAS system of the form (5.1b),

that is, sufficient conditions to ensure GUAS for the system (5.1).

Consider now the nonlinear time-varying system in similar form as (5.1)

Σ1 : x1 = f 1(t,x1) + g(t,x1,x2)x2 (5.3)

Σ2 : x2 = f 2(t,x2) (5.4)

where the second term in (5.3) can be interpreted as a disturbance which must be driven to zero

without destabilizing the x1-subsystem. The following theory is taken from Panteley and Loria

(2001)

Lemma 5.1. If the systems Σ

1 and Σ2 are GUAS and the solutions of Σ1 and Σ2 are globallyuniformly bounded then the complete system Σ1 ∪ Σ2 is GUAS.

The question that remains is whether or not the solutions of Σ1 and Σ2 are globally uni-

formly bounded. Depending on different cases according to the growth rates of f 1(t,x1) and

g(t,x1,x2) this will be established subsequently.

Consider the following assumptions

(A1) The subsystem Σ2 is GUAS.

(A2) The subsystem Σ1 is GUAS and there exists a known C1 Lyapunov function V (5.2)(t,x1)

V (t,x1

), α1

, α2 ∈ K∞

, a continuous nondecreasing function α4

, and a positive semidef-

inite function W (x1) such that

α1(x1) ≤ V (t,x1) ≤ α2(x1)

V (t,x1) ≤ −W (x1) ∂V

∂ x1

≤ α4(x1)

(A3) For each fixedx2 and t there exists a continuous function λ : R+ → R+ with lim λs→∞ =0 such that g(t,x1,x2) in (5.3) satisfies

∂V

∂ x1 g(t,x1,x2) ≤ λ(x1)W (x1)



5.1 Stability in cascaded systems 35

(A4) There exist continuous functions θ1, α5 : R+ → R+ satisfying

g(t,x1,x2) ≤ θ1(x2)α5(x1)

and a continuous nondecreasing function α6 : R+ → R+ and a constant a ≥ 0 such that

α6(a) > 0 and

α6(s) ≥ α4

α−11 (s)

α5

α−11 (s)

and ∞

a

ds

α6(s) = ∞

with α1, α4 as in (A2).

(A5) For each r > 0 there exist constants κ, η > 0 such that for all t ≥ 0 and all x2 < r

∂V ∂ x1

g(t,x1,x2) ≤ κW (x1) ∀x1 ≥ η

where W (x1) and V (t,x1) are defined as in (A2).

(A6) There exists a function φ ∈ K such that for all t0 ≥ 0 the solutions x2(t) of Σ2 satisfies ∞t0

x2 (t, t0,x2(t0)) dt ≤ φ(x2(t0))

Based on the assumptions above, some growth theorems can be stated. For proofs, the reader

should refer to Panteley and Loria (2001).

Theorem 5.1. If (A1), (A2) and (A3) hold, then the cascade Σ1 ∪ Σ2 is GUAS.

Theorem 5.2. If (A1), (A2), (A4) and (A5) hold, then the cascade Σ1 ∪ Σ2 is GUAS.

Theorem 5.3. If (A1), (A2), (A4) and (A6 ) hold, then the cascade Σ1 ∪ Σ2 is GUAS.

Remark 5.1.1. The growth theorems can often be useful under certain circumstances. For each

fixed x2, uniformly in t, these are proposed in Panteley and Loria (2001) to be

Theorem 5.1 : ”The function f 1(t,x1) grows faster than g(t,x1,x2) as x1 → ∞”

Theorem 5.2 : ”The function f 1(t,x1) majorizes g(t,x1,x2) as x1 → ∞”Theorem 5.3 : ”The function f 1(t,x1) grows slower than g(t,x1,x2) as functions of x1”

Corollary 5.1. (See Lefeber (2000)) If (A2) is satisfied with

α1(x1) = c1x12α4(x1) = c4x1

and continuous functions k1, k2 : R+ → R exist such that

g(t,x1,x2) ≤ k1(x2) + k2(x2)x1 (5.5)

and (A1), (A6 ) is satisfied, the the cascaded system Σ1 ∪ Σ2 is GUAS.




In addition to the growth theorems there exists an integrability criterion (Loria, 2001) stated as

Theorem 5.4. Assume that Σ

1 i GUAS and suppose that the trajectories of

Σ2 are uniformly

globally bounded. If moreover, (A6 ) holds, and (A7 ), (A8 ) below are satisfied, then the solutions

of the system Σ1, Σ2 are uniformly globally bounded. If furthermore, the system Σ2 is GUAS,

then so is the origin of the cascade Σ1 ∪ Σ2.

(A7) There exist constants c1, c2, η > 0 and a Lyapunov function function V (t,x1) for Σ1

such that V : R+×Rn → R+ is positive definite and radially unbounded, which satisfies ∂V

∂ x1

x1 ≤ c1V (t,x1) ∀ x1 ≥ η

∂V

∂ x1 ≤ c2 ∀ x1 ≤ η

(A8) There exist two continuous functions θ1, θ2 : R+ → R+ such that g(t,x1,x2) satisfies

g(t,x1,x2) ≤ θ1(x2) + θ2(x2)x1

For clarity of exposition, sofar it has been assumed that the interconnection term in (5.1b) can

be factorized as g(t,x1,x2)x2; in some cases, this may be unnecessarily restrictive. With an

abuse of notation, let us redefine g(t,x1,x2)x2 g(t,x1,x2), i.e. consider a cascaded system

of the form

Π1 : x1 = f 1(t,x1) + g(t,x1,x2) (5.6a)

Π2 : x2 = f 2(t,x2) (5.6b)

where g(t,x1,x2) satisfies

g(t,x1,x2) ≤ α5(x1)γ (x2) + α5(x1)γ (x2) (5.7)

where α5, α5, γ and γ are nondecreasing functions such that γ (s) → 0 as s → 0 and

α5(x1) ≤ c1α5(x1) for all x1 ≥ c.

