1. Statistical Analysis of Stochastic Multi-Robot
Boundary Coverage
GANESH P KUMAR & SPRING M BERMAN
FULTON SCHOOLS OF ENGINEERING
ARIZONA STATE UNIVERSITY
1

2. Why Stochastic Boundary Coverage?
2
Distributed sensing
J. Brandon, Digital Trends, 7/13/11
http://goo.gl/utjpbc
Targeted drug delivery
Sinha et al., Mol. Cancer. Ther. ’06
http://goo.gl/zXVHGw
Collective transportImaging cancer cells
Qin et al., Adv. Funct. Mater. ’12
http://goo.gl/ZhRYSb
Nanoscale Applications Macroscale Applications

3. What is Stochastic Boundary Coverage?
Robots:
 Occupy random positions along
boundaries
 Sense or communicate within a local
range
 Don’t have:
 Global position
 Map of environment
Multi-Robot Boundary Coverage
[T. Pavlic, S. Wilson, G. Kumar, and S. Berman, ISRR ’13]
3
Example Goal: Achieve target
allocation on boundary

4. Related Work
4
Design of collective behaviors in robotic swarms
 Active Self-Assembly [Napp, Burden, & Klavins, RSS’06, RSS’09]
 Product Assembly [Matthey, Berman, & Kumar, ICRA’09]
 Chain Formation [Evans, Mermoud, & Martinoli, ICRA’10]
 Distributed Manipulation [Martinoli, Easton, & Agassounon, IJRR ’04]
 Turbine Inspection [Correll & Martinoli, DARS ’07]
 Macroscopically model collective behavior
 Require prior knowledge of encounter rates

5. Related Work
5
Models of adsorption processes
 Langmuir adsorption (reversible) [Langmuir, J Amer Chem Soc, 1918]
 Random sequential adsorption (irreversible) [Talbot et al., Colloid Surface A ‘00]
 Robot attachments may be modelled as adsorption
 Attachment rates controlled to get desired coverage

6. Saturation: Densest Possible Stochastic Coverage
CLOSED
BOUNDARY
Distance ≤ 𝒅
OPEN BOUNDARY
6

7. Problem Statement
Let 𝑛 robots of diameter 2δ attach at uniformly random positions to a unit-
length closed boundary.
Given saturating distance 𝑑, find:
 Probability of Saturation 𝑝𝑠𝑎𝑡
 Distribution of robot positions
 Distribution of inter-robot distances
7
Not saturated Saturated
n = 14

8. Agenda
 Solve problem for point robots (𝛿 = 0)
 Explain extension for finite-sized robots (𝛿 > 0)
 Validate results with Monte Carlo simulations
8
Note: We’ll consider only closed unit-length boundaries from now on.

9. Point Robots: Saturation
CLOSED
BOUNDARY
𝑟1
𝑟2
𝑟𝑖
𝑟𝑛
𝑟1 𝑟2 𝑟𝑖 𝑟𝑛
𝑡1 = 0 𝑡 𝑛+1 = 1𝑡𝑖
Open boundary: 𝑰 = [𝟎, 𝟏]
 Robot 𝑟𝑖 attaches at 𝑡𝑖 with 𝑡1 = 0 and 𝑡 𝑛+1 = 1
 Saturation implies 𝑡𝑖+1 − 𝑡𝑖 ≤ 𝑑, 𝑖 = 1,2, … , 𝑛
9

10. Point Robots: Saturation
 Here 𝑡𝑖 is called the 𝑖 -th order statistic of a uniform parent
 (𝑡2, 𝑡3, … , 𝑡 𝑛) is a point in the event space
 Since 𝑡𝑖 ≤ 𝑡𝑖+1 , valid configurations lie in the event simplex
𝛀n = 𝑡2, 𝑡3, … , 𝑡 𝑛 : 0 ≤ 𝑡𝑖 ≤ 𝑡𝑖+1 ≤ 1 ⊂ ℝ 𝑛−1
𝑉𝑜𝑙 𝑛−1 𝛀n =
1
𝑛−1 !
Ω3
(0,0)
(0,1)
(1,0)
𝑡3
𝑡2
10

