mechanism and robot kinematics, part ii: …...n numerical algebraic geometry n builds on the...
Post on 18-Jul-2020
1 Views
Preview:
TRANSCRIPT
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
3
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
4
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
6
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
7
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.
8
Solution paths
n Implicitly defined by H(x(t);p(t)) = 0
Nongeneric
x
×
×
×
×Parameter space
tpstart
pfinal
9
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
10
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
11
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
13
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
14
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
15
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
16
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
18
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
19
Membership Test
20
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
21
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.
22
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
23
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
24
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
25
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
26
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
29
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
31
Example: 7-bar Structure
Problem:
Assemble these 7 pieces, as labeled.
32
Result for Generic Links
18 rigid structures
• 8 real, 10 complex for this set of links.
•All isolated – can be found with traditional homotopy
33
Special Links (Roberts Cognates)
Dimension 1:
6th degree four-bar motion
Dimension 0:
1 of 6 isolated (rigid) assemblies
34
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
35
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
36
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!
37
Further Reading
World Scientific 2005
38
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
top related