computational swarm intelligence: a conceptual journey
TRANSCRIPT
Computational Swarm Intelligence: A Conceptual Journey from Insects to
Robots
Swagatam Das1 and Tribeni Prasad Banerjee2
1Department of Electronics and Telecommunication Engineering,Jadavpur University, Kolkata 700032, India.
2Central Mechanical Engineering Research Institute,Durgapur - 713209, West Bengal, India.
Things to be Addressed
Introduction to Particle Swarm Optimization (PSO).Applications of PSO to mobile robot path planning, navigation and cooperative task performance.Introduction to Bacterial Foraging Optimization (BFO) Dynamics of computational chemotaxis in BFO anda scheme for adapting the chemotactic step-size.Application of BFOA to swarm robotics.
What is Optimization?
Optimization can be defined as the art of obtaining best policies to satisfy certain objectives, at the same time satisfying fixed requirements.- Gotfried
Unconstrained OptimizationExample: Maximize Z,
where Z= x12
x2
–x22x1 -2 x1
x2
Numerical Approach to Optimization
•
Steepest Descent/Gradient Descent Algorithm•
Problem: Minimize f(x1
, x2
, …, xn
)•
Approach: x1
:= x1
-
δf/ δx1
•
x2
:= x2 -
δf/ δx2
•
……
……. …….. ……
•
xn
:= xn
-
δf/ δxn
•
Loop through until δf/ δxi
, for all i, are zero.
•
Recursive difference equation of discrete gradient descent:
)()(.)()( nXXXfnXnX
1
Direction Of negative gradient
How a single agent can find global optima by following gradient descent?
But What about these multi-modal, noisy and even discontinuous functions?
Gradient based methods get trapped in a local minima or the Function itself may be non differentiable.
Way Out: MultiWay Out: Multi--Agent Optimization in Agent Optimization in Continuous SpaceContinuous Space
RandomlyInitialized Agents
Agents
Most Agents are nearGlobal Optima
After Convergence
Swarm intelligence
•
Collective system capable of accomplishing difficult tasks in dynamic and varied environments without any external guidance or control and with no central coordination.
•
Achieving a collective performance which could not normally be achieved by an individual acting alone.
•
Birds, wasps, bees and even bacteria are amongst the many organisms cooperatively form very complex patterns and display sophisticated kinds of cooperative behavior as a survival strategy.
•
Constituting a natural model particularly suited to distributed problem solving
Principles of Particle Swarm Optimization
Principles of Particle Swarm Optimization
Current direction
Direction of local maximum
global maximum
Resulting direction of motion
PSO: Starting Situation
Randomly Scattered Particles over the fitness landscape andtheir randomly oriented velocities
All Partic
les in
A close vicinity of th
e
Global optim
um The best Particle Conquering the Peak
Situation after a few iterations
Best Position found so far By the particle
Globally Best positionFound by the swarm
Definitions
Neighbourhoods
geograph ical social
Best Position found By the agent so far (Plb
)
Globally best or neighborhood best position found so far (Pgb
).
Current Position
Vi
(t)
Resultant VelocityVi
(t+1)
Particle Swarm OptimizationKennedy & Eberhart(1995)
Vi
(t+1)=φ.Vi
(t)+C1
.rand(0,1).(Plb
-Xi
(t))+C2
.rand(0,1).(Pgb
-Xi
(t)) Xi
(t+1)=Xi
(t)+Vi
(t+1)
C1.ran
d(0,1
).(Pl
b-Xi
(t))
C2.rand(0,1).(Pgb-Xi(t))
Psychosocial compromise
Here
I am!
The
best perf. of
my
neighbours
My
best perf.
x pg
pi
v
i-proximity
g-proximity
PSO algorithm
Initialize particles with random position and zero velocity
Evaluate fitness value
Compare & update fitness value with pbest
and gbest
Meet stopping criterion?
Update velocity and position
Start
End
YES
NO
pbest = the best solution (fitness) a particle has achieved so far.
gbest = the global best solution of all particles.