Let then V 2(t,x2) be a Lyapunov function for Π2, and consider the following corollary, which

under the assumptions of Theorems 5.2 and 5.3, establish GUAS of the cascade Π1, Π2.

Corollary 5.2. Consider the cascaded system Π1, Π2 satisfying (5.7) and suppose that (A2),

(A1) and (A4) hold with α6(V 1) = α4

α−11 (V 1)

α5

α−11 (V 1) , where we in (A2) use a Lya-

punov function V 1(t,x1,x2) V 1. Assume further that α5(x1) is majorised by the function

W (x1)α4(x1)

and the function γ (x2) satisfies either of the following

γ (x2) ≤ (V 2(t,x2))U (x2) (5.8)

where V 2(t,x2) ≤ −U (x2) with : R+ → R+ continuous; or there exists ϕ ∈ K such that ∞t0

γ (x2(t))dt ≤ ϕ(x2(t0)) (5.9)

Under these conditions the cascade Π1 ∪ Π2 is GUAS.



5.2 The underactuated satellite - a cascaded approach 37

Figure 5.2: A desired cascade structure

5.2 The underactuated satellite - a cascaded approach

Based on the theory concerning stability in cascades, and well known methods using Lyapunovdesign and backstepping, a novel idea of a control strategy will in the following be discussed

with the goal of achieving GUAS for the underactuated satellite system. The satellite model is

repeated here for clarity, that is

ω1 = τ 1 (5.10a)

ω2 = τ 2 (5.10b)

ω3 = ε ω1ω2 (5.10c)

w1 = ω3w2 + ω2w1w2 + ω1

2 (1 + w2

1 − w22) (5.10d)

w2 =

−ω3w1 + ω1w1w2 +

ω2

2

(1 + w22

−w21) (5.10e)

z = ω3 − ω1w2 + ω2w1 (5.10f)

A desired structure, at least in the general case, is shown in Figure 5.2. Usually a model is

not given in this form, hence various synthetic inputs and coordinate transformations have to

be utilized. This often contributes as a challenging task, and in many cases this is the most

durable effort when dealing with systems of this nature. Nevertheless, if we are able to trans-

form the system into such form, the theory from the previous section can be applied. This can

in many cases my easier than constructing Lyapunov functions for the complete system, which

in turn can be extremely difficult. In some sense, we can think of working with this structure

as a kind a ”separation principle”. This will be explained shortly.

From the general block diagram in Figure 5.2 it is evident that the Σ2 subsystem cannot be

a function of the states in the Σ1 subsystem, nor can Σk be a function of the states in the Σk−1.

This is inherent from the nature of cascaded systems. On the other hand, the output from the

Σ2 block works as a virtual input or disturbance to the Σ1 block. Furthermore, the Σ2 system

consists of the subsystems Σ3 ∪ Σ2, which clearly makes up an internal cascade by itself. The

same observation can be done recursively, until we end up with the first single block, Σk. If we

are able to split the system into such blocks, we can study each block separately, which gives

rise to the ”separation principle” as introduced above. Even though this is not completely true,

in the sense that a subsystem is coupled to the preceding subsystems through the connection

term, it is still a very powerful tool.

An informal outline of the procedure we are suggesting is given as follows;




i) Based on the original model in (5.10) we will try to transform it into a similar structure

as in Figure 5.2, where we want as many subsystems as possible.

ii) After doing necessary changes in coordinates and/or transformations we want z to be

controlled by ω3. We then use a kind of backstepping approach where we define ω3d.

iii) We want w1, w2 and ω3 to be controlled through ω1 and ω2. Clearly, z will be controlled

indirectly by these states as well. We will, as in the case of ω3, use a kind of backstepping

approach where we define ω1d and ω2d. From Proposition 4.2 we recall that a pure-state

feedback is not sufficient in obtaining asymptotic stability for the underactuated satellite.

We will therefor include some time-varying functions in ω1d and ω2d.

iv) Based on the theory on stability in cascades and Lyapunov functions, we will check

wether the closed-loop system is GUAS or not with our choice of synthetic inputs, and

consequently the final controller.

The rest of this chapter will consider each step as we try to characterize the promising approach

in achieving an asymptotically stabilizing controller, which works globally, uniformly in time.

Based on the discussion above, the initial step would be to split our system into as many

subsystems as possible. By looking at (5.10), it can easily be seen that the system is already

in a cascaded form, where (5.10a)-(5.10b) contribute as Σ2 and (5.10c)-(5.10f) make up the

subsystem Σ1. Nevertheless, it would be highly desired to obtain further separations, in senseof Figure 5.2. Numerous attempts, and many hours work, was done in trying to achieve this

goal, which unfortunately turned out to be a very difficult task, and most likely it is not even

possible. The reasons for this have to do with the nature of the highly coupled and nonlinear

differential equations, as well as our overall design strategy. This will be clearer after reading

the next paragraphs.

We again turn our attention to (5.10). As stated above, we want z to be controlled through

ω3. This can be done by introducing the following change of coordinates

ω3 = ω3

−ω3d (5.11)

where ω3d is the desired value of ω3. At this stage however, we only define the function

ω3d ψ(z), and consequently we get a new system equation, in place of (5.10c), that is

ω3 = ω1ω2 ⇒ ˙ω3 = ω1ω2 − ψ(z)z (5.12)

where we used the notation ∂ψ(z)

∂z ψ(z). Similar notation will be used throughout. Clearly,

by introducing ω3 we also get ψ(z) in both (5.10d) and (5.10e). Actually, ψ(z) is included in

every system equation except for (5.10a) and (5.10b). As mentioned earlier a subsystem can

not be a function of any states in any preceding subsystems. We were not able to obtain thisproperty for more than two subsystems, and consequently we were unfortunately not able to



5.2 The underactuated satellite - a cascaded approach 39

split up the system any further. Accepting this fact, we now rewrite the system in (5.10) as

