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Mechanism and Robot Kinematics, Part II:
Numerical Algebraic Geometry
C= harles Wampler
General Motors R&D Center
<= /span>
Including joint wo= rk with
Andrew Sommese, University of Notre Dame = ;
Jan Verschelde, Univ. Illinois Chicago <= /span>
Alexander Morgan, <= span style=3D'mso-tab-count:1;width:1.72%'> GM R&D
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Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Wampler: Numerical Algebraic Geometry
2&#= 13;
Outline
nZero-dimensional solution sets <= /div>
n= Numerical solution by polynomial continuation =
nRoot counts and homotopies
nParameter homotopies
nPositive-dimensional solution sets
n= Basic constructs
nWitness sets
nNumerical irreducible decomposition
n= Basic operations
nIntersection of algebraic sets <= /div>
nDeflation of nonreduced sets
n= Higher-level operations
nEquation-by-equation intersections
nFiber products
nExtracting real points from a complex set
nApplications
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Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Wampler: Numerical Algebraic Geometry
Numerical Algebraic Geometry
nPurpose
nNumerically represent & manipulate algebraic sets
nApproach
nNumerical continuation operating on witness sets
n
Basic Operations
<= span style=3D'font-size:86%'>n Witness generate
<= span style=3D'font-size:86%'>n Witness decomposition <= /span>
<= span style=3D'font-size:86%'>n Membership tests
<= span style=3D'font-size:86%'>n Intersection
<= span style=3D'font-size:86%'>n Deflation
Basic Constructs
n Witness sets
<= span style=3D'font-size:86%'>n Irreducible decomposition
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Why study polynomial systems?
nApplication areas
nEconomics & finance
nChemical equilibrium
nComputer-aided Geometric Design (CAGD)
nControl theory
nKinematics
nConstrained mechanical motion
nLinkages for motion constraint & transformation
nSuspensions, engines, swing panels, etc.
nComputer-controlled motion devices
nRobots, human-assist devices, etc.
------=_NextPart_01C6DF35.194A2C70 Content-Location: file:///C:/1E95224D/Wampler_NumAlgGeom_files/slide0130.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Wampler: Numerical Algebraic Geometry
Zero-Dimensional Sets
n Sol= ving by polynomial continuation
------=_NextPart_01C6DF35.194A2C70 Content-Location: file:///C:/1E95224D/Wampler_NumAlgGeom_files/slide0131.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Wampler: Numerical Algebraic Geometry
6&#= 13;
What is Continuation?
nFor some class of parameterized problems:
n<= span style=3D'font-size:86%'>H(x;p) =3D 0
nWant solutions at pfinal
nWe have solutions xstart,i for parameters pstart
n<= span style=3D'font-size:86%'>H(xstart,i;pst= art) =3D 0
nForm a parameter path
n<= span style=3D'font-size:86%'>p(t) =3D t pstart + (1-t) pfi= nal
nThis defines a homotopy
n<= span style=3D'font-size:86%'>H(x;p(t)) =3D 0
nNumerically follow solution path
n<= span style=3D'font-size:86%'>from t=3D1 to t=3D0
n
------=_NextPart_01C6DF35.194A2C70 Content-Location: file:///C:/1E95224D/Wampler_NumAlgGeom_files/slide0132.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Wampler: Numerical Algebraic Geometry
Example: Ellipse & Hyperbola
nWish to solve F(x,y)
<= span class=3DHB1B style=3D'position:absolute;left:-5.99%;top:.74em'>na1x2<= /span>+b1xy+c1y2+d1x+e1y+f1 =3D 0
na2x2<= /span>+b2xy+c2y2+d2x+e2y+f2 =3D 0
nKnow how to solve G(x,y)
<= span class=3DHB1B style=3D'position:absolute;left:-5.99%;top:.74em'>na1x2<= /span>+f1 =3D 0
<= span class=3DHB1B style=3D'position:absolute;left:-5.99%;top:.74em'>nc2y2<= /span>+f2 =3D 0
nHomotopy H(x,y,t)=3D0
nt(b1xy+c1y2<= /span>+d1x+e1y)+a1x2+f1 =3D 0
nt(a2x2<= /span>+b2xy+d2x+e2y)+c2y2+f2 =3D 0
nFollow 4 solution paths
nfrom t=3D0 to t=3D1.
