Kalman Filter. Predict: Update: x k k 1 = F k x k 1 k 1 + B k u k P k k 1 = F k P k 1 k 1 F T k + Q

Transcription

1 Kalman Filter

2 Kalman Filter Predict: x k k 1 = F k x k 1 k 1 + B k u k P k k 1 = F k P k 1 k 1 F T k + Q Update: K = P k k 1 Hk T (H k P k k 1 Hk T + R) 1 x k k = x k k 1 + K(z k H k x k k 1 ) P k k =(I KH k )P k k 1

3 Gaussian (Normal) Distribution Completely described by N(µ,!) Mean µ Standard deviation!, variance! 2 1 " 2# e$( x$ µ )2 / 2" 2

4 The Central Limit Theorem The sum of many random variables with the same mean, but with arbitrary conditional density functions, converges to a Gaussian density function. If a model omits many small unmodeled effects, then the resulting error should converge to a Gaussian density function.

5 Estimating a Value Suppose there is a constant value x. Distance to wall; angle to wall; etc. At time t 1, observe value z 1 with variance The optimal estimate is x ˆ (t 1 ) = z 1 with 2 variance " 1 " 1 2

6 A Second Observation At time t 2, observe value z 2 with variance " 2 2

7 Merged Evidence

8 Update Mean and Variance Weighted average of estimates. x ˆ (t ) = Az + Bz The weights come from the variances. Smaller variance = more certainty x ˆ (t ) = 2 # % $ % " 2 & # 2 ( " 2 +" '( z " 2 & + % 1 ( 1 " 2 + " 2 $ % 1 2 '( z 2 1 " 2 (t ) = " " 2 2 A + B =1

9 From Weighted Average to Predictor-Corrector Weighted average: x ˆ (t ) = Az + Bz = (1" K)z + Kz Predictor-corrector: x ˆ (t ) = z + K(z " z ) = ˆ x (t 1 ) + K(z 2 " ˆ x (t 1 )) This version can be applied recursively.

10 Predictor-Corrector Update best estimate given new data x ˆ (t ) = x ˆ (t ) + K(t )(z " x ˆ (t )) Update variance: K(t 2 ) = " 1 2 " 1 2 +" 2 2 " 2 (t 2 ) =" 2 (t 1 ) # K(t 2 )" 2 (t 1 ) = (1"K(t 2 ))# 2 (t 1 )

11 Static to Dynamic Now suppose x changes according to x = F(x,u,") = u +" (N (0,# " ))

12 Dynamic Prediction At t 2 we know x ˆ (t 2 ) " 2 (t 2 ) At t 3 after the change, before an observation. x ˆ (t " ) = x ˆ (t ) + u[t " t ] " 2 (t 3 # ) = " 2 (t 2 ) + " $ 2 [t 3 # t 2 ] Next, we correct this prediction with the observation at time t 3.

13 Dynamic Correction At time t 3 we observe z 3 with variance Combine prediction with observation. " 3 2 x ˆ (t ) = x ˆ (t " ) + K(t )(z " x ˆ (t " )) " 2 (t 3 ) = (1# K(t 3 ))" 2 (t 3 # ) K(t 3 ) = " 2 (t 3 # ) " 2 (t 3 # ) + " 3 2

14 Qualitative Properties x ˆ (t ) = x ˆ (t " ) + K(t )(z " x ˆ (t " )) K(t 3 ) = " 2 (t 3 # ) " 2 (t 3 # ) + " 3 2 Suppose measurement noise is large. Then K(t 3 ) approaches 0, and the measurement will be mostly ignored. Suppose prediction noise " 3 2 " 2 (t 3 # ) is large. Then K(t 3 ) approaches 1, and the measurement will dominate the estimate.

15 Kalman Filter Takes a stream of observations, and a dynamical model. At each step, a weighted average between prediction from the dynamical model correction from the observation. The Kalman gain K(t) is the weighting, based on the variances " 2 2 (t) and " # With time, K(t) and " 2 (t) tend to stabilize.

