This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 continuoustime 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 . Lets review this beast briefly. 1.2 Stochastic Processes Let ( , F ,P ) be a probability space. Definition 1. A stochastic process y ( t, ) is a rule for assigning to every a function y ( t, ). Equivalently, a stochastic process is a family of time functions parameterized by the outcomes . The domain of is the sample space, , and the domain of t is the set of real numbers, R . square Note that the dependence on is frequently omitted, and we usually use y ( t ) to denote the stochastic process y ( t, ) . 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 , y ( ,t ) is a deterministic function of time t . 1 Definition 2. The autocovariance 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 autocorrelation function R ( t 1 ,t 2 ) by R ( t 1 ,t 2 ) = E bracketleftbig y ( t 1 ) y T ( t 2 ) bracketrightbig . Definition 3 (Whiteness) . A stochastic process y ( t ) is called white if Cov ( y ( t 1 ) ,y ( t 2 )) = C ( t 1 ) ( t 1 t 2 ), where ( ) is the Diracdelta function. Meaning, the random variable y ( t 1 ) is uncorrelated with the random variable...
View Full
Document
 Fall '08
 Staff

Click to edit the document details