distance optimal target assignment in robotic networks...

Distance Optimal Target Assignment in Robotic Networks under Communication and Sensing

Constraints

Jingjin Yu Soon-Jo Chung Petros G. VoulgarisCSAIL @ MIT/MechE @ BU AE @ University of Illinois

Supported by:

The Stochastic Target Assignment Problem

𝑄 = 0,1 × [0,1]

𝑋 = {𝑥1, … , 𝑥𝑛}

𝑄 = 0,1 × [0,1]

𝑋 = {𝑥1, … , 𝑥𝑛}

𝑌 = {𝑦1, … , 𝑦𝑛}

𝑄 = 0,1 × [0,1]

𝑋 = {𝑥1, … , 𝑥𝑛}

𝑌 = {𝑦1, … , 𝑦𝑛}

Control: 𝑥𝑖 = 𝑢𝑖 , | 𝑢𝑖 | = 1

𝑄 = 0,1 × [0,1]

𝑋 = {𝑥1, … , 𝑥𝑛}

𝑌 = {𝑦1, … , 𝑦𝑛}

𝜎: permutation that pairs 𝑥𝑖 with 𝑦𝜎(𝑖)

𝑄 = 0,1 × [0,1]

𝑋 = {𝑥1, … , 𝑥𝑛}

𝑌 = {𝑦1, … , 𝑦𝑛}

min𝜎,{𝑢𝑖}

𝐷𝑛 =

| 𝑥𝑖(𝑡)|𝑑𝑡

𝜎: permutation that pairs 𝑥𝑖 with 𝑦𝜎(𝑖)

𝑄 = 0,1 × [0,1]

The Stochastic Target Assignment Problem, cont.

𝑟𝑠𝑒𝑛𝑠𝑒

𝑟𝑐𝑜𝑚𝑚

𝐺(𝑡)

Given 𝑟𝑠𝑒𝑛𝑠𝑒 and 𝑟𝑐𝑜𝑚𝑚, how can we guarantee distance optimality?

𝐺(𝑡)

Given 𝑟𝑠𝑒𝑛𝑠𝑒 and 𝑟𝑐𝑜𝑚𝑚, how can we guarantee distance optimality? Performance of decentralized, hierarchical strategies (algorithms)?

Related Work

Smith and Bullo, Monotonic target assignment for robotic networks, IEEE Trans. Automat. Control, vol. 54, no. 9, pp. 2042–2057, 2009

Related Work

Treleaven, Pavone, and Frazzoli, Asymptotically optimal algorithms for one-to-one pickup and delivery problems with applications to transportation systems, IEEE Trans. Automat Control, vol. 59, no. 9, pp. 2261–2276, 2013.

Related Work

Penrose, The longest edge of the random minimal spanning tree, Annals of Applied Probability, vol. 7, pp. 340–361, 1997.Penrose, Random Geometric Graphs, 2003

Related Work

Erdős and Rényi, On a classical problem of probability theory, Publ. Math. Inst. Hung. Acad. Sci., vol. Ser. A 6, pp. 215–220, 1961

Related Work

Erdős and Rényi, On a classical problem of probability theory, Publ. Math. Inst. Hung. Acad. Sci., vol. Ser. A 6, pp. 215–220, 1961

Karaman and Frazzoli, Sampling-based Algorithms for Optimal Motion Planning. Int. Journal of Robotics Research, vol. 30, no 7, pp. 846-894, 2011

Main Result

Distance optimality guarantee Necessary and sufficient condition for distance optimality (non-stochastic) Non-asymptotic 1 − 𝜖 probabilistic guarantee for 0 < 𝜖 < 1

𝑛 ≥

, 𝑟𝑠𝑒𝑛𝑠𝑒 <10𝑟𝑐𝑜𝑚𝑚

, 𝑟𝑠𝑒𝑛𝑠𝑒 ≥10𝑟𝑐𝑜𝑚𝑚

Tight asymptotic bounds for high-probability guarantee

Performance of decentralized, hierarchical strategies Upper bound on the distance cost for arbitrary robot/target distribution 𝑂(1) asymptotic optimality guarantee under the uniform distribution

Main Result

𝑛 ≥

Main Result

𝑛 ≥

Main Result

𝑛 ≥

Main Result

𝑛 ≥

𝑛 - number of robots

Distance Optimality Guarantee

Theorem (Necessary and Sufficient Conditions for Distance Optimality).Under sensing and communication constraints, distance optimality can be guaranteed if and only if at 𝑡 = 0,

