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CHAPTER 10 AHRS
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not a new attitude mechanization approach It is a vector convenient for modeling and correcting the attitude error Because the three attitude mechanization approaches are all theoretically equivalent, the attitude error representation can be used with any of the three mechanization approaches For error analysis, the transformation from the actual to the computed navigation frame can be represented via a multiplicative small angle transformation (I P) where the skew-symmetric matrix P has the representation 0 D E 0 N (1028) P = [ ] = D E 0 N With the above de nitions, we have that Rn b = (I P)Rn b (1029)
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The following relations are derived directly from eqn (1029) and will be useful in the subsequent analysis: Rb n Rb n Rn b Rb (I + P) n = Rb (I P) n = (I + P)Rn =
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(1030) (1031) (1032)
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Because Rn and Rn are both orthogonal, P = P, and (I P) 1 = (I+P) b b to rst order, eqns (1030 1032) are accurate to rst order
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Measurement Error Model
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For both the magnetometer and the accelerometer, a vector measurement is obtained in the body frame for which a value is known in the navigation frame Section 10511 will consider that measurement scenario in general, then Sections 10512 and 10513 will specialize the result to the accelerometer and magnetometer measurements 10511 General Analysis
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Let a vector r have the known value y = rn in navigation frame A measurement of y is available in body frame that is modeled as y b = rb + x y + y (1033)
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where xy represented sensor errors that will be calibrated and y represents other sensor errors that will be modeled as white noise The computed value of the measurement in navigation frame is yn = Rn yb xy b (1034)
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105 ERROR MODELS
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where xy is an estimate of xy if one is available The error model for yn is yn = yn yn
(1035) P)Rn b
r + xy + y xy = y (I n n n = Pr Rb ( xy + y ) + PRb ( xy + y ) = [rn ] Rn ( xy + y ) b (1036) where we have used the fact that Prn = [rn ] As de ned in eqn (B15), the matrix [rn ] is the skew symmetric matrix form of the vector rn The term PRn ( xy + y ) is dropped because it is second order in the b error quantities 10512 Accelerometer Analysis
By eqn (1010), the accelerometer measurement is ya = ab gb + xa + a ib (1037)
For the AHRS application, we are interested in using the gravity vector to estimate attitude We are not interested in the acceleration vector aib Therefore, the body frame acceleration ab is an error source, which can ib change rapidly Without additional information, ab cannot be estimated; ib therefore, we treat it as a measurement error by de ning a = a + ab so ib that (1038) ya = gb + xa + a From eqn (1038), the body frame estimate of the gravity vector is gb = xa ya (1039)
The measurement noise vector a is not white, nor is it stationary The measurement noise vector a is white and stationary Therefore, the strategy that is used in the proposed Kalman lter is to select time instants tai when aib (tai ) 0 and to use accelerometer aiding only at those time instants By this approach a (tai ) a (tai ) and the measurement noise at the instants of the accelerometer aiding can be modeled as white and stationary The method for de ning the times tai is de ned in Section 1054 and eqn (1070) Because the accelerometer measurement of eqn (1038) includes gb , the analysis of the residual measurement of eqn (1036) changes slightly, but the result is similar The residual gravity measurement in navigation frame de ned as gn = gn gn where gn = Rn gb The residual has the error model
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