Feedback Particle Filter and its Application to Coupled...

Feedback Particle Filterand its Application to Coupled Oscillators

Presentation atUniversity of Maryland, College Park, MD

Prashant Mehta

Dept. of Mechanical Science and Engineeringand the Coordinated Science Laboratory

University of Illinois at Urbana-Champaign

May 1, 2015

Bayesian Inference/FilteringMathematics of prediction: Bayes’ rule

Signal (hidden): X X ∼ P(X), (prior, known)

Feedback Particle Filter Prashant Mehta 2 / 34 Prashant Mehta



Observation: Y (known)





Observation model: P(Y|X) (known)






Problem: What is X ?







Solution

Bayes’ rule: P(X|Y)︸︷︷︸Posterior

∝ P(Y|X)P(X)︸︷︷︸Prior







Solution

Bayes’ rule: P(X|Y)︸︷︷︸Posterior

∝ P(Y|X)P(X)︸︷︷︸Prior

This talk is about implementing Bayes’ rule indynamic, nonlinear, non-Gaussian settings!


ApplicationsTarget state estimation








ApplicationsBayesian model of sensory signal processing


Nonlinear FilteringMathematical Problem

Signal model: dXt = a(Xt)dt+ dBt, X0 ∼ p∗0(·)

Posterior is an information state

P(Xt ∈ A|Z t) =∫

Ap∗(x, t)dx

E(Xt|Z t) =∫R

xp∗(x, t)dx

A. Bain and D. Crisan, Fundamentals of Stochastic Filtering. Springer, 2010.




Observation model: dZt = h(Xt)dt+ dWt


P(Xt ∈ A|Z t) =∫

Ap∗(x, t)dx

E(Xt|Z t) =∫R

xp∗(x, t)dx






Problem: What is Xt ? given obs. till time t =: Z t


P(Xt ∈ A|Z t) =∫

Ap∗(x, t)dx

E(Xt|Z t) =∫R

xp∗(x, t)dx







Answer in terms of posterior: P(Xt|Z t) =: p∗(x, t).


P(Xt ∈ A|Z t) =∫

Ap∗(x, t)dx

E(Xt|Z t) =∫R

xp∗(x, t)dx









P(Xt ∈ A|Z t) =∫

Ap∗(x, t)dx

E(Xt|Z t) =∫R

xp∗(x, t)dx









P(Xt ∈ A|Z t) =∫

Ap∗(x, t)dx

E(Xt|Z t) =∫R

xp∗(x, t)dx



Kalman filterSolution in linear Gaussian settings

dXt = αXt dt+ dBt (1)

dZt = γXt dt+ dWt (2)

Kalman filter: p∗ = N(Xt,Σt)

dXt = αXt dt + K(dZt− γXt dt)︸︷︷︸Update

Kalman Filter

Observation: dZt = γXt dt+ dWt

Prediction: dZt = γXt dt

Innov. error: dIt = dZt− dZt= dZt− γXt dt

Control: dUt = K dIt

Gain: Kalman gain

R. E. Kalman. A new approach to linear filtering and prediction problems. J. Basic Eng. (1961);R. E. Kalman and R. S. Bucy. New Results in Liner Filtering and Prediction Theory. J. Basic Eng. (1961).




dZt = γXt dt+ dWt (2)

Kalman filter: p∗ = N(Xt,Σt)


Kalman Filter





Gain: Kalman gain





dZt = γXt dt+ dWt (2)Kalman filter: p∗ = N(Xt,Σt)


Kalman Filter

-

+

Kalman Filter





Gain: Kalman gain







Kalman Filter

-

+

Kalman Filter





Gain: Kalman gain







Kalman Filter

-

+

Kalman Filter





Gain: Kalman gain







Kalman Filter

-

+

Kalman Filter





Gain: Kalman gain







Kalman Filter

-

+

Kalman Filter





Gain: Kalman gain







Kalman Filter

-

+

Kalman Filter





Gain: Kalman gain



Kalman filter

dXt = αXt dt︸︷︷︸Prediction

+ K(dZt− γXt dt)︸︷︷︸Update

Simple enough to be included in the first undergraduate course on control!


