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GROUND TESTING AND MODEL UPDATING FOR FLEXIBLE SPACE STRUCTURES

by

James MacKenzie Crawford

A thesis submitted in confonnity with the requirements for the degree of Doctor of PhiIosophy

Graduate Department of Aerospace Science and Engineering University of Toronto

Copyright O 200 L James MaciCemie Crawford

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Abstract

Ground Testing And Model Updating For Flexible Space Structures

'JY

James MacKenzie Crawford

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy

Graduate Department of Aerospace Science and Engineering University of Toronto

Aspects of the ground testing portion of the Dynamics [dentifkation and Control Ex-

periment (DICE) are explored. DICE is a pcoposed international Space Station (ES) exper-

iment whose goaI is to investigate System Identification (SI) and amtude and shape control

using a srnall free flying spacectdft, inside an ISS experimental module, that f e a ~ e s a rigid

central bus controlled using reaction wheels and an array of flexible appendages or 'ribs'

fitted with conuol moment gyroscope end-effectors. Extensive testing prior to launch is

required before a high quality O-g math model can be predicted and the benefits of on-orbit

system identification evaluated To this end, rwo technologies are developed in this thesis.

The 6rst is a passive mechanical suspension system that allows dynamitai ground testing in

the fùiiy deployed configuration, ptoviding support for the DICE bodies, a zero deflection

at-rest state, and high cornpliance for al1 degrees of freedom. Based on concepts h m the

literature combined with innovative design hprovements, a prototype is built and evalu-

ated, showing excellent performance in Iocal mode suppression and damping. It is then

successtùiiy used in tests with one of the prototype DICE ribs. The second technology is

a Model Updating technique that uses experimental data to fine tune the I-g math

model while retaining the ability to generalize the model to the O-g case. The technique

optimizes the parameters in the nonlinear mode1 by maicing direct use of t h e history data

h m muitiple ground test coniigurations and the weighted uncertainties of the parameters,

thereby maximizing the validity of the extrapolated on-orbit model. Aithough more com-

putationaiiy intensive than other SI methods, the MU technique aiiows noniinear models to

be updated, retains physicaiiy meanin@ mode1 parameters and state-space (a requirement

for the O-g model prediction), and d e n demonstrated with reaI expetkenta[ data performs

just as weii in reducing model m r as direct SI methods,

Acknowledgments

The author would like to acknowledge with gratitude the support of his doctoral cornmittee. Professor Peter C. Hughes for his experience. expertise, high standards and helpful critîdsm. Dr. Kieran A. Carroll for his unwavering vision and intuition, and Professor Peter C. Stangeby for his fiesh and unbiased views. This work was supported by NSERC. the University of Toronto, and CRESTech, and was sponsored by an industriai partner, Dynacon Enterprises Ltd. The DICE project was also sponsored by the Canadian Space Agency. The author would iike to personaiiy thank severai people who lent technical assistance during their tenure in the Spacecraft Dynamics and Control group at UTIAS. namely Andrew Men, Rob Bauer, Thierry Cherpiilod. and Rob Zee.

Contents

List of Tables vi

List of Figures vüi

1 Introduction 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The DICE Project 2

1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

13 'Ifiesis Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 ModelUpdating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Suspension System 8

. . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Suspension System Design 9

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Requkments 9

. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.12 Concept Options 11

. . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Nullspring Design 13

. . . . . . . . . . . . . . . . . . . . . . . 2 2 Suspension System Performance 16

3 System Mode1 24

. . . . . . . . . . . . . . . . . . . . . . . . 3.1 Ri% System and CMG Mode1 24

. . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Suspension System Model 28

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 The Paramemc Mode1 32

4 Modei Updating: Theory 35

. . . . . . . . . . . . . . . . . . . . . . . 4.1 Identifying a General Approach 36

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 ProbIem Statement 38

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 ReguIarization 39

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Sequentid Estimation 41

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 5 SearchToIerance 43

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Solution Uniqueness 46

. . . . . . . . . . . . . . . . . . . . . 4.7 Candidate ûptixnization Algorithms 48

. . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.1 lterative Methods 48

. . . . . . . . . . . . . . . . . . . . . . . . 4.7.2 Enmerative Methods 54

5 Model Updathg: Practice 60

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Experimentai Setup 61

. . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Suspension Mode1 Updating 62

. . . . . . . . . . . . . . . . . . . . . . . . . 5.3 CMG Sewo Mode1 Updating 66

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 RibModelUpdating 73

5.4.1 Simulated Rib Mode1 Updating . . . . . . . . . . . . . . . . . . . 75

. . . . . . . . . . . . . . . . 5.4.2 Step Relaxation Rib Model Uphting 81

. . . . . . . . . . . . . . . 5.4.3 Forced Response Rib Mode1 Updating 86

. . . . . . . . . . . . . . . . . . . 5.4.4 Estimating Damping Parameters 95

6 Conclusioos 101

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Thesis ConmIbutions 104

. . . . . . . . . . . . . . . 6.2 Beyond as Work: Testing the DICE Freeflyer 106

A Analysis and Design of the Nullspring Ceometry 114

List of Tables

. . . . . . . . . . . . . . . . . . . . . . . . 2.1 Final N u l l s p ~ g Design Values 16

. . . . . . . . . . . . . . . . . . . . . . . . 2.2 Caiibrated Nullspring Pmperties 22

. . . . . . . . . . . . . . . . . 2 3 Suspension Syaem Performance Cornparison 22

. . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Model Parameter Debitions 34

. . . . . . . . . . . . . . . . . . NulIspring Mode1 Parameters Cornparison 64

. . . . . . . . . . . . . . . . . . . . . . . . CMG Servo Update Parameters 67

. . . . . . . . . . . . . . . . . . . CMG Servo Mode1 UpdaMg Cornparison 70

. . . . . . . . . . . . . . . . . . . . . . . . Rib Systern Update Parameters 74

. . . . . . . . . . . . . . . . . . . . . . Simulated Updating P-Value Results 77

. . . . . . . . . . . . . . . . . . . . . . . Step Relaxation P-Value Resdk 84

. . . . . . . . . . . . . . . . . . . . forced Response Updating Cornparison 87

C P U T i e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

Modal Properties and Projections . . . . . . . . . . . . . . . . . . . . . . . 99

List of Figures

1 -1 DICE Freeflyer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 DemoDICE Prototype 4

. . . . . . . . . . . . . . . . . . . . . 2.1 Nullspring: Zeroing the Spring Rate 14

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Nuilspring Geometry 15

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Nullspring Design 17

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Main Spring Clamp 17

. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Side Spring and Load Cell 17

. . . . . . . . . . . . . . . . . . . . 2.6 Knife Edge Bearing and Tuning Head 17

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Nuilspring Lever 18

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Nullspring Tuning 18

. . . . . . . . . . . . . . . . . . . . . . . 2.9 Long Term Stifhess Stability Test 19

. . . . . . . . . . . . . . . . . . . . . 2 .IO Sampte Nullspring Evriluation Data. 20

. . . . . . . . . . . . . . . . . . . . . . . . 3.1 Layout of the DtCE Rib Mode1 25

. . . . . . . . . . . . . . . . . . . . . . . . . 3.2 DICE Rib Coordinate Systems 26

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 SuspendedCMG 27

. . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 CMG With Gimbal Tited 27

. . . . . . . . . . . . . . . 4.1 Picking Regularization Weight X Using LOrve 41

. . . . . . . . . . . . . . . . . . 4 2 Effect of Sequence Duration on Sensitivity 42

. . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 Picking the Search Tolerance 45

. . . . . . . . . 5.1 Example Cost Contours for the NulIspcïng Evaluation Tests 63

. . . . . . . . . . . . . . 5.2 Nullspring Mode1 Updating: P-Value Convergence 65

. . . . . . . . . . . . . . . . . 5.3 NuUspring Mode1 Updating: Data Correlation 65

. . . . . . . . . . . . . . . . . . . . . 5.4 Sensitivity of Current Limit Parameter 68

. . . . . . . . . . . . . 5.5 CMG Servo Mode1 Updating: P-Value and Cost, SA 69

. . . . . . . . . . . 5.6 CMG Servo Model Updaring: Response Correlation, SA 69

. . . . . . . . . . . . . . . . . 5.7 Simulated Updating: Siml P-Values and Cost 75

. . . . . . . . . . . . . . . 5.8 Simulated Updating: Siml Response Correlation 76

. . . . . . . . . . . . . . 5.9 Simuiated Linear Updating: Sim3 P-Value and Cost 78

. . . . . . . . . . . . . . . 5.10 Simulated Linear Updating: Simj Residd Emr 78

. . . . . . . . . . . . . . . . . . 5.1 1 Simulated Rib Updating: Noise Resiiience 79

. . . . . . . . . . 5.12 Step Relaxation Updating: VerticaI Test P-Vdue and Cost 83

. . . . . . . . 5.13 Step Relaxation UpdaMg: Vertical Test Response Correlation 83

. . . . . . . . . . . . . . 5.14 MU Using Various Data Sets: Comparing P.values 90

. . . . . . . . . 5.15 Multi-Configuration Updating: Run 9 RL Vertical Response 90

. . . . . . . 5.16 Multi-Configination Updating: Parameter and Cost Convergence 9L

. . . . . . . . . 5.17 MuIti-Configuration Updating: Run 9 RI Vertical Response 91

. . . . . . . . . . . . . . . . . . . . . . . . 5 . 18 Frequency Response Correlation 93

. . . . . . . . . . . . . . . . . . . . . . . . 5.19 Damping Parameter Oscillation 96

. . . . . . . . . . . . . . . . . . . . . 6.1 Flow of the Method Selection Process IO2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Nullspring Geome try. 115

for Kasey

Chapter 1

Introduction

In 1990 the Hubble Space Telescope (HST) began its mission in a cloud of contro-

versy and doubt. Even though this instrument continues to generate crisp images

of unimaginable places at unprecedented optical cesolution, its beginnings were

racked with technicd glitches. One problem, eclipsed in the media coverage by

the infamous optical tlaw, was an overlooked detail caiied "control-structure in-

teraction". Because the telescope requires very precise attitude control (better than

0.1 arcsecond accuracy) the fine lock attitude control system (ACS) must be of cor-

respondingly high bandwidth. The spaceaaft ais0 requires considerable power

and thedore was configureci with large deployable solar panels. These solar pan-

els are exated both internaiiy (by any slewing of the bus) and externally (by rapid

thermal gradients). Due to the overlap of the control and stmctural bandwidths,

the ACS further excited the solar panels dynamically which undermined the tele-

scope's pointhg performance.

This situation canies over into spacecraft large and srnaIl. The performance of

many satellites in operation today, and those envisioned for Future missions, have

pomting accuracy as the aïticai performance limitation. The key factor attributed

to controlstmdure interaction is the ratio of attitude control bandwidth to the

flexibIe stmcturaI bandwidth. if this ratio ne=, and partidarly if it exceeds,

unity this interaction will occur to the detriment of the ACS. However, a high

bandwidth ACS and low bandwidth structural flexibility cm coexist in harmony

on a spacecraft. This may be done by accounting for the flexibility in the ACS

design, but inevitably requires an accurate on-orbit math modei of the system, one

that is based on experimental dynamical response data from the very spacecraft

being controlied. The Dynamics, Identification and Control Experiment @ICI!)

addresses this issue [II. Tt is an investigation of the virtues of using models derived

from experiments done on-orbit and System Identification (SI) to design attitude

controlles, as compared to using ground based tests to extrapolate the on-orbit

system model.

Among other modifications, the attitude control problems of the HST were

solved by tuning the fine lock ACS algorithm based on actual experimental data

coiieded on-orbit This is a case where on-orbit Si was put to good use in gen-

erating an accurate on-orbit dynamicd model of the systern, which in turn was

used to redesign the control algorithm. As the Hubble nears the end of its tifteen

year mission, new exating projects such as the Next Generation Space Telescope

(NGm are pushing the performance boundaries of pointing contd even further,

making the experimental goals of DICE ail the more relevant. In particular, where

the chosen orbit of the NGST wiii not easily accommodate s e ~ c i n g missions, this

technology should be part of the operational plan of the spacecraft.

1.1 The DlCE Project

The main experimentai objective of the DICE project is to compare the performance

of attitude controiiers based on pre-launch designs with those designed using on-

orbit system identification. To this end, the DICE Freeflyer will be modeied and

tested on the ground- This test-veriûed modei will be used to design the suite

of prelaunch controilers. Once the W y e r is in space, deployed by the crew

and free-floating inside the experimentai module of the International Space Station

(ISS) , the on-orbit experhnentai phases of DICE continue thus:

Figure 1.1 : DICE Freefiyer.

Phase 1: Pre-launch conttoiier performance experiments.

b Phase 2: On-orbit ÇT experïments.

b Redesign: Telernetry is downIoaded to a ground station where researchers

rapidly (over several days) identify on-orbit models and redesign the con-

trollers.

b Phase 3: Contr011er performance exp"ments based on onsrbit SI.

A simiiar experimental sdieme was used in the Middeck Active Control Experi-

ment (MACE) by researchers at the Massachusetts Institute of Technology (MIT),

where the focus of the experiments was in attenuating flexible response of a beam

in heefau [2].

The DICE M y e r is the main test artide of the project, the concept for which

is shown in Figure 1.1, It is rniddeck locker stowable and has five deployed flexi-

ble appendages or nbs each 45 cm m length, five singIe axis control moment gyro

(CMG) rib tip torque actuators, and a 2û kg bus with an on-board computer, wire-

Iess communication, battery power, sensor suite, a 3-axis reaction wheel attitude

control system, and a 3-axis air blower stationkeeping system. The system is fully

Figure 12: DemoDICE Prototype.

tetherless and with the very gentle air blowers and large washout space in the IÇS,

can provide hi& quality O-g to a very low bandwidth for unconstrained attitude

conlroi experiments. Because of the ni flexibiiity and CMG actuatos, shape con-

ûol in the form of vibration attenuation can be done in conjunction with attitude

contro1.

At th& point in the project there have been two revisions of a Iaboratory proto-

type, the most ment caüed DemoDICE, shown in Figure 1 2 The purpose of this

prototype is for technology dernonstration and riskmitigation for the final freefiyer

design, and for conducting early research on a similar structure. Of course if aU the

research goah could be met with a ground prototype, the flight wouId h d y be

justified; in hct, DemoDICE is lacking in 3-axis attitude aosscoupluig with the n i

dynamics. However, it does feahue aii the main components of the DICE keefiyer,

except with only one axis of rotation and with only two ribs, both equipped with

CMGs. The bus rotates M y on a vertical ax& rotary air bearing and is M y teth-

erless The rib tips are constrained verticaüy by guy wires £rom the upper levei

of the bus so that they may oniy move in the horizonta1 plane. n i e rib assem-

bIies may a h be tested separateiy wherein 2-axis bending motion is ailowed, by

hanging them vertically from a fixed point or by mspending them horizontally.

f .2 Motivation

The research work associated with the DICE pmject will evenhdiy play a part in

improving the accuracy to which a spacecraft can point at a given target. bfuch

of the ground testing currently performed on spaceçraft is oriented toward system

functionality check-out and safety testing, and is not up to the task of providing a

better modei of the system from which better control performance can be derived.

The work herein is motivated by this deiiaency and seeks to pave the way to a

ground tesîkg campaign for the DICE freefiyer that is supportive of its other ex-

perimental goals. Because spacecraft large (expensive) and smaii (lower cost) are

affected by control-structure interaction, technologies develaped to mitigate this

problem are applicable aaoss the entire spacecraft development market. Within

the framework of the DICE pmject, the speciüc motivation of this work is to give

credibility: how can we measure the benefits of on-orbit systern identification un-

l e s we make the most of testing opportunities we rnight have pnor to launch?

1.3 Thesis Goals

The goals of th& work are to pave the way for testing the fully deployed DICE

freeflyer prior to launch, to aid in the design of high performance controilers to

be used in O-g. Specifically, this work resolves the two main areas of risk for the

ground testing that existed at the outset: the suspension system to be used in the

testing, and the method by which the system mode1 can be reconciled such that it

reproduces the 1-g experimental data accurately whiie permitting extrapolation to

0-5

1.4 Model Updating

Mode1 Updating (MU) is a speciai type of a broader class of problerns d e d Sys-

tem Identification (SI). It is a form of indirect SI in that it seeks to optimize parame-

ters which in turn are used to define a system modei whose stnidme is predefined,

such as a Finite Element (FE) model for example. The optimization is based on ac-

tuai experirnental data so the updated model wiii be better able to predict the real

system behavior. MU is of interest in many types of engineering because it can

be used to make improvements to any type of pararnetnc model of a real testabie

system. One is then free to apply the updated model in new configurations, with

different loading, etc., because of the tie to the physical rnodel space.

The other form of SI is, of course, direct SI. In the direct approach the experimen-

ta1 data are used to derive an input/output relationship with no physical model

basis, iimited only to the class of general linear statespace models l e s than some

prescriied order. One example is the Observer Kalman Fiter Identification/Eigen

Realization Algorithm (OKID/ERA) [3] which was found to be a suitable method

for generating the O-g DlCE mode& based on the on-orbit experirnental data af-

ter Phase 2 of the O-g DICE experiments. However, because there are no practi-

cal methods for recovering a physicai model basis h m an arbiûary one [29], the

lack of a physicai state space prevents one h m gen-g direct SI models to

new con6gurations. Direct SI cannot be used for predicting on-orbit models using

ground test data.

Related to SI are several other analysis types. in Design ûptimization a design

task is cast so as to d o w adjutment of the parameters until the best possible de-

sign is found based on some m e a m of "goodness-" This task can span mdtiple

disciplines within the same design problem. Damage Detection or Localization is

where measured data h m a p h t is compared to the plant modei output to pin-

point where the system is not performing reguiarly, indicatirtg either damage or a

need for maintenance. This is o h done on-line in reai time for aiticai failure re-

sponse. Aiso related are Inverse ProbIems where a system modd and experimental

data from sensors, which provide approximations for a limiteci &set of the model

states are used to estimate the inputs, initiai conditions and boundary conditions

of the system (41. These are ail related to ÇI because in some way the sensitivity

of a systern model, which is often rank defiaent, influences the determination of

quantities other than model output.

In the mode1 updating Iiterature [5] it appears as though most of the well-

estabiished technipes cannot be applied with fuIi satisfaction to the DICE system

where nonlinearities and time varying effects may w d be signiscant. Many spe-

ciaI tools and tri& are applicable though, so a new technique built from a blend

of these is developed in this thesis. The focus throughout this work, however, is on

acquiring working updated models from actual test data rather than d w e h g too

heaviiy on theoretical deveiopment, although theory is relieci upon to the extent

necessary to ensure updated models both reproduce experimental data and are

physically meaningful. One must take a practical approach to MU if the methods

are to be used in a pracîical situation.

Chapter 2

Suspension System

The DICE aght hardware wiii be depIoyed in its fight configuration during p u n d

testing. From an operationai point of view, this testing will pmvide the fight qual-

Scation and the gross modei verification necessary for the h a 1 safety review, as

weil as provide a valuable opporhrnity for trouble-shooting and training For re-

searctiers and aew. Additionally, £rom a research standpoint, the ground tests will

ailow mode1 updatuig on the DICE test artide as a whole.

Before DKE can decisivdy quantify the benefits of on-orbit SI, ground testing

opporhmities must be M y explored. On-orbit models predicted using advanced

p u n d testing techniques rnight provide as good a basis for controiler design, or

even better, as models reaiized from actuai on-orbit data, but the degree to which

this is true can not be determined without performing these advanceci ground

tests.

Researchers have devised methods of simulatirtg O-g for different types of test-

h g which indude the use of drop towers, parabalic Eght maneuvers m aircraft,

sub-orbital rocket ûights, neutral buoyancy tanks, flat fioors and air p u b , long

suspension cables, bungee corcis, electro-pneumatic devices, and spring media-

nisrns. It is desirabIe that the DICE ground tesis remain in the iaboratory where a

variety of long duration tests can be conducted cheaply. The baseIine test pian is

to employ a suspension systern to counteract the effects of gravity sufficiently to

CHAPTER 2. SUsPENSlON SYSTEM 9

d o w a nondeBected shape at rest (as DICE wili have in freefal) by pmviding in-

dependent compIiant support for al the major DICE bodies while aiiowing general

motion of all the degrees of freedom. The rnost convenient orientation h m spatial

and modeiing perspectives is with the plane of the nonddected n is perpdic-

ular to the gravity vector. Each point of suspension, one per rib tip and several

for the hub moduIe, will require an independent mechanism sized and tuned for

optimal performance of the s p d c body being supporteci.

