Multi-Robot SystemsCSCI 7000-006Wednesday, September 23, 2009NikolausCorrell

So far (Modeling)Deterministic models for deliberative systemsGradient-based controllers for reactive systemsGenerating controllers by performing gradient descent on a cost-functionFrom global to local optimization problems using Voronoi partitions

TodayMore gradient-based control: Shape formationFlockingIntroduction to hybrid systems

Gradient-based controlConvergence to minimal sets of a cost function over robot positionsMinimal sets can also be shapes or isocontoursMinimal sets can also be temporary and local

Gradient-based approach for shape formationGoal: distribute all robots along a 2D curveApplications: construction, perimeter surveillance“Minimum Set” given by an implicit function s(x,y)=0 on a 3D surfaceL. Chaimowicz, Michael, N., and V. Kumar, "Controlling Swarms of Robots Using Interpolated Implicit Functions" Proceedings of the 2005 IEEE International Conference on Robotics and Automation, pp. 2498-2503, Barcelona, Spain, April 2005.

Shape formation: ControllerLetf be a suitable convex function with the desired shape as isocontour with value 0Let qi=[xi,yi] be the robot positionLet vi=qi’ be the robot speed and ui=vi’ its accelerationLet Fc and Fr be forces repelling robots from each other

StabilityLyapunov candidate V(q,q’)>=0V(q,q’)<0Course Question: what did we not prove?

ProblemsWhat about the repulsive terms?What about too few robots?What about too many robots?Further readingM. A. Hsieh, V. Kumar and L. Chaimowicz. Decentralized Controllers for Shape Generation with Robotic Swarms. Robotica, Vol. 26, Issue 5, September 2008, pp 691-701.

Shape generationf could be a sum of Radial Basis Functions given a set of constraint pointsConstraintRBF i is centered around piFind set of weights wi so that all constraints are satisfied

From theory to practiceSimulationRobots get stuck in local minimaUnreachable shapes (inside of letter P, e.g.) depending on initial positionReal robotsNo local range and bearingConstraints non-holonomic

Example: Herding/FlockingAgents are attracted to their neighborsAgents are repelled by their neighborsAgents move voluntarily (random or informed)

ModelKinematic model:Artificial Potential fieldRandom noiseAgent-to-agent forceM. Schwager, C. Detweiler, I. Vasilescu, D. M. Anderson, D. Rus - Data-Driven Identification of Group Dynamics for Motion Prediction and Control, Journal of Field Robotics 25(6-7):305-324, 2008.

What can you do with this model?Numerical simulationInitialize positionsCalculate agent-to-agent interaction forces between all agentsUpdate positionsGradient controller?Yes! Only speed is updatedCan we formulate this as acost function?

Generalized Coverage ControlCost to service point qin Q:New: Team-based costMixing function: encodes collaborationNew cost functionQM. Schwager, A Gradient Optimization Approach to Adaptive Multi-Robot Control, Ph.D. Thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering, September, 2009.

Properties of the Mixing functionTells how information from different robots should be combined to sense at qCourse question: What happens for

Mixing functionForResults in standard Voronoi cost function (Monday)

Mixing functionLetCost functionResult:Q

From generalized coverage to flockingCost function Let agent-to-agent force beTake gradient:

Hybrid SystemsSo far: all robots behave according to the same dynamical systemHybrid systems: robot dynamics are a function of discrete statesLogicX’=f1(X)X’=f2(X)Logic

Example: Cow HerdingContinuous part*Artificial potential field:Far-field attractionNear-field repulsionGaussian noise added to force estimatesDiscrete partCows can be in two states: Grazing and Stressed.Different potential fields for each state*M. Schwager, C. Detweiler, I. Vasilescu, D. Anderson, and D. Rus, “Data-driven identification of group dynamics for motion prediction and control,” Journal of Field Robotics, 2008.

Behavioral HypothesisWe theoretically study the influence of two potential social effects:Animals tend to aggregate more when under stress due to a stimulusStress propagates within the herd [Butler, 2006]RRThese hypotheses are implemented in a hybrid dynamical model and tested in simulation.

System DescriptionCows and EnvironmentHereford and Hereford x BrangusUSDA experimental range, 466ha paddockSensorsGPSAccelerometerCommunication900Mhz radioActuatorsStereo headphonesElectrical stimulation

Formal descriptionState-space of agent iR4State transition probabilitiesControl input (stimulus)Stress propagationArtificial Potential fieldRandom noiseAgent-to-agent force

Simulation EnvironmentDynamical simulationExperimentInitial condition: N cows grazing inside a circular fence of 25m diameter (random distribution)Fence moves northwards with constant 20m/h (open loop) After 5h simulated time the experiment is stoppedInvestigate different values for a and RSpeed-up of about x15 between real experiment and dynamical simulation

Sample Result: Impact of Increased Gregarious Behavior during Stress50 simulations per data pointR= 0 mR= 5 mR= 10 mFor constant stimulus, a(x=S)>a(x=G) necessary condition for aggregation to work

Sample Result: Impact of Stress PropagationSuccess: >50% of population within fenceR= 0 mR= 5 mR= 10 mModerate stress propagation increases control performance, but potentially leads to instable systems

Hybrid SystemsAnalysis of individual dynamics, but unclear what state the other robots are inAnalysis of discrete dynamics, e.g. Markov chainVerification using numerical toolsOverviewGoebel, Rafal; Sanfelice, Ricardo G.; Teel, Andrew R. (2009), "Hybrid dynamical systems", IEEE Control Systems Magazine29 (2): 28–93

SummaryGradient descent approaches are a versatile tool forShape formationFlockingCoverageCommunity is moving unified theory for controller analysis and synthesisAnalysis of discrete-continuous systems still in its infancy

Next WeekDiscussion of course projects“develop”, “study”, “explore” are all words that should NOT be in your research objectiveformulate a hypothesis that leads to your methodProbabilistic Models for reactive and deliberative systemsAssignment of teams