Distance Optimal Target Assignment in Robotic Networks under Communication and Sensing Constraints

Transcription

1 Distance Optial Target Assignent in Robotic Networks under Counication and Sensing Constraints Jingjin Yu BU Soon-Jo Chung Petros G. Voulgaris University of Illinois Supported by:

2 The Stochastic Target Assignent Proble

3 The Stochastic Target Assignent Proble Q = 0, [0,]

4 The Stochastic Target Assignent Proble Q = 0, [0,] X = {x,, x n }

5 The Stochastic Target Assignent Proble Q = 0, [0,] X = {x,, x n } Y = {y,, y n }

6 The Stochastic Target Assignent Proble Q = 0, [0,] X = {x,, x n } Y = {y,, y n } Control: x i = u i, u i =

7 The Stochastic Target Assignent Proble Q = 0, [0,] X = {x,, x n } Y = {y,, y n } Control: x i = u i, u i = σ: perutation that pairs x i with y σ(i)

8 The Stochastic Target Assignent Proble Q = 0, [0,] X = {x,, x n } Y = {y,, y n } Control: x i = u i, u i = σ: perutation that pairs x i with y σ(i) in D n = σ,{u i } i x i (t) dt

9 The Stochastic Target Assignent Proble, cont. 3

10 The Stochastic Target Assignent Proble, cont. r sense 3

11 The Stochastic Target Assignent Proble, cont. r sense 3

12 The Stochastic Target Assignent Proble, cont. r sense 3

13 The Stochastic Target Assignent Proble, cont. r sense G(t) 3

14 The Stochastic Target Assignent Proble, cont. r sense G(t) Given r sense and, how can we guarantee distance optiality? 3

15 The Stochastic Target Assignent Proble, cont. r sense G(t) Given r sense and, how can we guarantee distance optiality? Perforance of decentralized, hierarchical strategies (algoriths)? 3

16 Related Work 4

17 Related Work Sith and Bullo, Monotonic target assignent for robotic networks, IEEE Trans. Autoat. Control, vol. 4, no. 9, pp , 009 4

18 Related Work Sith and Bullo, Monotonic target assignent for robotic networks, IEEE Trans. Autoat. Control, vol. 4, no. 9, pp , 009 Treleaven, Pavone, and Frazzoli, Asyptotically optial algoriths for one-to-one pickup and delivery probles with applications to transportation systes, IEEE Trans. Autoat Control, vol. 9, no. 9, pp. 6 76, 03. 4

19 Related Work Sith and Bullo, Monotonic target assignent for robotic networks, IEEE Trans. Autoat. Control, vol. 4, no. 9, pp , 009 Treleaven, Pavone, and Frazzoli, Asyptotically optial algoriths for one-to-one pickup and delivery probles with applications to transportation systes, IEEE Trans. Autoat Control, vol. 9, no. 9, pp. 6 76, 03. Penrose, The longest edge of the rando inial spanning tree, Annals of Applied Probability, vol. 7, pp , 997. Penrose, Rando Geoetric Graphs, 003 4

20 Related Work Sith and Bullo, Monotonic target assignent for robotic networks, IEEE Trans. Autoat. Control, vol. 4, no. 9, pp , 009 Treleaven, Pavone, and Frazzoli, Asyptotically optial algoriths for one-to-one pickup and delivery probles with applications to transportation systes, IEEE Trans. Autoat Control, vol. 9, no. 9, pp. 6 76, 03. Penrose, The longest edge of the rando inial spanning tree, Annals of Applied Probability, vol. 7, pp , 997. Penrose, Rando Geoetric Graphs, 003 Erdős and Rényi, On a classical proble of probability theory, Publ. Math. Inst. Hung. Acad. Sci., vol. Ser. A 6, pp. 0, 96 4