Zn preparing for future, more compiete, DICE ground testing, the goals of this

thesis indude testing a single ni with CMG end-effector in the horizontal sus-

pended cod3guration. This requires a prototype of a single eIement of the pro-

posed overaii suspension system. A concept for the suspension element was che

sen such that it would be straightforwant to =ale up to meet the payload capacity

of a hub supporthg eiement, and a detaded design was prepared and prototypeci.

Testing and analysis were carried out with the aid of this prototype. The remainder

of this chapter brieffy reviews the design and evduation of the suspension system

prototype.

2.1 Suspension System Design

The foczus now tunis to the design of a single eIement of the DICE suspension

system. This design is optimized for use with a DICE rib nispended hurizontaüy.

EssentiaiIy the medianism will provide a Iong wire h m which the payload wiii

hang, supporting ttie payload from above with Iow stiffness, additional mas, and

kinetic energy dissipation- More detaii on the design can be found in [6].

21.1 Requirements

The operational requirements for the suspension system are the foUowing:

b hyfoad Range: The mspension system eIements must be suitabie for pay-

Ioads in the 0.2 to 10 kg range.

CHAPTER 2. SUSPENSION SYSTEM

b Efective Smke: The maximum expected displacement of a rib tip will be gov-

emed by the excitation signals, typical control displacements, sensor ranges

and material limits. Given the c u m t estimates of aii these attributes, a use-

fui suspension excursion of f 10 cm was chosen.

b Safi Operation: Static or dynamic instability that could harm the hardware or

the operator must be prevented.

it Erne of Tuning: Where any set-up or tuning is required, the design m u t facil-

itate the task to aiiow qui& easy and accurate adjustment.

b Refiabiliy Low maintenance effort after instdation is required to reduce the

lime and cost of the test program.

b Low Cost: DICE is a low budget space project. Enormous jumps in cost codd

terminate tasks or jeopardize the entire project-

The research requirements are more performance related. Even though model

updating wiii take the suspension and gravitationai effects into account, the more

benign these effects are the greater chance of success in extrapolahg to the O-g

model. Therefore, the foiiowing additional requirements &O guided the design:

b Low Stifiess: This minimizes perturbation to the flexible modal frrquenaes

and maximizes separation between the quasi-rigid and the flexilde modes.

b Low Moving Mms: This minimizes perturbation to the payload mass ma&.

b Low Friction and Damping: This ensures a more fiight-like environment and

leads to a better modd of the system.

b Straighrfotward to Model: This dows better accuracy and simplifies extrapo-

lation to û-g behavior.

b Suppressed Local Modes.. Additional dynamic effects h m the suspension sys-

tem must be avoided to make the system benign. It is aIso desirable, although

not *ctly necessary, to be abIe to modei the suspension system without in-

troducing additional state variables to the overaii modei.

CHAPTER 2. SUSPENSION SYSTEM 11

b Puyload Wire Length: To reduce the penduiar stü£ness and quasi-rigid modal

freqyencies the wire from which the payload is suspended should be as Iong

as practical, given the constraints of the faciiity.

Additionally, there are suspension-system-related requirements on the DICE

freeflyer:

b Hard Point Locations: To reduce static defiection of the DiCE ribs, the payload

wire must attach near the body center of mass.

b Hard Point Srrength: There must be negligible hard point deflection with re-

spect to the payload chassis due to the load from the payload wire.

Hard Point Access: The payload wire must not interfere with any structures of

the test article throughout the range of motion.

2.1.2 Concept Options

Other researchers have developed suspension systems for a vanety of space struc-

tures speaalized for different types of tests [7]. The most comrnon and rudi-

mentary approach, used for dynamical and deployment testing, is to suspend the

spaceaaft and its appendages from compliant shock cords. The lengths of the

cords and the mass of the payIoad limit how low the plunging stiffness can be.

Also, to reduce the iii effects of lateral modes m these long cor& (sometimes caiIed

violin modes), fabric sleeves around the cords are used for passive damping. These

basic techniques are adequate only for gros validation of the s p a c d function-

ality and are not suitable for high fidelity system identification.

Suspension systems for more rigorous testing have also been developed. One

mechaniai approach [8] uses a form of a zero spring rate mechanh, or, as it will be

referred to herein, a nullspnng. These terms shouId be used with care because the

effective spring rate is not zero, but in fact nonlinear [9]. However, with the proper

design parameters it can have a very low and almost iinear effective spring rate

in the vicinity of the neutrai or zero deûection. The concept features positive and


negative springs acting in parael and tuned to ndify the effective stiffness. Large

off-the-sheIf uni& can be purchased h m a Company caiieâ AEC-Able Engineering

lnc. in Goleta, California, for a cost of approximately S30,ûOO (US) each. However

these are designed for payloads in the 50 to 250 kg range.

An alternative mechanical design features another type of zero-stiffness spring:

a nondrcular tuming disk on a torsion-spring-loaded shaft with the payIoad wire

wrapped on the disk periphery [IO]. The payIoad hangs from the wire and as the

disk turns, allowing the payload to move verticaiiy, the moment a m of the disk

changes. The spiral shape of the disk and the torsion spring d t s in a constant

suspension force (and zero stiffness) which exactly canceis out the weight of the

payload. When the payload is moved verticaily to a new position it stays there

as it wouid in freefaii. The moving components of this design require rnuch more

mass and the minimum number of bearings is large, so the concept does not sa le

down weii for smaii payIoads. As weiI, it is not convenient for hining the mecha-

nism for different payload masses. The inertial effects h m the dianging moment

a m of the non-circdar disk are inherentiy nonhear [Il] but attempts have been

made to account for this using a smaii DC motor on the disk shaft with adaptive

feedforward control[12]. However this concept reptesents mu& more effort than

the previous alternative for no advantage in performance.

An active eiedro-pneumatic design [13] uses a combination of an electro-linear

voice coii actuator and a pneumatic piston. The piston supports the weight of the

payIoad. Air is ported through large diameter lines to an extemal reservoir maicing

it a very low stiffness support A srnail amount of air is aiiowed to escape amund

the piston, acting as a linear air bearing, with a smaiI cornpressor on the reservoù

replenishing the air. The voice coi1 is activeiy controiIed to provide fine trim and

a user-set srnaii centering stiffness. It can a h be used to actively reduce the mas

&cts of the device by feeding forward an acceierometer signai. These devices are

commercially available h m CSA Engineering Inc. m Paio Alto, California, cost in


the range of $80,000 (US) each, and are generally sized for payloads of 25 to 300 kg.

The high cost and inappropriate payload ranges of the comercidy available

units preclude purdiasing a suspension system. The performance of the commer-

ad nullspring as compared to the electro-pneumatic unit suggests that both con-

cepts perform equaily m excursion range and stiffness [14] [15]. In terms of darnp-

ing and mas perturbation, the nullspring concept performs better. For added ex-

traneous modes, however, the niiilspring concept is more problernatic, aithough

additionai advantages the nulkpring has are that it is a passive system, it is straight-

forward to modei, and a custom design can easily be developed in-house. There-

fore a nullspring prototype has been developed with the design focus on making

improvements where others had found the concept to be lacking, partidarly in

friction and the suppression of local dynamics.

%me researches have explored reducing the pendular stiffness at each sus-

pension point by controüing the suspension mechanisrn positions in the horizon-

tal plane above the test artide, thus keeping the payload wires vertical [16] [lq. Another approach is to devise a vertical nuiispring hineci to cancel out the pen-

dular stiffness [18). However, these efforts are far more cornplex and expensive

than reducing vertical stiffness, and inûoduce more non-ideal behaviors than they

solve. Because pendular stifmess is very straightforward to mdel, no provisions

for reducing it are made other than using the maximum convenient payload wïre

length.

2.1.3 Nullspring ûesign

Kinematic and etastic dationships govern the effective plunging stiffness, or se-

cant stiffnessr of the nulIspring. Appendix A reviews the mathematics of the geo-

metric design. Equaiiy important to the design are the availability of components

and the Iessons gleaned h m the literaturer particularIy [8] and [IS]. The main

difference between the typical space stntcture suspension system found in the lit-


Figure 2.1 : NuIIspring: Zeroing the Spring Rate.

erature and the one developed for DICE is the payload mass. Smail amounts of

friction and suspension moving mas will have a large effed on the comparatively

light DICE payload. The design effort fe-used on optimizing a nullspring for the

DICE payload range and overcorning the inadequaaes experienced in the designs

found in the literature.

To produce a suspension systern that both supports a payload and has very low

stifhess, one can use positive and negative springs working in pardel. Figure 2.1

illustrates one such configuration (shown in a below-neutral position) where the

main spring supports the payload with a positive (or restoring) force and the side

spring and Iever form a negative (or propelling) sprùig that canceis the stiffness

of the main spring [15]. Other configurations e t [17] but do not sale down for

s m d payloads as wd. The design process is iterative. One first sources a suitable

main spMg and rnakes initial assumptions on ro, rm, and L (see Figure 2.2). The

relationships in Appendk A wiil yield the optimal side spring force and stiffness

and the effective secant stiffness over the range of a. One then retums to the initial

assumptions, making adjustrnents which yield a more srùtable side spring force or

secant stiffness, or to match the parameters with an available side spkg.

The nullspring is hmed by adjjting the side spring tension. Kienholz [15] sug-

gesk maxhkbg TM whidl mnimiizes the t-on m the side spring and thus the

CHAPTER 2 SUSPENSION SYSTEM

Figure 22: N u l I s p ~ g Geometry.

fiction in the Iever pivot. But this Ieads to lower frequency local modes in the side

spring and causes larger variation in the tension aaoss the excursion range; this is

undesirable for long-term tuning stability. However, by incorporating a knife-edge

bearing in the lever pivot, a large side tension does not sigruficantly increase fric-

tion. The tension (and ra) was designed to taise the side spring's k t mode well

above the excitation band for DICE to 50 Hz. With T ~ O significantly shortes than the

lever length ro, rather than longer as in ail the iiterature sources, this alIows a sin-

gie side spring instead of requinng two straddling the lever, reducïng compkxity,

local dynamics, friction and the rnoving mas .

Local side spring dynamics were further reduced by concentrathg the side

spring mass as far h m the lever as possible ( m g L) and using a shng

but Iight steel wire to connect the spring to the lever. The levei of excitation in

the side spring is also reduced by the srdi rm to ro ratio. To reduce friction the

numtïer of bearings m the system was reduced h m three, as in other designs (81

[15], to one by using wire clamping for the side spMg lever attachment Iever and

at the payload wire and main spring junction

Viriouç feahrres in the nuiispring design facilitate the tuning of the secant stiff-

ness. Clamps for the side and main springs aliow the circuiar aoss section springs


Table 2.1 : Final Nullspriag Design Values.

hperty Lever radius Side force radius Base length Main sprhg stiffness Side spring stifiess Nominal side spring tension

Value 0.498 rn 0.060 m 0.840 m 12.5 Nlrn 976 Nlm -56 N

to be rigidly ciamped anywhere dong their length aüowing adjustment of their

stiffness. The side spring is conneded to the lever with a stiff steel wire, wrapped

on a standard machine head from an acoustic steel string guitar. This can be turned

to adjust the side spring tension, measured with a load ceii built into its ciamp

base. The main spring mount can be adjusted verticaiiy so that once the payload

has been hung, the neutral position can be adjusted. The most effective way to set

up the device is to hang the payload and iteratively tune the side spring tension

and the neutrai position. The tension is set to the maximum level where the lever

stiii retunis to the neutral position, the locking damps are tightened, and then the

system cm be calibrated. The fùial geometry of the prototype that was buiit, evalu-

ated and used is summarized in Table 2.1. Figures 23 thmugh 2.7 depict the major

parts of the suspension design. For the actual testing, the moving parts of the n d -

spring and the payload itseif were isotated from the circuIating air gusts in the iab

using cardboard shieids.

2.2 Suspension System Performance

When the nuüspring is perfectly tuned so that the negative spring practically can-

cels out the positive main spring stiffness at the neutrai position, the secant stiff-

ness is very Iow near the neuûai position but changes suddenly after a certain

excursion. The Ieft plot in Figure 2.8, based on the noniinex reiationships in A p

pendix A, shows this tendency. Although it is desirable to rninimize the secmt

C m 2. SUSPENSION SYSTEM

main sprlng clamp

knife edge beadng

Wood wire

Figure 23: Nullspring Design. Figure 2.4: Main Spring Clamp.

Figure 2.5: Side Spring and Load Ceii. Figure 2.6: Knife Edge Bearing and Tuning Head.


Figure 2.7: Nuilspring Lever.

Od Perfectly Tuned

I

O 0.05 Excursion (m)

O - 4.05 O 0.05

Excursion (m)

Figure 2.8: Nuilspring Tuning.


Figure 2.9: Long Tenn Stifniess Stability Test.

stiffness, modeling and thus modek updatuig is far more straightforward if the

secant stiffness is essentially constant as with a linear spring. The Rght pbt in

Figure 2.8 shows what happens to the stilfness if the side spring force is reduced

by 1% front the perfedy bdanced state. Over the full excursion the stiffness is al-

most constant. At this secant stiffness of 0+12N/m appiied to the DICE n i with the

CMG, the k t flexible mode changes by ody 0.6% so the suspension device wodd

deariy be set at an adequately low &W. Obviously the stifhess is highly sen-

sitive to the tuning. However by using this tuning, which is easy to h d given

that the balance is at the point between staticaiiy stable and unstable, one can be

confident that the stiffness is almost constant over the excursion. Then the secant

stiffness can be determineci through cali'bration tests. Because of the hi& sensitiv-

ity to tuning it is not assumed the same m t stüfness can be set der adjusûnent

of the side spring tension. Remlibration would be required.

An experiment was conducted over several days where the plunging frequency

of the nullspring with a known payIoad was measured without changing the side

spring tension. The reSulk are shown in Figure 2.9. They show a standard de-

viation of about 3 . 6 O h (or appmximateiy k0.017 N/m) from the mean stiffness.

Although this may seem to be an mdesirably high deviaticm, it represents ody

about 0.0450 of the lateral s t ï h e s of a DICE rib. The change in the tension over

this period was bdow the known accuracy of the Ioad c d reade~ As discussed


Nullspring Calibration Data: Heavy 0.025 I

time (s)

Figure 2. IO: Sample NulIspring Evaluation Data.

above, this reflects the sensitivity of the secant stiffness to the side spring tension.

This ako suggests that it wilI be unCikely that the mode1 updating wiü result in a

niceiy converged solution for the secant stiffness given that it is such a relatively

s m d numbet Also evident from the long term stability test is that the maximum

deviation occurs early indicating that the mechanism setties into a tuning. For ail

the suspension nuis used in the nb system modei updating, a single tension setting

was used where it was ailowed to settie over several days first.

The caiiiration tests wese conducted by hanging a known cakirated m a s from

the payload wire. This mass featured a battery pack and a rib sensor target LED so

the same sensor and data a c w t i o n systern used for the DICE n i couid be em-

ployed. Two different payload masses, one called 'Tight' and one caiied 'Heavy',

were used to derive independent estunates for the same nuIispring parameters

with the same side spring timing. A very srnall step force was applied by hanging

a smd m a s fiom the payload by a nylon thread. When the system was at equi-

Iibrium, the smaii mass was reieased by melting the thread using a concentrateci

heat source, thereby imparting a virtud upward pomting step force equai to the

weight of the small mas. The payload was then dowed to osdate he iy up and


down with the target LED staying in the field of view of the n i delledion sensor.

The data coiiected were ideal for determining stiffness and damping c&cients of

the nuilspring knowing the mass of the payload. A sample nui is shown in Figure

210. With a fully free test üke this, the payload is susceptible to the air currents in

the lab, which could not be fully controlled and are evident in the data as irregular-

ities in the decay. However, the suspension systern is seen to operate successfuiiy

with very low stüfness and damping.

The frequency of the osdation was determined by caiculating the spectral

peak of the data. Stiffness was derived from this knowing the total mass of the

payload and the nuilspring moving components m, which was caIculated based

on an inertial soüd mode1 of the assembiy using CAD software, and aoss-checked

by weighrng the components. The effect of air drafts in the lab were removed from

the data to improve the estimate of the damping coefficient. Tliis was done by

filtering the data to a window dosely sunounding the resonant peak. A Iine was

fit to the log Hiibert transforrn of the resulting data. The dope of this iine indi-

cates the logarithrnic decay and the modal damping ratio. Because a very dose fit

was achieved, this indicates that iinear viscous damping is an appropriate mdei

to use. Based on the mass and stiffness the effective viscous damping coefficient

d, was caiculated. Table 2.2 sununarizes the properties found for the finaI tun-

ing and evaluation data sets using th& caliiration method. The apparently large

discrepancy in the damping estimates is to be expected. It is possiile this is due

to inaccuaaes in the stiffness or the moving mass estimates, but more likely it is

because this structure has very Iight damping so the estimate is higIily sensitive to

sIight variations in the dope of the the Iogarithmic decay. Estirnating such a srnaii

number accurateiy is bound to be troublesome. Later on, in Chapter 5, this same

effect is shown to be problematic for modd updating.

There is no indication of a suspension device in the literahue for such a s x d

payload as the DICE ri% system, but Table 23 compares the nullspring perfor-


Table 22: Cahïrated Nulispring Properties.

Table 2.3: Suspension System Performance Cornparison.

Mass Perturbation (%)

mance with another suspension system used by researchers at MIT [IS], an ele-

pneumatic device. As was indicated above, the nullspring suspension stiffbess

(and thus the quasi-cigid frequency) is adequately low for the DICE flexible struc-

ture. The M I ï device performs slightiy better in quasi-rigid frequency and mass

perturbation, but it does have the advantage of a much heavier payload. In damp-

ing the nullspring perfonns better than the equipment at MIT, even though it is

much more &cuit to reduce damping for srnaII payloads. In general the n d -

spring more than adequately satisfis the requirements, and measures up compa-

rably with highly reputabIe designs.

Internai dynamics not tested for speciûcaiiy were the side spring vioiin mode

and the main spring surging mode. For the violin mode, the side spring tension

and mass distriiution were designeci to raise the side spring mode weü out of

the test bandwidth. The surging mode is the internai axial dynamics of the main

spring, or effectively the second (and above) main spring &ai mode. This mode

is of concern because it can potenfiaIly be exated by an active payload. A simpte

axiai finite model of the main spring indicated thh modal freguency to be If3 Hz,

again above the nominal test bandwidth of 5 Hz. Had this effect been more domi-


nant and affecteci experimentd r e d k , one couid make amendments to either the

design or to the suspension model. The design changes might inciude using a main

spring of lower mas or length, thus inaeasing the surging modal frequency, or for

the purposes of model updating one codd incorporate a more comprehensive sus-

pension model with states associated wih the main spring. in this partidas case,

neither of these measures were judged necessary.

Chapter 3

System Model

A basic tool required for the DICE project is a gwd numericd model of the over-

ail system. It will be used for system performance verifkation, NASA safety re-

views, training, and controuer design. This mode1 encompasses the flexible and

rigid body dynamics, the sensors and achators, cornputer interfaces, control algo-

rithms, and even the real-tirne software. It is this model that will be updated using

data coilected during ground testing.

W~thïn the DICE team, considerable effort has been directeci toward modeling

the DICE system. The basic structure of the math mode1 was deveioped earIy in the

project by Phung [19] and revised by Zee and Crawford [20], and the sensors and

aduators were modeied by Choi [22]. Based on this math model a numerical modei

was prepared using the Xmath/SystemBuild software package. The author added

a model of the suspension system in preparation for the ground testing segment

of the project This chapter reviews the aspects of the modei that are devant to

modei updating. For m e r reading on the DICE simuiator see [22L and for more

detaiis on the suspension system model, see (233.

