Mechanism and Robot Kinematics, Part II: Numerical Algebraic Geometry

Transcription

1 Mechanism and Robot Kinematics, Part II: Numerical Algebraic Geometry Charles Wampler General Motors R&D Center Including joint work with Andrew Sommese, University of Notre Dame Jan Verschelde, Univ. Illinois Chicago Alexander Morgan, GM R&D

2 Outline Zero-dimensional solution sets Numerical solution by polynomial continuation Root counts and homotopies Parameter homotopies Positive-dimensional solution sets Basic constructs Witness sets Numerical irreducible decomposition Basic operations Intersection of algebraic sets Deflation of nonreduced sets Higher-level operations Equation-by-equation intersections Fiber products Extracting real points from a complex set Applications 2

3 Numerical Algebraic Geometry Purpose Numerically represent & manipulate algebraic sets Approach Numerical continuation operating on witness sets Basic Constructs Witness sets Irreducible decomposition Basic Operations Witness generate Witness decomposition Membership tests Intersection Deflation 3

4 Why study polynomial systems? Application areas Economics & finance Chemical equilibrium Computer-aided Geometric Design (CAGD) Control theory Kinematics Constrained mechanical motion Linkages for motion constraint & transformation Suspensions, engines, swing panels, etc. Computer-controlled motion devices Robots, human-assist devices, etc. 4

5 Zero-Dimensional Sets Solving by polynomial continuation

6 What is Continuation? For some class of parameterized problems: H(x;p) = 0 Want solutions at p final We have solutions x start,i for parameters p start H(x start,i ;p start ) = 0 Form a parameter path p(t) = t p start + (1-t) p final This defines a homotopy H(x;p(t)) = 0 Numerically follow solution path from t=1 to t=0 6

7 Example: Ellipse & Hyperbola Wish to solve F(x,y) a 1 x 2 +b 1 xy+c 1 y 2 +d 1 x+e 1 y+f 1 = 0 a 2 x 2 +b 2 xy+c 2 y 2 +d 2 x+e 2 y+f 2 = 0 Know how to solve G(x,y) a 1 x 2 +f 1 = 0 c 2 y 2 +f 2 = 0 Homotopy H(x,y,t)=0 t(b 1 xy+c 1 y 2 +d 1 x+e 1 y)+a 1 x 2 +f 1 = 0 t(a 2 x 2 +b 2 xy+d 2 x+e 2 y)+c 2 y 2 +f 2 = 0 Follow 4 solution paths from t=0 to t=1. 7

8 Solution paths Implicitly defined by H(x(t);p(t)) = 0 x Nongeneric Parameter space p start t p final 8

9 An Ill-Conceived Homotopy x Parameter space (real) p start p final Parameters for which H(x,p) has fewer solutions Q: How do we make sure this doesn t happen? A: Use complex space exceptions are complex co-dimension 1 = real codimension 2 General 1-dim parameter path miss exceptions with probability 1 9

10 Polynomial Structures Landmarks Landmarks (C) Start system solved via (A) or (B) initial run Polytopes (B) Start system all isolated (BKK) solutions multi-homogeneous Verschelde, solved via Garcia & Zangwill, Verlinden 77 & Cools, 94 Huber Morgan convex hulls, Drexler, & 77 Sturmfels, & Sommese, polytope theory total Gao parameterized degree & Li, 03 systems Polynomial Li, Chow, products Sauer & Yorke, 88 Mallet-Paret & Yorke, 78 Morgan & Sommese, 89 projective Morgan,Sommese space & Wampler, 95 Set Wright, structures 85 Verschelde (A) Start system Morgan, 86; & book, Cools, solved with linear algebra 10

11 Parameter Continuation initial parameter space target parameter space Start system easy in initial parameter space Root count may be much lower in target parameter space Initial run is 1-time investment for cheaper target runs 11

12 Positive-Dimensional Sets Basic Constructs Witness Sets Irreducible Decomposition

13 Slicing & the Witness Cascade Fundamental theorem of algebra A degree N square-free polynomial p(x,y)=0 hits a general horizontal line y=c in N isolated points Slicing theorem An degree N reduced algebraic set of dimension m in n variables hits a general (n-m)- dimensional linear space in N isolated points Witness generation algorithm Witness points at every dimension Relies on traditional homotopy properties to get all isolated solutions at each dimension Sommese & Wampler, 95 Sommese & Verschelde, 00 13

14 Witness Set Suppose A C n is pure-m-dimensional algebraic set that is a solution of F(x)=0 Witness set for A consists of: F(x) the system a system of polynomials (straight-line function) L(x) generic slicing plane a linear space of dimension (n-m) W = {x 1,..., x d } Witness points solution points of {F(x),L(x)}=0 d = degree of A 14

15 Decomposed Witness Set Pure-dimensional A={A 1,..., A k } where each A i is irreducible Decomposed witness set for A System, F(x) Slice, L(x) Decomposed witness point set W={W 1,..., W k }, where W i ={x 1,..., x di } is witness point set for A i d d k =d 15

