Astronautics Space Flight

Applied mathematics in integrated navigation systems by Robert M. Rogers

By Robert M. Rogers

He topic of built-in navigation platforms lined during this publication is designed for these at once concerned with the layout, integration, and attempt and overview of navigation structures. it really is assumed that the reader has a heritage in arithmetic, together with calculus. built-in navigation structures is the combo of an onboard navigation answer (position, pace, and angle) and self sufficient navigation facts (aids to navigation) to replace or right navigation ideas. during this e-book, this mixture is finished with Kalman filter out algorithms.

This presentation is segmented into elements. within the first half, components of easy arithmetic, kinematics, equations describing navigation systems/sensors and their errors types, aids to navigation, and Kalman filtering are constructed. distinct derivations are awarded and examples are given to assist within the figuring out of those components of built-in navigation platforms. difficulties are integrated to extend the appliance of the fabrics awarded.

The moment variation contains software program, extra history fabric and routines, and extra functions. chosen bankruptcy, part, and workout similar software program is supplied in a significant other CD-ROM to augment the training event of the reader. The incorporated software program has been built utilizing MATLAB/Simulink(TM) model 6.5 by way of The MathWorks, Inc. extra fabric contains: integrating navigation aides for a navigation system’s vertical axis; workouts that increase the scope of difficulties encountered in built-in navigation structures; and the overall challenge of angle decision and estimation even if for terrestrial or house functions. This variation presents a extra entire beginning for addressing the several points of built-in navigation structures.

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It will be shown later that the angle a, whose rotation direction is about the z axis (down) axis (shown in Fig. 4), satisfies the following differential equation: while the corresponding equation for the WA frames, whose rotation direction is Fig. 5 Wander azimuth for Fig. 2. COORDINATE SYSTEMS AND TRANSFORMATIONS Fig. 6 Wander azimuth for Fig. 3. about the z axis (up) axis (shown in Figs. 6), is Body Frame The body frame is rigidly attached to and defined within the vehicle carrying the navigation system.

4) where y: represents the transformed y axis after rotation 1. 5) Earth-Centered, Earth-Fixed to Navigation Frame (Wander Azimuth) This transformation matrix requires an additional transformation from the local geodetic to the navigation frame through the WA angle a , as shown in Fig. 6. 6) 0 Or, ca! c+ rh ca s@sh + sa ch -sa s4 sh 1 + sa! 7) 48 APPLIED MATHEMATICS IN INTEGRATED NAVIGATION SYSTEMS Earth-Centered, Earth-Fixed to Local Geodetic Refemng to Fig. 1, the transformation matrix from ECEF to local geodetic proceeds with the following sequence of three rotations: Rotation 1: A about z, Rotation 2: -4 about y: Rotation 3: align axes from up-east-north to north-east-down Or, Earth-Centered Inertial to Local Geodetic Frame The rotation from ECI to ECEF, as a result the Earth's rotation, is defined by the direction of the Earth's rotation.

In the rotated frame y, this vector is unchanged in length and is denoted as r y . It is desired to determine the component values associated with this vector in the rotated frame y. This frame is rotated with respect to the x frame by the angle @. 1 is redrawn in Fig. 2 with emphasis on a few triangle segments used to develop expressions for the vector components. Using simple trigonometry, the following relationships are established: XI =qcosq=> q=- B=qsin+=>B=- x1 cos 11/ x l sin + cos @ y1 -q = y sin @ = (x2 - B) sin + Fig.

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