Topological complexity of motion planning algorithms

Transcription

1 Topological complexity of motion planning algorithms Mark Grant School of Mathematics - University of Edinburgh 31st March 2011 Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

2 Overview 1 Configuration spaces 2 Topology 3 Topological complexity of motion planning 4 Methods of computation 1 Upper bounds 2 Lower bounds 5 Future work Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

3 Configuration spaces The set of attainable configurations of a mechanical system form a topological space X, the so-called configuration space or C-space of the system {points in X } {configurations of system} {continuous paths in X } { continuous motions of system} Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

4 Configuration spaces Obstacles in workspace appear thickened in C-space, where the robot is represented as a point Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

5 Configuration spaces Flying robot in 3D workspace may have X = SO(3) R 3 (orientation) (position) Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

6 Configuration spaces Configurations of robot arm given by angles θ i S 1 or [a i, b i ] and extensions q j [a j, b j ] X = S 1 S 1 S 1 [a 1, b 1 ] [a n, b n ] Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

7 Configuration spaces The configuration space of k particles of radius r in a box is X (k, r) [0, 1] 3k Its topology is not well understood, except in a few cases (high or low density, k = 3) Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

8 Topology Topological spaces A topological space is a set X with a collection of open subsets satisfying: X, X X are open U V open whenever U, V open λ U λ open whenever each U λ is open Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

9 Topology Topological spaces A topological space is a set X with a collection of open subsets satisfying: X, X X are open U V open whenever U, V open λ U λ open whenever each U λ is open Allows to formalize notions such as nearby points, continuity, connectedness, convergence,... Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

10 Topology Most topologies come from a measure of distance Metric spaces A metric on a set X is a function d : X X R satisfying 1 d(x, y) 0 and d(x, y) = 0 if and only if x = y 2 d(y, x) = d(x, y) 3 d(x, z) d(x, y) + d(y, z) (triangle inequality) The pair (X, d) is called a metric space Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

11 Topology For example, R n = {x = (x 1, x 2,..., x n ) x i R} has its Euclidean metric d(x, y) = (x 1 y 1 ) (x n y n ) 2, and any subset X R n inherits this metric Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

12 Topology The metric topology on (X, d) has open sets U X characterised by: For each x U, the open ball B(x, ε) = {y X d(x, y) < ε} is contained in U for some ε > 0 Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

13 Topology Continuity A function f : X Y between topological spaces is continuous if U open in Y implies f 1 (U) = {x X f (x) U} open in X The identity function Id X : X X, Id X (x) = x and the constant function X {x 0 } to the one-point space are continuous for any space X Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

14 Topology Intuitively, f takes nearby points to nearby points Metric continuity A function f : (X, d X ) (Y, d Y ) between metric spaces is continuous at x X if for every ε > 0 there exists δ > 0 such that f ( B(x, δ) ) B ( f (x), ε ), and f is continuous if it is continuous at all x X Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

15 Topology Homeomorphism Spaces X and Y are homeomorphic if there are continuous maps f : X Y and g : Y X such that f g = Id Y and g f = Id X Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

16 Topology If X and Y are spaces, there is a natural product topology on X Y = {(x, y) x X, y Y } Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

17 Topology If X and Y are spaces, there is a natural product topology on X Y = {(x, y) x X, y Y } The interval [a, b] = {x R a x b} inherits its topology from that of R Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

18 Topology Homotopy Two continuous maps f 0, f 1 : X Y are homotopic if there is a continuous map H : X [0, 1] Y such that f 0 (x) = H(x, 0) and f 1 (x) = H(x, 1) for all x X. Then H is a homotopy, and we write f 0 f 1 Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

19 Topology Homotopy Two continuous maps f 0, f 1 : X Y are homotopic if there is a continuous map H : X [0, 1] Y such that f 0 (x) = H(x, 0) and f 1 (x) = H(x, 1) for all x X. Then H is a homotopy, and we write f 0 f 1 A homotopy H : S 1 [0, 1] X between loops f 0, f 1 : S 1 X Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

20 Topology Homotopy of spaces Two spaces X, Y are homotopy equivalent, written X Y, if there are continuous maps f : X Y and g : Y X such that f g Id Y and g f Id X Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

21 Topology Contractibility X is contractible if X {x 0 } Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

22 Topology Contractibility X is contractible if X {x 0 } R 2 is contractible, but R 2 D is not Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

