1. Fault-Tolerant Formations of Mobile Robots
Ross Mead
CS Dept., University of Southern California, USA
Rob Long
Jerry B. Weinberg
Dept. of CS, Southern Illinois University Edwardsville, USA

2. Outline Conclusions Future Work Q&A Problem Statement Considerations
Approach
Algorithm
Implementation
Evaluation
Conclusions
Future Work
Q&A 2

3. Problem Statement swarm formation
How can a large collection of robots moving with no group organization coordinate to form a global structure?
swarm
formation 3

4. Considerations In related work on formations, robotic units know…
where they belong in the formation
who their neighbors are supposed to be
Criteria of Fredslund & Matarić 2002:
generality – conforming to a variety of formations
stability – maintaining the formation
dynamic switching capability – responding to commands for changes in its organization
robustness/scalability – responding to changes in group size
obstacle avoidance – dealing with large and small obstructions 4

5. Approach Utilize reactive robot control strategies
closely couple sensor input to actions
Treat the formation as a cellular automaton
lattice of computational units (cells)
each cell is in one of a given set of states
state transitions governed by a set of rules
complex emergent behavior from simplicity
It relies solely on communication with local robot neighbors, and hence has the advantage of eliminating the need for each robot to maintain information about the global or overall structure of the entire formation. 5

6. “Robot-Space” Cellular Automata
Each robot in the formation is represented as a cell ci in an n-dimensional automaton of N cells…
ci = {hi, si, F, S}
In keeping with a simple reactive control strategy, the approach of Mead et al. [18, 19] is to treat the robots in the formation as cells in a 1-dimensional robot-space cellular automaton [Fig. 13]. This distinguishes itself in that the actual robots that make up the global structure (i.e., not the structure itself) are represented by the cells. This approach overcomes many of the limitations inherent in a world-space automaton, eliminating the dependence on the environment and reducing the complexity of the automaton. 6

7. “Robot-Space” Cellular Automata
C ← automaton:
C = h1 U h2 U … U hN
= {c1, c2, …, cN}
Each robot in the formation is represented as a cell ci in an n-dimensional automaton of N cells…
ci = {hi, si, F, S}
neighborhood
hi = ci U { n neighbors }
n ≤ nmax
In keeping with a simple reactive control strategy, the approach of Mead et al. [18, 19] is to treat the robots in the formation as cells in a 1-dimensional robot-space cellular automaton [Fig. 13]. This distinguishes itself in that the actual robots that make up the global structure (i.e., not the structure itself) are represented by the cells. This approach overcomes many of the limitations inherent in a world-space automaton, eliminating the dependence on the environment and reducing the complexity of the automaton. 7

8. “Robot-Space” Cellular Automata
C ← automaton:
C = h1 U h2 U … U hN
= {c1, c2, …, cN}
Each robot in the formation is represented as a cell ci in an n-dimensional automaton of N cells…
ci = {hi, si, F, S}
state
si = {pi, ri,des, ri,act, Γi, Θi, ti}
(... described later ...)
In keeping with a simple reactive control strategy, the approach of Mead et al. [18, 19] is to treat the robots in the formation as cells in a 1-dimensional robot-space cellular automaton [Fig. 13]. This distinguishes itself in that the actual robots that make up the global structure (i.e., not the structure itself) are represented by the cells. This approach overcomes many of the limitations inherent in a world-space automaton, eliminating the dependence on the environment and reducing the complexity of the automaton. 8

9. “Robot-Space” Cellular Automata
C ← automaton:
C = h1 U h2 U … U hN
= {c1, c2, …, cN}
Each robot in the formation is represented as a cell ci in an n-dimensional automaton of N cells…
ci = {hi, si, F, S}
state transition
si = {pi, ri,des, ri,act, Γi, Θi, ti}
(... described later ...)
sit = S(... si-1t-1, sit-1, si+1t-1 ...)
t ← time step (counter)
In keeping with a simple reactive control strategy, the approach of Mead et al. [18, 19] is to treat the robots in the formation as cells in a 1-dimensional robot-space cellular automaton [Fig. 13]. This distinguishes itself in that the actual robots that make up the global structure (i.e., not the structure itself) are represented by the cells. This approach overcomes many of the limitations inherent in a world-space automaton, eliminating the dependence on the environment and reducing the complexity of the automaton. 9

