轻松掌握卡尔曼滤波追踪：《Tracking and Kalman Filtering Made Easy》简介

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processing orthonormal transformation methods to the DOLP approach used in

Section 5.3 for obtaining a polynomial ﬁt to data. The two methods are shown

to be essentially identical. The square-root Kalman ﬁlter, which is less sensitive

to round-off errors, is discussed in Section 14.5.

Up until now the deterministic part of the target model was assumed to be

time invariant. For example, if a polynomial ﬁt of degree m was used for the

target dynamics, the coefﬁcients of this polynomial ﬁt are constant with time.

Chapter 15 treats the case of time-varying target dynamics.

The Kalman and Bayes ﬁlters developed up until now depend on the

observation scheme being linear. This is not always the situation. For example,

if we are measuring the target range R and azimuth angle  but keep track of the

target using the east-north x, y coordinates of the target with a Kalman ﬁlter,

then errors in the measurement of R and  are not linearly related to the

resulting error in x and y because

x ¼ R cos  ð1Þ

and

y ¼ R sin  ð2Þ

where  is the target angle measured relative to the x axis. Section 16.2 shows

how to simply handle this situation. Basically what is done is to linearize

Eqs. (1) and (2) by using the ﬁrst terms of a Taylor expansion of the inverse

equations to (1) and (2) which are

R ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

þ y

ð3Þ

 ¼ tan

ð4Þ

Similarly the equations of motion have to be linear to apply the Kalman–

Bayes ﬁlters. Section 16.3 describes how a nonlinear equation of motion can be

linearized, again by using the ﬁrst term of a Taylor expansion of the nonlinear

equations of motion. The important example of linearization of the nonlinear

observation equations obtained when observing a target in spherical coordinates

(R, , ) while tracking it in rectangular (x, y, z) coordinates is given. The

example of the linearization of the nonlinear target dynamics equations

obtained when tracking a projectile in the atmosphere is detailed. Atmospheric

drag on the projectile is factored in.

In Chapter 17 the technique for linearizing the nonlinear observation

equations and dynamics target equations in order to apply the recursive Kalman

and Bayes ﬁlters is detailed. The application of these linearizations to a

nonlinear problem in order to handle the Kalman ﬁlter is called the extended

Kalman ﬁlter. It is also the ﬁlter Swerling originally developed (without the

xviii PREFACE

ACKNOWLEDGMENT

I would like to thank Fred Daum (Raytheon Company), who ﬁrst educated me

on the Kalman ﬁlter and encouraged me throughout this endeavor. I also would

like to thank Erwin Taenzer (formally of the Raytheon Company), from whose

many memos on the g–k and g–h–k ﬁlters I ﬁrst learned about these ﬁlters. I am

indebted to Barbara Rolinski (formerly of the Raytheon Company), who helped

give birth to this book by typing a good part of its ﬁrst draft, including its many

complicated equations. This she did with great enthusiasm and professionalism.

She would have ﬁnished it were it not that she had to leave to give birth to her

second child. I would also like to thank Barbara Rolinski for educating her

replacement on the typing of the text with its complicated equations using the

VAX Mass-11 Word Processor. I would like to thank Lisa Cirillo (formerly of

the Raytheon Company), Barbara Rolinski’s ﬁrst replacement, for typing the

remainder of the ﬁrst draft of the book. I am most appreciative of the help of

Richard P. Arcand, Jr. (Raytheon Company), who helped Barbara Rolinski and

Lisa Cirillo on difﬁcult points relative to the use of the VAX Mass-11 for typing

the manuscript and for educating Ann Marie Quinn (Raytheon Company) on the

use of the Mass-11 Word Processor. Richard Arcand, Jr. also meticulously made

the second-draft corrections for the ﬁrst part of the book. I am most grateful to

Ann Marie Quinn for retyping some of the sections of the book and making the

legion of corrections for the many successive drafts of the book. Thanks are also

due the Ofﬁce Automation Services (Raytheon Company) for helping to type

the second draft of the second part of the book. Sheryl Evans (Raytheon

Company) prepared many of the ﬁgures and tables for the book and for that I

am grateful. I am grateful to Richard Arcand, Jr. for converting the text to

Microsoft Word on the MAC. I am extremely grateful to Joanne Roche, who

completed the horrendous task of retyping the equations into Microsoft Word

xxi

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