## Abstract

This paper presents an approach to detect the index of incremental photoelectric encoder with shorter time for rotary inertial navigation system. The order of index detection and coarse alignment is exchanged in this approach and information from coarse alignment is used to calculate the direction of index. Then two cases of azimuth axis and four cases of horizontal axes are analyzed and corresponding solutions are designed. The paper examines the solutions through two experiments in a tri-axis rotary inertial navigation system, and the results demonstrate that the system can capture index pulse in the application of the presupposes schemes with shorter time.

© 2016 Optical Society of America

## 1. **Introduction**

Inertial navigation system (INS) is a key autonomous device for vehicles such as airplanes and missiles [1,2 ]. However, the error of INS will increase as time goes by [3]. Generally, there are three methods to resolve this problem.The first one is to improve the precision of inertial sensors such as gyroscopes [4–6 ] and accelerometers [7,8 ]. The second one is to integrate with other navigation information [9,10 ]. And the third one is to rotate the inertial measurement unit (IMU) regularly [11,12 ], which is commonly called rotary inertial navigation system (RINS) [13,14 ] and has become popular these years because of its high ratio of performance to price [15,16 ]. In RINS, angle measuring sensors are necessary for the realization of rotation [17]. An incremental photoelectric encoder is chosen as the sensor in many systems of this type for its advantages of shock, vibration, cost, and size [18]. In addition, an incremental encoder provides information by a quadrature encoder pulse (QEP) and QEP is characterized by the ability of anti-electromagnetic interference, high accuracy and fast response [19].

For an incremental photoelectric encoder, as it can only provide a relative angle, namely angular velocity, an index pulse generated at a fixed place is given to help define the absolute angle. Nevertheless, an index pulse will not be given until it rotates to the fixed place. This implies that it costs time to generate and detect the index pulse. However in many applications, short setup time is an important factor [20,21 ]. Taking the application in an airplane as an example, once the airplane receives an urgent task, there is little time before taking off, and therefore every second is precious. So it requires the INS utilized in this field to work normally as soon as it is turned on [22] and limits the application of RINS in these fields.

Normally to find the index pulse, the rotary mechanism rotates in the same direction because it does not know which direction the index is in. The rotation continues until the index pusle is generated and detected. If a rotary mechanism without any resistance rotates a full circle, the index pulse can be captured undoubtedly. Thus the maximum angle it rotates is $360\xb0$. By comparison, for a mechanism with limits, as shown in Fig. 1 , once the position where rotation starts is just in the reverse direction, the maximum angle can reach approximately $720\xb0$. Therefore, if we want to detect the index rapidly, we can get the specific direction and shorten the route.

The primary working procedures of INS are coarse alignment, fine alignment, and navigation [23]. In RINS, the key purpose of rotation is to improve the accuracy of fine alignment and navigation [24,25 ]. So it allows rotation to start working no later than fine alignment. Based on this, a novel method to find index pulse with the assistance of coarse alignment is developed, and the direction of index can be estimated.

## 2. Methods

When there is only relative angle, the rotary mechanism can still fix IMU where it stays at the beginning of work. And coarse alignment can be finished in the same way as ordinary INS. According to Eq. (1), Eq. (2), and Eq. (3), the attitude angles, including pitch angle and roll angle, and azimuth angle of IMU can be obtained.

Continuing with the airplane example, once it parks on the apron and shuts down, the airplane’s attitude angles and azimuth angle are invariant until its next task. When an airplane shuts down, storing of the airplane’s azimuth angle for the next task is operable. Since an apron is built horizontally, both the pitch angle and the roll angle of the airplane are almost zero. Hence, the original attitude angles and azimuth angle of the airplane are known conditions. Together with these angles of IMU obtained by the calculation proposed above, the positional relationship of the airplane and IMU is definite. Moreover, the rotary mechanism’s rotational axes except the azimuth axis are used to keep IMU horizontal, so the attitude angles of IMU are quite small. As shown in Fig. 2 where the small attitude angles of both airplane and IMU are ignored, there are three coordinates and one vector. Geographic coordinates are marked East and North. IMU coordinates are marked ${X}_{p}$ and ${Y}_{p}$. Airplane coordinates are marked ${X}_{b}$ and ${Y}_{b}$. The vector marked ${X}_{c}$ represents the index of incremental photoelectric encoder z. Meanwhile, ${\psi}_{m}$ is the azimuth angle of airplane at last shut down, ${\psi}_{p}$ is the present azimuth angle of IMU, ${\varphi}_{zc}$ is the angle between the index of encoder z and the axis x of airplane coordinates, and ${\varphi}_{zpc}$is the angle between the index of encoder z and the axis x of IMU coordinates. In these four physical quantities, ${\varphi}_{zc}$is determined once an encoder is mounted in RINS, and the other three vary with the motion of the airplane or IMU. However, the relationship between them is definitive. By Eq. (4), ${\varphi}_{zpc}$ can be calculated and the direction of index can be determined by the value of this angle.

