1

AM3 Task 1.4

Stochastic Hybrid Models for Aerial and Ground Vehicles

Sam BurdenMAST Annual ReviewUniversity of PennsylvaniaMarch 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 (http://eecs.berkeley.edu/~sburden/python)– 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-10-1.4a): 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 http://eecs.berkeley.edu/~sburden/python

• 2Q10 (A-10-1.4b): Hybrid System Identification– Creating geometric framework for ID problem– Applying framework to abstract mathematical models– Collecting trajectory data– Implementing estimation and identification

• 3Q10 (A-10-1.4c): Characterize System Noise• 4Q10 (A-10-1.4d): 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 systems1 H := (Q, D, F, R):

1. Bernardo et. al. 2007

• 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: