Multi-Player Pursuit Evasion Games, Learning, and Sensor Webs

1. Multi-Player Pursuit Evasion Games, Learning, and Sensor Webs Shankar Sastry University of California, Berkeley ATO Novel Approaches to Information Assurance And Control Theory Workshop February 5th, 2003.

2. UC Berkeley Pursuit-Evasion Game (PEG) Setup

3. UAV-UGV Coordinated PEG • Unknown Terrain • Not accurately mapped • Obstacles (moving & stationary) • Pursuers • Cooperative teams of UAVs & UGVs • UAV visually scans a limited region • UAVs rotorcraft-based • Networked communications • Evaders • Intelligent UGVs • Moves between & under obstacles • Actively avoids detection • Objective • Capture evader(s) in minimum time Pursuers Obstacles Evader

4. Cooperative Observation Problem Difficulty Obstacles Suppose at each instant in time, the location of all evaders is given. Optimal placement of pursuers in order that the maximum number of evaders are visible? -> NP-hard Reducible to Vertex Cover Problem withG=(V,E) Pursuers Evaders

5. A Two-step Solution: Exploration then Pursuit Exploration • Exploration followed by pursuit is notefficient • Sensors are imprecise • Worst-case assumptions on the trajectories of the evaders leads to very conservative results Pursuit ?

6. Probabilistic Framework Use a probabilistic framework to combine exploration and pursuit-evasion games. • Non-determinism comes from • Poorly mapped terrain • Noise and uncertainty in the sensors • Probabilistic models for the motion of the evaders • and the UAVs

7. Pursuit-Evasion Game Experimental Settings Multiple pursuers attempt capture of the evader(s) Pursuers can only move to adjacent empty cells Pursuers have perfect knowledge of current location Sensor model: false positives and negatives for evader detection Evader moves randomly to adjacent cells Unknown number of multiple evaders Sensor model for detection & tracking of targets Supervisory UAVs fly over obstacles & evaders, but cannot capture -> heterogeneous team Safety study of control policies

8. + y(t) ={v(t),e(t),o(t)} Sensor model Measurement step Evader motion model Prediction step Map Building: Map of Evaders At each t, the probability of evader state being x given the measurement histories is recursively updated.

9. Pursuit-Evasion Game Experiment Setup Waypoint Commands Pursuer: UAV Current position, vehicle status Evader location detected by vision system Current position, vehicle status Pursuer: UGVs Ground Command Post Evader: UGV

10. position of targets • position of obstacles • positions of agents Exogenous disturbance Strategy Planner Map Builder Hierarchy in Berkeley Platform Communications Network desired agents actions targets detected agents positions obstacles detected tactical planner Tactical Planner & Regulation Vehicle-level sensor fusion Uncertainty pervades every layer! obstacles detected trajectory planner state of agents regulation • obstacles • detected • targets • detected inertial positions height over terrain actuator positions • lin. accel. • ang. vel. control signals actuator encoders vision ultrasonic altimeter INS GPS Terrain UAV dynamics UGV dynamics Targets

11. POMGAME Representing and Managing Uncertainty • Uncertainty is introduced in various channels • Sensing -> unable to determine the current state of world • Prediction -> unable to infer the future state of world • Actuation ->unable to make the desired action to properly affect the state of world • Different types of uncertainty can be addressed by different approaches • Nondeterministic uncertainty : Robust Control • Probabilistic uncertainty : (Partially Observable) Markov Decision Processes • Adversarial uncertainty : Game Theory

12. Markov Games • Framework for sequential multiagent interaction in an Markov environment

13. Policy for Markov Games • The policy of agent i at time t is a mapping from the current state to probability distribution over its action set. • Agent i wants to maximize • the expected infinite sum of a reward that the agent will gain by executing the optimal policy starting from that state • where is the discount factor, and is the reward received at time t • Performance measure: • Every discounted Markov game has at least one stationary optimal policy, but not necessarily a deterministic one. • Special case : Markov decision processes (MDP) • Can be solved by dynamic programming

14. Partial Observation Markov Games (POMGame)

15. Policy for POMGames • The agent i wants to receive at least • Poorly understood: analysis exists only for very specially structured games such as a game with a complete information on one side • Special case : partially observable Markov decision processes (POMDP)

16. Optimal Pursuit Policy • Performance measure : capture time • Optimal policy m minimizes the cost

17. Persistent pursuit policies • Persistent pursuit policy gwith a periodT

18. Pursuit Policies • Greedy Policy • Pursuer moves to the cell with the highest probability of having an evader at the next instant • Strategic planner assigns more importance to local or immediate considerations • u(v): list of cells that are reachable from the current pursuers position v in a single time step.

