Using Multi-constellation GNSS and EGNOS for Bridge Deformation ...

Using Multi-constellation GNSS and EGNOS for Bridge Deformation ...

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Using Multi-constellation GNSS and EGNOS for Bridge Deformation Monitoring X. Meng a, *, N. Gogoi a, A. H. Dodson a, G. W. Roberts a, b, C. J. Brown c a Nottingham Geospatial Institute, The University of Nottingham, United Kingdom - b Faculty of Science and Engineering, The University of Nottingham Ningbo, China - .
  • gps time series of a few hours
  • top curve
  • egnos data access service
  • bridge monitoring
  • vertical bridge deflection
  • geostationary navigation overlay service
  • gnss
  • bridges
  • gps
  • world

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MANEUVERING AND CONTROL
OF MARINE VEHICLES
Michael S. Triantafyllou
Franz S. Hover
Department of Ocean Engineering
Massachusetts Institute of Technology
Cambridge, Massachusetts USA
Maneuvering and Control of Marine Vehicles
Latest Revision: November 5, 2003
◦c Michael S. Triantafyllou and Franz S. Hover Contents
1 KINEMATICS OF MOVING FRAMES 1

1.1 RotationofReferenceFrames . . . . .. . . . . . . . . . .. . . . . . . . . . 1

1.2 DifferentialRotations . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 2

1.3 RateofChangeofEulerAngles . . . .. . . . . . . . . . .. . . . . . . . . . 4

1.4 DeadReckoning. .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 5

2 VESSEL INERTIAL DYNAMICS 5

2.1 MomentumofaParticle . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 5

2.2 LinearMomentuminaMovingFrame . . . . . . . . . . .. . . . . . . . . . 6

2.3 Example:MassonaString . . . . . .. . . . . . . . . . .. . . . . . . . . . 7

2.3.1 MovingFrameAffixedtoMass . . . . . . . . . . .. . . . . . . . . . 8

2.3.2 RotatingFrameAttachedtoPivotPoint . . . . . .. . . . . . . . . . 8

2.3.3 StationaryFrame . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 8

2.4 AngularMomentum . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 9

2.5 Example:SpinningBook. . . . . . . .. . . . . . . . . . .. . . . . . . . . . 10

2.5.1 x-axis . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 11

2.5.2 y-axis . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 11

2.5.3 z-axis . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 12

2.6 ParallelAxisTheorem . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 12

2.7 BasisforSimulation . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 12

3 NONLINEAR COEFFICIENTS IN DETAIL 13

3.1 HelpfulFacts . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 14

3.2 NonlinearEquationsintheHorizontalPlane . . . . . . . .. . . . . . . . . . 15

3.2.1 FluidForce X . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 15

3.2.2 FluidForce Y . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 16

3.2.3 FluidMoment N . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 17

4 VESSEL DYNAMICS: LINEAR CASE 17

4.1 SurfaceVesselLinearModel . . . . . .. . . . . . . . . . .. . . . . . . . . . 17

4.2 StabilityoftheSway/YawSystem. . .. . . . . . . . . . .. . . . . . . . . . 18

4.3 BasicRudderActionintheSway/YawModel . . . . . . .. . . . . . . . . . 20

4.3.1 AddingYawDampingthroughFeedback . . . . . .. . . . . . . . . . 21

4.3.2 HeadingControlintheSway/YawModel. . . . . .. . . . . . . . . . 21

4.4 ResponseoftheVesseltoStepRudderInput. . . . . . . .. . . . . . . . . . 22

4.4.1 Phase1:AccelerationsDominate . . . . . . . . . .. . . . . . . . . . 22

4.4.2 Phase3:SteadyState . . . . .. . . . . . . . . . .. . . . . . . . . . 22

4.5 SummaryoftheLinearManeuveringModel . . . . . . . .. . . . . . . . . . 23

4.6 StabilityintheVerticalPlane . . . . .. . . . . . . . . . .. . . . . . . . . . 23

i
5 SIMILITUDE 23

5.1 UseofNondimensionalGroups . . . .. . . . . . . . . . .. . . . . . . . . . 23

5.2 CommonGroupsinMarineEngineering . . . . . . . . . .. . . . . . . . . . 25

5.3 SimilitudeinManeuvering . . . . . . .. . . . . . . . . . .. . . . . . . . . . 27

5.4 RollEquationSimilitude . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 29

6 CAPTIVE MEASUREMENTS 30

6.1 Towtank . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 30

6.2 RotatingArmDevice. . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 30

6.3 Planar-MotionMechanism . . . . . . .. . . . . . . . . . .. . . . . . . . . . 30

7 STANDARD MANEUVERING TESTS 33

7.1 Dieudonn´eSpiral .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 33

7.2 Zig-ZagManeuver.. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 33

7.3 Circleer. .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 34

7.3.1 DriftAngle . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 34

7.3.2 SpeedLoss . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 34

7.3.3 HeelAngle.. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 34