21 Related Work Sith and Bullo, Monotonic target assignent for robotic networks, IEEE Trans. Autoat. Control, vol. 4, no. 9, pp , 009 Treleaven, Pavone, and Frazzoli, Asyptotically optial algoriths for one-to-one pickup and delivery probles with applications to transportation systes, IEEE Trans. Autoat Control, vol. 9, no. 9, pp. 6 76, 03. Penrose, The longest edge of the rando inial spanning tree, Annals of Applied Probability, vol. 7, pp , 997. Penrose, Rando Geoetric Graphs, 003 Erdős and Rényi, On a classical proble of probability theory, Publ. Math. Inst. Hung. Acad. Sci., vol. Ser. A 6, pp. 0, 96 Karaan and Frazzoli, Sapling-based Algoriths for Optial Motion Planning. Int. Journal of Robotics Research, vol. 30, no 7, pp , 0 4

22 Main Result Distance optiality guarantee Necessary and sufficient condition for distance optiality (non-stochastic) Non-asyptotic ε probabilistic guarantee for 0 < ε < r sense log ε r sense, r sense < 0 n log ε, r sense 0 Tight asyptotic bounds for high-probability guarantee Perforance of decentralized, hierarchical strategies Upper bound on the distance cost for arbitrary robot/target distribution O() asyptotic optiality guarantee under the unifor distribution

23 Main Result Distance optiality guarantee Necessary and sufficient condition for distance optiality (non-stochastic) Non-asyptotic ε probabilistic guarantee for 0 < ε < r sense log ε r sense, r sense < 0 n log ε, r sense 0 Tight asyptotic bounds for high-probability guarantee Perforance of decentralized, hierarchical strategies Upper bound on the distance cost for arbitrary robot/target distribution O() asyptotic optiality guarantee under the unifor distribution 3

24 Main Result Distance optiality guarantee Necessary and sufficient condition for distance optiality (non-stochastic) Non-asyptotic ε probabilistic guarantee for 0 < ε < r sense log ε r sense, r sense < 0 n log ε, r sense 0 Tight asyptotic bounds for high-probability guarantee Perforance of decentralized, hierarchical strategies Upper bound on the distance cost for arbitrary robot/target distribution O() asyptotic optiality guarantee under the unifor distribution 4

25 Main Result Distance optiality guarantee Necessary and sufficient condition for distance optiality (non-stochastic) Non-asyptotic ε probabilistic guarantee for 0 < ε < r sense log ε r sense, r sense < 0 n log ε, r sense 0 Tight asyptotic bounds for high-probability guarantee Perforance of decentralized, hierarchical strategies Upper bound on the distance cost for arbitrary robot/target distribution O() asyptotic optiality guarantee under the unifor distribution

26 Main Result Distance optiality guarantee Necessary and sufficient condition for distance optiality (non-stochastic) Non-asyptotic ε probabilistic guarantee for 0 < ε < r sense log ε r sense, r sense < 0 n log ε, r sense 0 Tight asyptotic bounds for high-probability guarantee Perforance of decentralized, hierarchical strategies Upper bound on the distance cost for arbitrary robot/target distribution O() asyptotic optiality guarantee under the unifor distribution n - nuber of robots 6

27 Distance Optiality Guarantee Theore (Necessary and Sufficient Conditions for Distance Optiality). Under sensing and counication constraints, distance optiality can be guaranteed if and only if at t = 0,. Every robot can counicate with every other robot,. Each target is observable by soe robot. 6

28 Distance Optiality Guarantee Theore (Necessary and Sufficient Conditions for Distance Optiality). Under sensing and counication constraints, distance optiality can be guaranteed if and only if at t = 0,. Every robot can counicate with every other robot,. Each target is observable by soe robot. 6

29 Distance Optiality Guarantee Theore (Necessary and Sufficient Conditions for Distance Optiality). Under sensing and counication constraints, distance optiality can be guaranteed if and only if at t = 0,. Every robot can counicate with every other robot,. Each target is observable by soe robot. 6

