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The signal (t) di ers from zero due to noise a , biases xa , and acceleration aib Typical values of a and xa can be characterized based on speci cations from the accelerometer manufacturer Therefore, it is possible to use (t) as an indicator of the magnitude of aib The indicator (t) could be used in at least two ways 1 The indicator (t) could be used to in uence Ra An example might be 2 (1061) R a = a + R If R is an increasing function of , then the Kalman gain for the accelerometer corrections would decrease as (ie, the acceleration) increases 2 In an asynchronous measurement aiding approach, the indicator (t) could be used, together with additional conditions, to select time instants tai when the acceleration should be small Such time instants are expected to be appropriate for accelerometer aiding The second approach is somewhat more consistent with the stochastic character of the Kalman lter design The application presented in Section 108 uses the asynchronous aiding approach and computes Ra as in eqn (1061) The condition used to select the accelerometer aiding times is de ned in eqn (1070) Many alternative conditions could be de ned There is of course a tradeo If the conditions are made too stringent, then accelerometer aiding will rarely if ever occur and the tilt errors may grow with time If the conditions are made too loose, then the vehicle acceleration may corrupt the attitude estimate through the accelerometer aiding

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Given the initial estimates of Section 103, we can approximate the initial covariance matrix of the error state x = [ , xg , xa ] The initialization procedure assumed that the vehicle was stationary, non-accelerating, and non-rotating, over some initial period of time t [t1 , t2 ] with T = t2 t1 With the de nition of the error covariance matrix there is a con ict between the symbols used to indicate [ ] and error covariance In the balance of this chapter, P with no subscript or superscript denotes [ ] while any use of P to denote an error covariance will contain a subscript of a superscript See for example eqns (1063) or (1066) The initial values of the gyro biases are estimated by averaging the gyro outputs 1 T g u( )d (1062) xb (t2 ) = T 0

Refer to the text in this chapter if necessary A good score is at least 18 correct Answers are in the back of the book 1 The output of a rectifier circuit without a filter included is (a) 60 Hz ac (b) smooth dc (c) pulsating dc (d) 120 Hz ac 2 Which of the following components is not necessarily required in a power supply designed to produce 12-V dc output with 117-V rms ac input (a) The transformer (b) The filter (c) The rectifier (d) All of the above components are required

1 With b = 0 and T small relative to g so that xg can be considered ib constant over the initialization period, then

g E xb (t2 ) g var xb (t2 )

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