## Abstract

Benefiting from frame structure, RINS can improve the navigation accuracy by modulating the inertial sensor errors with proper rotation scheme. In the traditional motor control method, the measurements of the photoelectric encoder are always adopted to drive inertial measurement unit (IMU) to rotate. However, when carrier conducts heading motion, the inertial sensor errors may no longer be zero-mean in navigation coordinate. Meanwhile, some high-speed carriers like aircraft need to roll a certain angle to balance the centrifugal force during the heading motion, which may result in non-negligible coupling errors, caused by the FOG installation errors and scale factor errors. Moreover, the error parameters of FOG are susceptible to the temperature and magnetic field, and the pre-calibration is a time-consuming process which is difficult to completely suppress the FOG-related errors. In this paper, an improved motor control method with the measurements of FOG is proposed to address these problems, with which the outer frame can insulate the carrier’s roll motion and the inner frame can simultaneously achieve the rotary modulation on the basis of insulating the heading motion. The results of turntable experiments indicate that the navigation performance of dual-axis RINS has been significantly improved over the traditional method, which could still be maintained even with large FOG installation errors and scale factor errors, proving that the proposed method can relax the requirements for the accuracy of FOG-related errors.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Due to the characteristic of concealment and anti-interference, inertial navigation system (INS) is widely employed in military application, such as aircraft, missile and ship [1,2]. With the development of FOG, strap-down INS (SINS) is gradually replacing the traditional platform INS (PINS), because of its simple structure, low cost and high reliability [3]. In recent years, rotary modulation technology has been widely researched to further improve the navigation performance [4]. By rotating the inertial measurement unit (IMU), the gyro drifts and accelerometer biases perpendicular to the rotation axis will be modulated into sine and cosine components in navigation coordinate, so that the accumulation of navigation errors can be restricted [5–7]. Benefiting from the double frame structure, dual-axis RINS can modulate gyro drifts in all three directions under a proper rotation scheme [8]. In general, dual-axis RINS is working on the state of continuous rotation along IMU’s vertical axis most of timeand the vertical gyro drift can be restricted by alternating IMU direction pointing up and down [9]. Because the horizontal gyro drifts mainly appear on the velocity errors as an 84.4 minute Schuler periodic oscillation, while the vertical gyro drift mainly leads to 24 hours long periodic oscillation. Compared to vertical gyro, the horizontal gyro drifts have much greater influence on the positioning accuracy, especially in short-term (2-3 hours) navigation applications [10].

Rotation strategy and motor control method will directly affect the navigation performance in RINS [11]. In traditional method, the measurements of photoelectric encoder are always adopted as the feedback values to drive motor rotate. In this way, IMU can only achieve the rotary modulation relative to the shell of RINS [12]. However, when carrier rotates in the same direction as IMU, the modulated gyro drifts may not appear as zero-mean in navigation coordinate, thereby affecting the performance of rotary modulation. In [1], the author introduces the information measured by vertical gyro to control frame motor in Single-axis RINS (SRINS). By employing this method, the carrier’s heading motion can be effectively insulated, but this solution still has limitations. In actual application, the heading maneuver seldom exists independently; most high-speed carriers need to roll a certain angle to balance the centrifugal force, especially for aircraft and warship. The schematic diagram of aircraft in the turning motion is depicted in Fig. 1. It can be seen that the steeper the turning motion is, the larger the angles at which the aircraft needs to roll. Generally, when aircraft makes medium banked turn, the roll angle is around 20-30 degrees; when making steep banked turn, the roll angle can reach to 45 degrees.

If carrier’s roll motion is not insulated, the coupling errors caused by the FOG scale factor errors and installation errors will be motivated [13]. Since IMU rotates continuously along the vertical direction, the influence of the coupling errors may become more serious and complicated. On the other hand, the fiber ring is susceptible to the temperature field and magnetic field, so the error parameters of FOG are always unstable [14–16]. In the application of FOG INS, the pre-calibration on IMU parameters is always performed before navigation to ensure the positioning accuracy, but it is a time-consuming process which is not conducive to improving the carrier’s mobility [17–21]. In this paper, we designed an improved motor control method with the measurements of FOG for dual-axis RINS, with which the carrier’s heading and roll motion can be insulated simultaneously. Turntable navigation experiments demonstrate the navigation performance is significantly improved and the positioning accuracy can still be maintained even with large FOG installation errors and scale factor errors, proving that the proposed motor control method can relax the requirements for the accuracy of FOG-related errors.

