This document summarizes research on collective transport in autonomous multi-robot systems. It presents: 1) A model of collective transport behavior in desert ants using a stochastic hybrid system approach. 2) A stochastic controller for multi-robot boundary coverage that allocates robots around boundaries in a robust manner. 3) Analysis of the statistical properties of multi-robot configurations around single boundaries, including computing the probability of saturated configurations and position/distance distributions. Future work involves implementing algorithms on real robots for collective transport tasks and adding visual servoing capabilities.

1. Collective Transport in Autonomous
Multi-Robot Systems
ALGORITHMS, ANALYSIS &APPLICATIONS
GANESH P KUMAR
Advisor: Prof. Spring Berman
1

2. Motivation
Pheeno Robots
Search & Rescue Construction
Robot team transports heavy payload
No GPS or prior information about
environment
Robots sense and communicate
within limited range
2

3. Novel Contributions
Modelled collective transport in A. cockerelli as a Stochastic
Hybrid System (SHS)
Designed a stochastic controller for multi-robot boundary
coverage
Robust to environmental variations
May be made to mimic A. cockerelli behaviour
Computed statistical properties of multi-robot
configurations around single boundary
Devised fast algorithm for sampling saturated configurations
3

4. Outline
Model collective transport in Desert Ant A. Cockerelli
Design stochastic controller to allocate robots around
boundaries
Analyze properties of stochastic multi-robot configurations
around single boundary
Future Work
4

5. Modelling Collective Transport in A.
cockerelli
HSCC 2013
5

6. pSHS : Behavioural Model
FrontBack
Detached
F
BD
𝑟 𝐷𝐹, 𝑟𝐹𝐷 𝑟𝐹𝐵, 𝑟𝐵𝐹
𝑟 𝐷𝐵, 𝑟𝐵𝐷
 State vector 𝒙 = 𝑁𝐹, 𝑁 𝐵, 𝑁 𝐷, 𝑥 𝐿, 𝑣 𝐿
𝑇
 Behavioural states S = 𝐹, 𝐵, 𝐷
 𝑁𝑖∈𝑆 = population in state 𝑖
 Dynamical variables 𝑥 𝐿, 𝑣 𝐿
 Flow Equation 𝒙 = 0 0 0 𝑣 𝐿
𝐹
𝑚 𝐿
 6 reactions :𝑆𝑖 → 𝑆𝑗 in Chemical
Reaction Network
6

7. pSHS : Dynamical Model
 Front and back ants lift with net
force 𝐹𝑢𝑝
 Net normal force
𝐹𝑛 = 𝑚 𝐿 𝑔 − 𝐹𝑢𝑝
 Front ants pull with
proportional velocity regulation
𝐹𝑝 = 𝐾(𝑣 𝐿
𝑑
− 𝑣 𝐿)
LOAD
𝑚 𝐿 𝑔
𝑁 𝐹 𝐹𝑝𝜇𝐹𝑛
𝐹𝑟𝑜𝑛𝑡
Fup =
𝑁 𝐹 + 𝑁 𝐵 𝐹𝐿
𝐵𝑎𝑐𝑘
Load Dynamics
𝑥 𝐿 = 𝑣 𝐿
𝑣 𝐿 = 𝑁𝐹 𝐹𝑝 − 𝜇𝐹𝑛
7

8. Model Predictions vs. Averaged Data
8

9. Controller for Multi-Boundary
Coverage
ISRR 2013
ASME JDSMC 2014
Swarm Intelligence 2014
9

10. Achieve target allocation of robots around disks at steady state
Robots:
• Perform correlated random walks
• Local sensing and communication
• Can identify whether another robot is
bound or unbound
Disks:
• Randomly distributed throughout
environment
• Each type requires a different target
robot group size
Example: 3 robots per type-1 disk
1 robot per type-2 disk
Problem Statement
10