3.1 Rib System and CMG Model

F i p 3.1 illustrates the architecture of the DICE rib simulator [2a (which maks

use of normai flexible modes) and Figure 32 shows the rib syçtern hardware (with

CHAPTER 3. SYSTEM MODEL

Figure 3.1: Layout of the DICE Rib Model.

the CMG girnbal tilted) and the various coordinate frames. To calculate the modal

matrices in the model, the flexible ni is discretized using h i t e elements. Ten el-

ements are used and each of the resulting 11 nodw have six degrees of freedom

(DOF). Given the slendemess of the DICE rib, ciassical Bernoulli-Euler beam ele-

ments are appropriate for modeiing the Iateral beam stiffness. The rib mode1 aiso

includes axial stifhess, torsional stiffness, and geornetric stiffness when the rib is

oriented such that gravity induces an axiai load (vertical configuration). The beam

mas is modeled with a consistent m a s matrix and the CMG m a s matrix (at neu-

td gimbd tilt, calculated using measurements and a CAD rnodel) is added to the

nZÏ tip node degrees of freedom. The global stiffness and m a s matrices are then

decomposed mto normal eigenmodes. The flexible dynamics part of the simuIator

operates in modal space and makes use of the k t five modes of the rib -the first

two bending modes in each Iateral direction and the torsion mode. Linear damp-

ing is &O intepteci into the model as modal damping factors for each mode. The

rïb is fully constrained at its mot The motion of the rib tip is measured as the dis-

Figure 3.2: DICE Rib Coordinate Systems.

placement of the rib sensor target LED (y and t) Iocated on the CMG as seen by an

optical motion sensor.

The gyric torques arising h m the spinning CMG flywheei are handled in the

simulator in two pIaces:@cible gyiciry is a d a t e d with the rotation of the CMG

due to flexible rib motion and c o r n m ~ n d ~ c i t y is the gyric torque generated due to

the velocity of the gimbai servo actuator. When the gimbai is in the neutrai position

and the rib is not defiected, the CMG aywheei spins in an axis aligneci with the

ni. The gimbd system tilts the tiywheeI motor assembly a maximum of f 20" with

respect to the CMG chassis (and the ni tip kame FT) about the gimbai ax.is, which

remains nonnai to the spin axis and aligned with the rib tip y-axis. Thexdore, at

neutml tiIt, the gyric torque is aligned with the rib tip z-axis. The direction of the

g r i c torque vector is time varying with @bal tilt and its magnitude is a product

of the rate of tiIt (command) plus the rate of CMG rotation (flexible) and the stored

angdar momentum of the spinning fiywheei.

Figures 33 and 3.4 show photographs of the CMG. The CMG servo system is

modeleci as a DC motor with a potentiometer for position feedbadc and a simple

CHAPTER 3. SYSTEM MODEL

Figure 3.3 : Susptixied CMG. Figure 3.4: CMG With Girnbai Tilted.

Pm conh-ol system closing the servo loop, c o ~ e c t e d by a gearbox and a nonlin-

ear siide linkage to the gimbaled spin motor assembly, as in the adual physicaI

system. Logic system latency, current limit, and vdocity iimit parameters are in-

cluded in the model. The simulator outputs the servo position read by the interna1

potentiometer, which was usefui for updating the CMG servo mode1 as a separate

subsystem, this data being available h m the adual hardware.

The mass properties of the CMG are modeIed in two sections: the rib tip fixed

chassis with the servo motor, and everything that tilts with the gimbai d e d the

flywheel motor assembly (motor assembly for short). Because the mass center of

the motor assembly is aimost on the gimbal axis, the CMG mass matrix is approx-

imated as a tirne invariant rigid body with the gimbal in the neutral position as

mentioned above. The time-varying effect on the CMG mass matrix of the second

mass moment of the motor assembly as it rotates on the *bal is ignored because

it is a very smdi eflect. However, the command gyrïuty m d e i indudes a proof

m a s toque term asociated with the gimbai acderation and the motor assembly

rotational in&.

Servo motor specifications are not available h m the manufacher so reason-

able estimates were made, based on the measuxed motor winding resistance and

CHAP7ER 3. SYSTEM MODEL 28

the general size of the motor. Shdar estimates were made for the servo control

architecture and gains. These parameters were then made avaiiabte for modd up-

dating. The h a 1 updated model of the CMG wül likely not have accurate approx-

imations for the physical parameters, but the pole wili be phced to accurately

represent the servo behavior and nonlinear parameters such as the current ümit

wili be appropriatdy scded. In later revisions of the system the gimbal servo will

be czustom designed d e r than pu- as a black box so model updating will

correlate better to the physicd system.

3.2 Suspension System Model

For simdating the rib structure dynamics in its horizontal configuration, a model

of the suspension system is @d. The nullspring (Chapter 2), which is the ba-

sic elexnent of the suspension system, is inherently nonlinear but is designed to

operate in its rnost linear range. Originaily, certain noniineaxities that might par-

tiapate noticeably in the dynannics were rnodeled but were later disabled because

their contributions were so d that model updating could not successfuily tune

their -ated parameterç. Although this section only briefiy introduces the sus-

pension system modei, a more comprehemive document covers the subject [23j.

in it, the generai case of the full DZCE structure with 5 ribs and a bus module in

gened motion a l l supported by suspension elements is covered. For the present,

only the flexible motion of a rib is describeci. Aii equations are expressed in ma-

truc notation, that being the rnost convenient for implementing in software. The " notation represents the matrix aoss produd operation.

The suspension effects all irnpinge on the structure through the hard-point 10-

cated near the CMG m a s center on its chassis. Thedore it is assumeci that when

the suspended r û ~ tip dispIacement is zero, the rib is not ddected. The first stage

in the simutator resoIves the iocation and veiocity of the hard-point Coordinate

axes devant to the motion of the suspension as shown m Figure 3 2 are

3~ fixed to the ri% tip FE node

3t3 fixed to the n i base

The transformation matrix between them is Cm, and 38 is ground fixed. Symbois

w d for resolving the hard-point position and vdocity are

rhp vector from n i tip FE node to hard-point, expressed in 71.

drt translation of rib tip FE node, expressed in

Brt rotation of rib tip FE node, expressed in FB

d h ~ translation of hard point, expressed in FE

O ~ P rotation of hard point, expressed in FB

The position and rotation and the associated rates of the hard-point are simply

calculated as

where the rib tip motion is accessed from the rib dynamics in the simuiator.

The overall stïfEness d e d upon a body suspended by the nullspring is a corn-

bination of severai de&. 'Che swinging pendular stiffness in the two horizontal

translation directions is a bct ion of the payload weight and the suspension wire

Iength L,

The payload mass rn, is approximated as the CMG mass plus haif the nb and

cabling mass. The plunging stiffness or secant stiffness k, of the nullspring is a

hc t ion of the mechanism geometry, elasticity, and tuning. The goveniing r&-

tionships are detailed in Appendix A. The tuneci design features a virtualiy con-

stant km ove. the nuiispring excursion which is calibrated after the nuilspring is

fully set up. Finaiiy, the tonional stiffness of the suspension wire is a fundion of

the wire length, diameter dw, and material shear modulus G,

To caicdate s&ess forces and torques at the rib tip node due to the suspension

wire, diagonal coefficient matrices are defined. in translation,

and in rotation,

KR = diag (Ot 0, km,)

The suspension force f, and torque t, at the rib tip node are

fs = KT dhp

and

Initiaüy a comprehensive damping and Mction modei was devised for the n d -

spring. This induded separate viscous damping factors for the main and side

springs. The nullspring reiationships in Appendix A can be used ta relate the

nullspring extutsion rate to the rates of change of the Iengths of the main and side

springs. Coulomb friction in the knife edge bearing was also modeled using a

reset inkgrator technique [24]. However, because damping was so low and vis-

cous damping descriied the dissipative effeds very weil, the modei was reduced

CHAPTER 3. SYSTEM MODEL 3 1

in complexity to a single viscous dampuig factor. This effective and sirnpIe mode1

was much more cooperative in the mode1 updating efforts described in Chapter 5.

Acting only vertically, the damping force fd at the rib tip node due to the nuiIspring

is

while the damphtg torque at the rib tip node is

Because the masses of the moving parts of the suspension system are not zero,

they will have inertial effects as w& However, these effects wiil only be felt by

the payload when the hard-point moves vertidy, assuming the payload wire is

very h g compared with the ni tip deflection. One possible approach to mod-

e h g this is to feed back the vertical acceieration of the hard-point and multiply

this by the effective vertical nuilspring inertia, thereby generating the suspension

D'Alembert force. Unfortunateiy, this mates an algebraic Ioop inside the simula-

tor. The irnpliat system solver available in Xmath (the DASSL method) can easily

handIe this mathematicai difficulty but requires more than an order of magnitude

more processing time to finish a simulation. Thetefore this approach had to be

abandoned.

htead, the inertiai effects of the suspension are added to the mas aatrix be-

fore the modal mode1 is cdcuiated. This way thetie is no extra processirtg requVed

durhg simulation The general approach is to prepare an in@ matnx for the

ndspring by transforming the effective moving mas of the suspension mecha-

nism to the 13 tip node. Where m, is this effective mas, the inertial matrix is

To restrid this &ect to the verticai motion of the hard point, the foilowing niask is

applied to M, using element by element multiplication:

Finaiiy, the masked M, augments the flexible mass rnatrix at its rib tip node de-

grees of freedom.

The moving elernents of the nuilspring are the main s p ~ g , the lever, and the

side spring. The main spring is 6xed at its upper end and effectively moves with

the payload at its lower end, so assuming no local spring surging and a consistent

mass mode1 only 1/3 of the main spring mass is added to the payioad. The lever

rotates about the M e edge bearing but for small angles one can translate the

effective inertia to the payload wire attadunent point. The side spring rotates as

weil, but at a much lower rate (D as compared tu a in Figure 22), and because

the mass is concentrated dose to its center of rotation, its inertial contnïution is

extremely srnd and is ignored. The total effetive nuiispring mass m, is

where mm is the main spring mas, Il is the lever rotational inertia about the bear-

hg, and ro is the length of the Ievel, as depided in Figure 22

3.3 The Parametric Model

Parametenpng the modd in preparation for modd updating involves assigning

a scaling factor pi to each constant in the model, i = l...n, which may be devant

to the model updating. This scaiing factor is d e d the p-value. So if pi = 1, the

parameter would be left at the nominal initial estimate value. The cost functional

depends on the colurnn matrix of p-values, p. Also included m this process is de-

termining the uncertainty range for each parameter. This is done in a variety of

ways dependhg on the parameter. Statistical information could be derived h m a

series of measurements, published data might quote an uncertainty, or good engi-

neering judgment might give a likeiy range within which the true value would be

found.

It is important to pose the mode1 updating problem in a way that wiü lead to a

meaningful result. This begins with taking care in pararneterizing the model. For

example, for a mass matrix, it should be impossible to mate a non-positive definite

matrix while the parameters are within th& uncertainSr ranges. To ensure this,

they were parameterized in their prinaple axes as three prînciple moments and

three axis rotations, rather than the actual matrix values.

Many constants in the model were easily measurable with low uncertainty so

they were not parameterized for mode1 updating. Other parameters had very low

sensitivity so were not induded. However, many other uncertain parameters had

the possibility of conbnbuting significantly to the cost function and were made

available for model updating. They are listed in Table 3.1. As will be later shown,

most of these parameters do not come into play with the final model updating for

various reasons. in order to be a good candidate, the parameter must be w d -

behaved. Most parameters with sizabie sensitivity are weii-behaved, but some,

like parameters associated with damping in higher modes and second mass mo-

ments, contain features like singula~&~ or have a minimum far outside their un-

certainty range, which move significantly depending on the p-values of other pa-

rameters. A parameter need not be sûictly convex within its uncertainty range, but

if the generai behmor of a parameter is very sensitive to what the other p-values

are, this parameter is strongly ams-coupled and problematic for mode1 updating.

There is a N k that the model updating wiil not bring about physical meaning to

the final solution if key parameters are omitted, which is why aii the parameters

listed were investigated from many different reference p-value sets. The modei

pararneters chosen for updating wiü be discusseci further in Chapter 5.

Table 3.1 : Mode1 Parameter Definitions

Description total CMG mass rib tip to CMG CofG (x) pcornponent z-component CMG inertia, pria axis 1 prin. axis 2 prin. axis 3 prin. axis 1 rotation prin. axis 2 rotation prin. axis 3 rotation motor assembty mass rib tip to motor assy. Co= (z) y-component z-component motor assy. inertia, prin. axis 1 prin. axis 2 pria axis 3 prin. axis 1 mtation prin. axis 2 rotation prin. axis 3 rotation flywheel spin momentum rib Young's Modulus rib density y-bendimg mode 1 dampmg

2-bending mode 1 damping torsional mode damping y-bending mode 2 damping z-bending mode 2 damping rib iength rib radius rib Poisson's ratio rib tip to gimbal axis rib sensor target offset CMG actuator latency suspension secant stifhess main spring damping side spring damping step ment time offset n i geometric stitihess suspension system mass servo motor current limit servo motor speed 1 s t servo motor constant CMG girnbai bias servo proportiooal gain servo integral gain servo derivative gain horiz. spin correction

Chapter 4

Model Updating: Theory

Many papers and books have been written in ment years on model updating, and

specifically finite element (FE) model updating (MU). Frisweli and Mottershead

have summarized much of the Literature in a textbook and a survey paper [25] 151.

An issue of the j o d Mechanical S'stem and Signal Processing (1998, vol. 12, no.

1) was entirely devoted to the subject of model updating. MU is w d in a broad

range of disiplines from soi1 mechanics to structural identification. It has been ap-

plied using a diversity of data sources fiom radar imaging satellites to earthquake

acceiemmeters. It is such an extensively studied and accepted field that many

comrnerciaiiy available FE packages aheady indude some form of model updat-

ing. The impact that FE modeling has had on engineering analysis, design, and

product deveiopment, has been sustained by MU providing real wortd relevancy

to FE modeIs.

For the DICE pmject, two special requirements drive the choice of method for

updating the numericd modei, First, the method must be abIe to handle any nonlin-

ear or tirne-varying effeds in the system without simply linearizing them, because

these effects couid be significant. Prim experience with a similar structure cded

Daïsy [26] found that seemingly Iow magnitude nonlinear effects can easily corrupt

SI results. Second, the method must maintain a physicaiiy meaningfui model state

space so that, for example, physicai material parameters can be extraded from the

updating. In this way, the updated model can be reconfigured, other subsysterns

cari be added or taken away, alternative loading applied, and most importantly the

model can be used for predicting û-g behavior before the opportunity arises to do

O-g tests - hence the tie-in to ground tesiing.

4.1 Identifying a General Approach

Most of the typical or standani FE model updating methods that are featured in

current commercial FE software take advantage of speaal properties of linear FE

models. For structural dynamics, these techniques generally adjust linear m a s

and stiffhess matrices, either directly or through weighted elemental matrices or

physical properües, based on errors in modal parameters or frequency response

funciions (FRF). Modal and FRF data are appealing because they fully swnmarize

the behavior of a linear system and are a compact way to combine large amounts

of data together.

When using modal data as a basis for the model error measurement, the exper-

imentai data must be manipuiated so that mode shapes are orthogonal. Further-

more, the model degrees of &dom (DOF) must match 1:l with the experimentai

measwinent points, so either the model must be reduced using one of the many

known appmaches of mode1 redution, or the experirnental data expanded often

by using the FE model itsel€ in some way to fiii in the missing DOF [27]. &O,

damping is usuaiiy ignored for the entire process. When FRFs are used in MU, the

experirnental and model FRFs are compared in logarithmic d e . A distribution

of points in the test frequency range is chosen that wiü as& the model updating,

and this depends on damping, windowing, resonances and anti-resonances [28].

Again, damping becomes a computationai inconvenience and is typically ignored.

Neither modal nor FRF-based MU lends itseif to systems outside the linear M, K

type FE rnodels.

The o v d system model for the DICE hardware moves beyond the realm of

linear finite elements. The dynamidy flexlile portion of the DICE system, the

flexible ni system, is weii suited to FE modeling, but attached to the tips of each rib

is a Control Moment Gyro (CMG) featuring a rapidly spinning flywheel. This gives

rise to gync modes which can be described m the numericd model with complex

mode shapes or as a coupled dynamic effect ninnuig in paraiiel with the conven-

tional laterai beam modes. The system is Lime-varying, due to the gimbal rotation

during large commanded excursions from the neutd position, and noniinear, due

to several effects in the servo control system ($3.1). ûther more subtie noniinear

effects, such as sensor or actuator latency and f'riction, are part of the systern modei

as weii.

Frequency-based methods like modal or FRF updating unfortunately assume

the system is ünear- The only way to discern any nonlinear or time varyuig effects

in a meaningful way is to view them in the time domain. Direct systern identi-

fication such as OKID/ERA make use of time series data [3]. This method uses

OKIû to h d the time domain impulse response (in the t o m of Markov param-

eters) which is used by ERA to identify a bladc box state-space model of the sys-

tem. Elsewhere in the DICE project, this method wiii be used for analyzing actuai

on-orbit data for redesign of the on-orbit controUers. This method is appealing be-

cause of the easy handiing of inconvenient (though iinear) &ects like @ty. if

strong nodinear effects arise, this tends to Iead to very large observer and model

orders in an attempt to ünearize. An extension to OEûD/ERA calIed Augmented SI

[26] dows known nonlineer CUES to be identified without large order modeis. A

controller designed witti 0iCiDER.A-derived models wül perform weii provided

that the ÇI data was s i m k m magnitude and rich in the frequency band typical

for the contcolier.

R e d , however, the second requirement for îhe DKE MLI: the physical state-

space is required for generalizing the system configuration aftet updating- Retain-

ing a physical state-space, or rather transforming to the physicai cwrdinates h m

the arbitrary state-space of the ERAderived black box model, is practicdy irnpos-

sibIe because as Bauer [29] has discovered, one either requires M-state excitation

and measurement or no repeated mots and perfect noise-& data. Neither of these

is a reality with DICE or the majority of real ident5cation probiems.

Outside of direct SI, the idea of using time series data is vaiuable but few refer-

ences explore it. Ln one reference using tirne series data for FE mode1 updating [30],

Dohrmann uses the approach to update the parameters of a nonlinear Ieaf spring

model of a truck suspension. A cost h c t i o n is devised as sirnply the squared er-

ror between experimental time domain data and the simulated response h m an

integrated parametric model. The method was successful Ut çimuiation but did not

work weii using mal data mostly because of diffidties in performing the spedic

experiments examineci in the paper. However, this approach does meet the two

requirements for DICE ground testing: it handles norùinear effects and can retain

the physicai state space of the model.

4.2 Problem Statement

The cost function upon which the model updating is based is the squared norm

of the difference between the model-simdated time series data and a reference

time series of experirnental data, divided by the squared norm of the reference

series. This error is often c a W the midual. Where p is the column matrix of model

parameters with respect to an origin O and the modei G(p) gives rise to simulated

output y, the residuai cost is

LocalIy a cost function j can be approximated by the truncated Taylor series

1 f (p) = I(0) - bTp + 5 P=AP

where

A is cded the Hessian ma&. By dserentiating Equation 4.2 we h d that

This indicates that when an extremum is encountered at ii, where the gradient b

vanishes,

This is the basic optimization form. if we had an estimate of A and b, we couid

solve for p. We can not solve for A or b analytidy because they are very con-

voluted functions of the model G(p) and y,f. The m a t effective way to h d the

optimal value of p, which minimizes f, is to ernploy one of the many least squares

minimization procedures.

4.3 Reg ularization

A very powerful enhancement to mode1 updating is to make use of information

about the model parameters other than the most likely vaiues, namely the param-

eter uncerhhties. Regularization is a method of doing this whiie also improv-

ing the numericd conditioning of the optimization problem. In the ciassic form

A$ = b, A niight be rank defiaent or singular and A and b might contain noise.

Methods using singuiar value decomposition (ÇVD) can be applied to numerically

fiii the nuil space of A g d y without disrupting the cest of the matrix [311. This

luxury is not afforded in the present problem because we never actually calculate

A. However, that does not prevent the same g e n d prinaples from being applied.

One can set uncertainty ranges for each mode1 parameter. Because the pa-

rarneters represent physicdy meanin@ quantitics su& as Young's modulus or

CHAPTER 4. MODEL UPDATING: :THEORY 40

mass, reasonable uncertainty bounds can be set by using the statistics of any mea-

suremenk made, instrument precision, published uncertainties, or failing those

sources, engineering judgment.

