1. 1. Collective Transport in Autonomous Multi-Robot Systems ALGORITHMS, ANALYSIS &APPLICATIONS GANESH P KUMAR Advisor: Prof. Spring Berman 1
2. 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. 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. 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. 5. Modelling Collective Transport in A. cockerelli HSCC 2013 5
6. 6. pSHS : Behavioural Model FrontBack Detached F BD , , , State vector = , , , , Behavioural states S = , , = population in state Dynamical variables , Flow Equation = 0 0 0 6 reactions : in Chemical Reaction Network 6
7. 7. pSHS : Dynamical Model Front and back ants lift with net force Net normal force = Front ants pull with proportional velocity regulation = ( ) LOAD Fup = + Load Dynamics = = 7
8. 8. Model Predictions vs. Averaged Data 8
9. 9. Controller for Multi-Boundary Coverage ISRR 2013 ASME JDSMC 2014 Swarm Intelligence 2014 9
10. 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. 11. Microscopic Model Species: , , Design parameters: , Encounter rates: , 11
12. 12. Macroscopic Model Equilibrium Allocation ODEs 12
13. 13. Statistical Analysis of Stochastic Boundary Coverage ICRA 2014 IEEE-Trans. On Robotics (Submitted) 2015 0 = 0 +1 = 1 2 13
14. 14. Saturation dist Boundary of length identified with I 0, robots each of radius attach randomly to boundary Configuration is saturated iff all distances above are bounded above by = 0 = 14
15. 15. ProblemStatement dist Given Quadruple = (, , , ) Define random configuration. Compute probability of saturation . Compute pdfs of robot positions and inter-robot distances for random configurations. for random saturated configurations. = 0 = 15
16. 16. SaturationforPointrobots 0 = 0 +1 = 1 2 Point robots have = 0 Random configuration: robots attach to boundary uniformly randomly and independently Sort robot positions fixing two artificial robots at end- points, creating = t1, , and 0:+1 16
17. 17. PositionSimplex 0 = 0 3 = 1 2 samples from the th order statistic of a uniform parent pdf can be considered a point in Valid configurations form the position simplex { : :+} = = 0 T = 17
18. 18. ConceptofSlack 18 1 2 Define th slack as 1 Collect all slacks in slack vector 1 +1 T For any configuration, the sum of slacks equals : = 3
19. 19. Simplex-Hypercube Intersection 19 Valid slack vectors form a slack simplex +1 { : = } Saturated slack vectors form a hypercube +1 { : } Define favourable region by 1 2 2 1 2 2 2: 0 1, 2 1 + 2 = 1 (, 0) (0, ) ( , ) (, )
20. 20. Computing We have Using Inclusion Exclusion Principle, we have Here = is the maximum number of -separated robots that can attach to boundary Positions and slacks have scaled Beta pdfs: 20
21. 21. Small andLarge cases 1 = 2 = 2 = We need to determine PDFs of robot positions and slacks under saturation Define 3 parameters for (, , = 0, ): : = = max number of -separated robots = last slack in such a configuration , if 0 + 1, if = 0 = remaining number of robots +1 = more robots need to be placed 21
22. 22. Small andLarge cases If = 0, then no more robots need to be placed There are just enough robots to saturate This is the small case (: 1, : 2, : 0, : 0.4) with = 2, = 0.2, = 0 If > 0, we have the large case (: 1, : 2, : 0, : 0.6) with = 1, = 0.4, = 1 22
23. 23. Saturationforsmall forms a regular simplex, with vertices along the columns of: +1 (, : 0.2, : 0, : 2) 23
24. 24. Large case Now we have = + robots to place Now is a convex polytope with cospherical vertices Unlike in small case, no analytic expression for pdfs of saturated slacks and positions 24
25. 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 (: 1, : 0.6, : 0, : 2) 25
26. 26. GEOMSAMP Given , , = , , sample a random saturated slack vector Use QuickHull to partition into pyramids Compute = /( ) for each pyramid Choose a random pyramid with prob. Sample a point from 26
27. 27. REPSAMP: SamplingusingRepresentatives Address large case using results from small case Choose a saturated configuration of + 1 representatives This represents a sample from Corresponds to the + 1 nonzero elements in every vertex Choose intermediates randomly Saturation condition remains invariant! 1 2 +1 1 2 +1 Hollow circles are intermediates 0 27
28. 28. Future Work 28
29. 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. 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. 31