Localization for Group of Robots using Matrix Contractors

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1 for Group of Robots using Matrix Contractors Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, Manchester Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

2 Classical contractors Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

3 A (classical) contractor associated with a set X R n is an operator such that for all boxes [x] IR n : Contraction: C ([x]) [x], C : IR n IR n Completeness: C ([x]) X = [x] X Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

4 Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

5 A matrix contractor of a constraint Γ(A,B,...) is an operator C : IR m a n a IR m b n b IR m a n a IR m b n b... such that ([A 1 ],[B 1 ],...) = C ([A],[B],...) satisfies : Contraction: [A 1 ] [A], [B 1 ] [B],... Completeness: Γ(A,B,...),A [A], B [B],... implies that A [A 1 ], B [B 1 ],... Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

6 Example 1. The minimal contractor for the unary constraint S = S T is C ([S]) = [S] [S] T. where [S] IR n n. For instance ( ) ( [1,3] [2,4] [1,3] [2,4] C = [ 1, 3] [ 1, 1] [ 1, 3] [ 1, 1] = ) ( [1,3] [ 1,3] [2, 4] [ 1, 1] ( [1,3] [2,3] [2, 3] [ 1, 1] ). ) Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

7 Example 2. The minimal contractor for the ternary constraint A + B = C is [A] [C] [B] C plus [B] = [C] [A]. [C] [A] + [B] Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

8 Example 3. For A B = C or for R T = R 1, no minimal contractor exists in the literature, to our knowledge. Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

9 Example 4. The minimal contractor C associated to P R m m is a positive semi-definite matrix has been proposed in [3]. For instance: [ 7, 3] [ 1, 4] [ 5, 4] C [ 1, 4] [ 8, 3] [2, 9] = [ 5, 4] [2, 9] [4, 9] [0, 3] [ 1, 2] [ 4, 4] [ 1, 2] [0.4, 3] [2, 5.2] [ 4, 4] [2, 5.2] [4, 9] Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

10 Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

11 The set of angles A is not a lattice. Thus, we cannot define intervals of angles. Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

12 Bisectable abstract domains Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

13 Generalize interval algorithms with bisections on a Riemannian manifold M such a R, R n, a sphere, the Klein bottle, etc. Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

14 Is such a paving always possible? How to define intervals of M? Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

15 From the distance d (a,b) between a and b, we define the diameter w (X), X M. A bad family IM is a family of subsets of M which satisfies [2]: Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

16 1) IM is a Moore family (containing M), i.e., i,[a](i) IM i [a](i) IM 2) IM is equipped with a bisector, i.e., a function β : IM IM IM such that β ([x]) = {[a],[b]} : (i) [a] and [b] do not overlap, (ii) [a] and [b] cover [x] (iii) β minimizes max{w ([a]),w ([b])}. Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

17 Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

18 Embedding Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

19 The equation cos (α) = 1 where α is an angle has a unique solution: α = 2kπ, k Z. Angles form a Riemannian manifold. Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

20 Embedding. To avoid the ambiguity, we perform an embedding: ( ) cosα α sinα Or equivalenltly, using complex numbers: α cosα + i sinα Now, we introduce a pessimism (Embedding effect) Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

21 The interval angle [ π 2,π] is represented by the box [ 1,0] + i [0,1]. The sum can be done using the relation: cos(a + b) = cosa cosb sina sinb sin(a + b) = cosa sinb + sina cosb Other operations on angles can also be defined. A classical interval resolution can thus be done. Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

22 Example. Solve We take [a] = [b] = [ π 2,π]. a + b = 0,a [ π 2,π],b [π 2,π] Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

23 The forward contraction is cos(a + b) cos[a] cos[b] sin[a] sin[b] = cos ( [ π 2,π]) cos ( [ π 2,π]) sin ( [ π 2,π]) sin ( [ π 2,π]) = [ 1,0] [ 1,0] [0,1] [0,1] sin(a + b) cos[a] sin[b] + sin[a] cos[b] = [0,1] [0,1] = [ 1,1] = [ 1,0] [0,1] + [0,1] [ 1,0] = [ 1,0] + [ 1,0] = [ 2,0] Since 0 (1,0), the backward step yields (cos[a],sin[a]) = (cos[b],sin[b]) = ( 1,0). Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

24 Consider the equivalence relation α β β α 2π Z cos(α β) = 1 The set A of all angles is A = R = R 2πZ. Sets A,A A,A m n are Riemannian manifolds. Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

25 If α and β are angles and if ρ R, we can define α + β, α β and ρ α. Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

26 Arcs Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

27 An arc α is a connected subset of A. We have α = α, α with α A and α [0,π]. The set of all arcs is denoted by IA. We may define an arc arithmetic. For instance 1 0, 2 2 π = 1 0,π = 0,π. 2 2 Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

28 IA is not a Moore family. The smallest Moore family which contains IA is the unions of arcs. Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

29 Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

30 Note. It may be dangerous to deal with union of arcs. Example of Chabert[1]. With initial domains [x] = [y] = [1,9], { y = x 9(x 5) 2 = 16y an explosion of the number of intervals during the propagation occurs. Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

31 Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

32 Absolute. Estimate the pose of robots (x i,y i,θ i ) T,i {1,2,...}. Relative. Estimate the azimuth, the distances or the bearings. No absolute frame is needed. Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

33 Azimuth angles for three robots Azimuth is measured using a compass and goniometric sensors. Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

