Control and Decision Making in Uncertain Multi-agent Hierarchical Systems A Case Study in Learning and Approximate Dynamic Programming

PI Meeting August 1

st

, 2002

Shankar Sastry University of California, Berkeley

Outline

 Hierarchical architecture for multiagent operations  Confronting uncertainty  Partial observation Markov games (POMgame)  Model predictive techniques for dynamic replanning 1

Partial-observation Probabilistic Pursuit-Evasion Game (PEG) with 4 UGVs and 1 UAV

Fully autonomous operation 2

Hierarchy in Berkeley Platform

tactical planner

Strategy Planner

• position of targets • position of obstacles • positions of agents

Map Builder Communications Network

desired agents actions agents positions obstacles detected targets detected trajectory planner

Tactical Planner & Regulation

obstacles detected state of agents

Vehicle-level sensor fusion

regulation control signals ultrasonic altimeter height over terrain INS •lin. accel.

•ang. vel.

inertial actuator positions positions GPS actuator encoder s • obstacles detected • targets detected vision Uncertainty pervades every layer!

UAV dynamics Terrain

Exogenous disturbance

dynamics

3

Representing and Managing Uncertainty

 Uncertainty is introduced in various channels – Sensing -> unable to determine the current state of world – – Prediction Actuation of world -> unable to infer the future state of world -> unable to make the desired action to properly affect the state  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 5

Markov Games

 Framework for sequential multiagent interaction in an Markov environment 6

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 7

Partial Observation Markov Games (POMGame)

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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) 9

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

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

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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.

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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 28

Constructing A Policy Class

 Given a mission with specific goals, we – decompose the problem in terms of the functions for success and the means that are available that need to be achieved – 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 p for X 0 2 P for X, we can construct an equivalent policy such that they have the same value function, i.e. V X ( p ) = V X p 0 0 2 P 0 ( p 0 ) .  It suffices for us to find a good policy for the transformed POMDP X 0 .

 Value function can be approximated by a optimization techniques deterministic function , and m s samples are taken and reused to compute the value function for each candidate policy. --> Then we can use standard to 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.

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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 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 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 Grid size 10 by 10 20 by 20 1-Greedy pursuers (7.3, 4.8) (42.3, 19.2) Optimized pursuers (5.1, 2.7) (12.3, 4.3)  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

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

39

Benchmarking Trajectory Nonlinear, coupled dynamics are intrinsic characteristics in pirouette and nose-in circle trajectories .

Example PD controller fails to achieve nose-in circle type trajectories.

PD controller 40

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.

41

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

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Toughest Maneuvers for Rotorcraft UAVs

Nose-in During circling pirouette maneuver1 Heading kept the same maneuver2 maneuver3 •Any variation of the following maneuvers in x-y direction •Any combination of the following maneuvers 43

Demo of RL controller doing acrobatic maneuvers (Spring 02)

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More Acrobatic Maneuvers (Spring 02)

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From PEG to More Realistic Battlefield Scenarios

 Adversarial attack – Reds just do not evade, but also attack -> Blues cannot blindly pursue reds.

 Unknown number/capability of adversary -> Dynamic selection of the relevant red model from unstructured observation  Deconfliction between layers and teams  Increase number of feature -> Diversify possible solutions when the uncertainty is high 46

Why General-sum Games?

" All too often in OR dealing with military problems, war is viewed as a zero sum two-person game with perfect information. Here I must state as forcibly as I know that war is not a zero-sum two-person game with perfect information. Anybody who sincerely believes it is a fool. Anybody who reaches conclusions based on such an assumption and then tries to peddle these conclusions without revealing the quicksand they are constructed on is a charlatan....There is, in short, an urgent need to develop positive-sum game theory and to urge the acceptance of its precepts upon our leaders throughout the world." Joseph H. Engel, Retiring Presidential Address to the Operations Research Society of America, October 1969 47

General-sum Games

 Depending on the cooperation between the players, – – Noncooperative Cooperative  Depending on the least expected payoff that a player is willing to accept- Nash’s special/general bargaining solution  By restricting the blue and red policy class to be the finite size, we reduce the POMGame into the bimatrix game .