A second term is devWd in the cost function caiIed the regutarization cost. This

tenn refIects how far away from the most W y parameter set the point p in param-

eter space is. Making this term quaciratic inmases the Likeiihood that the Hessian

matri. is positive definite and improves the conditioning of the optimization prob-

lem, promoting solution uniqueness and speedy convergence. Although regular-

ization in this form is now established in MU, its origins are found in techniques

for solving inverse problems [32].

Based on the parameter uncertainties, the regularization cost is sensitive to

movement of the parameter toward the extremities of the uncertainty bounds. This

has the effect of dowing parameters whose uncertainty is relatively large to move

around more. Conversely, if a parameter is weii known, it will move les. No hard

ümits are placed on the parameters because this introduces sharp edges in the

cost function, requiring more complex and computationally expensive optimiza-

tion algorithrns such as hea r programming. Some mearchers [Xi] have aiso tcied

weighting the reguIarization cost by the parameter's inverse sensitivîty which pre-

vents parameters that do not strongly influence the cost function from oscillating.

A preferable method would be to not include these parameters in the updating at

a. For a paranieter set p which has an initial estimate po and uncertainty a, the

regularization cost is

f, =C (Pi -Poi)2 i= 1 4

The overaii cost function then becomes the surn of the original residual error cost

(Equation 4.1) and the regularization cost weighted by A,

One of the preferred methods of deàding on an appropriate value for X is by

CHAPTER 4. MODEL UPDATING: THEORY

25

2 .

- a O g 1.5, 3

;r - 1 . K Lambda Opt = 7dN5

8 2

0.5

Figure 4.1 : Picking Regularization Weight X Using L-Curve.

looking at its 'L-curve,' so called because of the distinctive knee or point of max-

imum curvature in the plot [33] [32]. This curve compares the magnitudes of the

residual and regularization costs at the minimized parameter set for various X in

logarithmic scaie. Figure 4.1 shows such an L-curve based on reai experimental

data h m updating the CMG servo modei (553). The highlighted point, showing

the optimal choice for A, represents a balance between confidence in the experï-

mental data and the parameter uncertainty, Le., where reguiarization just starts to

effect the location of the minimum in a meaningfui way relative to the residual. It

effectiveiy moves the Hessian matruc far enough fiom having very small eigenvai-

ues to withstand noise without domhating the cost function.

4.4 Sequential Estimation

For gradient descent optimization methods, where the algorithm can easiiy be-

come studc in a Iocal minimum, another important embellishment of the mode1 up-

CXWTER 4. M O D E UPDATING: THEORY

3 Second Sensiîiviiy 10 Second Sensitivity

Figure 4.2: Effect of Sequence Duration on Sensitivity.

dating algorithm is the use of sequentiai parameter estimation. Using this method

one builds up an estimate of a parameter over tirne as more data becomes available

[41. Kahan fdters use this concept to generate an on-going optimal estimation of

a nonstationary mode1 state. In the case of MU, sequential estimation is used to

estimate the stationary values of the mode1 parameters using progressively longer

sequences of data, and in this context, it is a new and innovative teduuque.

The advantage of using a short data sequence, other than the obvious saving

in computation, is that the convex region of the cost hc t i on tends to be wider

and the initiai guess cm be hrther from the optimal solution without the risk of

the gradient descent algorithm getting trappeci in a local minimum. Likewise the

quadratic assurnption is valid over a wider range. Figure 4 2 iilustrates this with

an example, where on the left the cost is caidated over a range of parameter

values using 3 seconds of reference data, and on the cight using 10 seconds. How-

ever, longer data sequences resuit in a statisticaily more relevant estimation of the

optimal parameter set The advantages of both cases c m be accessed by starüng

the updatmg with an initial p-value guess and a short data set and repeating the

updating with progressively more data. Each new step begins with the soIution

derived h m the previous triai. This is repeated until the solution has settIed to a

converged state or when the overail residual cost has converged. Using a geomet-

ricaliy increasing sequence length seems to optimize the computational effiaency

of the method whiIe maintainhg a Iow risk of gelting trapped at an inappropriate

local minllnum,

In general the rate of convergence will differ between parameters. Some may

not converge at al before aii the available data are used or before the rninimiza-

tion problem becomes intractable computationdy. The set of rninimized p-value

curves over the range of sequence durations is a gaod indication of parameter be-

havior and convergence. Note that one can not use a weighted average of the opti-

mizing parameter sets over multiple iterations if the updating problem is coupled,

as it generaiiy is.

4.5 Search Tolerance

There is a fundamental limitation in the accuracy that can be expected from any

dgonthm searching for the minimum of a given cost functional, and this is deter-

mined by the round-off error in the function. For a simple cost funciion using a

few floating point calculations, this error might be alrnost as srnail as the machine

precision of the cornputer cm, which is 10''5 for double precision mathematics. if

many Eloaüng point operations are used, however, each time the function is com-

puted these round-off errors accumulate.

To cast some iight on round-off error, we review the analysis found in [34]. Re-

ferring to Equation 42, if a wiicpe minimum exists, the Hessian ma& A must be

positive definite, so its eigenvaiues are positive, XI >_ Xz 2 ... 2 A, > O. The Taylor

series appmxhates the cost function as a convex quadratic in the neighborhood

of a unique minimum. Due to rounding error, any point p wiii have an m r ep

lower bounded by r,, so the best which can be expeded ignoring (for now) the

accumulated round-off in the caldation of j (p) is

where f l( f ) indicates the fioating point caldation off, which, as a squared nom,

is always positive. If P is a minimum and 6 is the largest number where

for some direction u, then any muUmiz;ition algorithm can oniy get within d of the

true minimum with certainty. If we pi& the eigenvedor un, correspondhg to An,

the smaUest eigenvalue of A, substitute the point + 8u, into Equation 42, and

apply the bounds found in Equation 4.8, we see that at best

We cannot use this expression to quantify the overail search tolerance because it ig-

nores the worst source of round-off error, namdy the accumulating fioating point

e m r built up over the thousands of operations used to caldate the cost hct ion

and the hnited resolution of sampled data. However it does demonstrate the de-

pendency on the shape of the cost function near the minimum. Specificaiiy~ if one

or more eigenvalues of A are srnaIl compared to the others in the system, which

corresponds to Little change in the cost function dong some directions (desaibed

as narrow d e y s ) , then finding the bue minimum value is difficuit and the accu-

racy to which the minimum can be found is degradeci.

The overwhelming sources of error in the caIcuIation off are the accumdated

roundsff error of the integratm and the limiteci resolution at which experimen-

ta1 data is available. Data are saved in ascii text format to 5 signiscant digits so,

for example, typicai maximum displacements are recorded accurateiy to within

fO.005 mm. More irnportantIy, data are sampkd at 12-bits over the 140mm rib

sensor fieid of view, or to j=0.017mm. To obtaÏn simuIated mode1 data for the cal-

culation of f, the modd G(p) is numeridy integrated. Romd-off mors in each

CHAPTER 4. MODEL CPDA?WG: :THEORY

Figure 4.3: Picking the Search Tolerance.

tecursive integration step will cause some drift in the simulated solution, increas-

ing the error. By picking an optimal integration aigorithm and hrning its operating

parameters appropriately integration drift can be aimost eliminated, but the sheer

number of operations required to caldate the cost function cannot be reduced.

Because the combined effects of these m r sources are diffidt to quantify ac-

curately, the optimal search tolerance is established by running a series of updating

tests while varying the tolerance* A value is picked which balances the e m r d a -

tive to the solution found at the h e s t tolerance, set a s the estimated cost function

caldation accuracy (explained below), and the computational expense. One such

experiment using actual data £rom the DICE nb updating is depicted in Etgure 43.

The trend shows that in general the finer search tolerance values will find a lower

minimum but at p a t e r computational expense while coarser vaiues wilI lead the

algorithm to finish too early. However it is conceivable that a coarse tolerance

could fînd a very good solution just through lu& convers cri^ a very fine toler-

ance setting is more susceptibIe to srnail region local minima which couid yieid a

C m 4. MODEL UPDATING: ZïiEORY 46

poor solution Therefore several tests are typicdy conducteci. In Figure 4.3 it a p

pears that search tolerances greater than 0.001 start to degrade the solution. To be

safe, the finai value picked h m these results was a bit lower than this, at 0.0002

Assurning that near a minimum the cost function can be approximated as a

polynootial, the caldation accuracy can be estimated by evaiuating the funcüonai

at many points dong an axis close to and surroundhg the minimum. A polyno-

mial is fit to the data using least squares and the accuracy is estimated as the stan-

dard deviation of the error in the caicuiated points relative to the polynomial. For

the cases studied herein, a fifth order fit appeared to remove any structure From

the enor aiIowing an appropnate measure of the random cost fundion caldation

error. As mentioned above, this was used as a lower bound for the search tolerance

to use in the model updating.

4.6 Solution Uniqueness

There are two related aspects to the issue of solution uniqueness. First, does the

cost fundion exhiiit multiple minima or a minimum manifold on which the cal-

cuiated cost is equal? Second, do there exist multiple distinct p-value sets which,

when applied to the model, generate an equivaient output? The former can be a

resuIt of the latter.

Even with extremdy good data and very stable robust optimization techniques,

a unique global minimum a n not be found if it does not exist This depends

as much on how the problem is posed as it does on the experimental data and

the model. Take for example a system of a mass hanging by a spring. If two

parameters are sought, m a s and stiffhess, and the free viiration frequency aione is

used as a basis for the modei updating, the solution will not be unique because the

frequency is oniy sensitive to a change in the ratio of k to m. In the [l, 21 direction

of the parameter space the cost Function is flat For systems with more than one

mode Janter states in 1351 that when MU is based on modal data, frequencies and

mode shapes are not enough to update a mode1 and ensure a unicpe physicaliy

meanin@ solution, The modal mass, whïch affects the participation factor of

each mode in the system response, must also be used.

Solution uniqueness is promoted by making use of as much information as

possible. Ça for the hanging mass example, if a test is devised where a hown

step force is applied and the response magnitude is used in the cost function, then

the stiffiiess and mass can be uniquely identified. Further, if initiai guesses for

the mas and stiffness values and theh uncertainties, as weii as the uncertainty of

the step force, cm be made use of, the uniqueness and physical signiûcance of the

solution can be further assured. In essence, this example is not output equivalent

so by stmcturing the cost hct ion to use aii the available information one avoids

the issue of solution uniqueness.

W~th the exception of strictly convex problems, in most practical cases it cannat

be proven that a unique gIobai minimum exists. Unfortunately, it is usually im-

possibIe to prove convexity [36]. Simply looking at the sensitivities of most of the

parameters in the present problem disproves global convexity. Previously it was

stated that near an aiieged minimum the cost can be approximated as a quadratic

(Equation 4.2). There the gradient is very srnaii, so if the Hessian matrix A is posi-

tive definite, the cost is convex in ai i directions and the minimum is !ocaiiy unique-

If the Hessian cannot be caidated it cannot be pmven that the point is not simpIy

a stationary point. However, depending on the method used to arrive at the zero-

gradient point, it might Iikefy be a minimum. One can also increase the IikeIihood

of a minimum being giobaI by supplying the best possibIe first guess, but even for

statistidy based optimizers this is not guaranteed. Noise in the cost hc t ion can

make a flat region, where the eigenvaiues of A are dose to zero, seem iike the min-

imum is not unique to practicai optimization aigorithms. Reguhization deviates

this probhn.

The same is true for output equivaience. Nthough Becket al [3;1 offkr a method

of iteratively searching the parameter space for a manifold of output equivaient pa-

rameter sets for a linear structural model, it is computationally intensive, requires

accurate knowledge of the mas, and noise-£ree data. Additionally, it has only been

demonstrated on very simple models. Output equivdence is a consequence of the

model of the system, and for DICE, the structure of the numerical model is outside

the scope of this thesis. However, in the spirit of rnitigating solution uniqueness

and output equivdence issues, all pnor knowledge of the parametes is uicorpo-

rated through uncertainty weighted regularization, and a dive* of data sets and

test configurations is used.

4.7 Candidate Optimization Algorithms

The model updating problem is an exercise in optimization and an algorithm that

suits the probkm must be chosen. The available classes of optimization algorittuns

are enonnous, with many promising candidates (361. First, the problem m u t be

dassified to narrow down the avaiiable options. The problem at hand, namely up-

dating the DICE model, is deterministic (in that the parameters are well dehed

and unders td and the model states and outputs are ail available), Ehite dimen-

sional, and unconstrained. The regularization cost is not a hard constraint and

does not affect the topologicd dassification of the problem. It is aIso a continuous

problem, given that the computationai precision is small enough that continuous

principles apply. With this dass of problem, the two choices of algorithm types are

itemrive and enurnerative.

4.7.1 lterative Methods

Eterative methods use a scheme based on the local shape of the cost surface to pick

a sequence of points that will lead to a minimum. They are fast but are usually

attracted to local minima. For most aigorithms of this type, the basic tool is the

iine rnic~irnirer~ The job of this dgorithm is to find the minimum of the cost function

CHAPTER 4. MODEL UPDATING: ZXEORY 49

in a given direction in the parameter space. The first step is to bracket the mini-

mum. A minimum of a function (dong a line) is bracketed by three points when

the function evaluations at the first and third points on the line yield higher costs

than at the second (middle) point. An algorithm for bracketing the minimum is

described in detail in [38]. To minimize the function in the bracketed intwal, the

Brent algorithm was found to be the most robust and efficient [34]. This method

uses multiple techniques, allowing one to take over when another has difficulty

The principal method is a parabolic fit using h e three bracket points. At each iter-

ation this leads to a new bracket which is d e r . When the parabolic fit procedure

starts to bounce around in a non-converging limit cycle, the Brent method resorts

to the Golden Section search method [38], which is another method for systemati-

cally reducing the bracket. Eventually the bracket is reduced in size until its width

is twice the desired tolerance of the minimization, the point midway between be-

ing the minimum abscissa returned.

The trick to using the line minimizer efficiently to optimize a multi-dimensional

cost function is in choosing the minimization directions. Iteratively minimizing

one parameter at a time orthogonally to the parameter space is one way to proceed.

Sometimes this works well, but when a narrow valley exists which is not orthog-

onal to the parameter space, as happens when the parameters are coupled, this

technique is very slow because the principal axis of the cost function is repeatedly

crossed. One can aIso pick the direction in which to minimize based on the cost

function gradient The simplest form of this is Steepest Descent where the direction

of the maximum gradient is sdected for each Line minimization. For a quadratic

function each successive iteration will necessarily be orthogonal, so again the min-

imum will be reached in a series of zigzags [39I. A much more efficient method

of choosing the directions is d e d the Conjugate Gradient Method- The rest of this

section elaborates on issues &mt to this and presents the best available iterative

methods for the optimization problem at hand.

CHAPTER 4. MODEL UPDATENG: E O R Y

Cost Function Gradient

Many weil reputed optimization aigorithms require the gradient of the cost func-

tion. This is hue of the gradient decent methods like Conjugate Gradient, and

quasi-Newtonian methods iike the Variable Meûic method. However, in the present

problem and in many reai world optimization applications, the cost gradient is

not directly available in a closed form so one must cesort to either a denirative-fie

method or make use of a numericaIIy calculated gradient.

The simplest method of detaminhg a numerical derivative is by 6nite ciiffer-

ence. To estimate the gradient dong a direction u, the one sided ciifference ap-

proach uses a single additional j calctdation a srnaii distance h h m the point of

interest p, approximating the gradient a s

This method is notonously Mccurate, so one usuaiiy uses a symmetric method

where the cost function must be caicuiated a smaii distance h on either side of the

point in question, giving a gradient estimate

For an n dimensional problem, this means 2n cost calculations.

As for a c m c y , roundsff errors in the cost hc t i on calculation (see 545) and

in h make a larger value of h more desirable, while the truncation error associateci

with the first order ônite ciifference gradient equation make a srnalier h desirable-

To compromise, the optimal vdue of h for ca id t i ng the gradient of f using the

symmetric finite ciifference method is appmximately [38]

{CL f"'

where is the accuracy with which f is calcuiated. The practid difficuity of this

is that numerid analysiç of the function is ~qyired before h can be pidced, and

it is different for each direction in parameter space. Also, as one moves through

parameter space, both f and f" change. The accuracy of V f and the stability of

the Conjugate Gradient Method have been found to be very sensitive to the choice

of h in triais using red data.

Other higher order methods and recursive methods of findhg the numerical

derivative of a function are far more computationaliy expensive. W1th the avail-

ability of effiaent derivative-free optimization methods th= is littIe benefit in pur-

suing numerical derivatives.

Conjugate Directions

When attempting to use a gradient method to move downhiU and hopefully arrive

at a local function minimum, there is no better set of directions to use than conju-

gate directions. This is by far the most effiaent way to search for a minimum when

the function is smooth and quadratic-like. Conjugate directions are organized so

that they are 'non-interfering' [381. If we rninimize a function dong a direction u,

the maximum gradient at the iine minimum is by definition perpendicuiar to u. To

pi& a new direction v in which to search that does not 'undo' the minimization

accompiished by the Iast step, the change in the gradient must stay perpendicular

to II. From Equation 4.4, the change m the gradient is

So for v to be conjugate to u,

When this reiationship holds for each pair in a direction set, the set is d e d the

conjtgate direction set, If the quadratic approximation of the function is perfect,

then successive line minimhtions dong the conjugate set will find the minimum

exactiy in n steps. If the cpdratic approximation is dose, then it might take

more than one trip through conjugate directions, but the minimum will converge

quadraticaiiy, meaning the number of converged sign%cant digits doubles each

step. Because of the definitions of the residuai and regularization costs (Equations

4.1 and 4.6), the quadratic approximation given by the tnuicated Taylor series is

very dose, especidy near a minimum.

Conjugate Gradient Method

Between the elegance of the conjugate direction set and the awkwardness of nu-

mencd derivatives, experience showed that awkwardness prevails for the present

updating problem using reai test data. Mthough very qui& convergence was

found with the conjugate gradient method, it required very carefd 'rehearsal' in

the numericd derivative caldations before suitable directions could be found in

which to search, making it an impractical method. Methods that require intuition

and experience tend to fa3 more frequently as the dimension of the problem gws

UP-

Powell's Method

A method that retains the use of conjugate directions while nicely sidestepping

the recpirement of expücit gradient calculation is Powell's Method. The details

are desaibed in [34]. In basic terms, Powell's method uses a recursive algorithm

to arrive upon a conjugate gradient direction set starting h m an initial guess (the

identity matrix works weil). For a problem of n parameters, n(n+ 1) iine minimiza-

tions will result in the exact minimum of a perfect pdrat ic , as weli as a matrix of

conjugate directions associated with the function. Several researchers have mated

embellishmenfs on the method which by itseIf can have numericd difficulties [381.

TypicaiIy the direction set becomes rank deficïent and fol& in upon itself aliowing

minimization in only a subspace of the panmeters, Heuristic d e s have been de-

vised for replaàng a redundant direction in the set with a new one that elimmates

CHAPTER 4. MODEL UPDATING: THEORY 53

the danger of rank defiaency at the expense of giving up quaciratic convergence,

although convergence is stili very rapid. Note that aithough a triai in the sequential

estimation sdieme provides a gwd starting estimate for the optimal p-value set for

the next ûiai, it does not provide a good starthg estunate for the conjugate direc-

tion set Much better optimization performance is achieved by beginning with the

identity matrix each trial.

Downhill Simplex Method

This method, developed by Nelder and Mead [40], uses a geometric object cailed a

simplex whose vertices lie on the cost surface. For a parameter space of dimension

n a simplex consists of n + 1 vertices, the interconnecting line segments between

the vertices, and the resulting polygonal faces. The aigorithm moves the vertices

so that the simplex waks downhill and h d s the minimum of the cost function.

The authors in [381 make the analogy that the simplex behaves like an amoeba

which slowly crawls dong the cost function to its lowest value. It requires no spe-

aal assumptions for the cost function and does not require gradient information.

Each vertex starts at an initiai point, and the one with the highest cost is reflected

through the opposite face of the sirnplex to a lower point The point is chosen to

maintain the same hyper-volume of the simplex. If no lower point is found, the

simplex expands itseif, and when the costs at aii the vertices are near the mini-

mum cost the simplex contracts. This method is slightly better at avoiding local

minima than gradient descent methods because if one or more vertices are outside

the region of attraction of the local minimum the simplex can escape. However it

is slower than conventional gradient based algorithms, particularly in cases where

the sensitivïty to di€fem~t parameters varies considerably.

4.7.2 Enumerative Methods

Enumerative methods tend to search the parameter space for the minimum cost

location using a distribution of test points, often selected at random. Typically it

takes rnany more cost evaiuations to arrive at the minimum than for iterative meth-

ods, but enumerative techniques do not get happeci in local minima. Once con-

verged, one can be reasonably sure that the aigorithm has Iocated the global min-

imum provided settings within the algorithm have been appropriaMy defined. It

is also possible to combine enmerative metfiods with iterative methods. Once the

region containing the giobal minhum has been idenlified using an enumerative

approach, fine tuning can proceed rapidy using a more tradition iterative methoci.

Several main tediniques exist in the enumerative class of optimization.

Genetic Algorithms

Genetics and evolutionary theory comprise medianisrnç by which selection d-

teria have a tendency to produce a species that is optimaily adapted for a given

environment By posing the minimization problem appropriately, one can make

use of an evolutionary analogy to successfully search for a global minimum of an

arbitrary cost function [41]. The parameters axe disczetized and coded into binary

strings which represent chromosomes, themess of which is measufed by the cost

function. Each iteration represenk a generation where reproduction is governeci by a

set of simple operatos induding cmssowr and murarion. These opentors are prob-

ab&süc in nature and create a compromise between distributhg genetic informa-

tion through the popuiation, random recurrence of lost genetic information, and

sumival of the fittest. Further operators indude assigneci elire status, which d o w s

the best kw chromosomes to be passed to the next generation unchanged, and new

blood (sic) which is random information repIacing the wost chromosomes each

genera tion.

In pradice, it has been f o n d that genetic algorithms work best when the pa-

rameters are not strongly coupled. Furthermore, due to computationai expense

one is ümited to a coarse resolution in the model parameters. For these reasons

genetic algorithms were not fully pursued for model updating the DICE structure.

Neural Networks

Neural networks are hding an increasingly wide range of applicability and have

already been proven viable as a means for heat condudion model updating. Aigo-

rithrns based on neural network techniques are partidarly good at generalizing

in that they can predict output based on unseen data. The structure of a n e d net-

work, as its name suggests, is modeled after an array of biological neurons. SimpIe

operators, or neurons, with multiple inputs and a single output are organized in a

network of weighted connections. They are often Iaid out in Iayers with an input

layer, possibly one or more hidden iayers (not expiicitly accessed), and an output

layer. One architecture used for model updating makes use of an input iayer of

radial bas5 function (RBF) neurons and an output Iayer of ünear, or bias, neurons

[W. RBF neurons have a high probability of firing when the inputs, coliected in a

vector, point to near the center of the neuron in its parameter space. The output of

a firing RBF neuron is the sum of its input weights. The outputs from the layer of

RBF neurons feed into a layer of ünear bias neurons whose outputs are sirnply the

sum of the inputs to the neuron plus a bias.

The training algorithm must set the centers of the RBF neurons, the biases of

the linear neurons, and the weights of the connections between the RBF and bias

layers based on the training data. These data are derived fiom the model to be

updated. The modei is altered by adjusting its parameters and the response of the

modei is calculated, be it modal data, FRFs, etc. Training data is organized in an

inverse configuration to the model, so for eadi data set to be used for trauiing the

neural network, the input is the caidated modei response and the output is the

adjustment to the parameters. In this way, the net learns (by tuning the weights)

what adjustments to make in the model so that it can match a given response-

Once learning converges, using the net to update the model involves presenting it

with measured data h m an adral experiment. The output from the net is then

the updated parameter set the model would need to reproduce this experimentai

data.

The major advantages of k g neural networks in model updating are that

they are very robust to noisy data, they sidestep the problem of having to match

the experimental measurement space with the rnodel coordinate space, and an ar-

bitrary parameter set can be assigned. ln the context of updating the DICE system

model, with the way the problem has been posed the latter two advantages are

already taken care of, and ftom an engineering standpoint, the sensors and test

equipment have been designed to be as noiseless as possible. The major disad-

vantage of n e d networks is the vast computationai cost, although most of this

is in calcuiating the model response. Given t h , neural networks were not further

pursued.

Simulated Annealing

A wideIy accepted enmerative optimization technique is the method of Simu-

Iated Annealung (SA). This method is analogous to another of nature's minimiza-

tion algorithms, that of thermodynamic cooling of a material. At a high tempera-

hm, the atoms in the material move relatively freely with respect to one another.

As the temperature drops, thermal mobiliîy is reduced and the atoms eventuaüy

lock into a hzen state. If the cooling pcocess is done gradually enough that ther-

mal equilibrium is virtually rnaintained, the material forms a perfect crystal where

the atoms are lined up in a lattice billions wide in ai i directions. This state repre-

sents the minimum en- of the system. If cooIing happens rapidly as in quench-

ing, the finai state is more amorphous and of higher enetgy. The iterative opti-

mization technicpes introduced in 54.7.1 are like the quenching example where

we greediiy go downhiii as far we can to quickly h d the nearest minimum. With

the gradual cooling approach we go downhill if we can, but sometimes go uphill.

Eventuaiiy we find the global minimum.

In 1877, Boltzmann describeci how fluids in thermal equilibrium behave as a

probabiiity distribution, aptly cded the Boltzmann distribution, where the proba-

biiity of the system being in a state s is @en by [41]

where S is the set of aii states, k is a constant of nature caiied the Boltzmann con-

stant which relates temperature to energy, T is the temperature, and E, is the en-

ergy of the system at state S. The numericd evaluation of the Boltzmann distribu-

tion is irrelevant here but it does suggest that a system in equrlibrium at T has its

energy probabilistically distributeci among al1 states. Even at a low temperature

there is a non-zero probabiIity of the system being in a high energy state. Thir is

what dows the annealing pmcess to 6nd the global minimum energy state rather

than getting stuck in a local r n i . u m .

In 1953, Metropolis [43] used this in an algorithm to describe how thermal equi-

libriurn is reached, the generai scheme of w M is the foiiowing. At an initial state

sl, if one randorniy selects a new state 3% the probability of accepting this new

state is

IF Es, < E,,, or m other words if the cost at the new point is Iess than the cost at the

oId pomt, the probability is p a t e r than unity so the new state is always accepted.

However, if Es2 > Es, it is sometimes accepted, more so when the temperature is

hi$. It is convenient to recast the acceptance criterion thus: accept the new state

where u is a unifody distributed random number between O and 1.

This procedure is cded the Metropolis aigorithm and desaibes how a system

reaches thermal equilibriurn at a temperature T. in 1983, Kirkpatridc et al. [44]

used the Metropolis aigorithm for optimization, The concept of scheduling the

temperature transitions was introduced and the SA algorithm was complete. The

overail algorithrn requires the foilowing:

b a configurable system (paramererized model),

a method of randomly picking new system states to try (parameret- sels),

L an energy hct ion to measure the ment of the state (cosrfinction), and

b a temperature parameter T and a scheduie for graduaiiy lowering it.

In schedding the ternperature transitions, one starts with a high temperature

and reduces it by discrete amounts as equilibrium is reached at each step. Even-

tuaüy new states are no longer accepted and the system has frozen, hopefuiiy at

the global minimum. Picking To and the scheduie requires some trial and error.

Although one can devise a scheme for deciding when a temperature transition

c m occur, it was found that iterating through the Metropoiis algorithm 1ûh to

1% times, where n is the number of parameters, was suffisent to mach qui-

librium at each temperature. T was Iowered by a factor p each transition, where

0.9 < p < 0.95. And finaiiy 20 to 30 temperature transitions were usudy enough,

although this can be monitored by Iooking at the cost.

The trickiest component in getüng SA to opîimize the cost function effectively

is the random step algorithm. Of four methods describeci in (411, two gave some

what promisùig results. in lme ad@ment, a parameter was picked at random and

changed to a new value randomly dishibuteci within its uncertainty range. The

other, normal a&sment, smiply added a n o d y distributed n-dimensional step

to the current state. This technique required an additional parameter: the vari-

ance of the step. This was picked as a fraction of the uncertainty range in each

dimension. The success of these schemes was spotty at best The main diffidty of

random walk schemes like these is that when local downhill moves exist, it almost

aiways picks an uphiU move. In narrow vaiieys and near the global minimum this

problem is most pronounced.

To ciraunvent this deficiency, an adaptation of the Downhiil Simplex algorithm

(discussed in $4.7.1) is suggested by Press et al. [38]. To merge the Metropoh ai-

gorithm with the simplex approach, a logarithmically distriiuted random nurn-

ber, proportional to T, is added to each vertex, and a sirnilady distributed number

is subrracred from each new point tried as a replacement vertex. As in the usual

Metropolis algorithm, aii true downhill states are accepted as well as some uphill

ones. The same strategies for moving the vertices are employed: reflections, ex-

pansions and contractions. When T + 0, the aigorithm converges to the d o w M

simplex algorithm. This method offers much better success in the application to

modei updating, and with much less triai and error. It is successfully used in some

of the runs presented in Chapter 5.

Chapter 5

Model Updating: Practice

Theoretical methods of extracting the desired information h m data is ortiy the

Erst step in updating the model. Many tessons are learned once the methods are

used on data h m reai experiments. In g e n d it was found in the present research

that model updating is more successful when the parameters being updated have

a dWct iink to the data used in calcdating the cost fundion. For example, in the

case of the DICE n i with a CMG actuator, ahhough the current Mt in îhe CMG

servo motor used to tilt the gimbai does dtimately affect the rib ddection, the n i

stifhess does so in a much more direct way, and in a way that the system modei is

far better at predicting. Consequentiy updating the rib stiffness parameter using

rib deflection data shows more pIausibIe and weli behaved parameter convergence

than updating the servo motor current M t using rib deflection data.

With this in mind, various subsystems are ideai candidates for sub-mode1 up-

dating. Using this approach, smaüer groups of mode1 parameters can be updated

using data or experùnents pertainnig to the particular subsystem. The overaii sys-

tem can then be updated by assembling the best estimates for the subsystem pa-

rameters in the overail modei, and updating whatever parameters remain to be

tuned. In some cases this Ieads to l e s parameter coupling and m general a much

d e r and more tractable optimization problern. In order to make this work the

subçystems must be naturally disjoint and the divisions made appropriateiy to

mitigate ermrs in patching the systems back together.

Two opportunities for subsystem model updating exist in the DICE rib system:

the suspension system and the CMG servo system. In the case of the suspension

system some of the parameters are weli suited for both the subsystem updating

and updating within the context of the overall rib system model, so this subsystem

updating is mainly an exercise in dernonstrating some of the charaderistics of the

approach. However, in the case of the CMG servo system, sub-mode1 updating is

necessary to adùeve plausible estimates for the servo parameters that are required

before forced response n i modei updating can proceed.

Certain settings, such as the regularization weight X and the search tolerance,

have to be established before model updating can successfuiiy be implemented.

These parameters tend to be s p d c to the model being updated, so are evaiu-

ated for each mb-mode1 updating problem separately, based on the methods in-

troduced in the previous chapter, 54.3 and $45.

5.1 Experimental Setup

The experimentai portion of this work makes use of a prototype DICE ni nibsys-

tem, shown in Figure 32, induding a CMG actuator (Figure 33) and a nullspring

prototype (Figures 2 3 to 2.7). The ni is a 0.125 inch diameter aluminum rod 18

inches in length, It is rigïdly constrained at the mot and free to osdate in both

lateral directions at the CMG end. Two different test conQurations are useci: the

vertical with the rïb hanging downward, and the horizontal with the CMG sus-

pended using the nullspring.

For the nulIspring modei updating nuis, the calibration data gathered during

the evaiuation phase of the suspension system (see 522) were used. The input to

the systern is the smaii step force, and the output is the free vertical oscillation of

the test mass as measured by the opticai sensor.

CHAPTER 5. MODEL UPDATING: PRACI1CE 62

The CMG uses a s m d commerciaiiy available servo motor and a siide linkage

to actuate the gimbal (Figure 3.4). The system nuis under cornputer control, which

issues position commands to the servo and sampIes the servo position and the

lateral (y and z axes) position of the n i tip using the optical rib deflection sensor.

In the CMG servo model updating, the input to the system is a commanded servo

position trajedory, and the output is the servo angle. In the overall rib system

updating, two input methods are used: step relaxation of the rib tip (and zero

input afte. the step event), and a commanded trajectory to the servo. The output

is the n i tip y and t deflections. The sampüng rate used for al1 testing was lQHz,

and generaiiy tests lasted 2 to 3 minutes.

5.2 Suspension Model Updating

A simulation model was built with five adjustable parameters: stiffness k,, vis-

cous damping coefficient d,, total mass (m, + payload mass), instant when the

step was applied t, and the magnitude of the step force f,t. Note that although

the model is hear, updating t, is analogous to updating a Iatency parameter -a

noniinear effect. To effectiveiy investigate the modei updating approach with real

data in a way that would be useful for more comprehensive m later, no prior

data conditionhg such as filtering (as was done during the nullspring calibraLion)

was employed.

Example cost contour plots in two parameters (Figure 5.1) indicate the chal-

lenges of using gradient descent optimization algorithms in this case. In the ks

vs. m, plot (left) coupling renders a narrow valley with an alniost flat Eiwr; in the

k, vs. dm plot (right), the different scales between the ordinate and abscissa illus-

trate the disparity in the sensitivities of the two parameters. However the srnooth

nature of the cost function in aii parameters for the nullspring evaluation data al-

lowed the Poweii algorithm to successfdy h d a minimum solution in aii trials.

Updating trials with varying X showed that a suitable value was on the order of

CHAFTER 5. MODE. UPDATING: PRACïïCE

Figure 5.1 : Example Cost Contours for the Nullspring Eduation Tests.

5.0 x 10'~; however, it was also found that the parameters converged nicely within

their uncertainty regions even with no regularization at all. in cases k e th& it is

preferable to suppress reguiarization so that the optimization can focus on residual

error reduction.

To compare the results of the model updating with those of the calibration in

§22, modei updating was fint used to estimate the parameters for the Light and

Heavy data separately. To h d m, the measured payioad m a s was subtracted

h m the converged total mas. Ideaiiy values for k,, dm and fa should be com-

mon for the two data sets, so a third model updating nin was performed using

the combined residual cost from both data sets, grouping the cornmon parame-

ters together. In the combined case the average nullspring mass is reported. Table

5.1 summarizes the remlts where 230 s of test data are used. In general gwd cor-

espondence in the parameter estirnates are found between MU and calibration

results. ûniy a consistent p-value for damping is elusive.

One r e m n damping was t r i e to identify was the disturbance environment

in the [ab. Recaii Figure 2.10 where some irreguianties in the data mdicated the

effects of air drafts in the Iaboratory where the tesis were conducteci. For un-

constrained experiments, the lighter the payload and the softer the nuilspring the

more the imperfections dominate- Darnping is also cüfkuit to accurately identify

Table 5.1: Nullspring Model Parameters Cornparison.

Model Update Tests

because it is obviously very low. Figure 52 illustrates the behavior of the p-values

Light 0.9148 0.00563 0.0643 3.861

0.0108

Calibration Tests

0.9055 0.9072

during an MU m. As the arnount of data used in the optimization is inaeased, the

0,00599 0.0647~

damping parameter decreases because the imperfections in the data which cause

Heavy 0.9015 0.00666 0.0633

4.063 0.00976

0.00419 0.0647~

damping over-estimation are averaged out. It takes much longer to reach a con-

Combined 0.9149 0.00626 0.0638 3.759 4.064 0.0103

verged state than for the other parameters, which converge almost immediately.

Making use of the more comprehensive damping and friction madel describeci

in 531 resdts in very poor parameter convergence. The damping i. so low and

viscous-like it makes sense to model it with a single simple viscous damping pa-

To iliustrate the effects of the sequential estimation appmch, Figure 53 shows

the comlation of the model data to the experimental data More and after updat-

hg. in the upper cuve where the modd assumes the initial parameter guesses,

after about 70 seconds the model is out of phase by more than 90" with the refer-

ence data, so starting an updating run with that much data could easily result in

hding a local minimum. To be d e , the sequentid estimation process was started

with 20 s of data. The middle cunre shows the cordation after 60 s of data have

b e n used to update the parameters. Notice how the deviation between the data

becomes noticeabIe only after 60 S. The lower m e represents a triai where d the

data is useci. We have confidence that the modei has been corredly updated be-

Figure 51: Nuilspring Model Updating: P-Value Convergence.

!nM P.value Set Heavy Paykad nmii

lm- P4iUue Set 6Oa of üatr

Figure 5.3: Nuiispring Mode1 Updating: Data Correlation.

CHAPTER 5. MODEL LIPDAZïlVG: PRACTICE 66

cause it correlates closely to the experimentai data and the model parameters are

within their uncertainty ranges.

in general, with the n u l l s p ~ g evaluation data it is found that the MU tech-

niques used so far show encouraging results. Damping is the moçt difûcuit pa-

rameter to pinpoint, but with extended data sets it converges. Regularization is

not necessary for this weii-posed example, and sequential estimation is success-

fui in avoiding local minima and demonstrating p-due convergence. The Powd

aigorithm, the gradient descent-based optimization technique, is suitable for this

example.

5.3 CMG Servo Model Updating

The CMG servo requires forced response experiments to proceed with updating

its model. The input to the system is the commanded servo angIe and the output

is the angle measurement from the potentiometer resolving circuit input signals

were generated by creating a unifonnly distri'buted random signal, filtering and

scaling the signal to a suitable range. The Eiiter limited the bandwidth to beIow

the Nyquist frequency and applied a first order roll-off within the pass band to

pinken the noise. This created a fiat response in girnbai rate which is proportiona1

to the gyric torque. These signals were also appropriate for the ri3 model updating

where it is desirable to avoid exciting higher modes in the structure, there being no

anti-aliaçing filters on the data acquisition system. Also prepared was a brief but

energetic bandlimited pulse, much Like a 'chirp' signai except based on a random

input rather than a sine sweep. Aithough this signal was not rich enough in its

spectral density for updating the model, it served to create additional data sets for

evaluating the pvalue solutions. All these bandlimited signals are inadequate for

exercising some of the saturation parameters in the servo systern so an additionai

signal featuring a series of steps was mated.

In total there are eight parameters available for updating, listeci in Table 5 1

Table 53: CMG Servo Update Parameters.

servo motor speed limit servo motor constant CMG actuator latency CMG gimbal bias servo propomonal gain servo integral gain servo derivative gain

Name ili,

Experimentation and sensitivity analysis showed that only some of the parame-

ters are suitable to include in the updating. The servo modei and its parameters

are more nonlinear than the rib dynamical mode[, and coupling between some of

the parameters is strong. For example, the motor constant Km sets the generai scale

of the dynamics in the system. Therefore this parameter couples strongiy with ilh,

u:,.,, Gp, Gi, and Gd. As was mentioned in $3.1, there are no manufacturer's spea-

fications for this black-box device, so the absolute scale of the system is unknown

and cannot be known given the operating regimes available for testing. The rd-

ative sa le between parameters is important because it deûnes where the motor

poles are and this can be tuned using model updating. S ice Km is so problem-

atic due to i b wide-spread coupling within the model, and since its abdute value

would have no physical meaning no matter how successfui the modei updating, it

makes no sense including it in the modei updating. By adjusting the other parame-

ters a suitable representation of the senro system can be derived. Mer parameters

not induded are Gi and wh because their sensitivities are extremeiy low compared

to the others and thus do not participate rneaningfdy.

Description servo motor current limit

Initial experiments with using the Poweü aigorithm to update the servo model

were disappointing. SeveraI surveys with single and multiple data sets showed

that by using a random distribution of starting points within the uncertainty ranges,

Figure 5.4: Sensitivity of Current tirnit Parameter.

very different optimal solutions with different residuai error levels were found.

Obviously the cost function is riddled with local minima. Another difLidty Pow-

ell has is in its quadratic assumption. 1ts conjugate direction set estimate, which

is iteratively built up, is skewed by the jagged nature of the cost mitivities, es-

pecidy in the c u m t M t case as depicted in Figure 5.4. AIso note in this figure

the aimost fiat cost relationship h m p-value 0.5 to 4. Obviously when the cur-

rent Mt is overestimated it no longer participates in the mode1 because the b i t

is never reached. The slight curve in the sensitivity is a result of regulanzation

which helps in keeping the parameter d o m in its effective range.