19. Persistent Pursuit Policies for unconstrained motion Theorem 1, for unconstrained motion • The greedy policy is persistent. ->The probability of the capture time being finite is equal to one ->The expected value of the capture time is finite

20. Persistent Pursuit Policies for constrained motion Assumptions • For any • Theorem 2, for constrained motion • There is an admissible pursuit policy that is persistent on the average with period

21. Experimental Results: Pursuit Evasion Games with 4UGVs (Spring’ 01)

22. Experimental Results: Pursuit Evasion Games with 4UGVs and 1 UAV (Spring’ 01)

23. Pursuit-Evasion Game Experiment • PEG with four UGVs • Global-Max pursuit policy • Simulated camera view • (radius 7.5m with 50degree conic view) • Pursuer=0.3m/s Evader=0.5m/s MAX

24. Pursuit-Evasion Game Experiment • PEG with four UGVs • Global-Max pursuit policy • Simulated camera view • (radius 7.5m with 50degree conic view) • Pursuer=0.3m/s Evader=0.5m/s MAX

25. Experimental Results: Evaluation of Policies for different visibility Capture time of greedy and glo-max for the different region of visibility of pursuers 3 Pursuers with trapezoidal or omni-directional view Randomly moving evader • Global max policy performs better than greedy, since the greedy policy selects movements based only on local considerations. • Both policies perform better with the trapezoidal view, since the camera rotates fast enough to compensate the narrow field of view.

26. Experimental Results: Evader’s Speed vs. Intelligence Capture time for different speeds and levels of intelligence of the evader 3 Pursuers with trapezoidal view & global maximum policy Max speed of pursuers: 0.3 m/s • Having a more intelligent evader increases the capture time • Harder to capture an intelligent evader at a higher speed • The capture time of a fast random evader is shorter than that of a slower random evader, when the speed of evader is only slightly higher than that of pursuers.

27. Game-theoretic Policy Search Paradigm • Solving very small games with partial information, or games with full information, are sometimes computationally tractable • Many interesting games including pursuit-evasion are a large game with partial information, and finding optimal solutions is well outside the capability of current algorithms • Approximate solution is not necessarily bad. There might be simple policies with satisfactory performances -> Choose a good policy from a restricted class of policies ! • We can find approximately optimalsolutions from restricted classes, using a sparse sampling and a provably convergent policy search algorithm

28. Constructing A Policy Class • Given a mission with specific goals, we • decompose the problem in terms of the functions that need to be achieved for success and the means that are available • analyze how a human team would solve the problem • determine a list of important factors that complicate task performance such as safety or physical constraints • Maximize aerial coverage, • Stay within a communications range, • Penalize actions that lead an agent to a danger zone, • Maximize the explored region, • Minimize fuel usage, …

29. Policy Representation • Quantitize the above features and define a feature vector that consists of the estimate of above quantities for each action given agents’ history • Estimate the ‘goodness’ of each action by constructing where is the weighting vector to be learned . • Choose an action that maximizes . • Or choose a randomized action according to the distribution Degree of Exploration

30. Policy Search Paradigm • Searching for optimal policies is very difficult, even though there might be simple policies with satisfactory performances. • Choose a good policy from a restricted class of policies ! • Policy Search Problem

31. PEGASUS (Ng & Jordan, 00) • Given a POMDP , • Assuming a deterministic simulator, we can construct an equivalent POMDP with deterministic transitions . • For each policy p2P for X, we can construct an equivalent policy p02P0 for X0 such that they have the same value function, i.e. VX (p) = VX0 (p0) . • It suffices for us to find a good policy for the transformed POMDP X0 . • Value function can be approximated byadeterministic function, and ms samples aretaken and reused to compute the value function for each candidate policy. --> Then we can use standard optimization techniquesto search for approximately optimal policy.

