The Guaranteeing Estimation Method to Calibrate a Gyro Unit

Abstract

This paper is devoted to the guaranteeing estimation method with application to the calibration problem of a gyro unit. Mathematical models are constructed to describe the kinematics of the gyro unit on a test bench. The applicability limits and errors of the models are investigated. A numerical solution procedure is developed for guaranteeing estimation problems based on their reduction to l₁-approximation problems.

APPENDIX

Proof of Lemma 1. On the time interval T the rotation occurs about a fixed direction. Therefore, the rotation matrix D_cir is decomposed by averaging as follows:

$${{\bar {D}}_{{{\text{cir}}}}} = {{\bar {D}}_{{{\text{cir1}}}}} + {{\bar {D}}_{{{\text{cir2}}}}},\quad {{\bar {D}}_{{{\text{cir1}}}}} = \left( {\begin{array}{*{20}{c}} 0&0&0 \\ 0&0&0 \\ 0&0&1 \end{array}} \right),\quad {{\bar {D}}_{{{\text{cir2}}}}} = \left( {\begin{array}{*{20}{c}} {{{c}_{1}}}&{ - {{c}_{2}}}&0 \\ {{{c}_{2}}}&{{{c}_{1}}}&0 \\ 0&0&0 \end{array}} \right),$$

(A.1)

where c_i is the result of the time averaging of the functions sin ψ(t) and cos ψ(t).

Due to formulas (6), (7), and (A.1), in the course of averaging, the vector u_z is represented as the sum of two terms, one proportional to w and the other orthogonal to D(0)w:

$$\begin{gathered} {{{\bar {u}}}_{z}} = \bar {D}{{u}_{x}} = D{\kern 1pt} '{\kern 1pt} {{{\bar {D}}}_{{{\text{cir1}}}}}{{D}_{{{\text{fix}}}}}{{u}_{x}} + D{\kern 1pt} '{\kern 1pt} {{{\bar {D}}}_{{{\text{cir2}}}}}{{D}_{{{\text{fix}}}}}{{u}_{x}} \\ = D{\kern 1pt} '\left( \begin{gathered} {{0}_{{1 \times 3}}} \\ {{0}_{{1 \times 3}}} \\ {{w}^{{\text{T}}}} \\ \end{gathered} \right){{u}_{x}} + {{u}^{ \bot }} = d_{3}^{'}{{w}^{{\text{T}}}}{{u}_{x}} + {{u}^{ \bot }} = {{D}_{{\operatorname{init} }}}({{I}_{3}} + \hat {\beta })w{{w}^{{\text{T}}}}{{u}_{x}} + {{u}^{ \bot }}, \\ \end{gathered} $$

(A.2)

where u^⊥ = $D{\kern 1pt} '{\kern 1pt} {{\bar {D}}_{{{\text{cir2}}}}}{{D}_{{{\text{fix}}}}}{{u}_{x}}$.

The orthogonality of u^⊥ to the direction D(0)w = D_init(I₃ + $\hat {\beta }$)w can be established using formulas (2) and (6): D(0)w = D′D_cir(0)D_fixw = D'(0, 0, 1)^T; the corresponding scalar product is explicitly calculated as

$${{w}^{{\text{T}}}}D{{(0)}^{{\text{T}}}}{{u}^{ \bot }} = (0,\,\,0,\,\,1)D{\kern 1pt} {{'}^{{\text{T}}}}D{\kern 1pt} '\left( {\begin{array}{*{20}{c}} {{{c}_{1}}}&{ - {{c}_{2}}}&0 \\ {{{c}_{2}}}&{{{c}_{1}}}&0 \\ 0&0&0 \end{array}} \right){{D}_{{{\text{fix}}}}}{{u}_{x}} = {{(0,\,\,0,\,\,1)}^{{\text{T}}}}\left( {\begin{array}{*{20}{c}} {{{c}_{1}}}&{ - {{c}_{2}}}&0 \\ {{{c}_{2}}}&{{{c}_{1}}}&0 \\ 0&0&0 \end{array}} \right){{D}_{{{\text{fix}}}}}{{u}_{x}} = 0.$$