Kalman filter



Kalman Filter

-

+



Kalman filter



Kalman Filter

-

+

This illustrates the key features of feedback control:

1 Use error to obtain control (dUt = K dIt)

2 Negative gain feedback serves to reduce error (K =γ

σ2W︸︷︷︸

SNR

Σt)



Pretty Formulae in MathematicsMore often than not, these are simply stated

Euler’s identity

eiπ =−1

Euler’s formula

v− e+ f = 2

Pythagoras theorem

x2 + y2 = z2

Kenneth Chang. What Makes an Equation Beautiful? in The New York Times on October 24, 2004


Filtering ProblemNonlinear Model: Kushner-Stratonovich PDE

Signal & Observations dXt = a(Xt)dt+ dBt, (1)

dZt = h(Xt)dt+ dWt (2)

Posterior distribution p∗ is a solution of a stochastic PDE:

dp∗ = L †(p∗)dt+1

σ2W(h− h)(dZt− hdt)p∗

where h = E[h(Xt)|Zt] =∫

h(x)p∗(x, t)dx

L †(p∗) =− ∂ (p∗ ·a(x))∂x

+12

∂ 2p∗

∂x2

R. L. Stratonovich. Conditional Markov Processes. Theory Probab. Appl. (1960);H. J. Kushner. On the differential equations satisfied by conditional probability densities of Markov processes. SIAM J. Control (1964).






dp∗ = L †(p∗)dt+1



h(x)p∗(x, t)dx

L †(p∗) =− ∂ (p∗ ·a(x))∂x

+12

∂ 2p∗

∂x2







dp∗ = L †(p∗)dt+1



h(x)p∗(x, t)dx

L †(p∗) =− ∂ (p∗ ·a(x))∂x

+12

∂ 2p∗

∂x2

No closed-form solution in general. Closure problem.



Particle FilterAn algorithm to solve nonlinear filtering problem

Approximate posterior in terms of particles p∗(x, t) =1N

N

∑i=1

δXit(x)

Algorithm outline

1 Initialization at time 0: Xi0 ∼ p∗0(·)

2 At each discrete time step:Importance sampling (Bayes update step)Resampling (for variance reduction)

J. E. Handschin and D. Q. Mayne. Monte-Carlo techniques to estimate conditional expectation in nonlinear filtering. Int. J. Control (1969);N. Gordon, D. Salmond, A. Smith. Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE Proc. F Radar Signal Process (1993);J. Xiong. Particle approximation to filtering problems in continuous time. The Oxford handbook of nonlinear filtering (2011);A. Budhiraja, L. Chen, C. Lee. A survey of numerical methods for nonlinear filtering problems. Physica D (2007).




N

∑i=1

δXit(x)

Algorithm outline







N

∑i=1

δXit(x)

Algorithm outline







N

∑i=1

δXit(x)

Algorithm outline



e.g. dZt = Xt dt+ small noise





N

∑i=1

δXit(x)

Algorithm outline







N

∑i=1

δXit(x)

Algorithm outline







N

∑i=1

δXit(x)

Algorithm outline



Innovation error, feedback? And most importantly, is this pretty?



Feedback Particle FilterA control-oriented approach

Signal & Observations dXt = a(Xt)dt+ dBt (1)


Motivation: Work of Huang, Caines and Malhame on Mean-field games (IEEE TAC 2007).Related approaches: D. Crisan and J. Xiong, Approximate McKean-Vlasov representations for a class of SPDEs. Stochastics (2009);S. K. Mitter and N. J. Newton. A variational approach to nonlinear estimation. SIAM J. Control Optimiz. (2003);F. Daum and J. Huang. Generalized particle flow for nonlinear filters. Proc. SPIE (2010);S. Reich, A dynamical systems framework for intermittent data assimilation. BIT Numer. Math. (2011).





Controlled system (N particles):

dXit = a(Xi

t)dt+ dBit + dUi

t︸︷︷︸mean-field control

, i = 1, ...,N (3)

BitN

i=1 are ind. standard white noises.






Controlled system (N particles):

dXit = a(Xi

t)dt+ dBit + dUi

t︸︷︷︸mean-field control

, i = 1, ...,N (3)

BitN

i=1 are ind. standard white noises.

Variational approach:

1. Gradient flow construction:

Nonlinear filter is shown to be a gradient flow (steepest descent)

2. Optimal transport:

Derivation of the feedback particle filter



Update StepHow does feedback particle filter implement Bayes’ rule?

Feedback particle filter Kalman filter

Observation: dZt = h(Xt)dt+ dWt dZt = γXt dt+ dWt

Prediction: dZit =

h(Xit )+h2 dt dZt = γXt dt

h = 1N ∑

Ni=1 h(Xi

t)

Innov. error: dIit = dZt− dZi

t dIt = dZt− dZt

= dZt− h(Xit )+h2 dt = dZt− γXt dt

Control: dUit = K(Xi

t) dIit dUt = K dIt

Gain: K is a solution of a linear BVP K is the Kalman gain

Main Result: FPF is an exact algorithm (in the mean-field, N→ ∞, limit).

