1 AM3 Task 1.4 Stochastic Hybrid Models for Aerial and Ground Vehicles Sam Burden MAST Annual Review University of Pennsylvania March 8-9, 2010

2 Overview Sam Burden – Ph.D. Student, UC Berkeley – Advised by Prof. Shankar Sastry Collaborators: – Aaron Hoover (Micromechanics, UC Berkeley) – Prof. Robert Full (Micromechanics, UC Berkeley) – Dr. Shai Revzen (Autonomy, UPenn) Goal: develop general principles for hybrid system identification, apply to MAST platforms – High-fidelity 3D motion capture system – Theoretical principles for identification of stochastic hybrid systems – Practical implementation of ID technique to model MAST platforms – Apply model to aid control and improve design of the platforms Expected Results at End of Fiscal Year: model for terrestrial platform, controller to execute beacon following behavior Stochastic Hybrid Models for Aerial and Ground Vehicles

3 Technical Relevance High-fidelity 3D Motion Capture – Camera calibration via bundle adjustment (Argyros, 2009) – Free / open source software written in Python System Identification – Linear systems (Ljung, 1987) – Continuum dynamics / PDEs (Tomlin, 2006) – Piecewise affine hybrid systems (Niessen, 2005) – Stochastic hybrid systems (Lygeros, 2008) State Estimation – Linear systems (Kalman Filter; Rauch, Tung, Striebel, 1965) – Nonlinear Systems (Unscented K.F.; Julier and Uhlmann, 1995) We are using nonlinear geometric theory to identify stochastic hybrid dynamics

4 Relevance to MAST Experimental tools ( ) – High-fidelity 3D motion capture – State estimation for stochastic nonlinear and hybrid systems Collecting kinematic data is hard; we’ll make it easier Theoretical / Modeling tools – Unified theoretical framework for studying stochastic nonlinear and hybrid systems – System identification in this general framework Identification is hard; we’ve simplified it Experimental outcomes – Develop empirically-validated model for terrestrial platform – Execute useful low-level behavior, e.g. beacon following

5 Technical Accomplishments 1Q10 (A a): Experimentation – Evaluated fidelity of VICON; inadequate for present application – Developed high-speed camera calibration software suite – Implemented nonlinear state estimation software suite – Tools available at 2Q10 (A b): Hybrid System Identification – Creating geometric framework for ID problem – Applying framework to abstract mathematical models – Collecting trajectory data – Implementing estimation and identification 3Q10 (A c): Characterize System Noise 4Q10 (A d): Control MAST Platform

6 HS: Intuitive Picture Natural abstraction for running, flapping, & climbing robots

7 HS ID: Illustrative Example Inelastic Bouncing Ball Velocity vector `jumps’ discontinuously Kalman filter, particle filter will fail to estimate state

8 HS ID: Illustrative Example Inelastic Bouncing Ball How can we identify the dynamics? We can solve this problem in general – i.e. for running, flapping, climbing robots

9 Collaborations Weekly Meetings – Aaron Hoover (Prof. Ronald Fearing, Micromechanics, UC Berkeley) Development of terrestrial platform Experimental design and execution – Prof. Robert Full (Micromechanics, UC Berkeley) Experiment design and resources (force platform, HS cameras) – Dr. Shai Revzen (Prof. George Pappas, Autonomy, UPenn) Software for motion capture and state estimation Theory for hybrid system identification Monthly Meetings: Berkeley MAST Group – Autonomy, Micromechanics, Integration, Microelectronics Possible new collaborative efforts / experiments – Identification for other MAST platforms with hybrid dynamics – Implement controllers, improve design using identified dynamics

10 Future Plans 3Q—4Q 2010: Plans – Develop, validate open-loop model for terrestrial robot – Characterize dynamical uncertainty with respect to model – Design control scheme using stochastic hybrid model 2011—2013: Ideas, Goals – Improve estimation and identification for hybrid systems using e.g. particle filters in abstract geometric spaces – Model the effect of varying terrain and morphology esp. as a means to decrease dynamical uncertainty – Closed-loop model for robot dynamics by explicitly considering dynamical effect of control effort

11 Discussion & Questions Metrics – Presenting Hybrid System ID work at HSCC in April, 2010 (HSCC: Hybrid Systems, Computation, and Control) Thank you for your time Collaborators – Aaron Hoover (Prof. Ronald Fearing) – Prof. Robert Full – Dr. Shai Revzen (Prof. George Pappas) Support – ARL MAST (Autonomy Center) – NSF GRF – Prof. Shankar Sastry Acknowledgements

12 Technical Slides Hybrid System Formal Framework HS Identification Problem Statement HS ID Intuitive Example HS ID Recap

13 HS: Formal Framework Consider hybrid dynamical systems 1 H := (Q, D, F, R) : 1. Bernardo et. al Nice properties – Determinism – Existence & Uniqueness – Structural Stability

14 Given output from the discrete-time stochastic model estimate the state. HS ID: Problem Statement

15 HS ID: Illustrative Example Inelastic Bouncing Ball Velocity is discontinuous when ball bounces Kalman filter, Particle filter will give poor estimates after the bounce We can solve this problem We can identify general hybrid dynamical systems

16 HS ID: Recap We consider a general class of hybrid systems We wish to estimate the state in the presence of uncertainty and noisy measurements We can solve this problem in general – i.e. for walking, flapping, climbing robots ex: