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 F max v is bounded by v max

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, F max, v max, 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 = Gm i m j /r p Repulsive if r < R Attractive if r > R Local rule: r < 1.5 R F < F max

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 60 o Average error: 5.6 o 75 o

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 = 1200G = 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 (F max ) Assume a central particle moves left: Four particles exert F = G / R p This force projected on the x-axis: √3/2 F cohesion = 2 √3 G / R p F cohesion = F fragment when G = G t Predicts very well: Within 6% while G ranges from 87 to G t is independent of n Knowing G t helps design systems F max F push F pull F cohesion

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 R1.5R

Useful values for parameter G Unclustered formation: G < G t Nicest formations: G = G V (max V, derivation of V = 0) Smallest formation (maximally clustered): G = G max

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 = Gm i m j /r p

Evaluating Square Lattices Again, join a particle with line segments to two of its neighbours Angle should be multiple of 90 o 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 4.6 o

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, G v does depend on N, but weakly (G v 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?