7.3.4 HeelinginSubmarineswithSails . . . . . . . . . .. . . . . . . . . . 35

8 STREAMLINED BODIES 35

8.1 NominalDragForce . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 35

8.2 MunkMoment . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 35

8.3 SeparationMoment. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 36

8.4 NetEffects:AerodynamicCenter . . .. . . . . . . . . . .. . . . . . . . . . 37

8.5 RoleofFinsinMovingtheAerodynamicCenter . . . . . .. . . . . . . . . . 37

8.6 AggregateEffectsofBodyandFins . .. . . . . . . . . . .. . . . . . . . . . 38

8.7 Coefficients Z , M , Z,and M foraSlenderBody. . . .. . . . . . . . . . 39
w w q q
9 SLENDER-BODY THEORY 39

9.1 Introduction. . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 39

9.2 KinematicsFollowingtheFluid . . . .. . . . . . . . . . .. . . . . . . . . . 40

9.3 DerivativeFollowingtheFluid. . . . .. . . . . . . . . . .. . . . . . . . . . 41

9.4 DifferentialForceontheBody . . . . .. . . . . . . . . . .. . . . . . . . . . 41

9.5 TotalForceonaVessel. . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 42

9.6 TotalMomentonaVessel . . . . . . .. . . . . . . . . . .. . . . . . . . . . 43

9.7 RelationtoWingLift. . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 44

9.8 Convention:HydrodynamicMassMatrix A . . . . . . . .. . . . . . . . . . 44

10 PRACTICAL LIFT CALCULATIONS 44

10.1 CharacteristicsofLift-ProducingMechanisms . . . . . . .. . . . . . . . . . 44

10.2 Jorgensen’sFormulas . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 45

10.3 Hoerner’sData:Notation . . . . . . .. . . . . . . . . . .. . . . . . . . . . 46

10.4 Slender-BodyTheoryvs.Experiment .. . . . . . . . . . .. . . . . . . . . . 47

10.5dyApproximationforFinLift. . . . . . . . . .. . . . . . . . . . 48

ii
11 FINS AND LIFTING SURFACES 49

11.1 OriginofLift . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 49

11.2 Three-DimensionalEffects:FiniteLength . . . . . . . . . .. . . . . . . . . . 49

11.3 RingFins . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 50

12 PROPELLERS AND PROPULSION 50

12.1 Introduction. . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 50

12.2 SteadyPropulsionofVessels . . . . . .. . . . . . . . . . .. . . . . . . . . . 51

12.2.1 BasicCharacteristics . . . . . .. . . . . . . . . . .. . . . . . . . . . 52

12.2.2 SolutionforSteadyConditions . . . . . . . . . . .. . . . . . . . . . 54

12.2.3 Engine/MotorModels . . . . .. . . . . . . . . . .. . . . . . . . . . 54

12.3 UnsteadyPropulsionModels. . . . . .. . . . . . . . . . .. . . . . . . . . . 56

12.3.1 One-StateModel:Yoergeret al. . . . . . . . . . . .. . . . . . . . . . 56

12.3.2 Two-StateModel:Healeyet al. . . . . . . . . . . .. . . . . . . . . . 56

13 ELECTRIC MOTORS 57

13.1 BasicRelations . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 57

13.1.1 Concepts. .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 57

13.1.2 Faraday’sLaw. . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 58

13.1.3 Ampere’sLaw. . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 58

13.1.4 Force. . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 58

13.2 DCMotors . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 58

13.2.1 PermanentFieldMagnets . . .. . . . . . . . . . .. . . . . . . . . . 59

13.2.2 ShuntorIndependentFieldWindings. . . . . . . .. . . . . . . . . . 60

13.2.3 SeriesWindings. . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 60

13.3 Three-PhaseSynchronousMotor. . . .. . . . . . . . . . .. . . . . . . . . . 61

13.4InductionMotor . . . . .. . . . . . . . . . .. . . . . . . . . . 62

14 TOWING OF VEHICLES 64

14.1 Statics . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 65

14.1.1 ForceBalance . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 65

14.1.2 CriticalAngle . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 67

14.2 LinearizedDynamics . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 68

14.2.1 Derivation .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 68

14.2.2 DampedAxialMotion . . . . .. . . . . . . . . . .. . . . . . . . . . 70

14.3 CableStrumming .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 72

14.4 VehicleDesign. . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 73

15 TRANSFER FUNCTIONS & STABILITY 73

15.1 PartialFractions .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 73

15.2 PartialFractions:UniquePoles . . . .. . . . . . . . . . .. . . . . . . . . . 74

15.3 Example:PartialFractionswithUniqueRealPoles . . . .. . . . . . . . . . 74

15.4 PartialFractions:Complex-ConjugatePoles . . . . . . . .. . . . . . . . . . 75

15.5 Example:PartialFractionswithComplexPoles . . . . . .. . . . . . . . . . 75

15.6 StabilityinLinearSystems. . . . . . .. . . . . . . . . . .. . . . . . . . . . 75

iii
15.7 Stability�� PolesinLHP . . . . . .. . . . . . . . . . .. . . . . . . . . . 76

