MCE- 2.6 Identification of Stochastic Hybrid System Models Shankar Sastry Sam Burden UC Berkeley

1. MCE-2.6 Identification of Stochastic Hybrid System Models Shankar Sastry Sam Burden UC Berkeley

2. MCE-2.6 Overview Identification of Stochastic Hybrid System Models • Sam Burden • PhD Candidate, UC Berkeley • Advised by Prof. Shankar Sastry • Collaborators: • Prof. Ronald Fearing (MCE, UC Berkeley) • Prof. Robert Full (MCE, UC Berkeley) • Prof. Daniel Goldman (MCE, GATech) • Goal: model reduction and system identification tools for hybrid models of terrestrial MAST platforms • Theorem: reduction of N-DOF polypeds to common 3-DOF model • Algorithm: scalable identification of detailed & reduced models

3. Technical Relevance:Dynamics of Terrestrial Locomotion OctoRoACH designed by Andrew Pullin, Prof. Ronald Fearing

4. Technical Relevance:Reduction and Identification of Polyped Dynamics physical system robot, animal system identification polyped models 10—100 DOF model reduction reduced model < 10 DOF

5. Reduction of Polyped Dynamics physical system robot, animal system identification polyped models 10—100 DOF model reduction reduced model < 10 DOF

6. Reduction of Polyped Dynamics m l, k, β • Detailed morphology • Multiple limbs • Multiple joints per limb • Mass in every limb segment • Precedence in literature • Multiple massless limbs (Kukillaya et al. 2009) • Realistic disturbances • Fractured terrain (Sponberg and Full 2008) • Granular media (Goldman et al. 2009) M, I M I m l : body mass : moment of inertia : leg mass : leg length k β : leg stiffness : leg damping

7. Reduction of Polyped Dynamics • Theorem:polypedmodel reduces to 3-DOF model • Let H = (D, F, G, R) be hybrid system with periodic orbit g • Then there exists reduced system (M, G) and embedding • Dynamics of H are approximated by (M, G) • (Burden, Revzen, Sastry2013 (in preparation) )

8. Identification of Polyped Dynamics physical system robot, animal system identification polyped models 10—100 DOF model reduction reduced model < 10 DOF

9. Identification of Polyped Dynamics • Mathematical models are necessarily approximations • Model parameters must be identified & validated using empirical data • Identification problem for hybrid system H = (D, F, G, R): • Challenging to solve for terrestrial locomotion: circulating limbs introduce nonlinearities in dynamics & transitions impact of limb with substrate introduces discontinuities in state

10. Lateral perturbation experiment real-time

11. Lateral perturbation experiment Platform accelerates laterally at 0.6 ± 0.1 g in a 0.1 sec interval providing a 50 ± 3 cm/sec specific impulse, then maintains velocity. camera diffuser mirror magnetic lock animal motion cart Cockroach running speed: 36 ± 8cm/sec Stride frequency: 12.6 ± 2.9 Hz (~80ms per stride) trackway cart motion pulley rail mass cable elastic ground Revzen, Burden, Moore, Mongeau, & Full, Biol. Cyber. (to appear) 2013

12. Lateral perturbation experiment Measured: • Heading, body orientation • Linear, rotational velocity • Distal tarsal (foot) position • Cart acceleration induces equal & opposite animal acceleration

13. Mechanical self-stabilization Animal Lateral Leg Spring (LLS) Schmitt & Holmes 2000 3 legs act as one • Cart acceleration induces equal & opposite animal acceleration • Apply measured acceleration directly to model Quantitative predictions for purely mechanical feedback

14. Mechanical self-stabilization Animal Lateral Leg Spring (LLS) Schmitt & Holmes 2000 • Apply measured acceleration directly to model • Cart acceleration induces equal & opposite animal acceleration Inertial Disc

15. Result: LLS Fits Recovery for >100ms Animal Inertial Disc Lateral Leg Spring (LLS)

16. Technical Accomplishments:Reduction and Identification of Polyped Dynamics physical system robot, animal system identification polyped models 10—100 DOF model reduction reduced model < 10 DOF