ω1 = τ 1 (5.13a)ω2 = τ 2 (5.13b)

˙ω3 = ω1ω2 − ψ (ψ + ω3 − ω1w2 + ω2w1) (5.13c)

w1 = (ψ + ω3) w2 + ω2w1w2 + ω1

2 (1 + w2

1 − w22) (5.13d)

w2 = − (ψ + ω3) w1 + ω1w1w2 + ω2

2 (1 + w2

2 − w21) (5.13e)

z = ψ + ω3 − ω1w2 + ω2w1 (5.13f)

where we used (5.12) and we dropped the argument in ψ for simplicity. This gives us the two

(temporary) subsystems

Σ1 : z1 = f 1(t, z1) + g(t, z1, z2)z2 (5.14)Σ2 : z2 = f 2(t, z2,u) (5.15)

where we have used z2 = [ω1, ω2]T, z1 = [ω3, w1, w2, z]T, u = [τ 1, τ 2]T and we have that

f 1(t, z1) =

− (ψ + ω3) ψ

(ψ + ω3) w2

− (ψ + ω3) w1

ψ + ω3

, f 2(t, z2,u) =

τ 1τ 2

(5.16)

g(t, z1, z2) =

12 ω2 + ψw2

12 ω1 − ψw1

12

1 + w2

1 − w22

w1w2

w1w2 1

2

1 + w2

2 − w21

−w2 w1

(5.17)

The direct approach would here be to find a control u that renders Σ1 and Σ2 GUAS, while

still satisfying the conditions discussed earlier. However, we first introduce some synthetic

inputs such that we can conclude GUAS for

Σ1 alone, when neglecting the connection term,

i.e. z2

≡ 0. The synthetic inputs, together with ψ(z), are to be discussed subsequently, so at

this point we only define the coordinate changes and desired values as

ω1 = ω1 − ω1d, ω1d φ1(t, ω3, w1, w2) (5.18)

ω2 = ω2 − ω2d, ω2d φ2(t, ω3, w1, w2) (5.19)

After doing necessary manipulations on Σ1 and Σ2 we get the final cascaded system

Σ1 : x1 = f 1(t,x1) + g(t,x1,x2)x2 (5.20)

Σ2 : x2 = f 2(t,x2, u) (5.21)

where we have used x2 = [ω1, ω2]T, x1 = [ω3, w1, w2, z]T, u = [τ 1, τ 2]T and we have




f 1

(t,x1

) =

φ1φ2 − ψ (ψ + ω3 − φ1w2 + φ2w1)

(ψ + ω3) w2 + φ2w1w2 + φ12 1 + w2

1

−w22− (ψ + ω3) w1 + φ1w1w2 + φ22 1 + w22 − w21

ψ + ω3 − φ1w2 + φ2w1

, f

2(t,x

2, u) = τ 1

− dφ1dt

τ 2 − dφ2dt (5.22)

g(t,x1,x2) =

12 ω2 + ψw2 + φ2

12 ω1 − ψw1 + φ1

12

1 + w2

1 − w22

w1w2

w1w2 1

2

1 + w2

2 − w21

−w2 w1

(5.23)

where we dropped the arguments of ψ, φ1 and φ2 for simplicity. Also note that

dφi

dt =

∂φi

∂t +

∂φi

∂ ω3 ˙ω3 +

∂φi

∂ w1 w1 +

∂φi

∂ w2 w2, i = 1, 2 (5.24)

Finding the synthetic inputs ψ(z), φ1(t, ω3, w1, w2) and φ2(t, ω3, w1, w2) such that Σ1 is ren-

dered GUAS, when neglecting the connection term between Σ1 and Σ2, makes up the next

natural step in the procedure. Note that neglecting the connection term can be interpreted as if

Σ2 is already rendered GUAS by the real inputs, hence x2 ≡ 0.

The final, and in some sense the more direct outline of the strategy, is described in the next

section. It briefly considers how to find the synthetic inputs, the real control, and also how a

potential Lyapunov proof for showing GUAS for the closed loop system is likely to occur.

5.3 Control design

As stated above, in this section we will try do give an outline of how to find the synthetic inputs,

and consequently the final control, such that the underactuated satellite system in (5.20)-(5.21)

is rendered GUAS.

We first start by looking at Σ2. Obviously the final control will be of the form

τ 1 = dφ1

dt + σ1(ω1) (5.25)

τ 2

= dφ2

dt + σ

2(ω

2) (5.26)

where we defined dφidt in (5.24). The first term in both (5.25) and (5.26) cancels out the dy-

namics of φ1 and φ2 in Σ2. This has to be done in order for the system to be a valid cascade,

as in (5.20)-(5.21). Accepting this fact, it is trivial to show that by making good choices for

σ1(ω1) and σ2(ω2) the solutions of Σ2 will converge to zero in exponential time, that is, Σ2

will be rendered GES, and hence also GUAS. This can for instance be the case if we choose

σi(ωi) = −ki ωi, i = 1, 2, and use a quadratic Lyapunov function to prove stability. Without

any additional assumptions we already can conclude that the subsystem Σ2 will not cause us

any trouble in the further design.

The rest of the design can be split into to parts. We first start by looking at Σ1, when neglect-ing the connection term. We can do this based on the assumption that Σ2 is already rendered



5.3 Control design 41

GUAS, which was in fact shown above to be the case. For clarity, we define Σ1 Σ1 : x2 ≡ 0,

that is

Σ1 : x1 =

φ1φ2 − ψ (ψ + ω3 − φ1w2 + φ2w1)

(ψ + ω3) w2 + φ2w1w2 + φ12

1 + w2

1 − w22

− (ψ + ω3) w1 + φ1w1w2 + φ2

2

1 + w2

2 − w21

ψ + ω3 − φ1w2 + φ2w1

(5.27)

where we used (5.22). We now need to find ψ, φ1 and φ2 such that Σ1 is rendered GUAS.

Obviously, this is nontrivial, and in fact it is considered to be a rather difficult task. Some

guidelines are nevertheless given in the following.

At the time of writing the work of completing the above mentioned task is not yet solved.

Some progress has been made however, and a promising Lyapunov candidate along the trajec-

tories of w1 and w2 in Σ1 is defined as V 1 V 1(w1, w2), where

V 1(w1, w2) = ln(1 + w21 + w2

2) (5.28)

which is positive definite and V 1(0, 0) = 0, clearly. The derivative can easily be found to be

V 1 = φ1 w1 + φ2 w2 (5.29)

The latter is in some sense a strong result, and in fact it resembles a quadratic Lyapunov func-

tion V = xTP x, where x = [w1, w2]T , ˙x = [φ1, φ2]T and P > 0. It can easily be seen that we

can choose φ1 and φ2 to make V 1 negative definite. However, even though we would like the

solution to be this simple, we have to remember that Σ1 consists of two more states, in which

both φ1 and φ2 are included. The strong coupling is in fact what makes the stabilization of the

Σ1 system as hard as it is. In this report we choose to end the discussion of stabilizing thissubsystem by outlining what is left to do in the approach. A summary is given as

i) Construct one or more Lyapunov candidates along all trajectories in the Σ1 subsystem.

As mentioned above, V 1 is a potential candidate along w1 and w2. We also need to

include ω3 and z in some additional Lyapunov functions.

ii) After finding potential Lyapunov candidates, we must determine ψ(z), φ1(t, ω3, w1, w2)and φ2(t, ω3, w1, w2) such that we can conclude GUAS for Σ

1, according to Theorem

A.1 and Corollary A.1. The functions φ1 and φ2 are chosen to be time-varying, as a

result of Proposition 4.2. Typically they will including a periodic Sine function.

iii) When checking wether or not Theorem A.1 and Corollary A.1 are satisfied we can inmany cases use mathematical manipulation, accompanied by different inequalities as

described in Appendix A.1.

iv) If the derivative of the Lyapunov functions turn out to be negative semi-definite due to

for instance sin2(t) terms, Strict Lyapunov functions will be tried constructed, according

to Appendix A.3.

Finally, when starting on the last part we now that Σ2, and hopefully also Σ1, is rendered

GUAS. To finalize the conclusion on wether or not the complete system Σ1 ∪ Σ2 obtains

the same property as well, we need to utilize the theory about stability in cascaded systems,

which was discussed in the beginning of this chapter. In short, we need to check wether or not

our assumptions in neglecting the connection term in Σ1 was correct, hence verifying if theconnection term satisfies some necessary conditions or not.



Chapter 6

Conclusions

A detailed and general model for an underactuated satellite has been derived, including topics

related to disturbances and different actuators. In sense of representing the kinematics, the

relatively new (w, z)-parametrization was chosen due to its interesting and favorable proper-

ties compared with other minimal attitude representations. As for the dynamics, these were

described by the well known Newton-Euler equations of motion.

After taking numerous simplifications into account, several properties of the underactuated

satellite have been presented. Among others, the most important being the fact that the model

does not satisfy Brockett’s necessary condition, hence making attitude stabilization impossible

in sense of using a continuous time-invariant pure-state feedback. This was also shown to be

applicable for the discontinuous case as well.

Based on the different model properties that were revealed, and the fact that complete three-

axis stabilization still is an open problem in the literature, in sense of an underactuated rigid

spacecraft, a novel approach has been proposed. Unlike previously work however, a strategy

utilizing theory related to nonlinear systems in cascades has been exploited.

At the time of writing, the work of achieving a globally uniformly asymptotically stabiliz-

ing controller in sense of the proposed strategy has not yet been completed. Nevertheless, the

problem has been reduced from stabilizing the entire system to finding some functions which

makes a subsystem globally uniformly asymptotically stable instead. From the theory of cas-

caded systems it was also shown that some additional growth conditions must be satisfied.

By combining well known methods like Lyapunov design, backstepping and theory related

to cascaded systems, the proposed strategy seems highly potential, and hopefully it will make

a contribution in the problem of achieving complete three-axis stabilization in sense of an un-

deractuated satellite.

6.1 Further work

Even though the report is by itself self-standing, it should be considered a prestudy of a poten-

tial procedure of achieving complete three-axis stabilization. After finishing this report there isdefinitely more to do on the subject. The first natural steps would be as follows;



44 Conclusions

• Complete the work of finding Lyapunov candidates, and necessary functions to conclude

GUAS for the subsystem denoted by Σ1 in Section 5.3.

• Verify that the the complete system, consisting of the two cascades Σ1 and Σ2 satisfies

the conditions related to stability in cascaded systems.

• If both tasks above are satisfied, conclude GUAS for the complete system.

• Implement the closed-loop system in for instance MATLAB, and execute necessary sim-

ulations to verify the theory.

If one are able to complete these tasks, it will be possible to go even further in the sense of;

• Perform extensive simulations, using a benchmark model and the proposed time-varying

control laws.

• Include disturbance torques like gravity gradient and aerodynamic drag in the model.

• Reaction wheels have not been considered in this work. Results in Crouch (1984) states

that attitude stabilization is not possible with momentum exchange devices. What is the

best that can be achieved with such devices?

• Tracking and robust attitude control of underactuated spacecrafts are still open problems.



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State-of-the-Art and Open Problems. Journal of the Astronautical Sciences, 48(2):pp. 337–

358.

Tsiotras, P. and Longuski, J. (1995). A New Parameterization of the Attitude Kinematics.

Journal of the Astronautical Sciences, 43(3):pp. 243–262.

Tsiotras, P. and Longuski, J. M. (1994). Spin-axis stabilization of symmetric spacecraft with

two control torques. Systems and Control Letters, 23:pp. 395–402.

Tsiotras, P. and Longuski, J. M. (1996). Comments on a new parametrization of the attitude

kinematics. In Astrodynamics Specialists Conference.