Comparing with ${\varphi}_{zpc}$, the calculation of the similar angles for encoder x and y is more complex. As shown in Fig. 3 , the positional relationship between IMU coordinates and airplane coordinates is related to the position where rotation along axis z stops, in spite of that the attitude angles are close to zero. So, different from the azimuth angle, the attitude angles of IMU and airplane cannot be added or subtracted directly, and there should be a transformation in advance. ${{\theta}^{\prime}}_{p}$ and ${{\gamma}^{\prime}}_{p}$ are the new angles after transformation in Fig. 3. And the transformation is given by Eq. (5).

Then taking axis x for example, the angular relationship is given in Fig. 4 . The vector marked ${Y}_{c}$ represents the index of incremental photoelectric encoder x. ${\theta}_{b}$ is the pitch angle of airplane. ${\varphi}_{xc}$ is the angle between the index of encoder x and the axis y of airplane coordinates. ${{\theta}^{\prime}}_{p}$ is the new angle in Eq. (5), and ${\varphi}_{xpc}$ is the present angle between the index of encoder x and the axis y of IMU coordinates. Analyzing this figure, the relationship between these angles can be described as Eq. (6). Considering that ${\theta}_{b}$ is close to zero, Eq. (6) can be simplified as Eq. (7).

Similarly, the angle ${\varphi}_{ypc}$ between the index of encoder y and the axis z of IMU coordinates can be derived as Eq. (8).

where ${\varphi}_{yc}$ is the angle between the index of encoder y and the axis z of airplane coordinates, and ${{\gamma}^{\prime}}_{p}$ is the new angle in Eq. (5).So far, the values of the angles characterizing the distance to index have been obtained and what’s needed next is the determination of rotation direction. The values of ${\varphi}_{zpc}$, ${\varphi}_{xpc}$, and ${\varphi}_{ypc}$ in Eq. (4), Eq. (7) and Eq. (8) are just approximations, in particular the error amounts of ${\varphi}_{xpc}$ and ${\varphi}_{ypc}$can run up to several degrees possibly. Thus in order to make sure the index can be detected, we assign small angles to axis x and y as the reference instead of angle zero, while the determination for axis z is still made referring to zero. Again in the application of a conventional airplane, the small angle is selected as $10\xb0$, because it’s impossible for an airplane parking on the apron to have a big attitude angle. With regard to other applications, the small angle can be set as another value. Based on the analysis, the scheme to detect index pulse is designed as follows.

For axis z, it rotates forward if ${\varphi}_{zpc}$ is positive and backward if ${\varphi}_{zpc}$ is negative. For the convenience of experiments in the following, the two cases are named Case A and Case B thereafter, respectively. For axes x and y, there are more steps and cases. As the treatments of axes x and y are the same, axis x is used as an example here.

As shown in Fig. 5 , if ${\varphi}_{xpc}\le -10\xb0$, it rotates backward, and it rotates forward if ${\varphi}_{xpc}\ge 10\xb0$. For the case that $-10\xb0<{\varphi}_{xpc}<0\xb0$, it rotates forward until ${\varphi}_{xpc}=-10\xb0$, and then rotates backward. If $0\xb0\le {\varphi}_{xpc}<10\xb0$, it rotates backward until ${\varphi}_{xpc}=10\xb0$, and then rotates forward. Once the rotation starts, there is no stopping until index pulse is detected. Likewise, the four cases are named from Case 1 to Case 4 .

## 3. Experiments and discussion

To demonstrate the effectiveness of this novel approach, two experiments are carried out in a tri-axis RINS with fiber optic gyro. During the experiments, the system is mounted on a turntable to imitate the attitudes and azimuth of an airplane, as shown in Fig. 6 where the inner structure of this RINS is also given in the form of schematic diagram. The three axes of this system are axis z, x, and y going from inside to outside, and the value of angles between the indexes of photoelectric encoders and the axes of airplane coordinates are ${\varphi}_{zc}=-147.14\xb0$, ${\varphi}_{xc}=1.84\xb0$, and ${\varphi}_{yc}=2.47\xb0$ which were calibrated accurately ahead of time. The range that axes x and y can rotate is only $\pm 180\xb0$ because of a resist block while axis z can rotate without a hindrance. So, with a traditional method, the maximum angle that axis z rotates before it detects index pulse is $360\xb0$, and the similar angles of axis x and y can reach $720\xb0$ as analyzed at the beginning of this paper.

