synchronization and dimensionality reduction in networks ......networkfrontiers 2015 7 (c) s. revzen...

NetworkFrontiers 2015(c) S. Revzen

Synchronization and Dimensionality Reduction in Networks of Hybrid Phase

Oscillators: A Perspective from Legged

Locomotion

ShaiRevzen

NetworkFrontiers 2015(c) S. Revzen

Outline Locomotion and dimensionality reduction

Data Driven Floquet Analysis

Gaits as networks of synchronized oscillators

Hybrid oscillators and synchronization

Horses, Dogs, and Robots

NetworkFrontiers 2015(c) S. Revzen

Outline Locomotion and dimensionality reduction

Estimating oscillator phase

Gaits as networks of synchronized oscillators

Legged locomotion → hybrid oscillators

Core results for hybrid oscillator networks

Data driven models of legged locomotion

Revzen & Kvalheim SPIE-DSS 2015

doi: 10.1117/12.2178007

(Section 1 of SPIE-DSS paper)

NetworkFrontiers 2015 4(c) S. Revzen

Locomotion: a bundle of fun Base space: body configuration

Fibre: 3D position and orientation SE(3)Periodic motions on the base space give rise to translation

along a subgroup of the SE(3) fibre

Bundle projection

State-space trajectoryInitial to Final state

Periodic body motions

Zero section / Base spaceof body configurations

Locomotion bundlewith fibre SE(3)

Computing Reduced Equations for Robotic Systems with Constraints and SymmetriesOstrowski; IEEE Tr. Robot. & Autom.; vol 15 no 1 pp 111, 1999

(Section 1 of SPIE-DSS paper)

NetworkFrontiers 2015 5(c) S. Revzen

ANCHOR

TEMPLATE

Collapse Dimensions

(synergies,symmetries)

Add Degrees of Freedom(legs,joints,muscles)

Templates and Anchors

Full & Koditschek (1999)

Animals have many DOF, but move “as if” they have only a few. Animals limit pose to a behaviourally relevant family of postures

(Section 1 of SPIE-DSS paper)

NetworkFrontiers 2015 6(c) S. Revzen

Oscillator theory & Templates Provable

reductions Data Driven

tools

Phillip Holmes, Princeton John Gukenheimer, Cornell

NetworkFrontiers 2015 7(c) S. Revzen

Template AnchorMathematical

Models

Seipel, Holmes and Full, 2004

Schmitt and Holmes, 2000

Schmitt, Garcia, Razo, Holmes and Full, 2004

Schmitt and Holmes, 2003

The Dynamics of Legged Locomotion: Models, Analyses, and Challenges Holmes, Full, Koditschek and Guckenheimer

SIAM Reviews, 2006, 48, 207-304

Dynamic Mechanical Models

Lateral Leg Spring+leg with damping,

muscle model +individual legs,neuron models +detailed animal

morphology

(Section 1 of SPIE-DSS paper)

NetworkFrontiers 2015 8(c) S. Revzen

Templates and Time Constants

1st template direction

2nd template direction

non-templatenon-template(pose error)(pose error)

directiondirection

Limit Cycle

Template withSlow recovery

Posture error,Fast recovery

State space

(c) 2006-2012 Shai Revzen 9

Is a cockroach an oscillator?

(c) 2006-2012 Shai Revzen 10

The Counterfactual Cockroach

(c) 2006-2012 Shai Revzen 11

Modeling a Cockroach as an Oscillator

Revzen, et.al. in “Progress in Motor Control” Sternad (ed.) (2008)

NetworkFrontiers 2015(c) S. Revzen

Outline Locomotion and dimensionality reduction

Data Driven Floquet Analysis

Gaits as networks of synchronized oscillators

Hybrid oscillators and synchronization

Horses, Dogs, and Robots

NetworkFrontiers 2015 13(c) S. Revzen

CorrectionSeries

Better Phase Estimation from Data Need an algorithm:

IN: multivariate data OUT: phase estimate

Traditional method: interpolate events times

“Phaser” (2008) accurate only on cycle

Multivariate data

proto-phase

proto-phase

Noiseestimates

CorrectionSeries

Correctionw/ Fourier

Series

Embedding

PCA

Correction w/ Fourier Series

proto-phase

Phase estimate

Revzen & Guckenheimer, Estimating the phase of synchronized oscillators, Phys Rev E, 2008, 78, 051907