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Proceedings of the 35th IEEE Decision and Control, pp. 495–496.

Tsiotras, P. and Luo, J. (2000). Control of underactuated spacecraft with bounded inputs.

Automatica, 36:pp. 1153–1169.

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Wertz, J. R. (ed.) (1999). Space mission analysis and design. Kluwer Academic Publishers,

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Wisniewski, R. and Blanke, M. (1999). Fully magnetic attitude control for spacecraft subject

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19:pp. 1–9.



Appendix A

Theory

A.1 Mathematical preliminaries

Definition A.1. A continuous function γ : [0, a) → [0, ∞) is said to belong to class K if is it

strictly increasing and γ (0) = 0. It is said to belong to class K∞ if a = ∞ and γ (r) → ∞ as

r → ∞.

Definition A.2. The norm x of a vector x is a real-valued function with the properties

• x ≥ 0 for all x ∈ Rn, with x = 0 if and only if x = 0.

• x + y ≤ x + y, for all x, y ∈ Rn.

• αx = |α|x, for all α ∈ R and x ∈ Rn

.

The second property is the triangle inequality. Also worth mentioning is that all these properties

are valid for all p-norms given as

x p = (|x1| p + · · · + |xn| p)1/p , 1 ≤ p < ∞ (A.1)

In the following some other important inequalities are defined. Together with the triangle in-

equality these can be very useful when developing Lyapunov proofs.

An important result concerning p-norms is the H older inequality

nk=1

|akbk| ≤ nk=1

|ak| p1/p nk=1

|bk|q1/q

, 1 p

+ 1q

= 1 (A.2)

If we let p = q = 2 in (A.2) we get the Cauchy-Schwartz inequality, that is nk=1

akbk

2

≤

nk=1

ak2

nk=1

bk2

(A.3)

Another important relation is the Minkowski inequality, which for a general p − norm in fact

can be shown to be the triangle inequality, written as

nk=1

|ak + bk| p1/p

≤ nk=1

|ak| p1/p

+ nk=1

|bk| p1/p

(A.4)



50 Theory

Finally we include Young’s inequality, which is deduced from Young’s integral. This inequality

plays an important role in the theory of inequalities. In it’s most general form it is given as

ab ≤ a p

p +

bq

q , where a, b ≥ 0, p > 1,

1

p +

1

q = 1 (A.5)

For arbitrary real numbers a and b when then have

ab ≤ a2

2 +

b2

2 (A.6)

from which we can deduce

ab = λa b

λ ≤ λ2

2 a2 +

1

2λ2b2 (A.7)

where λ = 0 is an arbitrary real number. By defining κ

2λ2

we get the useful form of Young’s inequality

ab ≤ κa2 + 1

4κb2 (A.8)

A.2 Lyapunov stability

Consider the non-autonomous system

x = f (x, t) (A.9)

where f : [0,

∞)

×D

→Rn.

Theorem A.1. (Khalil (1996 ), Theorem 3.8) Let x = 0 be an equilibrium for (A.9) and D ⊂Rn be a domain containing x = 0. Let V : [0, ∞) × D → R be a continuously differentiable

function such that

W 1(x) ≤ V (x, t) ≤ W 2(x) (A.10a)

∂V (x, t)

∂t +

∂ V (x, t)

∂x f (x, t) ≤ −W 3(x) (A.10b)

∀ t ≥ t0, ∀ x ∈ D where W 1(x), W 2(x) are continuous positive definite functions and W 3(x)is a continuous positive semidefinite function on D. Then, x = 0 is locally uniformly asymp-

totically stable.

Corollary A.1. Suppose that all the assumption of Theorem A.1 are satisfied globally (for all

x ∈ Rn) and W 1(x) is radially unbounded. Then, x = 0 is globally uniformly asymptotically

stable.

Definition A.3. A Lyapunov function satisfying (A.10) is called a strict Lyapunov function

A.3 Strict Lyapunov functions for time-varying systems

Consider the time varying system

x = f (x, t) (A.11)

with x ∈ Rn and f (x, t) is a nonlinear function periodic in time of period T > 0.



A.4 Controllability 51

Assumption A.1. A Lyapunov function V (x, t), periodic in time and of period T > 0, a

positive definite function W (x) and a nonnegative function p(t), periodic and of period T ,

such that ∂V

∂t (x, t) +

∂ V

∂x (x, t)f (x, t) ≤ − p(t)W (x) (A.12)

and two functions αi(·), i = 1, 2 of class K∞ such that

α1(|x|) ≤ V (x, t) ≤ α2(|x|) (A.13)

are known.

Assumption A.2. The constant T 0 p(s)ds is strictly positive.

Theorem A.2. ( Mazenc, 2003) If Assumptions A.1 and A.2 are satisfied by the system (A.11) ,

one can determine the explicit expressions of a continuously differentiable function Γ(·) of class K∞ and of a positive definite function λ(·) continuously differentiable, with a positive

first derivate, such that the function

U (x, t) = Γ

V (x, t)

+ P (t)λ

V (x, t)

(A.14)

with

P (t) = −t

T 0

p(s) ds + T

t0

p(s) ds (A.15)

is a strict Lyapunov function for system (A.11).

A.4 Controllability

The following presentation is based on Sussmann (1987).

Consider the nonlinear affine system Ψ

x = f (x) +mi=1

gi(x)ui u = (u1, . . . , um) ∈ Rm (A.16)

where x = (x1, . . . , xn) are local coordinates for a C∞ manifold M , and f, g1, . . . , gm are C∞vector fields on M . Assume that f (0) = 0.

Before presenting the Sussmann sufficient condition for small-time local controllability, some

definitions are needed;

If q ∈ M is of the form x(T ) for some trajectory such that x(0) = p, then q is said to be

reachable from p in time T . The set of all q that are reachable from p in time T for the system

Ψ is the time T reachable set from p and will be denoted by Reach(Ψ, ≤ T, p). Furthermore

Reach(Ψ, ≤ T, p) =

0≤t≤T

Reach(Ψ, t , p) (A.17)

for T ≥ 0, The system Ψ is small-time locally controllable (STLC) from p if p is an interiorpoint of Reach(Ψ, ≤ T, p) for all T > 0.