34 Azimuth contractor Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

35 Azimuth matrix is the matrix of azimuth angles α ij between i th robot and j th robot: 0 α 12 α 1n. A = α α(n 1)n. α n1 α n (n 1) 0 By convention, α ii = 0. Note that A ( R 2πZ) n 2 which is a Riemannian manifold. Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

36 This uncertainty can be represented under the form of an interval matrix. 0 [α 12 ] [α 1n ]. [A] = [α 21 ] [ ] [ α(n 1)n). ] [α n1 ] αn (n 1) 0 where [α ij ] IA. Recall that IA is not a Moore family. Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

37 For i,j,k i j k, azimuth (A) α ij α ji π (α ij α ik ) + (α jk α ji ) + (α ki α kj ) π (α ij α ki ) (α ji α jk ) + (α kj α ki ) This can used to build an azimuth contractor C az ([A]). Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

38 Distance contractor Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

39 The distance matrix associated with n points is 0 d 12 d 1n. D = d d(n 1)n d n1 d n (n 1) 0 To build distance contractor consistent with the constraint D is a distance matrix, we consider constraints such as: { distance(d) d ij = d ji d ij d ik + d kj Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

40 Distance-Azimuth contractor Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

41 The localization in terms of azimuth angles are done with respect to composition of azimuth C az ([A]) and distance C d ([D]) contractors. We can also use mixed constraints such as distaz(d,a) { sin(α ik α ij ) d ij = sin(α ki α kj ) d kj Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

42 With five robots Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

43 A localization problem with five robots with respect to azimuth and distance is considered. Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

44 Take: [A] = 0 [2, 2.1] [2.8, 3] [2.5, 3.3] [2.1, 2.2] [5.2, 5.3] 0 [1.9, 2] [3.1, 3.2] [1.6, 1.7] [0.1, 0.2] [1.2, 1.3] 0 [2.6, 2.7] [1.3, 1.5] [5.5, 6.5] [6.2, 6.3] [0.5, 0.6] 0 [0.7, 0.8] [0.9, 0.1] [1.4, 1.5] [1.7, 1.8] [2.2, 2.4] 0 [D] = 0 [9.9, 10] [8, 10] [22, 31] [10, 13] [9.8, 10] 0 [9, 12] [19, 22] [15, 20] [7, 9] [8, 11] 0 [19, 21] [7, 10] [23, 32] [20, 22] [20, 21] 0 [26, 28] [10, 15] [15, 22] [8, 10] [25, 30] 0 Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

45 The contracted azimuth matrix is 0 [2.0, 2.1] [2.9, 3] [2.5, 3.3] [2.1, 2.2] [5.2, 5.24] 0 [1.9, 2.0] [3.1, 3.2] [1.6, 1.7] [0.1, 0.2] [1.2, 1.3] 0 [2.6, 2.7] [1.3, 1.4] [5.6, 6.4] [6.2, 6.3] [0.5, 0.6] 0 [0.7, 0.8]. [0.9, 1.0] [1.4, 1.5] [1.7, 1.8] [2.3, 2.4] 0 Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

46 Pose of five robots with azimuth measurements Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

47 The contractions with respect to azimuth and distance are: 0 [2.0, 2.1] [2.94, 3] [2.6, 2.9] [2.1, 2.2] [5.2, 5.24] 0 [1.9, 1.94] [3.1, 3.2] [1.6, 1.7] [A] = [0.14, 0.2] [1.2, 1.3] 0 [2.6, 2.7] [1.3, 1.4] [5.90, 6.0] [6.2, 6.3] [0.5, 0.54] 0 [0.7, 0.8] [0.94, 1.0] [1.4, 1.5] [1.7, 1.8] [2.3, 2.4] 0 [D] = 0 [9.9, 10.1] [8.1, 9] [25, 30.6] [10.6, 13] [9.9, 10.1] 0 [10.3, 11] [20.4, 22] [16.7, 20] [8.1, 9] [10.3, 11] 0 [20, 21] [8, 10] [25.1, 30.6] [20.4, 22] [20, 21] 0 [26, 28] [10.6, 13] [16.7, 20] [8, 10] [26, 28] 0 Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

48 Pose of five robots before contraction Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

49 One contraction with respect to azimuth Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

50 After two contractions with respect to azimuth Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

51 Many contractions with respect to azimuth Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

52 Many contractions with respect to the distance Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

53 Many contractions with respect to both distance and azimuth Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

54 With 12 robots Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

55 A localization problem with 12 robots with respect to azimuth and distance is considered. Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

56 With 12 robots, before contraction Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

57 After contractions Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

58 Gilles Chabert. Techniques d intervalles pour la résolution de systèmes d équations. PhD dissertation, Nice, France, L. Jaulin, B. Desrochers, and D. Massé. Bisectable Abstract Domains for the resolution of equations involving complex numbers. Reliable Computing, 23:35 46, L. Jaulin and D. Henrion. Contracting optimally an interval matrix without loosing any positive semi-definite matrix is a tractable problem. Reliable Computing, 11(1):1 17, Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

59 Acknowledgement Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

60 The authors are thankful to the Science and Engineering Research Board, Department of Science and Technology, New Delhi, India for the funding to carry out the present research. N. R. Mahato is also supported by Raman-Charpak Fellowship 2016, Indo-French Centre for the Promotion of Advanced Research, New Delhi, India. Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for Manchester Group of Robots using Matrix Contractors

Introduction to Computational Manifolds and Applications

IMPA - Instituto de Matemática Pura e Aplicada, Rio de Janeiro, RJ, Brazil Introduction to Computational Manifolds and Applications Part - Constructions Prof. Marcelo Ferreira Siqueira mfsiqueira@dimap.ufrn.br