Multi-Robot Systems

CSCI 7000-006Friday, September 11, 2009

Nikolaus Correll

So far

• Reactive vs. Deliberative Algorithms• Both approaches are probabilistic for noisy

sensors and actuators• Robustness/Deterministic behavior can be

increased by– Combining different sensors– Information exchange– Actively validating hypothesis– Redundancy

Today

• Exact and approximative algorithms• Centralized vs. Distributed Systems• Market-based algorithms

Exact Algorithms

• Find always the best solution• Search the entire solution space

– Determine what “best” means (fitness function)– Enumerate all solutions– Pick best solution

• Some problems: dynamic programming• Finding the best solution can be very time-

consuming/impossible for NP-hard problems

Example: Traveling Salesman Problem

• Traveling Salesman Problem• Find the shortest route connecting n cities• Never visit any city twice• Computational representation: sequence• Brute force algorithm: calculate length of

all possible permutations• 60 cities -> 4.2 * 10^81 permutations • NP hard, exact better than brute-force

solutions exist (e.g. dynamic programming)

Course Question

• Come up with a reactive algorithm for solving the TSP. Hint: ants.

Reactive Algorithm for the TSP

• Use a population of ant-like agents starting at random cities

• Each ant randomly select a city that it has not yet visited on this tour (repeat until all cities are visited)

• Each ant calculates the length of this path and deploys an inverse amount of “pheromones” on the path

• In following iterations, ants are programmed to select paths from city i to city j with a higher likelihood

• Algorithm converges to a local optimum

Lessons from this example

• Exact problems can be very hard to solve• Also “pure” CS offers a wide range of

algorithmic solutions• The design problem trades off provable

optimality with speed• In robotics algorithmic choice is constrained

by sensors, actuators, computation and communication

Coverage example (Wednesday)

• Exact algorithm for single robot

• Approximative algorithm for multiple robots

• Robots might find the optimal solution

• Worst case: every robot covers everything

Course Question

• Come up with an exact algorithm for covering M cells with N robots as fast as possible.

• Hints:– The problem reduces to allocate a subset of cells

to each robot to minimize the maximum number of cells allocated to one robot.

– Identify sub-problems / algorithms

Possible Solution

• Enumerate all possible sets of allocations

• Calculate the cost of each allocation

• Cost: TSP path over all cells

• NP-Hard

Stirling numbers of the 2nd kindfor 3 and 4 cells and up to 4 robots.© Mathworld

Centralized vs. Distributed Algorithms

• Finding the best solution requires knowing all parameters of the system

• Usually requires “leader” or centralized agent• Course Question: What problems do you

expect in a centralized system?

Centralized Systems

• Information needs to be sent to a central unit• Commands need to be sent to each robot• Problems

– Information get lost both ways– Process needs to be repeated when individuals fail– Individual failure needs to be detected– …

How to distribute an algorithm?

• Smart way: using the optimal substructure of the problem (dynamic programming)

• Not all problems can be efficiently distributed• Robust: Every robot solves the whole problem

for the entire team– Problem: ambiguous solutions– Resolution: conflict resolution rules, e.g. lower id

goes first– Example: Market-based task allocation

Market-based task allocation

• Tasks are offered by auctioneer• Every robot bids with the cost that it would

need to do the task• Robot with the lowest cost gets the job• Simplest auction: greedy, non-optimal

ordering• Variations: bidding on all possible

permutations

Example: Box Pushing

• Two tasks: watch the box, push the box

• Three robots, only one can watch the box

• Watch the box requires LMS

• Watcher auctions off “push left” and “push right” tasks

"Sold!: Auction methods for multi-robot coordination".Brian P. Gerkey and Maja J Mataric´. IEEE Transactions on Robotics and Automation, Special Issue on Multi-robot Systems, 18(5):758-768, October 2002.

Example: Coverage

• Robots calculate cost for covering a blade by solving the TSP

• Sequential bidding approximates near optimal

• Deterministic bid evaluation allows for decentralized auction-closing

• Re-Allocation upon errorP. Amstutz, N. Correll, and A. Martinoli. Distributed Boundary Coverage with a Team of Networked Miniature Robots using a Robust Market-Based Algorithm. Annals of Mathematics and Artifcial Intelligence. Special Issue on Coverage, Exploration, and Search, Gal Kaminka and Amir Shapiro, editors, 52(2-4):307-333, 2009.

Re-Auctioning example

Robot 1 “slips”

Round # Robot 1 Robot 2

Bids during auction

1 1 on 5 0 on 7

2 1 on 5 1 on 8

3 2 on 8 1 on 8

Nikolaus Correll 199/20/2007

Results

DFS/A* No collaboration

DFS/A* Information exchange

Market-based coordination

Summary

• The better you plan, the better the performance• Noise requires you to re-plan all the time• Feasible algorithms determined by robot

capabilities: sensors, actuators, computation and communication

• Algorithmic complexity exponential for NP hard problems

• Potentially very high cost for marginal improvements!

Outlook

• Control-based approaches (in two weeks)• Modeling: examining resource trade-offs on

paper (in three weeks)• Next week: building week