AM3 Task 1.3 Navigation Using Spatio-Temporal Gaussian Processes Songhwai Oh

1. AM3 Task 1.3 Navigation Using Spatio-Temporal Gaussian Processes Songhwai Oh (Presented by Sam Burden) MAST Annual Review University of Pennsylvania March 8-9, 2010

2. Overview Navigation Using Spatio-Temporal Gaussian Processes • Songhwai Oh • Professor, Seoul National University • Subcontracted under Prof. Shankar Sastry (UC Berkeley) • Goal: control and navigation in unstructured and uncertain environments • Model environment as a Gaussian Process (GP) • Learn parameters for the GP • Efficiently incorporate new data • Use GP for control and navigation • Expected Results at End of Fiscal Year: multiple agents learn GP model, navigate new environments

3. Technical Relevance • Current techniques in GP regression • Offline • Gradient descent for parameter tuning • Incorporating new observations requires full computation • Centralized • Estimation decoupled from control and navigation • No natural way to integrate data from multiple agents • Our approach to GPs • Real-time • Distributed • Integrate estimation with navigation

4. Relevance to MAST • Our approach to GPs • Real-time • Distributed • Integrate estimation with navigation Enables MAST platforms to navigate in unstructured and uncertain environments

5. Technical Accomplishments • 1Q09 (A-09-1.3a): Spatio-temporal GPs: done • 2Q09 (A-09-1.3b): Integrate GP with MPC: ongoing • 3Q09 (A-09-1.3c): Assumed-Density GP • Developed a distributed GP learning algorithm instead • 4Q09 (A-09-1.3d): Assumed-Density GP with MPC • Ongoing; developed bio-inspired navigation strategies [2] • 1Q10 (A-10-1.3a): Approximation • Developed rigorous approximation for truncated Gaussian Process • 2Q10 (A-10-1.3b): Multi-Agent Integration • Share observations among agents, coordinate navigation • 3Q10 (A-10-1.3c): Learning GP Kernel • 4Q10 (A-10-1.3d): Implement on MAST Platform

6. Gaussian Process Estimation Unknown / uncertain environment modeled as a random process specified by mean and covariance Gaussian Process Regression provides straightforward but costly way to estimate process How can we reduce GP computation without degrading performance?

7. Truncated Gaussian Process Approximation Error Truncation Size Idea: recent measurements are more informative Formalize, provide bounds to quantify tradeoff between accuracy and computation time

8. Truncated GPs Recap Approximation Error Truncation Size Effective estimation tool Low-computation approximations Bounds on approximation accuracy We are closer to putting GPs on MAST platforms

9. Collaborations and Future Plans • Prof. Herbert Tanner (Autonomy, UDelaware) • Model Predictive Navigation using GPs • Helps avoid local minima • Possible new collaborative efforts / experiments • Use GPs to map occupancy and wireless signal strength • Ideas going forward • Nonstationary GP kernels for mapping • Efficient parameter estimation for GPs • Mixture-of-GPs for multiscale

10. Discussion & Questions Thank you for your time • Metrics • [1] Jongeun Choi, Songhwai Oh, and Roberto Horowitz, Distributed Learning and Cooperative Control for Multi-Agent Systems. Automatica, vol. 45, no. 12, pp. 2802-2814, Dec. 2009. • [2] Jongeun Choi, Joonho Lee, and Songhwai Oh, Navigation Strategies for Swarm Intelligence using Spatio-Temporal Gaussian Processes. Submitted to Neural Computing and Applications. • [3] Yunfei Xu, Jongeun Choi and Songhwai Oh, Mobile Sensor Network Coordination Using Gaussian Processes with Truncated Observations. Submitted to IEEE Transactions on Robotics.

11. Technical Slides Gaussian Processes Spatio-Temporal Conditional Distribution Dynamics and Sensing Navigation Strategies Path Planning Switched Path Planning Truncated Gaussian Processes Navigation using Truncated Observations

12. Gaussian Process 1 [O'Callaghan, Ramos, Durrant-Whyte, 2009] 2 [Ferris, Fox, Lawrence, 2007] • A Gaussian process (GP) is a stochastic process. Any finite number of samples from a GP has a Gaussian distribution • Regression using Gaussian processes: • Widely used in geostatistics (Kriging), statistics, machine learning • GP can be used to estimate a field such as • mapping1 • radio signal strength (WiFi-SLAM2) • danger level or stealthiness • others: temperature, lighting, noise-level, etc.

13. Recap: Spatio-Temporal Gaussian Processes

14. Recap: Conditional Distribution

15. Recap: Agent Dynamics and Sensing

16. Recap: Navigation Strategies

17. Path Planning Flocking Consensus Navigation Goal

18. Switching Path Planning • Assumption: Stationary field, ¾max2 > k(s,s) • Algorithm: • Starts with exploration strategy • If (max(prediction error) < ²) Switch to exploitation (tracing) strategy • Theorem [2]: Under some smoothness conditions, with probability at least 1- ¾max2 / ²2, |z(s,t)-z*(s,¿|t)| < ² at ¿ > t+1, for all s.

19. Switching Path Planning Trajectories of agents and estimated field Field to be estimated

20. Switching Path Planning Predicted variance, RMS, maximum errors of agents performing switching path planning. RMS over simulation time (100 Monte Carlo runs)

21. Truncated Gaussian Processes • Motivation • Keeping all measurements up to time t requires large memory and computation time. • Most recent measurements are more informative. • Kernel function of spatio-temporal Kalman filter Difference in prediction error variance as a function of the truncation size. (Blue: ¾t2 = 5, red: ¾t2 = 10.)

22. Truncated Gaussian Processes Theorem [3]: