1. APPLICATION OF THE KALMAN FILTER
ROHULLAH LATIF
ALICIA FESSLER
MATHEMATICAL METHODS

2. Outline
 History
 The Kalman Filter
 UAV & The Kalman Filter
 Study Approach
 Problem Statement
 Results- Time Update
 Results- Measurement Update
 Results – 10 Iterations
 Results- Graphics
 Conclusion

3. History
 Developed by Rudolf E. Kalman
in 1960
 Operates by combining 2
methods: Prediction &
measurement
 Successfully used in the Apollo
navigational system
 Commonly used in tracking
systems in satellites, cell phones,
noise cancellation devices,
etc..
http://www.northropgrumman.com/AboutUs/OurHeritage/Pages/Inspace.
aspx
Navigation Photo Link Gps Picture Link

4. Kalman Filter
 Used in applications where
variables of interest can not be
directly measured
 Instead indirect measurements
are used to calculate the
desired parameters.
 Certain degree of error are
present with such analysis.
 Kalman Filter combines all
measurements data along with
previous knowledge to estimate
a desired variableFigure 1. Application of Kalman Filter
Courtesy of: http://www.cs.unc.edu/~welch/kalman/media/pdf/maybeck_ch1.pdf

5. UAV and the Kalman Filter
 Rely heavily on navigational
methods
 By estimating certain variables,
operator can determine the
location of the UAV
UAV Helicopter Image Link

6. Study Approach
[1] xt = Atxt−1 + B 𝑡u 𝑡 + wt
[2]Pt−1 = AtPt−1 + At + Qt
[3] Kt = PtHt HPtHt + R −1
4 xt = xt + Kt(zt − Hxt
5 Pt = (I − Kt + H Pt
Figure 2. Continuous Kalman Filter Cycle
Courtesy of: http://www.edn.com/design/analog/4413345/2/Using-adaptive-filtering-to-enhance-capacitive-sensing-of-buttons-sliders

7. Problem Statement
Estimate the true altitude &
velocity, without measurement
noise, of the UAV at each time
step:
Table 1: UAV’s Measured Altitude and Velocity
Time (seconds), Measured Altitude (meters), Measured Velocity
(meters/second)

8. Results- Time Update
1. Project the state ahead
𝑥0
−
= 𝐴 𝑥−1 =
1 1
0 1
150
−1
=
149
−1
2. Project the error covariance ahead
𝑃0
−
= 𝐴𝑃−1 𝐴 𝑇 =
1 1
0 1
∗ 𝐼 ∗
1 0
1 1
=
1 1
0 1
1 0
1 1
=
2 1
1 1

9. Results- Measurement Update
1. Compute the Kalman Gain
𝐾0 = 𝑃0
−
𝑃0
−
+ 𝑅 −1
=
2 1
1 1
2 1
1 1
+
1 0
0 1
−1
=
2 1
1 1
3 1
1 2
−1
=
0.6 0.2
0.2 0.4
2. Update the estimate via 𝑧𝑡
𝑥0 = 𝑥0
−
+ 𝐾0 𝑧0 − 𝑥0
−
=
149
−1
+
0.6 0.2
0.2 0.4
150.54
−0.47994
−
149
−1
=
149
−1
+
0.6 0.2
0.2 0.4
1.54
0.52006
=
149
−1
+
1.028
0.516
=
150.028
−0.4834
3. Update the error covariance
𝑃0 = 1 − 𝐾0 𝑃0
−
= 𝐼 −
0.6 0.2
0.2 0.4
2 1
1 1
=
.4 .8
.8 .6
2 1
1 1
=
1.6 1.2
2.2 1.4

10. Results- 10 Iterations
Table 2: Simulated Kalman filter with known state, shown as Actual Altitude (meters) and
Actual Velocity (meters/second), and the resulting Kalman estimate, shown as Estimated
Altitude (meters) and Estimated Velocity(meters/second). The error between the true value
and the estimated values are also shown as Altitude Difference (meters) and Velocity
Difference (meters/second).

11. Results- Graphics
Figure 5: Graphical representation of simulated Kalman Filter. (a) Actual altitude (orange line) plotted with altitude
estimated by the Kalman filter (blue line). (b) Error in altitude estimate as calculated by subtracting the estimated
altitude from the real simulated altitude. (c) Actual velocity (orange line) plotted with velocity estimated by the Kalman
filter (blue line). (d) Error in velocity estimate as calculated by subtracting the estimated velocity from the real simulated
velocity.

12. Conclusion
 Prediction and measurement
 Kalman filter is used as an
estimator for signals in the
presence of Gaussian noise.
 This filter provides a powerful
tool that is able to provide
estimation of past, present, and
even future states
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