48

From POMGame To Bimatrix Game

Bimatrix game usually has multiple Nash equilibria, with different values. 49

Elucidating Adversarial Intention

 The model posterior distribution can be used to predict the future observation, or select the model.  Then the blue team can employ the policy such that  Example Implemented : tracking unknown number of evaders with unknown dynamics with noisy sensors 50

Dynamic Bayesian Model Selection

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Dynamic Bayesian model selection (DBMS) is a generalized model selection approach to time series data of which the number of components can vary with time

•

If K is the number of the components at any instance and T is the length of the time series, then there are O(2 KT ) possible models which demands an efficient algorithm

•

The problem is formulated using Bayesian hierarchical modeling and solved using reversible jump MCMC methods suitably adapted.

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DBMS

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DBMS: Graphical Representation

      a – Dirichlet prior A – Transition matrix for m t d t – Dirichlet prior w t – component weights z t – allocation variable F – transition dynamics 53

DBMS

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DBMS: Multi-target Tracking Example

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+ True target trajectory Observation Estimated target position 56

+ True target trajectory Observation Estimated target position 57

Vision-based Landing of an Unmanned Aerial Vehicle

Berkeley Researchers: Rene Vidal, Omid Shakernia, Shankar Sastry 58

What we have accomplished

 Real-time motion estimation algorithms – Algorithms: Linear & Nonlinear two-view, Multi-view  Fully autonomous vision-based control/landing 59

Image Processing

60

Vision Monitoring Station

61

Vision System Hardware

 Ampro embedded Little Board PC – Pentium 233MHz running LINUX – Motion estimation, UAV high-level control – Pan/Tilt/Zoom camera tracks target  Motion estimation algorithms – Written C++ using LAPACK – Estimate relative position and orientation at 30 Hz – Sends control to navigation computer at 10 Hz UAV Pan/Tilt Camera Onboard Computer 62

Flight Control System Experiments Position+Heading Lock (Dec 1999) Landing scenario with SAS (Dec 1999) Position+Heading Lock (May 2000) Attitude control with mu-syn (July 2000)

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Semi-autonomous Landing (8/01)

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Autonomous Landing (3/02)

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Autonomous Landing (3/02)

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Multi-body Motion Estimation and Segmentation Vidal, Soatto, Sastry

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Multi-body Motion Estimation

 Motivation – Conflict Detection + Resolution + Formation Flight – Target Tracking  Given a set of image points and their flows obtain: – Number of independently moving objects – – – Segmentation Motion Structure : object to which each point belongs : rotation and translation of each object : depth of each point  Previous work – Orthographic projection camera (Costeira-Kanade’95) – Multiple points moving in straight line (Shashua-Levin’01)  This work considers full perspective motion projection, with multiple objects undergoing general  Motion not fooled by camouflage like other segmentation cues (texture, color, etc.) 68

Image Measurements

 Form optical flow matrices  n= feature points, m= frames  Optical flow measurements live in a six dimensional space 69

Factorization

 For one object one can factorize into motion and structure components  One can solve linearly for A and Z from 70

Multiple Moving Objects

 For multiple independently moving objects  Obtain number of independent motions 71

Segmentation

 Segmentation of the image points 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20 8 10 12 2 4 6 14 16 18 20 2 4 6 8 10 12 14 16 18 20 72

Experimental Results

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Experimental Results

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A Roadmap for Cooperative Operation of Autonomous Vehicles

 John Koo, Shannon Zelinski, Shankar Sastry  Department of EECS, UC Berkeley 76

Motivation

 Multiple Autonomous Vehicle Applications – Unmanned aerial vehicles perform mission collectively – Satellites for distributed sensing – Autonomous underwater vehicles performing exploration – Autonomous cars forming platoons on roads  Enabling Technologies – Hierarchical control of multi-agents – Distributed Sensing and Actuation – Computation – Communication – Embedded Software 77

Formation Flight of Aerial Vehicles

 Group Level – – – Formation Control Conflict Resolution Collision Avoidance  Vehicle Level – Vehicle Navigation – Envelope Protection

Design Challenges

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Different Levels of Centralization

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Multiple Modes of Operation

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Organization of Information Flow

q 1 q 2 q 3 78

Possible Formations for a UAV mission Line Formation Diamond Formation Loose Formation

79

Components of Formation Flight

 Formation Generation – Generate a set of feasible formations where each formation satisfies multiple constraints including vehicle dynamics, communication, and sensing capabilities.