15.8 GeneralStability .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 76

16 CONTROL FUNDAMENTALS 76

16.1 Introduction. . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 76

16.1.1 Plants,Inputs,andOutputs . .. . . . . . . . . . .. . . . . . . . . . 76

16.1.2 TheNeedforModeling. . . . .. . . . . . . . . . .. . . . . . . . . . 77

16.1.3 NonlinearControl. . . . . . . .. . . . . . . . . . .. . . . . . . . . . 77

16.2 RepresentingLinearSystems. . . . . .. . . . . . . . . . .. . . . . . . . . . 77

16.2.1 StandardState-SpaceForm . .. . . . . . . . . . .. . . . . . . . . . 77

16.2.2 ConvertingaModelintoaTransferFunction. . . . . . . 78

16.2.3 ConvaTransferFunctionintoaState-SpaceModel. . . . . . . 78

16.3 PIDControllers . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 79

16.4 Example:PIDControl . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 79

16.4.1 ProportionalOnly . . . . . . .. . . . . . . . . . .. . . . . . . . . . 80

16.4.2 Proportional-DerivativeOnly .. . . . . . . . . . .. . . . . . . . . . 80

16.4.3 Proportional-Integral-Derivative. . . . . . . . . . .. . . . . . . . . . 80

16.5 HeuristicTuning .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 81

16.6 BlockDiagramsofSystems. . . . . . .. . . . . . . . . . .. . . . . . . . . . 81

16.6.1 FundamentalFeedbackLoop. .. . . . . . . . . . .. . . . . . . . . . 81

16.6.2 BlockDiagrams:GeneralCase.. . . . . . . . . . .. . . . . . . . . . 81

16.6.3 PrimaryTransferFunctions . .. . . . . . . . . . .. . . . . . . . . . 82

17 MODAL ANALYSIS 83

17.1 Introduction. . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 83

17.2 MatrixExponential. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 83

17.2.1 Definition .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 83

17.2.2 ModalCanonicalForm . . . . .. . . . . . . . . . .. . . . . . . . . . 84

17.2.3 ModalDecompositionofResponse . . . . . . . . .. . . . . . . . . . 84

17.3 ForcedResponseandControllability .. . . . . . . . . . .. . . . . . . . . . 84

17.4 PlantOutputandObservability . . . .. . . . . . . . . . .. . . . . . . . . . 85

18 CONTROL SYSTEMS – LOOPSHAPING 86

18.1 Introduction. . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 86

18.2 RootsofStability–NyquistCriterion.. . . . . . . . . . .. . . . . . . . . . 86

18.2.1 MappingTheorem . . . . . . .. . . . . . . . . . .. . . . . . . . . . 87

18.2.2 NyquistCriterion . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 87

18.2.3 RobustnessontheNyquistPlot . . . . . . . . . . .. . . . . . . . . . 88

18.3 DesignforNominalPerformance. . . .. . . . . . . . . . .. . . . . . . . . . 89

18.4forRobustness . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 89

18.5 RobustPerformance . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 90

18.6 ImplicationsofBode’sIntegral. . . . .. . . . . . . . . . .. . . . . . . . . . 91

18.7 TheRecipeforLoopshaping . . . . . .. . . . . . . . . . .. . . . . . . . . . 91

iv
19 LINEAR QUADRATIC REGULATOR 92

19.1 Introduction. . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 92