30 Distance Optiality Guarantee Theore (Necessary and Sufficient Conditions for Distance Optiality). Under sensing and counication constraints, distance optiality can be guaranteed if and only if at t = 0,. Every robot can counicate with every other robot,. Each target is observable by soe robot. r sense 6

31 Non-Asyptotic Optiality Guarantee 7

32 Non-Asyptotic Optiality Guarantee Lea. Given and fixing 0 < ε <, G(0) is connected with probability at least ε if n log ε. 7

33 Non-Asyptotic Optiality Guarantee Lea. Given and fixing 0 < ε <, G(0) is connected with probability at least ε if n log ε. = / 7

34 Non-Asyptotic Optiality Guarantee Lea. Given and fixing 0 < ε <, G(0) is connected with probability at least ε if n log ε. q i = / 7

35 Non-Asyptotic Optiality Guarantee Lea. Given and fixing 0 < ε <, G(0) is connected with probability at least ε if n log ε. q i = / 7

36 Non-Asyptotic Optiality Guarantee Lea. Given and fixing 0 < ε <, G(0) is connected with probability at least ε if n log ε. n P n i = 0 = q i = / 7

37 Non-Asyptotic Optiality Guarantee Lea. Given and fixing 0 < ε <, G(0) is connected with probability at least ε if n log ε. n P n i = 0 = < e n q i = / 7

38 Non-Asyptotic Optiality Guarantee Lea. Given and fixing 0 < ε <, G(0) is connected with probability at least ε if n log ε. n P n i = 0 = < e n q i P E(n i = 0) P(n i = 0) i= i= = / 7

39 Non-Asyptotic Optiality Guarantee Lea. Given and fixing 0 < ε <, G(0) is connected with probability at least ε if n log ε. n P n i = 0 = < e n q i P E(n i = 0) P(n i = 0) < e n i= i= = / 7

40 Non-Asyptotic Optiality Guarantee Lea. Given and fixing 0 < ε <, G(0) is connected with probability at least ε if n log ε. n P n i = 0 = < e n q i P E(n i = 0) P(n i = 0) < e n = ε i= i= = / 7

41 Non-Asyptotic Optiality Guarantee Lea. Given and fixing 0 < ε <, G(0) is connected with probability at least ε if n log ε. Theore (Rando Geoetric Graphs [Penrose 97]). For n uniforly distributed nodes in the unit square, let G(0) be the counication graph for a given at t = 0. Then for any real nuber c, as n (i.e., 0), P G is connected πn log n c) = e ec. Theore [Xue & Kuar 04]. For n uniforly distributed nodes in the unit square, the network is asyptotically connected if and only if each node has Θ(log n) neighbors. 7

42 Non-Asyptotic Optiality Guarantee Lea. Given and fixing 0 < ε <, G(0) is connected with probability at least ε if n log ε. Theore (Rando Geoetric Graphs [Penrose 97]). For n uniforly distributed nodes in the unit square, let G(0) be the counication graph for a given at t = 0. Then for any real nuber c, as n (i.e., 0), P G is connected πn log n c) = e ec. Theore [Xue & Kuar 04]. For n uniforly distributed nodes in the unit square, the network is asyptotically connected if and only if each node has Θ(log n) neighbors. 7

43 Non-Asyptotic Optiality Guarantee, cont. Theore (Non-Asyptotic Bounds) Fixing 0 < ε <, robots can counicate with each other and all targets are observable at t = 0 with probability at least ε when r sense log ε r sense, r sense < 0 n log ε, r sense 0 8

44 Non-Asyptotic Optiality Guarantee, cont. Theore (Non-Asyptotic Bounds) Fixing 0 < ε <, robots can counicate with each other and all targets are observable at t = 0 with probability at least ε when r sense log ε r sense, r sense < 0 n log ε, r sense 0 n = Θ( log ) is sufficient and necessary for high probability asyptotic guarantee on the connectivity of G 0. 8