NetworkFrontiers 2015(c) S. Revzen

Estimation error distribution

Revzen & Guckenheimer, Estimating the phase of synchronized oscillators, Phys Rev E, 2008, 78, 051907

NetworkFrontiers 2015 15(c) S. Revzen

Phaser-NG performance

(Collaboration with S.Wilshin, RVC; C. Scott, UMich; J.M.Guckenheimer, Cornell)

Event phase Phaser Phaser-NG

NetworkFrontiers 2015 16(c) S. Revzen

2D Performance of Phaser-NG

NetworkFrontiers 2015 17(c) S. Revzen

8D Performance of Phaser-NG

NetworkFrontiers 2015 18(c) S. Revzen

Floquet Theory (pub. 1883) A look at linear time periodic systems

Solutions split:

Therefore:

P() a T-(demi)periodic function by construction

(Section 2 of SPIE-DSS paper)

NetworkFrontiers 2015 19(c) S. Revzen

Floquet Theory for Oscillators Oscillators linearize via by a periodic change

of coordinates

q

j

F()

=F ⋅e1

2−

⋅F−1⋅

statespace

return map section

orbitbasis ofinvariant

subspaces of return map

invariant manifoldsassociated

with Floquetbasis

Poincare section

(Section 2 of SPIE-DSS paper)

NetworkFrontiers 2015 20(c) S. Revzen

Finding the Dimension of a Template

● Model: Coupled Van der Pol osc.

● SDE integrator

Finding the dimension of slow dynamics in a rhythmic system; Revzen & Guckenheimer; J. Roy. Soc. Interface 2012, 9, 957-971

NetworkFrontiers 2015 21(c) S. Revzen

Time constants by noise level

Finding the dimension of slow dynamics in a rhythmic system; Revzen & Guckenheimer; J. Roy. Soc. Interface 2012, 9, 957-971

=0.09

=0.03

=0(no noise)

=0.01

NetworkFrontiers 2015 22(c) S. Revzen

Dimension of Cockroach Template

Finding the dimension of slow dynamics in a rhythmic system; Revzen & Guckenheimer; J. Roy. Soc. Interface 2012, 9, 957-971

NetworkFrontiers 2015 23(c) S. Revzen

Phase-driven models

from Revzen (2009)

NetworkFrontiers 2015 24(c) S. Revzen

Phase from one (horse) leg

NetworkFrontiers 2015 25(c) S. Revzen

Horse Gaits in terms of Phase

NetworkFrontiers 2015(c) S. Revzen

Outline Locomotion and dimensionality reduction

Data Driven Floquet Analysis

Gaits as networks of synchronized oscillators

Hybrid oscillators and synchronization

Horses, Dogs, and Robots

NetworkFrontiers 2015(c) S. Revzen

Phase and multiple oscillators

Each oscillator contributes a “circle”

Combined (product) space is a torus Tori are cubes with faces identified Tori are “flat” and allow for “translation”

T1

T2T3

NetworkFrontiers 2015(c) S. Revzen

Dog on flat terrain

walkflat (r)

trotflat (g)

gallopflat (b)

Ideal trot is greencircled point (s)

Ideal walk~ red circle pace

[p p p]

[0 p 0]

[p/2 p 3p/4]

trotflat (g)

NetworkFrontiers 2015(c) S. Revzen

Dog on rough terrain

Ideal walk

[0 p 0]

[p/2 p 3p/4]

walkrough (c)

galloprough (y)

trotrough (p)

NetworkFrontiers 2015(c) S. Revzen

Outline Locomotion and dimensionality reduction

Data Driven Floquet Analysis

Gaits as networks of synchronized oscillators

Hybrid oscillators and synchronization

Horses, Dogs, and Robots

NetworkFrontiers 2015 31(c) S. Revzen

Hybrid System as Linked Domains State-space is disjoint

union of domains

System “leaves” a domain at a guard

Carried by reset map into new domain

NOTE: dimension can change

(Section 3 of SPIE-DSS paper)

Burden, Revzen & Sastry, “Model Reduction Near Periodic Orbits of Hybrid Dynamical Systems” IEEE TAC 2015

NetworkFrontiers 2015 32(c) S. Revzen

RESULT: has invariant sub “manifold” Hybrid oscillators can

have finite time (irreversible) transients

Collapse to lower dimensional dynamics

natural “template”structure?

