cmg paper 97

7/24/2019 Cmg Paper 97

1/124

A Geometric Study ofSingle Gimbal Control Moment Gyros

Singularity Problems and Steering Law

Haruhisa Kurokawa

Mechanical Engineering Laboratory

Report of Mechanical Engineering Laboratory, No. 175, p.108, 1998.

7/24/2019 Cmg Paper 97

2/124

i

In this research, a geometric study of singularitycharacteristics and steering motion of single gimbal

Control Moment Gyros (CMGs) was carried out in order

to clarify singularity problems, to construct an effective

steering law, and to evaluate this laws performance.

Passability, as defined by differential geometry

clarified whether continuous steering motion is possible

in the neighborhood of a singular system state.

Topological study of general single gimbal CMGs

clarified conditions for continuous steering motion over

a wider range of angular momentum space. It was shown

that there are angular momentum vector trajectories such

that corresponding gimbal angles cannot be continuous.

If the command torque, as a function of time, results in

such a trajectory in the angular momentum space, any

steering law neither can follow the command exactly

nor can be effective.

A more detailed study of the symmetric pyramid type

of single gimbal CMGs clarified a more serious problemof continuous steering, that is, no steering law can follow

all command sequences inside a certain region of the

angular momentum space if the command is given in

real time. Based on this result, a candidate steering law

effective for rather small space was proposed and verified

not only analytically, but also using ground experiments

which simulated attitude control in space.

Similar evaluation of other steering laws and

comparison of various system configurations in terms

of the allowed angular momentum region and the

systems weight indicated that the pyramid type single

gimbal CMG system with the proposed steering law is

one of the most effective candidate torquer for attitude

control, having such advantages as a simple mechanism,

a simpler steering law, and a larger angular momentum

space.

A Geometric Study ofSingle Gimbal Control Moment Gyros

Singularity Problems and Steering Law

by

Haruhisa Kurokawa

Abstract

KeywordsAttitude control, Singularity, Momentum exchange device, Inverse kinematics, Steering law

7/24/2019 Cmg Paper 97

3/124

ii

This research work is a result of projects conducted

at the Mechanical Engineering Laboratory, Agency of

Industrial Technology and Science, Ministry of

International Trade and Industry, Japan. Related projects

are, Development of Attitude Control Equipment

(FY19821987), Attitude Control System for Large

Space Structures (FY19881993), and High Precision

Position and Attitude Control in Space (FY19931997).

I wish to acknowledge my debt to many people. Prof.

Nobuyuki Yajima of the Institute of Space and

Astronautical Science (ISAS) are earnestly thanked for

inspiring me with this theme, as well as for collaborations

during his tenure as a division head of our laboratory. I

would extend thanks to the late Prof. Toru Tanabe,

formerly of the University of Tokyo for his guidance in

the culmination of this work into a dissertation. In

finishing this work, the following professors guided me,

Assoc. Prof. Shinichi Nakasuka of the University Tokyo,

Prof. Hiroki Matsuo of ISAS, Prof. Shinji Suzuki, Prof.Yoshihiko Nakamura,Assoc. Prof. Ken Sasaki of the

University of Tokyo.

Many discussions with Dr. Shigeru Kokaji of our

laboratory proved invaluable. He patiently listened to

my abstract explanation of geometry and provided

valuable suggestions. Furthermore, he assisted me by

soldering and checking circuits, and reviewed this paper June 7, 1997

Acknowledgments

from cover-to-cover, providing constructive criticism.

I would also like to thank my colleague Akio Suzuki

who constructed most of the experimental apparatus, and

designed and installed controllers for the attitude control.

Prof. Tsuneo Yoshikawa of Kyoto University helped

me when we started the project of attitude control by

CMGs. Discussions held with Dr. Nazareth Bedrossian

and Dr. Joseph Paradiso of the Charles Stark Draper

Laboratory (CSDL) were invaluable. They gave me

valuable suggestions with various research papers in this

field.

Dr. Mark Lee Ford as a visiting researcher of our

laboratory spent his precious hours for me to correct

expressions in English.

I would like to thank all the above people, other

colleagues sharing other research projects, and the

Mechanical Engineering Laboratory (MEL) and the

directors especially the Director General Dr. Kenichi

Matuno and the former Department Head Dr. Kiyofumi

Matsuda for allowing me to continue this research.Finally, I thank my wife and daughters for their patience

particularly during some hectic months.

Haruhisa Kurokawa

7/24/2019 Cmg Paper 97

4/124

iii

Contents

Abstract ............................................................................................................................................................ i

Acknowledgments ........................................................................................................................................... ii

Terms ........................................................................................................................................................... viii

Nomenclature ................................................................................................................................................. ix

List of Figures ................................................................................................................................................. x

List of Tables ............................................................................................................................................... xiii

Chapter 1 Introduction .............................................................................................. 1

1.1 Research Background ..................................................................................................................................... 1

1.2 Scope of Discussion ........................................................................................................................................ 3

1.3 Outline of this Thesis ...................................................................................................................................... 4

Chapter 2 Characteristics of Control Moment Gyro Systems ............................... 5

2.1 CMG Unit Type ............................................................................................................................................. 5

2.2 System Configuration .................................................................................................................................... 5

2.2.1 Single Gimbal CMGs ............................................................................................................................ 6

2.2.2 Two Dimensional System and Twin Type System ................................................................................ 7

2.2.3 Configuration of Double Gimbal CMGs...............................................................................................7

2.3 Three Axis Attitude Control ........................................................................................................................... 7

2.3.1 Block Diagram ...................................................................................................................................... 8

2.3.2 CMG Steering Law ............................................................................................................................... 8

2.3.3 Momentum Management ...................................................................................................................... 8

2.3.4 Maneuver Command ............................................................................................................................. 8

2.3.5 Disturbance ........................................................................................................................................... 8

2.3.6 Angular Momentum Trajectory .............................................................................................................8

2.4 Comparison and Selection ............................................................................................................................. 9

2.4.1 Performance Index ................................................................................................................................ 92.4.2 Component Level Comparison ............................................................................................................. 9

2.4.3 System Level Comparison .................................................................................................................... 9

2.4.4 Work Space Size and Weight ................................................................................................................ 9

Chapter 3 General Formulation .............................................................................. 11

3.1 Angular Momentum and Torque ................................................................................................................... 11

3.2 Steering Law ................................................................................................................................................. 12

3.3 Singular Value Decomposition and I/O Ratio...............................................................................................12

3.4 Singularity ..................................................................................................................................................... 13

7/24/2019 Cmg Paper 97

5/124

iv

3.5 Singularity Avoidance ................................................................................................................................... 13

3.5.1 Gradient Method ................................................................................................................................. 14

3.5.2 Steering in Proximity to a Singular State .............................................................................................14

Chapter 4 Singular Surface and Passability .......................................................... 15

4.1 Singular Surface ........................................................................................................................................... 15

4.1.1 Continuous Mapping ........................................................................................................................... 15

4.1.2 Envelope .............................................................................................................................................. 16

4.1.3 Visualization Method of the Surface ...................................................................................................16

4.2 Differential Geometry .................................................................................................................................. 17

4.2.1 Tangent Space and Subspace............................................................................................................... 17

4.2.2 Gaussian Curvature ............................................................................................................................. 17

4.3 Passability .................................................................................................................................................... 18

4.3.1 Quadratic Form ................................................................................................................................... 18

4.3.2 Signature of Quadratic Form ............................................................................................................... 19

4.3.3 Passability and Singularity Avoidance ................................................................................................19

4.3.4 Discrimination ..................................................................................................................................... 20

