2. Control and Decision Making in Uncertain Multiagent Hierarchical Systems June 10 th, 2002 H. Jin Kim and Shankar Sastry University of California, Berkeley

3. 1 Outline  Hierarchical architecture for multiagent operations  Confronting uncertainty  Partial observation Markov games (POMgame)  Incorporating human intervention in control and decision making  Model predictive techniques for dynamic replanning

4. 2 Partial-observation Probabilistic Pursuit-Evasion Game (PEG) with 4 UGVs and 1 UAV Fully autonomous operation

5. 3 Uncertainty pervades every layer! Hierarchy in Berkeley Platform actuator positions inertial positions height over terrain obstacles detected targets detected control signals INSGPS ultrasonic altimeter vision state of agents obstacles detected targets detected obstacles detected agents positions desired agents actions Tactical Planner & Regulation Vehicle-level sensor fusion Strategy PlannerMap Builder position of targets position of obstacles positions of agents Communications Network tactical planner trajectory planner regulation lin. accel. ang. vel. Targets Exogenous disturbance UAV dynamics Terrain actuator encoder s UGV dynamics Impossible to build autonomous agents that can cope with all contingencies

6. 4 Human Interface Command Current Position, Vehicle Stats Evader location detected by Vision system Ground Station High degree of autonomy does not guarantee superior performance of overall system

7. 5 Lessons Learned and UAV/UGV Objective  To design semi-autonomous teams that deliver mission reliably under uncertainty and evaluate their performance  Scalable/replicable system aided by computationally tractable algorithms  Hierarchical architecture design and analysis –High-level decision making in a discrete space –Physical-layer control in a continuous space  Hierarchical decomposition requires tight interaction between layers to achieve cooperative behavior, to deconflict and to support constraints.  Confronting uncertainty arising from partially observable, dynamically changing environments and intelligent adversaries –Proper degree of autonomy to incorporate reliance on human intervention –Observability and directability, not excessive functionality

8. 6 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 POMGAME

9. 7 Matrix Games  Single-stage, 2-player games  A matrix specifies rewards for each player, for each action pair  In general, each player’s policy represents the probability distribution over its action set  Player1 wants to get at least  Optimal policy can be solved by linear programming

10. 8 Markov Games  Framework for sequential multiagent interaction in an Markov environment

11. 9 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

12. 10 Partial Observation Markov Games (POMGame)

13. 11 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)

14. 12 Acting under Partial Observations  Memory-free policies (mapping from observation to action or probability distribution over action sets) are not satisfactory.  In order to behave truly effectively we need to use memory of previous actions and observations to disambiguate the current state.  The state estimate, or belief state –Posterior probability distribution over states = the likelihood the world is actually in the state x, at time t, given the agent’s past experience (I.e. actions and observation histories). A priori human input on the initial state of world

15. 13 Updating Belief State –Can be updated recursively using the estimated world model and Bayes’ rule. New info on the state of world New info on prediction

16. 14 Pursuit-Evasion Games  Consider approach in Hespanha, Kim and Sastry –Multiple pursuers catching one single evader –Pursuers can only move to adjacent empty cells –Pursuers have perfect knowledge of current location –Sensor model: false positives (p) and negatives (q) for evader detection –Evader moves randomly to adjacent cells  Extensions in Rashid and Kim –Multiple evaders: assuming each one is recognized individually –Supervisory agents: can “fly” over obstacles and evaders, cannot capture –Sensor model for obstacle detection as well

17. 15 BEAR Pursuit-Evasion Scenario Evade!

18. 16 Problem Formulation

19. 17  Performance measure : capture time  Optimal policy  minimizes the cost Optimal Pursuit Policy

20. 18  cost-to-go for policy , when the pursuers start with Y t = Y and a conditional distribution  for the state x(t)  cost of policy  Optimal Pursuit Policy

21. 19 Persistent pursuit policies  Optimization using dynamic programming is computationally intensive.  Persistent pursuit policy g

22. 20 Persistent pursuit policies  Persistent pursuit policy g with a period T

23. 21 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.

