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

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