19.2 Full-StateFeedback. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 93

19.3 TheMaximumPrinciple . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 93

19.4 GradientMethodSolutionfortheGeneralCase . . . . . .. . . . . . . . . . 94

19.5 LQRSolution . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 95

19.6 OptimalFull-StateFeedback. . . . . .. . . . . . . . . . .. . . . . . . . . . 96

19.7 PropertiesandUseoftheLQR . . . .. . . . . . . . . . .. . . . . . . . . . 96

19.8 ProofoftheGainandPhaseMargins .. . . . . . . . . . .. . . . . . . . . . 97

20 KALMAN FILTER 98

20.1 Introduction. . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 98

20.2 ProblemStatement . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 98

˙
20.3 Step1:AnEquationforΓ . . . . . . .. . . . . . . . . . .. . . . . . . . . . 99

20.4 Step2: H asaFunctionofΓ . . . . .. . . . . . . . . . .. . . . . . . . . . 100

20.5 PropertiesoftheSolution . . . . . . .. . . . . . . . . . .. . . . . . . . . . 101

20.6 CombinationofLQRandKF . . . . .. . . . . . . . . . .. . . . . . . . . . 102

20.7 ProofsoftheIntermediateResults. . .. . . . . . . . . . .. . . . . . . . . . 103

T 20.7.1 Proofthat E(e W e)=trace(ΓW ) . . . . . . . . .. . . . . . . . . . 103

� T T 20.7.2 Proofthat trace(−ΦHCΓ)= −Φ ΓC . . . . .. . . . . . . . . . 104

�H
� T T T 20.7.3 Proofthat trace(−ΦΓC H )=−ΦΓC . . . .. . . . . . . . . . 104

�H
20.7.4 ProofoftheSeparationPrinciple . . . . . . . . . .. . . . . . . . . . 105

21 LOOP TRANSFER RECOVERY 105

21.1 Introduction. . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 105

21.2 ASpecialPropertyoftheLQRSolution . . . . . . . . . .. . . . . . . . . . 106

21.3 TheLoopTransferRecoveryResult . .. . . . . . . . . . .. . . . . . . . . . 107

21.4 UsageoftheLoopTransferRecovery .. . . . . . . . . . .. . . . . . . . . . 108

21.5 ThreeLemmas . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 109

22 APPENDIX 1: MATH FACTS 110

22.1 Vectors. . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 110

22.1.1 Definition .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 110

22.1.2 VectorMagnitude. . . . . . . .. . . . . . . . . . .. . . . . . . . . . 111

22.1.3 VectorDotorInnerProduct. .. . . . . . . . . . .. . . . . . . . . . 111

22.1.4 VectorCrossProduct. . . . . .. . . . . . . . . . .. . . . . . . . . . 112

22.2 Matrices . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 112

22.2.1 Definition .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 112

22.2.2 MultiplyingaVectorbyaMatrix . . . . . . . . . .. . . . . . . . . . 112

22.2.3 aMatrixbya. . . . . . . . . .. . . . . . . . . . 113

22.2.4 CommonMatrices . . . . . . .. . . . . . . . . . .. . . . . . . . . . 113

22.2.5 Transpose .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 114

22.2.6 Determinant. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 114

22.2.7 Inverse. . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 115

22.2.8 Trace. . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . 115

v 22.2.9 EigenvaluesandEigenvectors . . . . . . . . . . . . . . . . . . . . . . 115

