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For an AHRS application in which position and velocity information are in not available, n = 0 In this case, eqn (1046) is mechanized as b Rn = Rn b b ib (1052)
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Following a method similar to that of Section 10521, to derive the dynamic model for P, set the derivative of eqn (1032) equal to the left side of eqn (1046): PRn + (I + P)Rn b b which simpli es as follows, PRn + (I + P)Rn b b = (I + P)Rn ( b b + b ) in ib b ib n b n = (I + P)Rb + (I + P)Rb ib Rn b b in b b n n b b P = Rb ib Rn Rb in Rn (1054) = (I + P)Rn ( b b ) ib in b (1053)
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to rst order In vector form (using eqns (223) and (B15)), eqn (1054) is (1055) = Rn b n b ib in where b is the error in the gyro measurement of the body-frame inertialib ib relative angular rate vector By its de nition as b = b b , the ib ib
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ib de nition of b in eqn (1025), and the de nition of the gyro measurement in eqn (108), the gyro measurement error is modeled as b = xg g ib with xg = Fg xg + g Substituting eqn (1056) into eqn (1055) yields = Rn xg Rn g Rn n b b b in (1057) (1056)
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The vector n , derived on p 52 and reprinted in eqn (1051), is the in rotation rate of the navigation frame relative to the inertial frame represented in navigation frame Eqn (1057) shows that the lack of position and velocity information a ects the AHRS accuracy by the accumulation of attitude error at rates related to and In the description that follows, n in is treated as a white noise term driving the error state model In fact, this term is not white or even stationary Improved handling of this term could lead to improved performance The accumulated error is intended to be estimated and removed by the accelerometer and magnetometer aiding sensors
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Based on the above analysis, with the error state vector de ned as x = [ , xg , xa ] , we combine eqns (109), (1011), and (1057) to obtain the dynamic model for the state vector as 0 Rn 0 b xg = 0 0 xg Fg xa xa 0 0 Fa n Rn 0 Rn 0 in b b g + 0 I 0 0 g 0 0 0 I a We combine eqns (1041) and (1043) to obtain the error state measurement models as gn m
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Ha x + Rn a b Hm x Rn m b
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(1058) (1059)
with Ha and Hm de ned by eqns (1042) and (1045), respectively
CHAPTER 10 AHRS
Measurement Noise Covariance
The Kalman lter measurement update computations require the variance of the discrete-time aiding measurements This is straightforward for the magnetometer measurements:
2 Rm = var( m ) = m
The accelerometer aiding is more interesting In the subsequent discussion, let Ra denote the accelerometer measurement variance that will be used in the computation of the Kalman lter gains used for accelerometer aiding Note that Ra is not necessarily the same as var( a ), which is not known The challenge in using the accelerometer measurements for AHRS aiding is to limit the possibilities for the body frame acceleration ab to cause ib inaccuracy in the attitude estimate Consider the signal (t) = ya (t) ge (1060)
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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