mechanism and robot kinematics, part ii: …...n numerical algebraic geometry n builds on the...
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Mechanism and Robot Kinematics, Part II:
Numerical Algebraic Geometry
Charles WamplerGeneral Motors R&D Center
Including joint work with
Andrew Sommese, University of Notre DameJan Verschelde, Univ. Illinois ChicagoAlexander Morgan, GM R&D
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Outline
n Zero-dimensional solution setsn Numerical solution by polynomial continuation
n Root counts and homotopiesn Parameter homotopies
n Positive-dimensional solution setsn Basic constructs
n Witness setsn Numerical irreducible decomposition
n Basic operationsn Intersection of algebraic setsn Deflation of nonreduced sets
n Higher-level operationsn Equation-by-equation intersectionsn Fiber productsn Extracting real points from a complex set
n Applications
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Numerical Algebraic Geometry
n Purposen Numerically represent & manipulate algebraic sets
n Approachn Numerical continuation operating on witness sets
Basic Operationsn Witness generaten Witness decompositionn Membership testsn Intersectionn Deflation
Basic Constructsn Witness setsn Irreducible decomposition
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Why study polynomial systems?
n Application areasn Economics & financen Chemical equilibrium n Computer-aided Geometric Design (CAGD)n Control theoryn Kinematics
n Constrained mechanical motionn Linkages for motion constraint & transformation
n Suspensions, engines, swing panels, etc.
n Computer-controlled motion devicesn Robots, human-assist devices, etc.
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Zero-Dimensional Sets
n Solving by polynomial continuation
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What is Continuation?
n For some class of parameterized problems:n H(x;p) = 0
n Want solutions at pfinal
n We have solutions xstart,i for parameters pstartn H(xstart,i;pstart) = 0
n Form a parameter pathn p(t) = t pstart + (1-t) pfinal
n This defines a homotopyn H(x;p(t)) = 0
n Numerically follow solution path n from t=1 to t=0
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Example: Ellipse & Hyperbola
n Wish to solve F(x,y)n a1x2+b1xy+c1y2+d1x+e1y+f1 = 0n a2x2+b2xy+c2y2+d2x+e2y+f2 = 0
n Know how to solve G(x,y)n a1x2+f1 = 0n c2y2+f2 = 0
n Homotopy H(x,y,t)=0n t(b1xy+c1y2+d1x+e1y)+a1x2+f1 = 0n t(a2x2+b2xy+d2x+e2y)+c2y2+f2 = 0
n Follow 4 solution paths n from t=0 to t=1.
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Solution paths
n Implicitly defined by H(x(t);p(t)) = 0
Nongeneric
x
×
×
×
×Parameter space
tpstart
pfinal
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An Ill-Conceived Homotopy
n Q: How do we make sure this doesn’t happen?n A: Use complex space
n exceptions are complex co-dimension 1 = real codimension 2
n General 1-dim parameter path miss exceptions with probability 1
Parameters for which H(x,p) has fewer
solutions
x
××
×Parameter space
(real)
pstart
pfinal
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Polynomial Structures
(A) Start system solved with 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, ‘77nDrexler, ‘77
n total degreenChow, Mallet-Paret & Yorke, ‘78
n projective spacenWright, ‘85nMorgan, ‘86; book, ‘87
Landmarksnmulti-homogeneous
nMorgan & Sommese, ‘87nparameterized systems
nLi, Sauer & Yorke, ‘88nMorgan & Sommese, ‘89
LandmarksnPolytopes (BKK)
nVerschelde, Verlinden & Cools, ‘94nHuber & Sturmfels, ’95nGao & Li, ’03
nPolynomial productsn Morgan,Sommese & Wampler,’95
nSet structuresnVerschelde & Cools, ‘94
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Parameter Continuation
initial parameter
space
target parameter
space
n Start system easy in initial parameter spacen Root count may be much lower in target parameter spacen Initial run is 1-time investment for cheaper target runs
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Positive-Dimensional Sets
nBasic ConstructsnWitness SetsnIrreducible Decomposition
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Slicing & the Witness Cascade
n Fundamental theorem of algebran A degree N square-free
polynomial p(x,y)=0 hits a general horizontal line y=c in Nisolated points
n Slicing theoremn An degree N reduced algebraic
set of dimension m in n variables hits a general (n-m)-dimensional linear space in Nisolated points
n Witness generation algorithmn Witness points at every
dimensionn Relies on traditional homotopy
properties to get all isolated solutions at each dimension
Sommese & Wampler, ’95Sommese & Verschelde, ’00
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Witness Set
n Suppose A∈Cn is pure-m-dimensional algebraic set that is a solution of F(x)=0
n Witness set for A consists of:n F(x) ð the system
n a system of polynomials (straight-line function)n L(x) ð generic slicing plane
n a linear space of dimension (n-m)n W = {x1,..., xd} ð “Witness points”
n solution points of {F(x),L(x)}=0n d = degree of A
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Decomposed Witness Set
n Pure-dimensional A={A1,..., Ak}n where each Ai is irreducible
n Decomposed witness set for An System, F(x)n Slice, L(x)n Decomposed witness point set
n W={W1,..., Wk}, n where Wi={x1,..., xdi} is witness point set for Ai
n d1+...+dk=d
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Irreducible Decomposition
n Mixed-dimensional A={A0,...,Ak}n where each Ai is pure-i-dimensionaln Ai={Ai1,...,Aiki}, each Aij irreducible
n Decomposed witness set for An System, F(x)n Slice, L(x)n Decomposed witness point set
n W={W0,..., Wk}, Wi={Wi1,...,Wiki}, n where Wij={x1,..., xdi} is witness point set for Aij
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Basic Operations
nIrreducible DecompositionnWitness generatenWitness decomposition
nMembership testsnIntersectionnDeflation
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Irreducible Decomposition
n Witness Generation Algorithm n gives points organized by dimensionn may include “junk” points
n Witness Classify n eliminates junkn groups points by irreducible components
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Membership Test
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Irreducible Decomposition
n Step 1: eliminate junk pointsn They lie on higher-dimensional sets
n Use membership testn A local dimension test would be better!