Yang, Mehta and Meyn. Feedback Particle Filter. IEEE TAC (2013).





Prediction: dZit =


h = 1N ∑

Ni=1 h(Xi

t)


t dIt = dZt− dZt



t) dIit dUt = K dIt








Prediction: dZit =


h = 1N ∑

Ni=1 h(Xi

t)


t dIt = dZt− dZt



t) dIit dUt = K dIt








Prediction: dZit =


h = 1N ∑

Ni=1 h(Xi

t)


t dIt = dZt− dZt



t) dIit dUt = K dIt








Prediction: dZit =


h = 1N ∑

Ni=1 h(Xi

t)


t dIt = dZt− dZt



t) dIit dUt = K dIt





Variance ReductionFiltering for a linear model.

Mean-square error:1T

∫ T

0

(Σ(N)t −Σt

Σt

)2

dt

102 103

10−3

10−2

10−1

N (number of particles)

Bootstrap (BPF)

Feedback (FPF)

MSE


The Next Few SlidesResults 1 and 2


1 Nonlinear filter is shown to be a gradient flow


1 Derivation of feedback particle filter

Poisson’s equation is central to both 1 and 2

Details appear in: Laugesen, Mehta, Meyn and Raginsky. Poisson’s equation in nonlinear filtering. SIAM J. Control Optimiz. (2015);Also see: Yang, Mehta and Meyn. Feedback particle filter. IEEE Trans. Automat. Control (2013);Yang, Laugesen, Mehta and Meyn. Multivariable feedback particle filter. Automatica (To Appear).


The Next Few SlidesResults 1 and 2


1 Nonlinear filter is shown to be a gradient flow


1 Derivation of feedback particle filter

Poisson’s equation is central to both 1 and 2

Details appear in: Laugesen, Mehta, Meyn and Raginsky. Poisson’s equation in nonlinear filtering. SIAM J. Control Optimiz. (2015);Also see: Yang, Mehta and Meyn. Feedback particle filter. IEEE Trans. Automat. Control (2013);Yang, Laugesen, Mehta and Meyn. Multivariable feedback particle filter. Automatica (To Appear).


Poisson’s EquationReview of various mathematical forms

Strong form: − ∇ ·∇︸︷︷︸Laplacian

.= ∇2

φ(x) = h(x)− c on domain Ω




.= ∇2


Weak form:∫

∇φ ·∇ψ dx =∫(h− c)ψ dx ∀ test fns. ψ ∈ H1




.= ∇2


Weak form:∫


Generalization:∫

∇φ ·∇ψ p(x)dx =∫(h− c)ψ p(x)dx ∀ψ ∈ H1




.= ∇2


Weak form:∫


Generalization:∫


or: Ep[∇φ ·∇ψ] = Ep[(h− h)ψ] ∀ψ ∈ H1




.= ∇2


Weak form:∫


Generalization:∫


or: Ep[∇φ ·∇ψ] = Ep[(h− h)ψ] ∀ψ ∈ H1

where h = Ep[h] =∫

h(x)p(x)dx




.= ∇2


Weak form:∫


Generalization:∫


or: Ep[∇φ ·∇ψ] = Ep[(h− h)ψ] ∀ψ ∈ H1

where h = Ep[h] =∫

h(x)p(x)dx

Strong form: −∇ · (p(x)∇φ)(x) = (h(x)− c)p(x)


Poisson’s Equation in PhysicsThis equation is fundamental to many fields!

Electric potential: − 14π

∇2φ = ρ ← charge density

Gravitational potential:1

4πG∇

2φ = ρ ←mass density

Temperature: κ∇2φ = q← heat-flux density

Walter Strauss. Partial Differential Equations. Wiley (1992).


Gradient flowAn elementary example

Time stepping procedure

x∗(t+∆t) = arg miny

12|y− x∗(t)|2 +∆t h(y)





12|y− x∗(t)|2 +∆t h(y)

Calculus 101: x∗(t+∆t) = x∗(t)−∆t ∇h(x∗(t+∆t))





12|y− x∗(t)|2 +∆t h(y)

Calculus 101: x∗(t+∆t) = x∗(t)−∆t ∇h(x∗(t+∆t))

Cont. limit:dx∗

dt=−∇h(x∗).





