September 23, Modeling of Gradient-Based Controllers II

Multi-Robot Systems

CSCI 7000-006Wednesday, September 23, 2009

Nikolaus Correll

So far (Modeling)

• Deterministic models for deliberative systems• Gradient-based controllers for reactive

systems– Generating controllers by performing gradient

descent on a cost-function– From global to local optimization problems using

Voronoi partitions

Today

• More gradient-based control: – Shape formation– Flocking

• Introduction to hybrid systems

Gradient-based control

• Convergence to minimal sets of a cost function over robot positions

• Minimal sets can also be shapes or isocontours

• Minimal sets can also be temporary and local

Gradient-based approach for shape formation

• Goal: distribute all robots along a 2D curve

• Applications: 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: Controller

• Let f be a suitable convex function with the desired shape as isocontour with value 0

• Let qi=[xi,yi] be the robot position

• Let vi=qi’ be the robot speed and ui=vi’ its acceleration

• Let Fc and Fr be forces repelling robots from each other

Stability

• Lyapunov candidate V(q,q’)>=0

• V(q,q’)<0• Course Question: what did we not prove?

Problems

• What about the repulsive terms?

• What about too few robots?• What about too many

robots?• Further reading

M. 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 generation

• f could be a sum of Radial Basis Functions given a set of constraint points

• Constraint• RBF i is centered around pi

• Find set of weights wi so that all constraints are satisfied

From theory to practice• Simulation

– Robots get stuck in local minima– Unreachable shapes (inside of letter P, e.g.) depending on initial

position• Real robots

– No local range and bearing– Constraints non-holonomic

Example: Herding/Flocking

• Agents are attracted to their neighbors

• Agents are repelled by their neighbors

• Agents move voluntarily (random or informed)

Model

• Kinematic model:

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x i = (x,y, ˙ x , ˙ y )Artificial Potential field

Agent-to-agent forceRandom noise

M. 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 simulation– Initialize positions– Calculate agent-to-agent interaction forces

between all agents– Update positions

• Gradient controller?– Yes! Only speed is updated– Can we formulate this as a

cost function?

Generalized Coverage Control

• Cost to service point q in Q:• New: Team-based cost

• Mixing function: encodes collaboration

• New cost function

Q

M. 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 function

• Tells how information from different robots should be combined to sense at q

• Course question: What happens for

€

α → −∞

Mixing function

• For

• Results in standard Voronoi cost function (Monday)

€

α → −∞

€

minf i

Mixing function

• Let

• Cost function

• Result:

Q

From generalized coverage to flocking

• Cost function

• Let agent-to-agent force be

• Take gradient:

Hybrid Systems

• So far: all robots behave according to the same dynamical system

• Hybrid systems: robot dynamics are a function of discrete states

X’=f1(X) X’=f2(X)

Logic

Logic

Example: Cow Herding

• Continuous part*– Artificial potential field:

• Far-field attraction• Near-field repulsion

– Gaussian noise added to force estimates

• Discrete part– Cows 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 Hypothesis

We theoretically study the influence of two potential social effects:

1. Animals tend to aggregate more when under stress due to a stimulus

2. Stress propagates within the herd [Butler, 2006]

R

R

These hypotheses are implemented in a hybrid dynamical model and tested in simulation.

System Description• Cows and Environment

– Hereford and Hereford x Brangus

– USDA experimental range, 466ha paddock

• Sensors– GPS– Accelerometer

• Communication– 900Mhz radio

• Actuators– Stereo headphones– Electrical stimulation

Formal descriptionState-space of agent i

State transition probabilities

Stress propagationControl input (stimulus)

Artificial Potential field

Agent-to-agent forceRandom noise

R4

Simulation Environment

• Dynamical simulation• Experiment

– Initial 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 stopped

• Investigate different values for a and R

• Speed-up of about x15 between real experiment and dynamical simulation

Sample Result: Impact of Increased Gregarious Behavior during Stress

For constant stimulus, a(x=S)>a(x=G) necessary condition for aggregation to work

R= 0 mR= 5 m

R= 10 m50 simulations per data point

Sample Result: Impact of Stress Propagation

R= 0 mR= 5 m

R= 10 m

Success: >50% of population within fence

Moderate stress propagation increases control performance, but potentially leads to instable systems

Hybrid Systems

• Analysis of individual dynamics, but unclear what state the other robots are in

• Analysis of discrete dynamics, e.g. Markov chain

• Verification using numerical tools

• OverviewGoebel, Rafal; Sanfelice, Ricardo G.; Teel, Andrew R. (2009), "Hybrid dynamical systems", IEEE Control Systems Magazine 29 (2): 28–93

Summary

• Gradient descent approaches are a versatile tool for– Shape formation– Flocking– Coverage

• Community is moving unified theory for controller analysis and synthesis

• Analysis of discrete-continuous systems still in its infancy

Next Week

• Discussion of course projects– “develop”, “study”, “explore” are all words that

should NOT be in your research objective– formulate a hypothesis that leads to your method

• Probabilistic Models for reactive and deliberative systems

• Assignment of teams