Using the component c₁ as an example, we explain the idea of estimating from above the result of the time averaging of the function cos ψ(t). Consider the continuous case of averaging:

$${{c}_{1}} = \frac{1}{T}\int\limits_0^{\text{T}} {\cos \psi (t)dt.} $$

The dynamics of the angle ψ are described by a differential equation and constraints on the functions on its right-hand side:

$$\frac{{d\psi (t)}}{{dt}} = s + \varepsilon (t),\quad \psi (0) = {{\psi }_{0}},\quad \left| {\varepsilon (t)} \right|\;\leqslant \;{{\varepsilon }_{{\max }}},\quad s + \varepsilon (t) > 0.$$

The change of variables t = t(ψ), $\epsilon$(ψ) = ε(t(ψ)), |$\epsilon$(ψ)| $\leqslant$ ε_max, in the integral yields

$$\int\limits_0^{\text{T}} {\cos \psi (t)dt} = \int\limits_{{{\psi }_{0}}}^{\psi (T)} {\frac{{\cos \psi }}{{s + \epsilon (\psi )}}d\psi .} $$

This integral can be written as the sum of integrals on the half-periods of the function cosψ (intervals where the function has a fixed sign) and two integrals corresponding to the time intervals at the beginning and end of the interval [ψ₀, ψ(T)]. For example, if ψ₀ < π/2, this interval is represented as follows:

$$[{{\psi }_{0}},\psi (T)] = [{{\psi }_{0}},\pi {\text{/}}2] \cup [\pi {\text{/}}2,3\pi {\text{/}}2] \cup [3\pi {\text{/}}2,5\pi {\text{/}}2] \cup \ldots \cup [\pi {\text{/}}2 + 2\pi {{n}_{{{\text{cir}}}}},\psi (T)],$$

where n_cir is the number of complete revolutions of the system about the axis of rotation and the length of the last interval does not exceed π, i.e., π/2 + 2πn_cir $\leqslant$ ψ(T) $\leqslant$ 3π/2 + 2πn_cir.

The integrand on each such interval has a fixed sign, and the maximum value of the integrand (hence, that of the integral) is achieved at $\epsilon$(ψ) = –sgn(cos ψ)ε_max:

$$\int {\frac{{\cos \psi }}{{s + \epsilon (\psi )}}d\psi } \;\leqslant \;\int {\frac{{\cos \psi }}{{{{{\min }}_{{\left| \epsilon \right|\;\leqslant \;{{\varepsilon }_{{\max }}}}}}(s + \epsilon )}}d\psi } = \int {\frac{{\cos \psi }}{{s - \operatorname{sgn} (\cos \psi ){{\epsilon }_{{\max }}}}}d\psi .} $$

Therefore, each integral can be estimated bilaterally (from below and above):

$$\left| {\int\limits_{{{\psi }_{0}}}^{\pi /2} {\frac{{\cos \psi }}{{s + \epsilon (\psi )}}d\psi } } \right|\;\leqslant \;\frac{2}{{s - {{\varepsilon }_{{\max }}}}},\quad \left| {\int\limits_{\pi /2 + 2\pi {{n}_{{{\text{cir}}}}}}^{\psi (T)} {\frac{{\cos \psi }}{{s + \epsilon (\psi )}}d\psi } } \right|\;\leqslant \;\frac{2}{{s - {{\varepsilon }_{{\max }}}}},$$

$$\frac{{ - 2}}{{s - {{\varepsilon }_{{\max }}}}}\;\leqslant \;\int\limits_{\pi /2}^{3\pi /2} {\frac{{\cos \psi }}{{s + \epsilon (\psi )}}d\psi } \;\leqslant \;\frac{{ - 2}}{{s + {{\varepsilon }_{{\max }}}}},\quad \frac{2}{{s + {{\varepsilon }_{{\max }}}}}\;\leqslant \;\int\limits_{3\pi /2}^{5\pi /2} {\frac{{\cos \psi }}{{s + \epsilon (\psi )}}d\psi } \;\leqslant \;\frac{2}{{s - {{\varepsilon }_{{\max }}}}}.$$