10. “Robot-Space” Cellular Automata
C ← automaton:
C = h1 U h2 U … U hN
= {c1, c2, …, cN}
Each robot in the formation is represented as a cell ci in an n-dimensional automaton of N cells…
ci = {hi, si, F, S}
formation
F = {f’(x), R, Φ, pseed}
f’(x) ← description
R ← robot separation
Φ ← relative heading
pseed ← start position
In keeping with a simple reactive control strategy, the approach of Mead et al. [18, 19] is to treat the robots in the formation as cells in a 1-dimensional robot-space cellular automaton [Fig. 13]. This distinguishes itself in that the actual robots that make up the global structure (i.e., not the structure itself) are represented by the cells. This approach overcomes many of the limitations inherent in a world-space automaton, eliminating the dependence on the environment and reducing the complexity of the automaton. 10

11. Algorithm – Formation Definition
F is sent to some robot, designating it as the seed cell cseed...
cseed is not a leader, but rather an initiator of the coordination process
For purposes of calculating desired relationships, each cell ci considers itself to be at some formation-relative position pi:
pi = [ xi f(xi) ]T
In the case of cseed, this position pseed is given…
f(x) = a x2
Note that cseed is not a leader, but rather an initiator of the coordination process.
cseed
pseed 11

12. Algorithm – Desired Relationships
The desired relationship ri→j,des from ci to some neighbor cj is determined by calculating a vector v from pi to the intersection f(vx) and a circle centered at pi with radius R:
R2 = (vx – pi,x)2 + (f(vx) – pi,y)2 ri→j,des = [ vx f(vx) ]T
Relationships are rotated by –Φ to account for robot heading...
f(x) = a x2 R –v +v
pseed
desired relationship
with left neighbor ci-1
ri→i-1,des
ri→i+1,des
desired relationship
with right neighbor ci+1 12

13. Algorithm – Desired Relationships
The desired relationship ri→j,des from ci to some neighbor cj produces a unique formation-relative position pj:
pj = pi + ri→j,des rj→i,des = –ri→j,des
(for stability)
f(x) = a x2
pi-1
pi+1
pseed
ri→i-1,des
ri→i+1,des 13

14. Algorithm – Desired Relationships
ci announces an auction for pj–denoted A(pj)–iff:
||pi – pseed|| < ||pj – pseed|| Γi ≈ 0
(for densest packing) (for stability)
f(x) = a x2
pi-1
pi+1
pseed 14

15. Algorithm – Desired Relationships
A robot that receives an auction message for pj will announce a bid B(pj) iff n = 0 (i.e., it has no neighborhood and, thus, is not yet part of the automaton) or:
n < nmax pj is closer to pseed
(for repair) (for densest packing)
f(x) = a x2
pi-1
pi+1
pseed 15

16. Algorithm – Desired Relationships
A robot that receives an auction message for pj will announce a bid B(pj):
B(pj) = E d + X n
d ← distance from robot to pj (weighted by energy cost modifier E)
n ← number of existing neighbors (weighted by relation cost modifier X)
f(x) = a x2
pi-1
pi+1
pseed 16

17. Algorithm – Desired Relationships
After a period of time, ci announces the winning bidder based on the minimum bid B(pj)…
F and ri→j,des are communicated locally within the neighborhood.
Each neighbor cj repeats the process, but considers itself to be at its own unique formation-relative position pj.
f(x) = a x2
pi-1
pi+1
pseed 17

18. Algorithm – Desired Relationships
Propagate changes in neighborhoods in succession.
Calculated relationships generate a connected graph that produces the shape of the formation.
f(x) = a x2 18

19. Algorithm – Actual Relationships
Using sensor readings, robots calculate an actual relationship ri→j,act with each neighbor cj.
State of all cells in hi governs robot movement:
rotational error Θi and translational error Γi
relationships based on relative coordinate systems 19

20. Algorithm – Extended Definition20

21. Algorithm – Extended Definition21

22. Implementation 22

23. Evaluation – Generality
The generality of a system refers to its ability to conform to a variety of different formations.
Analysis from various trials and experiments has suggested a classification of the formations that can currently be produced…
Analysis from various trials and experiments suggests that the algorithm is extremely general and has warranted a classification of the formations that can currently be produced: 23

24. Evaluation – Generality
Non-formation (swarm)
Explicit formation
Straight line formation
Function-based formation
Branching formation
Lattice formation 24