n
n
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Solution paths
nImplicitly defined by H(x(t);p(t))= =3D 0
Nongeneric
´= <= i>
´<= i>
´= <= i>
´<= i>
Parameter space
t
pstar= t
pfina= l
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An Ill-Conceived Homotopy
nQ: How do we make sure this doesn’t happen?&= #13;
nA: Use complex space
n exceptions are
ncomplex co-dimension 1 =3D real codimension 2 =
nGeneral 1-dim parameter path miss exceptions with probability 1
Parameters for which H(x,p) has fewer = solutions
´= <= i>
´= <= i>
´<= i>
Parameter s= pace (real)
pstar= t
pfina= l
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Polynomial Structures
(A) Start system solved with <= span style=3D'position:absolute;top:80.48%;left:14.59%;width:73.72%;height:5.92= %'>linear algebra
(B) Start system solved via convex hulls, polytope theory
(C) Start system solved via (A) or (B) initial run
Landmarks
n all isolated solutions =
nGarcia & Zangwill, ‘77
nDrexler, ‘77
n total degree
nChow, Mallet-Paret & Yorke, ‘78
n projective space
nWright, ‘85
nMorgan, ‘86; book, ‘87
= Landmarks
nmulti= -homogeneous
nMorgan & Sommese, ‘87
nparam= eterized systems
nLi, Sauer & Yorke, ‘88
nMorgan & Sommese, ‘89
Landmarks
nPolyt= opes (BKK)
nVerschelde, Verlinden & Cools, ‘94
nHuber & Sturmfels, ’95
nGao & Li, ’03
nPolyn= omial products
n Morgan,Sommese & Wampler,’95
nSet structures
nVerschelde & Cools, ‘94
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Parameter Continuation
initial parameter space
target parameter space
nStart system easy in initial parameter space <= /span>
nRoot count may be much lower in target parameter space
nInitial run is 1-time investment for cheaper targe= t runs
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Positive-Dimensional Sets
nBasic Constructs
nWitness Sets
nIrreducible Decomposition
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13
Slicing & the Witness Cascade
nFundamental theorem of algebra
nA degree N <= /i>square-free polynomial p(x,y)=3D0 hits a gener= al horizontal line y=3Dc in N isolated points
nSlicing theorem
nAn degree N <= /i>reduced algebraic set of dimension m in n = variables hits a general (n-m)-dimensional linear space in N = isolated points
nWitness generation algorithm
nWitness points at every dimension
n= Relies on traditional homotopy properties to get all isolated solutions at each dimension
= <= /span>
= <= /span>
Sommese & Wampler, ’95
Sommese & Verschelde, ’00
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Witness Set
nSuppose AÎCn<= /span> is pure-m-dimensional algebraic set that is a solution of F(x)=3D0
nWitness set for A consists of:
nF(x) ð the system
na system of polynom= ials (straight-line function)
nL(x) ð generic slicing plane
na linear space of dimension (n-m)
nW =3D {x1,..., xd<= /span>} ð “Witness points”
nsolution points of {F(x),L(x)}=3D0
nd =3D degree of A
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Decomposed Witness Set
nPure-dimensional A=3D{A1,..., Ak}
nwhere each Ai is irreducible
nDecomposed witness set for A
nSystem, F(x)
nSlice, L(x)
nDecomposed witness point set
nW=3D{W1,..., Wk},
nwhere Wi=3D{x1,..., xdi<= /span>} is witness point set for Ai
nd1+...+dk=3Dd
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Irreducible Decomposition
nMixed-dimensional A=3D{A0,...,Ak}
nwhere each Ai is pure-i-dimensional <= /div>
nAi=3D{Ai1,...,Aiki<= /span>}, each Aij irreducible&#= 13;
nDecomposed witness set for A
nSystem, F(x)
nSlice, L(x)
nDecomposed witness point set
nW=3D{W0,..., Wk}, Wi<= /span>=3D{Wi1= ,...,Wik= i<= /span>},
nwhere Wij=3D{x1,..., xdi= } is witness point set for Aij
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Basic Operations
nIrreducible Decomposition
nWitness generate
nWitness decomposition
nMembership tests
nIntersection
nDeflation
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18
Irreducible Decomposition
nWitness Generation Algorithm
ngives points organized by dimension
nmay include “junk” points <= /span>
nWitness Classify
neliminates junk
ngroups points by irreducible components
n
n
= <= /span>
= <= /span>
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Membership Test
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Irreducible Decomposition
nStep 1: eliminate junk points
nThey lie on higher-dimensional sets
nUse membership test
nA local dimension test would be better!
nStep 2: break the rest into components
nMonodromy finds points that are connected
nLike the membership test, but around a closed path= in the space of slicing planes
nLinear trace verifies that groups are complete
nExhaustive trace testing is feasible on small sets
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Linear Traces
Sasaki, 2001
Rupprecht, 2004
Sommese, Verschelde & Wampler, 2002
nTrack witness paths as slice translates parallel to = itself.
nCentroid of witness points for an algebraic set = must move on a line.
