View
451
Download
1
Category
Preview:
Citation preview
APPLICATION OF THE KALMAN FILTER
ROHULLAH LATIF
ALICIA FESSLER
MATHEMATICAL METHODS
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
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
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
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
Study Approach
[1] xt = Atxt−1 + B𝑡u𝑡 + wt
[2]Pt−1 = AtPt−1 + At + Qt
[3] Kt = PtHt HPtH
t + 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
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)
Results- Time Update
1. Project the state ahead
𝑥0− = 𝐴 𝑥−1 =
1 10 1
150−1
=149−1
2. Project the error covariance ahead
𝑃0− = 𝐴𝑃−1𝐴
𝑇 =1 10 1
∗ 𝐼 ∗1 01 1
=1 10 1
1 01 1
=2 11 1
Results- Measurement Update
1. Compute the Kalman Gain
𝐾0 = 𝑃0− 𝑃0
− + 𝑅 −1 =2 11 1
2 11 1
+1 00 1
−1
=2 11 1
3 11 2
−1
=0.6 0.20.2 0.4
2. Update the estimate via 𝑧𝑡
𝑥0 = 𝑥0− + 𝐾0 𝑧0 − 𝑥0
− =149−1
+0.6 0.20.2 0.4
150.54−0.47994
−149−1
=149−1
+0.6 0.20.2 0.4
1.540.52006
=149−1
+1.0280.516
=150.028−0.4834
3. Update the error covariance
𝑃0 = 1 − 𝐾0 𝑃0− = 𝐼 −
0.6 0.20.2 0.4
2 11 1
=.4 .8.8 .6
2 11 1
=1.6 1.22.2 1.4
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).
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.
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
World GPS Image Link
Recommended