To know moreFirst Mathematical Analysis of PSO:
Maurice Clerc, James Kennedy: The particle swarm - explosion, stability, and convergence in a multidimensional complex space. IEEE Trans. Evolutionary Computation 6(1): 58-73 (2002): Outstanding Paper Award in IEEE TEC.
1. J. Kennedy, R. Eberhart, and Y. Shi, Swarm Intelligence, Morgan Kaufmann (2001).
2. Maurice Clerc, Particle Swarm optimization, ISTE Publishing Company (February 24, 2006).
3. Swagatam Das and Amit Konar: Particle Swarm Optimization and Differential Evolution: Algorithms, Analysis and Applications, Springer verlag, Germany, (in Press, to appear in early 2009).
Books on PSO
Applications of PSO•
60 papers in different archival quality IEEE Transactions in last 5 years.
•
Two Special issues in IEEE Transactions on Evolutionary Computing.
•
Major Areas of Applications: Training Artificial Neural Networks Adaptive and Optimal Control Electrical Power Systems Digital Signal Processing Robotics Wireless Networking and Mobile Computing Bioinformatics
What is a robot?•
Hollywood’s imagination
R2-D2
Star Wars
3PO
What is a robot?•
By general agreement, a robot is:A programmable machine that imitates the actions or appearance of an intelligent creature–usually a human.
•
To qualify as a robot, a machine must be able to:1) Sensing and perception: get information from its surroundings2) Carry out different tasks: Locomotion or manipulation, do
something physical–such as move or manipulate objects3) Re-programmable: can do different things4) Function autonomously and/or interact with human beings
Types of Robots
Humanoid
Legged robots
Underwater robots
Wheeled mobile robotsAerial Robots
• Locomotion
Mobile Robot ExamplesHilare II
http://www.laas.fr/~matthieu/robots/
Sojourner Rover
NASA and JPL, Mars exploration
Path Planning of Mobile Robots
•
Problem Statement: Compute a collision-free path for a rigid or articulated object
(the robot) among static obstacles
•
Inputs:–
Geometry of robot and obstacles
–
Kinematics of robot (degrees of freedom)–
Initial and goal robot configurations (placements)
•
Outputs:–
Continuous sequence of collision-free robot configurations connecting the initial and goal configurations
A Swarm Intelligence Approach to Centralized Path Planning
Goal
Box
Robot Location
O
An
A1
A2
A3
PSO based Centralized Path Planning
xi
yi (xi , yi)
(xi , yi)
xi
yi
i
Vi
y
x
(xig, yig)
(xi, yi)
(xi, yi)
y
x
Obstacle
iivixix cos
iiviyiy sin
When
t =1, the the
next location of the robot
Let f
be an objective function that determines the length of the trajectory. For n
robots, the central server machine should minimize:
n
iigiigiiiii yyxxyyxxf
1
2222 }))()(())()(({
Adding constraints for collision avoidance
n
iigiiiigiiii yvyxvxvf
1
22 }})sin()cos{({
n
iigiiiigiiii yvyxvxvf
1
22 }))sin()cos(({
obsi
n
i
n
jijji drd
/))))((,(min(.
,
1 1
220
Distributed Path PlanningLet fi
be the constrained objective function for the i-th robot:
2)(2)(2)(2)( igyiyigxixiyiyixixif
n
ijjobsidrjid
,/))}(''(,{min(.
1220
Employ n PSOs
running in parallel, where the i-th
PSOattempts to minimize the function fi
in each iteration
if
Considerable speed-up and makes sense in real time.
Simulation Result for Path Planning of single robot through static obstacle with PSO
Comparison with Other Neural Algorithms
Back- Propagation
SOFM PSO-based Single robot path planning
Performance Evaluation Heuristics1. Traversal time (T): the time taken by the robot to move from a given starting position
to a fixed goal position in its workspace. 2. Traversed Distance (D): Let (xi
, yi
) & (xi-1
, yi-1
) be the current and the previous position of the robot in the map. Thus, the total distance of traversal for successive n positions, is computed by
nD =
(((xi - xi-1 )2 + (yi - yi-1 )2 )0.5) i=1
3. RMS Path Deviation (PD): The computation of the parameter P
for a given path P, thus can be formally defined as follows-
P
= [ (dij
)2
/N ]0.5
iShortest Path
Actual Path
( xi
, yi
)
PI = 1 T/ + 2 D/ + 3 p/
such that i = 1
and 0 < i < 1 for all i.
