KinematicSynthesisOctober6,2015MarkPlecnik

ClassifyingMechanismsSeveraldichotomies

SerialandParallel FewDOFSandManyDOFS

Planar/SphericalandSpatial RigidandCompliant

MechanismTrade-offsWorkspace Rigidity Designing

KinematicsNo. ofActuators

FlexibilityofMotion

ComplexityofMotion

Serial Large Low Simple Depends Depends Depends

Parallel Small High Complex Depends Depends Depends

FewDOF Small Depends Complex Few Little Less

ManyDOF Large Depends Simple Many Alot More

Serial,ManyDOF Parallel,ManyDOF Parallel,FewDOF Serial,FewDOF

ProblemsinKinematics

(x1, y1)(x2, y2)

l1

l2

l3

x3

y3

(X, Y, θ)

ϕ1

ϕ2

ϕ3

Dimensions

JointParameters

EndEffectorCoordinates ForwardKinematicsKnown:Dimensions, JointParametersSolvefor:EndEffectorCoordinates

InverseKinematicsKnown:Dimensions, EndEffectorCoordinatesSolvefor:JointParameters

SynthesisKnown:EndEffectorCoordinatesSolvefor:Dimensions, JointParameters

ChallengesinKinematics• Usingsweepinggeneralizations,howdifficult isittosolve

– forwardkinematics– inversekinematics– synthesis

overdifferenttypesofmechanisms?• Rankedonascaleof1to4with4being themostdifficult:

ForwardKinematics InverseKinematics Synthesis

Serial Parallel Serial Parallel Serial Parallel

Planar 1 2 2 1 3 3.5

Spherical 1 2 2 1 3 3.5

Spatial 1.5 2.5 2.5 1.5 3.5 4

Planar Spherical Spatial

SynthesisApproaches• Synthesisequationsarehardtosolvebecausealmostnothing isknownabout

themechanismbeforehand

SomeMethodsforSynthesis• Graphicalconstructions– 1soln perconstruction• Useatlases(libraries)(seehttp://www.saltire.com/LinkageAtlas/)• Evolutionaryalgorithms– multiplesolutions• Optimization– 1soln,goodstartingapproximationrequired• Samplingpotentialpivotlocations• Resultanteliminationmethods– allsolutions,limitedtosimpler

systems• Groebner Bases– allsolutions,limitedtosimplersystems• Intervalanalysis– allsolutionswithinaboxofusefulgeometric

parameters• Homotopy– allsolutions,canhandledegreesinthemillionsand

possiblygreaterwithveryrecentdevelopments

ConfigurationSpaceofaLinkageTerminology:Circuits- notdependentoninput linkspecificationBranches- dependentoninput linkspecification

Circuit1

Circuit2

Nobranches

Circuit1

Circuit2

Branch1

Branch2

Branch3Branch4

Singularities

Circuitandbranchescanleadtolinkagedefects

TypesofSynthesisProblemsa)Functiongeneration:setofinputanglesandoutputangles;b)Motiongeneration:setofpositionsandorientationsofaworkpiece;c)Pathgeneration:setofpointsalongatrajectoryintheworkpiece.

FunctionGeneration MotionGeneration PathGeneration

Aboveareexamplesoffunction,motion, andpathgeneration forplanarsix-barlinkages.Analogous problemsexistforsphericalandspatiallinkagesofallbars.

Givescontrolofmechanicaladvantage

ExamplesofFunctionGeneration

Measurementsfromstrokesurvivors

MechanismGeneratedtorque

Summation

TheBirdExampleTechnique• Spatialchainsareconstrainedbysix-barfunctiongenerators

Spatialchain4DOF

Functiongeneratorstocontroljointangles AsingleDOF

constrainedspatialchain

Goal:achieveaccuratebiomimeticmotion

ExamplesofMotionGenerationandPathGeneration

KinematicsandPolynomials• Kinematicsareintimatelylinkedwithpolynomialsbecausetheyarecomposedof

revoluteandprismaticjointswhichdescribecirclesandlinesinspace,whicharealgebraiccurves

• Theselinesandcirclescombinetodescribemorecomplexalgebraicsurfaces

PPS TS CS PRS

The Plane Sphere Elliptical CylinderCircular Cylinder

Hyperboloid

RPS

Right Torus

RRS

Torus

RRS

MaxNumberofPositionsFunction Motion Path

Four-bar 5 5 9

WattI 5 8 15

Watt II 9 5 9

Stephenson I 5 5 15

Stephenson II 11 5 15

Stephenson III 11 5 15

PolynomialsandComplexity• Linkagescanalwaysbeexpressedaspolynomials• Whennewlinksareadded,thecomplexityofsynthesisrapidly

increases

Four-bar

Six-bar

Degree6polynomial

curve

Degree18polynomial

curve

Degree6polynomialsystem

Degree264,241,152polynomialsystem

SynthesisProblems

Thedegreeofpolynomialsynthesisequationsincreasesrapidlywhenlinksareadded

Degree1polynomialcurve:

Degree2polynomialcurve:

WaystoModelKinematics• Planar

– Rotationmatrices,homogeneoustransforms,vectors– Planarquaternions– Complexnumbers

• Spherical– Rotationmatrices– Quaternions

• Spatial– Rotationmatrices,homogeneoustransforms,vectors– Dualquaternions

• Allmethodscreateequivalentsystems,althoughtheymightlookdifferent.Differentconveniencesaremadeavailablebyhowkinematicsaremodelled

PlanarKinematicsWithComplexNumbers

θ

𝑎"𝑎# +

𝑏"𝑏#

=𝑎" + 𝑏"𝑎# + 𝑏#

cos𝜃 − sin 𝜃sin 𝜃 cos𝜃

𝑎"𝑎# =

𝑎" cos𝜃 − 𝑎# sin𝜃𝑎" sin 𝜃 + 𝑎# cos𝜃

(ax + iay) + (bx + iby) = (ax+bx) + i(ay+by)

eiθ(ax + iay) = (cosθ + isinθ)(ax + iay)= (axcosθ – aysinθ) + i(axsinθ + aycosθ)

a

b

x

y

x

y

Re

Im

Re

Im

a