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

2

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.

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

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

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

Higher-Level Algorithms

n Equation-by-equation intersectionsn Finding the real points in a complex componentn Finding sets of exceptional dimension

28

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

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