32. Performance Guarantee & Scalability • Theorem • We are guaranteed to have a policy with the value close enough to the optimal value in the class P.

33. Acting under Partial Observations • Computing the value function is very difficult under partial observations. • Naïve approaches for dealing with partial observations: • State-free deterministic policy : mapping from observation to action • Ignores partial observability (i.e., treat observations as if they were the states of the environment) • Finding an optimal mapping is NP-hard. Even the best policy can have very poor performance or can cause a trap. • State-free stochastic policy : mapping from observation to probability distribution over action • Finding an optimal mapping is still NP-hard. • Agents still cannot learn from the reward or penalty received in the past.

34. Example:Abstraction of Pursuit-Evasion Game • Consider a partial-observation stochastic pursuit-evasion game in a 2-D grid world, between (heterogeneous) teams of ne evaders and np pursuers . • At each time t, • Each evader and pursuer, located at and respectively, • takes the observation over its visibility region • updates the belief state • chooses action from • Goal: capture of the evader, or survival

35. Example: Policy Feature • Maximize collective aerial coverage -> maximize the distance between agents where is the location of pursuer that will be landed by taking action from • Try to visit an unexplored region with high possibility of detecting an evader where is a position arrived by the action that maximizes the evader map value along the frontier

36. Example: Policy Feature (Continued) • Prioritize actions that are more compatible with the dynamics of agents • Policy representation

37. Benchmarking Experiments • Performance of two pursuit policies compared in terms of capture time • Experiment 1 : two pursuers against the evader who moves greedily with respect to the pursuers’ location • Experiment 2 : When we supposed the position of evader at each step is detected by the sensor network with only 10% accuracy, two optimized pursuers took 24.1 steps, while the one-step greedy pursuers took over 146 steps in average to capture the evader in 30 by 30 grid.

38. The State-of-the-Art

39. What pursuers really see

40. Tiny OS (TOS) Jason Hill, Robert Szewczyk, Alec Woo, David Culler • TinyOS • Ad hoc networking

41. Smart Dust, Dot Motes, MICA Motes Dot motes, MICA motes and smart dust

42. 1. Field of wireless sensor nodes • Ad hoc, rather than engineered placement • At least two potential modes of observation • Acoustic, magnetic, RF

43. 2. Subset of more powerful assets • Gateway nodes with pan-tilt camera • Limited instantaneous field of view

44. 3. Set of objects moving through

45. 4. Track a distinguished object

46. Sensor net increases visibility

47. Potential Issues in current PEG Cameras have small range GPS jamming, unbounded error of INS, noisy ultrasonic sensors Communication among pursuers may be difficult over a large area Unmanned vehicles are expensive It is unrealistic to employ many number of unmanned vehicles to cover a large region to be monitored. A smart evader is difficult to catch Benefits from sensor network Large sensing coverage Location aware sensor network provide pursuers with additional position information Network can relay information among pursuers Sensor network is cheap and can reduce number of pursuers without compromising capture time Sensibly reduce exploration of the environment A wide, distributed network is more difficult to compromise What a sensor network can do for PEG Overall Performance can be dramatically increased by lowering capture time, by increasing fault tolerance and making the pursuer team resilient to security attacks

48. Strategy Planner Map Builder Vehicles coordination layer Pursuers’ communication infrastructure Tactical Planner & Regulation Vehicle-level sensor fusion Control Signals to pursuer Single vehicle estimation and control layer Nest Sensorweb vision GPS Sensor information layer Physical Platform Used

49. Pursuit Evasion Games using sensor webs • Self organization of motes into a sensorweb • Creation of a communication infrastructure • Self-localization • Synchronization • Tracking of evaders’ by pursuers’ team • Evaders’ position and velocity estimation by sensor network • Communication of sensors’ estimates to ground pursuers • Design of a pursuit strategy • Coordination of ground & aerial pursuers • Network maintenance • Robustness • Security

50. Closed-loop at many levels • Within a node • Algorithms adapt to available energy, physical measurements, network condition • Across the network • discovery and routing, transmission protocols are energy aware and depend on application requirements • Within the middleware components • synchronization, scheduling, localization • On the vehicle • direction, stability, probabilistic map building • Among the vehicles • competitive, hidden Markov decision processes Used