11. Concept of Slack
 Define
Total slack 𝑠 = length of curve = 1
Individual slack 𝑠𝑖 = 𝑡𝑖+1 − 𝑡𝑖
 Individual slacks (𝑠1, 𝑠2, … , 𝑠 𝑛) determine slack space
𝑠1 𝑠2 𝑠3
𝑠 = 1
𝑠1 + 𝑠2 + 𝑠3 = 𝑠
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12. Geometric Interpretation of 𝒑 𝒔𝒂𝒕
 Valid slacks lie on the regular slack simplex
𝐒 𝐧 = { 𝑠1, 𝑠2, … , 𝑠 𝑛 : 0 ≤ 𝑠𝑖, ∑𝑠𝑖 = 𝑠} ⊂ ℝ 𝑛−1
 Saturated configurations lie in the saturated hypercube:
𝐇 𝐧 = { 𝑠1, 𝑠2, … , 𝑠 𝑛 : 0 ≤ 𝑠𝑖 ≤ 𝑑} ⊂ ℝ 𝑛
 We have 𝒑 𝒔𝒂𝒕 =
𝑽𝒐𝒍 𝒏−𝟏 𝐒 𝐧∩𝐇 𝒏
𝑽𝒐𝒍 𝒏−𝟏(𝐒 𝒏) 𝐇3
(0,0,0)
(𝑑, 𝑑, 𝑑)
𝐒3
(0, 𝑠, 0) (𝑠, 0,0)
(0,0, 𝑠)
𝑠 2
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13. Computing Simplex Volume
 The volume of 𝑛 dimensional regular simplex 𝐑 of side 𝑎 is
𝑉𝑜𝑙 𝑛(𝐑) =
𝑎
2
𝑛
.
𝑛+1
𝑛!
 Since 𝐒 𝑛 is regular with side 𝑠 2,
𝑉𝑜𝑙 𝑛−1 𝐒 𝑛 = 𝑠 𝑛−1.
𝑛
𝑛−1 !
 E.g. 𝑉𝑜𝑙2 𝐒 𝟑 =
𝑠2 3
2
𝑝𝑠𝑎𝑡 =
𝑉𝑜𝑙 𝑛−1 𝐒 𝐧 ∩ 𝐇 𝒏
𝑽𝒐𝒍 𝒏−𝟏(𝐒 𝒏)
13

14. Computing Intersection Volume
 Not straightforward at all, requires indirect attack!
 Let 𝐴 𝑘 be a 𝑘-element subset of slack axes {𝑠1, 𝑠2, … , 𝑠 𝑛}.
 Consider a sub-region of 𝐒 𝑛 on which slacks in 𝐴 𝑘 are unsaturated.
𝐸𝑥𝑡(𝐴 𝑘) = 𝑠1, 𝑠2, … , 𝑠 𝑛 : 𝑠𝑖 ∈ 𝐴 𝑘 𝑠𝑖 ≥ 𝑑 ∩ 𝐒 𝑛
 Then 𝐸𝑥𝑡(𝐴 𝑘) is a regular simplex !
𝑉𝑜𝑙 𝑛−1 𝐸𝑥𝑡(𝐴 𝑘) =
𝑠 − 𝑘𝑑 𝑛−1 𝑛
𝑛 − 1 !
𝑝𝑠𝑎𝑡 =
𝑉𝑜𝑙 𝑛−1 𝐒 𝐧 ∩ 𝐇 𝒏
𝑉𝑜𝑙 𝑛−1(𝐒 𝑛)
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15. Computing Intersection Volume
 Note that unsaturated region 𝐒 𝑛𝐇 𝑛 = 𝐴 𝑘
𝐸𝑥𝑡(𝐴 𝑘) for all
nonempty 𝐴 𝑘 ⊆ {𝑠1, 𝑠2, … , 𝑠 𝑛}
 Its volume can be computed by the Inclusion-Exclusion Principle
𝑉𝑜𝑙 𝐵1 ∪ 𝐵2 … ∪ 𝐵𝑛 =
∑𝑉𝑜𝑙 𝐵𝑖 − ∑𝑉𝑜𝑙 𝐵𝑖 ∩ 𝐵𝑗 + ⋯ + −1 𝑛 𝑉𝑜𝑙(𝐵1 ∩ 𝐵2 … ∩ 𝐵𝑛)
 Writing 𝑝𝑠𝑎𝑡 in this form gives our result!
𝒑 𝒔𝒂𝒕 = 𝟏 − ∑ 𝟏≤𝒌≤𝑲 −𝟏 𝒌−𝟏( 𝒏
𝒌
) 𝟏 −
𝒌𝒅
𝒔
𝒏−𝟏
where 𝐾 =
𝑠
𝑑
.
15
𝑝𝑠𝑎𝑡 = 1 −
𝑉𝑜𝑙 𝑛−1 𝐒 𝐧𝐇 𝒏
𝑉𝑜𝑙 𝑛−1(𝐒 𝐧)