4.4 Internal Impassable Surface ......................................................................................................................... 21

4.4.1 Impassable Surface of an Independent Type System .......................................................................... 21

4.4.2 Impassable Surface of a Multiple Type System .................................................................................. 21

4.4.3 Minimum System ................................................................................................................................ 22

Chapter 5 Inverse Kinematics ................................................................................. 23

5.1 Manifold....................................................................................................................................................... 23

5.2 Manifold Path .............................................................................................................................................. 24

5.3 Domain and Equivalence Class ................................................................................................................... 24

5.4 Terminal Class and Domain Type ................................................................................................................ 25

5.5 Class Connection ......................................................................................................................................... 25

5.5.1 Type 2 Domain .................................................................................................................................... 25

5.5.2 Type 1 Domain .................................................................................................................................... 26

5.5.3 Class Connection Rules....................................................................................................................... 27

5.5.4 Continuous Steering over Domains ....................................................................................................28

5.5.5 Manifold Selection .............................................................................................................................. 28

5.5.6 Discussion of the Critical Point...........................................................................................................29

5.6 Topological Problem ..................................................................................................................................... 29

Chapter 6 Pyramid Type CMG System ................................................................... 31

6.1 System Definition ........................................................................................................................................ 31

6.2 Symmetry ..................................................................................................................................................... 31

6.3 Singular Manifold for the H Origin ............................................................................................................. 33

7/24/2019 Cmg Paper 97

6/124

v

6.4 Singular Surface Geometry .......................................................................................................................... 35

Chapter 7 Global Problem, Steering Law Exactness and Proposal ................... 41

7.1 Global Problem ............................................................................................................................................ 41

7.1.1 Control Along the z Axis ..................................................................................................................... 41

7.1.2 Global problem.................................................................................................................................... 45

7.1.3 Details of the Problem ......................................................................................................................... 45

7.1.4 Possible Solutions ............................................................................................................................... 47

7.2 Steering Law with Error .............................................................................................................................. 47

7.2.1 Geometrical Meaning .......................................................................................................................... 47

7.2.2 Exactness of Control ........................................................................................................................... 48

7.3 Path Planning ............................................................................................................................................... 49

7.4 Preferred Gimbal Angle ............................................................................................................................... 49

7.5 Exact Steering Law ...................................................................................................................................... 51

7.5.1 Workspace Restriction......................................................................................................................... 51

7.5.2 Repeatability and Unique Inversion ....................................................................................................51

7.5.3 Constrained Control ............................................................................................................................ 52

7.5.4 Reduced Workspace ............................................................................................................................ 52

7.5.5 Characteristics of Constrained Control ...............................................................................................54

Chapter 8 Ground Experiments .............................................................................. 57

8.1 Attitude Control ........................................................................................................................................... 578.1.1 Dynamics............................................................................................................................................. 57

8.1.2 Exact Linearization ............................................................................................................................. 57

8.1.3 Control Method ................................................................................................................................... 58

8.2 Experimental Facility and Procedure ...........................................................................................................58

8.2.1 Facility ................................................................................................................................................. 58

8.2.2 Design of Control Command Sequence ..............................................................................................59

8.2.3 Experimental Procedure ...................................................................................................................... 59

8.3 Experimental Results ................................................................................................................................... 60

8.3.1 Attitude Keeping under Constant Disturbance .................................................................................... 60

8.3.2 Rotation About the z Axis ................................................................................................................... 64

8.3.3 Maneuver after Momentum Accumulation .........................................................................................67

8.3.4 Mode Selection and Switching.............................................................................................................69

8.4 Summary of Experiments ............................................................................................................................ 69

Chapter 9 Evaluation ............................................................................................... 71

9.1 Conditions for Comparison .......................................................................................................................... 71

9.2 Spherical Workspace .................................................................................................................................... 71

9.3 Evaluation by Weight ................................................................................................................................... 72

7/24/2019 Cmg Paper 97

7/124

vi

9.4 Ellipsoidal Workspace ................................................................................................................................. 73

9.5 Summary of Evaluation ................................................................................................................................ 75

Chapter 10 Conclusions .......................................................................................... 77

Appendix A Double Gimbal CMG System .............................................................. 79

A.1 General Formulation ................................................................................................................................... 79

A.2 Singularity ................................................................................................................................................... 79

A.3 Steering Law and Null Motion ................................................................................................................... 80

A.4 Passability ................................................................................................................................................... 80

A.4.1 Two Unit System ................................................................................................................................ 80

A.4.2 Three Unit System .............................................................................................................................. 81

A.5 Workspace ................................................................................................................................................... 81

Appendix B Proofs of Theories ............................................................................... 83

B.1 Basis of Tangent Spaces .............................................................................................................................. 83

B.2 Gaussian Curvature ..................................................................................................................................... 83

B.3 Inverse Mapping Theory ............................................................................................................................. 84

B.4 Impassable condition for two negative signs ..............................................................................................85

Appendix C Internal Impassability of Multiple Type Systems .............................. 87

C.1 Roof Type System M(2, 2) .......................................................................................................................... 87

C.1.1 Evaluation of Singular Surface (2) .....................................................................................................87




C.1.5 Conclusion .......................................................................................................................................... 88

C.2 M(3, 2): M(2, 2)+1 ...................................................................................................................................... 88

C.2.1 Condition (3) of M(2,2) ...................................................................................................................... 88

C.2.2 Condition (5) of M(2,2) ...................................................................................................................... 89

C.3 M(3, 3): M(2, 2)+2 ...................................................................................................................................... 89

C.4 M(2, 2, 1): M(2, 2)+1 .................................................................................................................................. 89

C.5 M(2, 2, 2): M(2, 2)+2 .................................................................................................................................. 89

C.6 Minimum System ........................................................................................................................................ 89

Appendix D Six and Five Unit Systems .................................................................. 91

D.1 Symmetric Six Unit System S(6) ................................................................................................................. 91

D.1.1 System Definition ................................................................................................................................ 91

7/24/2019 Cmg Paper 97

8/124

vii

D.1.2 Fault Management ............................................................................................................................... 91

D.1.3 Four out of Six Control ....................................................................................................................... 92

D.2 Five Unit Skew System ................................................................................................................................ 92

Appendix E Specification of Experimental Apparatus and Experimental Procedure

. ................................................................................................... 95

E.1 Experimental Apparatus .............................................................................................................................. 95

E.2 Specifications .............................................................................................................................................. 97

E.3 Attitude Control System .............................................................................................................................. 97

E.4 Steering Law Implementation ..................................................................................................................... 99

E.5 Code Size and Calculation Time ................................................................................................................. 99

E.6 Parameter Estimation .................................................................................................................................. 99

Appendix F General kinematics ............................................................................ 101

F.1 Analogy with a Spatial Link Mechanism...................................................................................................101

F.2 Spatial Link Mechanism Kinematics ......................................................................................................... 101

F.3 Singularity .................................................................................................................................................. 102

F.4 Passability .................................................................................................................................................. 102

References ............................................................................................................... 105

7/24/2019 Cmg Paper 97

9/124

viii

Terms

Class : A set of manifolds which correspond to a certain

domain and are equivalent to each other.

Domain : A region in the angular momentum space which

is surrounded by singular surfaces and does not

contain any singular surface.

Double gimbal CMG : Fig. 21

Gimbal vector : A unit vector of gimbal direction.

Independent type : A single gimbal CMG system without

parallel gimbal direction pair.

Manifold : A connected subspace of gimbal angle space

whose element corresponds to the same total

angular momentum.

