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In this paper, we examine the effect of changing the temperature points on MEMS-based inertial sensor random error. We collect static data under different temperature points using a MEMS-based inertial sensor mounted inside a thermal chamber. Rigorous stochastic models, namely Autoregressive-based Gauss-Markov (AR-based GM) models are developed to describe the random error behaviour. The proposed AR-based GM model is initially applied to short stationary inertial data to develop the stochastic model parameters (correlation times). It is shown that the stochastic model parameters of a MEMS-based inertial unit, namely the ADIS16364, are temperature dependent. In addition, field kinematic test data collected at about 17 °C are used to test the performance of the stochastic models at different temperature points in the filtering stage using Unscented Kalman Filter (UKF). It is shown that the stochastic model developed at 20 °C provides a more accurate inertial navigation solution than the ones obtained from the stochastic models developed at −40 °C, −20 °C, 0 °C, +40 °C, and +60 °C. The temperature dependence of the stochastic model is significant and should be considered at all times to obtain optimal navigation solution for MEMS-based INS/GPS integration.

The performance of an integrated Global Positioning System (GPS)/Inertial Navigation System (INS) is mainly characterised by the ability of the INS to bridge GPS outages. In recent years, a promising technology namely, Micro-Electro-Mechanical Systems (MEMS)-based inertial sensors, has been developed, which can provide a low-cost navigation solution when integrated with GPS. MEMS systems are commonly fabricated using silicon, which possesses significant electrical and mechanical advantages over other materials [

The inertial sensor errors can be divided into two types: deterministic (systematic) errors and random errors [

The deterministic error sources include bias and scale factor errors, which can be removed by specific calibration procedures in a laboratory environment. Park and Gao [

On the other hand, the inertial sensor random errors primarily include the sensor noise, which consists of two parts, a high and a low frequency component. The high frequency component has white noise characteristics, while the low-frequency component is characterised by correlated noise [

A specific shortcoming in most of the above investigations is the disregard of the stochastic variation of these errors, which is of significant importance, and has not yet been investigated at different temperature points. The GPS/INS integrated system accuracy is significantly affected by the stochastic characteristics of the inertial navigation system [

This paper examines the effect of changing the temperature points on the MEMS inertial sensor noise models using for the first time a rigorous Autoregressive-based Gauss-Markov process (AR-based GM). In this work we collect static data sets under different temperature points using a MEMS-based IMU, namely the ADIS16364 [

In this section we briefly describe the Gauss-Markov (GM), and Autoregressive (AR) models and then we derive the AR-based GM model. For more details on stochastic modelling of inertial sensor errors see El-Diasty and Pagiatakis [

Gauss-Markov (GM) random processes are stationary processes that have exponential autocorrelation functions. GM processes are important because they represent a large number of physical processes with reasonable accuracy, while they exhibit a relatively simple mathematical formulation [_{c}, the model is described by the following continuous equation of time [

The autocorrelation function (see _{c}^{2} is the variance at zero time shift (τ = 0). The most important characteristic of the GM process is that it can represent bounded uncertainty, which means that any correlation coefficient at any time shift is less or equal to the correlation coefficient at zero time shift _{c} (correlation time) and

The discrete time model of GM process can be written as [

Thus, the discrete-time first-order GM model can take the form of _{k}

To avoid the problem of inaccurate modelling of inertial sensor random errors due to inaccurate autocorrelation function determination, we can apply another method for estimating inertial sensor errors, as introduced by Nassar [

An AR process of order _{k}_{k-i}_{i}_{k}_{k}

If we have a first-order AR model, then the discrete form will be [_{k}

To take the advantage of both, the AR and GM models, we choose the first-order GM model [_{c}_{1} and not from the autocorrelation function approximation. This is possible by combining

If we take the natural logarithm of both sides, then:

Therefore, the correlation time can be estimated from first-order AR model parameter _{1} as follows:

In this paper, we call this process AR-based GM model.

To examine the performance of the six stochastic models to be developed from the above static tests at different temperature points, dual frequency GPS data from a Trimble BD950 receiver and inertial data from the ADIS16364 IMU were collected on July 15, 2008 in Hamilton Harbour, Ontario, onboard the hydrographic surveying vessel “Merlin”, owned by the Canadian Hydrographic Service of the Department of Fisheries and Oceans. The kinematic test temperature was about 17 °C during the entire test time span.