The rest of the paper is organized as follows. Section 2 introduces the configuration of dual-axis RINS and defines the coordinate systems. Section 3 performs the error analysis of traditional motor control method under carrier’s roll motion, and introduces the improved motor control method in detail. The results of turntable experiments are discussed in Section 4, followed by the conclusion in Section 5.

## 2. The Specification of dual-axis RINS

#### 2.1 Configuration of dual-axis RINS

As shown in Fig. 2, the dual-axis RINS mainly contains a double-frame structure (outer frame and inner frame) and an inertial measurement unit (IMU). On each side of the frames, a brushless DC torque motor is equipped to drive the frame rotate, a photoelectric encoder is used for collecting information on rotation angles. When the outputs of two photoelectric encoders are zero, the outer frame and inner frame respectively coincide with the roll and azimuth axis of RINS. The IMU is mounted inside the inner frame, three fiber optic gyros (FOGs) and three quartz flexible accelerometers (QFAs) are equipped, together with the relative circuit boards. The device specification of dual-axis RINS is listed in Table 1.

#### 2.2 Definition of coordinates and symbols

Relevant coordinates and several symbols referred in this paper are defined as follows.

Navigation coordinate system (n,$O\text{-}{x}_{n}{y}_{n}{z}_{n}$): n-coordinate sets a reference coordinate to operate the navigation algorithm. $O{x}_{n}$and $O{y}_{n}$ are two horizontal axes respectively pointing to East and North directions, $O{z}_{n}$ points to skyward direction. The three axes constitute a right-handed orthogonal coordinate.

Body coordinate system (b,$O\text{-}{x}_{b}{y}_{b}{z}_{b}$): b-coordinate is used to describe shell of dual-axis RINS. $O{x}_{b}$(pitch axis) points to the rightward direction, ${\theta}_{b}$ is pitch angle; $O{y}_{b}$ (roll axis) is horizontal axis pointing to forward direction, ${\gamma}_{b}$ represents roll angle; $O{z}_{n}$(azimuth axis) constitutes a right-handed orthogonal coordinate together with $O{x}_{b}$and $O{y}_{b}$, ${\psi}_{b}$ is azimuth angle.

IMU coordinate system (p,$O\text{-}{x}_{p}{y}_{p}{z}_{p}$): p-coordinate is used to describe the IMU platform. ${x}_{g},{y}_{g},{z}_{g}$ represent the sensitive axes of three gyros. $O{z}_{p}$ coincides with the inner frame shaft, then $O{x}_{p}$ is defined as the projection of ${x}_{p}$ in the vertical plane of $O{z}_{p}$, and $O{y}_{p}$ constitutes a right-handed orthogonal coordinate together with $O{z}_{p}$and $O{x}_{p}$. The spatial relationship between p-coordinate and sensitive axes of gyros is depicted in Fig. 3. The installation errors of gyros can be described as five small angles ${\alpha}_{gy}^{Z},{\beta}_{gx}^{Y},{\beta}_{gy}^{X},{\delta}_{gz}^{X},{\delta}_{gz}^{Y}$.

Hence, the cosine matrix required to transform the gyro sensitive axes to p-coordinate can be described as,

Instructions coordinate system (a,$O\text{-}{x}_{a}{y}_{a}{z}_{a}$): a-coordinate represents a special p-coordinate when the output of inner encoder is zero, which is introduced to calculate the motor control instructions. ${C}_{a}^{n}$ could be obtained by the following equations.

## 3. Improved motor control method for dual-axis RINS

#### 3.1 Error analysis of traditional motor control method under carrier’s roll motion

In order to simplify the analysis, the flight is considered as a short-term navigation, meaning that the influence of vertical gyro drift can be ignored. Hence, the IMU needs to rotate along the vertical axis to suppress the horizontal sensor errors, similar with the rotation scheme of Single-axis RINS. In traditional motor control method, the angular information measured by photoelectric encoder is always used as feedback for motor closed-loop control. The representative rotation scheme during navigation could be described as follows, the outer frame locks at zero, and the inner frame rotates bi-directionally with the angular rate of 6 °/s. Hence, the rotation of IMU is conducted relative to the system shell, and the outer frame is always perpendicular to the mounting bracket. When the aircraft turns according to Fig. 1, the movement state of dual-axis RINS is given in Fig. 4.