16 Simplifications We have only discussed a one-dimensional system. Most applications are higher dimensional. We have assumed the state variable is observable. In general, sense data give indirect evidence. x = F(x,u," ) = u +" 1 1 z = G(x," ) = x +" 2 2 We will discuss the more complex case next.

17 Up To Higher Dimensions Our previous Kalman Filter discussion was of a simple one-dimensional model. Now we go up to higher dimensions: State vector: x " # n Sense vector: z " # m u " # l Motor vector: First, a little statistics.

18 Expectations Let x be a random variable. The expected value E[x] is the mean: E[x] = " x p(x) dx # x = 1 N N $ x i 1 The probability-weighted mean of all possible values. The sample mean approaches it. Expected value of a vector x is by component. E[x] = x = [x 1,Lx n ] T

19 Variance and Covariance The variance is E[ (x-e[x]) 2 ] " 2 = E[(x # x ) 2 ] = 1 N N $ 1 (x i # x ) 2 Covariance matrix is E[ (x-e[x])(x-e[x]) T ] C ij = 1 N N # k =1 (x ik " x i )(x jk " x j ) Divide by N!1 to make the sample variance an unbiased estimator for the population variance.

20 Biased and Unbiased Estimators Strictly speaking, the sample variance " 2 = E[(x # x ) 2 ] = 1 N N $ 1 (x i # x ) 2 is a biased estimate of the population variance. An unbiased estimator is: s 2 = 1 N #(x i " x ) 2 N "1 1 But: If the difference between N and N!1 ever matters to you, then you are probably up to no good anyway [Press, et al]

21 Covariance Matrix Along the diagonal, C ii are variances. Off-diagonal C ij are essentially correlations. # C = " 2 C C % 1,1 1 1,2 1,N C C = " 2 % 2,1 2,2 2 % O M % $ C N,1 L C = " 2 N,N N & ( ( ( ( '

22 Independent Variation x and y are Gaussian random variables (N=100) Generated with! x =1! y =3 Covariance matrix: C xy = " % $ ' # &

23 c and d are random variables. Generated with c=x+y d=x-y Covariance matrix: Dependent Variation C cd = # "7.93& % ( $ " '

24 Discrete Kalman Filter Estimate the state x " # n of a linear stochastic difference equation x k = Ax k "1 + Bu k + w k "1 process noise w is drawn from N(0,Q), with covariance matrix Q. with a measurement z k = Hx k + v k z " # m measurement noise v is drawn from N(0,R), with covariance matrix R. A, Q are nxn. B is nxl. R is mxm. H is mxn.

25 Estimates and Errors x ˆ k " # n is the estimated state at time-step k. x ˆ " k # $ n after prediction, before observation. Errors: e " " k = x k " x ˆ k e k = x k " x ˆ k Error covariance matrices: P k " = E[e k " e k " T ] P k = E[e k e k T ] Kalman Filter s task is to update ˆ x k P k

26 Time Update (Predictor) Update expected value of x x ˆ " k = Aˆ x k "1 + Bu k Update error covariance matrix P P " k = AP k "1 A T + Q Previous statements were simplified versions of the same idea: x ˆ (t " ) = x ˆ (t ) + u[t " t ] " 2 (t 3 # ) = " 2 (t 2 ) + " $ 2 [t 3 # t 2 ]

27 Measurement Update (Corrector) Update expected value x ˆ k = x ˆ " k + K k (z k " Hˆ x " k ) " z k " Hˆ x k innovation is Update error covariance matrix P k = (I"K k H)P k " Compare with previous form x ˆ (t ) = x ˆ (t " ) + K(t )(z " x ˆ (t " )) " 2 (t 3 ) = (1# K(t 3 ))" 2 (t 3 # )

28 The Kalman Gain The optimal Kalman gain K k is K k = P " k H T (HP " k H T + R) "1 P " = k H T HP " k H T + R Compare with previous form K(t 3 ) = " 2 (t 3 # ) " 2 (t 3 # ) + " 3 2

29 Extended Kalman Filter Suppose the state-evolution and measurement equations are non-linear: x k = f (x k "1,u k ) + w k "1 z k = h(x k ) + v k process noise w is drawn from N(0,Q), with covariance matrix Q. measurement noise v is drawn from N(0,R), with covariance matrix R.

