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

2. MCE-2.6 2 MCE-2.6 Overview 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 Identification of Stochastic Hybrid System Models

3. MCE-2.6 3 Technical Relevance: Dynamics of Terrestrial Locomotion OctoRoACH designed by Andrew Pullin, Prof. Ronald Fearing

4. MCE-2.6 4 physical system robot, animal polyped models 10—100 DOF reduced model < 10 DOF system identification model reduction Technical Relevance: Reduction and Identification of Polyped Dynamics

5. MCE-2.6 5 physical system robot, animal polyped models 10—100 DOF reduced model < 10 DOF system identification model reduction Reduction of Polyped Dynamics

6. MCE-2.6 6 Reduction of Polyped Dynamics 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) : body mass : moment of inertia : leg mass : leg length MImlMIml : leg stiffness : leg damping kβkβ m M, I l, k, β

7. MCE-2.6 7 Reduction of Polyped Dynamics Theorem: polyped model reduces to 3-DOF model – Let H = (D, F, G, R) be hybrid system with periodic orbit  – Then there exists reduced system (M, G) and embedding – Dynamics of H are approximated by (M, G) – (Burden, Revzen, Sastry 2013 (in preparation) )

8. MCE-2.6 8 physical system robot, animal polyped models 10—100 DOF reduced model < 10 DOF system identification model reduction Identification of Polyped Dynamics

9. MCE-2.6 9 Identification of Polyped Dynamics a)circulating limbs introduce nonlinearities in dynamics & transitions b)impact of limb with substrate introduces discontinuities in state 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:

10. MCE-2.6 10 Lateral perturbation experiment real-time

11. MCE-2.6 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. Cockroach running speed: 36 ± 8 cm/sec Stride frequency: 12.6 ± 2.9 Hz ( ~ 80ms per stride) trackway camera diffuser mirror magnetic lock animal motion cart cart motion rail pulley mass cable elastic ground Revzen, Burden, Moore, Mongeau, & Full, Biol. Cyber. (to appear) 2013

12. MCE-2.6 12 Lateral perturbation experiment Measured: 1.Heading, body orientation 2.Linear, rotational velocity 3.Distal tarsal (foot) position Cart acceleration induces equal & opposite animal acceleration

13. MCE-2.6 13 3 legs act as one Mechanical self-stabilization Cart acceleration induces equal & opposite animal acceleration AnimalLateral Leg Spring (LLS) Apply measured acceleration directly to model Quantitative predictions for purely mechanical feedback Schmitt & Holmes 2000

14. MCE-2.6 14 Lateral Leg Spring (LLS) Apply measured acceleration directly to model Inertial Disc Cart acceleration induces equal & opposite animal acceleration Animal Schmitt & Holmes 2000 Mechanical self-stabilization

15. MCE-2.6 15 Result: LLS Fits Recovery for >100ms AnimalInertial DiscLateral Leg Spring (LLS)

16. MCE-2.6 16 physical system robot, animal polyped models 10—100 DOF reduced model < 10 DOF system identification model reduction Technical Accomplishments: Reduction and Identification of Polyped Dynamics