By Esmat Bekir

The rising expertise of very reasonably cheap inertial sensors is offered for navigation as by no means ahead of. The ebook lays the analytical origin for realizing and enforcing the navigation equations. It starts off through demystifying the imperative subject of the body rotation utilizing such algorithms because the quaternions, the rotation vector and the Euler angles. After constructing navigation equations, the publication introduces the computational matters and discusses the actual elements which are tied to imposing those equations. The publication then explains alignment ideas. creation to fashionable Navigation platforms deals an effective set of rules for polar navigation. It additionally exhibits tips to improve the functionality of the inertial process while aided through the worldwide Positioning procedure. it truly is a suitable textbook for senior undergraduate and graduate scholars in aeronautical and electric engineering. it will possibly even be used as a reference e-book for practitioners within the box.

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1 Introduction As discussed in Chapter 2, the direction cosine matrix (DCM) is the tool that transforms vectors from one coordinate frame to another. The elements of this matrix are the direction cosines of the principal axes of one frame to the other. However to construct the DCM from the direction cosines is not only unattractive, but also inconvenient. One reason is that the orthonormality property is lost and with it the length and orientation of the transformed vector. To preserve this property one can use the rotation matrix to construct the DCM.

The difference between them can be illustrated graphically by the example in Fig. 3. Fig. 3a depicts a twodimensional vector v in the x-y plane. Fig. 3b depicts the same vector (stationary) while the x-y frame rotates an angle I about the z-axis. Fig. 3c depicts the same vector as it rotates an angle I about the z-axis and the frame is stationary. As will be evident shortly, we call the matrix that rotates the frame, as in Fig. 3b, an outer transformation matrix, and the matrix that rotates the vector, as in Fig.

X-axis: Eq. 18) Using Eq. 27) we see that it corresponds to the DCM in Eq. 1). C x I 0 ª1 «0 c I « «¬0 s I 0º s I »» c I »¼ Thus this quaternion will perform a rotation I about the x-axis 2. y-axis: Eq. 19) which corresponds to the DCM given in Eq. 2) C y ș ªcș 0 sș º «0 1 0 »» « «¬sș 0 cș »¼ 3. z-axis: Eq. 20) which corresponds to the DCM given in Eq. 7 45 Conversion between Forms From the above we have four forms for expressing coordinate transformations: the DCM, the Euler angles, the rotation vector and the quaternion.