As a result, the absolute value of the integral on the averaging interval admits the following upper bound:

$$\left| {\int\limits_{{{\psi }_{0}}}^{\psi (T)} {\frac{{\cos \psi }}{{s + \varepsilon (\psi )}}d\psi } } \right|\;\leqslant \;\left| {\int\limits_{{{\psi }_{0}}}^{\pi /2} {\frac{{\cos \psi }}{{s + \varepsilon (\psi )}}d\psi } } \right| + \left| {\int\limits_{\pi /2 + 2\pi {{n}_{{{\text{cir}}}}}}^{\psi (T)} {\frac{{\cos \psi }}{{s + \varepsilon (\psi )}}d\psi } } \right|$$

$$ + \;\left| {\sum\limits_{j = 1}^{{{n}_{{{\text{cir}}}}}} {\left( {\int\limits_{\pi /2}^{3\pi /2} {\frac{{\cos \psi }}{{s + \varepsilon (\psi )}}d\psi } + \int\limits_{3\pi /2}^{5\pi /2} {\frac{{\cos \psi }}{{s + \varepsilon (\psi )}}d\psi } } \right)} } \right|$$

$$\leqslant \;\frac{4}{{s - {{\varepsilon }_{{\max }}}}} + \left| {\sum\limits_{j = 1}^{{{n}_{{{\text{cir}}}}}} {\frac{{ - 2}}{{s + {{\varepsilon }_{{\max }}}}}} + \frac{2}{{s - {{\varepsilon }_{{\max }}}}}} \right|\;\leqslant \;\frac{4}{{s - {{\varepsilon }_{{\max }}}}} + \frac{{{{n}_{{{\text{cir}}}}}4{{\varepsilon }_{{\max }}}}}{{(s + {{\varepsilon }_{{\max }}})(s - {{\varepsilon }_{{\max }}})}}.$$

The angular rate and the number of complete revolutions of the system are related by

$$sT = 2\pi \,{{n}_{{{\text{cir}}}}} + \Delta \psi $$

for some Δψ $\leqslant$ 2π. Consequently,

$$\left| {{{c}_{1}}} \right| = \left| {\frac{1}{T}\int\limits_0^{\text{T}} {\cos \psi (t)dt} } \right|\;\leqslant \;\frac{4}{{T(s - {{\varepsilon }_{{\max }}})}} + \frac{{{{n}_{{{\text{cir}}}}}4{{\varepsilon }_{{\max }}}}}{{T({{s}^{2}} - \varepsilon _{{\max }}^{2})}}$$

$$ = \frac{4}{{T(s - {{\varepsilon }_{{\max }}})}} + \frac{{(sT - \Delta \psi )4{{\varepsilon }_{{\max }}}}}{{2\pi sTs(1 - \varepsilon _{{\max }}^{2}{\text{/}}{{s}^{2}})}} = \frac{4}{{T(s - {{\varepsilon }_{{\max }}})}} + \frac{{2(1 - \Delta \psi {\text{/}}(sT))}}{{\pi (1 - \varepsilon _{{\max }}^{2}{\text{/}}{{s}^{2}})}}\frac{{{{\varepsilon }_{{\max }}}}}{s}.$$

Thus, we obtain

$$\left| {{{c}_{1}}} \right|\;\leqslant \;\frac{4}{{T(s - {{\varepsilon }_{{\max }}})}} + C\frac{{{{\varepsilon }_{{\max }}}}}{s},$$

where the parameter C = $\frac{2}{{\pi (1 - \varepsilon _{{\max }}^{2}{\text{/}}{{s}^{2}})}}$ is an upper bound for the fraction $\frac{{2(1 - \Delta \psi {\text{/}}(sT))}}{{\pi (1 - \varepsilon _{{\max }}^{2}{\text{/}}{{s}^{2}})}}$.

Proof of Proposition 1. We transform the integrand of the objective function into problem (18) by substituting formulas (12) and (13) with the additional notation ${v}$ = ${v}$(s, y) = D_init(sy + yy^Tu_x):

$${{\Phi }^{{\text{T}}}}z = {{\Phi }^{{\text{T}}}}\left( {\Gamma {{D}_{{\operatorname{init} }}}(sy + y{{y}^{{\text{T}}}}{{u}_{x}}) + {{\nu }_{0}} + r + \delta \nu {\kern 1pt} '} \right)$$