Manifold equivalence : Two manifolds corresponding

to a certain domain are equivalent if there is an

angular momentum path which corresponds to a

continuous manifold path between these two

manifolds.

Multiple type : A single gimbal CMG system composed

of groups each of which elements possess

identical gimbal direction.

Null motion : Gimbal angle motion which keeps the

angular momentum vector constant.

Single gimbal CMG : Fig. 21

Singular surface : A surface formed by the total angular

momentum vector point, H, which corresponds

to singular point.

Singular vector : A unit vector to the plane spanned by

all torque vectors when the system is singular.

Skew type : A single gimbal CMG system with gimbal

directions axially symmetric about one direction.

Symmetric type : A single gimbal CMG system with

gimbal directions arranged normal to surfaces of

a regular polyhedron.

Torque vector : A unit vector of a component CMG to

which direction the CMG can generate an output

torque.

Workspace: Allowed region of the angular momentumvector of a CMG system.

7/24/2019 Cmg Paper 97

10/124

ix

Symbol Definition Section number

: Skew angle of the symmetric pyramid type

system 6.1

: Euler parameter of satellite orientation 8.1.1

* : Vector part of 8.1.1

B: Strip like surface of impassable surface called

branch 6.4

c* : = cos 6.1

ci: = gi hi. Torque vector 3.1

C: Jacobian matrix of the kinematic function,

H= f () 3.1

dS S. 4.2.1

dN N 4.2.1

dT T 4.2.1

D : Domain in the Hspace surrounded by singular

surfaces 5.3

: = {i}. Sign parameter of the singular surface.

4.1.1

gi: Gimbal vector 3.1

G : Equivalence class in a domain. 5.3

hi: Normalized angular momentum vector 3.1

H: = hi.= f(). Total angular momentum vector.

3.1

: Gaussian curvature of the singular surface.4.2.2

LA: Segment included by a manifold of H=(0,0,0)t

6.3

Nomenclature

Symbol Definition Section number

M(2, 2): Roof type system 2.2.1

Mi : Manifold 5.1

MSj : Singular manifold 5.1

n : Number of CMG units in the system 3.1

pi : = 1 / (uhi) 4.1.3

P: Diagonal matrix ofpi. 4.1.3

i: Gimbal angle of ith CMG unit 3.1

: =(i.). A state variable of the system. Point of n

dimensional torus T(n) 3.1

S: Singularly constrained tangent space of the

space (two dimensional). 4.2.1

N: Null space of C (n2 dimensional). 4.2.1

T: Complementary subspace of N (two

dimensional). 4.2.1

rg: Symmetric transformation in the space. 6.2

Rg: Symmetric transformation in the Hspace. 6.2

s* : = sin 6.1

S(n) : Symmetric type single gimbal CMG system.

2.2.1

S: A region of the singular surface of sign . 4.1.1

T: Total output torque of the system 3.1

u: Singular vector. Unit vector normal to all

torque vectors. 3.4

: Gimbal rate vector. Time derivative of . 3.1

N: Null motion, 3.2

7/24/2019 Cmg Paper 97

11/124

x

Chapter 2

21 Two types of CMG units

22 Configurations of single gimbal CMGs

23 Twin type system

24 Block diagram of three axis att itude

control

Chapter 3

31 Orthonormal vectors of a CMG unit

32 Gimbal angle and vectors

33 Input Output ratio

34 Singularity condition and singular vector

35 Typical vector arrangement for a 2D

system

36 Steering at a singular condition

Chapter 4

41 Vectors at a singularity condition

42 Examples of the singular surfaces for the

pyramid type system.

43 Envelope of a roof type system M(2, 2).

44 Cross sections of a singular surface of the


45 Infinitesimal motion from a singular point

of 2D system.

46 Second order infinitesimal motion from

singular surface.

47 Possible motions in both direction of uat

a singular point.

48 Local shape of an impassable singular

surface.

49 Impassable surface of S(6)

410 Impassable surface of Skew(5) with skew

angle = 0.6 rad.

411 Impassable surface of another Skew(5),

with skew angle = 1.2 rad.

412 Impassable surface of S(4).

Chapter 5

51 Manifolds in the neighborhood of a

singular point.

52 Continuous change of manifolds.

53 An example of a continuous manifold

path.

54 Relations between Hspace, manifold

space and space.

55 Domains and manifolds of the pyramid

type system 56 Class connection graph around domains

57 An illustration of class connection rule

(1).

58 An illustration of class connection rule

(2).

59 An illustration of motion by the gradient

method.

510 Manifold relations around critical point

Chapter 6

61 Schematic of a pyramid type system

62 Transformation in Hspace and in space

63 Line segments for singular manifold

64 Definition of the cross sectional plane and

the distance d

65 Saddle like part of the envelope

66 Cross sections of singular surface

67 Internal impassable singular surface

68 Analytical line on an impassable surface

69 E quilateral parallel hexahedron of

impassable branches

610 Overall structure of impassable branches

611 Internal impassable surface with envelope

cutaway

612 Cross section through the xz plane

613 Cross section through the xy plane

Chapter 7

71 Candidate of workspace 72 Cross section nearly crossing P

List of Figures

7/24/2019 Cmg Paper 97

12/124

xi

73 Manifold bifurcat ion and termination

from DA

74 Simplified class connection diagram

around domain DA

75 Manifolds of eight domains around the z

axis

76 Singular manifold of a point U on the z

axis

77 Manifold of Hnear the origin

78 Continuos change of manifold for H

nearly along the z axis

79 Manifold connection over several

domains

710 Cross sections of domains

711 Possible motion following an example of

singular surface

712 Illustration of Htrajectory of the CMG

system for the example maneuver

713 Avoidance of an impassable surface

714 Problems of movement on an impassable

surface

715 Change in manifolds for Hmoving along

the x axis

716 Estimation of reduced workspace for

exact steering

717 D is co nt in ui ty i n t he m ax im um o f

det(CCt)

718 Cross section of possible workspace by

constrained steering law

719 Reduced workspace of the constrained

system

720 Reduced workspace of three modes

Chapter 8

81 Experimental test rig showing the centermount suspending mechanism

82 Target trajectory

83 Block diagram of the control system

84 Results of Experiment A

85 Results of Experiment B

86 Results of Experiment C

87 Results of Experiment D

88 Results of Experiment E

89 Results by Experiment F

810 Results of Experiment G 811 Results of Experiment H

812 Command sequence of Experiment J

813 Results of Experiment J

Chapter 9

91 System configurations for comparison

92 Spherical workspace size for var ious

system configurations

93 Trade-off between workspace size and

system weight

94 Definition of ellipsoidal workspace

95 Average radius vs. skew angle

96 Workspace radius as a function of aspect

ratio

97 Combined plot of radii as a function of

aspect ratio 98 Converted weight as a function of aspect

ratio

99 Radius as a function of aspect ratio for a

degraded system with one faulty unit

Appendix A

A1 Vectors and variables relevant to a double

gimbal CMG

A2 Vectors at singularity conditions

A3 Infinitesimal motion at a singular point

of condition (b)

Appendix D

D1 Envelopes of S(6) and degraded systems

D2 Four unit subsystem of MIR type system

D3 Restricted workspace of a constrained

MIR-type system

D4 Concept of singularity avoidance by an

additional torquer

Appendix E

E1 Experimental apparatus

E2 Block diagram of experimental apparatus

E3 Three axis gimbal mechanism

E4 Single gimbal CMG

E5 Balance adjuster

E6 Onboard computer

E7 Block diagram of the model matching

controller. E8 Block diagram of the tracking controller.

