M-estimator for the 3D symmetric Helmert coordinate transformation

Abstract

The M-estimator for the 3D symmetric Helmert coordinate transformation problem is developed. Small-angle rotation assumption is abandoned. The direction cosine matrix or the quaternion is used to represent the rotation. The $3 \times 1$ multiplicative error vector is defined to represent the rotation estimation error. An analytical solution can be employed to provide the initial approximate for iteration, if the outliers are not large. The iteration is carried out using the iterative reweighted least-squares scheme. In each iteration after the first one, the measurement equation is linearized using the available parameter estimates, the reweighting matrix is constructed using the residuals obtained in the previous iteration, and then the parameter estimates with their variance-covariance matrix are calculated. The influence functions of a single pseudo-measurement on the least-squares estimator and on the M-estimator are derived to theoretically show the robustness. In the solution process, the parameter is rescaled in order to improve the numerical stability. Monte Carlo experiments are conducted to check the developed method. Different cases to investigate whether the assumed stochastic model is correct are considered. The results with the simulated data slightly deviating from the true model are used to show the developed method’s statistical efficacy at the assumed stochastic model, its robustness against the deviations from the assumed stochastic model, and the validity of the estimated variance-covariance matrix no matter whether the assumed stochastic model is correct or not.

Appendix: An analytical solution of coordinate transformation

The coordinate transformation can be solved analytically if the coordinates at both reference frames are stochastically independent, as can be found in the geodetic literature (Grafarend and Awange 2003; Shen et al. 2006; Chang 2015). The computation steps are given as follows:

Calculate the barycenter coordinates (in both frames)

$$\begin{aligned} \bar{{{\varvec{x}}}}=\frac{{{1}}}{n}\sum _{i=1}^n {{\varvec{x}}_i } ,\bar{{{\varvec{y}}}}=\frac{{{1}}}{n}\sum _{i=1}^n {{\varvec{y}}_i } . \end{aligned}$$

(40)

Calculate the relative coordinates (to the barycenter)

$$\begin{aligned} \Delta {\varvec{x}}_i ={\varvec{x}}_i -\bar{{{\varvec{x}}}},\Delta {\varvec{y}}_i ={\varvec{y}}_i -\bar{{{\varvec{y}}}}. \end{aligned}$$

(41)

Calculate the DCM profile matrix

$$\begin{aligned} {\varvec{H}}=\sum _{i=1}^n {\Delta {\varvec{y}}_i \Delta {\varvec{x}}_i^T } . \end{aligned}$$

(42)

Calculate the singular value decomposition

$$\begin{aligned} \left( {{\varvec{{U,D,V}}}} \right) =\hbox {svd}\left[ {\varvec{{H}}} \right] ,s.t.,{\varvec{{H}}}={\varvec{{UDV}}}^{T} \end{aligned}$$

(43)

where ${{\varvec{U}}}$ and ${{\varvec{V}}}$ are orthogonal matrices, ${{\varvec{D}}}$ is diagonal with the diagonal elements being the singular values in the decreasing order. Calculate the DCM estimate

$$\begin{aligned} \hat{{{\varvec{R}}}}={\varvec{U}}\left[ {{\begin{array}{lll} 1&{}\quad 0&{}\quad 0 \\ 0&{}\quad 1&{} \quad 0 \\ 0&{} \quad 0&{} \quad {\det \left[ {\varvec{U}} \right] \det \left[ {\varvec{V}} \right] } \\ \end{array} }} \right] {\varvec{V}}^{T}. \end{aligned}$$

(44)

Calculate the quaternion profile matrix

(45)

with $\kappa =\sum _{i=1}^n {\Delta {\varvec{y}}_i^T \Delta {\varvec{x}}_i } =\hbox {trace}\left[ {\varvec{H}} \right]$ and ${\varvec{z}}=\sum \limits _{i=1}^n \left( \Delta {\varvec{x}}_i \times \Delta {\varvec{y}}_i \right)$. Calculate the largest eigenvalue and the corresponding eigenvector

$$\begin{aligned} \left( {\rho _{\max } ,{\varvec{q}}_{\max } } \right) =\hbox {eig}\left[ {\varvec{K}} \right] ,s.t.,{\varvec{Kq}}_{\max } =\rho _{\max } {\varvec{q}}_{\max } . \end{aligned}$$

(46)

Calculate the quaternion estimate

$$\begin{aligned} \hat{{{\varvec{q}}}}={\varvec{q}}_{\max } . \end{aligned}$$

(47)

Calculate the cross inertia moment

$$\begin{aligned} c=\hbox {trace}\left[ {\hat{{{\varvec{R}}}}{\varvec{H}}^{T}} \right] , \end{aligned}$$

(48)

or

$$\begin{aligned} c=\rho _{\max } . \end{aligned}$$

(49)

Calculate the scale estimate

$$\begin{aligned} \hat{{s}}=\frac{-d+f}{2c\sigma _x^2 } \end{aligned}$$

(50)

with $f=\sqrt{d^{2}+4\sigma _x^2 \sigma _y^2 c^{2}}$, $d=\sigma _y^2 a-\sigma _x^2 b$, $a=\sum \nolimits _{i=1}^n {\Delta {\varvec{x}}_i^T \Delta {\varvec{x}}_i }$, and $b=\sum \nolimits _{i=1}^n {\Delta {\varvec{y}}_i^T \Delta {\varvec{y}}_i }$. Calculate the translation estimate

$$\begin{aligned} \hat{{{\varvec{t}}}}=\bar{{{\varvec{y}}}}-\hat{{s}}\hat{{{\varvec{R}}}}\bar{{{\varvec{x}}}}. \end{aligned}$$

(51)

The analytical solutions with the DCM or the quaternions are summarized as follows:

Algorithm 2

(DCM): (40), (41), (42) $\rightarrow$ (43), (44) $\rightarrow$ (48), (50) $\rightarrow$ (51).

Algorithm 3

(quaternions): (40), (41), (42), (45) $\rightarrow$ (46), (47) $\rightarrow$ (49), (50) $\rightarrow$ (51).

The detailed proofs of the two algorithms, i.e., the derivation of the solutions, can be found in (Grafarend and Awange 2003; Chang 2015) for the DCM case, for example, and (Shen et al. 2006) for the quaternions case.