By Farrell, Jay
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Additional resources for Aided Navigation
To characterize the performance of the navigation system, we must start with a characterization of the accuracy of the instruments that are used as inputs to the kinematic model of eqn. 1). 4) a ˜(t) = a(t) − b(t) − ω1 (t). The signals ω1 and b represent imperfections in the sensor that can be modeled as stochastic processes. The signal ω1 represents random measurement noise. The quantity b represents a sensor bias. This bias is unknown and varies slowly with time. The sensor manufacturer provides the following model of the bias time variation ˙ = −λb b(t) + ωb (t) b(t) where λb is a positive constant and ωb is another a random process.
Similarly, underwater applications are denied access to satellite signals; however, a variety of acoustic ranging sensor systems exist. Understanding of the use of GPS as an aiding signal enables extension to underwater acoustic ranging systems. However, the speed of sound in water is signiﬁcantly slower and more variable than the speed of light in Earth’s atmosphere; therefore, depending on the desired level of performance characterization of the speed of sound in water and accommodation of the time of travel of the ranging signal can be critical.
Let the vector v denote the directed line segment from P1 to P2 . Relative to φ1 , v can be described as ⎡ ⎤1 x2 − x1 v = ⎣ y2 − y1 ⎦ z2 − z1 ⎤1 ⎡ ⎤1 x2 x1 = ⎣ y2 ⎦ − ⎣ y1 ⎦ z2 z1 ⎛⎡ ⎤1 ⎡ x2 xO ⎜ = ⎝⎣ yO ⎦ + R12 ⎣ y2 zO z2 ⎛⎡ ⎤1 ⎡ x1 xO ⎜ − ⎝⎣ yO ⎦ + R12 ⎣ y1 zO z1 ⎡ ⎤2 ⎞ ⎦ ⎟ ⎠ ⎤2 ⎞ ⎦ ⎟ ⎠ CHAPTER 2. REFERENCE FRAMES 40 ⎛⎡ ⎤2 ⎡ ⎤2 ⎞ x1 x2 ⎜ ⎟ = R12 ⎝⎣ y2 ⎦ − ⎣ y1 ⎦ ⎠ z2 z1 v1 = R12 v2 . 20) Eqn. 20) is the vector transformation between coordinate systems. This relation is valid for any vector quantity.
Aided Navigation by Farrell, Jay