The six static data sets were collected at a sampling rate of about 200 Hz at six different temperature points ranging from −40 °C to 60 °C for a period of 3 hours, which were then used to develop the six AR-based GM stochastic models (i.e., AR-based GM at −40 °C, −20 °C, 0 °C, +20 °C, +40 °C, and +60 °C) described in Section 3, for three gyro and three accelerometer bias errors. The correlation times for the AR-based GM model were estimated using

The UKF estimator is used to filter the kinematic data of ADIS16364 IMU mounted aboard vessel collected at +17 °C with three GPS receivers (one stationery base GPS receiver on land and two GPS receivers aboard vessel) to estimate three INS/GPS positions, three INS/GPS velocities, and three INS/GPS attitudes. The six AR-based GM stochastic model parameters (correlation times) developed in Sub-section 4.3 at different temperature points are implemented in the UKF estimator to find the stochastic model that provides the best navigation solution. To test the performance of the models, we estimate the INS-only solutions for northing, easting and heading during eight, 100 s GPS artificial outages using the UKF estimator in the prediction mode. The “true” northing and easting are estimated from two GPS receivers in differential mode (one base station GPS receiver and one rover GPS receiver aboard the vessel) whereas, the “true” heading is estimated from the two GPS antennas aboard vessel, separated by 2.37 m, as mentioned before.

Now, we estimate the overall root-mean-square (RMS) error for northing and easting for all eight, 100 s GPS outages.

In

To this end, we conclude that in order to have an optimal navigation solution, we should include in the processing stage of MEMS-based INS/GPS integration different stochastic model parameters at different temperature points with +20 °C interval and we should use the temperature-dependent stochastic model nearest to the real sensor temperature during the test.

This paper investigated the effect of changing the temperature points on the MEMS inertial sensor noise models using an AR-based GM model. The AR-based GM model estimation was achieved by using static data sets collected under different temperature points using ADIS16364 MEMS-based IMU, and showed that the estimated correlation times of an AR-based GM model for gyro and accelerometer biases are temperature-dependent. In addition, the AR-based GM models developed from stationary data sets collected at different temperature points were implemented in the UKF estimator to process and integrate inertial and GPS data collected in kinematic mode with the same inertial unit at +17 °C. The overall RMS error results from the UKF filter estimation of northing, easting and heading of eight GPS outages when compared with the “true” GPS-based position, and heading showed that the stochastic model should be developed from stationary data collected at or near the same temperature point at which the field test data were collected (i.e., stochastic model developed at +20 °C in this paper is the best model with real world kinematic data collected at 17 °C in this paper).

This research was supported partially by the Natural Sciences and Engineering Research Council (NSERC) of Canada and by the GEOIDE National Centre of Excellence (Canada). The data used in this paper were collected at the Space Instrumentation Laboratory (SIL) of York University lead by Prof. Ben Quine.

Autocorrelation function of the first-order Gauss-Markov process.

Autoregressive (AR) structure.

AR-based GM stochastic modelling steps.

The thermal/vacuum chamber and the position of the IMU during testing.

Field system used to collect kinematic data.

GPS test trajectory (blue) used to develop the model and GPS artificial outages (red) to test the model.

The three steps used to develop, test, and validate the AR-based GM model at different temperature points.

Correlation times for the three different gyros (X gyro, Y gyro, and Z gyro) at different temperature points.

Correlation times for the three accelerometers (X acc, Y acc, and Z acc) at different temperature points.

INS-only northing from kinematic test data collected at +17 °C during GPS outage#5 with AR-based GM model developed at different temperature points.

INS-only easting from kinematic test data collected at +17 °C during GPS outage#5 with AR-based GM model developed at different temperature points.

INS-only heading solution from kinematic test data collected at +17 °C during GPS outage#5 with AR-based GM model developed at different temperature points.

Overall RMS errors in northing using kinematic test data collected at +17 °C and the AR-based GM model developed at different temperature points.

Overall RMS errors in easting using kinematic test data collected at +17 °C and the AR-based GM model developed at different temperature points.

Overall RMS errors of the heading solution using kinematic test data collected at +17 °C and the AR-based GM model developed at different temperature points.

ADIS16364 IMU specifications [

| |

Initial bias error | ±3 °/s |

In-run bias stability | 0.007 °/s |

Bias temperature coefficient | ±0.01 °/s/ °C |

Angular Random Walk | 2 °/√h |

| |

| |

Initial bias error | ±8 mg |

In-run bias stability | 0.1 mg |

Bias temperature coefficient | ±0.0.05 mg/ °C |

Velocity Random Walk | 0.12 m/s/√h |