30 The Jacobian Matrix For a scalar function y=f(x), "y = f #(x)"x For a vector function y=f(x), "y = J"x = #)f 1 #"y 1 & (x) L )f & 1 % (x)( % ( %)x 1 )x n ( % M ( % ( = % M M (* % $ "y n ' )f n (x) L )f ( % n (x)( $ %)x 1 )x n '( #"x 1 & % ( % M ( % ( $ "x n '

31 Linearize the Non-Linear Let A be the Jacobian of f with respect to x. A ij = "f i "x j (x k#1,u k ) Let H be the Jacobian of h with respect to x. H ij = "h i "x j (x k ) Then the Kalman Filter equations are almost the same as before!

32 EKF Update Equations Predictor step: Kalman gain: Corrector step: x ˆ " k = f (ˆ x k "1,u k ) P k " = AP k "1 A T + Q K k = P k " H T (HP k " H T + R) "1 x ˆ k = x ˆ " k + K k (z k " h(ˆ x " k )) P k = (I"K k H)P k "

33 Linearized Motion Model for a Robot Y y v x R ω G X

34 Linearized Motion Model for a Robot Y y R v x From a robot-centric perspective, the velocities look like this: ω G X

35 Linearized Motion Model for a Robot Y y R v x From a robot-centric perspective, the velocities look like this: G X ω From the global perspective, the velocities look like this:

36 Linearized Motion Model for a Robot Y y R v x From a robot-centric perspective, the velocities look like this: G X ω From the global perspective, the velocities look like this: The discrete time state estimate (including noise) looks like this:

37 Linearized Motion Model for a Robot Y y R v x From a robot-centric perspective, the velocities look like this: G X ω From the global perspective, the velocities look like this: The discrete time state estimate (including noise) looks like this: Problem! We don t know linear and rotational velocity errors. The state estimate will rapidly diverge if this is the only source of information!

38 Linearized Motion Model for a Robot Now, we have to compute the covariance matrix propagation equations.

39 Linearized Motion Model for a Robot Now, we have to compute the covariance matrix propagation equations. The indirect Kalman filter derives the pose equations from the estimated error of the state:

40 Linearized Motion Model for a Robot Now, we have to compute the covariance matrix propagation equations. The indirect Kalman filter derives the pose equations from the estimated error of the state: In order to linearize the system, the following small-angle assumptions are made:

41 Linearized Motion Model for a Robot From the error-state propagation equation, we can obtain the State propagation and noise input functions F and G : From these values, we can easily compute the standard covariance propagation equation:

42 Sensor Model for a Robot with a Perfect Map Y y R x z L G X

43 Sensor Model for a Robot with a Perfect Map Y y R x z L From the robot, the measurement looks like this: G X

44 Sensor Model for a Robot with a Perfect Map Y G y R x z X L From the robot, the measurement looks like this: From a global perspective, the measurement looks like:

45 Sensor Model for a Robot with a Perfect Map Y G y R x z X L From the robot, the measurement looks like this: From a global perspective, the measurement looks like: The measurement equation is nonlinear and must also be linearized!

46 Sensor Model for a Robot with a Perfect Map Now, we have to compute the linearized sensor function. Once again, we make use of the indirect Kalman filter where the error in the reading must be estimated.

47 Sensor Model for a Robot with a Perfect Map Now, we have to compute the linearized sensor function. Once again, we make use of the indirect Kalman filter where the error in the reading must be estimated. In order to linearize the system, the following small-angle assumptions are made:

48 Sensor Model for a Robot with a Perfect Map Now, we have to compute the linearized sensor function. Once again, we make use of the indirect Kalman filter where the error in the reading must be estimated. In order to linearize the system, the following small-angle assumptions are made: The final expression for the error in the sensor reading is:

49 Slides Slides were taken from: 395Tfall05/resources/ lectures/kalman.ppt