7/24/2019 Cmg Paper 97

13/124

xii

E9 Block diagram of the gradient method.

E10 Block diagram of the constrained method.

Appendix F

F1 Analogy to a parallel link mechanism

7/24/2019 Cmg Paper 97

14/124

xiii

List of Tables

Chapter 2

21 Component Level Comparison

22 System Level Comparison

Chapter 6

61 Symmetric Transformations

62 Segment Transformation Rule

Chapter 8

81 Condition and Results of Experiments (1)

82 Condition and Results of Experiments (2)

Appendix E

E1 Specification of experimental apparatus

E2 Code size and calculation time of process

Appendix F

F1 Sim ilar ity between CMGs and lin k

mechanism

7/24/2019 Cmg Paper 97

15/124

xiv

7/24/2019 Cmg Paper 97

16/124

1

2. Characteristics of Control Moment Gyro Systems

A Control Moment Gyro (CMG) is a torque generator

for attitude control of an artificial satellite in space. It

rivals a reaction wheel in its high output torque and rapid

response. It is therefore used for large manned satellites,

such as a space station, and is also a candidate torquer

for a space robot.

There are two types of CMGs, single gimbal and

double gimbal. Though single gimbal CMGs are better

in terms of mechanical simplicity and higher output

torque than double gimbal CMGs, the control of single

gimbal CMGs has inherent and serious singularity

problem. At a singularity condition, a CMG system

cannot produce a three axis torque. Despite various

efforts to overcome this problem, the problem still

remains, especially in the case of the pyramid type CMG

system.

This research aims to elucidate this singularity

problem. Detailed study of the pyramid type systemleads to a global problem of singularity. The final

objective of this work involves evaluation of various

steering laws and the proposal of an effective steering

law. As all the geometric studies are either theoretical

or analytical and based on computer calculations, ground

experiments were carried out to support those results.

1.1 Research Background

Research of CMG systems started in the mid 1960s.

This was intended for later application to the large

satellite of the USA, Skylab, and its high precision

component, Apollo Telescope Mount (ATM)1, 2,3).

The studies included hardware studies of a gyro bearing

and gyro motor, and software studies for attitude control

and CMG steering control. Evaluation of various types

and configurations was made in terms of weight and

power consumption4). At that time, an onboard computer

lacked the ability to perform real time matrix inversion

calculation. One of the candidates was a twin type

system made of two single gimbal CMGs driven inopposite directions. Control of this system requires only

simple calculation5). If another system was chosen, a

simple computation scheme was required using an analog

circuit. For example, a method using an approximation

with some feedback was proposed6, 7, 8). For the three

double gimbal CMG system9)applied to the Skylab, an

approximated inverse using the transposed Jacobian was

used10). This CMG system successfully completed its

mission, though one of the CMGs became nonfunctional

during the flight11). After that, studies of double gimbal

CMGs have continued for eventual application to the

space shuttle and the space station Freedom which is

now called ISS12, 13).

Another CMG type, i.e., a single gimbal one, was

studied for use in satellites such as the High Energy

Astronomical Observatory (HEAO) and the Large

Space Telescope (LST). One of the configurations

intensively studied was a pyramid type, which consists

of four single gimbal CMGs in a skew configuration.Comparing six different independently developed

steering laws indicated that an exact inverse calculation

was necessary14). It was also observed from various

simulations that the singularity problem could not be

ignored. It was concluded that some sort of singularity

avoidance control using system redundancy was required

for this type system.

A roof type system, which is another four unit system

of single gimbal CMGs, was also a candidate for the

HEAO. As its mathematical formulation is simpler than

that of the pyramid type, singularity avoidance was

originally included in a steering law15). An improvement

of this law involved a new approach in which the nature

of numerical calculation and discrete time control were

utilized16).

Singularity avoidance has been studied for all CMG

types. This was a simple matter for double gimbal CMG

systems17)20). Typically used was a gradient method,

which maximized a certain objective function by using

redundancy21). While this method was effective in the

evaluation of double gimbal CMG systems, it was not

successful for single gimbal CMG systems. For

Chapter 1

Introduction

7/24/2019 Cmg Paper 97

17/124

2

Technical Report of Mechanical Engineering Laboratory No.175

example, optimization of a redundant variable resulted

in discontinuity16) or an optimized value became

singular15)in the case of a roof type system. In the case

of a pyramid type system, various problems were found

in computer simulations even when a gradient method

was used.Margulies was the first to formulate a theory of

singularity and control22). His paper included geometric

theory of a singular surface, a generalized solution of

the output equation and null motion, and the possibility

of singularity avoidance for a general single gimbal CMG

system. Also, some problems of the gradient method

were pointed out using an example of a two dimensional

system.

Works by the Russian researcher, Tokar, were

published in the same year, and included a description

of the singular surface shape23), the size of the

workspace24)and some considerations of the gimbal

limits25). In his next paper26), passability of a singular

surface was introduced. It was made clear that a system

such as a pyramid type has an impassable surface inside

its workspace. Moreover, the problems of steering near

such an impassable surface were described. In spite of

those important results, his work was not widely

received, because the original papers were published in

Russian. Even though an English translation appeared,

several terms were used for a CMG, such as gyroforce,gyro stabilizer and gyrodyne. His conclusion was

that a system with no less that six units would provide

an adequately sized workspace including no impassable

surfaces. After this work, a six unit symmetric system

was designed for the Russian space station MIR27).

Some years later after Tokars studies, Kurokawa

formulated passability again28)in terms of the geometric

theory given by Margulies. Most of these results

coincided with Tokars work. In addition, the existence

of impassability in the roof type system was clarified29)

and a discrimination method using the surface curvaturewas presented30,31,32,33). In the last paper, the theory

was expanded to a general system including a double

gimbal CMG system. It was made clear that multiple

systems of no less than six units do not have any internal

impassable surface, while any system of less that six

units must have such a surface. Various configurations,

even containing faulty units, were compared with regard

to their workspace size as an extension of Tokars work.

Along with these theoretical and general research

works, intensive efforts continued to find an effective

steering law regarding the passability problem as a local

problem. Most of these dealt with the pyramid type

system. The reason this type was selected was because

a six unit system was considered too large and too

complicated. Many proposals suggested a type of

gradient method34, 35, 36). The method utilized for the

four unit subsystem of the MIR was also of this

kind27). Another method used global optimization28),and nearly all methods showed some problems in

computer simulations.

Passability is defined locally and its problem reported

first was a kind of local problem28). Later, Bauer showed

difficulty in steering as a global problem37). He found

two different command sequences, both of which could

not be realized by the same steering method. After this,

Vadali proposed a method to overcome this problem

using a preferred state38). Finally, the problem by Bauer

was formulated exactly, stating that no steering law can

follow an arbitrary command sequence inside certain

wide region of the workspace39). Under this limitation,

an effective method was proposed.

The research described above dealt with exact

control, but other research has also been carried out. One

research effort permitted an error in the output if required.

Generalized inverse Jacobian22)minimizes the error.

Extension of this method, called the SR inverse method,

was first proposed for control of a manipulator and later

applied to CMG control40,41). Another research type

dealt with path planning. If the command sequence inthe near future is given, steering can be planned

beforehand which realizes not only singularity avoidance

but also some degree of optimization42, 43, 44). In one

of the research papers43), some paths were chosen by

Kurokawa in consideration of impassable surfaces. Since

all these tended to take a heuristic approach, evaluation

was made by computer simulation considering attitude

control of a given satellite.

More realistic studies have also been made which

dealt with attitude control using a CMG system,

considering disturbance and other torquers. The largestproblem may be a precision control using a CMG system.

