Surface Geometry and the Haptic Rendering of Rigid Point Contacts

202 Pages·2013·4.15 MB·English

by  Kevin Casey Walker

Surface Geometry and the Haptic
Rendering of Rigid Point Contacts
by
Kevin Casey Walker
Athesis
presentedtotheUniversityofWaterloo
infulfilmentofthe
thesisrequirementforthedegreeof
DoctorofPhilosophy
in
ElectricalandComputerEngineering
Waterloo,Ontario,Canada,2013
(cid:13)c KevinCaseyWalker2013
IherebydeclarethatIamthesoleauthorofthisthesis. Thisisatruecopyofthethesis,including
anyrequiredfinalrevisions,asacceptedbymyexaminers.
Iunderstandthatmythesismaybemadeelectronicallyavailabletothepublic.
ii
Abstract
This thesis examines the haptic rendering of rigid point contacts in virtual simulations. The
haptic renderers generate force feedback so that the operator can interact with the virtual scenes
in a realistic way. They must be able to recreate the physical phenomena experienced in the
real world without displaying any haptic artifacts. The existing renderers are decomposed into a
projection function and a regulation scheme. It is shown that the pop-through artifact, whereby
thevirtualtoolinstantaneouslyjumpsbetweentwodistantsurfacepoints,iscausedwheneverthe
operatorencountersasingularitywithintherenderer’sprojectionfunction. Thiswaswellknown
for the minimum distance based renderers, but it is shown here that such singularities arise with
theconstraintbasedrenderersaswell.
A new projection function is designed to minimize the existence of singularities within the
model. When paired with an appropriate regulation scheme, this forms the proposed mapping
renderer. The new projection is calculated by mapping the model onto a canonical shape where
the haptic problem is trivial, e.g. a circle in the case of a 2D model of genus zero, which avoids
pop-throughonsmoothmodels. Thehapticproblemisthenrecastasavirtualconstraintproblem,
where the traditional regulation schemes, designed originally for planar surfaces, are shown to
introduce a velocity dependent error on curved surfaces that can distort the model’s rendering
andtocoupletheregulationtowardsanddynamicsalongtheconstraint. Setstabilizationcontrol,
based on feedback linearizing the haptic device with respect to a virtual output consisting of
coordinates transversal and tangential to the model surface, is proposed as an alternative. It is
showntobeabletodecouplethesystemintotransversalandtangentialsubsystemsthatcanthen
bemadeasymptoticallystableandassignedarbitrarydynamics,respectively.
iii
Acknowledgements
I would like to start by thanking my supervisor, Dr. David Wang, for his guidance during
my years as his student. He was always available when help was needed and has provided
much needed advice on many occasions. Dr. Christopher Nielsen also had a significant impact
on my work and I learned much from co-authoring a couple of papers with him. Dr. Spiro
Karigiannis was instrumental in introducing me to the area of differential geometry, guiding
me in how it relates to the haptics problem. I would like to thank the members of my PhD
Examining Committee Drs. Shahin Sirouspour, Glenn Heppler, Daniel Davison and Dana Kulic´
fortheirfeedbackandguidance. Theirinterestinmyworkprovidedmuchneededmotivation.
IwouldalsoliketothankJohnDaly,MichaelTribou,AndreHladio,ReginaLeung,Roop&
Willem Petersen, John Tang and Nizar Messaoudi for their support. The companionship of my
fellowgradstudentshasbeeninvaluableandIcherishthefriendshipsthathavedeveloped.
The support from my family has been overwhelming. My mother and father have always
shown the utmost respect for higher education and consistently encouraged me to pursue my
abilities to their furthest reaches. My brother Martin, his wife Carrie, and their children, Josh,
Chelsea and Katelyn, have patiently listened as I explained what it was that I was working on,
iv
offering their encouragement and enthusiasm. My in-laws decided to show their support in a
very tangible way, providing a shower of gifts as chapters were completed in the writing phase,
alwaysprovidingsomethingtolookforwardto. Thisincludesbasketsofbeer,sweetsandgames
fromAndrew,Erin,Aidan,Claire,andMichaelJardine(top,leftinphotosonthepreviouspage),
Yvonne, Jacob, Cecilia, Patrick and Elizabeth Simpson (top, middle), Peter, John, Catherine,
Thomas and Robert Jardine (top,right), and Claire, Michael and Maria Connell (bottom, mid-
dle). The heads of the clan, Lawrence and Janet Jardine, topped it off by presenting me with an
engravedboxcontainingavintagewine(bottom,right)andacelebratorydinnertomarkthethe-
siscompletion. Theirenthusiasmandprayersforthisundertakingstrengthenedmyperseverance
tomakeitthroughtotheend,
None of this would have been possible without my wife, Mary-Elizabeth. From our initial
decision to move from Halifax to Waterloo so that I could start my PhD, to encouraging me
through the final days of writing, her love and support were invaluable throughout this whole
process. Herhelp,includinggiftsofencouragement(whiskeystonesatbottom,left),perspective
whenIwastoowrappedupinmywork,andprayersforthegracesthatIneeded,wasinstrumental
infinishingthislongrunningwork.
v
Dedication
Tomydarlingwifeand,aboveall,toGod,whoinvitedmetodiscoverHisTruth.
vi
Contents
ListofTables xi
ListofFigures xvii
1 Introduction 1