25. Evaluation – Stability
A system’s ability to maintain formation (once established) dictates its stability.
To test the control algorithm against this principle, we manipulated one or many robots in a variety of different formations, changing both position and orientation…
To test the control algorithm against this principle, we manipulated one or many robots (either via remote control or by hand) in a variety of different formations, changing both its position and orientation. 25

26. Evaluation – Dynamic Switching
Dynamic switching capabilities refer to the ability of the system to respond to an operator’s commands for changes in formation organization.
To manipulate the formation, a human operator sends one of a variety of commands to any single robot…
propagates changes in the automaton
causes a chain reaction in neighbors
results in a global transformation
To manipulate the formation, a human operator need only send one of a variety of commands to any single robot. This propagates changes in the automaton, causing a chain reaction in neighbors, which then change states accordingly, resulting in a global transformation of the formation. Commands include formation translation, rotation, scaling, resizing, and change [6]. The ease with which an operator can manipulate formations with potentially large numbers of robots is a hallmark of this system. The current implementation, however, presents a limitation in formation transitions that require a change in the size of the neighborhood—recall that neighborhoods are currently specified manually (i.e., they are static) and, thus, cannot accommodate such scenarios. This limitation is clear when an operator commands a change from, say, a function-based formation (e.g., a parabola; two neighbors) to a lattice formation (e.g., a hexagon; six neighbors). Section V describes a method for dynamic neighborhoods to allow such a transition. 26

27. Evaluation – Robustness/Scalability
Evaluating robustness/scalability considers the ability of a system to respond to changes in group size.
Algorithm is independent of the number of robots.
Robots can be reassigned to new tasks or exhibit failure.
As numbers begin to dwindle or the task changes, other robots may join the ranks to increase the numbers.
Relying only on local information within an established neighborhood, it can be shown that the algorithm is independent of the number of robots; however, in the current implementation, the number of robots must be given at startup so that neighborhood of each robot can be initialized (i.e., all neighborhoods are specified manually). While this number can, indeed, be any value that corresponds to the true number of robots in the system, once set, this implementation is limited to only this number. This is not acceptable for most real-world applications. It is quite common for robots to be reassigned to different tasks or exhibit failure; likewise, as numbers begin to dwindle or the task changes, it is possible that other robot units join the ranks to increase these numbers. A robust formation control system must be tolerant of this. In Section V, we propose a method to address dynamic neighborhoods, thus, overcoming groups of static size. 27

28. Evaluation – Robustness/Scalability
Relying only on local information within an established neighborhood, it can be shown that the algorithm is independent of the number of robots; however, in the current implementation, the number of robots must be given at startup so that neighborhood of each robot can be initialized (i.e., all neighborhoods are specified manually). While this number can, indeed, be any value that corresponds to the true number of robots in the system, once set, this implementation is limited to only this number. This is not acceptable for most real-world applications. It is quite common for robots to be reassigned to different tasks or exhibit failure; likewise, as numbers begin to dwindle or the task changes, it is possible that other robot units join the ranks to increase these numbers. A robust formation control system must be tolerant of this. In Section V, we propose a method to address dynamic neighborhoods, thus, overcoming groups of static size. 28

29. Conclusions
Presented a distributed cellular automata-based formation control architecture capable of controlling large numbers of robots.
Discussed a distributed auctioning method to allow robot formation to reconfigure autonomously.
Examined the architecture with respect to necessary characteristics to handle real-world events. 29

30. Questions?
For more information, please visit or see the following papers:
Mead, R. & Weinberg, J.B. (2008). A Distributed Control Algorithm for Robots in Grid Formations. To appear in the Proceedings of the Robot Competition and Exhibition of The 23rd National Conference on Artificial Intelligence (AAAI-08), Chicago, Illinois.
Mead, R. & Weinberg, J.B. (2008). 2-Dimensional Cellular Automata Approach for Robot Grid Formations. To appear in Student Abstracts and Poster Program of The 23rd National Conference on Artificial Intelligence (AAAI-08). Chicago, Illinois.
Mead, R., Weinberg, J.B., & Croxell, J.R. (2007). A Demonstration of a Robot Formation Control Algorithm and Platform. To appear in the Proceedings of the Robot Competition and Exhibition of The 22nd National Conference on Artificial Intelligence (AAAI-07), Vancouver, British Columbia.
Mead, R., Weinberg, J.B., & Croxell, J.R. (2007). An Implementation of Robot Formations using Local Interactions. In the Proceedings of The 22nd National Conference on Artificial Intelligence (AAAI-07), Vancouver, British Columbia. 30