TABLE B2

Symbols Introduced in Sections 2 and 3

sensor observations and rotations
ωBgyroscope observations in the B-frame
aBaccelerometer observations in the B-frame
mBmagnetometer observations in the B-frame
Embedded Imageangular rate around the z-axis of the N- or L-frame
gNgravity vector in the N-frame
hNEMF vector in the N-frame
hDmagnetic declination
φ, θ, ψEuler angles roll, pitch, and yaw describing rotation from N-into the B-frame
Embedded Imagerotation matrix describing rotation from N-into the B-frame
sensor white noise (index marks sensor)
σstandard deviation of white noise component (index marks quantity)
heading propagation
ψ0initial heading
ψκlast updated absolute heading
κ|kaccumulated heading change
k or κtime indices indicating epochs or timestamps
dttime interval
bias-EKF
Δsnmatrix representing sensor scale and non-orthogonality of magnetometer
Δsimatrix representing soft-iron effects of platform in the magnetometer
Embedded Imagevector representing magnetic anomalies
Embedded Imagevector representing hard-iron effects of platform in the magnetometer
Embedded Imagevector representing sensor bias of the magnetometer
Embedded Imagevector subsuming slow-varying systematic deviations of the magnetometer
Embedded Imagevector subsuming fast-varying systematic deviations of the magnetometer
Embedded Image, Embedded Imagepredicted and estimated Embedded Image (EKF)
ζδsystem noise of Embedded Image (EKF)
wmfilter innovation of bias-EKF
Embedded Image, Embedded ImageEKF residuals (index marks quantity) with VCM
TG, TG, c (α)global test value and its critical value
αtype I error of global test
Embedded Imagechi-square distribution (index indicates three DoFs)
λnon-centrality parameter of chi-square distribution
H0, HAnull and alternative hypothesis
E{}expectation of random variable
heading update
dtW, u, nW, ulength of time window for updating ψ and number of values contained
dtW, 0, nW ,0length of time window for controlling ψ0 and number of values contained
nT, PTnumber of TG,i values exceeding TG,c (α) and corresponding percentage
nm, pmnumber of Embedded Image fulfilling conditions c1 and c2 and corresponding percentage
c1condition that the magnitudes of Embedded Image and hN must be the same
c2condition that the vertical components of Embedded Image and hN must be the same
γpreset threshold of maximum heading change in time window
γψpreset threshold for difference in propagated and magnetometer heading
γmpreset threshold of Embedded Image values that must fulfill c1 and c2
γσmpreset multiplier for σm acting as a threshold for c1 and c2