Misalignment Error Computation

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The measurement model is given by

ykm=Dmkm+o ykm=DRmbRkbnmkn+o

Form a small rotation perturbation on the left of the “base” alignment rotation.

ykm=Dexp(ϕ)R¯mbmkb+o

According to the relative orientation of the sensors, the “base” alignment rotation can be approximated by

R¯mb=[100001010]

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Compute the partial derivative of the measurement model w.r.t. the angle perturbation.

ykmϕ=Dϕ(exp(ϕ)R¯mbmkb)

Compute the derivative along the directions of each basis ϕi1i.

ykmϕi=limh0Dexp(h1i)R¯mbmkbR¯mbmkbh=limh0D(I+h1i)R¯mbmkbR¯mbmkbh=limh0Dh1i(R¯mbmkb)h=D1i(R¯mbmkb)=D(R¯mbmkb)1i

Stacking the three directional derivatives together yields

ykmϕ=D(R¯mbmkb)

Note that, as described here, the model coefficients D cannot be obtained directly. So it is approximated by

D=D~VD~R^(R¯mb)T

The final result is

ykmϕ=D~R^(R¯mb)T(R¯mbmkb)