 Formation Initialization – Given an initial and a final formation for a group of autonomous vehicles, formation initialization problem is to generate collision-free and feasible trajectories and to derive control laws for the vehicles to track the given trajectories simultaneously in finite time.  Formation Control – Formation control of multiple autonomous vehicles focus on the control of individual agents to keep them in a formation, while satisfying their dynamic equations and inter agent formation constraints, for an underlying communication protocol being deployed. 80

Components of Formation Flight

 Formation Generation – Generate a set of feasible formations and each formation satisfies multiple constraints including vehicle dynamics, communication, and sensing capabilities.

Leader Trajectory Formation Constraints + Dynamic Constraints

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Components of Formation Flight

 Formation Initialization – Given an initial and a final formation for a group of autonomous vehicles, formation initialization problem is to generate collision-free and feasible trajectories and to derive control laws for the vehicles to track the given trajectories simultaneously in finite time.

Diamond Formation Line Formation

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Components of Formation Flight

 Formation Control – Formation control of multiple autonomous vehicles focus on the control of individual agents to keep them in a formation, while satisfying their dynamic equations and inter-agent formation constraints, for an underlying communication protocol being deployed. 83

Formation Initialization Virtual vehicles Actual vehicles

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Elements Of Formation Flight

 Information Resources – Wireless network – Global Positioning System – Inertial Navigation System – Radar System (Local and Active) – Vision System (Local and Passive) 85

Loose Formation Flight

 GPS provides global positioning information to vehicles  Wireless network is used to distribute information between vehicles  Navigation computer on each vehicle calculates relative orientation, distance and velocities

GPS signals Wireless Network

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Tight Formation Flight

 Vision system equipped with omni-directional camera can track neighboring vehicles  Structure from motion algorithms running on vision system provides estimates of relative orientation, distance and velocities to navigation computer 87

Hybrid Control Design for Formation Flight

Construct a Formation Mode Graph by considering dynamic and formation constraints.

For each formation, information about the formation is computed offline and is stored in each node of the graph. Feasible transition between formations are specified by edges.

Given an initial formation, any feasible formations can be efficiently searched on the graph. 88

Back Up Slides

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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 90

(Nonlinear) Model Predictive Control

 Find that minimizes  Common choice 91

Planning of Feasible Trajectories

 State saturation  Collision avoidance  Magnitude of each cost element authority of layers represents the priority of tasks/functionality, or the 92

Hierarchy in Berkeley Platform

tactical planner

Strategy Planner

• position of targets • position of obstacles • positions of agents

Map Builder Communications Network

desired agents actions agents positions obstacles detected targets detected trajectory planner

Tactical Planner & Regulation

obstacles detected state of agents

Vehicle-level sensor fusion

regulation control signals ultrasonic altimeter height over terrain INS •lin. accel.

•ang. vel.

inertial actuator positions positions GPS actuator encoder s • obstacles detected • targets detected vision

UAV dynamics Terrain

Exogenous disturbance

dynamics

93

Cooperative Path Planning & Control Example :

Three UAVs are given straight line trajectories that will lead to collision. NMPPC dynamically replans and tracks the safe trajectory of H1 and H2 under input/state constraints.

H1 H0 H2

Trajectories followed by 3 UAVs

Coordination based on priority |Lin. Vel.| < 16.7ft/s |Ang| < pi/6 rad |Control Inputs| < 1 Constraints supported 94

Unifying Trajectory Generation and Tracking Control

 Nonlinear Model Predictive Planning & Control control into a single problem, using ideas from combines trajectory planning and – 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 95

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. 96