Misalignment Error Computation

Date: December 02, 2024

The measurement model is given by

y_{k}^{m} = D m_{k}^{m} + o

y_{k}^{m} = D R^{m b} R_{k}^{b n} m_{k}^{n} + o

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

y_{k}^{m} = D \exp (ϕ^{\land}) {\bar{R}}^{m b} m_{k}^{b} + o

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

{\bar{R}}^{m b} = [\begin{matrix} - 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix}]

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

\frac{\partial y_{k}^{m}}{\partial ϕ} = D \frac{\partial}{\partial ϕ} (\exp (ϕ^{\land}) {\bar{R}}^{m b} m_{k}^{b})

Compute the derivative along the directions of each basis $ϕ_{i} 1_{i}$ .

\begin{aligned} \frac{\partial y_{k}^{m}}{\partial ϕ_{i}} & = lim_{h \to 0} D \frac{\exp (h 1_{i}^{\land}) {\bar{R}}^{m b} m_{k}^{b} - {\bar{R}}^{m b} m_{k}^{b}}{h} \\ = lim_{h \to 0} D \frac{(I + h 1_{i}^{\land}) {\bar{R}}^{m b} m_{k}^{b} - {\bar{R}}^{m b} m_{k}^{b}}{h} \\ = lim_{h \to 0} D \frac{h 1_{i}^{\land} ({\bar{R}}^{m b} m_{k}^{b})}{h} \\ = D 1_{i}^{\land} ({\bar{R}}^{m b} m_{k}^{b}) \\ = - D ({\bar{R}}^{m b} m_{k}^{b})^{\land} 1_{i} \end{aligned}

Stacking the three directional derivatives together yields

\frac{\partial y_{k}^{m}}{\partial ϕ} = - D ({\bar{R}}^{m b} m_{k}^{b})^{\land}

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

D = \tilde{D} V \approx \tilde{D} \hat{R} ({\bar{R}}^{m b})^{T}

The final result is

\frac{\partial y_{k}^{m}}{\partial ϕ} = - \tilde{D} \hat{R} ({\bar{R}}^{m b})^{T} ({\bar{R}}^{m b} m_{k}^{b})^{\land}