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

2


2

𝑄 = 0,1 × [0,1]


2

𝑋 = {𝑥1, … , 𝑥𝑛}

𝑄 = 0,1 × [0,1]


2

𝑋 = {𝑥1, … , 𝑥𝑛}

𝑌 = {𝑦1, … , 𝑦𝑛}

𝑄 = 0,1 × [0,1]


2

𝑋 = {𝑥1, … , 𝑥𝑛}

𝑌 = {𝑦1, … , 𝑦𝑛}

Control: 𝑥𝑖 = 𝑢𝑖 , | 𝑢𝑖 | = 1

𝑄 = 0,1 × [0,1]


2

𝑋 = {𝑥1, … , 𝑥𝑛}

𝑌 = {𝑦1, … , 𝑦𝑛}


𝜎: permutation that pairs 𝑥𝑖 with 𝑦𝜎(𝑖)

𝑄 = 0,1 × [0,1]


2

𝑋 = {𝑥1, … , 𝑥𝑛}

𝑌 = {𝑦1, … , 𝑦𝑛}


min𝜎,{𝑢𝑖}

𝐷𝑛 =

𝑖

| 𝑥𝑖(𝑡)|𝑑𝑡

𝜎: permutation that pairs 𝑥𝑖 with 𝑦𝜎(𝑖)

𝑄 = 0,1 × [0,1]

The Stochastic Target Assignment Problem, cont.

3


3

𝑟𝑠𝑒𝑛𝑠𝑒


3



3


𝑟𝑐𝑜𝑚𝑚


3



𝐺(𝑡)


3



𝐺(𝑡)

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


3



𝐺(𝑡)

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

Related Work

4

Related Work

4

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

Related Work

4


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

4



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

4




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

4




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

22

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

𝑛 ≥

2


2

log1

𝜖

2


2

, 𝑟𝑠𝑒𝑛𝑠𝑒 <10𝑟𝑐𝑜𝑚𝑚

5

5


2

log1

𝜖

5


2

, 𝑟𝑠𝑒𝑛𝑠𝑒 ≥10𝑟𝑐𝑜𝑚𝑚

5

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

23


𝑛 ≥

2


2

log1

𝜖

2


2


5

5


2

log1

𝜖

5


2


5



Main Result

24


𝑛 ≥

2


2

log1

𝜖

2


2


5

5


2

log1

𝜖

5


2


5



Main Result

25


𝑛 ≥

2


2

log1

𝜖

2


2


5

5


2

log1

𝜖

5


2


5



Main Result

26


𝑛 ≥

2


2

log1

𝜖

2


2


5

5


2

log1

𝜖

5


2


5



𝑛 - number of robots

Distance Optimality Guarantee

6

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.


6





6





6





Non-Asymptotic Optimality Guarantee

7


7

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

𝑛 ≥5


2

log1

𝜖

5


2

.


7


𝑛 ≥5


2

log1

𝜖

5


2

.

1 2 …… 𝑚 = ⌈ 5/𝑟𝑐𝑜𝑚𝑚⌉


7


𝑛 ≥5


2

log1

𝜖

5


2

.


𝑞𝑖

1 2 …… 𝑚 = ⌈ 5/𝑟𝑐𝑜𝑚𝑚⌉


7


𝑛 ≥5


2

log1

𝜖

5


2

.


𝑞𝑖

1 2 …… 𝑚 = ⌈ 5/𝑟𝑐𝑜𝑚𝑚⌉


7


𝑛 ≥5


2

log1

𝜖

5


2

.


𝑞𝑖

𝑃 𝑛𝑖 = 0 = 1 −1

𝑚

𝑛

1 2 …… 𝑚 = ⌈ 5/𝑟𝑐𝑜𝑚𝑚⌉


7


𝑛 ≥5


2

log1

𝜖

5


2

.