45 An Ideal Hierarchical Strategy 9

46 An Ideal Hierarchical Strategy Ideal:, r sense as large as needed 9

47 An Ideal Hierarchical Strategy Ideal:, r sense as large as needed Hierarchical: The unit square is partitioned into sall squares (here, = 4) 9

48 An Ideal Hierarchical Strategy Ideal:, r sense as large as needed Hierarchical: The unit square is partitioned into sall squares (here, = 4) 9

49 An Ideal Hierarchical Strategy Ideal:, r sense as large as needed Hierarchical: The unit square is partitioned into sall squares (here, = 4) 9

50 An Ideal Hierarchical Strategy Ideal:, r sense as large as needed Hierarchical: The unit square is partitioned into sall squares (here, = 4) 9

51 An Ideal Hierarchical Strategy Ideal:, r sense as large as needed Hierarchical: The unit square is partitioned into sall squares (here, = 4) 9

52 Bounding Distance Cost at Lower Hierarchy 0

53 Bounding Distance Cost at Lower Hierarchy 0

54 Bounding Distance Cost at Lower Hierarchy q i 0

55 Bounding Distance Cost at Lower Hierarchy q i Theore [Talagrand 9] Let X = {x,, x n }, Y = {y,, y n } be two sets sapled i. i. d. fro the sae arbitrary distribution on 0,. Then E in σ n i= in which C is a universal constant. x i y σ i C n log n, 0

56 Bounding Distance Cost at Lower Hierarchy E D i C n i log n i q i Theore [Talagrand 9] Let X = {x,, x n }, Y = {y,, y n } be two sets sapled i. i. d. fro the sae arbitrary distribution on 0,. Then E in σ n i= in which C is a universal constant. x i y σ i C n log n, 0

57 Bounding Distance Cost at Lower Hierarchy E D i C n i log n i i= E[D i ] C i= n i log n i q i C i n i log i n i C n log n Theore [Talagrand 9] Let X = {x,, x n }, Y = {y,, y n } be two sets sapled i. i. d. fro the sae arbitrary distribution on 0,. Then E in σ n i= in which C is a universal constant. x i y σ i C n log n, 7

58 Bounding Distance Cost at Lower Hierarchy E D i C n i log n i i= E[D i ] C i= n i log n i q i C i n i log i n i C n log n Theore [Talagrand 9] Let X = {x,, x n }, Y = {y,, y n } be two sets sapled i. i. d. fro the sae arbitrary distribution on 0,. Then E in σ n i= in which C is a universal constant. x i y σ i C n log n, 8

59 Bounding Distance Cost at Higher Hierarchy q i

60 Bounding Distance Cost at Higher Hierarchy q i

61 Bounding Distance Cost at Higher Hierarchy P x j q i = P y j q i = p i q i

62 Bounding Distance Cost at Higher Hierarchy P x j q i = P y j q i = p i P x j q i, y j q i = P x j q i, y j q i = p i ( p i ) q i

63 Bounding Distance Cost at Higher Hierarchy P x j q i = P y j q i = p i P x j q i, y j q i = P x j q i, y j q i = p i ( p i ) q i Z j =, x j q i, y j q i, x j q i, y j q i, 0, otherwise

64 Bounding Distance Cost at Higher Hierarchy P x j q i = P y j q i = p i P x j q i, y j q i = P x j q i, y j q i = p i ( p i ) q i Z j =, x j q i, y j q i, x j q i, y j q i, S i = Z + + Z n 0, otherwise

65 Bounding Distance Cost at Higher Hierarchy P x j q i = P y j q i = p i P x j q i, y j q i = P x j q i, y j q i = p i ( p i ) q i Z j =, x j q i, y j q i, x j q i, y j q i, S i = Z + + Z n 0, otherwise E[S i ] = ne Z j = np i ( p i )

