Decentralized Control of a Quadrotor Aircraft Fleet to seek Information Claire J. Tomlin, Gabe Hoffmann, Maryam Kamgarpour, Robin Raffard, and Steven Waslander Electrical Engineering and Computer Sciences UC Berkeley Aeronautics and Astronautics Stanford University 14 February 2007 CHESS Winter Meeting
Planning operations for teams of aircraft Local objectives: Safety and efficiency with respect to vehicle s dynamics and actuation Global objectives: Region coverage, collision avoidance Constraints: Communication private information Computation performed on each vehicle (Source: Prof. Robin Murphy, Univ. South Florida, Katrina Search and Rescue)
Collision Avoidance
Region Surveillance (Source: Prof. Robin Murphy, Univ. South Florida, Katrina Search and Rescue) Problems: Information gathering, fast decomposition of team commands into actions for each vehicle
Testbed Quadrotor Design: Autonomous Control Wireless Full Onboard Sensing IMU, GPS, SODAR Stereo vision Laser range finder
Objectives: Safe and efficient control Automatic information gathering Constraints: Communication bandwidth Computational resources Collision avoidance A Mobile Sensor Network
Outline Decentralized optimization using dual decomposition Cooperative control to seek information Initial results with the testbed
Decentralized optimization based on dual decomposition Consider the following simple example: aircraft, each with dynamics Private (local) cost function: Global cost function:
Centralized Optimization
Dual Decomposition: Overview One primal problem leads to many dual problems The dual decomposition method solves one instance of these many dual problems breaks the primal problem into a set of small problems provides a lower bound on the global optimal solution If the problem is convex, dual decomposition returns the global optimal solution
Primal Problem Dual Decomposition: Overview 2 1 3 2 2 2 Decoupled Primal Problem 1 1 3 3 Dual Problem 2 2 2 1 1 3 3
Decentralized Optimization
Decentralized Algorithm Each vehicle picks a starting value for Repeat 1. Each vehicle communicates its values to the aircraft with which it shares constraints, 2. Each vehicle computes its optimal trajectory. 3. Each vehicle computes and updates Terminate when are small
after 5 iterations Decentralized results
Complexity Results Centralized Decentralized Algorithm Solve primal Apply dual decomposition Solution Memory Optimal solution if convex Optimal if convex Suboptimal, with duality gap given, otherwise Flop count Practice Communication with all agents, once Communication with a few agents, iteratively
Outline Decentralized optimization using dual decomposition Cooperative control to seek information Initial results with the testbed
Objectives: Safe and efficient control Automatic information gathering Constraints: Communication bandwidth Computational resources Collision avoidance A Mobile Sensor Network
Modeling Uncertainty to Increase Knowledge Target State: Observations: Target State Model: Vehicle States: Control Inputs: Motion Model: Sensor model: Use Bayes Rule to update the target state model, Minimize the expected future uncertainty,
Maximizing Information Ask a broad question Expected Observation Probability with a distinct answer Sensor Model
Bayes Rule using Particle Filters Uses available prior knowledge Allows multimodal posterior Permits nonlinear & non-gaussian models Particle Set Correction (weighting) Resampling Prediction Particle Set
Mutual Information from Particle Filters
Expanding the Mutual Information Single Node Decomposition for Distributed Optimization Pairwise Node Decomposition for Distributed Optimization
Distributed Optimization Program
Bearings Only Example Measure the direction to the target
Range Only Example Measure the distance to the target
Beacon Field Example Measure the field line orientation
4 Vehicles, Bearings-Only Results Mean and Quartile Error Bars for 1000 Trials 1 0.9 0.8 Pairwise Approximation Solo Approximation Probability of Target being < 1 unit from Estimate 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 2 4 6 8 10 12 14 16 18 20 Time Elapsed
Comparison of Pairwise Approx. Results 1 0.9 0.8 Mean and Quartile Error Bars for 1000 Trials Probability of Target being < 1 meter from Estimate 0.7 0.6 0.5 0.4 0.3 0.2 0.1 10 4 3 2 0 0 5 10 15 20 Time Time Elapsed
Development of the Testbed
Research Directions Communication models Decentralized optimization computation load distributed among resources allows for flexibility in shared information Implementation Generalizing to Other Applications Beacon tracking scenarios, RFID tracking Survey of disaster areas Highway accident discovery and monitoring Biological studies, animal monitoring Geological studies etc Embedded humans Fit into POMDP framework
Dynamics Two pairs of counterrotating blades One motor for each blade Dynamics decouple into sets of double integrators Controlling Altitude Controlling Roll/Pitch Controlling Yaw
Actuation Propellors Designed for Electric, Low Speed RC Planes Brushless DC Outrunner Motors Characteristics Brushless = No Commutator brush Outrunner = Coils connected to casing, not rotor Advantages over brushed DC motors Dissipates power through casing More coils = more torque = no gear box needed No wear Speed control inherent to use Less EMF noise
Sensing Videre Design Stereo-on-Chip (STOC) Camera SRI Small Vision System on FPGA 8 bit disparity 640x480 resolution 30 fps 140 grams Hokuyo URG-04LX LIDAR 4 m range 1 cm standard deviation 240 laser points / sweep 10 Hz sweeps 240 range 150 grams Avalanche Beacon Transceiver
LIDAR System