Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 3528146, 9 pages

http://dx.doi.org/10.1155/2016/3528146

## A Simplified Kalman Filter for Integrated Navigation System with Low-Dynamic Movement

^{1}School of Instrument Science & Engineering, Southeast University, Nanjing 210096, China^{2}Key Laboratory of Micro-Inertial Instrument and Advanced Navigation Technology, Ministry of Education, Nanjing 210096, China

Received 23 May 2016; Accepted 25 August 2016

Academic Editor: Francesco Braghin

Copyright © 2016 Xixiang Liu 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

In the integrated navigation system with inertial base, the update frequency of Strapdown Inertial Navigation System (SINS) is always higher than those of aided navigation systems; thus updating inconsistency among subsystems becomes an issue. The analysis indicates that the state transition matrix in Kalman filter is essentially a function of carrier motion. Based on this understanding, a simplified Kalman filter algorithm for integrated navigation is designed for those carriers with low-dynamic motions. With this simplified algorithm, when the filter is without aided information updating, only calculation and accumulation on state transition matrix are executed, and when the filter is with updating, normal time and measurement update are done based on the averaged state transition matrix. Thus the calculation load in the simplified algorithm will be significantly lessened. Furthermore, due to cumulative sum and average operation, more accurate state transition matrix and higher fusion accuracy will arrive for the smoothing effect on random noise of carrier motion parameters. Simulation and test results indicate that when the carrier is with a low-dynamic motion, the simplified algorithm can complete the data fusion of integrated system effectively with reduced computation load and suppressed oscillation amplitude of state vector error.

#### 1. Introduction

The integrated navigation system is widely used for various vehicles to provide speed, position, and (or) attitude. It is generally believed that the integrated system can give full play to each navigation system constructing integrated system and achieve advantages of complementary or (and) combination among each other [1–3]. Among many integrated navigation systems, the inertial integrated navigation system may be the most typical one [2–7]. The most commonly used integrated systems with inertial base are Strapdown Inertial Navigation System (SINS)/Global Navigation Satellite System (GNSS), SINS/Vision, SINS/earth field (such as terrain, magnetic, and gravity) integrated navigation, and so forth. Kalman filter (KF) and other improved filters based on the structure of KF, such as extended KF, unscented KF, and particle KF, are the main tools for data fusion in integrated systems [8–11]. When the models describing system and measurement processes are accurate and the statistical properties of noise are known, optimal estimation of the state vector can be obtained with KF for data fusion [8]. For convenience, all the above filters are named as Kalman-liking filters in the following text and only KF is analyzed.

In the integrated system, the problems of time and spatial inconsistency caused should be first solved before the information from different navigation systems can be used for data fusion [12, 13]. The time inconsistency is mainly caused by the inconsistency of clocks used in different systems while spatial inconsistency is caused by different installing position of each system. To cope with the first problem, a unified standard time signal is usually introduced to synchronize each system. For example, the second-pulse signal from GNSS is often used to synchronize the clock of SINS in the integrated system of SINS/GNSS [12]. To cope with the second one, lever-arm compensation methods are always used. For example, in the integrated SINS/GNSS, lever-arm length and the measurement from SINS are often used to construct and compensate lever-arm velocity when using velocity matching algorithm [13]. Thus, after unified time signal and lever-arm compensation method are introduced, the inconsistent update frequency among each system becomes an important issue. Due to different working mechanism, the data update frequency of each navigation system is quite different [1, 8]. For example, the update frequency of SINS is always 100 Hz or 200 Hz, even up to 1 kHz [1], and the frequency of GNSS is about 1 Hz~20 Hz [7], and the frequency of earth field navigation system always depends on the characteristic of earth field and there exists some uncertainty [2, 3]. In general, the update frequency of inertial navigation system always used as main system in the integrated system with inertial base is much higher than those of aided navigation systems.

From the aspects of system evolution and measurement correction, KF can be divided into time update and measurement update processes which provides a potential solution to the inconsistency problem of update frequency [1, 8]. When the update frequencies between the main and aided systems are consistent, time update and measurement update operation will be carried out successively. When those are inconsistent, only time update operation will be run without measurement update of aided system and both operations will be run with measurement update [8].

In theory, the separated operation of time update and measurement update can solve the inconsistency between main system and aided system in integrated system effectively. In the time update process of KF, the update calculation for state vector, error covariance of state vector, and some related variance, such as state transition matrix, will be executed. The calculation amount is relatively smaller than that of measurement update that needs inversion matrix operation. However, higher update frequency of SINS will bring large calculation load because the same update frequency as that of SINS is needed. When the frequency of SINS is 100 Hz, the calculation amount caused by time update is heavier than that of measurement update with 1 Hz measurement update frequency.

