A Multiposition Initial Alignment Method of Portable MIMU/FOG Compound Navigation System

In the future, a pedestrian navigation system (PNS) will be widely applied. In navigation system, an initial alignment is essential before starting navigation. There are two primary alignment methods, one is an inertial/magnetometer assisted alignment method, the other is a multiposition rotation modulation alignment method. However, magnetometer is susceptible to magnetic disturbance, and the large volume of rotary modulation system is not suitable for PNS. To solve these problems, a compound navigation system (CPNS) is proposed. It consists of a single-axis fiber-optic gyro (FOG) and a triaxial microelectromechanical system inertial measurement unit (MIMU) without modulation mechanism. CPNS is placed in different positions. According to the variation law of cosine function, FOG data in multiposition are used as fitting alignment, MEMS gyro tracks attitude changes between different positions. The experimental results demonstrate that initial alignment accuracy can be well calculated with several multiposition schemes under different conditions. This method could be a novel solution for PNS’s initial alignment.


I. INTRODUCTION
Because there is a need to develop weapons that do not rely on a satellite navigation system [1], the Defense Advanced Research Projects Agency launched the Micro-Technology for Positioning, Navigation and Timing program in 2010 [2], [3]. PNS is the key research direction in the program. In navigation systems, it is essential to determine the initial values of attitude angles (i.e., initial alignment) before starting navigation. High accuracy IMUs can perform initial alignment without other sensors in a less disturbed environment, but their weight and cost limit their use in PNS. Over last decades, the performance and accuracy level of MIMU have been continuously improved. Due to its outstanding advantages [4], [5], MIMU has become a research focus in the navigation field. For MIMU, the initial alignment cannot be directly achieved because microelectromechanical system (MEMS) sensors have high noise and instability characteristics [6], [7]. Many studies have been carried out to improve the MIMU performance to accomplish the initial The associate editor coordinating the review of this manuscript and approving it for publication was Ehsan Asadi . alignment of the MIMU [8]- [10]. Two primary methods can be applied to improve the performance of MIMU. One is an inertial/magnetometer assisted alignment method [11]. The other is a multiposition rotation modulation alignment method, in which a turntable is applied to modulate the rotary IMU.
In the inertial/magnetometer assisted alignment method, the attitude information is obtained by a tri-axial magnetometer that can measure a geomagnetic field. Online calibration or periodic calibration is needed before using low accuracy magnetometers. Thus, an integrated compensation model based on the least squares ellipsoid fitting is proposed to compensate the integrated error of the magnetometer [12], [13]. After calibrating the magnetometer, a nonlinear filtering data fusion method is discussed to perform stationary base alignment in [14]- [16]. However, the deterioration of magnetometer accuracy under dynamic attitude tracking, the method is not suitable for the dynamic alignment. To solve the problem, Ludwig [17] used a genetic algorithm to find the optimal weight between the relative orientation from a gyro and the absolute orientation from an accelerometer/magnetometer. The advantage of the inertial/magnetometer assisted alignment method is short initial alignment time. However, the magnetometer accuracy is easily prone to disturbances from electrical and metal surroundings, and the initial alignment accuracy is low.
To improve the alignment accuracy, a multiposition rotation modulation alignment method, which uses an error compensation technique (i.e., rotating modulation technique), is proposed [18]- [20]. The rotating modulation technique can operate in two modes: multiposition rotating modulation and continuous rotation modulation. In [21]- [24], a multiposition rotation path was designed to reduce the alignment error that is caused by inertial sensor errors. The way to improve alignment accuracy is to increase the number of positions, but the alignment time increases at the same time. To reduce the alignment time, Collin [25] proposed a continuous rotation modulation method, and discussed the effect of rotating modulation on the stochastic error processes of a MEMS gyro. Several real-time continuous rotation modulation methods based on a gyro have been proposed in [26]- [28]. In this method, the inertial sensors error is suppressed by the modulation mechanism. Thus, the long-term alignment accuracy is considerably improved. However, the rotating modulation technique scheme inevitably increases the operation complexity, power supply, and the cost of the navigation system. The large volume of the system is not suitable for PNS.
The characteristics of various primary initial alignment methods are illustrated in Table 1. To overcome the shortcomings of primary methods, this paper presents a simple and rapid initial alignment method. The proposed method aims to achieve the following objectives: 1) A CPNS consisting of a MIMU and a single-axis FOG is proposed to realize the PNS; 2) A MEMS tracking strategy replaces the traditional rotation modulation system and achieves alignment accuracy in the system with small volume and light weight; 3) During the initial alignment process, the MEMS tracking system is isolated from external angular vibrations, which reduces the interference factors such as artificial misoperation.
In this study, the configuration of CPNS is developed. In addition, a multiposition cosine-fitting method aiming to calculate the initial alignment accuracy is introduced. During the process, the least squares method is applied to the cosinefitting method. Finally, the experimental results demonstrate that the initial alignment accuracy can be calculated with the several multiposition schemes under different conditions.