23 Topology Paths A path in a space X is a continuous function γ : [0, 1] X The path space of (X, d) is the metric space (X [0,1], d path ) where X [0,1] = {paths γ : [0, 1] X }, d path (γ, ω) = sup d(γ(t), ω(t)) t [0,1] Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

24 Topology Paths A path in a space X is a continuous function γ : [0, 1] X The path space of (X, d) is the metric space (X [0,1], d path ) where X [0,1] = {paths γ : [0, 1] X }, d path (γ, ω) = sup d(γ(t), ω(t)) t [0,1] Can talk about nearby paths Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

25 Topological complexity of motion planning Let X be a C-space Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

26 Topological complexity of motion planning Let X be a C-space A Motion Planning Algorithm in X takes as input a pair (A, B) X X and outputs a path γ : [0, 1] X from A to B Assumption: There always exists such a path (X is path-connected) Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

27 Topological complexity of motion planning Topological reformulation Consider the endpoint map π : X [0,1] X X, π(γ) = ( γ(0), γ(1) ). Then an MPA in X is a section of this, that is, a function s : X X X [0,1] such that π ( s(a, B) ) = (A, B) Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

28 Topological complexity of motion planning Topological reformulation Consider the endpoint map π : X [0,1] X X, π(γ) = ( γ(0), γ(1) ). Then an MPA in X is a section of this, that is, a function s : X X X [0,1] such that π ( s(a, B) ) = (A, B) Theorem (M. Farber) There exists a continuous section s as above if and only if X is contractible Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

29 Topological complexity of motion planning So, MPAs may have essential discontinuities, or instabilities, due to the topology of X. Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

30 Topological complexity of motion planning So, MPAs may have essential discontinuities, or instabilities, due to the topology of X. Our premise It is desirable to minimise these discontinuities, to produce optimally stable MPAs Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

31 Topological complexity of motion planning An MPA s : X X X [0,1] is called tame if there is a finite partition X X = F 1 F 2 F k such that 1 F i F j = for i = j 2 Each s i = s Fi : F i X [0,1] is continuous 3 Each F i is nice (ENR) Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

32 Topological complexity of motion planning Definition The Topological Complexity of the space X, denoted TC(X ), is the smallest k such that there exists a tame MPA on X with sets X X = F 1 F 2 F k Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

33 Topological complexity of motion planning Example: The n-sphere S n = {x R n+1 d(x, 0) = 1} is path-connected (n > 0) and not contractible Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

34 Topological complexity of motion planning Case 1: n odd There is a nowhere vanishing vector field v on S n Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

35 Topological complexity of motion planning Case 1: n odd There is a nowhere vanishing vector field v on S n F 1 = {(A, B) S n S n B = A}, s 1 = shortest path A to B Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

36 Topological complexity of motion planning Case 1: n odd There is a nowhere vanishing vector field v on S n F 1 = {(A, B) S n S n B = A}, s 1 = shortest path A to B F 2 = {(A, A)}, s 2 = equator in direction v(a) Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

37 Topological complexity of motion planning Case 2: n even Vector field v with two zeroes A 0, A 0 S n F 1 = {(A, B) S n S n B = A}, s 1 = shortest path A to B Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

38 Topological complexity of motion planning Case 2: n even Vector field v with two zeroes A 0, A 0 S n F 1 = {(A, B) S n S n B = A}, s 1 = shortest path A to B F 2 = {(A, A) A = A 0, A 0 }, s 2 = equator in direction v(a) Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

39 Topological complexity of motion planning Case 2: n even Vector field v with two zeroes A 0, A 0 S n F 1 = {(A, B) S n S n B = A}, s 1 = shortest path A to B F 2 = {(A, A) A = A 0, A 0 }, s 2 = equator in direction v(a) F 3 = {(A 0, A 0 ), ( A 0, A 0 )}, s 3 = any paths Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

40 Topological complexity of motion planning So TC(S n ) = { 2 if n odd 2 or 3 if n even Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

41 Topological complexity of motion planning The flying robot has C-space X = SO(3) R 3 Theorem (M. Farber) If X Y then TC(X ) = TC(Y ) (Homotopy Invariance) Since X SO(3), we have TC(X ) = TC(SO(3)) Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

42 Topological complexity of motion planning The Lusternik-Schnirelmann category cat(x ) of X is the smallest k such that X = U 1 U 2 U k where each U i X is open and U i X is homotopic to a constant Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

