Journal of Sensors

Volume 2017, Article ID 7161858, 11 pages

https://doi.org/10.1155/2017/7161858

## A Fast SINS Initial Alignment Method Based on RTS Forward and Backward Resolution

^{1}Key Laboratory for Urban Geomatics of National Administration of Surveying, Mapping and Geoinformation, Beijing 100044, China^{2}School of Geomatics and Urban Spatial Information, Beijing University of Civil Engineering and Architecture, Beijing 100044, China

Correspondence should be addressed to Jian Wang; moc.361@ecnauhnaijw

Received 30 November 2016; Revised 14 March 2017; Accepted 5 April 2017; Published 10 May 2017

Academic Editor: Dzung Dao

Copyright © 2017 Houzeng Han et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

For the strapdown inertial navigation system (SINS), the procedure of initial alignment is a necessity before the navigation can commence. On the quasi-stationary base, the self-alignment can be fulfilled with the high quality inertial sensors, and the fine alignment is usually executed to improve the alignment performance. Generally, fast estimating of heading misalignment is still a challenge due to the existence of gyro errors. An innovative data processing strategy called forward and backward resolution is proposed for INS initial alignment. The Rauch-Tung-Striebel (RTS) smoothing is applied to obtain the smoothed attitude estimates with the filter information provided by the forward data processing. The obtained attitudes are then treated as aiding measurements to implement the forward resolution with the repeated data set, the converged sensor biases are used as constraints, and the iterative processing is conducted to obtain the updated attitudes. Simulation studies have been conducted to validate the proposed algorithm. The results have shown that the alignment accuracy and convergence rate have been improved with the added RTS aided forward and backward resolution; more stable heading estimates can be obtained by calibrating with estimated gyro bias. A real test with a high quality inertial sensor was also carried out to validate the effectiveness of the proposed algorithm.

#### 1. Introduction

Navigation can be referring to a technique that involves the determination of position and orientation of a moving object. Traditionally, the Global Navigation Satellite System (GNSS) has been widely used for navigation applications in the outdoor environment [1]. However, the GNSS positioning performance quickly deteriorates in the challenging environments, like urban canyons and under foliage. On the other hand, the inertial navigation system (INS), which is a dead-reckoning based navigation system, is the only form of navigation that does not rely on external reference [2]. The INS can successively supply position, velocity, and attitude information in any environment. Consequently, it has been widely used in the fields of navigation and positioning such as vehicles, ships, aircraft, and guided missiles. Before an INS is ready to navigate, the initial navigation parameters need to be provided, among which the initial position and velocity can be easily obtained from external reference, such as GNSS [3–5]. The initial alignment, which refers to the attitude initialization, becomes the main factor that influences the navigation accuracy.

For the strapdown inertial navigation system (SINS), the purpose of alignment is to obtain the rotation matrix between the body frame and navigation frame. The initial alignment method usually involves two steps, namely, a coarse alignment utilizes analytic method followed by a fine alignment. The coarse alignment aims to obtain the initial attitudes with small misalignment angles; however, the coarse alignment is only feasible for the high quality inertial sensors. For the low cost INS, the initial alignment, especially for the heading, is still a challenge due to the existence of high sensor noise and bias. The heading information for the low cost INS is usually provided by the external reference, such as magnetic compass [6].

In broad sense, the fine INS initial alignment can be divided into three types [7], which are quasi-stationary alignment that assumes the INS is stationary with respect to the Earth [8, 9], GNSS alignment that uses position and velocity information to aid INS alignment [10, 11], and transfer alignment which uses position or velocity and sometimes attitude from the master INS [12, 13]. Generally, the Kalman filter (KF) is the most commonly used to calibrate position, velocity, and attitude based the difference of INS outputs and external reference [14]. In the alignment process, the alignment accuracy can be improved if the misalignment angles and gyro bias error can be effectively estimated. Considering the large initial attitude error uncertainties derived when using the low cost sensors, the unscented Kalman filter (UKF) is commonly used to handle large attitude errors [15, 16]. In addition, the filter has been introduced to improve the transfer alignment accuracy considering flexure deformation and lever-arm effect [17, 18].

In real environment, the output from the inertial sensor, especially for low or medium cost sensors, suffers from large sensor noise. Efforts have been made to reduce the noise effect. El-Sheimy et al. utilized wavelet denoising method to eliminate sensor noise in the initial alignment [19]. An IIR filter combined with a Kalman filter was used to attenuate the influence of sensor noise and outer disturbance [20]. A Vondrak low-pass filter, with the ability to reduce the high frequency noise in the force observation accurately, was applied to improve attitude accuracy [21]. When using the low-pass filter, the difficulty is to find universal filter parameters applicable for dynamic environment, and the online processing is another problem.

In order to fulfill the requirements of high accuracy and short time for initial alignment, the same set of sensor data can be used repeatedly to conduct initial alignment. The SINS mathematical platform is adjusted accordingly with updated compensation parameters [22], and a fast compass alignment is feasible based on iterative calculation [23]. The forward and backward processes can be effectively applied in INS alignment, and the analysis of forward and backward compass processes has been carried out using the error equations in frequency domain [24]. In this research, a quasi-stationary condition is considered, which is characterized as having bounded position and attitude movement such as that produced by wind gusts and passenger [7]. This is a typical situation for land vehicles. An innovative forward and backward data processing method is proposed to improve the alignment performance, an easily implemented Rauch-Tung-Striebel (RTS) smoother is applied to obtain smoothed attitudes in the backward resolution, and the obtained attitudes together with converged sensor biases are used as constraints in the repeated resolution process.