1.1 OriginalContributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 ThesisOverview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 HapticsBackground 7
2.1 HapticSimulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 ImpedanceDeviceBasedHapticSimulators . . . . . . . . . . . . . . . . . . . . 13
2.3 HapticThread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 PhysicsThread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.5 ModellingandControlofanImpedanceHapticDevice . . . . . . . . . . . . . . 18
2.5.1 RobotModellingintheJointSpace . . . . . . . . . . . . . . . . . . . . 19
2.5.2 RobotModellingintheOutputSpace . . . . . . . . . . . . . . . . . . . 20
2.5.3 RobotControlinCartesianCoordinates . . . . . . . . . . . . . . . . . . 21
2.5.4 DynamicModelBasedCompensation . . . . . . . . . . . . . . . . . . . 23
vii
2.5.5 StateSpaceHapticDeviceModels . . . . . . . . . . . . . . . . . . . . . 25
2.6 SystemLimitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 PerformanceCriteriaandExpectationsofaHapticSystem 38
3.1 EvaluationofHapticSimulators . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2 ContactBetweenRigidBodies . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2.1 ContactNormalForce . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.2 ImpulsiveCollisionForce . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2.3 CentripetalForce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4 HapticRendering 51
4.1 ProjectionFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.1.1 TestModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.1.2 MinimumDistanceProjection . . . . . . . . . . . . . . . . . . . . . . . 56
4.1.3 ConstraintProjection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2 RegulationSchemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3 AdditionalHapticEffects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.3.1 ForceShading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.3.2 Texture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.3.3 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3.4 ContactImpulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5 TheMappingRenderer 69
5.1 TheProjectionProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
viii
5.2 DifferentialGeometryandGridGeneration . . . . . . . . . . . . . . . . . . . . 71
5.2.1 CurveShorteningFlows . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.2.2 NumericalFormulationoftheCSFProblem . . . . . . . . . . . . . . . . 74
5.2.3 FlowBehaviourandVelocityFunctions . . . . . . . . . . . . . . . . . . 77
5.2.4 GenerationofTransversalLines . . . . . . . . . . . . . . . . . . . . . . 83
5.3 BuildingTransformationandProjectionFunctionsfromtheModelGrid . . . . . 84
5.4 RegulationwithCurvedLinesofProjection . . . . . . . . . . . . . . . . . . . . 86
6 ComparisonandEvaluationoftheHapticRenderers 89
6.1 TrajectoryBasedRendererComparison . . . . . . . . . . . . . . . . . . . . . . 90
6.1.1 EvaluationbyTestTrajectories . . . . . . . . . . . . . . . . . . . . . . . 90
6.1.2 DifferentialEquivalenceofRenderers . . . . . . . . . . . . . . . . . . . 91
6.1.3 Pop-through . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.1.4 AmbiguousInteractions . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.1.5 SummaryoftheProjectionFunction’sPerformance . . . . . . . . . . . . 106
6.2 ComputationalConsiderations . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.3 DynamicPropertiesoftheRenderers . . . . . . . . . . . . . . . . . . . . . . . . 109
6.3.1 FeedbackLinearization . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.3.2 PerformanceonaLineModel . . . . . . . . . . . . . . . . . . . . . . . 115
6.3.3 PerformanceonaCircleModel . . . . . . . . . . . . . . . . . . . . . . 119
6.3.4 SummaryoftheImpactoftheRegulationSchemeonPerformance . . . . 123
6.4 SummaryofRendererEvaluations . . . . . . . . . . . . . . . . . . . . . . . . . 124
ix
7 SetStabilizationandHaptics 126
7.1 VirtualConstraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.2 SetStabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.3 ApplicationofSetStabilizationtoaCurvein2D . . . . . . . . . . . . . . . . . 138
7.3.1 SetStabilizationontheLine . . . . . . . . . . . . . . . . . . . . . . . . 138
7.3.2 SetStabilizationontheCircle . . . . . . . . . . . . . . . . . . . . . . . 141
7.4 ApplicationofSetStabilizationtoaSurfacein3D . . . . . . . . . . . . . . . . . 146
7.4.1 ControlDesignforaSetStabilizationControllerontheSphere . . . . . . 149
7.4.2 EvaluationoftheSetStabilizationControllerontheSphere . . . . . . . 153
7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
8 Conclusions&FutureWork 157
8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
8.2 FutureWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
8.2.1 TransformationFunctionImprovements . . . . . . . . . . . . . . . . . . 160
8.2.2 SetStabilizationExtensions . . . . . . . . . . . . . . . . . . . . . . . . 161
8.2.3 DeviceInertiaasExperiencedbytheOperator . . . . . . . . . . . . . . 162
8.2.4 StabilityoftheHapticSystem . . . . . . . . . . . . . . . . . . . . . . . 164
Appendices 166
A ExperimentalPlatform 167
A.1 SystemComponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
References 170
x