II. CONFIGRATION OF CPNS
CPNS is composed of a MIMU and a single-axis FOG, as shown in Fig. 1. It is assumed that the MIMU body frame coincides with the east-north-up (ENU) frame, and the single-axis FOG is placed along the y-axis of the MIMU body frame. CPNS rotates along the z-axis. Because FOG is placed on the o-xy plane in the body frame, the FOG output ω F is the projection of Earth's rotation rate in the navigation frame, which can be described as [29] ω F = ω n ie cos θ where ω n ie is the north component of Earth's rotation rate in the navigation frame, and θ is the intersection angle between the actual position of FOG and the true north direction. Clearly, the FOG output is the cosine modulation of the Earth's rotation rate projection in the rotation plane.
There are two preconditions for the proposed initial alignment method used in CPNS: 1) the system has to be calibrated before the initial alignment; the calibration method and path design are described in [30]. After the installation error compensation, the outputs of FOG and MIMU are unified into the same coordinate system; 2) the FOG accuracy is much higher than that of the MEMS gyro accuracy; thus, the random error of FOG is not considered in (1).

III. MULTI-POSITION COSINE-FITTING METHOD A. PROBLEM FOMULATION
The multiposition method uses only the static location positions. Consider that the output value has the form of a cosine signal with a relative phase difference where y (i) is the output value; i is the position index; p is the amplitudes of the signal; q is the angular frequency given by q = 2π f f s , where f is the frequency of the signal and f s is the sampling frequency; x (i) is the input value; ε (i) is the residual error. ϕ is the phase difference between y (i) and x (i).

B. LEAST SQUARES METHOD
Equation (2) is identical to the general quadratic form of the cosine function written as To minimize the residual error, the minimum value for the objective function is calculated as where x i , and y i are the i th position sample. For i = 1, 2, · · · , N , the matrix form in (3) is expressed as The parameter vector is estimated by minimizing the error The parameter vector is estimated as The minimization problem is solved by the least squares method [31] C. COSINE-FITTING METHOD

1) PITCH ANGLE AND ROLL ANGLE CALCULATION
A transformation from the navigation frame to the body frame can be carried out as three successive rotations about different axes.
Rotate angle ϕ about the navigation frame z-axis; Rotate angle θ about the new frame y-axis; Rotate angle γ about the new frame x-axis; where ϕ, θ, and γ denote the heading angle, pitch angle, and roll angle, respectively.
MEMS triaxial accelerometers in MIMU are installed along the three axes of the body frame. The projection components of the accelerometer output in the body frame are measured as The accelerometer output in the navigation frame is The attitude matrix C b n from the navigation frame to the body frame can be described as According to (10) and (11), (12) -(14) can be derived.
The pitch angle and roll angle are [32] θ = arcsin f b y g (15) γ

2) O-XY PLANE IN THE BODY FRAME COMPONENTS OF THE EARTH'S ROTATION RATE
Earth's rotation rate in the body frame can be expressed as where ω b ie is the Earth's rotation rate in the body frame, ; ω n ie is the Earth's rotation rate in the navigation frame, L is the local latitude, ω n ie = 0 ω ie cos L ω ie sin L T , and ω ie is the Earth's rotation rate.
According to (11) and (15), the projection modulus of the Earth's rotation rate in the o-xy plane in the body frame is = ω ie cos L sin 2 γ cos 2 θ +sin 2 θ

3) COSINE-FITTING METHOD
The relationship between the FOG output and MIMU heading angle can be described as where ω F (i) is the FOG output; a is the projection modulus of the Earth's rotation rate in the o-xy plane in the body frame, which is calculated by (16); b is a constant value for unit conversion; ϕ M (i) is the track angle in different positions, it is obtained by MIMU attitude updating; i is the position index; ϕ is the initial heading angle. ε is the residual error.

D. INITIAL ALIGNMENT SCHEME
As shown in Fig. 2, CPNS is placed at different locations in the o-xy plane in the body frame. We propose an effective and simple method for the initial alignment, as shown in Fig. 3. The initial alignment scheme is described as follows: 1) Maintain a static state of the initial position, using a MEMS accelerometer to perform the alignment in the o-xy plane in the body frame. The pitch angle and roll angle are calculated by (15) and (16); 2) After the alignment, the projection modulus of the Earth's rotation rate in the o-xy plane in the body frame is calculated by (18); 3) CPNS is rotated around the z-axis in the body frame, and a multiposition static measurement is carried out. During the rotating process, the relative angle between multiposition is obtained by MEMS gyro attitude updating; 4) After the multiposition measurement, the absolute heading angle (angle to the north) ϕ of the initial position is calculated by fitting the alignment.