52 Theory

If F is a family of C∞ vector fields on a manifold M , then L(F ) denotes the Lie algebra

of vector fields generated by the elements of F . The family F is said to satisfy the Lie algebrarank condition (LARC) at p if L(F )( p) is the whole tangent space of M at p. If Ψ satisfies

LARC, the system is locally accessible.

Let Y = (Y 0, . . . , Y m) be a finite sequence of indeterminates. We let A(Y) denote the free

associative algebra over R generated by Y j , and L(Y) denote the Lie subalgebra of A(Y)generated by Y 0, . . . , Y m. We define Br(Y) to be the smallest subset of L(Y) that contains

Y 0, . . . , Y m and is closed under bracketing. Let f be denoted by g0. We are given a C∞ mani-

fold M and (m + 1)-tuple g = (g0, g1, . . . , gm) of C∞ vector fields on M . Each g j is therefor

a member of D(M ), the algebra of all partial differential operators P : C∞(M ) → C∞(M ),

where

C∞(M ) denotes the space of

C∞ real-valued functions on M . There are therefor a well

defined evaluation map Ev(g) : A(Y) → D(M ) obtained by ”substituting the g j for the Y j”,so that

Ev(g)

I

aI Y I

=I

aI gI (A.18)

where, if I = (i1, . . . , ik) then gI = gi1gi2 . . . gik . E v p denotes the evaluation map at p.

Define the degree δ θ of the bracket B ∈ Br(Y) to be the sum

δ θ(B) = 1

θ

δ 0(B) +m

i=1

δ i(B) (A.19)

where δ i(B) is the number of times that Y i occurs in B. We say that a bracket B ∈ Br(Y) is

”bad” if δ 0(B) is odd and δ 1(B), . . . , δ m(B) are even.

Let β (B) be the symmetrization operator given by β (B) =

π∈S mπ(B) where π ∈ S m,

and S m is the group of permutations of {1, . . . , m}. For π ∈ S m, π is the automorphism of

L(Y) which maps Y 0 to Y 0 and Y i to Y π(i).

Theorem A.3. (Sussmann 1987) Consider a system

x = f (x) +mi=1

gi(x)ui x ∈ M |u| ≤ 1 (A.20)

and a point p ∈ M such that f ( p) = 0. Assume that g = (f, g1, . . . , gm) satisfies the LARC at

p. Assume that there is a θ ∈ [1, ∞] such that whenever B ∈ Br (Y) is a ”bad” bracket, then

there are brackets C 1, . . . , C k in Br (Y) such that

Ev p(g)(β (B)) =k

i=1

ξ iEv p(g)(C i) (A.21)

for some ξ 1, . . . , ξ k ∈ R , and δ θ(C i) < δ θ(B) for i = 1, . . . , m. Then (A.20) is STLC in p.



A.5 Stabilizability 53

A.5 Stabilizability

Consider the nonlinear control system Ψ

x = f (x,u) ∈ Rn u ∈ R

m (A.22)

The condition given in the next theorem is often referred to as Brockett’s necessary condition

or Brockett’s theorem. It is a necessary condition for stabilizability by continuous pure-state

feedback was presented. It was originally presented by Brockett (1985) for C1 pure-state feed-

back, and shown by Zabczyk (1989) to also hold for continuous pure-state feedback. It can be

formulated as follows (Aneke, 2003)

Theorem A.4. Assume that there exists a continuous pure-state feedback law u : Rn → Rm ,

that renders the origin of Ψ asymptotically stable. Then the function f : Rn

×Rm

→ Rn is

locally surjective, i.e., the function f maps an arbitrary neighborhood of (0,0) ∈ Rn × Rm

onto a neighborhood of 0 in Rn.

Remark A.5.1. If Brockett’s necessary condition is not satisfied for a system, the system can-

not be asymptotically stabilized by continuous dynamic feedback either, in the sense that any

extended system

x = f (x, z) ∈ Rn z = v ∈ R

m (A.23)

where v is a new control, and z is a new state variable, will still not satisfy the necessary

condition.

The following theorem shows a relation between stabilizability by means of discontinuous

pure-state feedback laws and stabilizability by means of the continuous time-varying and con-tinuous pure-state feedback laws.

Theorem A.5. (Coron and Rosier (1994)) Assume that Ψ can be locally (resp. globally)

asymptotically stabilized by means of a discontinuous feedback law. Then for ant T > 0, Ψcan be locally (resp. globally) stabilized by means of a continuous time-varying feedback law

of period T; If, moreover Ψ is an affine system, as in (4.8) , then Ψ can be locally (resp. globally)

asymptotically stabilized by means of a continuous feedbacl law (independent of t; u = u(x)).

It is proven in Coron (1992) that the Sussmann sufficient condition for STLC at p = 0 implies

that the origin of Ψ is ”locally continuously reachable in small time”. The following corollary

is then easily deduced from Coron (1995), theorem 1.4.

Corollary A.2. Assume that f in (A.22) is analytic and that n ≥ 4. If the system Ψ satisfies

Sussmann sufficient condition for STLC at the origin, then Ψ is locally stabilizable in small

time by means of almost smooth periodic time-varying feedback laws.



54 Theory



Appendix B

Newton-Euler equations of motion

Equations of motion for a rigid body can be derived by summing up the equations of motion

for individual mass elements dm with velocity v p. A rigid body B with a mass element dm is

shown in Figure B.1. The point c is the center of mass, while o is the point where we want to

express the equations of motion about. Later it will be assumed that the two points coincide.

The material in this chapter is based on Egeland and Gravdahl (2002).

dm

o c

Inertial frame

ro

r

rc

rg

rm

Figure B.1: Rigid body with mass element dm.