16 Irreducible Decomposition Mixed-dimensional A={A 0,...,A k } where each A i is pure-i-dimensional A i ={A i1,...,a ik i }, each A ij irreducible Decomposed witness set for A System, F(x) Slice, L(x) Decomposed witness point set W={W 0,..., W k }, W i ={W i1,...,w ik i }, where W ij ={x 1,..., x di } is witness point set for A ij 16

17 Basic Operations Irreducible Decomposition Witness generate Witness decomposition Membership tests Intersection Deflation

18 Irreducible Decomposition Witness Generation Algorithm gives points organized by dimension may include junk points Witness Classify eliminates junk groups points by irreducible components 18

19 Membership Test 19

20 Irreducible Decomposition Step 1: eliminate junk points They lie on higher-dimensional sets Use membership test A local dimension test would be better! Step 2: break the rest into components Monodromy finds points that are connected Like the membership test, but around a closed path in the space of slicing planes Linear trace verifies that groups are complete Exhaustive trace testing is feasible on small sets 20

21 Linear Traces Track witness paths as slice translates parallel to itself. Centroid of witness points for an algebraic set must move on a line. Sasaki, 2001 Rupprecht, 2004 Sommese, Verschelde & Wampler,

22 Intersecting Components Witness Cascade treats a system all at once Witness Classify breaks solution into its irreducible pieces What if we want to intersect two pieces found in this way? set A solution of F(x)=0 set B solution of G(x)=0 Find A B 22

23 Diagonal Homotopy for A B Given: Witness sets W A,W B for irreducibles A and B Find: Witness set for A Consider the set AxB It is a solution component of {F(x),G(y)}=0 AxB is irreducible Diagonal Homotopy finds irreducible decomposition of (AxB) {(x,y) x=y} B Start points (a i, b j ) from W A xw B Sommese, Vershelde & Wampler,

24 Deflation Some irreducible component of f -1 (0), say Z, may be nonreduced This makes path tracking on Z difficult How can we do monodromy, traces, etc? Wish to replace f(x) with some g(x) such that Z is a component of g -1 (0) Deflation generates a g(x,u) such that a component of g -1 (0) projects naturally one-to-one to Z 24

25 How to Deflate a Point Suppose z is an isolated root of square system f(x)=0 f J ( z) = ( z) is singular, say rank r<n x Append new equations fˆ ( x, u) : = J( x)( Bu n r n 1 random B R, b R New system has isolated root of lower multiplicity multiplicity m point can be deflated in (m-1) or fewer iterations Initial ideas: Ojika 1987 Algorithm: Leykin, Verschelde & Zhao 2004 See also, Dayton & Zeng b) = 0 25

26 How to Deflate a Component Slice to get a witness set A generic slice isolates a generic point Deflate the witness point The same deflation equations work on a Zariski open subset of the component Done! Sommese & Wampler

27 Higher-Level Algorithms Equation-by-equation intersections Finding the real points in a complex component Finding sets of exceptional dimension

28 Subsystem-by-Subsystem Intersection Solving A B on C n \Q A & B not irreducible 28

29 Equation-by-Equation Solving f 1 (x)=0 Co-dim 1 f 2 (x)=0 Co-dim 1 Co-dim 1,2 f 3 (x)=0 Co-dim 1 Co-dim 1,2,3 Diagonal homotopy Diagonal homotopy N equations, n variables Special case: N=n nonsingular solutions only initial results show promise Similar diagonal intersections Co-dim 1,2,...,N-1 f N (x)=0 Co-dim 1 Diagonal homotopy Final Result Co-dim 1,2,...,min(n,N) 29

30 Some Application Examples

31 Example: 7-bar Structure Problem: Assemble these 7 pieces, as labeled. 31

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 32

33 Special Links (Roberts Cognates) Dimension 1: 6 th degree four-bar motion Dimension 0: 1 of 6 isolated (rigid) assemblies 33

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 34

35 Finding Exceptional Mechanisms (S&W 2006, preprint) for high enough j, the j th fiber product j Π L PM = M P P M 43 j times contains an irreducible component that is the main component of the fiber product j Π P where Z is an exceptional mechanism in M Efficient algorithms for computing fiber products are under study More to come: Industrial Problems Seminar 9/29 Z 35

36 Extracting Real Points Numerical irreducible decomposition finds complex solution components Applications care about real solutions 0-dimensional components Just check the magnitude of imaginary parts Higher-dimensional components More difficult Real dimension = complex dimension # of real connected pieces can be high For the case of curves, two procedures required: Find singular points of self-conjugate complex components Find intersections of conjugate pairs of components Lu, Bates, Sommese, Wampler 2006 see next week s workshop! 36

37 Further Reading World Scientific

38 Summary Polynomials arise in applications Especially kinematics Continuation methods for isolated solutions Highly developed in 1980 s, 1990 s Numerical algebraic geometry Builds on the methods for isolated roots Treats positive-dimensional sets Witness sets are the key construct Open problems Local dimension test Multihomogeneous or BKK w/higher dimen l sets Real sets of higher dimension Efficient algorithm for exceptional sets 38