Since a CMG system can generate a large output torque

and its output resolution is critical for precision control,

various analyses and simulations have shown that

pointing control by a CMG system can result in a limit

cycle because of friction in gimbal motion45, 46, 47). In

spite of efforts such as improvement of motor control48)

and torque cancellation by additional reaction wheels49),

the problem of precision control has not been overcome.

For application to the space station, another studies were

carried out such as an effective combination of a CMG

and RCS50) and integration of CMGs and power

7/24/2019 Cmg Paper 97

18/124

3


storage51). In order to evaluate its attitude control

performance, not only numerical simulations, but also

some experiments using real mechanisms have been

made, such as a platform supported by a spherical air

bearing44, 52). The author also developed ground test

equipment using normal ball bearings53)and attemptedrobust attitude control using a CMG system54,55).

The motion of a CMG system with regard to the

motion of the angular momentum vector is similar to

the motion of a link mechanism22). Analysis of the

motion and control of such a mechanism has been widely

studied. Those results were, therefore, used for CMG

control40, 41). On the other hand, some researchers first

studied CMG control and then applied their results to a

robot control56, 57, 58). In spite of various researches in

robot kinematics59, 60, 61), generalized theory for

singularity and inverse kinematics has not been

formulated yet.

1.2 Scope of Discussion

This research effort deals with the following subjects:

(1) General formulation of an arbitrarily configured

CMG system, especially of single gimbal CMGs.

(2) Geometric study of the singularity problem of a

general single gimbal CMG system.

(3) Problem of exact and real-time steering of the

pyramid type CMG system.

(4) Proposal and evaluation of steering laws for the

pyramid type CMG system.

(5) Evaluation of various CMG systems.

The main purposes of this work are to clarify the

singularity problems, to construct an exact and strictly

real-time steering law, and to specify and evaluate its

performance. Among all, singularity problems are the

most important relating to the others. A singularity can

degrade a CMG system, even causing the system to loose

control, and this situation might be fatal for an artificial

satellite. Therefore, a CMG system must have

redundancy and it must be controlled to avoid

singularities by using an appropriate steering law.

Problems include whether such singularity avoidance is

globally possible and which steering law can realize such

control. Even if a steering law cannot avoid all the

singularities, the systems working range of the angular

momentum must be specified in which singularity

avoidance is strictly guaranteed because such

specification is necessary for designing the total attitudecontrol system. Thus, this work deals with CMG systems

alone, but it is made in consideration with the attitude

control of artificial satellites. Exactness and strict real

time feature of steering laws are essential for the real-

time attitude control.

For this aim, a geometric approach was taken. As

described above, there have been various research worksdealing with singularity and steering laws. Most used

computer simulations to evaluate their steering laws, for

lack of other methods. As simulations alone cannot

guarantee the performance of a system as nonlinear as a

CMG system, it is necessary to clarify the problem of

singularity by other means. A geometrical approach is a

more effective way of simplification and qualitative

comprehension. The theoretical portion of this work

aims for general formulation of singularity problems.

Under consideration of these general results,

extensive study was made for a specific type of system,

that is, the pyramid type. The reasons why this system

was chosen are:

1) A three-unit system does not need further study

because it has no redundancy. Systems with no less

than six units also do not need detailed study for

singularity avoidance, a fact described in more detail

in this work. Thus, four and five unit systems remain

for further study.

2) Most previous research works dealt with this

pyramid type system. Four units are the minimumhaving one degree of redundancy. The number of

units is important in the real situation. By a

simplified evaluation, a system with fewer units is

lighter for a given total storage of angular

momentum. Also, steering law calculation is less

complicated for a system with fewer units.

3) The pyramid type system has symmetry, which

enables easier analysis. Numerical data and

analytical expression of some geometric

characteristics can be reduced by using this

symmetry. This fact is useful for actualimplementation.

As geometric study is more qualitative rather than

quantitative, ground experiments were performed to

demonstrate the performance of the steering laws. Also

for evaluation, various system types are compared in

terms of the size of the possible angular momentum

vector operational space and the systems weight.

As mentioned above, specific studies of an attitude

control are beyond the scope of this work. Such studies

involve optimal maneuvering and angular momentum

management, which are possible only after the

1. Introduction

7/24/2019 Cmg Paper 97

19/124

7/24/2019 Cmg Paper 97

20/124

5


A control moment gyro (CMG) system is a torquer

for three axis attitude control of an artificial satellite.

There are two types of CMG units and various

configurations of three axis torquer systems. Designing

a CMG system therefore includes a process of selecting

a unit type and a system type defined by configuration.

Among two unit types and various system types, a

single gimbal CMG system of pyramid configuration is

mainly described in this work. For the simple

comparison, this chapter gives an outline of CMG system

characteristics with consideration paid to its use in an

attitude control system. The angular momentum

workspace, torque output, steering law and singularity

problems are the important factors for evaluation of a

CMG system.

2.1 CMG Unit Type

A CMG consists of a flywheel rotating at a constant

speed, one or two supporting gimbals, and motors which

drive the gimbals. A rotating flywheel possesses angular

momentum with a constant vector length. Gimbal

motion changes the direction of this vector and thus

generates a gyroeffect torque.There are two types of CMG units, as shown in Fig.

21, single gimbal and double gimbal. A single gimbal

CMG generates a one axis torque and a double gimbal

CMG generates a two axis torque. In both cases, the

direction of the output torque changes in accordance with

gimbal motion. For this reason, a system composed of

several units is usually required to obtain the desired

torque.

2.2 System Configuration

Typical system configurations will now be discussed.

The configuration is defined by a set of principal axes

of all the component CMG units, which are the gimbal

axes in the case of single gimbal CMGs and the outer

gimbal axes in the case of double gimbal CMGs. In the

following figures, these principal axes are indicated by

arrows denoted by gi .

The system of each configuration is named as a system

type such as twin type system or the pyramid type system.

Chapter 2

Characteristics of Control Moment Gyro

Systems

Fig. 21 Two types of CMG unit s

Flywheel

G yro Motor

(a) Single gimbal C MG

Gimbal Motor

GimbalMechanism

G yro E ffect Torque

Angular Moment um Vector

T

GyroMotor

InnerGimbalMotor

OuterGimbal

Inner Gimbal

Outer GimbalMotor

(b) Double gimbal CMG

Flywheel

7/24/2019 Cmg Paper 97

21/124

6


2.2.1 Single Gimbal CMGs

Typical single gimbal CMG systems have certain kinds

of symmetries, which can be classified into two types,

independent and multiple. They are somewhat

different in their mathematical description.

(1) Independent Type

Independent type CMGs have no parallel axis pairs.

Two categories of independent type CMGs, symmetric

types and skew types, have been mainly studied.

Symmetric Type Gim bal axe s are arr ang ed

symmetrically according to a regular polyhedron. There

are five regular polyhedrons with 4, 6, 8, 12 and 20

surfaces. Possible configurations of this type are three,

four, six and ten unit systems, because only surfaces not

parallel to each other are considered and because a

tetrahedron and hexahedron are complementary or

dual to each other. The three, four, six, and ten unit

systems are denoted as S(3), S(4), S(6) and S(10). The

four unit or S(4) system, shown in Fig. 22(a), is called

the symmetric pyramid type. Most of this work dealswith this type of system. An example of the six unit or

S(6) system, shown in Fig. 22(b), is now in use on the

Russian space station MIR.

Skew Type All individual units are arranged

in axial symmetry about a certain axis as depicted in

Fig. 22(c). Skew three and four unit systems of certain

skew angles are the same as the S(3) and the S(4).