Simuiated Anneaiing (SA), with the Simplex walk algorithm, was thezefore em-

ployed instead of Powd. In one attempt this method was able to fhd a better

minimum than the best Powd solution out of 20 attempts. SA does not make use

of secpentiai estimation, so the algorithm be@ with the full length of data rather

than using graduaiiy longer data sequences. Computationally it takes more cost

Fundion evaiuations than PoweiI, but for the CMG servo updating case its d t s

are more guaranteed.

Figure 55 shows a typicai evolution of the pvalues and objective hmction cost

CHAPTER 5. MODE üPDATZlVG: PRACiïCE

10 20 m 40 50

Temperature Transition

Figure 55: CMG Servo Mode1 Updating: P-Value and Cost, SA.

Iniüal Response Correlation 0.1

a05 @ - O 0 - m C = 4.05

I " Y I

Updated Response Correlation a1

B nos e - 0 z 4.05

tirne (s)

Figure 5.6: CMG Servo Mode1 Updating: Response Correlation, SA.

Table 53: CMG Servo Mode1 Updating Cornparison.

aaoss the temperature transitions of an SA run using four data sets. The p-values

settie weii except for ilm which moves in its low sensitivity region even though the

Run Method Cl SA

0KIWER.A C2 SA

Powell

cost appears relatively converged. This parameter is the most difficult to idenûfy,

but obviously has little bearing on the residual cost, so is less important

Mode1 Update index, I,

Figure 5.6 shows the response correlation between the modei and the reference

RI-A R1-B R2-A R2-B Steps CP-A CP-B 55.9 (492) 46.1 46.1 -22.8 -23 -6.1 12.2 (10.0) 10.0 18.5 -56.1 -28.1 -8.4 53.7 (48.4) 53.1 (54.1) 12.0 26.5 25.0 53.5 (48.2) 53.2 (54.3) 4.6 21.4 19.6

data More and after updating. Although there is some improvement, the model

starts off in relativeiy good agreement with the experimentai data. This is expected

Avg. 23.7 -6.0 39.0 36.4

given that the servo is a closed loop tracking systern. Of note is that the forced re-

sponse n i systern updating, desaibed later in $5.4.3, would not converge without

this updated CMG servo model, even though the CMG model updating only cut

the residual error by half. Considering that the initial parameter guesses are very

speculative but result in low residuai error, there is LttIe chance the updated pa-

rarneters bear much physicai meaning. However, for the DICE updating this can

be tolerated because it is expected that the CMG will behave the same whether ori-

ented ve r t idy horizontally, or even in O-g, so on-orbit modei extrapolation of the

ovedi system can stiU be caxried out with these ground test servo modei updating

Table 5 3 presents a cornparbon of a number of updating and system identi-

fication runs for the CMG servo. The rows represent models resuiting fimm nrns

using different methods and data sets, and the columns are experimental data sets

used to evaluate the success of the m. 'RI' and 'Rî' are the random excitation

signais, 'Q' is the chirp signal, 'Steps' is the input si@ consisting of the series

of steps, and 'A' and '8' denote different data acquisition trials for the same input

signal. The (bradceted) numbers in the table indicate which data sets are used in

the creation of that model, so for example Run CI SA is a simulated anneaiing run

that uses oniy a singie reference data set, nameiy RI, trial B. The 'Mode1 Update

Index', or I,, is defined as

where EO,, is the rms mode1 emr using the initial guess p-values as c o m p d to

the reference data set, and E,, is the mode1 error after updating (or of the iden-

tified rnodel in the ERA cases). In essence, the higher I , is the more effective the

updating is with respect to that refetence data set, to a maximum of 100°h, where

the new model matches the experimentd reference data perfectly. Generally 1, is

higher for data sets that were used in the updating. A negative 1, indicates that

the new mode1 is worse than the initiai guess mode1 with respect to that reference

data set. Note that 'cost Fundion' and 'm error' are d i f f m t Cost, used directiy

in the optimizer, is the squared enor nom between the ceference and model data

plus the weighted regularization cost, whiie the rms error is not squared, only in-

dudes the residual, and is used only as a measure of success after the model is

updated.

In all thme nms simulateci annealing performed the best of aii methods used.

In Run Cl, where oniy one data set was used, SA had trouble generaliang to the

Steps data but worked w d in C2 where two data sets were used, even though

Steps was not one of them. The Powd method m C2 appears to have worked

favorably, but r d th& is the best out of 20 attempts and does not perfom as

w& as the SA method whose first try is reported. Run C3 SA gave the best o v d

d t s .

A linear updating triai is presented for cornparison (C3, SA/lin). This was

carried out by iinearizing the simuiator and omitting the nonlinear parameters.

The run gives acceptable I , values though not as good as the noniinear run, but the

resulting parameters are not in keeping with the nonlinear results. in parti&,

the aduator iatency has a large effect in the noniinear results and is not utduded in

the iinear. Therefore, iinear parameters mch as the control gains try to compensate

making the servo dynamics artificidy slow.

The OKID/ERA results, generated by researdi coiiaborator Dr. R Bauer, rep

ment identifieci iinear state-space modeis of the system. The method uses OKID

to identify the Markov parameters of the system (linear impulse response), and

ERA to find the best state-space mode1 based on a Ieast squares fit. These results

dernonstrate the limitations of this method for such nonlinear systems. There is an

extension to OKTD/ERA caiied Augmented SI which accounts for known noniin-

ear effects, but that was not incorporated in this work [45].

A combination of SA and the Poweii methods was also investigated. The main

advantage here would be to Save the computationai expense of h e tuning the

minimum using SA, resorting to the more rapid Poweii method when the region

of the global minimum is identified. The difficuity Les in diagnosing when to

switch methods. Once the cost has more or less converged to a minimum using

SA, it is found that the Powell algorithm does little to improve the soiution and

requires rnany cost caldations in its attempt Stopping the SA algorithm earker

is risky not knowing if the global minimum is near at hand. Not surprisingly the

very en-ment which requVes the use of SA in the first place precludes the use

of the Powell algorithm as a cornpanion-

Conclusions that can be drawn h m the CMG servo updating are the foilowing:

b The Poweii algorithm wu, with considerable coaxing, generate a reasonabie

solution in this case, but SA performs the task much more readiIy and with

less rehearsai.

b Using a nonlinear mode1 and parameters is benefiaal to the updating in pro-

viding better residuaï reduction and a more physically meaningful solution.

b Model updating outperforms OKID/ERA in this CMG servo case.

The best performing updated parameter set, that of Run C3 SA in Table 5.3, was

then used in the rib system madel updating m, where they were not hrther

adjusted,

5.4 Rib Model Updating

There were experimental data h m two configurations available for updating the

overd rib system model: vertical and horizontal. The mode1 accounted for the

differences in these configurations: a tende axial stiffening load due to gravity in

the vertical codigura tion, and the suspension effects (including penduiar stiffness)

in the horizontal. Upda ting could use data from one or the other configura lion or

both sirnultaneously, or mdtiple data sets from each. This section sununarizes

results h m extensive experimentation with MU applied to the rib system.

The same bandlimited random exci ta tion signals tha t were introduced with the

CMG servo mode1 updating (55.3) were a h used for MU in the overd rib systern.

In addition, relaxation testing was done by dispiacing and carelully rdeasing the

ni tip, allowing it to osdate freely. The Latter method did not employ the CMG al-

though the flywheel was spinning. These step relaxation tests were used to isolate

the updating probtern to the rib parameters spe&caiiy before CMG servo updat-

ing had been performed. The experimental data were also integrated into multiple

data set nuis with the forced response data.

The parameters pidwl for the nt system qdating, listed in Table 5.4, were a

srnaii subset of the total set of paramet& constants in the modei üsted in Table

3.1. The cost function was practkally msensitive to many of the parameters within

th& uncertainty intervais. %me, such as the individual second moment i n d

Table 5.4: Rib System Update Parameters.

total CMG mass z-component, n i tip to CMG C O S vector z-component, n i tip to motor ass. CofG vector fiywheei spin momenturn rib Young's ModuIus y-bending mode 1 damping z-bendiig mode 1 damping suspension secant stifiess step event time ofiet (step relaxation ody) suspension system mass

parameters for the CMG and motor assembly featured singuiarities. Evidently the

experimental data were sdicient for updating the more dominant effects such as

overd mas, but not for the more minute details in the model. The updating ef-

forts only cansidered the y and z bending modes of the system; higher resonant

Frequenaes were above the sampling rate of 10Hz. Some parameters were cou-

pIed in unfavorable ways like the z-axis modal damping and suspension system

damping in the horizontal configuration tests. These two were directiy inversely

coupled and therefore could not be independentiy updated, so the average sus-

pension damping estimate from the evduation tesk and nullspring updating was

used and not included in any M e r updating. Three types of updating were

investigated: updating with simulatecl data, updating using real step relaxation

data, and forced response updating using real data.

It shotdd be mentioned that, in the rib response correlation pIok in this b p -

ter, oniy the y data are shown because the z mvariably showed the same general

trends. Data h m both axes are used in the nns ermr calculation. Also, in some

of the cost plots, the duration used in the caldation i. given m the abscissa la-

beL This was the duration used for the generation of the plot, as disîind from the

duration of the data sequences used in the updating.

Data Sequenœ Duration (s)

Figure 5.7: Simulated Updating: Siml P-Values and Cost.

5.4.1 Simulated Rib Mode1 Updating

A series of sirnulated updating runs was conducteci to observe the performance of

the modd updating dgorithm and speQficaily to answer two questions: if nonlin-

ear effects were more dominant m the response, would nonlinear updating per-

form better than linearized updating? And, Do there exkt output-equivaient solu-

tions for the parametric model? An input signai was generated by multiplying one

of the band-Limited random signals used in the actual tests by a factor of four. The

sensor field of view and Ch4G gimbal angIe iimits were disabfed in the simdator

and a hown parameter perturbation p, was introduced, The system response

was simulateci for this ph, in the vertical and horizontal configurations and the

data saved in the same file format as aduai experimentaI data. These simulateci

data were used as derence data sets for MU nuis, whose objective was to identify

h u e -

Initiai Y Response Conelaiion 0.m i

trme (s)

Figure 5.8: Simulated Updating: Siml Response Correlation.

Using the nonlinear model in the vertical configuration only, and with the regu-

larization weight X set to zero, the Powell algorithm fond Pahl (Table 55). P-value

convergence was achieved a h using oniy a s m d amount of data; in fact no ap-

preaable change in p or the cost occurred after using only 9 seconds of data, as

shown in Figure 5.7. Looking at the appeamnce of the response correlation before

and after the updating, the plots in Figure SS show the model output and refer-

ence data together, the top showhg the cordation before model updating and the

bottom after. It is interesthg that even though only 25s of data were used in the

final trial of this updating run, the model response shows very good correlation

with the reference data up to the full 60s shown on the plot, even though p,,

did not converge exactiy to pm. From the perspective of the optimization, this

solution is a perfectiy valid minimizing p-vdue set, In this case as the Line in pa-

rameter space h m p-1 ta is traverseci the cost does reduce overaii but onty

on the sale of the cost ca id t ion error so the trend is masked in no*. One can

condude h m this that to the toIerance tu whïch the cost fimction can be dm-

lated, there exisfs a manifoId which indudes ppt, and p,,~ on which the madel is

Table 5.5: Simulated Updating P-Value Results.

(Powell) 0.83954 0.93338 1.16190 1 .O7926 0.78475 -

0.20295 -

(iinear) 0.84 154 0.83935 1.231 19 0.90408 0.53600 -

0.17384 -

(multi) 0.78740 0.88828 1.08133 1.10432 0.80765 0.90227 0.20032 0.19878

km 0.95 - - 0.90826 - mis emr 2.15~ 1.74~ 3.35~ 4.97~ IO-^ 1.10~ IO-'' 5.41 x IO-"

virtudy output-equivalent.

When regularization is used to encourage the fimi solution towanl p, using

the optimal value of X as derived using the L-curve technique, p , ~ is found. This

p-value set is very dose to p, because it was known prior to updating and used

for centering the parameter uncertainties, so in this sense regularization is some-

thing of an artificiai way to adiieve the proper minimum. It does demonstrate,

however, that with prior knowledge of the likely solution and the relative uncer-

tainties, regularization can be used effectively to ûnd a physicaüy meaningful op-

timal solution. Again, because of the nature of the manifold and the caldation

noise, the exad solution is not found, but neither is the rms error signrficantly dif-

When a linearized version of the model is updated, the identiûed modei pararn-

eters p~ are a littie further h m than the parameters derived h m nonlinear

MU, PM. Ftrrthermore, the parameters take Ionger to settIe and the cost takes

longer to converge (Figure 59). Looking at a plot of the residual error before and

after updating the iinear model @gure 5.10) one can see that after updatmg with

25 s of data the residual is low h m O s to 25 s but higher after 25 s, whereas for the

Figure 5.9: Simulated Linear Updating: Sim3 P-VaIue and Cost.

t 0 1.5 - O

1

a - m 0.5

S O

Initial Y Rasidual

- O 5 1D 15 M

Data Sequem bomtion (s)

Figure 5.10: Simdated Lmeat Updating: Sim3 Residual Error.

Final Y Residual

- 0.04

E o.m - 5 0

4.02 ei

-0.04

Final RMS h 0.00496898

0-

O 10 30 60

üme (s)

Relative Noise Levd

Figure 5.1 1: Simulated Rib Updating: Noise Resilience.

noniinear case the final tesiduai error is virtually zero for all the data. This indi-

cates that the nonlinear modei updating is better able to generalize the model to

data not used in the updating because the model is more appropriate for the data.

in the multi-configuration case, simulateci data from vertical and horizontal

conûguration tests are used sinidtaneously to update the nonlinear model (Sim4).

Here the solution p,& is much doser to p, than p,, indicating that using multi-

ple conûgurations reduces the output-equivalent manifold.

Finally, in test Sim& simdated annealing with the simplex walk algorithm is

used. This approach found the solution with the lowest error. However, p a is

not significantly closer to pi, than p-1, giving further evidence to the presence

of output-ecpivalent solutions. The aigorithm muid have been anowed to work

away until p, was found, but had it been unknown, one wodd have stopped it

earlier when the error converged near the caiculation tolerance of the system.

Robustness to noise is a property that my algorithm making use of real experi-

mental data must have if it is to be of practicai use. For this study the assurnptions

are that noise in the system is additive, zero mean, constant variance, uncorrelated,

and Gaussian. The Poweii algorithm, with its strong dependency on the shape of

the cost function, is far more suspectable to noise, but even so it is found to be

very noise robust. Figure 5.11 shows p-value behavior of simulated updating nuis

based on the Poweii algorithm where the reference data are subject to an increasing

intensity of random Gaussian noise. This behavior shows some variation in p- as

the noise is increased, but not to the point where the solution does not converge

to, or at least cesemble, pm. For the final triai the rms noise level is over 20% that

of the reference signal, representing a much higher level than would be present in

most practicd experiments. The Iower plot in Figure 5.11 shows the relative e m r

in p,i, as compared to p, for Poweii runs using two different amounts of data. By

and large, when more data are used in the updating, the error due to noise is Iess-

From the simuiated modei updating examples cited, these main condusions

can be drawn:

b the solution being sought is not unique,

b nonhear updating does heip to some degree,

b multiple configurations of data and regularization promote uniqueness in

the solution, and

b the method is resiiient to noisy experimental data.

Outputsquivalent solutions are a major concern in MU. Knowing they ewt, one

has no idea how physicaiiy meaningful the optimizing sdution is without making

use of the uncertainty in the initiai parameter estimates. Fortunately cegulariza-

tion provides a technique for domg just that Furthemore, the output-eqirivaience

manifold is reduced by using muItipIe test configurations and data sets in the u p

dating.

It is no surprise that a diversity of tests Ieads to better estimates of the modei.

Two other identification techniques in the iiterature make use of the same concept.

The Perturbed Boundary Condition (PBC) method [46] is a modal identification

scheme where a structure is dynamically tested with altered boundary stiffness

and mas. By accounting for these perturbations in the model, they can be re-

moved to make several estimates of the unperturbed modes. S i a r l y , in Multiple

Boundary Condition Testç [47l, many separate portions of a very large

flexible structure are physically isolated using rigid supports, and tested. The re-

sulting corrected FE modeIs d the separate portions are patched together into a

model of the overall structure, and the boundary conditions are removed to pre-

dict the O-g modes.

In the present MU method, with the help of uncertahty-weighted regdariza-

Lion and multiple test configurations, problems associated with output-equivalence

and solution uniqueness are significantly reduced. Obviously one will still not

have 100% certainty in any solution coming h m the model updating proces.

What one will have is bener certainty in the model parameters and a model that

bertet predicts the experimental data.

5.4.2 Step Relaxation Rib Model Updating

ûynanucal testing and model updating of the rib systern without the CMG king

used to actively mate the system but wiih the flywhed spinning offers the oppor-

tunity to isolate the mode1 updating to the flexible gyric portions of the system.

&O, if only the y and z first bending modes are exated, this creates very clean

data for the main modal components of the rib motion. This is an important step

toward the full system updating.

These experiments offer some unique challenges. Tests were performed on the

n i system in both the vertical (with gravity ioad) and horizontal (with the suspen-

sion system) configurations. By restricting the initiai deflectïon to the 6rst bending

modes, caldation of the initial model state (in modal space) was possible, since

there was only a single sensor location for the rib defkction. In the cost function

calculation the model can then be propagated based on this initial state and a zero

input Experiments were conducted by attaching a very light nyIon shing to the

CMG near its center of mass and deforming the ni to a suitably Iarge excursion

by pullUig on th& string. The shape of the rib in this initial state approximates the

shape of the k t laterd bending mode. The end of the string was fastened rigidly

to a ground base, the n i allowed to come to rest, and a hot soldering iron used to

@&y melt ttuough the nylon, aiiowing the rib to swing free and osciliate.

One difficulty in using test data from the step relaxation tests lies in timing

the step event. In the interest of obtaining experimental data with the minimum

of hardware modification, the step event was introduced manualIy rather than

synchronized using the data acquisition computer. Therefore the step occurred

some lime between two data points, inaeasing the complexity in estimakg the

initial conditions at the h s t data point after the step. This pmblem was solved

using MU.

Two possible methods come to rnind. One could use the model updating pro-

cedure to search Eor appropriate velocities at the k t data point after the event.

This requires two parameters: one per coordinate. Alternatively, one couId search

for the offset p e d d r , ~ between the event and the k t data point This requires

ody one parameter. This second approach was irnpIemented successfully.

It was found that the Powell algorithm worked very well for these rum. Ran-

domiy pIaced starts redted in alI optima1 solutions being grouped very closdy

together. Figure 5.12 shows the p -due and cost convergence for a nin using ver-

tical configuration data, whiIe Eigure 5.13 shows the resulting model condation.

The horizontai test had similar trends. These resuits are encouraging h m an op-

timization perspective in that the residuai error was effectively reduced.

However, doser study of the resulting pvalues, as listed m Table 5.6, mdicate

that as was found in the simulated ni model updating, seemingiy good residual

0.035 ' J 2 4 6 6 10 12 ?8 16

Data Sequence Duration (s)

Figure 5.12: Step ReIaxation Updating: Vertical Test P-Value and Cost.

Initial Y Response Carrelation 0.015,

Updated Y Response CornLation - 0.015 E - a01 - 5 am5 E O 8 4.00s - % 4.M " 4.015

O 5 10 15

tirne (s)

Figure 5-13: Step Relaxation Updating: Vdcai Test R q o n s e Correlation.

Parameter

Table 5.6: Step Relaxation P-Value Results.

Separate Tests Vertical Horizontal O. 1667 0.9825 0.8215 0.8919 13338 1.0019 0.9984 1 .O002 0.9650 1.0019 0.3222 0.355 1 0.0466 0.0386

0.9998

Multi-Codïg. Vertical Horizontal 0.166? 0.9825'

0.7138 0.7728 1 .O062 0.33 12

O. 1957 0.2884 0.1957 3.2884

0.6563 1.2694

6.36~ IO-" 1.38~ IO-=

t~tationary for multi-config tests

redution does not necessady mean that the p-values found are fuily consistent

For example, it is not expected that the modal damping ratios should be as dif-

ferent between the y and z axes as was found in both the (separate) vertical and

horizonta1 m. The cabling running out the ri3 to the CMG will cause some dif-

ference, but not likely to the extent observed. More will be said about estirnating

the damping in $5.44, but for the multi-configuration test employing both verti-

cal and horizontal data sets simultaneously, the y and z damping ratio parameters

were lumped together (i.e., updated as a single parameter) for each configuration.

