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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Micro Electro Mechanical System (MEMS)-based inertial sensors have made possible the development of a civilian land vehicle navigation system by offering a low-cost solution. However, the accurate modeling of the MEMS sensor errors is one of the most challenging tasks in the design of low-cost navigation systems. These sensors exhibit significant errors like biases, drift, noises; which are negligible for higher grade units. Different conventional techniques utilizing the Gauss Markov model and neural network method have been previously utilized to model the errors. However, Gauss Markov model works unsatisfactorily in the case of MEMS units due to the presence of high inherent sensor errors. On the other hand, modeling the random drift utilizing Neural Network (NN) is time consuming, thereby affecting its real-time implementation. We overcome these existing drawbacks by developing an enhanced Support Vector Machine (SVM) based error model. Unlike NN, SVMs do not suffer from local minimisation or over-fitting problems and delivers a reliable global solution. Experimental results proved that the proposed SVM approach reduced the noise standard deviation by 10–35% for gyroscopes and 61–76% for accelerometers. Further, positional error drifts under static conditions improved by 41% and 80% in comparison to NN and GM approaches.

Micro Electro Mechanical System (MEMS)-based inertial sensors (accelerometers and gyroscopes) have been embraced by the auto industry in their quest to improve performance, reduce cost and to enhance the reliability of the vehicles [

The inertial sensor errors are divided into two main parts: systematic/deterministic and dynamic/random. The deterministic error sources mainly include the biases and the scale factor errors, which remain constant during a run and can be removed by specific calibration procedures in a laboratory environment [

Thus, before deploying MEMS based accelerometers and gyroscopes for vehicular navigation, an accurate determination of systematic and random errors is required to ensure the acceptable performance. The deterministic errors can be estimated using different lab calibration procedures as explained in [

The paper has been divided into five sections. Section 2 covers the conventional approaches of modeling the MEMS sensor errors. Section 3 explains the working of support vector regression. Experimental setup and the calibration results obtained using conventional and the proposed approaches are detailed in Section 4, along with their impact on the navigation solution accuracy. Finally, Section 5 concludes the paper.

There are numbers of errors like bias, scale factor, cross-axis sensitivity or misalignment, noise and temperature drifts that affect the performance of inertial sensors [

_{turn-on}_{temp}_{in-run}

The scale factor is the ratio of a change in output to a change in the intended input to be measured [

Thus, the static portion consisting of turn-on to turn-on biases, temperature dependent biases and a deterministic portion of in-run biases are evaluated using specific lab calibration procedure as discussed later in Section 2.2, whereas the remaining portion of in-run errors (also known as random errors) are estimated using stochastic processes such as GM process and Allan variance methodology. Typically, modeling the random portion of in-run biases using the above mentioned processes involves cumbersome mathematical calculations and is often inaccurate [

Crista inertial sensor is a small 3-axis MEMS device that consists of three single-axis ADXRS 610 Analog devices gyroscopes and a tri-axis KXD94-2802 accelerometer by Kionix [

The manufacturer passes the raw data through a low-pass filter to remove high frequency noise components before data is sampled by a 16-bit analog-to-digital converter. Die temperatures are also measured on each gyroscope and compared to a 10-point temperature calibration table stored in the memory (EEPROM) of the IMU. This in-built procedure corrects the measured sensor outputs by removing any temperature-induced errors and non-linearities. IMU also stores correction matrices to account for any misalignment errors and linear acceleration sensitivities so as to maximize its off-shelf accuracy. The user controls both the data update rate and the over-sample averaging output rate of the unit.

The following section briefly explains the existing methods for estimating the static biases using a six-position static test method for accelerometer and a simple averaging method for gyroscopes. Thermal calibration method for quantifying the drifts under varying temperature is given next, followed by the process to evaluate GM model parameters.

The six-position static and rate tests are among the most commonly used calibration methods [

Thermal calibration is required to compensate for the thermal drifts of MEMS IMUs. There are two main approaches for thermal testing,

The autocorrelation function of a discrete signal is the product of the random signal with a time-shifted version of itself. Therefore, it is widely used for characterization of correlated and slowly drifting noises. If

On applying forward Euler integration,

Therefore:

Random walk is another stochastic process obtained when white noise is integrated. In a mechanization process (a process of converting gyroscope and accelerometer data into navigation parameters), both gyroscope and accelerometer signals, which contains white noise components, are integrated to obtain change in angles and velocities. Hence, angles and velocities are corrupted with these integrated white noise components, called angular random walk and velocity random walk, which are usually obtained through the Allan variance method.