22.2.10ModalDecomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 117

22.2.11SingularValue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

22.3 LaplaceTransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

22.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

22.3.2 Convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

22.3.3 ConvolutionTheorem. . . . . . . . . . . . . . . . . . . . . . . . . . . 119

22.3.4 SolutionofDifferentialEquationsbyLaplaceTransform . . . . . . . 121

22.4 BackgroundfortheMappingTheorem . . . . . . . . . . . . . . . . . . . . . 121

23 APPENDIX 2: ADDED MASS VIA LAGRANGIAN DYNAMICS 124

23.1 KineticEnergyoftheFluid . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

23.2 Kirchhoff’sRelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

23.3 FluidInertiaTerms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

23.4 DerivationofKirchhoff’sRelations . . . . . . . . . . . . . . . . . . . . . . . 127

23.5 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

23.5.1 FreeversusColumnVector. . . . . . . . . . . . . . . . . . . . . . . . 130

23.5.2 DerivativeofaScalarwithRespecttoaVector . . . . . . . . . . . . 130

23.5.3 DotandCrossProduct. . . . . . . . . . . . . . . . . . . . . . . . . . 130

24 APPENDIX 3: LQR VIA DYNAMIC PROGRAMMING 130

24.1 ExampleintheCaseofDiscreteStates . . . . . . . . . . . . . . . . . . . . . 131

24.2 DynamicProgrammingandFull-StateFeedback . . . . . . . . . . . . . . . . 132

25 Further Robustness of the LQR 133

25.1 Tools. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

25.1.1 Lyapunov’sSecondMethod . . . . . . . . . . . . . . . . . . . . . . . 134

25.1.2 MatrixInequalityDefinition . . . . . . . . . . . . . . . . . . . . . . . 134

25.1.3 Frankliny . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

25.1.4 SchurComplement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

25.1.5 ProofofSchurComplementSign . . . . . . . . . . . . . . . . . . . . 135

25.1.6 SchurtofaNine-BlockMatrix . . . . . . . . . . . . . . . 135

25.1.7 QuadraticOptimizationwithaLinearConstraint . . . . . . . . . . . 136

25.2 CommentsonLinearMatrixInequalities(LMI’s) . . . . . . . . . . . . . . . 136

25.3 ParametricUncertaintyin A and B Matrices. . . . . . . . . . . . . . . . . . 137

25.3.1 GeneralCase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

25.3.2 UncertaintyinB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

25.3.3 tyinA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

25.3.4 AandBPerturbationsasanLMI. . . . . . . . . . . . . . . . . . . . 141

25.4 InputNonlinearities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

vi
1
1 KINEMATICS OF MOVING FRAMES
1.1 Rotation of Reference Frames
We denote through a subscript the specific reference system of a vector. Let a vector ex­
pressed in the inertial frame be denoted as γx, and in a body-reference frame γx. For the b
moment, we assume that the origins of these frames are coincident, but that the body
frame has a different angular orientation. The angular orientation has several well-known
descriptions, including the Euler angles and the Euler parameters (quaternions). The former
method involves successive rotations about the principle axes, and has a solid link with the
intuitive notions of roll, pitch, and yaw. One of the problems with Euler angles is that for
certain specific values the transformation exhibits discontinuities. Quaternions present a
more elegant and robust method, but with more abstraction. We will develop the equations
of motion using Euler angles.
Tape three pencils together to form a right-handed three-dimensional coordinate system.
Successively rotating the system about three of its own principal axes, it is easy to see
that any possible orientation can be achieved. For example, consider the sequence of [yaw,
pitch, roll]: starting from an orientation identical to some inertial frame, rotate the movable
system about its yaw axis, then about the new pitch axis, then about the newer still roll
axis. Needless to say, there are many valid Euler angle rotation sets possible to reach a given
orientation; some of them might use the same axis twice.
z=z’
z’’ z’’’
y’’’ x’
x y
x’’=x’’’ y’=y’’
Figure 1: Successive application of three Euler angles transforms the original coordinate
frame into an arbitrary orientation.
A first question is: what is the coordinate of a point fixed in inertial space, referenced to
a rotated body frame? The transformation takes the form of a 3×3 matrix, which we now
derive through successive rotations of the three Euler angles. Before the first rotation, the
0 body-referenced coordinate matches that of the inertial frame: γx = γx. Now rotate the b
movable frame yaw axis (z) through an angle δ. We have
� ⎭
cos δ sin δ 0