Neural Net Used
Travers
al Time(sec)
Travers
al Distanc
e(inch)
Mean Path Deviation(inch)
Performance Index
G
S
PSOBPRBF
167.4183.3177.8
119.2117.7116.4
27.332.030.6
0.2860.3070.363
G
S
PSOBPRBF
348.51396.11067.3
172.7194.5186.2
54.165.954.7
0.2580.4790.466
G
S
PSOBPRBF
248.4383.2344.8
185.2210.5205.9
60.374.773.3
0.2420.3960.308
G
S
PSOBPRBF
239.4244.2422.9
150.5159.8157.8
6.67.26.5
0.2950.3080.397
S
G
PSOBPRBF
210.5221.2218.7
146.2147.5146.9
10.411.511.9
0.2650.2700.463
G
S
PSOBPRBF
305.4461.8459.5
173.6195.9192.9
10.811.314.8
0.1840.3950.393
G
S
PSO BPRBF
259.6455.6434.5
143.6140.7137.2
10.811.011.2
0.2720.3450.383
PerformanceComparison
Simulation Results for Distributed Path Planning
Final configuration of the world-map When number of robot=8 and number
of obstacle=5.Final configuration of the world-map
When number of robot=4 and number of obstacle=8.
Cooperative Task-Driven Robot Behavior Using Particle Swarm Optimization
•
The problem of group behavior learning is basically one of learning how to collectively perform a given task.
•
Solving this problem requires group robots to continuously gain experience from their local interaction with an environment. A group behavior will be generated from group experience following a series of environmental state transitions.
•
Given a group of n
individual robots capable of sensing and changing their own local positions and orientations with respect to a box and a desired goal location in their workspace, the objective is to develop a PSO-based mechanism such that the robot group can gradually acquire a coordinated movement based on a series of environmental state.
Target
Box
β
Each robot i exerts a vector force Fi
making some angle βi with the straight line connecting the CG of the box and the goal right at their point of contacIf the line of action of the box does not pass through the CG of the box, thethere will be a torque acting on the box due to each of the robots given by,
iii lFJ .
Particle Representation:
Fi
βi
Ji
Code for robot i
Robot 1’s code Robot 2’s code Robot 3’s code θ
10-dimensional Position vector of a particlefor 3 cooperative robots
ii
iFs cos3
1.1
13
12
i
iJs
4321 )..( maximize sssS f
cos13 s
3 components of fitness function
The fitness function
In the above definition of the fitness function S1 implies that the robots must haveto maximize the projection of the pushing forces along the shortest possible path between the box and the goal. S2 implies that the rotation of the box should be discouraged during collective pushing and S3 measures how much the block moves correctly along the desired goal direction.
Three snapshots of collective box pushing by 3 mobile robots
At t = 0 At t = 28 At t = 55
S. Das and A. Konar, Multi-robot path-planning and cooperative Box Pushing with Particle Swarm Optimization, to appear in IEEE Transactionson Automatic Control.
The trajectory of a cubic box collectively pushed by
three group robots.
Goal
A Video Clip on Experimental Run
Cooperation of 2 Hex-crawler in carrying a log
Some More Experimental Results on PSO-Assisted Path Planning
1) Navigation By Boebot
2) Path finding by Boebot
Cooperation Between 2 Robots with PSO-assisted Multi-agent Q-learning
WEBOTS•
WEBOTS
S. Das and A. Konar, PSO-assisted multi-agent Q-learning for cooperative load transport and grabage cleaning by mobile robots, Engineering Applications of AI, Elsevier Science, 2009.
Bacterial Foraging Bacterial Foraging Optimization (BFOA)Optimization (BFOA)
(K. M. Passino, 2002)
K. M. Passino, Biomimicry of Bacterial Foraging for Distributed Optimization andControl, IEEE Control Systems Magazine, 52-67, (2002).