In this paper, the system state equation of integrated system is analyzed, and the analysis indicates that the time update process in KF is essentially the update of parameters related to state matrix, while the variables of state transition matrix are the motion functions of the carrier. When the vehicle is with low-dynamic motion, the changes of motion parameters are slow, and then the changes in time update are slow. In this paper, low-dynamic motion is defined as follows: a vehicle is with a constant velocity in the same direction and (or) with a perturbation of acceleration but without constant acceleration. Based on this understanding, a simplified KF for those carriers with low-dynamic motion is presented. In this simplified algorithm, only the update and summation on state transition matrix without the standard time update process are carried out when there is no measurement update. The case of integrated system of master/slave INS (M/S INS) is studied, and the simulation and turntable tests indicate that the simplified algorithm can fulfill data fusion with reduced studied, and the simulation and turntable tests indicate that the simplified algorithm can fulfill data fusion with reduced calculation but suppressed error oscillation of state vector, when the carrier is with a constant velocity or with a perturbation of acceleration.

The rest of this paper is organized as follows. The integrated system of M/S INS for ship condition is taken as example, and system and measurement equations are studied in Section 2. In Section 3, KF is introduced to fulfill data fusion for M/S INS integration. In Section 4 the time update process is analyzed and the simplified KF is designed and verified with simulation for those vehicles with low-dynamic motion. In Section 5, the results from turntable test are given and the conclusion is in Section 6.

#### 2. Integration System of M/S INS for Ship

As mobile platforms, ships are usually equipped with gimbal inertial navigation system or gimbal compass/log and other high accuracy navigation equipment to provide the ship motion information including speed, position, and attitude. With the excitation of wind wave, temperature difference, and so forth, deflection of ship deck will be generated. In this case, the navigation information from gimbal inertial navigation system cannot be used for weapons and (or) observation equipment because these types of equipment are installed at the head, tail, and highest position while MINS are at the center of ship [14]. Aiming to solve this problem, SINS with middle or low accuracy sensors are introduced and installed at the basis of the above-mentioned equipment to provide information excluding deck deformation. Generally, the accuracy of gimbal system is always higher than that of SINS with one order of magnitude. In order to use higher accuracy information from gimbal system for SINS including initial alignment and error correction, SINS and gimbal system are always integrated [14–16]. In this case, the gimbal system is defined as Master INS (MINS) while SINS is as Slave INS (SINS). In the rest of this paper, Strapdown INS is used as Slave INS and the difference is not distinguished.

In the M/S INS integrated system, MINS can provide various navigation information including velocity, position, and attitude with a lower update frequency than that of SINS. Note that when gyrocompass is used as MINS, the velocity and position can be provided with log. In the integration of M/S INS, matching methods of velocity plus attitude and velocity plus heading are always used [14–16]. The purpose of this paper is to find a simplified KF algorithm; thus only velocity plus heading is selected for analysis.

##### 2.1. System State Equation

In the integrated system of M/S INS, errors of SINS can be selected as state variables to construct state vector and navigation information from MINS can be regarded as aided information to construct measurement vector. The error of velocity, position, and attitude of SINS and the bias of gyroscope are all observable variables with velocity plus heading matching. Observability analysis on integrated navigation system is a complex and big issue but it will not be discussed because of the purpose of this paper. Here the error of velocity, misalignment angles, and bias of gyroscope are selected to construct state vector to simplify the analysis. The state vector of M/S INS integrated system is as follows [16]:where and are the east and north velocity errors, respectively, , and are the misalignment angles of pitch, roll and yaw, respectively, and , and are gyro bias along -, and -axis respectively.

The system state equation can be constructed as follows [1, 8]: where is the system state transition matrix, is the system process noise, and is the noise interference input matrix. According to the error propagation equations of velocity error, misalignment angle, and gyro bias, the system state transition matrix can be expressed as follows:where and are east and north velocity, respectively, and are angular velocity and radius of the earth, respectively, is the geographical latitude of the carrier, , and denote the projection of the accelerometer measured data in navigation frame along east, north, and up direction, respectively, and are the elements of direct cosine matrix (DCM) of SINS.

##### 2.2. System Measurement Equation

Take the differences of velocity and yaw between MINS and SINS as the measurement vector, where , and are the east and north velocity and heading from SINS, respectively, and , , and are those from MINS, respectively.

The system measurement equation can be constructed as follows [1, 8]:where is the measurement matrix and is the measurement noise. According to the relationship between the measurement vector and the state vector, the measurement matrix can be expressed as

According to (2), (5), and the KF introduced in Section 3, the data fusion between the MINS and SINS can be fulfilled.

#### 3. Data Fusion with Kalman Filter

Kalman filter is an optimal filtering algorithm based on iterative calculation. In order to facilitate the iterative calculation of computer, the continuous system as (2) and (5) should be discretized and can be expressed as [1, 8]where is the th iterative update, is the state transition matrix, and is the noise input transition matrix.

When the system is accurately modeled and process noise and measurement noise are both white noises without correlation between each other, optimal estimation of state vector can be obtained by KF. If the statistic characteristics of and can be assumed as and , respectively, the iterative update of KF can be expressed as shown in Figure 1, where is the estimation of state vector, is the one-step predictive value of state vector, is the error covariance matrix of state vector, is the one-step predictive value of error covariance matrix of state vector, and is the filter gain matrix. From the calculation process, as shown in Figure 1, the filtering process can be divided into the loops of gain calculation and filter calculation. From the update process, the filtering process can be divided into the time update and measurement update stages.