1 Hybrid Estimation (Filt ering) Problem Conditional Dependencies - - PDF document

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1 Hybrid Estimation (Filt ering) Problem Conditional Dependencies - - PDF document

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Motivation method that uses efficient Gaussian representation Multi-Modal Particle Filtering Life support systems BIO-Plex courtesy NASA JPL for Hybrid S ystems NASA robotic missions MSL Embedded systems: with Autonomous Mode Transitions

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Multi-Modal Particle Filtering

for Hybrid S ystems with Autonomous Mode Transitions

S tanislav Funiak, Brian Williams

MIT Space Systems and Artificial Intelligence Laboratories Cambridge MA, USA

method that uses efficient Gaussian representation transitions between modes that depend on continuous state

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Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions

Motivation

NASA robotic missions MSL

courtesy NASA JPL

Legged bipeds, intelligent assistants

courtesy MIT LEG Laboratory

Life support systems BIO-Plex Embedded systems:

  • Continuous and discrete behavior
  • Highly complex artifacts
  • Need for autonomous, robust operation

Hybrid model, estimate state from observations Extract diagnosis from subtle symptoms

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Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions

Outline

Hybrid modeling, Hybrid estimation problem Prior work: multi-modal methods, particle filtering Multi-modal particle filtering Derivation using Rao-Blackwellised particle filtering Unification with prior multi-modal methods Discussion Experimental results Approximations in the filter

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Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions

S imple Hybrid S yst em: Acrobatic Robot

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Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions

Hybrid Discrete/ Continuous Model

Probabilistic Hybrid Automata (Hofbaur, Williams)

state x:

xd discrete state (mode) with finite domain xc continuous state

input/output: ud discrete command

uc continuous command yc continuous output (observation)

dynamics:

T 1. probabilistic transitions between modes

set of transitions & guard conditions over and

F

  • 2. discrete-time dynamics for each mode

white Gaussian noise non-linear functions mode-dependent

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Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions

Hybrid Discrete/ Continuous Model

Probabilistic Hybrid Automaton for acrobatic robot

state x:

xd xc

input/output: ud

uc T1 yc

+ noise dynamics: m0,f m0,ok m1,ok m1,f θ1> 0.7: p= 0.02 θ1< 0.7: p= 0.01

p= 0.01

θ1> 0.7: p= 0.02 θ1< 0.7: p= 0.01

p= 0.01 T1> 0: p= 0.01 T1> 0: p= 0.01

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Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions

Hybrid Estimation (Filt ering) Problem

Task:

Given a PHA model, initial distribution , and sequence

  • f command inputs and observations

estimate xd,t and xc,t approximate posterior distribution

PHA

continuous input

uc,0:t

discrete input

ud,0:t

continuous output (observation)

yc,1:t

m0,f m0,ok m1,ok m1,f

p= 0.01 θ1> 0.7: p= 0.02 θ1< 0.7: p= 0.01

p= 0.01 T1> 0: p= 0.01 T1> 0: p= 0.01

θ1> 0.7: p= 0.02 θ1< 0.7: p= 0.01

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Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions

Conditional Dependencies in PHA

autonomous mode transitions: conditioned on continuous state coupling between discrete and continuous state the process is not Markov

m0,f m0,ok m1,ok m1,f

p= 0.01 θ1> 0.7: p= 0.02 θ1< 0.7: p= 0.01

p= 0.01 T1> 0: p= 0.01 T1> 0: p= 0.01

θ1> 0.7: p= 0.02 θ1< 0.7: p= 0.01

PHA autonomous transition Hybrid DBN of PHA models

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Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions

Background: Multi-Modal Estimation

  • e.g. IMM, HME, GPB
  • represent efficiently as mixture of Gaussians

Beam filter (HME), (Hofbaur, Williams):

  • Tracks leading set of likely trajectories
  • For each, estimates continuous state
  • Interleaves mode trajectory tracking

and continuous state estimation

nonlinearities and merging ) bias

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Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions

Background: Particle Filt ering

  • Verma et al., 2001 Dearden and Clancy 2002, Koutsoukos et al. 2002
  • Represent posterior distribution by samples evolved probabilistically
  • Sample the whole state space ) suffer from state space explosion

(except Freitas, 2002)

Initialization step Importance sampling step Selection step (1) (2) (3)

sample the initial distribution evolve each particle according to proposal distribution q evaluate importance weights resample particle according to their importance weights

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Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions

Reduce size of sampled space by factoring out

subspace with analytical solution (Murphy, Russell)

Divide state variables x into two sets, r and s:

sample r, analytically solve for s I f able to compute analytically,

  • nly need to sample r

Background: Rao-Blackwellised Particle Filtering

particle filter analytical solution

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Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions

Outline

Hybrid modeling, Hybrid estimation problem Prior work: multi-modal methods, particle filtering Multi-modal particle filtering Derivation using Rao-Blackwellised particle filtering Unification with prior multi-modal methods Discussion Experimental results Approximations in the filter

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Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions

Multi-modal Particle Filt er – Concept

  • Combines particle filtering with multi-modal estimation
  • Derived with Rao-Blackwellised PF + approximations
  • 1. Particle filter:

sample mode trajectories, not whole state space

  • analogous to HME (H&W)
  • 2. Analytical solution:

For each sample trajectory, estimate continuous state with Kalman Filter Use continuous estimates from time t in particle filter at time t+ 1

mok,0 mok,0 mok,0 mok,1 mok,1 mok,1 mf,1 mok,1 mok,0 mok,0

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Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions

Multi-modal Particle Filt er – Algorithm

mok,0 (1) Initialization step (2) Importance sampling step (3) Selection (resampling) step mok,0 mok,1 mok,1 mok,0 mok,1 mf,0 mf,1 (4) Exact (Kalman Filtering) step draw samples from the initial distribution over the modes initialize the corresponding continuous state estimates evolve each sample trajectory according to the transition model and previous continuous estimates determine transition & observ- ation model for each sample update continuous estimates mok,0 mok,0 mok,0 mok,1 mok,1 mok,1 mok,1 mf,1 mok,1 mok,1 mok,0 mok,0 mok,0 mok,1 mok,1 mok,1

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Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions

Proposal Distribution (Import ance S

ampling S t ep)

  • Conditioned on entire trajectory and prior observations

Computed efficiently from previous continuous estimate

(expand in xc,t-1) same for all xc,t-1 satisfying guard c

0.7 θ1 mean of estimated θ1 guard boundary probability

  • f guard c
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Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions
  • PF: need to let
  • Simplifies to
  • still hard to evaluate (non-standard)

ignore autonomous transitions in second term approximate weight using the prediction step of Kalman Filter

Importance Weights (Importance S

ampling S t ep)

mok,0 mok,0 mok,0 mok,1 mok,1 mok,1 mf,1 mok,1 mok,0 mok,0

y1 y2

expand in xc,t conditional independences

  • bservation likelihood

non-standard

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Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions

Unification with Prior Mult i-Modal Methods

  • Hybrid prediction & update equations (Hofbaur, Williams):
  • Hybrid transition function

$ proposal distribution

Hybrid observation function

$ weight function

Direct correspondence RBPF derives hybrid prediction & update equations Joint improvements Meaningful merging of the two methods

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Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions

Outline

Hybrid modeling, Hybrid estimation problem Prior work: multi-modal methods, particle filtering Multi-modal particle filtering Derivation using Rao-Blackwellised particle filtering Unification with prior multi-modal methods Discussion Experimental results Approximations in the filter

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Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions

Experiment al Result s

robot captures a ball

Distribution across modes

robot far to the right

Filtered vs. true

filtered actual

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Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions

Approximations in the Filter

Errors introduced by approximations:

  • nly tracking subset of mode trajectories
  • resampling in selection step reduces

information

  • non-linearities in continuous dynamics
  • hard non-linearities due to autonomous

transitions

Remedies (partial):

merge continuous

estimates

  • ptimal resampling

(Crisan)

UKF (Wan, Merwe) ? mok,0 mok,0 mok,0 mok,1 mok,1 mok,1 mok,1 mf,1 mok,1 mok,1 mok,0 mok,0 mok,0 mok,1 mok,1 mok,1 Selection step:

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Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions

Hard Non-linearities in Post erior

  • Exhibited in weight computation

t= 1: mok,0 mok,0 t= 2: mok,0 mok,0 mok,1

) errors in predictions at t ¸ 2

true (desired): 0.7 estimated: m0,f m0,ok m1,ok m1,f

p= 0.01

θ1> 0.7:

p= 0.01 p= 0.01 T1> 0: p= 0.01 T1> 0: p= 0.01

θ1> 0.7:

p= 0.01

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Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions

S ummary

Efficient sampling algorithm for hybrid models with

autonomous transitions

Unification of RBPF with prior multi-modal methods Explicit representation: understand approximations

Future work:

Experimental understanding of theoretical issues Implement proposed improvements Beam search vs. particle filtering

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Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions

References

  • Hofbaur, M. W., and Williams, B. C. (2002). Mode estimation of

probabilistic hybrid systems. In: Hybrid Systems: Computation and Control, HSCC 2002.

  • Freitas, N. (2002) Rao-blackwellised particle filtering for fault
  • diagnosis. IEEE Aerospace.
  • Doucet, Freitas, Gordon (2001). Sequential Monte Carlo Methods in
  • Pratcice. Springer-Verlag.
  • Dearden, R. and D. Clancy (2002). Particle filters for real-time fault

detection in planetary rovers. In: DX-02.

  • Murphy, K., and S. Russell (2001). Rao-Blackwellised particle

filtering for dynamic Bayesian networks. In: Sequential Monte Carlo Methods in Practice.

  • Crisan, D. (2001). Particle Filters - A Theoretical Perspective. In:

Sequential Monte Carlo Methods in Practice

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Backup slides

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Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions

Experiment al Result s

Filtered θ2

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Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions

More Future Work

Merge beam search and particle filtering potential to outperform search and PF alone Hybrid dynamic Bayesian networks to model more tightly coupled systems, natural

environments

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Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions

Beam Filt er (HME)

Assume that components fail “independently” Enumerate k best trajectories using A* search

any-time algorithm

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Equations for Acrobot