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
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