(2) Multiple Type

In this type some number of individual units possess

X Y

Z

g1

g2

g3g4

h4

1

4

h1

h2

h3

2

3

(a) Pyramid type S(4)

Fig. 22 Configura tions of single gimba l CMG s

g2

g1

g3

g4

g5

g6

h1

h6

h5

h4

h3

h2

(b) Symmetric type S(6)

g1

g1

g1

g2

g2

g2

(d) Mult iple ty pe M(3, 3)

11

12

13

21

22

23

g1

h1

g2

g3

g4

g5

g6 h2

h3h4

h5

h6

2n

(c) Skew ty pe

7/24/2019 Cmg Paper 97

22/124

7/24/2019 Cmg Paper 97

23/124

8


attitude control.

2.3.1 Block Diagram

A functional block diagram of a three axis attitude

control is shown in Fig. 24. Most of the blocks are thesame when either reaction wheels or CMGs are used.

The attitude and rotational velocity commands are

generated by a maneuver command generator denoted

by A in Fig. 24. The command and sensor information

are the inputs to the vehicle control law block, B. This

block calculates the torque necessary for control. The

next block, C, shows the CMG steering law which

calculates the CMG motion for the torque calculated by

block B. In this manner the actual CMG system is driven

and an output torque to the satellite is generated. The

blocks relating to CMG control are the CMG steering

law, C, and the momentum management block, D. Those

two blocks are described first in the following sections.

Then, relating subjects, i.e., maneuver commands,

disturbances and the motion of angular momentum

vector will be explained.

2.3.2 CMG Steering Law

The steering law block computes a set of gimbal angle

rates which produce the required torque. The steeringlaw is usually realized in two parts, one being simply a

solution to a linear equation and the other for singularity

avoidance by using system redundancy.

This block is usually designed independent of the

particulars of the total attitude control system. This

implies that the vehicle control law (B in Fig. 24) is

designed under the assumption that the output of the

CMG system corresponds exactly to the command. The

CMG steering law must satisfy this requirement. The

meaning of this exactness is described in a later chapter.

2.3.3 Momentum Management

A CMG and a reaction wheel are called momentum

exchange devices because they dont actually produce

angular momentum but rather exchange it with the

satellite. Such torquers have limits to their accumulation

of angular momentum, because the rotational speed of a

flywheel is limited. Therefore, another type torquer is

needed when it becomes necessary to offload excess

accumulated momentum. This unloading is usually done

by gas jets or magnetic torquers. The unloading process

must be carefully managed by the momentum

management control block, D, because such torquers

have their own limitations, i.e., a gas jet does not have

enough resolution and it have a limit of storage, and a

magnetic torquers output depends on orbit position.

For effective management of angular momentum, the

space of allowed angular momentum of a CMG systemmust be defined beforehand. This space is termed

workspace in this paper. The workspace must be

included by the possible angular momentum space of

the CMG itself. Moreover, a simple shaped space such

as a sphere tends to result in more simplified

management.

2.3.4 Maneuver Command

The command issued by a maneuver command

generator depends on the mode of operation. Typical

operational modes are pointing, maneuvering, scanning

and tracking. In the pointing mode, precision is of

primary importance and is affected by disturbances,

torque response and resolution. The speed of

maneuvering as well as momentum accumulation while

pointing is a matter of workspace size of the torquer.

2.3.5 Disturbance

The time dependence of disturbances vary accordingto orbit parameters and a mission type, such as earth

pointing or inertial pointing. In any case, a disturbance

may have cyclic terms and offset terms. The following

function is an example of disturbance used for the

simulation of HEAO with a pyramid type CMG

system14);

Tg= (Txsin t, Ty(cos t1), Tzsin t)t,

where denotes orbital angular rate. Because there isan offset in the y direction, angular momentum will be

accumulated in this direction while pointing.

2.3.6 Angular Momentum Trajectory

The size and shape of the workspace determines the

maximum accumulation of disturbances or the maximum

speed of maneuvering. A disturbance or a maneuvering

command can be expressed as a function of time by a

trajectory of the angular momentum vector of the

satellite. Since the total angular momentum of the system

is equal to the time integral of the disturbance, the angular

momentum trajectory of a CMG system can be expressed

using the spacecrafts momentum and disturbance. The

7/24/2019 Cmg Paper 97

24/124

9


Table 22 System Level Comparison

Torque Weight Steering Law Singularity

Reaction Wheel 1 1 simple none

Double Gimbal CMG 100 2 not simple slightSingle Gimbal CMG 1000 2 most complex serious

Table 21 Component Level Comparison

Angular Momentum Torque

Reaction Wheel 1 to 1000 1

Double Gimbal CMG 1000 to 3000 100

Single Gimbal CMG 10 to 2000 1000

workspace of a CMG system must include any possible

angular momentum trajectory when the unloading

torquers are not operating.

2.4 Comparison and Selection

CMG systems and a reaction wheel system are all

examples of the same type of torquers. In order to design

an attitude control system, some sort of selection criteria

is needed. By using the following performance indices,

a brief comparison will be made, first at the component

level then at the system level.

2.4.1 Performance Index

The performance of a CMG systems depends not onlyon elements of hardware design, such as the CMG unit

type and the system configuration, but also on the design

of the steering law. These factors all affect the maximum

workspace and the magnitude of the output torque, two

nonscalar performance indices. Another performance

index is the steering law complexity, which affects the

attitude control cycle time and the capacity of an onboard

computer.

2.4.2 Component Level Comparison

Table 21clarifies the main differences among thesethree torquers64). A reaction wheel has only one motor

which is used not only for accumulation of angular

momentum but also for generation of torque. On the

other hand, the CMGs use either two or three motors,

one for accumulation of angular momentum and the

others for torque generation. Since the torque of a motor

depends on its speed and the same maximum torque

cannot be generated over the motors working speed

range, both angular momentum and output torque of a

reaction wheel are much smaller than for CMGs.

Size and weight of a CMG depends on the size of the

flywheel and complexity of the mechanism. A double

gimbal CMG is the most complicated at the unit level,

but less so at the system level because this unit generates

a two axis torque.

Maximum output torque is much different. A single

gimbal CMG can produce more output torque than a

double gimbal CMG. The reason is as follows. The

output torque of a single gimbal CMG appears on the

flywheel and is then transferred directly to the satellite

across the gimbal bearings. The output torque can be

much larger than the gimbal motor torque required to

drive the gimbal. This is called torque amplification.By contrast, some part of the output torque of a double

gimbal CMG must be balanced by the gimbal motors.

Thus, in this case, the output toque is limited by the motor

torque limit.

2.4.3 System Level Comparison

Table 22shows a system level comparison for thethree types of torquers being compared. Difference in

the first two indices, torque and weight, are derived from

component level differences. The other two indices

relate to each other. The steering law of any reaction

wheel system is linear and no singularity problems arise.

Steering law complexity and singularity problems of

CMG systems, especially single gimbal CMGs, can be

serious and thus form the main subject of the present

work.

2.4.4 Work Space Size and Weight

The size and shape of the maximum workspace are

not compared in the above table because they depend

on the number of units and system configuration.