24. 22 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

25. 23 Persistent Pursuit Policies for constrained motion Assumptions 1.For any 2.Theorem 2, for constrained motion  There is an admissible pursuit policy that is persistent on the average with period

26. 24 Experimental Results: Pursuit Evasion Games with 4UGVs (Spring’ 01)

27. 25 Experimental Results: Pursuit Evasion Games with 4UGVs and 1 UAV (Spring’ 01)

28. 26 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

29. 27 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

30. 28 Experimental Results: Evaluation of Policies for different visibility  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. 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

31. 29 Experimental Results: Evader’s Speed vs. Intelligence 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. 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

32. 30 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 optimal solutions from restricted classes, using a sparse sampling and a provably convergent policy search algorithm

33. 31 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, …

34. 32 Policy Representation  Quantize 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

35. 33 Policy Learning  Policy parameters are learned using standard techniques such as gradient descent algorithm to maximize the long-term reward  Given a POMDP, and assuming that we have a deterministic simulative model, we can approximate a value for a specific policy by building a set of trajectory trees with depth  m s is independent of the size of the state space or the complexity of the transition distribution [Ng, Jordan00] Computational tractability

36. 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 n e evaders and n p 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

37. 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

38. 36  Prioritize actions that are more compatible with the dynamics of agents  Policy representation Example: Policy Feature (Continued)

39. 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. Grid size1-Greedy pursuersOptimized pursuers 10 by 10(7.3, 4.8)(5.1, 2.7) 20 by 20(42.3, 19.2)(12.3, 4.3)

40. 38 Incorporating Human Intervention  Given the POMDP formalism, informational inputs affect only initializing or updating the belief state, and does not affect the procedure of computing (approximately) optimal actions.  When a part of the system is commanded to take specific actions, it may overrule internally chosen actions and simultaneously communicate its modified status to the rest of the system, which then in turn adapts to coordinate their own actions as well.  A human command in the form of mission objectives can be expressed as a change to the reward function, that causes the system to modify or dynamically replan its actions to achieve it. The importance of a goal is specified by changing the magnitude of the rewards.

41. 39 Coordination under Multiple Sources of Commands  When different humans or layers specify multiple, possibly conflicting goals or actions, how the system can prioritize or resolve them ?  Different entities are a priori assigned different degrees of authority  If there are enough resources to resolve an important conflict, we may give operators the option of explicitly coordinating their goals  Surge in coordination demand when the situation deviates from textbook cases: can the overall system adapt real-time?  Intermediate, cooperative modes of interaction (vs. traditional human interrupt of full manual form) is desirable  Transparent, event-based display to highlight changes (vs. current data-oriented display)  Anticipatory reasoning (not just information on history) should be supported.

42. 40 Deconfliction between Layers Each UAV is given a waypoint by high- level planner Shortest trajectories to the waypoints may lead collision How to dynamically replan the trajectory for the UAVs subject to input saturation and state constraints

43. 41 (Nonlinear) Model Predictive Control  Find that minimizes  Common choice

44. 42 Planning of Feasible Trajectories  State saturation  Collision avoidance  Magnitude of each cost element represents the priority of tasks/functionality, or the authority of layers

45. 43 Hierarchy in Berkeley Platform actuator positions inertial positions height over terrain obstacles detected targets detected control signals INSGPS ultrasonic altimeter vision state of agents obstacles detected targets detected obstacles detected agents positions desired agents actions Tactical Planner & Regulation Vehicle-level sensor fusion Strategy PlannerMap Builder position of targets position of obstacles positions of agents Communications Network tactical planner trajectory planner regulation lin. accel. ang. vel. Targets Exogenous disturbance UAV dynamics Terrain actuator encoder s UGV dynamics

46. 44 H1 H2 H0 Cooperative Path Planning & Control Trajectories followed by 3 UAVs Coordination based on priority Example: Three UAVs are given straight line trajectories that will lead to collision. |Lin. Vel.| < 16.7ft/s |Ang| < pi/6 rad |Control Inputs| < 1 Constraints supported NMPPC dynamically replans and tracks the safe trajectory of H1 and H2 under input/state constraints.

47. 45 Summary  Decomposition of complex multiagent operation problems requires tighter interaction between subsystems and human intervention  Partial observation Markov games provides a mathematical representation of a hierarchical multiagent system operating under adversarial and environmental uncertainty  Policy class framework provides a setup for including human experience  Policy search methods and sparse sampling produce computationally tractable algorithms to generate approximate solutions to partially observable Markov games.  Human input can/should be incorporated, either a priori or on-the-fly, into various factors such as reward functions, feature vector elements, transition rules, action priority  Model predictive (receding horizon) techniques can be used for dynamic replanning to deconflict/coordinate between vehicles, layers or subtasks