1. Every robot can communicate with every other robot, 2. Each target is observable by some robot.

Non-Asymptotic Optimality Guarantee

Lemma. Given 𝑟𝑐𝑜𝑚𝑚 and fixing 0 < 𝜖 < 1, 𝐺(0) is connected with probability at least 1 − 𝜖 if

𝑛 ≥5

1 2 …… 𝑚 = ⌈ 5/𝑟𝑐𝑜𝑚𝑚⌉

𝑛 ≥5

𝑞𝑖

1 2 …… 𝑚 = ⌈ 5/𝑟𝑐𝑜𝑚𝑚⌉

𝑛 ≥5

𝑞𝑖

1 2 …… 𝑚 = ⌈ 5/𝑟𝑐𝑜𝑚𝑚⌉

𝑛 ≥5

𝑞𝑖

𝑃 𝑛𝑖 = 0 = 1 −1

1 2 …… 𝑚 = ⌈ 5/𝑟𝑐𝑜𝑚𝑚⌉

𝑛 ≥5

𝑞𝑖

𝑃 𝑛𝑖 = 0 = 1 −1

1 2 …… 𝑚 = ⌈ 5/𝑟𝑐𝑜𝑚𝑚⌉

< 𝑒−𝑛𝑚

𝑛 ≥5

𝑞𝑖

𝑃 𝑛𝑖 = 0 = 1 −1

1 2 …… 𝑚 = ⌈ 5/𝑟𝑐𝑜𝑚𝑚⌉

𝑖=1

𝐸(𝑛𝑖 = 0) ≤

𝑖=1

𝑃(𝑛𝑖 = 0)

< 𝑒−𝑛𝑚

𝑛 ≥5

𝑞𝑖

𝑃 𝑛𝑖 = 0 = 1 −1

1 2 …… 𝑚 = ⌈ 5/𝑟𝑐𝑜𝑚𝑚⌉

𝑖=1

𝐸(𝑛𝑖 = 0) ≤

𝑖=1

𝑃(𝑛𝑖 = 0)

< 𝑒−𝑛𝑚

< 𝑚𝑒−𝑛𝑚

𝑛 ≥5

𝑞𝑖

𝑃 𝑛𝑖 = 0 = 1 −1

1 2 …… 𝑚 = ⌈ 5/𝑟𝑐𝑜𝑚𝑚⌉

𝑖=1

𝐸(𝑛𝑖 = 0) ≤

𝑖=1

𝑃(𝑛𝑖 = 0)

< 𝑒−𝑛𝑚

< 𝑚𝑒−𝑛𝑚 = 𝜖

𝑛 ≥5

Theorem (Random Geometric Graphs [Penrose ‘97]). For 𝑛 uniformly distributed nodes in the unit square, let 𝐺(0) be the communication graph for a given 𝑟𝑐𝑜𝑚𝑚 at 𝑡 = 0. Then for any real number 𝑐, as 𝑛 → ∞ (i.e., 𝑟𝑐𝑜𝑚𝑚 → 0),

𝑃 𝐺 𝑖𝑠 𝑐𝑜𝑛𝑛𝑒𝑐𝑡𝑒𝑑 𝜋𝑛𝑟𝑐𝑜𝑚𝑚2 − log 𝑛 ≤ 𝑐) = 𝑒−𝑒

Theorem [Xue & Kumar ‘04]. For 𝑛 uniformly distributed nodes in the unit square, the network is asymptotically connected if and only if each node has Θ(log 𝑛) neighbors.

𝑛 ≥5

Theorem (Random Geometric Graphs [Penrose ‘97]). For 𝑛 uniformly distributed nodes in the unit square, let 𝐺(0) be the communication graph for a given 𝑟𝑐𝑜𝑚𝑚 at 𝑡 = 0. Then for any real number 𝑐, as 𝑛 → ∞ (i.e., 𝑟𝑐𝑜𝑚𝑚 → 0),

𝑃 𝐺 𝑖𝑠 𝑐𝑜𝑛𝑛𝑒𝑐𝑡𝑒𝑑 𝜋𝑛𝑟𝑐𝑜𝑚𝑚2 − log 𝑛 ≤ 𝑐) = 𝑒−𝑒

Theorem [Xue & Kumar ‘04]. For 𝑛 uniformly distributed nodes in the unit square, the network is asymptotically connected if and only if each node has Θ(log 𝑛) neighbors.

Non-Asymptotic Optimality Guarantee, cont.

