1 / 26

State Estimation and Kalman Filtering

State Estimation and Kalman Filtering. CS B659 Spring 2013 Kris Hauser. Motivation. Observing a stream of data Monitoring (of people, computer systems, etc ) Surveillance, tracking Finance & economics Science Questions: Modeling & forecasting Handling partial and noisy observations.

zarita
Download Presentation

State Estimation and Kalman Filtering

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. State Estimation and Kalman Filtering CS B659 Spring 2013 Kris Hauser

  2. Motivation • Observing a stream of data • Monitoring (of people, computer systems, etc) • Surveillance, tracking • Finance & economics • Science • Questions: • Modeling & forecasting • Handling partial and noisy observations

  3. Markov Chains X0 X1 X2 X3 Observe X1 X0 independent of X2, X3, … P(Xt|Xt-1) known as transition model Sequence of probabilistic state variables X0,X1,X2,… E.g., robot’s position, target’s position and velocity, …

  4. Inference in MC [Incremental approach] • Prediction: the probability of future state? • P(Xt) = Sx0,…,xt-1P (X0,…,Xt) = Sx0,…,xt-1P (X0) Px1,…,xt P(Xi|Xi-1)= Sxt-1P(Xt|Xt-1) P(Xt-1) • “Blurs” over time, and approaches stationary distribution as t grows • Limited prediction power • Rate of blurring known as mixing time

  5. X0 X1 X2 X3 Modeling Partial Observability Hidden state variables Observed variables O1 O2 O3 P(Ot|Xt) called the observation model (or sensor model) Hidden Markov Model (HMM)

  6. Filtering Query variable Unknown X0 X1 X2 Distribution given O1 O2 • Name comes from signal processing • Goal: Compute the probability distribution over current state given observations up to this point Known

  7. Filtering Query variable Unknown X0 X1 X2 Distribution given O1 O2 • Name comes from signal processing • Goal: Compute the probability distribution over current state given observations up to this point • P(Xt|o1:t) = Sxt-1P(xt-1|o1:t-1)P(Xt|xt-1,ot) • P(Xt|Xt-1,ot) = P(ot|Xt-1,Xt)P(Xt|Xt-1)/P(ot|Xt-1) = a P(ot|Xt)P(Xt|Xt-1) Known

  8. Kalman Filtering • In a nutshell • Efficient probabilistic filtering in continuous state spaces • Linear Gaussian transition and observation models • Ubiquitous for state tracking with noisy sensors, e.g. radar, GPS, cameras

  9. X0 X1 X2 X3 Hidden Markov Model for Robot Localization • Use observations + transition dynamics to get a better idea of where the robot is at time t Hidden state variables Observed variables z1 z2 z3 Predict – observe – predict – observe…

  10. X0 X1 X2 X3 Hidden Markov Model for Robot Localization • Use observations + transition dynamics to get a better idea of where the robot is at time t • Maintain a belief state bt over time • bt(x) = P(Xt=x|z1:t) Hidden state variables Observed variables z1 z2 z3 Predict – observe – predict – observe…

  11. Bayesian Filtering with Belief States • Compute bt, given ztand prior belief bt • Recursive filtering equation

  12. Bayesian Filtering with Belief States • Compute bt, given ztand prior belief bt • Recursive filtering equation Predict P(Xt|z1:t-1) using dynamics alone Update via the observation zt

  13. In Continuous State Spaces… • Compute bt, given zt and prior belief bt • Continuous filtering equation

  14. In Continuous State Spaces… • Compute bt, given zt and prior belief bt • Continuous filtering equation • How to evaluate this integral? • How to calculate Z? • How to even represent a belief state?

  15. Key Representational Decisions • Pick a method for representing distributions • Discrete: tables • Continuous: fixed parameterized classes vs. particle-based techniques • Devise methods to perform key calculations (marginalization, conditioning) on the representation • Exact or approximate?

  16. Gaussian Distribution • Mean m, standard deviation s • Distribution is denoted N(m,s) • If X ~ N(m,s), then • With a normalization factor

  17. Linear Gaussian Transition Model for Moving 1D Point • Consider position and velocity xt, vt • Time step h • Without noise xt+1 = xt + h vtvt+1 = vt • With Gaussian noise of std s1 P(xt+1|xt)  exp(-(xt+1 – (xt + h vt))2/(2s12) i.e. Xt+1 ~ N(xt + h vt, s1)

  18. vh s1 Linear Gaussian Transition Model • If prior on position is Gaussian, then the posterior is also Gaussian N(m,s)  N(m+vh,s+s1)

  19. Linear Gaussian Observation Model • Position observation zt • Gaussian noise of std s2 zt ~ N(xt,s2)

  20. Observation probability Linear Gaussian Observation Model • If prior on position is Gaussian, then the posterior is also Gaussian Posterior probability Position prior •  (s2z+s22m)/(s2+s22) s2s2s22/(s2+s22)

  21. Multivariate Gaussians X~ N(m,S) • Multivariate analog in N-D space • Mean (vector) m, covariance (matrix) S • With a normalization factor

  22. Multivariate Linear Gaussian Process • A linear transformation + multivariate Gaussian noise • If prior state distribution is Gaussian, then posterior state distribution is Gaussian • If we observe one component of a Gaussian, then its posterior is also Gaussian y = A x + e e ~ N(m,S)

  23. Multivariate Computations • Linear transformations of gaussians • If x ~ N(m,S), y = A x + b • Then y ~ N(Am+b, ASAT) • Consequence • If x ~ N(mx,Sx), y ~ N(my,Sy), z=x+y • Then z ~ N(mx+my,Sx+Sy) • Conditional of gaussian • If [x1,x2] ~ N([m1 m2],[S11,S12;S21,S22]) • Then on observing x2=z, we havex1 ~ N(m1-S12S22-1(z-m2), S11-S12S22-1S21)

  24. Presentation

  25. Next time • Principles Ch. 9 • Rekleitis (2004)

More Related