48. 46 Unifying Trajectory Generation and Tracking Control  Nonlinear Model Predictive Planning & Control combines trajectory planning and control into a single problem, using ideas from –Potential-field based navigation (real-time path planning) –Nonlinear model predictive control (optimal control of nonlinear multi-input, multi- output systems with input/state constraints)  We incorporate a tracking performance, potential function, state constraints into the cost function to minimize, and use gradient-descent for on-line optimization.  Removes feasibility issues by considering the UAV dynamics from the trajectory planning  Robust to parameter uncertainties  Optimization can be done real-time

49. 47 Modeling and Control of UAVs  A single, computationally tractable model cannot capture nonlinear UAV dynamics throughout the large flight envelope.  Real control systems are partially observed (noise, hidden variables).  It is impossible to have data for all parts of the high-dimensional state-space. -> Model and Control algorithm must be robust to unmodeled dynamics and noise and handle MIMO nonlinearity. Observation: Linear analysis and deterministic robust control techniques fail to do so.

50. 48 Modeling RUAV Dynamics Position Spatial velocities Angles Angular rates Servoinputs throttle longitudinal flapping lateral flapping main rotor collective pitch tail rotor collective pitch Body Velocities Angular rates Aerodynamic Analysis Coordinate Transformation Augmented Servodynamics Tractable Nonlinear Model

51. 49 Benchmarking Trajectory PD controller Example PD controller fails to achieve nose-in circle type trajectories. Nonlinear, coupled dynamics are intrinsic characteristics in pirouette and nose-in circle trajectories.

52. 50 Reinforcement Learning Policy Search Control Design 1.Aerodynamics/kinematics generates a model to identify. 2.Locally weighted Bayesian regression is used for nonlinear stochastic identification: we get the posterior distribution of parameters, and can easily simulate the posterior predictive distribution to check the fit and robustness. 3.A controller class is defined from the identification process and physical insights and we apply policy search algorithm. 4.We obtain approximately optimal controller parameters by reinforcement learning, I.e. training using the flight data and the reward function. 5.Considering the controller performance with a confidence interval of the identification process, we measure the safety and robustness of control system.

53. 51 Performance of RL Controller Manual vs. Autonomous Hover Assent & 360° x2 pirouette

54. 52 Demo of RL controller doing acrobatic maneuvers (Spring 02)

55. 53 pirouette maneuver2 maneuver1 maneuver3 Nose-in During circling Heading kept the same Any variation of the following maneuvers in x-y direction Any combination of the following maneuvers Set of Manuevers

56. 54 Video tape of Maneuvers

57. 55 Back Up Slides

58. 56 PEGASUS (Ng & Jordan, 00)  Given a POMDP,  Assuming a deterministic simulator, we can construct an equivalent POMDP with deterministic transitions.  For each policy  2  for  we can construct an equivalent policy  0 2  0 for  0 such that they have the same value function, i.e. V  (  ) = V  0 (  0 ).  It suffices for us to find a good policy for the transformed POMDP  0.  Value function can be approximated by a deterministic function, and m s samples are taken and reused to compute the value function for each candidate policy. --> Then we can use standard optimization techniques to search for approximately optimal policy.

59. 57 PEGASUS (Ng & Jordan, 00)  Given a POMDP,  Assuming a deterministic simulator, we can construct an equivalent POMDP with deterministic transitions.  For each policy  2  for  we can construct an equivalent policy  0 2  0 for  0 such that they have the same value function, i.e. V  (  ) = V  0 (  0 ).  It suffices for us to find a good policy for the transformed POMDP  0.  Value function can be approximated by a deterministic function, and m s samples are taken and reused to compute the value function for each candidate policy. --> Then we can use standard optimization techniques to search for approximately optimal policy.

60. 58 Performance Guarantee & Scalability  Theorem  We are guaranteed to have a policy with the value close enough to the optimal value in the class   Note that

61. 59 Markov Decision Process (MDP)  Framework for sequential decision making in the stationary environment

62. 60  The agent’s objective is to find an optimal policy, mapping from its interaction history to action sequence.  Optimal value function of a state of for a given MDP : –the expected infinite sum of a reward that agent will gain by executing the optimal policy starting from that state: where is the discount factor, and is the scalar reward received at time t.  Every MDP has at least one stationary deterministic optimal policy, i.e. that is optimal. Policy and Performance Metric

63. 61 Partially Observable Markov Decision Processes (POMDP)

64. 62 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.

65. 63 Policy Learning  Policy parameters are learned using standard techniques such as gradient descent to maximize the reward  Given a POMDP, and assuming that we have a deterministic simulative model, we can approximate a value for a specific policy by a set of trajectory trees with depth I.e.  Truncation  m s is independent of the size of the state space or the complexity of the transition distribution [Ng, Jordan00] Computational tractability

66. 64 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