Theorem (Non-Asymptotic Bounds) Fixing 0 < 𝜖 < 1, robots can communicate with each other and all targets are observable at 𝑡 = 0 with probability at least 1 − 𝜖 when

𝑛 ≥

Non-Asymptotic Optimality Guarantee, cont.

Theorem (Non-Asymptotic Bounds) Fixing 0 < 𝜖 < 1, robots can communicate with each other and all targets are observable at 𝑡 = 0 with probability at least 1 − 𝜖 when

𝑛 ≥

𝑛 = Θ(−1

𝑟𝑐𝑜𝑚𝑚2 log 𝑟𝑐𝑜𝑚𝑚) is sufficient and necessary for high probability asymptotic

guarantee on the connectivity of 𝐺 0 .

An Ideal Hierarchical Strategy

Ideal: 𝑟𝑐𝑜𝑚𝑚, 𝑟𝑠𝑒𝑛𝑠𝑒 as large as needed

Hierarchical: The unit square is partitioned into 𝑚 small squares(here, 𝑚 = 4)

Bounding Distance Cost at Lower Hierarchy

𝑞𝑖

Theorem [Talagrand ‘92] Let 𝑋 = {𝑥1, … , 𝑥𝑛}, 𝑌 = {𝑦1, … , 𝑦𝑛} be two sets sampled 𝑖. 𝑖. 𝑑. from the same arbitrary distribution on 0,1 2. Then

𝐸 min𝜎

𝑖=1

𝑥𝑖 − 𝑦𝜎 𝑖 ≤ 𝐶 𝑛 log 𝑛,

in which 𝐶 is a universal constant.

𝑞𝑖

𝐸 min𝜎

𝑖=1

𝑞𝑖

𝐸 𝐷𝑖 ≤𝐶

𝑚𝑛𝑖 log 𝑛𝑖

𝑖=1

𝐸[𝐷𝑖] ≤ 𝐶 𝑚

𝑖=1

≤ 𝐶 𝑚 𝑖 𝑛𝑖𝑚

log 𝑖 𝑛𝑖𝑚

≤ 𝐶 𝑛 log 𝑛

𝐸 min𝜎

𝑖=1

𝑞𝑖

𝑖=1

𝐸[𝐷𝑖] ≤ 𝐶 𝑚

𝑖=1

≤ 𝐶 𝑚 𝑖 𝑛𝑖𝑚

log 𝑖 𝑛𝑖𝑚

≤ 𝐶 𝑛 log 𝑛

𝐸 min𝜎

𝑖=1

𝑞𝑖

Bounding Distance Cost at Higher Hierarchy

𝑞𝑖

𝑃 𝑥𝑗 ∈ 𝑞𝑖 = 𝑃 𝑦𝑗 ∈ 𝑞𝑖 = 𝑝𝑖

𝑞𝑖

𝑃 𝑥𝑗 ∈ 𝑞𝑖 , 𝑦𝑗 ∉ 𝑞𝑖 = 𝑃 𝑥𝑗 ∉ 𝑞𝑖 , 𝑦𝑗 ∈ 𝑞𝑖 = 𝑝𝑖(1 − 𝑝𝑖)

𝑞𝑖

𝑍𝑗 =

−1, 𝑥𝑗 ∉ 𝑞𝑖 , 𝑦𝑗 ∈ 𝑞𝑖1, 𝑥𝑗 ∈ 𝑞𝑖 , 𝑦𝑗 ∉ 𝑞𝑖0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

𝑞𝑖

𝑍𝑗 =

, 𝑆𝑖 = 𝑍1 +⋯+ 𝑍𝑛

𝑞𝑖

𝑍𝑗 =

, 𝑆𝑖 = 𝑍1 +⋯+ 𝑍𝑛

𝐸[𝑆𝑖2] = 𝑛𝐸 𝑍𝑗

2 = 2𝑛𝑝𝑖(1 − 𝑝𝑖)

𝑞𝑖

𝑍𝑗 =

, 𝑆𝑖 = 𝑍1 +⋯+ 𝑍𝑛

2 = 2𝑛𝑝𝑖(1 − 𝑝𝑖)

𝐸 𝑆𝑖 = 𝐸 𝑆𝑖2 ≤ 𝐸 𝑆𝑖

𝑞𝑖

𝑍𝑗 =

, 𝑆𝑖 = 𝑍1 +⋯+ 𝑍𝑛

2 = 2𝑛𝑝𝑖(1 − 𝑝𝑖)

