TABLE OF CONTENTS This chapter presents a review of the mathematics necessary for the study of integrated navigation systems. The material presented is brief. The reader is encouraged to seek more in-depth information than that presented in this review.
The following subjects are presented in this chapter:
vector/matrix algebra.
vector/matrix calculus.
linearization techniques.
direction cosine matrices.
miscellaneous mathematical topics.
The basic algebraic vector and matrix operations, e.g., addition, subtraction, multiplication, and inversion (division), are summarized. Next, vector calculus, including differentiation of scalars and vectors with respect to scalars and vectors, is reviewed and applied to the development of a least-squares estimate. This is also the approach used later for developing the Kalman filter algorithm. The calculus of vectors continues with the linearization of nonlinear equations to obtain linearized forms of these equations. Linearized error forms of nonlinear navigation equations are desired for implementation in the Kalman filter algorithm.
The establishment of coordinate systems and transformations of vector components between different systems using direction cosine matrices is presented next. The dynamic equations that describe the evolution of direction cosine matrices are developed. These direction cosine matrix dynamic equations are used extensively in integrated navigation systems. Finally presented are miscellaneous mathematical topics, including the angular velocity addition theorem, used extensively in later developments of navigation state equations, and other matrix calculus operations.
In this chapter and subsequent chapters that compose Part 1, problems are included, most with step-by-step instructions. These problems expand on the material presented and are drawn from experiences in integrated navigation...