(Section 3 of SPIE-DSS paper)

Burden, Revzen & Sastry, “Model Reduction Near Periodic Orbits of Hybrid Dynamical Systems” IEEE TAC 2015

NetworkFrontiers 2015 33(c) S. Revzen

RESULT: the rest is “smoothable” Can be “stitched

together” into a smooth manifold

No uniquely hybrid behaviors exist!

(asymptotically)

(Section 3 of SPIE-DSS paper)

Burden, Revzen & Sastry, “Model Reduction Near Periodic Orbits of Hybrid Dynamical Systems” IEEE TAC 2015

NetworkFrontiers 2015 34(c) S. Revzen

Isolated Transitions ResultsFinite time arrival on an invariant sub-manifold

“Generic”: nearby systems are super-exponential

Sub-manifold is “smoothable”

(Section 3 of SPIE-DSS paper)

NetworkFrontiers 2015(c) S. Revzen

What do these gaits have in common?

(from TED talk, R.J.Full)

(Section 3 of SPIE-DSS paper)

(c) S. Revzen

Hybrid Systems as Piecewise Smooth State space is partitioned into domains

Domain boundaries are codim 1 manifolds

Flow is smooth in each domain

We define: Event-

Selected Systems

Vector fieldmonotone & transverseto guards

transitionmanifold

trajectory

velocityat a point

(Section 3 of SPIE-DSS paper)

(c) S. Revzen

Simultaneous Hybrid Transitions

Touchdown #1

Touchdown #2

Multiple Contact

(Section 3 of SPIE-DSS paper)

NetworkFrontiers 2015 38(c) S. Revzen

Multiple Contact Gaits

Rewrite with respect to touchdown times

Domains Dq are now indexed by {-1,+1}d

NetworkFrontiers 2015 39(c) S. Revzen

Whatever is new about dynamics appears in (nearly) piecewise constant systems

Multiple Contact Gaits

NetworkFrontiers 2015 40(c) S. Revzen

First order approximations

NetworkFrontiers 2015 41(c) S. Revzen

First order approximations

A continuous “first order model”

NetworkFrontiers 2015 42(c) S. Revzen

Intersecting Transitions : Key Results (1)

Impact times are defined and PCr

A unique PCr flow exists and is B-differentiable

Flows are piecewise smooth conjugate to flow boxes

NetworkFrontiers 2015 43(c) S. Revzen

Intersecting Transitions : Key Results (2)The dynamics are structurally stable

in the product smooth function topology

NetworkFrontiers 2015(c) S. Revzen

Outline Locomotion and dimensionality reduction

Data Driven Floquet Analysis

Gaits as networks of synchronized oscillators

Hybrid oscillators and synchronization

Horses, Dogs, and Robots

NetworkFrontiers 2015 45(c) S. Revzen

Independent touchdown,Synchronized liftoff

NetworkFrontiers 2015 46(c) S. Revzen

Horse trotting

NetworkFrontiers 2015 47(c) S. Revzen

Horses Local Lyapunov Exponent

NetworkFrontiers 2015 48(c) S. Revzen

Dog trotting

NetworkFrontiers 2015 49(c) S. Revzen

Dog Local Lyapunov Exponent

NetworkFrontiers 2015 50(c) S. Revzen

Comparison Horses vs. Dogs

NetworkFrontiers 2015 51(c) S. Revzen

Synchronizing a tripod(Section 3 of SPIE-DSS paper)

NetworkFrontiers 2015 52(c) S. Revzen

Controlling a RHex XRL robot has 6 legs

Gait is alternating tripods of support Need to synchronize tripods in antiphase

NetworkFrontiers 2015 53(c) S. Revzen

Piecewise Constant Control

NetworkFrontiers 2015 54(c) S. Revzen

Conclusions & Thanks Oscillator networks can model animal motion

We have tools to analyze such system models

Event Selected hybrid oscillator networks: Exhibit new forms of robust stability Do not have new bifurcations / qualitative dynamics

ARO #W911NF-14-1-0573 Morphologically Modulated Dynamics (Revzen);ARL MAST (Burden)

Sam Burden

http://purl.org/sburden/ECr-Yields-PCrhttp://purl.org/sburden/Hybrid-Oscillator-Reduction