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?