12|y− x∗(t)|2︸︷︷︸

metric

+ ∆t h(y)︸︷︷︸min





12|y− x∗(t)|2︸︷︷︸

metric

+ ∆t h(y)︸︷︷︸min

Calc. of Variation: 〈x∗(t+∆t),ψ〉= 〈x∗(t),ψ〉−∆t 〈∇h(x∗(t+∆t)),ψ〉





12|y− x∗(t)|2︸︷︷︸

metric

+ ∆t h(y)︸︷︷︸min

Calc. of Variation: 〈x∗(t+∆t),ψ〉= 〈x∗(t),ψ〉−∆t 〈∇h(x∗(t+∆t)),ψ〉

Cont. limit: 〈x∗(t),ψ〉= 〈x∗(0),ψ〉−∫ t

0〈∇h(x∗(s)),ψ〉ds.


Gradient Flow Interpretation of Heat Equation1998 paper of Jordon, Kinderlehrer and Otto


p∗t+∆t = arg minρ∈P

12

W22 (ρ,p

∗t )+∆t

∫ρ(x) lnρ(x)dx

Jordon, Kinderlehrer and Otto. The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal., 29 (1998)





12

W22 (ρ,p

∗t )+∆t

∫ρ(x) lnρ(x)dx

E-L equation: . . .






12

W22 (ρ,p

∗t )+∆t

∫ρ(x) lnρ(x)dx

E-L equation: . . .

Cont. limit:∂p∗

∂ t= ∇

2p∗



Gradient Flow for Nonlinear FilterConstruction via a time-stepping procedure

Signal & Observations dXt = 0,

dZt = h(Xt)dt+ dWt



D(ρ | p∗t )+∆t2

∫ρ(x)(Yt−h(x))2 dx,

where Yt :=Zt+∆t−Zt

∆t.

Laugesen, Mehta, Meyn and Raginsky. Poisson’s Equation in Nonlinear Filtering. SIAM J. Control Optim (2015)


Gradient Flow Interpretation of Nonlinear FilterConstruction via a time-stepping procedure



D(ρ | p∗t )+∆t2


S. K. Mitter and N. J. Newton. A variational approach to nonlinear estimation. SIAM J. Control Optimiz. (2003);





D(ρ | p∗t )+∆t2


E-L equation: . . .






D(ρ | p∗t )+∆t2


E-L equation: . . .

Cont. limit: dp∗ = (h− h)(dZt− hdt)p∗



Gradient Flow Interpretation of Nonlinear FilterPoisson’s equation?

E-L equation: Ep∗t+∆t[ψ] = Ep∗t [ψ]+Ep∗t+∆t

[(∆Zt−h∆t)∇h ·∇ς ]




[(∆Zt−h∆t)∇h ·∇ς ]

Poisson’s equation: ∇ · (p∗t (x)∇ς(x)) =−(ψ(x)− ψt)p∗t (x)




[(∆Zt−h∆t)∇h ·∇ς ]

Poisson’s equation: ∇ · (p∗t (x)∇ς(x)) =−(ψ(x)− ψt)p∗t (x)

Assumption: Spectral gap

For some λ0 > 0, and for all functions ψ ∈ H1 with Ep∗0[ψ] = 0,

∫|ψ(x)|2p∗0(x)dx≤ 1

λ0

∫|∇ψ(x)|2p∗0(x)dx. [PI(λ0)]


Gradient Flow Interpretation of Nonlinear FilterResult 1: Derivation of nonlinear filter

Result 1

Certain Technical conditions. The density p∗ is a weak solution of the nonlinear filter with priorp∗0. That is, for any test function ψ ∈ Cc(Rd),

〈ψ,p∗t 〉= 〈ψ,p∗0〉+∫ t

0〈(h− hs)(dZs− hs ds)ψ,p∗s 〉,

where 〈ψ,p∗t 〉.=∫

ψ(x)p∗(x, t)dx.


Optimal TransportResult 2: Derivation of feedback particle filter

Optimization problem

J(N)(s∗) .= min

s ∑tn

(Itn (stn

#(p∗tn ))−∆t2

Y2tn

),




J(N)(s∗) .= min

s ∑tn

(Itn (stn

#(p∗tn ))−∆t2

Y2tn

),

It(ρ).= D(ρ | p∗t )+

∆t2

∫ρ(x)(Yt−h(x))2 dx




J(N)(s∗) .= min

s ∑tn

(Itn (stn

#(p∗tn ))−∆t2

Y2tn

),

It(ρ).= D(ρ | p∗t )+

∆t2


st# : Optimal transport




J(N)(s∗) .= min

s ∑tn

(Itn (stn

#(p∗tn ))−∆t2

Y2tn

),

It(ρ).= D(ρ | p∗t )+

∆t2


st# : Optimal transport

st : dXit = u(Xi

t , t)dt+K(Xit , t)dZt


Feedback Particle FilterAlgorithm summary

Signal: dXt = a(Xt)dt+ dBt

Observations: dZt = h(Xt)dt+ dWt





Problem: Approximate the posterior distribution p∗(x, t).





