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where (t) = (t) (t) For a well designed instrument, the bias and noise terms should be determined completely by events within the instrument (ie, una ected by external elds or by the motion of the device) In addition, although the values of bg and may change with time in a nondeterministic manner, the characteristics of the change should be in a form that enables accurate quantitative characterization of the growth of the angle error The desire to make accurate quantitative statements about characteristics of nondeterministic signals motivates the introduction to and study of stochastic processes t Returning to Figure 42, the integrated gyro signal (t) = 0 r( )d is shown as the bottom graph of Figure 42 We have returned to this gure to state an important point During this experiment, the gyro was rotated by 90 deg, left stationary, rotated by 90 deg, and left stationary again This motion is performed by a human The erratic nature of the angular rate r is due to the human experimenter and is not the object of the stochastic analysis The objects of the stochastic analysis are the sensor imperfections bg and , and the error that they cause Although we do not know the true values of or , based on stochastic analysis we will be able to quantitatively predict the expected magnitude of as a function of time, see Exercises 419 This example is further discussed in Exercise 57

The previous discussion and examples have been provided to motivate the material presented in this chapter The primary objective for this chapter is to present methods to model and analyze the behavior of the navigation error state as a function of time The state space error models of such stochastic processes will be generalized versions of eqn (46): x(t) = F(t)x(t) + G(t)w(t) y(t) = H(t)x(t) + v(t) (47)

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where the vector x represents the error in the navigation state The vectors w and v are random variables determined in part by the navigation instruments The matrices F(t) and H(t) are determined by the design of the navigation system Such models are discussed in considerable detail in Sections 46 410 To understand and analyze such models requires knowledge of several additional concepts Therefore, the chapter rst de nes several concepts and quantities that have proven to be useful for analysis of stochastic processes The main topics to be discussed include: de nitions of important statistics, de nitions of theoretical concepts (eg, independence, random variables, stationarity), de nition of white and colored noise processes, and properties of linear systems with random inputs Readers already familiar with these topics should skip directly to Section 46, while only using Sections 42 45 for reference

3 Type the name you want to give the event in place of New Event 4 Use the From date and time controls to set the time at which you want the script to run

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This section presents various fundamental concepts related to probability and random variables that will be required to support subsequent discussions The discussion will focus on continuous random variables

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Let w represent a real, scalar random variable In the following, the notation P {w < W } denotes the probability that the random variable w is less than the real number W The distribution function of the random variable w is de ned as the function (48) Fw (W ) = P {w W } The function Fw (W ) is monotonically increasing in W , is continuous from the right with Fw ( ) = 0 and Fw ( ) = 1 The derivative of the distribution function is the density function of the random variable w and will

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dFw (W ) dW The density function has the interpretation that pw (W ) represents the probability that the random variable w assumes a value in the di erential range (W, W + dW ] Because the density is the derivative of a monotone function, pw (W ) 0 for any W Based on the above de nitions, for a scalar random variable, pw (W ) = P {W1 < w W2 } = Fw (W2 ) Fw (W1 ) =

E g(w) =

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(49)

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