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Improved Inertial Navigation With Cold Atom Interferometry

Abstract

This article discusses chances and challenges of using cold atom interferometers in inertial navigation. The error characteristics of the novel sensor are presented, as well as one option for an online estimation of the different readout errors. An extended Kalman filter framework is derived and analysed which uses the readout of the atom interferometer as observation in order to correct several systematic errors of a conventional IMU, allowing for an improved strapdown calculation in an arbitrary target system. The performance gain is discussed analytically based on the steady state variances of the filter, as well as on the example of a simulated scenario for Earth orbit satellites. The correction of the conventional IMU errors is further demonstrated in an experiment under laboratory conditions with a higher class sensor emulating an atom interferometer. While the application of the novel technology as a gyroscope is still limited, as pointed out in the paper, the presented framework yields options for a full six degree of freedom operation of the atom interferometer.

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ACKNOWLEDGMENTS

This work was sponsored by BMWi, project 50RK1957. The authors would like to thank [10] for providing the interferometer data for the experimental study in Section 5.1, as well as EXC-2123 QuantumFrontiers – 390837967 for providing a platform of exchange.

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Authors and Affiliations

Authors

Contributions

B.T. and N.W. wrote the first draft. B.T. developed the theory and performed the simulations. N.W. designed and performed the hexapod experiment. All authors discussed the results and contributed to the final manuscript.

Corresponding authors

Correspondence to B. Tennstedt, N. Weddig or S. Schön.

APPENDIX

APPENDIX

A System and Filter Equations

An error state EKF framework with full kinematic state vector will now be summarized.

The state vector contains 21 states:

$$X = \left[ {{{x}^{n}}\,{{v}^{n}}\,\theta \,\,\,{{b}_{a}}\,\,{{b}_{\boldsymbol{\omega} }}^{{}}\,\,\gamma \,\,\,l} \right].$$
(36)

Each element represents a \(3 \times 1\) vector. The first two elements contain the position and velocity triplets (in the n-frame). The third symbol is used to describe the roll, pitch and yaw angles of the system. They describe the orientation of the b-frame with regards to the n-frame. The remaining elements are the different systematic error terms of the system. The term ba is the accelerometer bias triplet and bω is the gyroscope bias triplet. Finally, γ stands for the misalignment between the (assumed) perfect CAI sensor frame and the sensor frame of the conventional IMU, while the last elements are the leverarm terms (offsets) between these frames.

The dynamic IMU behavior is modelled by the following system of differential equations. Terms which are constant with respect to the systematic error terms are omitted, as they are not relevant for the construction of the transition matrix:

$$\begin{gathered} {{{\dot {x}}}^{n}} = {{v}^{n}}, \\ {{{\dot {v}}}^{n}} = C_{b}^{n} \cdot {{{\hat {f}}}^{b}}, \\ {{{\hat {f}}}^{b}} = (I - \left[ {\gamma \times } \right])({{f}^{b}} - {{b}_{a}}) - (\hat {\boldsymbol{\omega} }_{{ib}}^{b} \times (\hat {\boldsymbol{\omega} }_{{ib}}^{b} \times l)), \\ \hat {\boldsymbol{\omega} }_{{ib}}^{b} = (I - \left[ {\gamma \times } \right])(\boldsymbol{\omega} _{{ib}}^{b} - {{b}_{\boldsymbol{\omega} }}), \\ \dot {C}_{b}^{n} = C_{b}^{n}\left[ {\boldsymbol{\omega} _{{nb}}^{b} \times } \right], \\ {{{\dot {b}}}_{a}} = 0, \\ {{{\dot {b}}}_{\boldsymbol{\omega} }} = 0, \\ \dot {\gamma } = 0, \\ l{\text{ = 0}} \\ \end{gathered} $$
(37)