16. Point Robots: PDFs of Robot Positions
 Joint pdf of robot positions is the uniform pdf
𝑓 𝑡2, 𝑡3, … , 𝑡 𝑛 = 𝑛 − 1 ! 𝟏Ω 𝑛
 To get 𝑓(𝑡𝑖), marginalize over remaining statistics by repeated integration
𝑓 𝑡𝑖 = 𝑛
𝑖
𝑡 𝑖−1 1 − 𝑡 𝑛−𝑖 𝟏𝐼 , 𝑖 = 2,3, … , 𝑛
 This is a Beta density, and simplifies to:
𝑓 𝑡𝑖 = 𝐵𝑒𝑡𝑎(𝑡|𝑖 − 1, 𝑛 + 1 − 𝑖)
16
𝐵𝑒𝑡𝑎 𝑡 𝑚, 𝑛
=
𝑚 + 𝑛 − 1 !
𝑚 − 1 ! 𝑛 − 1 !
𝑥 𝑚−1 1 − 𝑥 𝑛−1 𝟏𝐼

17. Point Robots: PDF of Slacks
 Define the domain comprised by all but the last slack:
D ≔ { 𝑠1, 𝑠2, … , 𝑠 𝑛−1 : 0 ≤ ∑𝑠𝑖 ≤ 1}
 The joint pdf of slacks is found to be uniform over D:
𝑓 𝑠1, 𝑠2, … , 𝑠 𝑛−1 = 𝑛 − 1 ! 𝟏D
 Marginalize over remaining slacks to get 𝑓(𝑠𝑖):
𝑓 𝑠𝑖 = 𝐵𝑒𝑡𝑎(𝑠𝑖|1, 𝑛 − 1)
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18. Extension to Finite-Sized Robots
 Robots attaching sequentially at uniformly random positions result in
Renyi’s Parking Problem scenario [Dvoretzky & Robbins ‘64]
 Resembles adsorption of particles on a surface
 We solve a different problem which extends point-robot analysis
 Robots coordinate to avoid overlaps using Slack Attach protocol
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19. Finite-Sized Robots: Slack Definitions
 Total Slack = length of boundary unoccupied after 𝑛 robots attach:
𝑠 ≔ 1 − 2𝛿𝑛
 Individual slack = closest distance between adjacent robots:
𝑠𝑖 ∶= 𝑡𝑖+1 − 𝑡𝑖 −2𝛿
𝑡1 𝑡2
𝑡 𝑛𝑡𝑖
2𝑛𝛿 𝑠 = 1 − 2𝑛𝛿
𝑡𝑖 𝑡𝑖+1
𝑠𝑖
… …
19

20. Slack Attach Protocol
Boundary
𝑟1
Step 1 All robots attach adjacent to
one another. The most clockwise one
is called 𝑟1
20

21. Slack Attach Protocol
Boundary
Step 2 First robot detaches and travels
around boundary to measure 𝑠, then
reattaches to original position
𝑟1
21

22. Slack Attach Protocol
Step 3 Robots collectively choose
𝑎𝑟𝑟 1 … 𝑛 − 1 uniformly randomly
from [0, 𝑠] and sort it in increasing
order. They compute individual slacks
𝑠𝑖 = 𝑎𝑟𝑟 𝑖 + 1 − 𝑎𝑟𝑟[𝑖]
𝑎𝑟𝑟[1]0 𝑎𝑟𝑟[2] 𝑎𝑟𝑟[3] 𝑠
𝑠1 𝑠2 𝑠3 𝑠4
22