$$ = {{\Phi }^{{\text{T}}}}\Gamma {v} + {{\Phi }^{{\text{T}}}}{{\nu }_{0}} + {{\Phi }^{{\text{T}}}}\delta \nu {\kern 1pt} '\; + {{\Phi }^{{\text{T}}}}{{D}_{{\operatorname{init} }}}\left( { - s(\hat {y}\alpha + \hat {y}\beta ) + \varepsilon y + {{y}^{{\text{T}}}}{{u}_{x}}(\hat {\alpha } + \hat {\beta })y + y{{y}^{{\text{T}}}}{{{\hat {u}}}_{x}}\alpha } \right)$$

$$ = {{({v} \otimes \Phi )}^{{\text{T}}}}\gamma + {{\Phi }^{{\text{T}}}}{{\nu }_{0}} + {{\Phi }^{{\text{T}}}}\delta \nu {\kern 1pt} '\; + {{\Phi }^{{\text{T}}}}{{D}_{{\operatorname{init} }}}\left( { - s(\hat {y}\alpha + \hat {y}\beta ) + \varepsilon y - {{y}^{{\text{T}}}}{{u}_{x}}(\hat {y}\alpha + \hat {y}\beta ) + y{{y}^{{\text{T}}}}{{{\hat {u}}}_{x}}\alpha } \right).$$

Hence,

$$\begin{gathered} {{\Phi }^{{\text{T}}}}z = {{({v} \otimes \Phi )}^{{\text{T}}}}\gamma + {{\Phi }^{{\text{T}}}}{{\nu }_{0}} + {{\Phi }^{{\text{T}}}}\delta \nu {\kern 1pt} '\; + \varepsilon {{\Phi }^{{\text{T}}}}{{D}_{{\operatorname{init} }}}y \\ + \;{{\Phi }^{{\text{T}}}}{{D}_{{\operatorname{init} }}}\left( { - s\hat {y} - {{y}^{{\text{T}}}}{{u}_{x}}\hat {y} + y{{y}^{{\text{T}}}}{{{\hat {u}}}_{x}}} \right)\alpha + {{\Phi }^{{\text{T}}}}{{D}_{{\operatorname{init} }}}\left( { - s\hat {y} - {{y}^{{\text{T}}}}{{u}_{x}}\hat {y}} \right)\beta . \\ \end{gathered} $$

(A.3)

These formulas involve, first, the properties of matrix operations

$${{\Phi }^{{\text{T}}}}\Gamma {v} = ({{\Phi }^{{\text{T}}}} \otimes {{{v}}^{{\text{T}}}})\gamma = {{({v} \otimes \Phi )}^{{\text{T}}}}\gamma ,\quad \hat {\alpha }y = - \hat {y}\alpha $$

and, second, the possibility of transferring the scalar product y^Tu_x to the other part of the corresponding multiplier group: $\hat {\alpha }y{{y}^{{\text{T}}}}{{u}_{x}}$ = $- {{y}^{{\text{T}}}}{{u}_{x}}\hat {y}\alpha$.

Let us define the matrices $C_{\alpha }^{'}$ and $C_{\beta }^{'}$:

$$C_{\alpha }^{'} = {{D}_{{\operatorname{init} }}}\left( { - s\hat {y} - {{y}^{{\text{T}}}}{{u}_{x}}\hat {y} + y{{y}^{{\text{T}}}}{{{\hat {u}}}_{x}}} \right),\quad C_{\beta }^{'} = {{D}_{{\operatorname{init} }}}\left( { - s\hat {y} - {{y}^{{\text{T}}}}{{u}_{x}}\hat {y}} \right).$$

Then the right-hand side of (A.3) is represented as a function that linearly depends on the variables q, α, β, ε, and $\delta \tilde {\nu }$:

$${{\Phi }^{{\text{T}}}}(y,s)z(y,s) = {{({v} \otimes \Phi )}^{{\text{T}}}}\gamma + {{\Phi }^{{\text{T}}}}{{\nu }_{0}} + {{\Phi }^{{\text{T}}}}\delta \nu {\kern 1pt} '\; + {{\Phi }^{{\text{T}}}}C_{\alpha }^{'}\alpha + {{\Phi }^{{\text{T}}}}C_{\beta }^{'}\beta + \varepsilon {{\Phi }^{{\text{T}}}}{{D}_{{\operatorname{init} }}}y.$$

(A.4)