Here, \({{\dot {x}}^{n}}\) and \({{\dot {v}}^{n}}\) are the derivatives of the position and velocity in the n-frame, \({{\hat {f}}^{b}}\) is the true specific acceleration, which consists of the specific force measured by the sensor \(({{f}^{b}})\). This measurement is corrected by the bias term of the accelerometer (ba), the misalignment (γ) and the pseudo-acceleration introduced by the leverarm (the double cross product term, which contains l). Finally, \(\hat {\boldsymbol{\omega} }_{{ib}}^{b}\) indicates the true angular rate, which is composed of the corrected measured angular rate, \(\boldsymbol{\omega} _{{ib}}^{b},\) as well as the gyroscope bias terms and misalignment matrix. The orientation \(C_{b}^{n}\) between n and b-frame is updated by the angular rates \(\boldsymbol{\omega} _{{nb}}^{b}\) that result from \(\boldsymbol{\omega} _{{ib}}^{b},\) reduced by Earth rate and transportation rate.

The resulting discrete-time solution of this system of differential equations is given by the following transition matrix. Note that the double cross product term in Eq. (37) has been simplified by eliminating off-diagonal products:

$$\Phi = \left( \begin{gathered} \begin{array}{*{20}{c}} {{{I}_{{3 \times 3}}}} \\ {{{0}_{{3 \times 3}}}} \\ {{{0}_{{3 \times 3}}}} \end{array}\,\,\,\,\begin{array}{*{20}{c}} {{{0}_{{3 \times 3}}}} \\ {{{I}_{{3 \times 3}}}} \\ {\,\,\,{{0}_{{3 \times 3}}}} \end{array}\begin{array}{*{20}{c}} {\Delta t{{{(I - \Gamma )}}_{\,}}} \\ {{{0}_{{3 \times 3}}}} \\ {{{I}_{{3 \times 3}}}} \end{array}\begin{array}{*{20}{c}} {{{0}_{{3 \times 3}}}} \\ {\Delta t(I - \Gamma )} \\ {{{0}_{{3 \times 3}}}} \end{array}\,\,\,\,\begin{array}{*{20}{c}} { - \Delta t\left[ {{{f}^{b}} \times } \right]\,} \\ { - \Delta t\left[ {{{\boldsymbol{\omega} }^{b}} \times } \right]\,} \\ {{{0}_{{3 \times 3}}}} \end{array}\,\,\,\begin{array}{*{20}{c}} {\Delta t\left[ {{{\boldsymbol{\omega} }^{b}} \times } \right]\,} \\ {{{0}_{{3 \times 3}}}} \\ {{{0}_{{3 \times 3}}}} \end{array} \hfill \\ \begin{array}{*{20}{c}} {{{0}_{{3 \times 3}}}} \\ {{{0}_{{3 \times 3}}}} \\ {{{0}_{{3 \times 3}}}} \end{array}\begin{array}{*{20}{c}} {\,\,\,\,\,\,{{0}_{{3 \times 3}}}\,\,\,\,\,\,} \\ {{{0}_{{3 \times 3}}}} \\ {{{0}_{{3 \times 3}}}} \end{array}\begin{array}{*{20}{c}} {{{0}_{{3 \times 3}}}} \\ {{{0}_{{3 \times 3}}}} \\ {{{0}_{{3 \times 3}}}} \end{array}\,\,\,\,\begin{array}{*{20}{c}} {\,\,\,\,\,{{I}_{{3 \times 3}}}} \\ {\,\,\,\,\,\,{{0}_{{3 \times 3}}}} \\ {\,\,\,\,\,\,{{0}_{{3 \times 3}}}} \end{array}\,\,\,\begin{array}{*{20}{c}} {\,\,\,\,\,\,\,\,\,\,\,\,{{0}_{{3 \times 3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}}}} \\ {{{I}_{{3 \times 3}}}} \\ {{{0}_{{3 \times 3}}}} \end{array}\,\,\,\,\begin{array}{*{20}{c}} {{{0}_{{3 \times 3}}}} \\ {{{0}_{{3 \times 3}}}} \\ {{{I}_{{3 \times 3}}}} \end{array} \hfill \\ \end{gathered} \right).$$
(38)

