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NetworkFrontiers 2015(c) S. Revzen
Synchronization and Dimensionality Reduction in Networks of Hybrid Phase
Oscillators: A Perspective from Legged
Locomotion
ShaiRevzen
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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
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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)
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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)
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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)
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NetworkFrontiers 2015 6(c) S. Revzen
Oscillator theory & Templates Provable
reductions Data Driven
tools
Phillip Holmes, Princeton John Gukenheimer, Cornell
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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)
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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
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(c) 2006-2012 Shai Revzen 9
Is a cockroach an oscillator?
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(c) 2006-2012 Shai Revzen 10
The Counterfactual Cockroach
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(c) 2006-2012 Shai Revzen 11
Modeling a Cockroach as an Oscillator
Revzen, et.al. in “Progress in Motor Control” Sternad (ed.) (2008)
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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
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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
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NetworkFrontiers 2015(c) S. Revzen
Estimation error distribution
Revzen & Guckenheimer, Estimating the phase of synchronized oscillators, Phys Rev E, 2008, 78, 051907
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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
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NetworkFrontiers 2015 16(c) S. Revzen
2D Performance of Phaser-NG
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NetworkFrontiers 2015 17(c) S. Revzen
8D Performance of Phaser-NG
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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)
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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)
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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
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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
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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
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NetworkFrontiers 2015 23(c) S. Revzen
Phase-driven models
from Revzen (2009)
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NetworkFrontiers 2015 24(c) S. Revzen
Phase from one (horse) leg
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NetworkFrontiers 2015 25(c) S. Revzen
Horse Gaits in terms of Phase
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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
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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
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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)
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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)
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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
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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
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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
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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
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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)
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NetworkFrontiers 2015(c) S. Revzen
What do these gaits have in common?
(from TED talk, R.J.Full)
(Section 3 of SPIE-DSS paper)
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(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)
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(c) S. Revzen
Simultaneous Hybrid Transitions
Touchdown #1
Touchdown #2
Multiple Contact
(Section 3 of SPIE-DSS paper)
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NetworkFrontiers 2015 38(c) S. Revzen
Multiple Contact Gaits
Rewrite with respect to touchdown times
Domains Dq are now indexed by {-1,+1}d
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NetworkFrontiers 2015 39(c) S. Revzen
Whatever is new about dynamics appears in (nearly) piecewise constant systems
Multiple Contact Gaits
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NetworkFrontiers 2015 40(c) S. Revzen
First order approximations
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NetworkFrontiers 2015 41(c) S. Revzen
First order approximations
A continuous “first order model”
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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
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NetworkFrontiers 2015 43(c) S. Revzen
Intersecting Transitions : Key Results (2)The dynamics are structurally stable
in the product smooth function topology
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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
![Page 45: Synchronization and Dimensionality Reduction in Networks ......NetworkFrontiers 2015 7 (c) S. Revzen Template Anchor Mathematical Models Seipel, Holmes and Full, 2004 Schmitt and Holmes,](https://reader033.vdocument.in/reader033/viewer/2022050601/5fa8de222f5eff08ac19fcf9/html5/thumbnails/45.jpg)
NetworkFrontiers 2015 45(c) S. Revzen
Independent touchdown,Synchronized liftoff
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NetworkFrontiers 2015 46(c) S. Revzen
Horse trotting
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NetworkFrontiers 2015 47(c) S. Revzen
Horses Local Lyapunov Exponent
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NetworkFrontiers 2015 48(c) S. Revzen
Dog trotting
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NetworkFrontiers 2015 49(c) S. Revzen
Dog Local Lyapunov Exponent
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NetworkFrontiers 2015 50(c) S. Revzen
Comparison Horses vs. Dogs
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NetworkFrontiers 2015 51(c) S. Revzen
Synchronizing a tripod(Section 3 of SPIE-DSS paper)
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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
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NetworkFrontiers 2015 53(c) S. Revzen
Piecewise Constant Control
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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