B.1 Translational motion

The translational equation of motion with reference to a point o can be written as

f o = mac. (B.1)

From Figure B.1 we have that rc = ro + rg, hence

vc = vo + ωib × rg, (B.2)

ac = ao + ωib × rg + ωib × ( ωib × rg), (B.3)

where we have used that rg is a constant in b.



56 Newton-Euler equations of motion

Combining (B.1) and (B.3) gives the force equation with reference to the point o:

f o = m ao + ωib × rg + ωib × ( ωib × rg) . (B.4)

The translational motion of a spacecraft can be controlled using thrusters. For a spacecraft in

orbit the motion is governed by the laws of orbital mechanics. Such a law is the restricted

two-body equation of motion:

a = −µ r

|r|3 (B.5)

where r is the spacecraft’s position and µ is the gravitational parameter for Earth. For more

details see a textbook in orbital mechanics, for instance Prussing and Conway (1993).

B.2 Angular motion

The Newton-Euler equations are derived from Euler’s First and Second Axioms:

f c = mac (B.6)

τ c = hc (B.7)

τ o = τ c + rg × f c (B.8)

where the angular momentum about c and o are defined as

hc = B

(r × v p)dm, (B.9)

ho =

B

( rd × v p)dm. (B.10)

By using that v p = vo + ωib × rd and rd = r + rg, (B.10) can be written as

ho = mrg × vo +

B

rd × (ωib × rd)dm. (B.11)

To simplify (B.11) the inertia dyadic

I o = B

−S 2(rd)dm (B.12)

is introduced. The angular momentum about o can then be written as

ho = mrg × vo + I o ωib. (B.13)

An alternative expression can be found by writing ho as

ho =

B

(r + rg) × v pdm

= hc + B(rg × v p)dm

= hc + rg × mrc, (B.14)



B.3 Model summary 57

where we have used that vc ≡ 1m

B v pdm.

Time differentiation of ho with respect to the inertial frame yields 1

ho = vc × mvo + rg × mvo + M o ωib + ωib × ( M o ωib). (B.15)

Equation (B.14) implies that

ho = hc + rg × mvc − vo × mvc, (B.16)

which combined with (B.15) gives

hc = τ c = rg × m(vo − vc) + M o ωib + ωib × ( M o ωib). (B.17)

Insertion of (B.17) in (B.8) and using (B.7) gives the angular equation of motion

τ o = rg × mao + M o ωib + ωib × ( M o ωib). (B.18)

B.3 Model summary

The equations (B.4) and (B.18) can be simplified by letting o coincide with the center of mass

c, meaning rg = 0 and M o = M c. The simplified equations are

f = ma, (B.19a)

τ = M ωib + ωib × ( M ωib), (B.19b)

where the subscript c has been dropped for convenience.

Writing the equations of motion in coordinate form in the b frame yields

mvb = f b, (B.20)

Mωbib + S(ωb

ib)Mωbib = τ b. (B.21)

At a first glance the translational and angular motion seems decoupled. A closer inspection

reveals that this is not the case. The reason is that disturbance torques, τ d, and forces, f d acting

on a spacecraft are usually dependent of the spacecraft’s position and attitude. However, forour purposes the translational and angular motion can be assumed decoupled.

1For more details about the derivation refer to Egeland and Gravdahl (2002)



58 Newton-Euler equations of motion



Appendix C

Mathematica printouts

Mathematica printouts of the calculations for checking wether or not the satellite model satis-

fies the LARC conditions and if it is STLC are given on the following pages.



H* Checking to see if accessiable, C = 8g1,g2,@f,g1D,@f,g2D,@f,@f,g1DD,@f,@f,g2DD< *L

X = MatrixForm @88w1<, 8w2<, 8w3<, 8 w1<, 8 w2<, 8z<<D

H * X = @w1 w2 w3 w1 w2 zD T, Xdot =f +g1u1 +g2u2 * L

f =880<, 80<, 8e* w1 * w2<, 8w3 * w2 +w2 * w1 * w2 +w1•2 *H1 + w1 ^ 2 - w2^2L<,

8- w3 * w1 +w1 * w1 * w2 +w2•2 *H1 - w1^2 + w2 ^2L<, 8w3 - w1 * w2 +w2 * w1<<

980<, 80<, 8e w1 w2<, 9 1€€€€2 H1 +w1

2 - w22L w1 +w1 w2 w2 +w2 w3=,

9w1 w2 w1 + 1€€€€2 H1 - w1

2 +w22L w2 - w1 w3=, 8- w2 w1 +w1 w2 +w3<=

MatrixForm @fD

i

k

j j j j j j j j j j j j j j j j j j j j j j

0

0e w1 w2

1€€€€2 H1 +w1

2 - w22L w1 +w1 w2 w2 +w2 w3

w1 w2 w1 + 1€€€€2 H1 - w1

2 +w22L w2 - w1 w3

- w2 w1 +w1 w2 +w3

y

{

zzzzzzzzzzzzzzzzzzzzzz

g1 =881<, 80<, 80<, 80<, 80<, 80<<

881<, 80<, 80<, 80<, 80<, 80<<

g2 =880<, 81<, 80<, 80<, 80<, 80<<

880<, 81<, 80<, 80<, 80<, 80<<

<<LinearAlgebra`MatrixManipulation`

H * - - - - - - - ¶X f - - -- -- - * L

Df = AppendRows@¶w1 f, ¶w2 f, ¶w3 f, ¶ w1 f, ¶ w2 f, ¶z fD

980, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0<, 8e w2, e w1, 0, 0, 0, 0<,

