trackingex

trackingex - Hnadout#2 supplemental notes for Application...

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Unformatted text preview: Hnadout #2 : supplemental notes for Application of Kalman filtering to Track a Moving Object Optimal Estimation (EML 6934, Section 6385), Fall 2009 University of Florida, Mechanical and Aerospace Engineering Prabir Barooah 1 Object Moving with “Constant” Velocity subject to Random Per- turbation 1.1 Dynamics in continuous-time A particle is moving in 2D. The four states of the particle are x 1 ( t ) : x- position x 2 ( t ) : x- velocity x 3 ( t ) : y- position x 4 ( t ) : y- velocity Say the particle is moving with “more or less” constant velocity. Meaning, that the force acting on the particle at every time t can be modeled as a zero mean random vector. In this case, the dynamics of particle motion can be written as ˙ x 1 ( t ) = x 2 ( t ) ˙ x 2 ( t ) = w 1 ( t ) ˙ x 3 ( t ) = x 3 ( t ) ˙ x 4 ( t ) = w 2 ( t ) where w 1 ( t ) and w 2 ( t ) are stochastic processes . Let’s review this beast briefly. 1.2 Stochastic Processes Let (Ω , F ,P ) be a probability space. Definition 1. A stochastic process y ( t,ω ) is a rule for assigning to every ω ∈ Ω a function y ( t,ω ). Equivalently, a stochastic process is a family of time functions parameterized by the outcomes ω . The domain of ω is the sample space, Ω, and the domain of t is the set of real numbers, R . square Note that the dependence on ω is frequently omitted, and we usually use y ( t ) to denote the stochastic process y ( t,ω ) . The following interpretations of stochastic process are useful: 1. For every fixed time t , y ( t ) is a random variable (or a random vector). 2. For every fixed ω , y ( ω,t ) is a deterministic function of time t . 1 Definition 2. The auto-covariance function C ( t 1 ,t 2 ) of a stochastic process y ( t ) is defined by C ( t 1 ,t 2 ) = Cov ( y ( t 1 ) ,y ( t 2 )) . and the auto-correlation function R ( t 1 ,t 2 ) by R ( t 1 ,t 2 ) = E bracketleftbig y ( t 1 ) y T ( t 2 ) bracketrightbig . Definition 3 (White-ness) . A stochastic process y ( t ) is called white if Cov ( y ( t 1 ) ,y ( t 2 )) = C ( t 1 ) δ ( t 1- t 2 ), where δ ( · ) is the Dirac-delta function. Meaning, the random variable y ( t 1 ) is uncorrelated with the random variable...
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trackingex - Hnadout#2 supplemental notes for Application...

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