The rest of the paper is organized as follows. In Section 2, the alignment model on the quasi-stationary base is constructed, and the effect of sensor bias on alignment accuracy is analyzed. In Section 3, after the INS alignment process is studied, an improved RTS aided forward and backward alignment method is developed to shorten the alignment time. In Section 4, two simulation tests and a real test are conducted for verification. Finally, some conclusions are given.

#### 2. Error Propagation Model for the Quasi-Stationary Alignment

##### 2.1. Coarse Alignment

Here we are mainly discussing the practical aspects of alignment with regard to land vehicle navigation. For the land vehicles, when the INS is in quasi-stationary condition, that is, low-vibration environment, the self-alignment can be fulfilled with inertial measurements. The coarse alignment usually involves two steps. In the first step, the INS is leveled by initializing the pitch () and roll () angles. For most inertial sensors, the accelerometer leveling can be completed with the knowledge of gravity information in the navigation frame. The relation of accelerometer measurements () with the local gravity () can be described as [2]withwhere is the rotation matrix from the navigation frame (*n*-), which is denoted as north, east, and down (NED) frame, to the body frame (*b*-), is heading angle, and is local gravity constant, which can be derived from the gravity model.

Equation (1) then can be written as

The pitch and roll angles can be calculated aswhere the subscripts , , and represent forward, right, and down direction of the body frame.

The accuracy of the attitude estimates is limited by the accelerometer error; the resulting attitude errors can be described aswhere represents accelerometer bias and is the misalignment error.

The INS tilt error is caused by the accelerometer error, for example, a milli-g accelerometer bias will induce 1 mrad error for roll and pitch estimates. Actually, promising attitude accuracy can be obtained for most of inertial sensors because the gravity is a relatively large quantity.

However, accurate self-alignment of the heading requires high quality inertial sensors. For the low cost inertial sensors, the heading is usually initialized using a magnetic compass. In the gyrocompassing process, the high quality gyros are used to sense the Earth’s rotation. The gyro angular rates () are related to the Earth’s rate bywith , , where and denote Earth’s rate and latitude, respectively; the latitude can be provided by GNSS. For quasi-stationary conditions, is a zero vector.

Equation (6) can be written as

The heading can be calculated aswhere the pitch and roll have already been calculated from leveling. The heading error is affected by gyro and accelerometer errors, which can be expressed aswhere is gyro bias.

It can be seen that the heading accuracy is largely affected by the east gyro errors. We have known that the Earth rate is about 15°/h; for the low cost gyros, it cannot align itself because the noise levels are near or higher than the Earth rate. The sensor data actually reflects the magnitude of misalignment; for example, at latitude 45°, in order to obtain a 1 mrad heading initialing accuracy, the gyro should be accurate to about 0.01°/h. During the coarse alignment process, the inertial measurements are smoothed to reduce the adverse effects from the random noise.

##### 2.2. Fine Alignment

After the coarse alignment has been completed on the stationary base, the attitude misalignments converge to small values. To achieve a more accurate initial attitude estimate, sequential measurements are used to carry out a self-alignment over a period of time. In the quasi-stationary condition, zero-velocity measurements are used in a Kalman filter during the initial alignment phase, which is used to calibrate velocity and attitude. Inertial sensor errors, such as accelerometer bias and gyro bias, are usually estimated together. However, the effects of inertial sensor errors cannot be fully separated from the attitude errors in a stationary condition.

The dynamic model for the small orientation errors can be expressed as

In addition, the velocity error model can be written as

The state vector is denoted by* x*where the subscripts* N*,* E*, and* D* denote the direction in the local navigation frame and is velocity error.

Then the system state equation can be expressed aswhere the dynamic matrix isand is the process noise, and are meridian radius and normal radius, respectively,* h* is the height of INS, and the inertial sensor errors are treated as random walk process.

On a stationary base, the measurement equation can be constructed aswhere the measurement matrix iswhere denotes the difference between the INS output and external measurements, and it can be expressed as . The zero-velocity update (ZUPT) is conducted for stationary INS. The symbol is the measurement noise vector and is identity matrix.

The error states are predicted bywhere is the a posteriori state vector at epoch and is the a priori state vector at epoch* k*; is the state transition matrix from epoch to epoch* k*, and it can be approximated as ; is the time increment, is the a priori covariance matrix of , and is the process noise covariance matrix. The correction step is provided as [25]where is the Kalman gain matrix, is the a posteriori state vector at epoch* k*, is the a posteriori covariance matrix of , and is the measurement noise covariance matrix. The estimated error states are fed back to update attitude estimation and compensate sensor output.

#### 3. Improved RTS Forward and Backward Based Alignment

After analyzing the model combined with (13) and (15), we can conclude that velocity errors are directly observed, which results in rapid estimation of horizontal orientation errors for that the accelerometer biases have little effects. The down orientation error, however, is indistinguishable from the east gyro bias; the two error states are correlated with each other. In other words, the misalignment angles are directly related with sensor data. It requires a period of time to obtain stable attitudes.

For SINS, when the vehicle body is stationary, the zero-velocity updates are carried out continuously. The calculated misalignment angles are used to update the rotation matrix periodically. For the ideal error-free inertial sensors, the measured sensor data stays the same. However, in real applications, the inertial sensors are usually contaminated by a variety of error sources, such as bias, scale factor, and random noise. A set of sensor data and reference measurements are required to conduct the initial alignment. In order to improve the alignment accuracy, the repeated SINS resolution can be conducted using the same data set.

In the quasi-stationary conditions, the INS is disturbed by external loadings, such as mechanical vibration, wind effects, and human activity; the attitudes are varied at a relative low level. In the repeated resolution process, a normal forward SINS resolution is applied first, then a backward SINS resolution will be applied, the obtained smoothing results are used as external constraints to conduct forward SINS resolution again, and finally the more accurate alignment estimates are expected to be achieved. The forward-backward SINS resolution process is shown in Figure 1.