9 1€€€€2 H1 +w1

2 - w22L, w1 w2, w2, w1 w1 +w2 w2, - w2 w1 +w1 w2 +w3, 0=,

9w1 w2, 1€€€€2 H1 - w1

2 +w22L, - w1, w2 w1 - w1 w2 - w3, w1 w1 +w2 w2, 0=, 8- w2, w1, 1, w2, - w1, 0<=

H * - - - - - - - ¶X g1 - - - - - - - * L

Dg1 = AppendRows@¶w1 g1, ¶w2 g1, ¶w3 g1, ¶ w1 g1, ¶ w2 g1, ¶z g1D

880, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0<,

80, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0<<

H * - - - - - - - ¶X g2 - - - - - - - * L

Dg2 = AppendRows@¶w1 g2, ¶w2 g2, ¶w3 g2, ¶ w1 g2, ¶ w2 g2, ¶z g2D

880, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0<,

80, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0<<

H * - - - - - - - @f, g1D - - - - - - - * L

ad1fg1 =Dg1.f - Df.g1

980<, 80<, 8- e w2<, 9 1€€€€2 H- 1 - w1

2 +w22L=, 8- w1 w2<, 8w2<=

60 Mathematica printouts



Dad1fg1 = AppendRows@¶w1 ad1fg1, ¶w2 ad1fg1, ¶w3 ad1fg1, ¶ w1 ad1fg1, ¶ w2 ad1fg1, ¶z ad1fg1D

880, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0<, 80, - e, 0, 0, 0, 0<,

80, 0, 0, - w1, w2, 0<, 80, 0, 0, - w2, - w1, 0<, 80, 0, 0, 0, 1, 0<<

H * - - - - - - - @f, g2D - - - - - - - * L

ad1fg2 =Dg2.f - Df.g2

980<, 80<, 8- e w1<, 8- w1 w2<, 9 1€€€€2 H- 1 +w1

2 - w22L=, 8- w1<=

Dad1fg2 = AppendRows@¶w1 ad1fg2, ¶w2 ad1fg2, ¶w3 ad1fg2, ¶ w1 ad1fg2, ¶ w2 ad1fg2, ¶z ad1fg2D

880, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0<, 8- e, 0, 0, 0, 0, 0<,

80, 0, 0, - w2, - w1, 0<, 80, 0, 0, w1, - w2, 0<, 80, 0, 0, - 1, 0, 0<<

H * - - - - - - - @g1, fD - - - - - - - * L

ad1g1f =Df.g1 - Dg1.f

980<, 80<, 8e w2<, 9 1€€€€2 H1 +w1

2 - w22L=, 8w1 w2<, 8- w2<=

Dad1g1f = AppendRows@¶w1 ad1g1f, ¶w2 ad1g1f, ¶w3 ad1g1f, ¶ w1 ad1g1f, ¶ w2 ad1g1f, ¶z ad1g1fD

880, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0<, 80, e, 0, 0, 0, 0<,

80, 0, 0, w1, - w2, 0<, 80, 0, 0, w2, w1, 0<, 80, 0, 0, 0, - 1, 0<<

H * - - - - - - - @g2, @f, g1DD - - - - - - - * L

ad1g2ad1fg1 =Dad1fg1.g2 - Dg2.ad1fg1

880<, 80<, 8- e<, 80<, 80<, 80<<

H * - - - - - - - @@g1, fD, g2D - - - - - - - * L

ad1g1ad1fg2 =Dg2.ad1g1f - Dad1g1f.g2

880<, 80<, 8- e<, 80<, 80<, 80<<

Dad1g1ad1fg2 = AppendRows@¶w1 ad1g1ad1fg2, ¶w2 ad1g1ad1fg2,

¶w3 ad1g1ad1fg2, ¶ w1 ad1g1ad1fg2, ¶ w2 ad1g1ad1fg2, ¶z ad1g1ad1fg2D

880, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0<,

80, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0<<

H * - - - - - - - @f, @@g1, fD, g2DD - - - - - - - * L

ad1fad1g1ad1fg2 =Dad1g1ad1fg2.f - Df.ad1g1ad1fg2

880<, 80<, 80<, 8w2 e<, 8- w1 e<, 8e<<

H * C = 8g1, g2, @f, g1D, @f, g2D, @f, @f, g1DD, @f, @f, g2 DD< * L

ControllabilityMatrix = AppendRows@g1, g2, ad1fg1, ad1fg2, ad1g2ad1fg1, ad1fad1g1ad1fg2D

981, 0, 0, 0, 0, 0<, 80, 1, 0, 0, 0, 0<,

80, 0, - e w2, - e w1, - e, 0<, 90, 0, 1€€€€2 H- 1 - w1

2 +w22L, - w1 w2, 0, w2 e=,

90, 0, - w1 w2, 1€€€€2 H- 1 +w1

2 - w22L, 0, - w1 e=, 80, 0, w2, - w1, 0, e<=

61



MatrixForm @ControllabilityMatrixD

i

k


1 0 0 0 0 0

0 1 0 0 0 0

0 0 - e w2 - e w1 - e 0

0 0 1€€€€2 H- 1 - w1

2 +w22L - w1 w2 0 w2 e

0 0 - w1 w2 1€€€€2 H- 1 +w1

2 - w22L 0 - w1 e

0 0 w2 - w1 0 e

y

{


DeterminantC = Simplify@Det@ControllabilityMatrixDD

- 1€€€€4 H1 +w1

2 +w22L

2e2

H * - - - - - - - Check "bad" brackets to conclude STLC -- -- -- - * L

H * - - - - - - - @g2, @f, g2DD - - - - - - - * L


880<, 80<, 80<, 80<, 80<, 80<<

H * - - - - - - - @g1, @f, g1DD - - - - - - - * L


880<, 80<, 80<, 80<, 80<, 80<<

MatrixForm @fD

i

k


0

0

e w1 w2

1€€€€2 H1 +w1

2 - w22L w1 +w1 w2 w2 +w2 w3

w1 w2 w1 + 1€€€€2 H1 - w1

2 +w22L w2 - w1 w3

- w2 w1 +w1 w2 +w3

y

{


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hegrenaes_techrep_2003

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