In order to analyze how the error parameters of FOG generate navigation errors, the changes of the platform declination angles need to be calculated under this condition. The space relationship between b-coordinate and p-coordinate is shown in Fig. 5, the angular rate of rolling motion is denoted as${\omega}_{r}$. During carrier’s roll motion, the outputs of three gyros could be expressed as,

And then, the horizontal platform declination angles can be obtained by integration as follows,

According to the error equations of INS [10], 10″ of platform declination angles could lead to 0.4m/s of velocity error in Schuler oscillation maximum value, thereby the coupling errors in Table 2 may seriously affect the positioning accuracy. Besides, the roll angle would return back to zero when aircraft finishes turning motion, which is equivalent to the carrier performing roll motion once again. If ${\phi}_{z1}$ and${\phi}_{z2}$ is defined as the output of inner encoder at the start and the end of turning motion respectively, the platform declination angles caused by FOG errors will be totally counteracted only when${\phi}_{z1}={\phi}_{z2}$. Since the inner frame keeps rotating, ${\phi}_{z2}$ can never exactly equivalent to${\phi}_{z1}$. Furthermore, if ${\phi}_{z1}$and${\phi}_{z2}$satisfy$\left|{\phi}_{z2}-{\phi}_{z1}\right|<{180}^{\circ}$, the platform declination angles will be partially offset, while if ${\phi}_{z1}$and${\phi}_{z2}$satisfy$\left|{\phi}_{z2}-{\phi}_{z1}\right|>{180}^{\circ}$, the platform declination angles will continue to enlarge at the end of carrier turn motion.

Besides, according to the analysis in [10], it is worth mentioning that the influence of accelerometer’s relative parameters on the navigation errors can be ignorable compared with gyro’s, so the error model on accelerometer is not provided here for brevity.

#### 3.2 An improved motor control method with measurements of FOG

The FOG can measure the carrier angular rate relative to the inertial space. If gyro information is introduced into motor control, the angular motion of carrier is possible to be insulated, thus the influence of coupling errors on navigation can be greatly reduced. The block diagram of the improved motor control method with measurements of FOG is given in Fig. 6.

A typical motor model is established in Fig. 6, ${M}_{F}$is the friction torque of motor shaft, ${C}_{M}$is torque factor of motor, ${C}_{e}$is counter-electromotive force factor, ${L}_{a}$is armature inductance, ${R}_{a}$is armature resistance, ${M}_{C}$ is drive torque, $J$is IMU’s moment of inertia. In RINS algorithm, ${\omega}_{ip}^{p}$ represents the outputs of FOG after the compensation of device errors, including installation errors, scale factor errors and gyro drifts. Then,${\omega}_{np}^{p}$ can be obtained as follows

*i*th row and the

*j*th column in the matrix of ${C}_{a}^{n}(i,j)$. Hence the deviation $\Delta \gamma $for PID controller model could be obtained as follows,

${\gamma}_{Target}$ represents the target position of the outer frame. In order to keep the IMU pointing in the vertical direction, ${\gamma}_{Target}$can be set as zero. The classical PID algorithm is adopted to calculate the control voltage ${U}_{out}$ of outer motor as follows,

By adopting the proposed motor control method, the spatial relationship of dual-axis RINS when aircraft turns is shown in Fig. 7.

When aircraft turns, the gray mounting bracket will follow the carrier’s roll motion, and the IMU can keep rotating along the skyward direction. In summary, by using the improved motor control method, dual-axis RINS can effectively insulate the roll and azimuth angular motion, inhibiting the generation of the coupling errors and achieving the optimal rotary modulation effect.

## 4. Experiments

In this section, turntable experiments are designed to verify the effectiveness of the improved motor control method. The experimental equipment is shown in Fig. 8. Dual-axis RINS is fixed to a hand-operated biaxial turntable, and roll axis of RINS is approximately coincident with the turntable’s outer frame shaft. The ${y}_{b}$ axis of RINS approximately points to the east, and the ${x}_{b}$ axis points to the southern direction. The dual-axis RINS is fed by 28V DC power supply and the experimental data is collected by a laptop at the frequency of 200Hz.

