EE 570: Location and Navigation

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EE 570: Location and Navigation On-Line Bayesian Tracking Aly El-Osery 1 Stephen Bruder 2 1 Electrical Engineering Department, New Mexico Tech Socorro, New Mexico, USA 2 Electrical and Computer Engineering Department, Embry-Riddle Aeronautical Univesity Prescott, Arizona, USA March 27, 2014 Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 1 / 25

Objective Sequentially estimate on-line the states of a system as it changes over time using observations that are corrupted with noise. Filtering: the time of the estimate coincides with the last measurement. Smoothing: the time of the estimate is within the span of the measurements. Prediction: the time of the estimate occurs after the last available measurement. Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 2 / 25

Given State-Space Equations x k = f k ( x k 1, w k 1 ) (1) z k = h k ( x k, v k ) (2) Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 3 / 25

Given State-Space Equations (n 1) state vector at time k x k = f k ( x k 1, w k 1 ) (1) z k = h k ( x k, v k ) (2) (m 1) measurement vector at time k Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 3 / 25

Given State-Space Equations Possibly non-linear function, f k : R n R nw R n x k = f k ( x k 1, w k 1 ) (1) z k = h k ( x k, v k ) (2) Possibly non-linear function, h k : R m R nv R m Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 3 / 25

Given State-Space Equations i.i.d state noise x k = f k ( x k 1, w k 1 ) (1) i.i.d measurement noise z k = h k ( x k, v k ) (2) Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 3 / 25

Given State-Space Equations x k = f k ( x k 1, w k 1 ) (1) z k = h k ( x k, v k ) (2) The state process is Markov chain, i.e., p( x k x 1,..., x k 1 ) = p( x k x k 1 ) and the distribution of z k conditional on the state x k is independent of previous state and measurement values, i.e., p( z k x 1:k, z 1:k 1 ) = p( z k x k ) Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 3 / 25

Objective Probabilistically estimate x k using previous measurement z 1:k. In other words, construct the pdf p( x k z 1:k ). Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 4 / 25

Objective Probabilistically estimate x k using previous measurement z 1:k. In other words, construct the pdf p( x k z 1:k ). Optimal MMSE Estimate E{ x k ˆ x k 2 z 1:k } = x k ˆ x k 2 p( x k z 1:k )d x k (3) in other words find the conditional mean ˆ x k = E{ x k z 1:k } = x k p( x k z 1:k )d x k (4) Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 4 / 25

Assumptions w k and v k are drawn from a Gaussian distribution, uncorrelated have zero mean and statistically independent. { E{ w k w T Q k i = k i } = (5) 0 i k { E{ v k v T R k i = k i } = (6) 0 i k { E{ w k v T i } = 0 i, k (7) Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 5 / 25

Assumptions f k and h k are both linear, e.g., the state-space system equations may be written as x k = Φ k 1 x k 1 + w k 1 (8) y k = H k x k + v k (9) Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 6 / 25

Assumptions f k and h k are both linear, e.g., the state-space system equations may be written as x k = Φ k 1 x k 1 + w k 1 (8) y k = H k x k + v k (9) (n n) transition matrix relating x k 1 to x k Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 6 / 25

Assumptions f k and h k are both linear, e.g., the state-space system equations may be written as x k = Φ k 1 x k 1 + w k 1 (8) y k = H k x k + v k (9) (m n) matrix provides noiseless connection between measurement and state vectors Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 6 / 25

State-Space Equations ˆ x k k 1 = Φ k 1ˆ xk 1 k 1 (10) P k k 1 = Q k 1 +Φ k 1 P k 1 k 1 Φ T k 1 (11) ˆ x k k = ˆ x k k 1 + K k ( z k H kˆ xk k 1 ) (12) P k k = (I K k H k )P k k 1 (13) Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 7 / 25

State-Space Equations ˆ x k k 1 = Φ k 1ˆ xk 1 k 1 (10) P k k 1 = Q k 1 +Φ k 1 P k 1 k 1 Φ T k 1 (11) ˆ x k k = ˆ x k k 1 + K k ( z k H kˆ xk k 1 ) (12) P k k = (I K k H k )P k k 1 (13) (n m) Kalman gain Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 7 / 25