Workspace size as a scalar value, and the weigh of the

CMG system can be roughly evaluated in terms of the

number of units. Lets consider similarly shaped

7/24/2019 Cmg Paper 97

25/124

10


flywheels of diameter dand thickness t. Similarity

implies td. Then, the weight and the size of maximumworkspace of an nunit system, denoted as WandH,

follow the following relation if the rotational rate of the

gyro is the same:

W n d2t n d3, (21)

H n (t d d2) dr n d5 . (22)

IfHis set constant, Wis given by;

W n d3 n 2/5 . (23)

This implies that the system with fewer units is lighter

but can still realize the same workspace size. Despite

the fact that other factors are ignored in estimating the

weight, it can generally be concluded that the systems

of less units have advantages in weight.

In this evaluation, it is assumed that the size of the

work space is proportional to the number of units by thesame multiplier for any system. From the comparison

in Chapter 9, this is almost true for systems of no less

than 6 units in the case of single gimbal CMGs. This,

however, is not true in the case of less that 6 units.

Therefore it is better to evaluate some configuration

composed of 4 to 6 units.

7/24/2019 Cmg Paper 97

26/124

11

3. General Formulation

Chapter 3

General Formulation

This chapter first defines vectors, variables and

parameters of a single gimbal CMG system in an

arbitrary configuration, after which a basic

mathematical description of several system

characteristics are made. These characteristics are the

kinematic equation, the steering law, the torque output

performance index, and singularity avoidance. The

shape of the maximum workspace and singularity

problem are described in the next chapter. Similar

descriptions for double gimbal systems are given in

Appendix A.

3.1 Angular Momentum and Torque

A generalized system is considered consisting of n

identically sized single gimbal CMG units. The number

nis not less than 3 to enable three axis control. The

system configuration is defined by the relative

arrangement of the gimbal directions. The system state

is defined by the set of all gimbal angles, each of which

are denoted by i. Three mutually orthogonal unitvectors are shown in Fig. 31and defined as follows:

gi: gimbal vector,

hi: normalized angular momentum vector,

ci: torque vector,

where

ci= hi/ i= gi hi . (31)

The gimbal vectors are constant while the others are

dependent upon the gimbal angle i. Once the initialvectors are defined as in Fig. 32, the other vectors are

obtained as follows;

hi = hi0cosi+ ci0sini ,

ci = hi0sini+ ci0cosi . (32)

The total angular momentum is the sum of all hi

multiplied by the units angular momentum value whichis denoted by h. In this work, Hdenotes the total angular

momentum without the multiplier h:

H= hi . (33)

This relation is simply written as a nonlinear mapping

from the set of ito H;

H= f() . (34)

The variable, =(1, 2, ..., n), is a point on an ndimensional torus denoted by T(n) which is the domain

of this mapping. The mapping range is a subspace of

the physical Euclidean space and is denoted byH. This

space is the maximum workspace.

By the analogy of this relation with a spatial link

mechanism, this relation will be called kinematics or

kinematic equation in this work (see Appendix F).

The output torque without the multiplier his obtained

by taking the time derivative as follows.

T= dH/ dt = hi/idi/dt . (35)

Any additional gyro effect torques generated by thesatellite motion are omitted because they are usually

h

c

g

Fig. 31 Orth onorma l vectors of a CMG unit

ci

ci0

hi

hi0

gi

i

Fig. 32 G imbal an gle an d vectors

7/24/2019 Cmg Paper 97

27/124

12


treated in the overall satellite system dynamics, which

includes the CMG system (see Chapter 8).

Because the total output torque is a sum of output of

each unit, it is also given as,

T = cii

= C , (36)

where

i= di/dt, and = (1, 2, ..... , n)t .

(37)

The variable iis the rotational rate of each gimbal.The vector is a component vector of a tangent spaceof T(n). The matrix Cis a Jacobian of Eq. 34 and is

given by,

C= (c1c2.... cn) . (38)

As the units angular momentum value is omitted in

Eqs. 35 and 36, the real output is obtained by

multiplying h.

3.2 Steering Law

The steering law functions to compute the gimbal

rates, , necessary to produce the desired torque, Tcom,

and is generally given as a solution of the linear equation

given in Eq. 36:

= Ct(CCt)1Tcom + (ICt(CCt)1C) k .

(39)

whereIis the n nidentity matrix and kis an arbitraryvector of nelements.

The first term has the minimum norm among all

solutions to the equation. The matrix Ct(CCt)1is called

apseudo-inversematrix. The second term, denoted by

N, is a solution of the homogeneous equation;

CN= 0 . (310)

This implies that the motion by this Ndoes not generatea torque (T) and keeps the angular momentum (H)

constant. In this sense, this term is called a null motion.

The null motion has n3 degrees of freedom because itis an element of the kernel of the linear transformation

represented by C.

An effective method of calculating a null motion is

given in Ref. 22. For example, a null motion of a four

unit system is generally given as,

N= ([c2c3c4], [c3c4c1],

[c3c1c2], [c1c2c3]) , (311)

where [a b c] denotes the vector triple product, a(bc).

3.3 Singular Value Decomposition and

I/O Ratio

The magnitude of the total output torque is not a

simple sum of the output of each unit. An elements of

each output, ici, normal to T cancels each other. Theratio of input and output norms, ||/|T|, can be evaluatedby a singular value of the matrix C.

The matrix Ccan be decomposed into a diagonal

matrix by two orthonormal matrices, Q(33) andR(nn) as follows;

QCR=

1

2

3

0 0 0 0

0 0 0 0

0 0 0 0

. .

. .

. . , (312)

where iis called a singular value of C. As shown inFig. 33, the maximum ratio of the input and output

norms is given by the radius of the ellipsoid whose

principal diameters are the singular values. Thus, the

Fig. 33 Input Output rat io

(b) Angular momentum ellipsoid

(a) Gimbal rat e

1

2

3 . .. n

n - sphere

||= 1

12

3

H

7/24/2019 Cmg Paper 97

28/124

7/24/2019 Cmg Paper 97

29/124

14


3.5.1 Gradient Method

A system containing more than three units possesses

null motion redundancy. Freedom in determining null

motion can realize singularity avoidance while keeping

the output torque exactly equal to the command. Thegradient method is a general method in which some

objective function is maximized. The following

formulation of a gradient method is taken from Ref. 21.

The objective function, W(), is chosen as acontinuous function of . It is zero in the singular stateand otherwise positive. The dependence of Won CMG

motion is:

W= ii , (318)

where

i= W/i . (319)

In order to obtain the objective function extremum,

the motion should be determined so that W ispositive. This Whas two parts, one given by thepseudo-inverse solution and the other by a null motion.

The first depends on the command torque Tcom, while

the latter depends on the selection of a null motion.

Though the first part cannot be changed, the latter can

be freely determined. The latter part is evaluated as

follows;

WN= t (ICt(CCt)1C) k . (320)

It is easily observed that the matrix (ICt(CCt)1C) is semi-positive symmetric. If the vector kis

selected as:

k= k, where k>0 , (321)

then WNbecomes a semi-positive quadratic form.Thus, the null motion by this kresults in non-negative

WN, so it is expected that singularity is avoided.Various objective functions have been proposed, suchas:

(1) (det(CCt))1/2, 21)

(2) min(i),36)

(3) min(1/|di|),

where diis a row vector of the matrix

Ct(CCt)1, 35)

(4) i,j|ci cj|2, 27).

This gradient method has been successful for double

gimbal CMG systems21). However, in the case of

c1

u

c2cn

TcomP ossible Output

Fig. 36 St eering at a singular condition

pyramid type single gimbal CMG systems, various

simulations showed that a gradient method is not

effective. Details of this problem is described in

Chapters 4, 5 and 7.