43 Topological complexity of motion planning The Lusternik-Schnirelmann category cat(x ) of X is the smallest k such that X = U 1 U 2 U k where each U i X is open and U i X is homotopic to a constant Theorem (M. Farber) If X is a topological group, then TC(X ) = cat(x ) It is known that cat(so(3)) = 4, therefore the flying robot has TC(X ) = 4 Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

44 Methods of computation: Upper bounds General dimensional upper bound TC(X ) 2 dim(x ) + 1 X is simply connected if any loop γ : S 1 X is homotopic to a constant If X is simply connected then TC(X ) dim(x ) + 1 Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

45 Methods of computation: Upper bounds Product formula TC(X Y ) TC(X ) + TC(Y ) 1 The robot arm has X = S 1 S 1 S 1 [a 1, b 1 ] [a n, b n ] (S 1 ) n Hence TC(X ) (n 1) = 2n (n 1) = n + 1 Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

46 Methods of computation: Lower bounds Lower bounds for TC(X ) come from Algebraic Topology AT is (roughly) the study of functors {Topological spaces} {Algebraic objects} X F (X ) {Continuous maps} {Homomorphisms} f : X Y F (f ): F (X ) F (Y ) or F (f ): F (Y ) F (X ) Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

47 Methods of computation: Lower bounds Cohomology is a functor {Topological spaces} {Graded rings} X H (X ) = H 0 (X ) + H 1 (X ) + H 2 (X ) + f : X Y H (f ): H (Y ) H (X ) Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

48 Methods of computation: Lower bounds Cohomology is a functor {Topological spaces} {Graded rings} X H (X ) = H 0 (X ) + H 1 (X ) + H 2 (X ) + f : X Y H (f ): H (Y ) H (X ) In particular, the diagonal map Δ: X X X given by Δ(x) = (x, x) induces a ring homomorphism H (X X ) H (Δ) H (X ) Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

49 Methods of computation: Lower bounds Theorem (M. Farber) If there are elements z 1, z 2,..., z n H (X X ) such that H (Δ)(z i ) = 0 and z 1 z 2 z n = 0, then TC(X ) > n Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

50 Methods of computation: Lower bounds Theorem (M. Farber) If there are elements z 1, z 2,..., z n H (X X ) such that H (Δ)(z i ) = 0 and z 1 z 2 z n = 0, then TC(X ) > n This allows us to show, for example, that TC(S n ) = { 2 if n odd 3 if n even TC(robot arm) = n + 1 Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

51 Methods of computation: Lower bounds Theorem (M. Farber) If there are elements z 1, z 2,..., z n H (X X ) such that H (Δ)(z i ) = 0 and z 1 z 2 z n = 0, then TC(X ) > n This allows us to show, for example, that TC(S n ) = { 2 if n odd 3 if n even TC(robot arm) = n + 1 The lower bound can be refined using cohomology operations, which impose more algebraic structure on cohomology Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

52 Future work Open questions TC =?? (4 or 5) Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

53 Future work Open questions TC =?? (4 or 5) The real projective n-space is RP n = {lines through 0 in R n+1 } TC(RP n ) = smallest dimension where RP n immerses =?? Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

54 Future work Navigation functions A navigation function on a Riemannian manifold X is a smooth function F : X X R satisfying various metric-like properties, such as 1 F(x, y) 0 2 F(x, y) = 0 if and only if x = y Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

55 Future work Navigation functions A navigation function on a Riemannian manifold X is a smooth function F : X X R satisfying various metric-like properties, such as 1 F(x, y) 0 2 F(x, y) = 0 if and only if x = y Can construct MPAs using the negative gradient flow grad(f) (direction of steepest descent) Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

56 Future work What to do when the topology of C-space is not known in advance? Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

57 Future work What to do when the topology of C-space is not known in advance? Treat X and TC(X ) as random variables and employ probablilistic techniques? How to learn X by sampling? Persistent homology? Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39

58 References 1 M. Farber, Topological complexity of motion planning, Discrete Comput. Geom. 29 (2003), M. Farber, Instabilities of robot motion, Topology Appl. 140 (2004), M. Farber, Topology of robot motion planning, in: Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology (P. Biran et al (eds.)) (2006), M. Farber, Invitation to Topological Robotics, EMS (2008) Mark Grant (University of Edinburgh) TC of MPAs 31st March / 39