Two short-term navigation experiments under the traditional and the improved motor control methods are orderly performed. Each test lasts about 2.5 hours, including system initialization, parsing coarse alignment, fine alignment and navigation. Alignment process is required to obtain the initial attitude of the dual-axis RINS, costing nearly 0.5 hours together with system initialization. It is worth mentioned that the standard deviations of the horizontal platform declination angles in alignment is less than 2″, and the vertical declination angles is less than 50″. According to INS error analysis, the alignment errors only affect the amplitude of the Schuler period, and the influence of alignment errors is negligible compared to the coupling errors. During the navigation, the inner frame rotates bi-directionally with the angular rate of 6°/s. The outer frame of turntable is operated to roll about 40° rapidly in every 15 minutes to simulate carrier’s roll motion. After staying still for a while, the turntable’s outer frame turns back to the original position.

By adopting different motor control methods, Fig. 9 shows the roll angle and outer encoder angle in once turntable roll motion. Since the vibration is inevitable during hand-operated turntable rolls, the output of outer encoder in Fig. 9(a) indicates the frame locking errors are less than 0.04° under disturbance condition. While in Fig. 9(b), the outer encoder angle and roll angle have the same amplitude and opposite direction, demonstrating that turntable roll motion has been effectively insulated by the outer frame of RINS.

Before the navigation experiments, the relative parameters of FOGs have been pre-calibrated, but there still exists residual errors. The value of residual errors cannot be obtained directly, which mainly includes two parts, the one is the calibration errors and the other part is time-varying random error caused by inertial sensor instability. The results under this situation is called “without extra sensor errors”, denoting as blue lines in Figs. 10 and 11. In each test, some extra sensor errors are added according to Table 2 through the offline navigation algorithm, from which we can analyze the influence of sensor errors on navigation performance by operating the two motor control methods. This situation is called “with extra sensor errors”, denoting as red line in the following figures.

As shown in Fig. 10(a), when turntable operates once roll motion, there is an inflection point on the velocity errors under the traditional motor control method. It indicates that the platform declination angle is motivated during the roll motion, resulting in a change in the slope of velocity errors. Moreover, the slope of velocity errors is even steeper after adding extra sensor errors, and the maximum value of northern velocity reaches 0.8 m/s. While in Fig. 10(b), it can be seen that the velocity errors are much smaller, the maximum value of northern velocity is only 0.22m/s. Besides, the inflection point has also disappeared, and there is no significant change in velocity errors after adding the extra sensor errors as well. In Fig. 11, similar conclusion can be drawn that the positioning errors has been significantly restricted after insulating the carrier’s roll motion, even with the addition of extra sensor errors. Above all, the experimental results demonstrate that the navigation performance can be highly improved by the insulation of carrier’s roll motion, which can still be maintained even with large FOG installation errors and scale factor errors.

## 5. Conclusion

According to the analysis in this paper, the drawbacks of traditional motor control method for RINS are mainly manifested in the following three aspects.

- (1) When carrier turns, the fiber optic gyro (FOG) drifts may no longer be zero-mean in navigation coordinate;
- (2) Carrier’s roll motion is often accompanied by heading motion, and the coupling errors will be motivated by FOG installation errors and scale factor errors, thereby deteriorating the navigation performance;
- (3) The error parameters of FOG are susceptible to the temperature and magnetic fields, while pre-calibration is a time-consuming process which is difficult to completely suppress the FOG-related errors.

In this paper, an improved motor control method with measurements of FOG is proposed to address the above problems for dual-axis RINS. Benefiting from the double-frame structure, the outer frame can insulate the carrier’s roll motion and the inner frame can simultaneously achieve the rotary modulation on the basis of insulating the heading motion. The results of turntable experiments indicate that the navigation performance is significantly improved over the traditional motor control method, which can still be maintained even with large installation errors and scale factor errors, indicating that the improving motor control method can relax the requirements for the accuracy of FOG-related errors. Furthermore, limited by the number of frames, dual-axis RINS can only insulate the angular motion in two directions. When the RINS is designed as triple or more frames structure, the angular motion in all three directions can be fully insulated, and achieving much better performance.

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