𝑞𝑖

𝑃 𝑛𝑖 = 0 = 1 −1

𝑚

𝑛

1 2 …… 𝑚 = ⌈ 5/𝑟𝑐𝑜𝑚𝑚⌉

< 𝑒−𝑛𝑚


7


𝑛 ≥5


2

log1

𝜖

5


2

.


𝑞𝑖

𝑃 𝑛𝑖 = 0 = 1 −1

𝑚

𝑛

1 2 …… 𝑚 = ⌈ 5/𝑟𝑐𝑜𝑚𝑚⌉

𝑃

𝑖=1

𝑚

𝐸(𝑛𝑖 = 0) ≤

𝑖=1

𝑚

𝑃(𝑛𝑖 = 0)

< 𝑒−𝑛𝑚


7


𝑛 ≥5


2

log1

𝜖

5


2

.


𝑞𝑖

𝑃 𝑛𝑖 = 0 = 1 −1

𝑚

𝑛

1 2 …… 𝑚 = ⌈ 5/𝑟𝑐𝑜𝑚𝑚⌉

𝑃

𝑖=1

𝑚

𝐸(𝑛𝑖 = 0) ≤

𝑖=1

𝑚

𝑃(𝑛𝑖 = 0)

< 𝑒−𝑛𝑚

< 𝑚𝑒−𝑛𝑚


7


𝑛 ≥5


2

log1

𝜖

5


2

.


𝑞𝑖

𝑃 𝑛𝑖 = 0 = 1 −1

𝑚

𝑛

1 2 …… 𝑚 = ⌈ 5/𝑟𝑐𝑜𝑚𝑚⌉

𝑃

𝑖=1

𝑚

𝐸(𝑛𝑖 = 0) ≤

𝑖=1

𝑚

𝑃(𝑛𝑖 = 0)

< 𝑒−𝑛𝑚

< 𝑚𝑒−𝑛𝑚 = 𝜖


7


𝑛 ≥5


2

log1

𝜖

5


2

.

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

𝑐.

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.


7


𝑛 ≥5


2

log1

𝜖

5


2

.

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

𝑐.

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.

8

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 ≥

2


2

log1

𝜖

2


2


5

5


2

log1

𝜖

5


2


5

Non-Asymptotic Optimality Guarantee, cont.

8

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 ≥

2


2

log1

𝜖

2


2


5

5


2

log1

𝜖

5


2


5

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

9


9

Ideal: 𝑟𝑐𝑜𝑚𝑚, 𝑟𝑠𝑒𝑛𝑠𝑒 as large as needed


9


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

1

𝑚


9



1

𝑚


9



1

𝑚


9



1

𝑚


9



1

𝑚

Bounding Distance Cost at Lower Hierarchy

10

1

𝑚


10

1

𝑚


10

𝑞𝑖

1

𝑚


10

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

𝑛

𝑥𝑖 − 𝑦𝜎 𝑖 ≤ 𝐶 𝑛 log 𝑛,

in which 𝐶 is a universal constant.