Allan variance is a method of representing root mean square random drift error as a function of averaged time [

For very low-cost MEMS IMUs, we are mainly concerned with the noise terms whose correlation time is much shorter than the sample time and contribute to the gyroscope Angle Random Walk (ARW) and accelerometer Velocity Random Walk (VRW) noise components, along with slowly drifting in-run biases. The values for ARW and VRW are obtained from (10):

RBFNN, one of the artificial intelligence approaches, is used for modeling the functional relationship corresponding to the given input-output sample pairs [

To model the random drift using existing RBFNN, we make use of the autocorrelation property. According to this property, the measurement system output at time instant

Thus, the RBFNN models the functional relationship using the input-output sample pairs as depicted by _{n}_{n}_{n}

Support vector machines as described in [

Given a set of input-output sample pairs {(_{1}, _{1}_{2}, _{2}_{n}, _{n})} the objective of Nu-SVR technique is to approximate the nonlinear relationship given in

In ^{T}^{(}*^{)} = _{i}_{j}_{i}_{j}_{i}^{T}_{j}

Usually there are four types of kernels used, namely, polynomial function, Radial Basis Function (RBF), sigmoid function and linear function. Selecting an appropriate kernel for a given problem improves the model prediction accuracy. In our study we selected an RBF kernel, as it delivers an acceptable accuracy and has less implementation difficulties [

Thus, given input-output training sample pairs, the Nu-SVR approach identifies the Lagrange multipliers

Laboratory tests were conducted on static data to identify the various error terms for Crista IMU. The manufacturer has already removed the fixed bias, temperature-induced errors and non-linearities from the raw IMU data and therefore they are not taken into account during MEMS error modeling. To model the remaining errors, three different approaches are considered in this paper which includes two traditional approaches: (a) determining static and dynamic biases through specific lab calibration procedures (as explained in Section 2.2.1–2.2.4); (b) An RBFNN method as described in Section 2.3; and (c) lastly, the proposed Nu-SVR approach. The detailed experimental processes and results of MEMS error modeling using the above three processes are described next.

We performed laboratory calibration of all the three gyroscopes and three accelerometers for low-cost Crista IMU. The turn-on to turn-on biases were calculated by averaging the static data collected for 1 minute for the gyroscopes (at 85 Hz) under stabilized room temperature for 10 runs taken at different times, over a period of 10 days (

The turn-on to turn-on bias obtained for the gyroscope and accelerometer is 0.23 °/s and 0.12 m/s^{2}. Next the in-run biases were calculated using 1 min of static data collected after 30 min intervals over a 90 min duration, when the IMU temperature had stabilized. The results obtained using three gyroscopes are shown in

For gyroscopes, simple averaging was performed for 1 min of data while for accelerometers, the six-position static test method was incorporated. The in-run bias obtained for gyroscope is 0.25 °/s while that specified by the manufacturer is less than 0.20 °/s. Similarly, for the accelerometer the obtained average value is 0.12 m/s^{2}, which is higher than the manufacturer value of 0.02 m/s^{2} at constant temperature. Further, to obtain the in-run biases under varying temperature conditions, we heated the IMU continuously using a heat gun (including a warm-up period). The internal temperature of the Crista IMU is observed for 1.5 h including the warm-up period where the temperature of the IMU continuously increases as per the Ramp method.

The temperature of the IMU is allowed to change from 20 °C to 75 °C which is the operating temperature range of the Crista IMU. The results obtained for the gyroscope drifts are illustrated in ^{2}. Further, it is illustrated in

Static data was collected for 12 h and was processed by Allan variance method to obtain velocity and angular random walk parameters. A log-log plot of

The initial downward slope of −1/2 indicates the IMU's primary error source is angular or velocity random walk as hypothesized. Further, effects of temperature changes on biases of gyroscope and accelerometer were determined by running the IMU continuously for a period of 12 h while the heat gun was occasionally used to vary the temperature of the IMU, including the warm-up period. The obtained ARW and VRW parameters are given in

IMU sensor errors are generally modeled by first-order Gauss Markov process that requires two parameters,

The correlation time (in sec) for each of the gyroscope and accelerometer is calculated by dividing the number of counts (_{c}

A total of 31,100 static IMU data was collected for duration of 6 min at a sampling frequency of 85 Hz. The collected dataset is then labeled as [_{1}, _{2}, _{3},…, _{31100}]. A part of the collected dataset is used to build the model and the rest is used to test the model. The matrix of training input is defined as [_{1}, _{2}, _{3},…, _{m}_{i}_{i}_{i}_{+1}, _{i}_{+2},…, _{i}_{+}_{n}_{−1}]^{T}_{n}_{+1}, _{n}_{+2}, _{n}_{+3},…,_{n}_{+}_{m}^{T}

The trained model was then tested using an independent set of data known as the testing set.