State-Space Equations ˆ x k k 1 = Φ k 1ˆ xk 1 k 1 (10) P k k 1 = Q k 1 +Φ k 1 P k 1 k 1 Φ T k 1 (11) ˆ x k k = ˆ x k k 1 + K k ( z k H kˆ xk k 1 ) (12) P k k = (I K k H k )P k k 1 (13) Measurement innovation Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 7 / 25

Kalman Gain K k = P k k 1 H T k ( H kp k k 1 H T k + R k ) 1 (14) Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 8 / 25

Kalman Gain K k = P k k 1 H T k ( H kp k k 1 H T k + R k ) 1 (14) Covariance of the innovation term Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 8 / 25

Kalman filter data flow Initial estimate (ˆ x 0 and P 0 ) Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 9 / 25

Kalman filter data flow Initial estimate (ˆ x 0 and P 0 ) Compute Kalman gain K k = P k k 1 H T k (H k P k k 1 HT k + R k ) 1 Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 9 / 25

Kalman filter data flow Initial estimate (ˆ x 0 and P 0 ) Compute Kalman gain K k = P k k 1 H T k (H k P k k 1 HT k + R k ) 1 Update estimate with measurement z k ˆ x k k = ˆ x k k 1 + K k ( z k H kˆ xk k 1 ) Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 9 / 25

Kalman filter data flow Initial estimate (ˆ x 0 and P 0 ) Compute Kalman gain K k = P k k 1 H T k (H k P k k 1 HT k + R k ) 1 Update estimate with measurement z k ˆ x k k = ˆ x k k 1 + K k ( z k H kˆ xk k 1 ) Update error covariance P k k = P k k 1 K k H k P k k 1 Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 9 / 25

Kalman filter data flow Initial estimate (ˆ x 0 and P 0 ) Compute Kalman gain K k = P k k 1 H T k (H k P k k 1 HT k + R k ) 1 Project ahead ˆ x k k 1 = Φ k 1ˆ xk 1 k 1 P k k 1 = Q k 1 + Φ k 1 P k 1 k 1 Φ T k 1 Update estimate with measurement z k ˆ x k k = ˆ x k k 1 + K k ( z k H kˆ xk k 1 ) k = k + 1 Update error covariance P k k = P k k 1 K k H k P k k 1 Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 9 / 25

Kalman filter data flow Initial estimate (ˆ x 0 and P 0 ) Compute Kalman gain K k = P k k 1 H T k (H k P k k 1 HT k + R k ) 1 Project ahead ˆ x k k 1 = Φ k 1ˆ xk 1 k 1 P k k 1 = Q k 1 + Φ k 1 P k 1 k 1 Φ T k 1 Update estimate with measurement z k ˆ x k k = ˆ x k k 1 + K k ( z k H kˆ xk k 1 ) k = k + 1 Update error covariance P k k = P k k 1 K k H k P k k 1 Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 9 / 25

Sequential Processing If R is a block matrix, i.e., R = diag(r 1, R 2,...,R r ). The R i has dimensions p i p i. Then, we can sequentially process the measurements as: For i = 1, 2,...,r K i = P i 1 (H i ) T (H i P i 1 (H i ) T + R i ) 1 (15) ˆ x i k k = ˆ x i k k + Ki ( z i k H iˆ x i 1 k k ) (16) P i = (I K i H i )P i 1 (17) where ˆ x 0 k k = ˆ x k k 1, P 0 = P 0 k k 1 and Hi is p i n corresponding to the rows of H corresponding the measurement being processed. Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 10 / 25

Observability The system is observable if the observability matrix H(k n+1) H(k n 2)Φ(k n+1) O(k) =. H(k)Φ(k 1)...Φ(k n+1) (18) where n is the number of states, has a rank of n. The rank of O is a binary indicator and does not provide a measure of how close the system is to being unobservable, hence, is prone to numerical ill-conditioning. Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 11 / 25

A Better Observability Measure In addition to the computation of the rank of O(k), compute the Singular Value Decomposition (SVD) of O(k) as O = UΣV (19) and observe the diagonal values of the matrix Σ. Using this approach it is possible to monitor the variations in the system observability due to changes in system dynamics. Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 12 / 25