It was found that lurnping the damping parameters had very Little effect on the

other nondamping parameters in the model. The p-value set found for the multi-

configuration m was again considerably different h m the separate vertical and

horizontal m. Retall that the lesson iearned from the simuiated updating results

was that the rnulti-co~guration nms will generate more physidy meaningfuI

results-

Aithough it was found that the Powd aigorithm anived at the same optimal

solution h m many starting pomts within the uncertainty range of the parameter

space, the problem codd very weii be iii-posed if a non-unique solution manifold

exists, and through happenstance different data sets favor different solutions. This

was suggested in the simulateci updating resuits. However, it should be kept in

mind that a defective model basis would also explain these shortcomings. This

was not the case in the simulateci updating, where by defmition the madei i s a

perfed match.

In an attempt to isolate any incorrect asmmptions or poor modeling choices

in t h original simdator, extensive experimentation (numerical and physical) was

conducted. This experimentation focused on the areas of the mode1 which are

different between the configurations (such as the suspension eEfects and the rib

geornetric stihess), and on assumptions of commonality between the configura-

tions (such as flywheel spin rate). Also investigated more dosely was the choice

of parameters, so again the sensitivities of all the model parameters were closely

studied. Aithough rnodei dektiveness and parameter choice cm never be nùed

out, it is likdy that the discrepanaes seen in the p-value sets are a cesult of the

non-uniqueness of the optimal solution as previously seen in the sùnulated model

upda ting.

For now, these resuits show the success of the Powell optimization algorithm.

More physically meaningful p-value sets wiii resuit h m using more data, nameiy

the forced response experimental data, as desaibed in the next section (95.43).

There, the step relaxation data sets will be used in conjunction with the forced

response data, adding more diversity to the resource of reference data. Tt should

aiso be noted that already h4U has gone further than the OKID/ERA style of SI

by offering resuits that can be mtinized against physicd meaning rather than

just enjoying the impressive residuai enor reduction and then calling an end to the

investigation,

5.4.3 Forced Response Rib Model Updating

Runs where the overall rib model is updated in separate and combined contigu-

rations are the most conclusive in dernonstrating the practical benefits of the MU

scheme. Sigle configuration updating results are also compared with direct SI

models generated using the OKID/ERA method. Table 5.7 summarizes the n i u p

dating results concisely. As in Table 53 in the previous section, for each nui, the

data sets used (4) and the level of success in reducing the nns error is indicated.

The experimental reference data sets, available in the vertical and horizontal con-

figurations, are 'RI' and 'R2' (random excitation), '8' (quasi-chirp exatation), and

'EK' (step relaxation tests desaibed in 55-42). To reduce the dazzling complexity

of the table, oniy the average Model Update index I, (defined in Equation 5.1)

for each configuration is reporteci, that being suf£iaent to desaibe the trends. Ad-

ditionally the tabIe reports the 'pe Error' which is defined as the Eudidean dis-

tance in parameter space that the resulting p,, is from a reference solution, taken

as what the author beiieves to be the most physicaiiy meaningful noniinear MU

solution found, that of Run 9 Poweii. By studying this error, one has a measure of

how meanin~ul the solution is; the lower the value, the better. This is distinct from

I, which measures how effective p, is when used with the model to predict the

physicai experimenlal response.

Runs 1,2 and 3 are a recap of the step relaxation results from 55.42 The main

üend obvious h m these runs is that using oniy a single configuration of exper-

imental data, the solution does not generalize weil to the other configuration. in

Run 1 for instance, using only vertical data resulted in a better vertical modei but

a worse horizontal one. In Run 3, which uses a combination of data hom both

con6gurations, I, is favorable for both configurations and the solution is more

physically meanin@

For singIe configuration tests based on forced response data, Runs 4 to 7, MU

using Poweil and SA is compared to direct ÇI using OKID/ERA. For each nm each

Table 5.7: Forced Response Updating Cornparison.

Data Sc Vertical

PoweIi

PoweIl

t~eference Solution

ts Used -- Horizontal Average I , Pmin

RI R2 CF RL Vert. Horiz. Error

4 PowelI SA ERA

5 Powell SA

ERA 6 Poweii

SA ERA

7 Powell SA

ERA

J J J J J J J J J J J J J J

J J J J J J J J J

J

64.72 -27.28 65.90 -29.41 68.65 -20.33 68.37 -35.19 69.17 -35.17 65.72 -28.06 -31.97 64.56 -31.28 63.71 -66.25 62.1 1 -28.40 63.12 -37.61 62.66 -78.20 57.19 61.85 62.28 70.78 63.92 71.23 66.75

0.857 1.040 d a 0.914 0.966 d a 0.857 0.872 da 1.015 0.902 da 0.385 0.000' 0.272

of the three methodç uses exactly the same arnount of data from the same physical

experiments. Here one can see that MU using either method performs on par with

ERA in terms of residual error reduction. Aiso, the lack of cross configuration

generaiization is further emphasized. It is immediately obvious that neither ERA

nor MU are successful at predicting the mode[ of one configuration using only

data from the other. For ERA this is expected. However, for MU it was expected

that this Limitation wodd be overcome. These red ts are further evidence that

the solution is non-unique, as was identified as a possibiiity from the simulated

updating experiments desaibed in $5.4.1. In those experiments, because the tme

p-value set was known, reguiarization could be used to draw the solution to it.

in real tests this luxury does not exist. Uncertainty intervals are set to capture

the likely p-value range, but if non-unique solutions exist weli within that range,

regularization cannot be w d to distinguish them reliably. Therefore, we resort

to a diversity of test conhgurations to reduce the ambiguity in the optimization

problem.

AIso evident in the cornparison between the Poweii and SA methods for Runs

4 to 7 is that neither method is a dear winner in tenns of residual error reduction.

The average I, for the configuration h m which the derence data was drawn

for each run was within a few percent between the two methods. Because the

Powell method was more economid in computation, it was chosen for the Iarger

runs involving more data. This is in conûast to the CMG servo model updating

where SA was the better pehnner due to the noisy and coupied nature of the cost

function.

In the MU nms, ciifferences in the structure of the model exist when differ-

ent model configurations are used. For example, geomeûic &es is added to

the ri% to account for t d e stress for the vertical experiments, and suspension

systern effects are inciuded for the horizontai. In this way the model generaIizes

h m one configuration to the other, and theoreticaily mode1 parameters identified

using data from one configuration should be valid for both configurations simui-

taneously. This cm not be done using OKID/ERA because it is not referenced to

a physicaliy based model. if one uses ERA with combined data sets from both

configurations the algorithni will not converge weii on a low order model that re-

sembles either configuration Therefore, it is not possible to compare ERA with

MU in multiple configurations.

The multi-configuration tests (Runs 3,8 and 9) al1 show lower p,, ertor than

the single configuration tests. In k t , most of the error is in the damping and

suspension pararneters which have reiatively low sensitivîty so were difficuit to

idenûfy. An instructive comparison can be made between Runs 1,2,4 and 6 (where

four different data sets are used separately), Runs 3 and 8 (where the same data sets

are used in multi-configuration pairs), and findiy Run 9 (where aii four sets are

used together). Figure 5.14 graphidy depicts the minimiPng p-values using the

nodinear Powd rnethod for each nui. in the four separate runs, the p-values are

quite disparate although as listed in Table 5.7 each yields a good I , with respect

to its own configuration Runs 3 and 8, although not sharing common data sets,

yield minimizing p-value sets which are much more similar, especiaiiy in the first

five pararneters which dominate in sensitivity. The pvalues h m these also agree

well with those of the ail-encompassing Run 9.

Figure 5.15 shows the Run 9 (Poweii) response condation for step relaxation

data in the verticai test configuration. The k t 16 s of data are plotted so one

c m compare with Figure 5.13 (i5.42) which shows response correlation for Run

1 where only the vertical step relaxation data are used. Obviously in the mdti-

configuration case the response cordation is not quite as good. However, the

overail performance in generaking to both conügurations as illustrated by I, is

far better.

Figures 5.16 and 5.17 illustrate the MU Run 9 Powell results. Depicting p-value

convergence and R1 verticai response correlation respectively these plots show

Figure 5.14: MU Using Various Data Sets: Comparing P-Values.

Initial Y Response Carrelation 1

CI

E 0.01 - g 0 . w

E O 3 rn 4.005 2 e o r u

tirne (s)

Figure 5.15: Multi-Cou.ûguration Updating: Run 9 RL Vertical Response.

Data Sequence Duration (s)

Figure 5-16: Multi-Configuration Updating: Parsuneter and Cos Convergence.

Initial Y Response Correlation O.Ot5i

time (s)

Figure 5.17: MuIti-Coufiguration Updating: Run 9 RI Vertical Response.

CHAPTER 5. MODEL UPDATING: PRACT7CE 92

how a typical successful model updating run looks. The rms error in forced re-

sponse data sets is usually larger than in step relaxation data, even though forced

response data are better for model identification because the full measurement

bandwidth can be exercised.

Model updating is constrained in that the model structure is set From the outset

and a very reduced number of parameters are adjusted. The model may or may

not include some nonlinearities. Direct SI methods like ERA are not constrained

in that they search the set of all linear models less than a maximum order. To es-

tablish how much difference the nonlinearities in the MU tests make, Run 9 was

executed on a linearized version of the simulator. CMG servo parameters from

a linearized servo updating run were incorporated, and the resulting rib model

actually had higher I , values (Table 5.7) than the nonlinear case, although the re-

sults are dose enough to be considered roughly equivalent performance. This is

consistent with the observation that, in terms of optimization, ERA, where it could

be applied, performed comparably with nonlinear MU. Evidently for the typical

levels of r i i deflection the dynamics are almost linear and can be well represented

in a physically meaningful linear model. Even though the Linearized servo updat-

ing discussed in $5.3 did not perform as well as the nonlinear, this did not detract

from the rib system updating.

Looking at the initial and updated system response in the frequency domain,

Figure 5.18 shows Frequency Response Function (FRF) correlation for Run 9 before

and after updating. These FRFs are averaged using horizontal y data from two

different random inputs (R1 and R2). In each of the two graphs in the figure, the

upper trace is the model response and the lower trace is the actual experimental

data, separated by two decades for clarity. Before updating, the general shape

and roU-off of the model are somewhat reminixent of the experimental reference

data but the resonant peaks are not well reproduced. However, for the updated

modd, many of the detailed features are virtually duplicated and the pIacement

Figure 5.18: Frequency Response Comlatian.

and width of the resonant peaks are very wd matched. The correlation is quite

accurate up to 2 Hz, or 20% of the sampling rate. This bandwidth performance is

ais0 typicd of direct SI mettiods such as OKZD/ERA [3]. Note that the kquency

separation of the peaks is a d t of the gyndty in the system.

Janter states in [35] that in mat practicd MLT experiments, an inaeased cor-

relation of a model in the test hquency band wilI result in reduced correlation

above it. Therefore, MU must succeçsfully span the frequency band in which the

model will be used -nmely the proposed control bandwidth for the system - if the model is to be used to design conmuerS. In the case of Run 9, one can assume

that the modei could be used to design a control system suitable up to 2K2, but

above that performance can be expected to drop off sigrdïcantly.

Run Method 1 Powell 2 Powell 3 Powell 4 Powell

SA 5 Poweii

SA 6 Powell

SA 7 Powell

SA 8 Powell . - - -

9 Powell

Table 5.8: CPU Tirne. Cost Evaluations Total C'PU (min.)

6586 414.9 9976 610.8 12899 1361.4 15623 2 194.3 1 1250 4125.0 14242 3634.7 15000 10500.0 18126 2769.7 17500 6416.7 17333 5 166.9 2 1000 14700.0 21 190 6597.3 15872 5140.8

One area where OKID/ERA outperfonns MU hands down is in computational

effiaenq Table 5.8 iists the CPü times consumed for the MU nuis studied in th&

section as observed using a 600MHz Pentiurn II procesor. The vast majority of

the computational effort was in simulating the mode1 repeatedly. in the Poweii

nuis, which used sequential estimation, most of the cost evaluations were in the

eariy stages of the run with shorter simulations, so were far more efficient than the

simulateci annealing nuis, where aU simulations spanned the entire duration of

the experimental data used. The amount of B U time inaeased with the number

of data sets used and the number of parameters included in the updating The

OKID/ERA nuis reporteci in this section al1 took about 1 to 2 minutes of CPU time

on the same ciass and speed of processot, while the Powd and SA nuis with which

they were compared took as much as 35 and 1 0 2 days respectively. The Iongest

Powell nui, induciing four data sets, nine parameters and a maximum of 40 s of

refefence data took 4.6 days. NeedIess to say, a simulateci anneaiing run was not

conducted for this run! Obviously the MU techniqyes are not suitabIe for the on-

orbit controIler redesign phase of the DICE experiment, where tumaround time is

a aiticai factor.

The final use of the updated model derived h m this entire exercise is to assist

in designing controIiers for the on-orbit systern. 7'herefore the mode1 needs to

be projected from the 1% case to O-g. By remaining m the physical coordinate

space this is easy to accomplish. The gravity-reiated parts of the model are sirnply

rernoved (axial loading in the rib and the suspension system effeds). The only

wrinkie is in projecting the modal damping estimates, discussed below in $5.4.4.

From the rib system MU experiments severai condusions can be drawn:

Multiple configuration nuis produce a more physicaiiy meaningful solution

while single configuration nins appear to suffer h m non-uniqueness.

b Even though MU works well in cases where nonlinearities are not insignif-

icant, nonlinear MU is not necessary for the actud DICE rib system where

the dyriamics at typical defiection levelç can be described well with a linear

model.

b In the frequency dornain, MU is accurate up to appmximately 20% of the

sampiing rate, which iç similar to the performance of direct Si methods such

as OKID/ERA*

b For the rib system, Powell and simulateci annealing optimize with equal ef-

fectiveness, while Powell is the more efficient.

b Staying in the physical cwrdinate space with the MU techniques presented

herein, as opposed the using direct SI and thereby d e r i n g the loss of the

physical coordinates, cos& three to four orders of magnitude m computa-

tional expense.

5.4.4 Estimating Damping Parameters

From the initial investigations through to the final M model updating, damping

parameters proved the most reiuctant to converge. The subtlety m identifying m-

dividual damping parameters for each mode is lost in the practicalities. In the

I 5 10 15 20

Temperature Transition

Figure 5.19: Damping Parameter Oscillation.

nb system, gyric cross-couphg of the y and t motion, cornbined with the low

damping IeveIs, made them difficult to distinguish between. in the SA run shown

in Figure 5.19 for example, the darnping parameters osda te inversdy with littie

sign of convergence or irnprovement to the cost. One damping parameter actuaiiy

dips negative on occasion which is certainly cause for doubt in the cesults. Legiti-

mate uncertainty in these parameters prevents regularization h mitigating this

problem without artificiaiiy increasing A, which biases the overd solution. The to-

tal amount of damping, which is dictated by the general decay of the motion over

several periods, converges, but the distriiution between the axes does not

More stabIe damping estimates were found by Iumping the parameters from

the two axes together, s a d c i n g the more comprehensive analysis in favor of sim-

pliaty. This assumption is based on the observation that the damping shouid be

appmximately the same a m s the y and r axes, the structure being symrnetnc in

that manner. Figure 5.16 iiiustrates th& the two damping parameters G and < ~ ( h ) ,

which for this nin represent the combined y and z damping ratios in the verti-

cal and horizontal configurations respective@ convexged as smoothly as the other

parameters, and at very simiIar values.

It was originally argued in Chapter 3 that the simple linear viscous damping

model was Che most appropriate way to modei damping in the flexiiIe rib and

the suspension system. One might revisit that argument in light of the ciiffiad-

ties in estimatirtg these parameters. Other linear damping modeis exist and could

have been used h m the outset, one exampie bbeing the Golla-Hughes-McTavish

(GHM) method [a]. It is ideal for modehg viscoeIasticity in space structures.

The method incorporates viscoektic finite elements which a~ derived from stan-

dard structurai elements and can be tuned to reproduce measured modal frequen-

cies and shapes over a wide bandwidth very accurately. However the problems

associated in the damphg estimates herein are more k l y the result of the very

low damping levels; identifjing parameters which are inherentiy srnali using non-

ided data is problematic, and is hdher complicated by the gyric coupling. It is

certainly not an issue of bandwidth because the DICE rib mode1 updating effort

focused on a limiteci £requency range, encompassing only the first bendimg modes

of the stnicture.

The other diallenge that darnphg offered was in projecting the O-g damping

estimates from the updated 1% model. Is damping higher or lower in space?

Many space structures with deployed appendages indude mechanisms with ei-

ther hinged joints or joints that are assumed rigid but in hct are not. When testing

Ïs done on the ground, the structure is loaded h m the gravitational weight of

the components, so fiction tends to lock up the joints. This reduces the apparent

damping of the structure. En space these Ioads are not present so the joints are

free to move and rub, causing friction and kinetic energy dissipation, or eEectiveIy

higher damping than observeci in ground based tests. However with the DICE

structure the ribs are designed to be cioser to the ided case. They are very rigidly

clamped to the bus structure and there are no articulated joints. In the simulator

damping is modeled as a ratio of the damping factor to the criticai damping fac-

tor, or modal damping. As the modal frequencies change between the ground and

onsrbit configurations, the modal damping ratios will diange even though the

mechanism and magnitude by whidi damping occurs do not.

To illustrate tius consider the DICE rib system that was tested as a standard

m a s (m), stilfness (k) and darnping (c) structure with one bending mode only in

each axis. The undamped frequency is given by w = m, the critical darnping

factor is defined as c, = 2 6 , and the damping ratio is < = c/c,. Thiç damping

ratio can be expressed as ( = c!2wm by manipulahg the denfütions. To projed

the on-orbit damping ratios of the k t bending modes the ground test results cm

be used. Using first the vertical configuration tests, the mass rra for both cases

would be the same, and it is aiso assumed that the mechanism causing damping

is the same, so the viscous damping factor c wodd be cornmon as weii. The ratio

of the on-orbit modal damping Co (we wish to solve for) to the vertical test modal

damping (; found during modd updaüng is

where w,, the onab i t modal fiequency, is determined h m the updated mode1 by

rernoving the gravity related effects and recalculating the modes, and wv is found

from the vertical gound test experîmentai data. This works with both the y and r

modes. Damping does have the effect of Iowering the natural hquency where the

damped fnquency is = w d m , but for the s m d values of C in this systern

the damped and undamped hquenaes are essentially the same-

The same thing can be done using the horizontal tests to produce a second

estimate- For the y bending mode, again the m a s and viscous damping factors are

assumeci to be the sarne between the ground and on-orbit cases, so

Table 5.9: Modal Propemes and Projections.

Mode 1 Freq. (Hz) 1 Daqing Ratio Vertical Ground Tests

Based on vertical mh. Based on horizontal results.

y bending z bending

For the z mode, the suspension system perturbs the mas and the damping. Where

cz is the n i induced viscous damping factor in the z axis, and na, and c, are

the contributions to the mass and damping of the suspension system during the

ground testing,

Now the problem becornes one of estimating c,. One c m borrow an estirnate of

c, using modal damping equivalence from the verticai test case (as was done for

Table 5.9) asswning again that the viscous damping mechanism and magnitude do

not change between ground based vertical configuration and on-orbit tests, but it

hardly produces a second independent <, estimate. This suggests that damping is

ciifficuit to pmject accuratdy from ground tests where damping is perturbed. The

procedure can be cast using stiffness as a reference instead of mas, because modal

equivalence c m aIso be expressed as C = o+r/2k, but k is perturbed in both axes of

both ground test configurations, so alI 6 estimates wiii be less accurate.