As shown in

Corresponding to the RBFNN approach, six Nu-SVR models are developed to model the random errors associated with each of the six inertial sensors. For training the Nu-SVR model, the data is arranged in a similar pattern as described in Section 4.2, with the training input and the output vector defined as _{i}_{i}_{i}_{+1}, _{i}_{+2},…, _{i}_{+}_{n}_{−1}]^{T}_{n}_{+1}, _{n}_{+2}, _{n}_{+3},…, _{n}_{+}_{m}^{T}

The accuracy of these trained models using the identified model parameters are then evaluated using the independent set of data known as the testing set. The Nu-SVR technique in this paper is implemented using LibSVM software [

From

For the GM model, IMU raw data compensation is carried out using the parameters identified in Section 4.1. Ideally, since the IMU is static, the drifts in the positional and velocity components should be zero. Any deviation from its original value shows the presence of errors. Thus larger drift reflects ineffectiveness of the approach in compensating the errors. It is clearly illustrated in

MEMS sensors are lightweight and low-cost but have large errors as compared to other higher grade inertial sensors. This paper introduces an enhanced approach based on a Nu-SVR technique to model the MEMS errors. A total of 31,100 samples of static IMU data was collected and divided into training and testing sets. The training sets are used to develop the RBFNN and Nu-SVR models, which were later tested on the independent dataset.

It was found that the Nu-SVR approach performs much better than the RBFNN method in terms of training time and exhibits smaller standard deviation of noises. For the Nu-SVR method, the training time is 10 times faster than the RBFNN method for the gyroscopes. This drastic reduction in the training time is a very beneficial factor for the real-time implementation of the algorithm. Further, our Nu-SVR method reduced the noise standard deviation by 76% when compared with the existing RBFNN approach. The reduction in standard deviations leads to improved navigation solution accuracy. Experimental results under static conditions indicated that Nu-SVR method has successfully enhanced the positional accuracy by 41% in comparison to the RBFNN method. Moreover, in comparison to the generally used GM method, an impressive improvement of 80% is exhibited by the proposed Nu-SVR approach. In future, we will apply our Nu-SVR approach to dynamic data under varied road dynamics and severe environmental conditions.

Financial support from the Air Force Office of Scientific Research (Grant No. FA9550-01-1-0519) and from the EECS Department, University of Toledo, in the form of graduate assistantship is thanked.

Gyroscope turn-on to turn-on biases.

Gyroscope in-run biases (const. temp.).

Gyroscope in-run biases (vary temp.)

Allan variance for gyroscopes.

Allan variance for accelerometers.

Autocorrelation sequence: (

Gyroscope X output, red: uncompensated, blue: compensated.

Gyroscope X output, red: uncompensated; blue: compensated using Nu-SVR approach.

Position drifts by GM, RBFNN and Nu-SVR methods.

Velocity drifts by GM, RBFNN and Nu-SVR approaches.

Manufacturer specifications for Crista IMU.

5.20 cm × 3.93 cm × 2.54 cm | |||

36.8 g | |||

±300 °/s | ±10 g | ||

<0.2 °/s | <0.0245 m/s^{2} | ||

<0.6 °/s | <0.500 m/s^{2} | ||

<0.75 °/s | <0.3 m/s^{2} | ||

<7.5 °/s (before cal.) | <2.5 m/s^{2} | ||

100 Hz | 40 Hz | ||

3 °/√h | 0.06 m/s/√h |

Manufacturer and lab calibrated noise (ARW and VRW) values.

3.5 °/√h | 3 °/√h | 3.50 °/√h | |

0.10 m/s/√h | 0.06 m/s/√h | 1.24 m/s/√h |

Gauss Markov parameter for accelerometers and gyroscopes.

^{2}_{·}°/s) | ||
---|---|---|

7.38 | 0.0022 | |

7.34 | 0.0036 | |

7.36 | 0.0022 | |

5.22 | 0.0183 | |

7.44 | 0.0195 | |

7.71 | 0.0227 |

Performance of gyroscope X with varying number of inputs.

0.2332 | 0.0552 | 2.8621 | |

0.2332 | 0.0413 | 1.0500 | |

0.2332 | 0.0600 | 2.2000 | |

0.2332 | 0.0767 | 2.6329 |

Standard deviation of the raw and compensated datasets by RBFNN model along with the training time.

^{2}) |
||||||
---|---|---|---|---|---|---|

0.2332 | 0.2703 | 0.2311 | 0.0060 | 0.0074 | 0.0079 | |

0.0413 | 0.0621 | 0.0516 | 0.0035 | 0.0047 | 0.0054 | |

1.0500 | 1.5900 | 1.3200 | 0.3035 | 0.2806 | 0.2966 |

Optimal values of parameters identified for training Nu-SVR.

10 | 10 | 10 | 10 | 5 | 10 | |

0.05 | 0.10 | 0.10 | 5 | 10 | 5 |

Standard deviation of the raw and compensated datasets.

^{2}) |
||||||
---|---|---|---|---|---|---|

0.2332 | 0.2703 | 0.2311 | 0.0060 | 0.0074 | 0.0079 | |

0.0369 | 0.0403 | 0.0352 | 9.8e-04 | 0.0011 | 0.0021 | |

0.1185 | 0.2647 | 0.2295 | 0.1433 | 0.1967 | 0.1917 |