2 ⇒ 𝐸[ 𝑆𝑖 ] ≤ 2𝑛𝑝𝑖 1 − 𝑝𝑖

𝑖=1

𝐸[ 𝑆𝑖 ] =

𝑖=1

2𝑛𝑝𝑖 1 − 𝑝𝑖 = 𝑚 2𝑛

𝑖=1

𝑚𝑝𝑖 1 − 𝑝𝑖

≤ 𝑚 2𝑛 𝑖=1𝑚 𝑝𝑖

𝑚1 − 𝑖=1

𝑚 𝑝𝑖

𝑚= 2𝑛 𝑚 − 1

𝑍𝑗 =

, 𝑆𝑖 = 𝑍1 +⋯+ 𝑍𝑛

2 = 2𝑛𝑝𝑖(1 − 𝑝𝑖)

2 ⇒ 𝐸[ 𝑆𝑖 ] ≤ 2𝑛𝑝𝑖 1 − 𝑝𝑖

𝑞𝑖

𝑍𝑗 =

, 𝑆𝑖 = 𝑍1 +⋯+ 𝑍𝑛

2 = 2𝑛𝑝𝑖(1 − 𝑝𝑖)

2 ⇒ 𝐸[ 𝑆𝑖 ] ≤ 2𝑛𝑝𝑖 1 − 𝑝𝑖

𝑖=1

𝐸[ 𝑆𝑖 ] =

𝑖=1

2𝑛𝑝𝑖 1 − 𝑝𝑖 = 𝑚 2𝑛

𝑖=1

𝑚𝑝𝑖 1 − 𝑝𝑖

≤ 𝑚 2𝑛 𝑖=1𝑚 𝑝𝑖

𝑚1 − 𝑖=1

𝑚 𝑝𝑖

𝑚= 2𝑛 𝑚 − 1

𝑞𝑖

Bounds on Distance Optimality

Theorem (Performance Upper-Bound of Ideal Hierarchical Strategies) Let𝐷𝑛 be the total distance of an ideal hierarchical strategy with ℎ hierarchies and 𝑚𝑖 regions at hierarchy 𝑖, then for arbitrary distribution on 0,1 2,

𝐸 𝐷𝑛 ≤ 𝐶 𝑛 log 𝑛 + 2 𝑛

𝑖=1

ℎ−1𝑚𝑖+1

𝑚𝑖.

𝑖=1

ℎ−1𝑚𝑖+1

𝑚𝑖.

Theorem [Ajtai et al. ‘84]. Under the uniform distribution, with high

probability, 𝐶1 𝑛 log 𝑛 ≤ 𝐷𝑛∗ ≤ 𝐶2 𝑛 log 𝑛.

𝑖=1

ℎ−1𝑚𝑖+1

𝑚𝑖.

Theorem [Ajtai et al. ‘84]. Under the uniform distribution, with high

probability, 𝐶1 𝑛 log 𝑛 ≤ 𝐷𝑛∗ ≤ 𝐶2 𝑛 log 𝑛.

Corollary. With uniform distribution, fixing ℎ and {𝑚𝑖}, as 𝑛 → ∞,

𝐸[𝐷𝑛]

𝐸[ 𝐷𝑛∗]→ 𝑂 1 .

Corollary. With uniform distribution, fixing ℎ and {𝑚𝑖}, as 𝑛 → ∞,

𝐸[𝐷𝑛]

𝐸[ 𝐷𝑛∗]→ 𝑂 1 .

A two-level ideal hierarchical strategy

Incorporating Arbitrary 𝑟𝑐𝑜𝑚𝑚 and 𝑟𝑠𝑒𝑛𝑠𝑒

Two-level decentralized hierarchical strategy

Two-level ideal hierarchical strategy

Incorporating Arbitrary 𝑟𝑐𝑜𝑚𝑚 and 𝑟𝑠𝑒𝑛𝑠𝑒

Two-level decentralized hierarchical strategy

Two-level ideal hierarchical strategy

Arbitrary 𝑟𝑠𝑒𝑛𝑠𝑒 can also be handled similarly.

Summary of Contribution

Guarantee on the distance optimality of the stochastic target assignment problem Necessary and sufficient condition for optimality Non-asymptotic probabilistic bounds Asymptotically tight bounds for high-probability guarantee

Performance of decentralized hierarchical strategies General upper bounds for arbitrary distributions 𝑂(1) approximation algorithm for the uniform distribution

Important takeaway: locally optimal behavior leads to near globallyoptimal behavior

distance optimal target assignment in robotic networks...

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