3.5.2 Steering in Proximity to a Singular State

There is no solution to Eq. 36 in a singular state

except when Tcomis orthogonal to the singular vector

u. Even when Tcomis normal to u, the solution is not

given by Eq. 39 because the linear equation is

mathematically singular. A generalized solution can be

obtained which is the exact solution when Tcomi s

normal to uotherwise minimizes the output error. The

minimum error is realized when the output is equal to

the projection of the torque command onto the plane

normal to the singular direction (Fig. 36). Such motion

is given as22):

= Ct(CCt+ k uut)1Tcom . (322)

Derivation of this is explained by supposing that there

is a virtual CMG unit whose torque vector cequals u.

Another method called the SR (Singularity Robust)

inverse steering law is proposed as a smooth extension

of this41). This method minimizes the weighted sum of

the input norm, ||, and the norm of the error. The SR

solution is given as:

= Ct(CCt+ W)1Tcom,

where Wis a nnmatrix . (323)

In both methods, the solution is zero if the command,

Tcom, is either zero or parallel to the udirection. This

method, therefore, cannot always guarantee avoidance

of a singular state nor can it escape from one. Moreover,

this kind of control is effective only if the attitude control

is not totally degraded by the error in torque. Details

are described in Section 7.2.

7/24/2019 Cmg Paper 97

30/124

15

4. Singular Surface and Passability

Angular momentum vectors in a singular condition

form a smooth surface which includes the angular

momentum envelope. This chapter first summarizes the

geometric theory of the singular surface of a general

single gimbal CMG system by following the research

work in Ref. 22. It includes a definition of a singular

surface, a mapping from a sphere to the surface, and

techniques for drawing the surface by computer

calculation. By using these techniques, the workspace

is visualized for various system configurations. Also,

geometric characteristics such as Gaussian curvature of

a singular surface is defined.

The passability of a singular surface is then defined.

The existence of an impassable surface explains why

most steering laws fail to generate output starting from

certain initial states. A gradient method works well for

avoiding passable singular points but not for avoiding

impassable ones.The passability can be determined by the curvature

of the singular surface. It is demonstrated that any

independent type system has an internal impassable

surface while multiple type systems of no less than six

units have no internal impassable surfaces.

4.1 Singular Surface

4.1.1 Continuous Mapping

Lets examine all the singular points and their H

vectors. First, an independent type system is assumed

in the following discussion.

The torque vectors, ci, satisfy the condition given by

Eq.(316) when the system is singular. On each singular

point, a singular vector uis defined. As a reverse relation

of this, singular points are obtained from a given uvector.

Given any singular vector u, there are two

possibilities of singularity condition for each unit as hSand hSin Fig. 41. The two cases are distinguished by

the following sign variable;

i= sign( uhi) . (41)

Thus there are 2ncombinations of singular points

for the given direction u. This combination is denoted

by or by a set of signs, such as {+ + + ... +}.For the given singular direction uand the given set

of signs, each torque vector in the singular condition is

determined by:

cSi = igi u/ |gi u | . (42)

From this point, variables subscripted by S denote

singular point values. The total angular momentum HSis obtained as follows:

HS= i(gi u) gi/ |gi u | . (43)

This defines a continuous mapping from uto HSwhile the iare fixed as parameters. The domain of uis

a unit sphere except gi direction, because thedenominator of Eq. 43 is zero when u= gi . Thus HSwith fixed iform a two dimensional surface with ucovering this sphere. This surface is denoted as S. If

all the iare reversed and the vector uis changed to u,HSremains the same. This implies that the surface of

{i} and the surface of all the ireversed are identical.For example S{ + +}is the same as S{+ }. One may

thus suppose that no less than half of the iare positive.Thus, the number of different surfaces is 2n1.

In case that u= gi, any state of this ith unit satisfies

Chapter 4

Singular Surface and Passability

Fig. 41 Vectors at a singularit y condition

g

u hS

hS

cS

= 1

= 1

cS

7/24/2019 Cmg Paper 97

31/124


16

the singular condition. As the vector hirotates about gi,

these singular Hform a unit circle which appears as a

hole or a window of the surface Sas shown in Fig. 42

(b). As there is a hole for each gi, or gi, direction, thesurface has 2nholes in total. Surfaces of different {i}

are connected by these unit circles (for example, C1inFig. 42(b) and (c)). Thus all the surfaces form a closed

surface. This closed surface is called a singular surface.

It may be noted that the same kind of continuous mapping

is defined from uto Swith all the Sforming a twodimensional surface in the ndimensional torus of . Sucha surface, however, is not termed a singular surface in

this paper.

An independent type system is assumed in the above

discussion. In the case of a multiple type system, the

number of different singular surface is 2m1where mis

the number of groups. Each surface has 2mholes of

diameter of several values which is determined by the

number of units in a group and sign . In case that u=gi, any state of units of this group satisfies the singularcondition. Thus, all singular Hof this uform a circular

plate which fills the hole. Another singular surface of

different sign connects to this plate by a circle of different

diameter.

4.1.2 Envelope

The angular momentum envelope, which is the

border of the maximum workspace, is most definitely

singular. The surface corresponding to all ipositive isclearly a part of the envelope. Surfaces with one negative

sign which is connected to this surface by the holes share

the envelope surface in the case of an independent type

system.

The envelope of a multiple type system consists of a

singular surface of all positive signs and circular plate

which fills 2mholes22). The one negative sign surfaces

do not share the envelope surface and is fully internal.The singular surface of a M(2, 2) roof type system shown

in Fig. 43is part of an envelope of all positive signs.

There are four circular holes of diameter 2. The circular

plates filling these four circles share the envelope. The

singular surface of one negative sign is connected at the

center of these plates.

4.1.3 Visualization Method of the Surface

The singular surface and envelope are visualized by

taking at each lattice points of the unit sphere andcalculating the angular momentum using Eq. 43.

g1

Unit Circle C1

Envelope

x y

z

(b) Singular surface of all sign positive denoted by S{++++}

Un it Circle C 1 En velope P ortion

Interna l Portion

Unit Circle C 2

(c) Singula r sur face of one minus sign denoted byS{+++}

(a) La tt ice points of the un it sphere of vector u

g1 g2

g3 g4

x y

z

Fig. 42 Exa mples of th e singula r surfa ces for th e


Ea ch dot of Figs. (b) & (c) corresponds to t he la tt ice

point of Fig. (a). The un it circle indicat ed by C 1connect s

two s ingular sur faces S{+ + + + }& S{ + + + }. Other

circles of the sur face S{ + + + }, C 2for exam ple, a re con-

nections to other singula r surfa ces such as S{ + + }

7/24/2019 Cmg Paper 97

32/124

17

4. Singular Surface and Passability

Figures 42and 43 are such examples. A singular

surface and an envelope may also be visualized using

various cross sections. The following inverse mapping

theory22)is available to obtain a cross section of the

singular surface.

Inverse Mapping Theory

Suppose that is constrained singular and Vis anarbitrary vector normal to u. If the differential dH

along the singular surface satisfies,

dH= Vu , (44)

then the differential of uis given by

du= ( CPCtV) u , (45)

where is the Gaussian curvature of the singularsurface, which is described in Section 4.2.2. The

matrix Pis a diagonal matrix whose nonzero element

Piiis given by:

Pii=pi= 1 / (uhi) . (46)

Using this theory, a cross section of the singular

surface is calculated by the following procedure. First,

obtain a singular point on the cross sectional plane and

its u vector by some means. Second, obtain dH on the

intersection of the surface tangential plane and the cross

sectional plane. Third, obtain Vby Eq. 44 and duby

Eq. 45 after which dis obtained by the relation d=

picidu(Appendix B). Finally, Hon the cross sectionalplane is obtai

cmg paper 97

Documents