23. Slack Attach Protocol
Boundary
Step 4 Robots choose positions
𝑡𝑖 = 𝑡𝑖−1 + 𝑠𝑖−1 + 2𝛿
with 𝑡1 ≔ 0
𝑡1
𝑡2
𝑡3
𝑡4
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24. Finite-Sized Robots: Saturation
 Saturation = “no robot can enter between two adjacent ones”
𝑑 ≔ 4𝛿 𝑠𝑖 ≤ 2𝛿
 Simplex-Hypercube intersection gives us
𝑝𝑠𝑎𝑡 = 1 − ∑ 𝑘=1
𝐾
−1 𝑘−1( 𝑛
𝑘
) 1 −
2𝑘𝛿
𝑠
𝑛−1
where 𝐾 = ⌊
1−2𝑛𝛿
2𝛿
⌋
24

25. Finite-Sized Robots: Robot Positions & Slacks
 Slacks have the scaled Beta pdf
𝑓 𝑠𝑖 = 𝑠. 𝐵𝑒𝑡𝑎(𝑡|1, 𝑛 − 1)
 Order statistics non-trivial!
𝑓 𝑡1 = 0
𝑓 𝑡2 = 𝑠. 𝐵𝑒𝑡𝑎 𝑡 1, 𝑛 − 1 + 2𝛿
𝑡3 = 𝑡2 + 𝑠2 + 2𝛿
 We don’t know correlations between 𝑡2 and 𝑠2 nor the joint pdf of
order statistics!
25

26. Results: Comparison with Monte Carlo Simulations
 Analytical solutions for 𝑝𝑠𝑎𝑡 compared with averages of 20,000 Monte Carlo
Trials
 Analytical solutions for order statistics and slack densities compared with
averages of 5,000 Monte Carlo Trials
26

27. Validating 𝒑 𝒔𝒂𝒕 for point robots
27
𝑛𝑑
𝑝𝑠𝑎𝑡
𝑝𝑠𝑎𝑡
𝑑
𝑛
Analytical Monte Carlo
𝑝𝑠𝑎𝑡 ↑ as 𝑛, 𝑑 ↑
# of robots
Sat. distance

28. Validating 𝒑 𝒔𝒂𝒕 for finite-sized robots
28
𝑝𝑠𝑎𝑡
𝛿
Analytical Monte Carlo
𝛿
# of robots
Robot radius 𝑛
𝑝𝑠𝑎𝑡
𝑝𝑠𝑎𝑡 ↑ as 𝑛, 𝛿 ↑; high (𝑛, 𝛿) unphysical
𝑛

29. Validating Robot Position PDF for Point Robots
Frequency plot of
5000 samples of 𝑡2
fit to a 𝐵𝑒𝑡𝑎(𝑡|1,4)
density for 𝑛 = 5
point robots
29
𝒕 𝟐 𝑡5𝑡3𝑡1 = 0 𝑡6 = 1𝑡4

30. Validating Slack PDF for Point Robots
Frequency plot of
5000 samples of
𝑠2 = 𝑡3 − 𝑡2 fit to
a 𝐵𝑒𝑡𝑎(𝑡|1,4)
density for 𝑛 = 5
point robots
30
𝑡1 = 0 𝑡2 𝑡5𝑡3
𝒔 𝟐
𝑡6 = 1𝑡4

31. Conclusion
 Posed the problem of computing saturation probability
 Analytically determined 𝑝𝑠𝑎𝑡 and order statistics for point robots
 Extended analysis to finite sized robots, explaining limitations
 Validated results with Monte Carlo simulations
31

32. Future Work
 Extend finite-sized robots case to asynchronous attachment
 Attachment over time, without coordination
 Enable analysis of our prior simulations [ISRR ’13]
 Determine order statistics and slack pdfs given saturation
𝑓 𝑡𝑖 sat , 𝑓(𝑠𝑖|sat)
 Develop distributed controllers for multi-robot transport
 Goal : achieve robustness of ant food retrieval
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[G. Kumar, A. Buffin, T. Pavlic, S.Pratt, and S. Berman, HSCC’13]

33. Acknowledgements
Sean Wilson, Theodore Pavlic, Ruben Gameros :
for valuable discussions on paper and presentation
33
The Autonomous Collective Systems Lab
Front: Prof. Spring Berman
From left to right:
• Dr. Theodore Pavlic (Postdoc collaborator)
• Ruben Gameros (Master’s student)
• Karthik Elamvazhuthi (Master’s student)
• Ganesh Kumar (PhD student)
• Sean Wilson (PhD student)