Remarks Kalman filter is optimal under the aforementioned assumptions, and it is also an unbiased and minimum variance estimate. If the Gaussian assumptions is not true, Kalman filter is biased and not minimum variance. Observability is dynamics dependent. The error covariance update may be implemented using the Joseph form which provides a more stable solution due to the guaranteed symmetry. P k k = (I K k H k )P k k 1 (I K k H k ) T + K k R k K T k (20) Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 13 / 25

System Model To obtain the state vector estimate ˆ x(t) x(t) = F(t) x(t)+g(t) w(t) (21) E{ x(t)} = tˆ x(t) = F(t)ˆ x(t) (22) Solving the above equation over the interval t τ s, t ˆ x(t) = e ( t t τs F(t )dt )ˆ x(t τs ) (23) where F k 1 is the average of F at times t and t τ s. Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 14 / 25

System Model Discretization As shown in the Kalman filter equations the state vector estimate is given by ˆ x k k 1 = Φ k 1ˆ xk 1 k 1 Therefore, Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 15 / 25

System Model Discretization As shown in the Kalman filter equations the state vector estimate is given by ˆ x k k 1 = Φ k 1ˆ xk 1 k 1 Therefore, Φ k 1 = e F k 1τ s I+F k 1 τ s (24) where F k 1 is the average of F at times t and t τ s, and first order approximation is used. Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 15 / 25

Discrete Covariance Matrix Q k Assuming white noise, small time step, G is constant over the integration period, and the trapezoidal integration Q k 1 1 2 [ ] Φ k 1 G k 1 Q(t k 1 )G T k 1 ΦT k 1 + G k 1Q(t k 1 )G T k 1 τ s (25) where E{ w(η) w T (ζ)} = Q(η)δ(η ζ) (26) Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 16 / 25

Linearized System F k = f( x) x, x=ˆ xk k 1 H k = h( x) x (27) x=ˆ xk k 1 where f( x) x = f 1 f 1 x 1 f 2 f 2 x 1. x 2 x 2,.... h( x) x = h 1 h 1 x 1 h 2 h 2 x 1. x 2 x 2 (28).... Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 17 / 25

First Order Markov Noise State Equation ḃ(t) = 1 T c b(t)+w(t) (29) Autocorrelation Function E{b(t)b(t +τ)} = σ 2 BI e τ /Tc (30) where E{w(t)w(t +τ)} = Q(t)δ(t τ) (31) and T c is the correlation time. Q(t) = 2σ2 BI T c (32) Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 18 / 25

Discrete First Order Markov Noise State Equation System Covariance Matrix b k = e 1 Tc τs b k 1 + w k 1 (33) Q = σ 2 BI [1 e 2 Tc τs ] (34) Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 19 / 25

Autocorrelation of 1st order Markov R b (τ) = σ 2 BI e τ /Tc σ 2 BI σ 2 e τ T c Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 20 / 25

Small Correlation Time T c = 0.01 R b (τ) τ Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 21 / 25

Small Correlation Time T c = 0.01 3 2 R b (τ) 1 b(t) 0-1 τ -2-3 0 0.2 0.4 0.6 0.8 1 Time (sec) Measured Actual Estimated Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 21 / 25

Larger Correlation Time T c = 0.1 R n (τ) τ Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 22 / 25

Larger Correlation Time T c = 0.1 3 2 R n (τ) 1 b(t) 0-1 τ -2-3 0 0.2 0.4 0.6 0.8 1 Time (sec) Measured Actual Estimated Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 22 / 25

Unscented Kalman Filter (UKF) Propagates carefully chosen sample points (using unscented transformation) through the true non-linear system, and therefore captures the posterior mean and covariance accurately to the second order. Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 23 / 25

Particle Filter A Monte Carlo based method. It allows for a complete representation of the state distribution function. Unlike EKF and UKF, particle filters do not require the Gaussian assumptions. Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 24 / 25

Bayesian Filtering: From Kalman Filters to Particle Filters, and Beyond, by Zhe Chen Aly El-Osery, Stephen Bruder (NMT,ERAU) EE 570: Location and Navigation March 27, 2014 25 / 25

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