Table 5.9 iists estimates for the pmjected on-orbit modal kquenaes and damp-

ing ratios based on verticai and horizontal ground tests of the rib system and the

1.014 1.0 15

0.00588 0.00588

Horizontal Ground Tests 0.00436 0.00436

y bending ,z bending

0.792 0.645

On-ûrbit Projections y bending r bending

0.672 0.673

0.0088Sf 0.005 14t 0.00888t 0.00371 f

CHAPTER 5. MODE.. IIPDATING: PRACïïCE 100

updated modd of Run 9 (Poweii) describeri in $5.43. Many assurnptions were

made in arriving at these projections, not the least of which is that the damping

estimates from the updated ground mode1 are accurate in the 6rst place, an as-

sumption that is suspect considering how inconsistent the results are over several

updating nins (see Figure 5.14 for example). This does refiect, however, that (as ex-

pected) the O-g system will have ver- low damping. Whether it is slightiy higher or

slightiy lower in this case is next to impossible to predict with certainty because the

û-g damping projections vary h m the ground based results by amounts similar

to the standard deviation of the different p u n d based estimates.

Chapter 6

Conclusions

This document has explored innovations in two major areas associated with the

testing of spacecraft on the ground:

b Development of a mechanicd suspension system that compensatecg for grav-

ity.

b Deveiopment of a model updating approadi that ailows extrapolation to a

0-g parameterized model from 1-g test data.

The thesis leans more t o w d practicalities than remaining within the conlines

of theoreticai development, aithough that too is visited to the extent necessary to

produce useful resuits.

The requirements for the suspension systern have been formalized and, based

on these requirements, a suitalde concept for the design has been identïfied - that of the nullspring mechanism. This concept has been deveioped into a novel

desip. 2nd 2 I?&y I?~Ec&E-I p m t n e e has been built shed to support a prototype

DICE n i in its horizontal configuration Testing and evaiuation has shown that the

nuiispring complies with aii the requirements and performs to a levei comparabIe

with suspension hardware developed at other cesearch institutes, although there

is no evidence in the Iiterature of prior suspension systems designed for payIoads

asdastheDICEribstruchrrechrre

Figure 6.1: Flow of the Method Selection Rocess.

CHAPTER6. CONCLUSIONS 1 03

In the model updating phase of the work, the probkm has been classiüed and a

general approach identifid. Figure 6.1 depicts how the classification of the prob-

lem relates to the method used in updatmg the modei. The problern has been

broken d o m into sub-modeis that have been successfully ugdatd, Numericd

experimentation with the Poweil optimization aigorithm, an iterative method that

is both derivative k and based on the conjugate gradient approach, has shown

success in optimizing the nulIspring modd. These tests also illustrate that with

very iightly damped structures, long data sets are required to h d a converged

darnping estimate. Updating the noniinear CMG servo model has k e n more chal-

Ienging because it feahues a much more jagged cost terrain. A Simulatecl An-

nealing method, using a walk algorithm based on the Siplex method, has been

successfuiiy employed for this, whde Powd has failed to find a fuiiy optimized

solution. Because the nonlinearities are quite significant, hearized MU and direct

SI using OKlD/ERA have not been as successful as MU with the noniinear CMG

servo modeL

To move to the DICE n i rnodei, parameters derived h m the CMG servo up-

dating have been incorporateci in the full system model and have been subjed to

no M e r updating. A reduced subset of parameters suitable for updating the rib

mode1 have been identifieci and many runs with both the Powd and SA methods

have been carried out. In generai, both methods work equaüy weII in optimizuig

the rib system modd, but the Powd method is more computationaiiy efficient.

Updating using simuiated refemce data has mdicated that for very Iarge rib de-

flectïons, nonlinear MU outperforms iinearized MU. However, when using data

front real ph- experiments, hearized model updating performs as well as

nonlinearized updating indicating that, for typicai rib defiection ranges, the lin-

earized mode1 is a suitable approximation.

The simdation MU d t s have aisa indiateci that the dut ion is non-unique

owing to output-equivalent p-due sets. This adds to the challenge of hding

CHAPTER 6. CONCLUSIONS 104

physically meanin@ results. Fortunately, these simulateci MU runs have fur-

ther shown that by making use of prior knowledge of the parameter uncertainties

h u g h regularization, and by using a diversity of data h m different nuis and

different systern conligurations, the non-uniqueness problem is significantly re-

duced. When trymg these techniques with reai data, reguiarization and multiple

data sets and configurations give consistent and physicdy meaningfui updated

models. This ability to generalize to different systern configurations is the most

significant advantage of MU over direct SI approaches such as OKID/ERA. The

most notable disadvantage is in computation, where the MU methods presented

herein are profoundly more expensive than direct SI.

Darnping offers speaai chalienges in obtaining consistent and converged esti-

mates. Light damping leveh and gyric coupling in the rib bending have made it

impossible to distinguish damping ratios for each mode. However, by Iumping the

parameters for each axis together, better convergence and consistency have been

demonstrated. This is a reasonable assurnption to make given the symmetry in the

actual structure.

Finaliy, exiracting the O-g model from the 1-g updated models is straightfor-

ward to do when the physical state vanable space is retained in the updated model,

except in the case of the darnping ratio estimates. It has been established that the

presence of the suspension system makes it ai l the more dif6cuIt to project onsrbit

damping ratios fmm the ground tests, aithough most of the difficuities stem h m

the very low Ievels of damping in the system.

6.1 Thesis Contributions

An important part of the originality in this bork lies in the development of a novel

suspension system for testing flmdble space structures on the ground, a suspen-

sion system whose stiffness, damping and inertial effects are as benign as practi-

caUy possible. Literature sources have yieided the nuilspring concept, but go on to

point out the major drawbacks, maidy in the area of local suspension modes and

frictior~ ûriginal contributions overcome these detîaencies as foiIows:

b Local dynarnics from within the suspension system have been pushed out

of the test bandwidth intended for the system by rnaking taitored choices in

the seleciion of the basic geometry and the configuration of the components,

choices con- to suggestions in uime sources.

An ancillary design effort has ailowed the number of bearings in the mecha-

nisrn to be reduced to one, where a Me-edge bearing serves, aü but elimi-

nating fiction and damping in the suspension system.

A second and more important source of originality is in the modd updating

appmach. AH of the individuai tooIs used in the modei updating and optimiza-

tion have been drawn diredy €rom sources in the iiterature. However, the manner

in which these toois have been assembled for updating such a general nonlinear

model and the successful demonstration of the method on a flexiile gyric test arti-

de are unique. Here speaficaiiy, the contributions are the foiiowing:

b The use of time series data has been successfully appiied to the problem of

updating a nonlinear system model.

b Sequential estimation has been successfuily applied to the model updating,

pemitting the use of Iarge amounts of data in estimating the stationary modei

parameters, without the risk of getüng trapped at inappropriate local min-

ima.

The method in this thesis benefits from the use of uncertainty-weighted reg-

uIarization to reduce the pmb1ems of iU-posedness and non-uniqueness, by

aiiowing uncertainty information about the parameters to be integrated into

the process.

The simuitaneous use of test data from multiple configurations, made possi-

ble by the unique architecture of the overail method, has been demonstrated

to reduce the difficulties associateci with a non-unique optimal solution.

CHAPTER 6. CONCLUSIONS 1 06

b The overall method has been successfully demonstrated on a real system.

6.2 Beyond This Work: Testing the DlCE Freeflyer

The main implication from the tests conducteci is that, for the structure being stud-

ied, multiple test co~gurations are necessary to rnitigate the problems associated

with solution non-uniqueness. It has also been found that doing substructural

testing greatly improves the resuits. It will be difficult to condud tests on the full

DICE freefiyer in multiple configurations, but substructural rib tests can definitely

be accomplished and in multiple configurations, helping to divers* the updated

model.

It wiii also be difficuit to project the on-orbit damping ratios for the dïfkrent

modes in the overall tests of the freeflyer. in 85.4.4 it was demonstrated that the

presence of the suspension system compiicated this calculation for a singIe mode,

and this will only be more difficuit for many modes. Subsûuctural tests are not

adequate for estimating the modal damping for the overd DICE structure. It has

been shown in this thesis that an estimated substrutAural modal damping ratio

is valid only for the specific mode tested, and does not directiy apply to a dif-

ferent mode, even the same mode shape at a different freguency. Similarly, the

modes of an assembly of substructures often do not resemble the modes of the

separate substructures (due to coupling or altered boundary conditions) so damp-

h g ratios estimated h m subsûucturai testing cannot be duectIy appiied m the

overall assembly. Further work needs to be done in this area before reliable on-

orbit damping estimates can be made from ground test data, but th& WU require

experimentation on the f d y suspended DICE Ereefiyer, once it becomes available.

Alternatively a simplistic though largeiy appropriate approximation is to assume

that the damping is unchanged between the 1-g and O-g cases, the d t s h m

the DICE rib O-g damping projections mdicating that the systern on-orbit wiIl have

similarly low damping leveis as the 1-g case.

CHAPTER 6. CONCLUSIONS

However, the general pmess established herein provides a path whereby the

overail DICE model can be updated and projected to O-g. The suspension system

prototype has proved to be very successful in all aspects of testing a single rib and

promises to be successful in the more arnbitious task of testing the entire DICE

structure. The model updating scheme, based on the approach of reconciling t h e

domain data, will ais0 work very weU for the M y deployed sûucture. What this

simple and intuitively based technique costs in computational effort is more than

justiEied since a physical model is maintained, whidi is straightforward to diagnose

for physical relevance and to project to O-g. There are no constraints on the struc-

ture of the model, and uncertaùities in the parametes are easily mcorporated. In

summary, the objectives of the thesis have successfdiy been met.

[l] P. C. Hughes, K. A. C m U , R E. Zee, G. Vukovilti. Dynamics, identifica-

tion and control experiments for the space shuttie middeck. CAS1 Journal,

45(2):1&131, June 1999.

[2] S. Grocott, J- How, D. Miller, D. MacMartin, K. Liu. Robust control design

and irnplementation on the Middeck Active Control Experiment. Journal of

Guidance, Conrrol, and Dynamics, 17(6):1163-1170, Nov-Dec 1994.

[3] J. N. Juang. Appied System Identifîcaion. Prentice Hd, 1994.

[4] H. R. Busby D. M. Trujiio. Practical Inverse Analysis in Engineering. CRC Press,

New York, 1997.

[5] J.E. Mottetshead, ML Frisweii. Modei updating in structurai dparnics: A

nirvey. Journalof Sound and Pîbration, 167(2):347-375,1993.

[6] J. M. Crawford. Fractical Suspension System Design. Technical Memorandurn

DICE-PIC-1, University of Toronto institute for Aerospace Studies, May 1997.

[7] D. W. Sparks, J. N. Juang. Survey of experiments and experimentai faciiities

for control of Ilexiile structures. Joumal of Guidance, Conm!, and Dynamia,

15(4):801-816, Jdy-August 1992

[8] G. C- Foss. A low stiffness suspension system for free-free modal testmg. In

I l th International Modal Anaiysis Conference, volume 2, pages 1402-1408,1993.

[9] S. E. Woodard, /. M. Housner. Nonlinear behavior of a passive zero-spring-

rate suspension system. Journal of Guidance and Control, 14(1):84-89, Jan-Feb.

1991.

[ I O ] L F- Yang, M. S. Chew, J. N. Juang. Banddrive suspension mechanism design

for ground-based testing of flexible space structures. Journal of Mechanical

Design, 1173134442, March 1995-

[ I l ] M . C. Reaves, M. S. Chew, J. N. Juang, S. H. Chiu. Dynarucs and control of

a large displacement suspension system for ground testing of flexible space

strubires. In AL44, Aerospace Design Conference, February 1992

[12] L. F. Yang, R. C. Tzeng, C. P. Kuo. Adaptive conbol of a zero-stiffness suspen-

sion system. Journal of Guidance. Conrrol, and Dynamics, 19(4):794-800, July-

August 2996.

[13] D. A. Kienhoiz. A pneumatic/electnc suspension system for simulating on-

orbit conditions. In Winter ASME Annual Meeting, November 1990.

[14] V. M . Cooley, A. A. Giunta. Laboratory evaluation of two advanced suspen-

sion system devices for ground vibration testing of large space structures. in

33rd R U A Structures, Structurai Dynamics, and Marerials Conference, volume 3,

pages 170Cb1710,1992.

[15] D. A. Kienhoiz, E. F. Crawley, T. J. Harvey. Very low hqyency suspension

systems for dynamic testing. In 30th A M Structures, S m c t u ~ l Dynamics, and

Materials Conference, volume 1, pages 327436,1989-

[16] Z. Pronk, P. van Woerkom. Hat Boor facilities in support of configurable space

structures development In 46th fnrernational Astronomical Congress, October

1995.

[17] R. R. Gold, 1. P. Friedman, W. H. R d , W. L. Hallauer. Suspension systerns

for ground testing of large space structures. Contraa Report NASA-CR-4325,

LangIey Research Center, ûctober 1990.

[18] V. M. Cooley, J. N. Juang, P. Ghaemmaghnmi- Design of ground test suspen-

sion systems for vdcat ion of flexibte space structures. In Dynarnics and Con-

tml of hrge Sîructures; Pmceedings of r h ~ S&h VPf&SU/ALZA Symposium, pages

713-725, June-Juiy 1987.

[19] N. Phung. Derivation of DICE Dynamical Model. Technical Memorandum

DICE-PN-5, University of Toronto lnstitute for Aerospace Studies, April1996.

[20] R. E. Zee, J. M. Crawford. Corrections to Phung's model. Technicai Memo-

randum DICE-REZ-3, University of Toronto Institute for Aerospace Studies,

October 1996.

[21] E. Choi. The Modehg and System Identification oE the Dynamics, Identifica-

tion, and Control Experiment. Master's thesis, University of Toronto Institute

for Aerospace Studies, 1997.

[22] R. E. Zee. Simulator Development for the Dynamcs, Identication, and Con-

trot Experiment Technicai Report DICE-4, University of Toronto Institute for

Aerospace Studies, June 1997.

[23] J. M. Crawford. Suspension Systern Xmath Model. Tedinical Memorandum

DICE-JMC-2, University of Toronto Institute for Aerospace Studies, October

1997.

[24] D. A. Haessig, B. Friedland. On the Modeling and Simulation of Friction.

Journal of Dynamic Systems. Measrirement, and Control, 113:3%362, Septernber

1991.

[251 M. 1. Friswell, J. E. Mottedead. Finite EZement Model Upduting in Structural

Dynamics. Kiuwer Acad-c Publishets, 1995.

REFERENCES 111

[26] T. Hong, K. A. Carmii, P. C. Hughes. Laboratory Results On System IdentiEica-

tion For Flexible Spacecraft in Tenth VPI&SU Symposium on Structural Dpamics

and Conml, pages 487499,1995.

[ 2 n T. W. Lim. Analytical Modd Irnprovement Using Measured Modes and Sub-

matrices. AlAA Journal, 29(6):1015-l018,1991.

[28] M.J. Raddiffe, N.A.J. Lieven An improved method for parameter selection

in h i t e element mode1 updating. ïhe Aemnautical Journal, pages 321-329,

June/ Juiy 1998.

[29) R J. Bauer. Calcuiating Mas, Stifhess and Damping h m Identifieci

State-Space Matrices. Technical report, University of Toronto Institute for

Aeroçpace Studies, March 1997. Final Report: Topics in Spacecraft Dynam-

ics and Control.

[30] CA. Dohrmann, T.G. Came. M& refinement using transient response. In

16th fnternational Modal Analysis Conference, volume 1, pages 630436,1998.

[31] Z Ligmi, H. Baiqing. A New Reguiarïzation Approach in Stm-1 System

Identification. In Joint Pmceedings in Aemnauiics and Asrronaun'cs, pages 89-93,

1993.

(321 H. R Busby D. M. Tmjao. Optimal regularization of the inverse-heat con-

ductim problem using the !-curve. International Journal of Numerical Methodr

in Hear and FZuid Flow, 4:447452,1994.

[33] H. Ahmadian, J.E. Mottershead, ML Frisweii. Reguiatisation Methods For

Finite EIement Madel Updating. Mechanical Systems and Signal Processing,

12(1):47-64, 1998.

[34 Richard P. Brent- Algorithnisforh!inimization fithout Derbatives. Prentice-Hall,

1973.

[35] T. Janter, P Sas. Uniqueness aspects of model-updating procedures. AM4

Journul, 28(3)53û-543, March 1990.

[36] SC. Garg. Numerical Minunization Methods for Functionals: Cornparison

and Extensions. Technicd Report UTIAS#209, University of Toronto Institute

for Aerospace Studies, JuIy 1977.

[37J LS. Katafygiotis, J.L. Be& Updating Models and Their Uncertainties. II:

Model Identifiability. Jotrnal of Engineering Mechanics, 224(4):463-467, April

1998.

[38] W. H. Press, S. A. Teukosky, W. T. Vetterling, B. P. Flannery. Numerical Recipes

In C. 2nd Ed, Cambridge, 2nd edition, 1992.

[39] J. Hertz, A. Krogh, R. Palmer. Introduction ro the Theoy of Neural Cornpuration.

Addison Wesley, 1991.

[40] J.A. Nelder, R Mead. A Simplex Method for Fundion Minimization. The

Cornputer Journal, 7:308-323,1965.

[41] RJ. Levin, NAJ. Leiven. Dynamic Finite Element Model Updating Using

Simtdated Annealing and Genetic Algorithms. Mechanical Sysrerns and Signal

Processing, 12(1)91-120,1998.

[a] Rd. Levh, NAJ- Lieven. Dynamic Finite Element Updating Using Neural

Networks. Journal of Sound and Vibration, 210(5)59347,1998.

[43J N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A, H. Tder. Equations

of state dcuiations by fast computing madiines. Journal of Chemical Physics,

21:2087-1092,1953.

[441 S. Kirkpatrick? CD. Gelatt, V'chi. Optimization by simulateci anneaiing.

Science, 220(4598):671-680, May 1983.

[45] R J. Bauer, P. C. Hughes. Augmented system identification of a control mo-

ment gyro. CQSIJoumal, 46(2):87-94, June 2000.

144 P. Barney, M. S t Pierre, D. Brown. Identification of a modal mode1 utilipng a

perturbed boundary condition test method. In 10th International Modal Analysis

Con$~nce, volume 2, pages 1221-2231,1992.

[471 B K Wada, C.P. Kuo, RJ. Glaser. Multiple boundary condition tests

0 for vedcation of large space structures. In Proceedings of the

AUALASME/MSi'ASCE 27th Structures. Smcturai Dynamics and Materials Con-

ference, New York, May 1986. ABA.

1481 D.J. McTavish, P.C Hughes. Modehg of ünear viscoeIastic space structures.

Journal of Vibration and Acoustics, 115103-110, J m e 1993.

Appendix A

Analysis and Design of the Nullspring Geometry

The devant geometry is shown in Figure AS. The exact solution of several of

the geometric values, in terms of the angle a away Erorn the neutrai (horizontal)

position are as foilcws:

However, these expressions are unwieldy to use in optimizing the design Based

on a third-order Taylor expansion,

APPEMIK A. ANALYSlS AND DESIGN OF THE NULLSPRING GEOMETRY 1 15

Figure A. 1 : Nullspring Geornetry.

tf the initial elongation of tite side spring at the neutral position (a = O) is denoted

60, the side s p h g force at any angle a is

Similarly, for a given excursion 2, the main spring force is

Considering the static moment baIance of the Iever about its pivot point, the restor-

hg force J, is

The overail effective vertical stiffness of the mechanism, cded the secan1 s@hess,

is defined as

Substituting the previous approximate solutions into this d a t i o n r d t s in the

f0Uowing:

Note that the stiffness is a function of a*, One can now solve for the value of the

side spring neutrai elongation do which renders the stif£ness k, zero to a third

order approximation:

With do, one can solve for the op timal tide srring stifbess using the fact that when

a = O, k, = O and f, = ks& as wd. Applying these conditions, and substituthg

the approximation for r2(a) into the moment balance Equation A.9, the foIIowing

is tnie:

(A. 1 3)

Designing and analyzing now becomes a relativeiy simple procedw:

b speafy the main spring stiffness,

b specify the geometric parameten ro, rm and L,

b solve for the optimal side spring elongation at a = 0, bo (Equation A.12),

it soIve for the optimal side spring stiffness k, (Equation A.13),

solve f x the side spring load f, at any angle (Equation A.7), and

b solve for the effective secant stiffness of the nullspring, k,, at any angIe ei-

ther the approximate solution in Equation A.11 or the exact solution

using the exact geometrical properbes and Equations A.7, A.9, and A.lO.

Design is an iterative process of adjusting the main spring rate and the lever ge

ometry.. solving for the optimal side spring properties, and then evaluating the

theoretical performance.

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