The derivative of the system function with respect to the inputs, denoted \(B = \partial f(x,u){\text{/}}\partial u,\) is also needed to compute the process noise matrix:

$$B = \left( \begin{gathered} \begin{array}{*{20}{c}} {{{I}_{{3 \times 3}}}}&{ - 2\boldsymbol{\omega} _{{ib}}^{b}} \end{array} \times l \hfill \\ \begin{array}{*{20}{c}} {{{0}_{{3 \times 3}}}}&{{{I}_{{3 \times 3}}}} \end{array} \hfill \\ \begin{array}{*{20}{c}} {{{0}_{{12 \times 3}}}}&{{{0}_{{12 \times 3}}}} \end{array} \hfill \\ \end{gathered} \right).$$
(39)

The process noise matrix is then given by the convolution of the noise variances in Qu over the integration time in the system Ф. By assuming a time invariant system it can be simplified to:

$${{Q}_{k}} = {{\Phi }_{k}}{{B}_{k}}{{Q}_{u}}B_{k}^{T}\Phi _{k}^{T}\Delta t.$$
(40)

Additional uncertainties for the remaining states are also added on the main diagonal of the resulting matrix Qk in order to stabilize the filter.

The filter boundary equations are set as follows:

$$\begin{gathered} {{x}_{0}} = 0, \hfill \\ {{w}_{k}} \sim \mathcal{N}(0,{{Q}_{k}}), \hfill \\ {{v}_{k}} \sim \mathcal{N}(0,{{R}_{k}}), \hfill \\ E\left[ {{{w}_{k}}w_{j}^{T}} \right] = {{Q}_{k}}{{\delta }_{{k - j}}}, \hfill \\ E\left[ {{{v}_{k}}v_{j}^{T}} \right] = {{R}_{k}}{{\delta }_{{k - j}}}, \hfill \\ E\left[ {{{v}_{k}}w_{j}^{T}} \right] = 0. \hfill \\ \end{gathered} $$
(41)

The initial state vector (x0) at t = 0 is set to 0, and the underlying noise processes of the input variables (\({{w}_{k}}\) with variance covariance matrix Qk) and measurements (\({{v}_{k}}\) with variance covariance matrix Rk) are assumed to be white.

The prediction step follows the standard error state Kalman filter form:

$$\begin{gathered} \hat {x}_{k}^{ - } = f(\hat {x}_{{k - 1}}^{ + },{{u}_{k}}), \\ P_{k}^{ - } = {{\Phi }_{k}} \cdot P_{{k - 1}}^{ + } \cdot \Phi _{k}^{T} + {{Q}_{k}}). \\ \end{gathered} $$
(42)

Here, the subscript k denotes the current iteration step. Additionally, a superscript “+” marks a filtered variable, while a “–” marks a predicted variable. Furthermore, a “hat” above the letter is used to express an estimated variable. P is the variance covariance matrix of the state, δx is the error of the state vector, and assumed to be zero after a filter step has been performed.

The filtering step is then carried out as in a normal EKF:

$$\begin{gathered} {{K}_{k}} = P_{k}^{ - } \cdot H_{k}^{T} \cdot {{({{H}_{k}} \cdot P_{k}^{ - } \cdot H_{k}^{T} + {{R}_{k}})}^{{ - 1}}}, \\ \delta \hat {x}_{k}^{ + } = {{K}_{k}} \cdot ({{p}_{k}} - h(\hat {x}_{k}^{ - })), \\ \hat {x}_{k}^{ + } = \hat {x}_{k}^{ - } + \delta \hat {x}_{k}^{ + }, \\ {{L}_{k}} = (I - {{K}_{k}} \cdot {{H}_{k}}), \\ P_{k}^{ + } = {{L}_{k}} \cdot P_{k}^{ - } \cdot L_{k}^{T} + {{K}_{k}} \cdot {{R}_{k}} \cdot K_{k}^{T}. \\ \end{gathered} $$
(43)

Kk is the Kalman gain matrix, and Hk the observation matrix, which has already been introduced for the different CAI operation modes in the paper for accelerometer and gyroscope bias, respectively. The observation equations for the leverarms and misalignment can be found in [17]. The resulting innovation is then added to the predicted state of the system.

