1. Kalman Filter Example

2. Rudolf E. Kalman b. 1930 Hungary Kalman Filter NASA Ames 1960
National Medal of Science (2009)

3. Actions and Observations Through Time
Belief(xt)
(using all evidence to date)
Get Measurement
Elapse Time
Belief'(xt)
(without latest evidence)

4. Kalman Data: Our estimate: x1..t = x1, x2, x3...xt Our measurement:
z1..t = z1, z2, z3...zt
Our action:
u1..t = u1, u2, u3...ut

5. Kalman Example
"Understanding the Basis of the Kalman Filter Via a Simple and Intuitive Derivation" Ramsey Faragher

6. Initial Estimate

7. Elapse Time

8. Get Measurement

9. New Belief

10. Kalman: Assume a Linear System
xt = Ft xt-1 + Bt ut + wt
xt = state (estimate)
ut = action
Ft = state transition matrix
Bt = control input matrix
wt = noise wt ~ N(0,Qt)

11. Kalman: Measurements zt = Htxt + vt zt = measurements
Ht = Transformation matrix
vt = observational noise vt ~ N(0, Rt)

12. Kalman: Belief Gaussian Distribution x0 = initial mean P0 = Variance
N(x0, P0)

13. Kalman Example
Action:
ft/m
State: xt vt

14. Kalman Update State: Action: xt = xt-1 + vt-1△t + 0.5 (ft/m) * △t2
vt = vt-1 + (ft/m)△t
x't
v't =
1 △t
0 1
xt-1
vt-1
△t2/2 △t
ft/m

15. Kalman: Elapse Time Update
x't = Ftxt-1 + Btut
P't = FtPt-1 FTt + Qt

16. Kalman: Measurement Update
Kalman Gain (how much to correct estimate)
Kt = P'tHTt (HtP'tHTt + Rt)-1
New Belief:
xt = x't + Kt(zt-Htx't)
Pt = P't - KtHtP't

17. Simple Kalman Filter Example
xt = just position (meters)
ut = 1 m/s
wt ~ N(0,0.1)
vt ~ N(0,1.0)
x0 = 4.5 P2 = 2.0
x't = xt-1 + ut + wt (F=1, B=1)
zt = xt + vt (H=1)

18. t=0
x0=4.5
P2=2

19. t=1
x1 = x0 + 1 = 5.5
P'1 = F0P0 FT0 + Q0
= 1 * 2 *
= 2.1

20. t=1 z1=5.673 K1 = P'tHTt (HtP'tHTt + Rt)-1 = (2.1 * 1)/(1*2.1*1+1.0)
= .677
x1 = x'1 + K1(z1-H1x'1)
= ( *5.5) =
Pt = P't - KtHtP't
= * 1 * 2.1 =

21. Kalman Filters
Must be linear system
Stored as a gaussian