Calibration of a 3D Sensor under Its Orientation Constraint

Abstract

Three-dimensional (3D) sensors usually require a calibration procedure. In some cases, scale factor errors depend on the signs of the projections of the vector input signal onto the sensitivity axes of the sensor. To eliminate the ambiguity of scale factor errors, the angular positions of the sensor can be restricted so that the corresponding projections have a definite sign. This paper presents an analytical solution of the optimal calibration problem for a 3D sensor under a constraint on its angular positions.

APPENDIX

We begin with proving the inequality $\det \mathcal{H}$ ≠ 0 in (33). Let ${{\mathcal{H}}_{\delta }}$ be the numerical image of the matrix $\mathcal{H}$: ${{\mathcal{H}}_{\delta }}$ = $\mathcal{H}$ + $\delta \mathcal{H}$. Also, let B denote the inverse of the matrix $\mathcal{H}$ calculated approximately; the matrix B is precisely known. Then $B{{\mathcal{H}}_{\delta }}$ + Δm = I + $\Delta \mathcal{H}$, where Δm is the error matrix when multiplying the matrices B and $\mathcal{H}$, and Δ$\mathcal{H}$ is the known error characterizing the inversion accuracy. From these equalities it follows that

$$B\mathcal{H} = I + \Delta \mathcal{H} - \Delta m - B\delta \mathcal{H}.$$

(A.1)

Direct calculations of the matrix Δ$\mathcal{H}$ show that its elements satisfy the inequality |(Δ$\mathcal{H}$)_ij| $\leqslant$ 10^–14, i, j = 1, …, 9. Let the elements Δm and δ$\mathcal{H}$ obey the constraints

$$\left| {\Delta {{m}_{{ij}}}} \right|\;\leqslant \;\epsilon ,\quad \left| {\delta {{\mathcal{H}}_{{ij}}}} \right|\;\leqslant \;\epsilon ,\quad i,j = 1, \ldots ,9,\quad {\text{where}}\;\;\epsilon \ll 1.$$

Then $\left| {{{{(B\delta \mathcal{H})}}_{{ij}}}} \right|\;\leqslant \;\epsilon \sum\nolimits_{s = 1}^9 {\left| {{{B}_{{is}}}} \right|}$ and, consequently,

$$\left| {{{{(\Delta \mathcal{H} - \Delta m - B\delta \mathcal{H})}}_{{ij}}}} \right|\;\leqslant \;{{10}^{{ - 14}}} + \epsilon \left[ {1 + \sum\limits_{s = 1}^9 {\left| {{{B}_{{is}}}} \right|} } \right]\quad i,j = 1, \ldots ,9.$$

The elements of the known matrix B belong to the intervals 0.1 < |B_ij| < 12. Therefore, |(Δ$\mathcal{H}$ – Δm – $B\delta \mathcal{H}$)_ij| $\leqslant$ 109$\epsilon$ + 10^–14. Assume that $\epsilon$ $\leqslant$ 10^–5; this can be ensured by modern computing means. Then the matrix I + Δ$\mathcal{H}$ – Δm – Bδ$\mathcal{H}$ in (A.1) is diagonally dominant and, hence, nonsingular by the Levy–Desplanques theorem [31]. As a result, the same property applies to the matrices $\mathcal{H}$ and B.

Consider the case a = a₍₁₎. According to (A.1),

$$\Phi = {{\mathcal{H}}^{{ - 1}}}a = {{(I + \Delta \mathcal{H} - \Delta m - B\delta \mathcal{H})}^{{ - 1}}}Ba = (I + W)Ba = {{\Phi }_{{{\text{calc}}}}} + \Delta \Phi ,$$

where Φ_calc = Ba is the calculated value of Φ, ΔΦ = WBa is the error of calculations, and

$$\left\| W \right\|\;\leqslant \;\frac{{\left\| {\Delta \mathcal{H} - \Delta m - B\delta \mathcal{H}} \right\|}}{{1 - \left\| {\Delta \mathcal{H} - \Delta m - B\delta \mathcal{H}} \right\|}},$$

with ||W|| standing for the spectral norm of W. Then ||ΔΦ|| $\leqslant$ ||W|| ||Ba||, where ||ΔΦ|| and ||Ba|| are the Euclidean norms of the corresponding vectors. Obviously,

$$\left\| W \right\|\;\leqslant \;\frac{{{{{\left\| {\Delta \mathcal{H}} \right\|}}_{{\text{F}}}} + {{{\left\| {\Delta m + B\delta \mathcal{H}} \right\|}}_{{\text{F}}}}}}{{1 - {{{\left\| {\Delta \mathcal{H}} \right\|}}_{{\text{F}}}} - {{{\left\| {\Delta m + B\delta \mathcal{H}} \right\|}}_{{\text{F}}}}}};$$

here, the subscript F indicates the Frobenius norm as a majorant for the spectral norm. (It is more difficult to estimate the error of calculations for the spectral norm.) Therefore,

$$\left\| W \right\|\;\leqslant \;\frac{{3\epsilon R + {{{10}}^{{ - 13}}}}}{{1 - 3\epsilon R - {{{10}}^{{ - 13}}}}},\quad R = \sqrt {\sum\limits_{i = 1}^9 {{{{\left[ {1 + \sum\limits_{s = 1}^9 {\left| {{{B}_{{is}}}} \right|} } \right]}}^{2}}} } .$$

Let the accuracy of calculating the value R not exceed $\epsilon$. In view of ||Ba|| $\leqslant$ 16, we have

$$\left\| W \right\|\;\leqslant \;\frac{{429\epsilon + 3{{\epsilon }^{2}} + {{{10}}^{{ - 13}}}}}{{1 - (429\epsilon + 3{{\epsilon }^{2}} + {{{10}}^{{ - 13}}})}}\quad {\text{and}}\quad \left\| {\Delta \Phi } \right\|\;\leqslant \;\frac{{0.069 \times {{{10}}^{5}}\epsilon + 3{{\epsilon }^{2}} + 2 \times {{{10}}^{{ - 12}}}}}{{1 - (0.005 \times {{{10}}^{5}}\epsilon + 3{{\epsilon }^{2}} + {{{10}}^{{ - 12}}})}}.$$

Thus, ||ΔΦ|| $\leqslant$ 0.07 for $\epsilon$ = 10^–5 and ||ΔΦ|| $\leqslant$ 0.007 for $\epsilon$ = 10^–6.

Since the known elements of the vector Ba = Ba₍₁₎ are not less than 1.8 in absolute value, we have proved that approximate calculations surely establish the signs of the components of the vector Φ. For ν = 4 and ν = 7, the considerations are similar.