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Distributed, Physics-based Control of Swarms of Vehicles
W. M. Spears, D.F. Spears,
J. C. Hamann and R. Heil (2000)
Presentation by Herke van Hoof
Contents
Introduction Approaches to distributed control systems Hexagonal lattices Square lattices Properties and behaviour Implementation on real robots Conclusions Future work Discussion
Introduction
Design robust network of autonomous vehicles
Monitoring a physical region Primitive agents Simple effectors Simple actuators Simple, local rules necessary
Introduction
Complex behavior in physical systems Artificial Physics (AP) Self-organisation Fault-tolerance Self-repair Scalability
Approaches to distributed control systems Swarm intelligence
Inspiration from biology, for instance ant foraging
Behaviour-based Behaviours are composed of primitive sub-behaviors
Physics-based Agents or obstacles exert virtual ‘forces’ on one
another Potential Fields Flocking Artificial Physics
Potential fields
Typically one robot Environment exerts forces
Robot can be ‘trapped’ Has difficulties with narrow
corridors Obstacles can induce unstable
motion (Koren, 1991) Calculating potential field
can be computationally expensive.
Flocking
Models life-like motion in swarms Complex behaviour is created
from simple local rules: avoid, align, center
Aligning demands much from sensors, computationally expensive
Not really a physics-based method.
What parameters to use?
Artificial Physics
Agents modelled as particles Particles have a position and a mass At each discrete time step, position and velocity
change:
F is the sum of forces on the particle, including friction, bounded by Fmax
v is bounded by vmax
x v t vF tm
Artificial physics
Artificial physics don’t describe low-level behavior
This makes AP platform independent Specification for sensing and acting may
be different for different platforms Friction, discrete time steps, Fmax, vmax,
may model real-world constraints
Artificial physics vs other methods
Potential fields Forces are very local. This makes computation faster. Multiple agents exert a force on each other
Boids Mathematical analysis enables finding of ‘useful’
parameter settings No ‘aligning’ – aims at preserving formation
Behavior-based Compare to behavior based with ‘cluster’ and ‘avoid’
behavior.
Designing Hexagonal Lattice
Hexagon seems complicated shape
But each neighbour is R from centre
F = Gmimj /rp
Repulsive if r < R Attractive if r > R Local rule: r < 1.5 R F < Fmax
Evaluating Lattice Quality
Local hexagons, some global flaws
Clusters of robots: Robustness
Measuring quality: Connect particle to other
particles, lines should cross at multiple of 60o
Average error: 5.6o
75o
Observing a phase transition
Cluster size is expected to decrease linearly with G
Instead, cluster size is relatively constant, untill a threshold value of G is reached
Similar to phase transition G = 1200 G = 600
Why is there a phase transition?
At G = 1200 (left), particles are attracted to middle
At G = 600 (right) particles are pushed away from middle
There are 6 possible ‘escape paths’
When is there a phase transition?
Middle particles repel each other (Fmax ) Assume a central particle moves left: Four particles exert F = G / Rp
This force projected on the x-axis: √3/2 Fcohesion = 2 √3 G / Rp
Fcohesion = Ffragment when G = G t
Predicts very well: Within 6% while G ranges from 87 to 291 000
G t is independent of n Knowing G t helps design systems
G t
F max Rp
2 3
Fmax
Fpush
Fpull
Fcohesion
Conservation of energy
PE converts into HE as particles find their position
High PE means more work is done by the system
Calculating Potential Energy
PE provides momentum: overcome local minima
Initial position: N particles at same location
Requires PE = N * (N-1) * V V = work needed to get a
particle to same position as another
Work = Force * distance
R’
F repulsive, V increases
F attractive, V decreases
R 1.5R
Useful values for parameter G
Unclustered formation: G < Gt
Nicest formations: G = GV (max V, derivation of V = 0)
Smallest formation (maximally clustered):G = Gmax
Robustness
Lattice structure is robust Removal of particles does
not change location of potential wells
Self-repairable Robust to ‘gusts of wind’
as well
N = 99
N = 49
Designing square lattice (1)
Creating a square lattice seems difficult
Half of a particle’s neighbours are at distance R, half at √2 * R
Simple trick: Introduce two ‘kinds’ of particles (different ‘spin’)
Designing Square Lattices (2)
For every neighbouring particle, sense its spin and distance r
Normalise r to r / √2 if particles have like spin
Then calculate force: F = Gmimj /rp
Evaluating Square Lattices
Again, join a particle with line segments to two of its neighbours
Angle should be multiple of 90o
Average error = 12.7 Suboptimal: global
flaw exists
Repairing flaws
To repair flaws, we have to get out of local optimum
Introduce some noise Particles may change
spin Still some flaws, but
error from 12.8 to 4.6o
Phase transition and energy
The same phase transition as with hexagons can be observed
Values for G can be calculated in analogous fashion
This time, Gv does depend on N, but weakly (Gv is 1466 for 200 particles and 1456 for 20 particles)
Properties and behaviour of lattices
Perfect lattices and transformation Other formations in 2d and 3d Dynamic behaviors
Perfect Lattices and Transformations Lattices can transform between squares
and hexagons by ignoring or taking into account spin
By adding an ordering attribute (m,n) ‘perfect’ lattices can be created (which can also be transformed)
In that case F is attractive instead of repulsive when ordering is wrong
Transforming
Transforming perfect lattices
Other formations in 2d and 3d
Air vehicles need 3d formations Layers of hexagons, pyramids, cubes, ... Find formations by playing with parameters Some formations best build per particle, or by
transformation
Dynamic Behaviors
Task 1: Approaching a goal Obstacles need to be avoided Goals are attractive, obstacles repulsive Obstacles only sensed locally Unclustered formations (low G) behave
like a fluid and perform better
Dynamic Behaviors
Task 2: Surveillance or Perimeter defense
Particles repel each other and are repelled by boundary to fill a space
Particles attracted by inner and outer boundary to fill a perimeter
Robust to removal of particles as excess particles can take over
Implementation of AP on robots Simple and cheap robots ‘can turn on a dime’ IR sensors Scan environment First derivative filter Width filter List of robot heading, distance
Implementation of AP on robots
Cycles of sensing, computation, motion Seven robots create a hexagon Robots find correct position in 7 cycles Move toward light source: 1 foot in 13
cycles Very slow: 22 seconds per cycle New localization technology will be faster
Implementation of AP on robots
Pictures taken at: Start 2 minutes 15 minutes 30 minutes
Conclusions
AP satisfies requirements for distributed control system (fault-tolerance, self-repair, self-organization)
AP enables designer to predict useful parameter values
AP is more efficient than ‘potential fields’ AP ‘middle level’ of control architecture
Future work
Genetic algorithms to design force laws Analyzing all aspects of AP Using trilateration for localization Which force laws guarantee optimality? Transition to more robots, like air vehicles
Better sensing and interaction with environment Velocity matching? More computing power
Discussion
Sensing and acting is very slow and inflexible
Evaluation of lattices seem to be based on one trial
What makes a hexagon or square pattern better than for instance the ‘surveillance’ pattern?