### Abstract

Stochastic modeling is a challenging task for low-cost sensors whose errors can have complex spectral structures. This makes the tuning process of the INS/GNSS Kalman filter often sensitive and difficult. For example, first-order Gauss-Markov processes are very often used in inertial sensor models. But the estimation of their parameters is a non-trivial task if the error structure is mixed with other types of noises. Such an estimation is often attempted by computing and analyzing Allan variance plots. This contribution demonstrates solving situations when the estimation of error parameters by graphical interpretation is rather difficult. The novel strategy performs direct estimation of these parameters by means of the expectation-maximization (EM) algorithm. The algorithm results are first analyzed with a critical and practical point of view using simulations with typically encountered error signals. These simulations show that the EM algorithm seems to perform better than the Allan variance and offers a procedure to estimate first-order Gauss-Markov processes mixed with other types of noises. At the same time, the conducted tests revealed limits of this approach that are related to the convergence and stability issues. Suggestions are given to circumvent or mitigate these problems when complexity of error structure is 'reasonable'. This work also highlights the fact that the suggested approach via EM algorithm and the Allan variance may not be able to estimate the parameters of complex error models reasonably well and shows the need for new estimation procedures to be developed in this context. Finally, an empirical scenario is presented to support the former findings. There, the positive effect of using the more sophisticated EM-based error modeling on a filtered trajectory is highlighted.

Original language | English (US) |
---|---|

Article number | 085204 |

Journal | Measurement Science and Technology |

Volume | 22 |

Issue number | 8 |

DOIs | |

State | Published - Aug 2011 |

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### All Science Journal Classification (ASJC) codes

- Instrumentation
- Engineering (miscellaneous)
- Applied Mathematics

### Cite this

*Measurement Science and Technology*,

*22*(8), [085204]. https://doi.org/10.1088/0957-0233/22/8/085204

}

*Measurement Science and Technology*, vol. 22, no. 8, 085204. https://doi.org/10.1088/0957-0233/22/8/085204

**Constrained expectation-maximization algorithm for stochastic inertial error modeling : Study of feasibility.** / Stebler, Yannick; Guerrier, Stéphane; Skaloud, Jan; Victoria-Feser, Maria Pia.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Constrained expectation-maximization algorithm for stochastic inertial error modeling

T2 - Study of feasibility

AU - Stebler, Yannick

AU - Guerrier, Stéphane

AU - Skaloud, Jan

AU - Victoria-Feser, Maria Pia

PY - 2011/8

Y1 - 2011/8

N2 - Stochastic modeling is a challenging task for low-cost sensors whose errors can have complex spectral structures. This makes the tuning process of the INS/GNSS Kalman filter often sensitive and difficult. For example, first-order Gauss-Markov processes are very often used in inertial sensor models. But the estimation of their parameters is a non-trivial task if the error structure is mixed with other types of noises. Such an estimation is often attempted by computing and analyzing Allan variance plots. This contribution demonstrates solving situations when the estimation of error parameters by graphical interpretation is rather difficult. The novel strategy performs direct estimation of these parameters by means of the expectation-maximization (EM) algorithm. The algorithm results are first analyzed with a critical and practical point of view using simulations with typically encountered error signals. These simulations show that the EM algorithm seems to perform better than the Allan variance and offers a procedure to estimate first-order Gauss-Markov processes mixed with other types of noises. At the same time, the conducted tests revealed limits of this approach that are related to the convergence and stability issues. Suggestions are given to circumvent or mitigate these problems when complexity of error structure is 'reasonable'. This work also highlights the fact that the suggested approach via EM algorithm and the Allan variance may not be able to estimate the parameters of complex error models reasonably well and shows the need for new estimation procedures to be developed in this context. Finally, an empirical scenario is presented to support the former findings. There, the positive effect of using the more sophisticated EM-based error modeling on a filtered trajectory is highlighted.

AB - Stochastic modeling is a challenging task for low-cost sensors whose errors can have complex spectral structures. This makes the tuning process of the INS/GNSS Kalman filter often sensitive and difficult. For example, first-order Gauss-Markov processes are very often used in inertial sensor models. But the estimation of their parameters is a non-trivial task if the error structure is mixed with other types of noises. Such an estimation is often attempted by computing and analyzing Allan variance plots. This contribution demonstrates solving situations when the estimation of error parameters by graphical interpretation is rather difficult. The novel strategy performs direct estimation of these parameters by means of the expectation-maximization (EM) algorithm. The algorithm results are first analyzed with a critical and practical point of view using simulations with typically encountered error signals. These simulations show that the EM algorithm seems to perform better than the Allan variance and offers a procedure to estimate first-order Gauss-Markov processes mixed with other types of noises. At the same time, the conducted tests revealed limits of this approach that are related to the convergence and stability issues. Suggestions are given to circumvent or mitigate these problems when complexity of error structure is 'reasonable'. This work also highlights the fact that the suggested approach via EM algorithm and the Allan variance may not be able to estimate the parameters of complex error models reasonably well and shows the need for new estimation procedures to be developed in this context. Finally, an empirical scenario is presented to support the former findings. There, the positive effect of using the more sophisticated EM-based error modeling on a filtered trajectory is highlighted.

UR - http://www.scopus.com/inward/record.url?scp=79960770197&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79960770197&partnerID=8YFLogxK

U2 - 10.1088/0957-0233/22/8/085204

DO - 10.1088/0957-0233/22/8/085204

M3 - Article

AN - SCOPUS:79960770197

VL - 22

JO - Measurement Science and Technology

JF - Measurement Science and Technology

SN - 0957-0233

IS - 8

M1 - 085204

ER -