66 Bounding Distance Cost at Higher Hierarchy P x j q i = P y j q i = p i P x j q i, y j q i = P x j q i, y j q i = p i ( p i ) q i Z j =, x j q i, y j q i, x j q i, y j q i, S i = Z + + Z n 0, otherwise E[S i ] = ne Z j = np i ( p i ) E S i = E S i E S i

67 Bounding Distance Cost at Higher Hierarchy P x j q i = P y j q i = p i P x j q i, y j q i = P x j q i, y j q i = p i ( p i ) q i Z j =, x j q i, y j q i, x j q i, y j q i, S i = Z + + Z n 0, otherwise E[S i ] = ne Z j = np i ( p i ) E S i = E S i E S i E[ S i ] np i p i

68 Bounding Distance Cost at Higher Hierarchy P x j q i = P y j q i = p i P x j q i, y j q i = P x j q i, y j q i = p i ( p i ) q i Z j =, x j q i, y j q i, x j q i, y j q i, S i = Z + + Z n 0, otherwise E[S i ] = ne Z j = np i ( p i ) E S i = E S i E S i E[ S i ] np i p i i= E[ S i ] = i= np i p i = n i= p i p i n i= p i i= p i = n 68

69 Bounding Distance Cost at Higher Hierarchy P x j q i = P y j q i = p i P x j q i, y j q i = P x j q i, y j q i = p i ( p i ) q i Z j =, x j q i, y j q i, x j q i, y j q i, S i = Z + + Z n 0, otherwise E[S i ] = ne Z j = np i ( p i ) E S i = E S i E S i E[ S i ] np i p i i= E[ S i ] = i= np i p i = n i= p i p i n i= p i i= p i = n 69

70 Bounds on Distance Optiality Theore (Perforance Upper-Bound of Ideal Hierarchical Strategies) Let D n be the total distance of an ideal hierarchical strategy with h hierarchies and i regions at hierarchy i, then for arbitrary distribution on 0,, E D n C n log n + n h i= i+ i.

71 Bounds on Distance Optiality Theore (Perforance Upper-Bound of Ideal Hierarchical Strategies) Let D n be the total distance of an ideal hierarchical strategy with h hierarchies and i regions at hierarchy i, then for arbitrary distribution on 0,, E D n C n log n + n h i= i+ i. Theore [Ajtai et al. 84]. Under the unifor distribution, with high probability, C n log n D n C n log n.

72 Bounds on Distance Optiality Theore (Perforance Upper-Bound of Ideal Hierarchical Strategies) Let D n be the total distance of an ideal hierarchical strategy with h hierarchies and i regions at hierarchy i, then for arbitrary distribution on 0,, E D n C n log n + n h i= i+ i. Theore [Ajtai et al. 84]. Under the unifor distribution, with high probability, C n log n D n C n log n. Corollary. With unifor distribution, fixing h and { i }, as n, E[D n ] E[ D n ] O.

73 Bounds on Distance Optiality n - nuber of robots A two-level ideal hierarchical strategy Corollary. With unifor distribution, fixing h and { i }, as n, E[D n ] E[ D n ] O.

74 Incorporating Arbitrary and r sense 3

75 Incorporating Arbitrary and r sense 3

76 Incorporating Arbitrary and r sense n - nuber of robots Two-level ideal hierarchical strategy n - nuber of robots Two-level decentralized hierarchical strategy 3

77 Incorporating Arbitrary and r sense n - nuber of robots Two-level ideal hierarchical strategy n - nuber of robots Two-level decentralized hierarchical strategy Arbitrary r sense can also be handled siilarly. 3

78 Suary of Contribution Guarantee on the distance optiality of the stochastic target assignent proble Necessary and sufficient condition for optiality Non-asyptotic probabilistic bounds Asyptotically tight bounds for high-probability guarantee Perforance of decentralized hierarchical strategies General upper bounds for arbitrary distributions O() approxiation algorith for the unifor distribution Iportant takeaway: locally optial behavior leads to near globally optial behavior