A. INITIAL ALIGNMENT SCHEME
The hardware design of CPNS is shown in Fig. 4. The MIMU uses ADIS16445 [33] which is produced by Analog Devices, Inc., and the single-axis FOG uses a homebuilt device. The typical performance parameters for CPNS are shown in Table 2. Numerical experiments are developed. Each sensor output is sampled at a rate of 100 Hz. The system is placed in any initial position and static data is collected for 1 minute. Then, the system is rotated to the next arbitrary position for one circle, and data is collected for 1 minute in each position, for a total of N positions altogether. Of note, each quadrant requires N /4 positions to ensure that each of them has an output data of the corresponding position.
The definition of the experiment under different condition is shown in Table 3:   The initial alignment accuracies in sixteen-position are shown in Table 5.
Where RMS is the root mean squared error. The sixteen-position cosine-fitting results are shown in Fig. 5.
In Fig. 5, the x-axis represents the MIMU calculation angle, the y-axis represents the FOG output, and the red line is the result of the multiposition cosine-fitting method. Blue dots are distributed arbitrary in each quadrant in Case 1.
Twelve-position, eight-position, and four-position Initial alignment accuracies are shown in Table 6.
The initial alignment accuracy for different cases is summarized in Table 7.
As shown in Table 7, the initial alignment accuracy is improved when the position increases. For the positions with arbitrary angles, the sixteen-position initial alignment accuracy is improved by 59.2% compared with the fourposition initial alignment accuracy. In more east-west directions and less north-south directions, the sixteenposition initial alignment accuracy is improved by 31.2% compared with the four-position initial alignment accuracy. In the east-west directions, the sixteen-position initial alignment accuracy is improved by 39.8% compared with the fourposition initial alignment accuracy. The experimental results show that for same position number, the more east-west directions have higher initial alignment accuracy. The initial alignment accuracy will be improved with an increase in eastwest directions. Similarly, the initial alignment accuracy will improve with the number of positions.
To verify the effectiveness of the proposed method, the initial alignment accuracy of three methods is compared. Method 1 is the auxiliary alignment method of the magnetometer. The total RMS noise of the magnetometer is 0.5mG [34]. Method 2 is the rotation modulation alignment method with motor in the hardware, and the gyro accuracy is 0.2 • /h (1σ ) [35]. Method 3 is the proposed method without   a motor. Using Case 3 of the proposed method as an example, the comparison results are shown in Table 8.
In Table 8, the initial alignment accuracy is improved by 46% compared with the achieved by the magnetometer methods and improved by 73% compared with the rotation modulation alignment method. The result proves the validity of the proposed method.

B. DISCUSSION
In this section, a series of experiments are performed to validate the feasibility and effectiveness of the proposed method based on CPNS, and the experimental results are discussed in detail. The results prove the superior performance of the proposed method, which successfully improves the initial alignment accuracy. The main advantages of the proposed method are listed below.
1) CPNS has a simple configuration, small size and light weight; 2) The proposed multiposition cosine-fitting method is effective and simple for PNS; 3) Experiments under different conditions are employed to verify the theoretical deduction. Therefore, the requirements for the position can be reduced.

V. CONCLUSION
The initial alignment is an essential procedure to achieve high-accuracy initial attitude determination for PNS. Aiming to achieve an effective method for the initial alignment, this paper presents two important issues: the configuration of CPNS and the multiposition fitting method. By combining the small size of MIMU and the high accuracy of FOG, a compound configuration of MIMU/FOG is proposed. To fit the FOG output and MIMU attitude information in the approximate o-xy plane, a multiposition cosine-fitting method based on the least squares method is proposed to realize the initial alignment. Experiments under different conditions verify the feasibility and effectiveness of the proposed method.
XIAOWEN CAI received the M.S. degree in control theory and control engineering from Xiangtan University, in 2008. She is currently pursuing the Ph.D. degree in optical engineering with Beihang University, Beijing, China. Her research interests include related to inertial navigation and integrated navigation.
CHUNXI ZHANG received the Ph.D. degree in optical engineering from Zhejiang University, Zhejiang, China, in 1996. He is currently a Professor with Beihang University, Beijing, China. His research interests include optical fiber sensing technology, inertial navigation, and integrated navigation.
YANQIANG YANG received the B.S. degree in measurement-control technology and instrumentation from the North University of China, Taiyuan, China. He is currently pursuing the Ph.D. degree with Beihang University, Beijing, China. His research interests include related to inertial navigation and integrated navigation.
SHUANG GAO received the Ph.D. degree in optical engineering from Beihang University, Beijing, China, in 2008. She is currently a Lecturer in optical engineering with Beihang University. Her research interests include inertial navigation and inclinometer for petroleum exploration wells.
RUOYU ZHANG received the B.S. degree in automation from the China University of Petroleum. She is currently pursuing the M.S. degree in optical engineering with the Beijing University of Aeronautics and Astronautics. Her research interests include calibration technology and inertial navigation.