⎛ ⎞1 0 0 γx
=
⎝ − sin δ cos δ 0 ⎠γx = R(δ)γx
.
(1)
b b b
0 0 1 2 1 KINEMATICS OF MOVING FRAMES
Rotation about the z-axis does not change the z-coordinate of the point; the other axes are
modified according to basic trigonometry. Now apply the second rotation, pitch about the
new y-axis by the angle χ:
� ⎭
cos χ 0 − sin χ
⎛ ⎞2 1 1 γx =
0 1 0 γx = R(χ)γx
. (2) ⎝ ⎠b b b
sin χ 0 cos χ
Finally, rotate the body system an angle ω about its newest x-axis:
� ⎭
1 0 0

⎛ ⎞3 2 2 γx =
⎝ 0 cos ω sin ω ⎠γx = R(ω)γx
. (3) b b b
0 − sin ω cos ω
This represents the location of the original point, in the fully-transformed body-reference
3 3 frame, i.e., γx .We will use the notation γx instead of γx from here on. The three independent
b b b
rotations can be cascaded through matrix multiplication (order matters!):
γx = R(ω)R(χ)R(δ)γx (4) b
� ⎭
cχcδ cχsδ −sχ
⎛ ⎞
= ⎝ −cωsδ + sωsχcδ cωcδ + sωsχsδ sωcχ ⎠γx

sωsδ + cωsχcδ −sωcδ + cωsχsδ cωcχ
= R(δ, χ, ω)γx.
All of the transformation matrices, including R(δ, χ, ω), are orthonormal: their inverse is
equivalent to their transpose. Additionally, we should note that the rotation matrix R
is universal to all representations of orientation, including quaternions. The roles of the
trigonometric functions, as written, are specific to Euler angles, and to the order in which
we performed the rotations.
In the case that the movable (body) reference frame has a different origin than the inertial
frame, we have
T γx = γx + R γx , (5) 0 b
where γx is the location of the moving origin, expressed in inertial coordinates. 0
1.2 Differential Rotations
Now consider small rotations from one frame to another; using the small angle assumption
to ignore higher-order terms gives
� ⎭
1 ζδ −ζχ
⎛ ⎞
R � ⎝ −ζδ 1 ζω ⎠ (6)
ζχ −ζω 1 1.2 Differential Rotations 3

� ⎭
0 ζδ −ζχ
⎛ ⎞
= ⎝ −ζδ 0 ζω ⎠+ I , 3×3
ζχ −ζω 0

where I donotes the identity matrix. R comprises the identity plus a part equal to the 3×3
γ γ(negative) cross-product operator [−ζE×], where ζE = [ζω, ζχ, ζδ], the vector of Euler
angles ordered with the axes [x, y, z]. Small rotations are completely decoupled; the order of
−1 T −1 γthe small rotations does not matter. Since R = R , we have also R = I + ζE×;3×3
γγx = γx − ζE × γx (7) b
γγx = γx + ζE × γx . (8) b b
We now fix the point of interest on the body, instead of in inertial space, calling its location
in the body frame γr (radius). The differential rotations occur over a time step ζt, so that
we can write the location of the point before and after the rotation, with respect to the first
frame as follows:
γx(t) = γr (9)
T γγx(t + ζt) = R γr = γr + ζE × γr.
Dividing by the differential time step gives
γζγx ζE
= × γr (10)
ζt ζt
= �γ× γr,
γwhere the rotation rate vector � � dE/dt because the Euler angles for this infinitesimal
rotation are small and decoupled. This same cross-product relationship can be derived in
the second frame as well:
γγx(t) = Rγr = γr − ζE × γr (11) b
γx(t + ζt) = γr. b
such that
γζγx ζE b
= × γr (12)
ζt ζt
= �γ× γr,
On a rotating body whose origin point is fixed, the time rate of change of a constant radius
vector is the cross-product of the rotation rate vector �γ and the radius vector itself. The
resultant derivative is in the moving body frame.