Problem: Approximate the posterior distribution p∗(x, t).

FPF Algo.: dXit = a(Xi

t)dt+ dBit

+K(Xi, t)(

dZt−12(h(Xi

t)+ ht)dt)

︸︷︷︸FPF control


Boundary Value ProblemEuler-Lagrange equation for the variational problem

Multi-dimensional boundary value problem

Gain Fn.: K = ∇φ

∇ · (p∇φ) =−(h− h)p

solved at each time-step.




Gain Fn.: K = ∇φ

∇ · (p∇φ) =−(h− h)p


Linear case:




Gain Fn.: K = ∇φ

∇ · (p∇φ) =−(h− h)p


Linear case:




Gain Fn.: K = ∇φ

∇ · (p∇φ) =−(h− h)p


Linear case: Nonlinear case:




Gain Fn.: K = ∇φ

∇ · (p∇φ) =−(h− h)p






Gain Fn.: K = ∇φ

∇ · (p∇φ) =−(h− h)p




Summary

Kalman Filter

Kalman Filter

-

+

Innovation Error:

dIt = dZt−h(X)dt

Gain Function:

K = Kalman Gain

Feedback Particle Filter


-

+

Innovation Error:

dIit = dZt−

12(h(Xi

t)+ ht)

dt

Gain Function:

K is solution of a linear BVP.



Summary

Kalman Filter

Kalman Filter

-

+

Innovation Error:

dIt = dZt−h(X)dt

Gain Function:

K = Kalman Gain



-

+

Innovation Error:

dIit = dZt−

12(h(Xi

t)+ ht)

dt

Gain Function:




Summary

Kalman Filter

Kalman Filter

-

+

Innovation Error:

dIt = dZt−h(X)dt

Gain Function:

K = Kalman Gain



-

+

Innovation Error:

dIit = dZt−

12(h(Xi

t)+ ht)

dt

Gain Function:




Coupled OscillatorsKuramoto model

dθit =

(ωi +

κ

N

N

∑j=1

sin(θ jt −θ

it )

)dt+σ dξ

it , i = 1, . . . ,N

ωi taken from distribution g(ω) over [1− γ,1+ γ]

γ — measures the heterogeneity of the population

κ — measures the strength of coupling

Y. Kuramoto. Self-entrainment of a population of coupled nonlinear oscillators (1975);Strogatz and Mirollo. Stability of incoherence in a population of coupled oscillators. J. Stat. Phy. (1991);N. Kopell and G. B. Ermentrout. Symmetry and phaselocking in chains of weakly coupled oscillators. Commun. Pure Appl. Math. (1986)



dθit =

(ωi +

κ

N

N

∑j=1

sin(θ jt −θ

it )

)dt+σ dξ

it , i = 1, . . . ,N



κ — measures the strength of coupling 1- 1+1




dθit =

(ωi +

κ

N

N

∑j=1

sin(θ jt −θ

it )

)dt+σ dξ

it , i = 1, . . . ,N







dθit =

(ωi +

κ

N

N

∑j=1

sin(θ jt −θ

it )

)dt+σ dξ

it , i = 1, . . . ,N




0 0.1 0.20.1

0.2

0.3

Incoherence

Synchrony



Hodgkin-Huxley type Neuron modelNormal form reduction

CdVdt

=−gT ·m2∞(V) ·h · (V−ET )

−gh · r · (V−Eh)− . . . . . .

dhdt

=h∞(V)−h

τh(V)

drdt

=r∞(V)− r

τr(V)2000 2200 2400 2600 2800 3000 3200 3400 3600 3800 4000

−150

−100

−50

0

50

100

Voltage

time

Neural spike train

J. Guckenheimer. Isochrons and phaseless sets. J. Math. Biol. (1975);Brown, Moehlis and Holmes. On the phase reduction and response dynamics of neural oscillator populations. Neural Computation (2004);E. M. Izhikevich. Dynamical Systems in Neuroscience. in Chapter 10. The MIT Press (2006).