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Intersecting Components
nWitness Cascade = ;
ntreats a system all at once&#= 13;
nWitness Classify= 3;
nbreaks solution into its irreducible pieces
nWhat if we want to intersect two pieces found in this way?
nset A solution of F(x)=3D0= 3;
nset B solution of G(x)=3D0= 3;
nFind A   B
n
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Diagonal Homotopy for A   B
nConsider the set AxB
n<= span style=3D'mso-hansi-font-family:Tahoma;font-size:71%'>It is a solution comp= onent of {F(x),G(y)}=3D0
n<= span style=3D'mso-hansi-font-family:Tahoma;font-size:71%'>AxB is irreducible &#= 13;
nDiagonal Homotopy fin= ds irreducible decomposition of
n<= span style=3D'mso-hansi-font-family:Tahoma;font-size:71%'>(AxB)    {(x,y) | x=3Dy} <= /span>
n<= span style=3D'mso-hansi-font-family:Tahoma;font-size:71%'>Start points (= ai, bj) from WAxWB<= /span> 
n
Sommese, Vershelde & Wample= r, 2004
nGiven:
nWitness sets WA,WB for irreducibles A and B
nFind:
nWitness set for A    B
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Deflation
nSome irreducible component of f-1(0), say Z, may be nonreduced
nThis makes path tracking on Z difficult
nHow can we do monodromy, traces, etc?
nWish to replace f(x) with some g(x) such that       is a component of g-1(0)
nDeflation generates a g(x,u) such that         a component of g-1(0) projects naturally one-to-one to
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How to Deflate a Point
nSuppos= e z is an isolated root of square system f(x)=3D0
n        = ;       is singular, say rank r<n
nAppend new equations
n
nNew system has isolated root of lower multiplicity
nmultip= licity m point can be deflated in (m-1) or fewer iterations
nInitial ideas: Ojika 1987
nAlgorithm: Leykin, Verschelde & Zhao 2004
nSee also, Dayton & Zeng 2006
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How to Deflate a Component
nSlice to get a witness set
nA generic slice isolates a generic point
nDeflate the witness point
nThe same deflation equations work on a Zariski open subset of the component
nDone!
nSommese & Wampler 2005
n
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Higher-Level Algorithms
n Equation-by-equation intersections=
n Finding the real points in a compl= ex component
n Findi= ng sets of exceptional dimension
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28
Subsystem-by-Subsystem Intersectio= n
Solving A   B     <= span style=3D'position:absolute;top:61.75%;left:2.62%;width:22.28%;height:5.5%'= >on Cn\Q
<= span style=3D'font-family:Tahoma'>A & B not= = irreducible
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Equation-by-Equation Solving
f1(x)=3D0 à Co-dim 1
f2(x)=3D0 à Co-dim 1
f3(x)=3D0 à Co-dim 1
Diagonal homotopy
Co-dim 1,2
Diagonal homotopy
Co-dim 1,2,3
Co-dim 1,2,...,N-1
fN(x)=3D0 à Co-dim 1
Diagonal homotopy
Co-dim 1,2,...,min(n,N)<= /div>
Final Result
Similar diagonal intersectio= ns
Special case:
N=3Dn
nonsingular solutions only
initial results show promise
N equations, n variables
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Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlh1AAcAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAADS ABsAgAAAADMzzAKQjI+py+0Po5y02osD2Lz7D4biSJbmiabqyq5GC8fyTNd2+976zvf+nPsJh8Ri LWhMKpdMJPMJjR410qr1anJit1yptgsOE7/istlGPqvXqjT7DQe543T4vI4/3/N88L4PePUXSAg1 WIiYdJjIKLTYCLnzGElJM1mJiUOVyelz2QlK8hlK+pFxipqqusraOlEAADs= ------=_NextPart_01C6DF35.194A2C70 Content-Location: file:///C:/1E95224D/Wampler_NumAlgGeom_files/slide0162.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Wampler: Numerical Algebraic Geometry
Some Application Examples
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