Kalman Filter Notes

Prateek Tandon

Generic Problem

• Imagine watching a small bird flying through a dense jungle.

• You glimpse intermittent flashes of motion.• You want to guess where the bird is and

where it may be in the next time step.• Bird’s state might be 6-dimensional:[x,y,z,x’,y’,z’] – three variables for position and

three for velocity.

Kalman Filter

Xk = Fk xk-1 + Bk uk + wk (state update)Zk = Hkxk + vk (measurement update)

Xk – current stateXk-1 – last stateUk – control inputWk ~ N(0,Qk), represents process noise distributed via multivariate

zero-mean normal distribution with covariance Qk

Vk ~ N(0,Rk), represents observation nose distributed via multivariate zero-mean normal distribution with covariance Rk

Fk – state transition modelBk – control input modelHk – observation model

Kalman Filter Algorithm

PREDICT:

UPDATE:

Predicted State

Predicted Covariance

Innovation and Measurement Residual

Innovation on Covariance

Optimal Kalman Gain

Updated State Estimate

Updated Covariance Estimate

Applications

• Radar tracking of planes/missles/navigation• Smoothing time series data

– Stock market– People tracking / hand tracking / etc– Sensor Data

• GPS Location Data smoothing application

Particle Filter AlgorithmFunction PARTICLE-FILTERING(e,N,dbn) returns a set of samples for the next time

stepInputs: e, the new incoming evidence

N, the number of samples to be maintained Dbn, a DBN with prior P(X0), transition model P(X1|X0), sensor model P(E1|X1)

Persistent: S, a vector of samples of size N, initially generated from P(X0)

Local variables: W, a vector of weights of size N

For i=1 to N doS[i] sample from P(X1 | X0 = S[i])

W[i} P(E | X1 = S[i])

S WEIGHTED-SAMPLE-WITH-REPLACEMENT(N,S,W)Return S

Particle Filter Example

Rain0Rain0 Rain1Rain1

Umbrella1Umbrella1

P(R0)

0.7

R0 P(R1)

T 0.7

F 0.3

R1 P(U1)

T 0.9

F 0.2

Particle Filter Example

Raint Raint+1Raint+1 Raint+1

(a) Propagate (b) Weight,

[Not Umbrella observed.]

(c) Resample

References

• "Kalman Filter." . WIKIPEDIA, 13 APRIL 2013. Web. 13 Apr 2013. <http://en.wikipedia.org/wiki/Kalman_filter>.

• Russell, Stuart, and Peter Norvig. Artificial Intelligence: A Modern Approach. 3rd. New Jersey: Pearson Education Inc., 2010. Print.