CdVdt

=−gT ·m2∞(V) ·h · (V−ET )

−gh · r · (V−Eh)− . . . . . .

dhdt

=h∞(V)−h

τh(V)

drdt

=r∞(V)− r

τr(V)2000 2200 2400 2600 2800 3000 3200 3400 3600 3800 4000

−150

−100

−50

0

50

100

Voltage

time

Neural spike train




CdVdt

=−gT ·m2∞(V) ·h · (V−ET )

−gh · r · (V−Eh)− . . . . . .

dhdt

=h∞(V)−h

τh(V)

drdt

=r∞(V)− r

τr(V)2000 2200 2400 2600 2800 3000 3200 3400 3600 3800 4000

−150

−100

−50

0

50

100

Voltage

time

Neural spike train

−100

−50

0

50

100

0

0.2

0.4

0.6

0.8

10

0.1

0.2

0.3

0.4

Vh

r

Limit cyle

r

h v




CdVdt

=−gT ·m2∞(V) ·h · (V−ET )

−gh · r · (V−Eh)− . . . . . .

dhdt

=h∞(V)−h

τh(V)

drdt

=r∞(V)− r

τr(V)2000 2200 2400 2600 2800 3000 3200 3400 3600 3800 4000

−150

−100

−50

0

50

100

Voltage

time

Neural spike train

−100

−50

0

50

100

0

0.2

0.4

0.6

0.8

10

0.1

0.2

0.3

0.4

Vh

r

Limit cyle

r

h v

Normal form reduction−−−−−−−−−−−−−→

θi = ωi +ui ·Φ(θi)



Gait CycleSignal model

Stance phase Swing phase

Model (Noisy oscillator)

dθt = ω0 dt︸︷︷︸natural frequency

+ noise






+ noise






+ noise






+ noise






+ noise






+ noise






+ noise






+ noise


Simulation ResultsSolution of the Estimation of Gait Cycle Problem

feedback particle filter

dynamics estimate

noisymeasurements

[Click to play the movie]

Tilton, Hsiao-Wecksler and Mehta. Filtering with rhythms: Application to estimation of gait cycle. American Control Conference (2012).


Geometric ControlLocomotion Systems

shape variables: x1,x2group variable: ψ

3-body system

P. S. Krishnaprasad. Geometric phases and optimal reconfiguration for multibody systems. UM Tech. Report (1990);P. S. Krishnaprasad. Motion control and coupled oscillators. Procs. of Symp. on Motion, Control & Geometry. Natl. Acad. Sciences (1995);R. Brockett. Pattern generation and the control of nonlinear systems. IEEE TAC (2003);S. Kelly and R. Murray. Geometric phases and robotic locomotion. J. Robotic Systems (1995); R. Murray and S. Sastry. Nonholonomic motion planning:Steering using sinusoids. IEEE TAC (1993); J. Blair and T. Iwasaki. Optimal gaits for mechanical rectifier systems. IEEE TAC (2011);




3-body system





3-body system





3-body system

x = f (x, x,τ)

τ: torque input





3-body system

x = f (x, x,τ)

τ: torque input

ψ = a1(x)x1 +a2(x)x2

Reconstruction equation



Control of Locomotion Gaits2-body System

ψ = f (x)x

A. Taghvaei, S. Hutchinson and P. G. Mehta. A coupled-oscillators-based control architecture for locomotory gaits. IEEE CDC (2014).



ψ = f (x)x = f (θ)




ψ = f (x)x = f (θ ,u)




ψ = f (x)x = f (θ ,u)

minu[0,T]

E[ψ(T)−ψ(0)︸︷︷︸

Geom. phase

+1

2ε

∫ T

0u(t)2dt

]



2-body system, Simulation Result

Particles True Phase

t

−π3

0

π3

x

x(t) Observation: y(t)

0 10 20 30 40 50 60 70 80t

−1.0

−0.8

−0.6

−0.4

−0.2

0.0

0.2

q1

open loop: q1(t) close loop: q1(t)

↑

→

→

↓

[Click to play the movie]



AcknowledgementStudents in red

1. Feedback particle filter:

Tao Yang, Rick Laugesen, Sean Meyn, Max Raginsky

2. Coupled oscillators for estimation:

Adam Tilton, Shane Ghiotto, Liz Hsiao-Wecksler

3. Coupled oscillators for control:

Amirhossein Taghvaei, Seth Hutchinson

Research supported by NSF


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Feedback Particle Filter and its Application to Coupled...

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