n Step 2: break the rest into componentsn Monodromy finds points that are connected
n Like the membership test, but around a closed path in the space of slicing planes
n Linear trace verifies that groups are completen Exhaustive trace testing is feasible on small sets
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Linear Traces
Sasaki, 2001Rupprecht, 2004Sommese, 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
n Witness Cascaden treats a system all at once
n Witness Classifyn breaks solution into its irreducible pieces
n What if we want to intersect two pieces found in this way?n set A solution of F(x)=0n set B solution of G(x)=0n Find A B
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Diagonal Homotopy for A B
n Consider the set AxBn It is a solution component of {F(x),G(y)}=0n AxB is irreducible
n Diagonal Homotopy finds irreducible decomposition ofn (AxB) {(x,y) | x=y}n Start points (ai, bj) from WAxWB Sommese, Vershelde & Wampler, 2004
n Given: n Witness sets WA,WB for irreducibles A and B
n Find:n Witness set for A B
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Deflation
n Some irreducible component of f-1(0), say Z, may be nonreducedn This makes path tracking on Z difficult
n How can we do monodromy, traces, etc?
n Wish to replace f(x) with some g(x) such that is a component of g-1(0)
n Deflation generates a g(x,u) such that a component of g-1(0) projects naturally one-to-one to
Z
Z
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How to Deflate a Point
n Suppose z is an isolated root of square system f(x)=0n is singular, say rank r<nn Append new equations
n New system has isolated root of lower multiplicityn multiplicity m point can be deflated in (m-1) or
fewer iterationsn Initial ideas: Ojika 1987n Algorithm: Leykin, Verschelde & Zhao 2004n See also, Dayton & Zeng 2006
)()( zxf
zJ∂∂
=
1, random
0))((:),(ˆ×× ∈∈
=+=nrn bB
bBuxJuxf
RR
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How to Deflate a Component
n Slice to get a witness setn A generic slice isolates a generic point
n Deflate the witness pointn The same deflation equations work on a
Zariski open subset of the componentn Done!
n Sommese & Wampler 2005
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Higher-Level Algorithms
n Equation-by-equation intersectionsn Finding the real points in a complex componentn Finding sets of exceptional dimension
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Subsystem-by-Subsystem Intersection
Solving A B on Cn\Q
A & B notirreducible
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Equation-by-Equation Solving
f1(x)=0 à Co-dim 1
f2(x)=0 à Co-dim 1
f3(x)=0 à Co-dim 1
Diagonal homotopy
Co-dim 1,2Diagonal homotopy
Co-dim 1,2,3
Co-dim 1,2,...,N-1
fN(x)=0 à Co-dim 1
Diagonal homotopy Co-dim 1,2,...,min(n,N)
Final Result
Similar diagonal intersections
•Special case:•N=n
•nonsingular solutions only
•initial results show promise
N equations, n variables
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Some Application Examples
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Example: 7-bar Structure
Problem:
Assemble these 7 pieces, as labeled.
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Result for Generic Links
18 rigid structures
• 8 real, 10 complex for this set of links.
•All isolated – can be found with traditional homotopy
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Special Links (Roberts Cognates)
Dimension 1:
6th degree four-bar motion
Dimension 0:
1 of 6 isolated (rigid) assemblies
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Example: Griffis-Duffy Platform
Special Stewart-Gough platform
Studied by:
Husty & Karger, 2000
Degree 28 motion curve (in Study coordinates)
• if legs are equal & plates congruent:
•factors as 6+(6+6+6)+4
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Finding Exceptional Mechanisms
n (S&W 2006, preprint) n for high enough j, the j th fiber product
contains an irreducible component that is the main component of the fiber product
where Z is an exceptional mechanism in Mn Efficient algorithms for computing fiber products
are under studyn More to come: Industrial Problems Seminar 9/29
44 344 21 L timesj
PPjP MMM ××=Π
ZjPΠ
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Extracting Real Points
n Numerical irreducible decompositionn finds complex solution components
n Applications care about real solutionsn 0-dimensional components
n Just check the magnitude of imaginary parts
n Higher-dimensional componentsn More difficultn Real dimension = complex dimensionn # of real connected pieces can be highn For the case of curves, two procedures required:
n Find singular points of self-conjugate complex componentsn Find intersections of conjugate pairs of componentsn Lu, Bates, Sommese, Wampler 2006n see next week’s workshop!
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Further Reading
World Scientific 2005
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Summaryn Polynomials arise in applications
n Especially kinematicsn Continuation methods for isolated solutions
n Highly developed in 1980’s, 1990’sn Numerical algebraic geometry
n Builds on the methods for isolated rootsn Treats positive-dimensional setsn Witness sets are the key construct
n Open problemsn Local dimension testn Multihomogeneous or BKK w/higher dimen’l setsn Real sets of higher dimensionn Efficient algorithm for exceptional sets