𝑞𝑖

1

𝑚


10


𝐸 min𝜎

𝑖=1

𝑛



𝑞𝑖

𝐸 𝐷𝑖 ≤𝐶

𝑚𝑛𝑖 log 𝑛𝑖

1

𝑚

𝑖=1

𝑚

𝐸[𝐷𝑖] ≤ 𝐶 𝑚

𝑖=1

𝑚1


≤ 𝐶 𝑚 𝑖 𝑛𝑖𝑚

log 𝑖 𝑛𝑖𝑚

≤ 𝐶 𝑛 log 𝑛


57


𝐸 min𝜎

𝑖=1

𝑛



𝑞𝑖



1

𝑚

𝑖=1

𝑚

𝐸[𝐷𝑖] ≤ 𝐶 𝑚

𝑖=1

𝑚1


≤ 𝐶 𝑚 𝑖 𝑛𝑖𝑚

log 𝑖 𝑛𝑖𝑚

≤ 𝐶 𝑛 log 𝑛


58


𝐸 min𝜎

𝑖=1

𝑛



𝑞𝑖



1

𝑚

Bounding Distance Cost at Higher Hierarchy

11

𝑞𝑖


11

𝑞𝑖


11

𝑞𝑖

𝑃 𝑥𝑗 ∈ 𝑞𝑖 = 𝑃 𝑦𝑗 ∈ 𝑞𝑖 = 𝑝𝑖


11

𝑞𝑖


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


11

𝑞𝑖



𝑍𝑗 =

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

,


11

𝑞𝑖



𝑍𝑗 =


, 𝑆𝑖 = 𝑍1 +⋯+ 𝑍𝑛


11

𝑞𝑖



𝑍𝑗 =


, 𝑆𝑖 = 𝑍1 +⋯+ 𝑍𝑛

𝐸[𝑆𝑖2] = 𝑛𝐸 𝑍𝑗

2 = 2𝑛𝑝𝑖(1 − 𝑝𝑖)


11

𝑞𝑖



𝑍𝑗 =


, 𝑆𝑖 = 𝑍1 +⋯+ 𝑍𝑛


2 = 2𝑛𝑝𝑖(1 − 𝑝𝑖)

𝐸 𝑆𝑖 = 𝐸 𝑆𝑖2 ≤ 𝐸 𝑆𝑖

2


11

𝑞𝑖



𝑍𝑗 =


, 𝑆𝑖 = 𝑍1 +⋯+ 𝑍𝑛


2 = 2𝑛𝑝𝑖(1 − 𝑝𝑖)


2 ⇒ 𝐸[ 𝑆𝑖 ] ≤ 2𝑛𝑝𝑖 1 − 𝑝𝑖

𝑖=1

𝑚

𝐸[ 𝑆𝑖 ] =

𝑖=1

𝑚

2𝑛𝑝𝑖 1 − 𝑝𝑖 = 𝑚 2𝑛

𝑖=1

𝑚1

𝑚𝑝𝑖 1 − 𝑝𝑖

≤ 𝑚 2𝑛 𝑖=1𝑚 𝑝𝑖

𝑚1 − 𝑖=1

𝑚 𝑝𝑖

𝑚= 2𝑛 𝑚 − 1


68



𝑍𝑗 =


, 𝑆𝑖 = 𝑍1 +⋯+ 𝑍𝑛


2 = 2𝑛𝑝𝑖(1 − 𝑝𝑖)


2 ⇒ 𝐸[ 𝑆𝑖 ] ≤ 2𝑛𝑝𝑖 1 − 𝑝𝑖

𝑞𝑖


69



𝑍𝑗 =


, 𝑆𝑖 = 𝑍1 +⋯+ 𝑍𝑛


2 = 2𝑛𝑝𝑖(1 − 𝑝𝑖)


2 ⇒ 𝐸[ 𝑆𝑖 ] ≤ 2𝑛𝑝𝑖 1 − 𝑝𝑖

𝑖=1

𝑚

𝐸[ 𝑆𝑖 ] =

𝑖=1

𝑚

2𝑛𝑝𝑖 1 − 𝑝𝑖 = 𝑚 2𝑛

𝑖=1

𝑚1

𝑚𝑝𝑖 1 − 𝑝𝑖

≤ 𝑚 2𝑛 𝑖=1𝑚 𝑝𝑖

𝑚1 − 𝑖=1

𝑚 𝑝𝑖

𝑚= 2𝑛 𝑚 − 1

𝑞𝑖

Bounds on Distance Optimality

12

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.


12



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.


12



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 .


12

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

13


13


13

Two-level decentralized hierarchical strategy


Two-level ideal hierarchical strategy

2



13

Two-level decentralized hierarchical strategy


Two-level ideal hierarchical strategy

2


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

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Distance Optimal Target Assignment in Robotic Networks...

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