11. Microscopic Model
Species: 𝑟, 𝑈, 𝐵
Design parameters: 𝑝 𝑢, 𝑝 𝑏
Encounter rates: 𝑒 𝑏, 𝑒 𝑢
11

12. Macroscopic Model
Equilibrium Allocation
𝐈𝐝𝐞𝐚𝐥 𝐚𝐥𝐥𝐨𝐜𝐚𝐭𝐢𝐨𝐧
ODEs
12

13. Statistical Analysis of Stochastic
Boundary Coverage
ICRA 2014
IEEE-Trans. On Robotics (Submitted) 2015
𝑡0 = 0 𝑡 𝑛+1 = 𝑠
𝑡1 𝑡2 𝑡 𝑛
13

14. Saturation
dist ≤ 𝑑
 Boundary of length 𝑠 identified with 𝑠I ≔ 0, 𝑠
 𝑛 robots each of radius 𝛿 attach randomly to boundary
 Configuration is saturated iff all distances above are
bounded above by 𝑑
𝑡 = 0 𝑡 = 𝑠
14

15. ProblemStatement
dist ≤ 𝑑
Given Quadruple 𝑸 = (𝒔, 𝒏, 𝜹, 𝒅)
 Define random configuration.
 Compute probability of saturation 𝑝𝑠𝑎𝑡.
 Compute pdfs of robot positions and inter-robot distances
 for random configurations.
 for random saturated configurations.
𝑡 = 0 𝑡 = 𝑠
15

16. SaturationforPointrobots
𝑡0 = 0 𝑡 𝑛+1 = 𝑠
𝑡1 𝑡2
𝑡 𝑛
 Point robots have 𝛿 = 0
 Random configuration: robots attach to boundary
uniformly randomly and independently
 Sort robot positions fixing two artificial robots at end-
points, creating 𝒕 = t1, … , 𝑡 𝑛
𝑻
and 𝒕0:𝑛+1
16

17. PositionSimplex
𝑡0 = 0 𝑡3 = 𝑠
𝑡1 𝑡2
 𝑡𝑖 samples from the 𝑖th order statistic of a uniform parent
pdf
 𝒕 can be considered a point in ℝ 𝑛
 Valid configurations form the position simplex
≔ {𝒕 ∈ ℝ 𝑛 ∶ 𝟎 ≤ 𝑡 ≤ 𝑠𝟏 ∧ 𝒕 𝟎:𝒏 ≤ 𝒕 𝟏:𝒏+𝟏}
𝒕 = 𝟎
𝒕 = 0 𝑠 T
𝒕 = 𝑠𝟏
17

18. ConceptofSlack
18
𝑠1 𝑠2
 Define 𝑖th slack as 𝑠𝑖 ≔ 𝑡𝑖 − 𝑡𝑖−1
 Collect all slacks in slack vector 𝒔 ≔ 𝑠1 … 𝑠 𝑛+1
T
 For any configuration, the sum of slacks equals 𝑠: 𝟏 𝑻 𝒔 = 𝑠
𝑠3

19. Simplex-Hypercube Intersection
19
 Valid slack vectors form a slack simplex
𝐒 𝑛+1 ≔ {𝒔 ∈ ℝ 𝑛: 𝒔 ≥ 𝟎 ∧ 𝟏 𝐓 𝒔 = 𝑠}
 Saturated slack vectors form a hypercube
𝐇 𝑛+1≔ {𝒔 ∈ ℝ 𝑛
: 𝟎 ≤ 𝒔 ≤ 𝑑𝟏 𝐓
}
 Define favourable region by
𝑠1 𝑠2
𝑠2
𝑠1
𝐔2
𝐔2
𝐇2: 0 ≤ 𝑠1, 𝑠2 ≤ 𝑑
𝑠1 + 𝑠2 = 𝑠
𝑡1
(𝑠, 0)
(0, 𝑠)
(𝑠 − 𝑑, 𝑑)
(𝑑, 𝑠 − 𝑑)

20. Computing 𝒑𝒔𝒂𝒕
 We have
 Using Inclusion Exclusion Principle, we have
Here 𝐾 =
𝑠
𝑑
is the maximum number of 𝑑-separated
robots that can attach to boundary
 Positions and slacks have scaled Beta pdfs:
20