Substituting formula (A.4) into the original objective functional (19) yields

$$\begin{gathered} I(\Phi ) = \mathop {\sup }\limits_{(q,\alpha ,\beta ,\varepsilon ,\delta \nu ') \in \mathcal{B}'} \left| {l(\Phi ) - {{a}^{{\text{T}}}}q} \right| \\ = \mathop {\sup }\limits_{(q,\alpha ,\beta ,\varepsilon ,\delta \nu ') \in \mathcal{B}'} \left| {\int {\left( {{{{({v} \otimes \Phi )}}^{{\text{T}}}}\gamma + {{\Phi }^{{\text{T}}}}{{\nu }_{0}} + {{\Phi }^{{\text{T}}}}\delta \nu {\kern 1pt} '\; + {{\Phi }^{{\text{T}}}}C_{\alpha }^{'}\alpha + {{\Phi }^{{\text{T}}}}C_{\beta }^{'}\beta + \varepsilon {{\Phi }^{{\text{T}}}}{{D}_{{\operatorname{init} }}}y} \right)dyds - {{a}^{{\text{T}}}}q} } \right|. \\ \end{gathered} $$

Since q = col(γ, ν₀), the function l(Φ) – a^Tq linear depends on q, and the multiplier at q is

$$\left( \begin{gathered} \int {{v} \otimes \Phi dyds} \\ \int {\Phi dyds} \\ \end{gathered} \right) - a.$$

Therefore, if condition (22) is violated, we have ${{\sup }_{{q \in {{{\mathbf{R}}}^{{12}}}}}}$|l(Φ) – a^Tq| = +∞ for a fixed Φ and arbitrary admissible α, β, ε, and δν'. Consequently,

$$\mathop {\sup }\limits_{(q,\alpha ,\beta ,\varepsilon ,\delta \nu ') \in \mathcal{B}'} \left| {l(\Phi ) - {{a}^{{\text{T}}}}q} \right| = \mathop {\sup }\limits_{(q,\alpha ,\beta ,\varepsilon ,\delta \nu ') \in \mathcal{B}'} \left| {\int {\left( {{{\Phi }^{{\text{T}}}}\delta \tilde {\nu } + {{\Phi }^{{\text{T}}}}C_{\alpha }^{'}\alpha + {{\Phi }^{{\text{T}}}}C_{\beta }^{'}\beta + \varepsilon {{\Phi }^{{\text{T}}}}{{D}_{{\operatorname{init} }}}y} \right)dyds} } \right|.$$

In other words, it is necessary to maximize the absolute value of a linear function where each term depends on only one variable not figuring in the other terms. This means that the maximum can be found independently in each of the variables. For a fixed Φ, the maximum is determined in an explicit form:

$$\mathop {\sup }\limits_{\alpha :\left| {{{\alpha }_{i}}} \right|\leqslant {{\alpha }_{{\max }}}} \int {{{\Phi }^{{\text{T}}}}C_{\alpha }^{'}\alpha dyds} = \mathop {\sup }\limits_{\alpha :\left| {{{\alpha }_{i}}} \right|\leqslant {{\alpha }_{{\max }}}} \int {\left( {\sum\limits_{i = 1}^3 {{{{(C{{{_{\alpha }^{'}}}^{{\text{T}}}}\Phi )}}_{i}}{{\alpha }_{i}}} } \right)dyds} $$

$$ = \sum\limits_{i = 1}^3 {\mathop {\sup }\limits_{{{\alpha }_{i}}:\left| {{{\alpha }_{i}}} \right|\leqslant {{\alpha }_{{\max }}}} } \int {{{{(C{{{_{\alpha }^{'}}}^{{\text{T}}}}\Phi )}}_{i}}{{\alpha }_{i}}dyds} $$

$$ = \int {\left( {\sum\limits_{i = 1}^3 {{{\alpha }_{{\max }}}\operatorname{sgn} ({{{(C{{{_{\alpha }^{'}}}^{{\text{T}}}}\Phi )}}_{i}}){{{(C{{{_{\alpha }^{'}}}^{{\text{T}}}}\Phi )}}_{i}}} } \right)dyds} = \int {{{{\left\| {{{C}_{\alpha }}\Phi } \right\|}}_{1}}dyds.} $$

A similar chain of considerations applies to the other terms in the objective functional (18). Thus, the explicit calculation of the supremum of the original objective functional finally leads to the optimization problem (21)–(22).