Foraging TheoryAnimals search for and obtain nutrients to maximize:
where E is the energy obtained per T time.
Foraging constraints: Physiology, predators/prey,environment
Evolution optimizes foraging
TE
Search/foraging strategies, use collective intelligence to find maximum amount of nutrients
avoiding noxious substances and predators
Foraging behavior of E. ColiE. coli: Diameter: 1µm, Length: 2µm
Sensors/actuators/controller, an autonomous underwater vehicle – “nanotechnologist’s dream” – from E. Coli
to nanobots.
Swimming, tumbling, and chemotactic
behavior of E. coli.
A bacterial swarm foraging on a fitness landscape
Bacterial Foraging Optimization: Imitating the behavior of a group of E. Coli To search for maxima/minima on a function surface.
•Chemotaxis
•Reproduction
•Elimination Dispersal
•Swarming
Major steps of BFOA
Chemotaxis
This process simulates the movement of an E.coli cell through swimming and tumbling via flagella.
)()()()(),,(),,1(
iiiiClkjlkj
T
ii
i(j,k,l) represents i-th
bacterium at j-th
chemotactic, k-th
reproductive and l-th
elimination dispersal step. C(i) is the size of the step taken in the random direction specified by the tumble (run length unit). Where
indicates a unit length vector in the random direction.
Reproduction
The least healthy bacteria eventually die while each of the healthier bacteria (those yielding
higher value of fitness function) asexually split into two bacteria which are placed in the same location. This keeps the swarm size constant.
Elimination-Dispersal
In BFOA, to simulate Gradual or sudden changes in the local environment, where the bacterial population lives,
some bacteria are liquidated at random with a very small probability while the new replacements are
randomly initialized over the search space.
Swarming
In a bacterial swarm there is always a tendency of the bacteria to group with some cell to cell signaling in between.
p
m
imm
s
irepellantrepellant
p
m
imm
s
itattractattrac
icc
wh
wdlkjJ
1
2
1
1
2
1tantan
)])(exp([
)])(exp([)),,((
Jcc is the swarming function.
Bacteria trajectories, Generation=1
1
2
0 10 20 300
10
20
30Bacteria trajectories, Generation=2
1
2
0 10 20 300
10
20
30
Bacteria trajectories, Generation=3
1
2
0 10 20 300
10
20
30Bacteria trajectories, Generation=4
1
2
0 10 20 300
10
20
30
BFOA working on two-dimensional sphere function
Modeling the Dynamics of BFOA
•
Velocity of a bacterium is moving on a single-dimensional fitness landscape
where G
is the gradient at position of bacterium28
2
CGkCVb
Swagatam Das, Sambarta Dasgupta, Arijit Biswas, Ajith Abraham, andAmit Konar, On Stability of Chemotactic Dynamics in Bacterial Foraging Optimization Algorithm, IEEE Transactions on SMC, Part –
A, (Accepted, 2008).
)(txxx
fdtdxStriking Similarity
with Classical Gradient Descent
Oscillation problem•
When bacterium is very close to optima
gradient becomes very small•
Then momentum term becomes
predominant•
velocity does not vanish at optima,
oscillation occurs
Sambarta Dasgupta, Swagatam Das, Ajith Abraham, and Arijit Biswas, Adaptive Computational Chemotaxis in Bacterial Foraging Optimization: An Analysis, IEEE Transactions on Evolutionary Computing
(in Press) 2008.
Adaptive Chemotaxis to avoid oscillation
• We adapt step size according to the
relation
•
Step size vanishes when bacterium goes close to optima.
•
No oscillation
)(11
|)(||)(|
JJ
JC
Applications of BFOA
Distributed intelligent control: swarm robotics
Optimization of real power loss and voltage stability and distribution static compensator
Design of active power filters
Pattern recognition: Shape Extraction from images
Learning and Neural network problems
* 5 application papers in different IEEE Transactions in last 2 years.
Vehicular swarms formation with BFOA... formation/pattern/group (satellites, aircraft, ground/undersea vehicles).
Simulation of Robo-swarm with BFOA
Simulation of Robo-swarm with Adaptive BFOA
Thank You !!!!!!