21. Small andLarge 𝒏cases
𝑡1 = 𝑑 𝑡2 = 2𝑑 𝑡 𝐾 = 𝐾𝑑
 We need to determine PDFs of robot positions and slacks
under saturation
 Define 3 parameters for 𝑄(𝑠, 𝑛, 𝛿 = 0, 𝑑):
 𝐾: = ⌊
𝑠
𝑑
⌋ = max number of 𝑑-separated robots
 𝑙 ≔ 𝑠 − 𝐾𝑑 = last slack in such a configuration
 𝑧 ≔
𝑛 − 𝐾 , if 𝑙 ≠ 0
𝑛 − 𝐾 + 1, if 𝑙 = 0
= remaining number of robots
𝑠 𝐾+1 = 𝑙
𝑧 more robots
need to be placed
21

22. Small andLarge 𝒏cases
 If 𝑧 = 0, then no more robots need to be placed
 There are just enough robots to saturate
 This is the small 𝑛 case
𝑄(𝑠: 1, 𝑛: 2, 𝛿: 0, 𝑑: 0.4) with 𝐾 = 2, 𝑙 = 0.2, 𝑧 = 0
 If 𝑧 > 0, we have the large 𝒏 case
𝑄(𝑠: 1, 𝑛: 2, 𝛿: 0, 𝑑: 0.6) with 𝐾 = 1, 𝑙 = 0.4, 𝑧 = 1
22

23. Saturationforsmall 𝒏
 forms a regular simplex, with vertices along the
columns of:
𝐒 𝑛+1
𝑄(𝑠, 𝑑: 0.2𝑠, 𝛿: 0, 𝑛: 2)
23

24. Large 𝒏 case
 Now we have 𝑛 = 𝐾 + 𝑧 robots to place
 Now is a convex polytope with cospherical vertices
 Unlike in small 𝑛 case, no analytic expression for pdfs of
saturated slacks and positions
24

25. Shape of
 Vertices are permutations of
 Pyramids are formed by adjoining centroid to facets
 Vertices of Base facets have zeros at identical locations
 Vertices of Connecting facets do not
𝑄(𝑠: 1, 𝑑: 0.6, 𝛿: 0, 𝑛: 2)
25

26. GEOMSAMP
Given 𝑄 𝑠, 𝑑, 𝜹 = 𝟎, 𝑛 , sample a random saturated slack
vector
 Use QuickHull to partition into pyramids
 Compute 𝑝𝑖 = 𝑉𝑜𝑙 𝐏𝑖 /𝑉𝑜𝑙( ) for each pyramid 𝐏𝑖
 Choose a random pyramid 𝐏𝑖 with prob. 𝑝𝑖
 Sample a point from 𝐏𝑖
26

27. REPSAMP: SamplingusingRepresentatives
 Address large 𝒏 case using results from small 𝒏 case
 Choose a saturated configuration of 𝐾 + 1 representatives
 This represents a sample from
 Corresponds to the 𝐾 + 1 nonzero elements in every vertex
Choose 𝑧 intermediates randomly
 Saturation condition remains invariant!
𝑠1
𝑟𝑒𝑝
𝑠2
𝑟𝑒𝑝 𝑠 𝐾+1
𝑟𝑒𝑝
𝑠1
𝑟𝑒𝑝
𝑠2
𝑟𝑒𝑝
𝑠 𝐾+1
𝑟𝑒𝑝
Hollow circles are
intermediates
0 ≤ 𝑠𝑖
𝑟𝑒𝑝
≤ 𝑑
27

28. Future Work
28

29. Pheeno Robot
 Developed as component of collective transport testbed
 Differentially driven base, with R-P-R manipulator arm
and 1 DOF gripper
 RPi Model B+ directing an Arduino Micro Pro
 RPi camera, IR Sensors, Wi-fi Adapter, LEDs
29
Total cost ~ $400

30. Timeline
30
 Complete internship at Mayfield Robotics by 15th Aug
 Implementing and serializing random attachment
algorithm using Pheenos (by 30th Sep)
 Submit journal paper by 31st Oct
 Visual servoing for manipulation (Late fall)
 Dissertation Writing (from 1st Nov)
 Final Defence (by late January 2016)

31. 31