B Derivation of the Linear CAI Phase Shift

The CAI phase shift as resulting from the atom position at the mid-point line is given by Eq. (44).

$$\phi = k \odot \left( {q(T) - 2q\left( {\frac{T}{2}} \right)} \right).$$
(44)

The differential equation for the velocity of the atoms yields

$$\dot {v} = f_{{ib}}^{{*b}} - 2\boldsymbol{\omega} _{{ib}}^{{*b}} \times \bar {v}.$$
(45)

Double integration leads to the position q.

$$q(t) = \int {\int\limits_0^t {\dot {v}{{d}^{2}}t} .} $$
(46)

Note that in contrast to Eq. (4) a different symbol is used for the atom velocity at the cross product, \(\bar {v}.\) This is necessary because in order to solve the integration analytically, a constant value of \(v\) is needed, which is obviously in contradiction to the differential equation. This newly introduced \(\bar {v}\) represents the atom velocity at the start of the measurement cycle. One typical value for the velocity is 2.8 m/s for the counterpropagating atom interferometer pair.

The integration yields the following results for the different position terms

$${{q}_{f}}\left( {\frac{T}{2}} \right) = \int {\int\limits_0^{\frac{T}{2}} {f_{{ib}}^{{*b}}} {{d}^{2}}t} = \frac{1}{8}f_{{ib}}^{{*b}}{{T}^{2}},$$
(47)
$${{q}_{f}}(T) = \int {\int\limits_0^T {f_{{ib}}^{{*b}}} {{d}^{2}}t} = \frac{1}{2}f_{{ib}}^{{*b}}{{T}^{2}},$$
(48)
$${{q}_{\boldsymbol{\omega} }}\left( {\frac{T}{2}} \right) = \int {\int\limits_0^{\frac{T}{2}} {2\boldsymbol{\omega} _{{ib}}^{{*b}}} } \times \bar {v}{{d}^{2}}t = \frac{1}{4}\boldsymbol{\omega} _{{ib}}^{{*b}} \times \bar {v}{{T}^{2}},$$
(49)
$${{q}_{\boldsymbol{\omega} }}(T) = \int {\int\limits_0^T {2\boldsymbol{\omega} _{{ib}}^{{*b}}} } \times \bar {v}{{d}^{2}}t = \boldsymbol{\omega} _{{ib}}^{{*b}} \times \bar {v}{{T}^{2}},$$
(50)

and in sum:

$$\phi = k \odot \left( {({{q}_{f}}(T) + {{q}_{\boldsymbol{\omega} }}(T)) - 2\left( {{{q}_{f}}\left( {\frac{T}{2}} \right) + {{q}_{\boldsymbol{\omega} }}\left( {\frac{T}{2}} \right)} \right)} \right),$$
(51)
$$ = k \odot \left( {\frac{1}{4}f_{{ib}}^{{*b}} - \frac{1}{2}\boldsymbol{\omega} _{{ib}}^{{*b}} \times \bar {v}} \right){{T}^{2}},$$
(52)

whereas \(f_{{ib}}^{{*b}}\) and \(\boldsymbol{\omega} _{{ib}}^{{*b}}\) may include the respective error models of the IMU.

One last word of caution. This linearised model of the phase shift in the scope of the EKF is only used to calculate the measurement sensitivity in matrix H and contains some simplifications that were explained before. In order to improve the convergence of the filter it may be reasonable to enhance the phase shift model based on the actual trajectory and dynamics the sensor is going to be used for, i.e., to include also higher order terms that result from the change of the acceleration and the turn rates during the measurement cycle.

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Tennstedt, B., Weddig, N. & Schön, S. Improved Inertial Navigation With Cold Atom Interferometry. Gyroscopy Navig. 12, 294–307 (2021). https://doi.org/10.1134/S207510872104009X

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  • DOI: https://doi.org/10.1134/S207510872104009X

Keywords:

  • inertial navigation
  • cold atom interferometry
  • hybridization
  • extended Kalman filter