The turntable is set horizontally, which represent the zero degree attitude angles of an airplane. Meanwhile, the azimuth axis of the turntable is adjusted until the axis y of RINS roughly points north, so the azimuth angle of the airplane is about zero and ${\psi}_{m}=0\xb0$. In the first experiment, the original angles of axes x, y, and z are set at about $-10\xb0$, $15\xb0$, and $-90\xb0$ respectively so that they rotate to detect index pulse by the methods in Cases 2, 1, and B. In the second experiment, the original angles of axis x, y and z are set at about $0\xb0$, $6\xb0$ and $180\xb0$, respectively, and they rotate by the methods in Cases 4, 3, and A. The rotation speed is set at $6\xb0/s$ for axis z and $1\xb0/s$ for axis x and y in all of the experiments.

Figure 7 shows the angles of the three axes in the first experiment. The time from $0s$ to $30s$ is used for the coarse alignment of RINS, and according to the calculations from Eq. (1), Eq. (2), and Eq. (3), ${\theta}_{p}=-14.82\xb0$, ${\gamma}_{p}=-9.89\xb0$, and ${\psi}_{p}=90.36\xb0$. Other results include ${\varphi}_{zpc}=-56.78\xb0$ according to Eq. (4), ${{\theta}^{\prime}}_{p}=-9.80\xb0$, ${{\gamma}^{\prime}}_{p}=14.88\xb0$ according to Eq. (5), ${\varphi}_{xpc}=11.64\xb0$ according to Eq. (7), and ${\varphi}_{ypc}=-12.41\xb0$ according to Eq. (8). So by the method in this paper, axis x should follow Case 2, axis y should follow Case 1, and axis z should follow Case B. The steps at the end of the three curves are the places finishing the index detections. The new index detections cost less time. For instance, with a traditional method, axis y would start rotating from $15\xb0$ and would not change its rotation direction from forward to backward until it touched the resist block which lies at $180\xb0$. Then it would rotate backward and stop at $2.47\xb0$ where the index of encoder y lies. In total, the angle it rotates is nearly $\left|180\xb0-15\xb0\right|+\left|2.47\xb0-180\xb0\right|=342.53\xb0$. Comparably, the angle in this new method is only $12.68\xb0$ as shown in Fig. 7.

Figure 8 shows the angles of the three axes in the second experiment. According to Eq. (1), Eq. (2), and Eq. (3), ${\theta}_{p}=0.19\xb0$, ${\gamma}_{p}=-5.51\xb0$, and ${\psi}_{p}=180.90\xb0$. Also, ${\varphi}_{zpc}=33.76\xb0$ according to Eq. (4), ${{\theta}^{\prime}}_{p}=-0.10\xb0$, ${{\gamma}^{\prime}}_{p}=5.51\xb0$ according to Eq. (5), ${\varphi}_{xpc}=1.94\xb0$ according to Eq. (7), and ${\varphi}_{ypc}=-3.04\xb0$ according to Eq. (8). So axis x should follow Case 4, axis y should follow Case 3, and axis z should follow Case A. Both axes x and y rotate to $10\xb0$ or $-10\xb0$ in reference to the position where the index lies at the beginning, and then rotates in reverse. Like the first experiment, the rotation angle that index detection costs here is merely scores of degrees, and this process can be finished in a short time. In fact, Case 4 indicates that the fastest way of detection in this situation is to rotate forward directly. However, this novel method here is designed to save the holistic time rather than shorten the time for every case. As a result, the time that index detection costs may increase in several cases, but the increment is small. The time decreases in other cases, but the decrement is large.

## 4. Conclusion

In conclusion, an approach to shorten the time to detect the index of incremental photoelectric encoder in RINS is designed. Based on the attitude angles and azimuth angle in coarse alignment, different strategies are given for index detection. In addition, two experiments, involving all of the cases listed, are designed and implemented to validate the effectiveness of this approach. According to the analysis and experiment result, only scores of degrees are needed here instead of the hundreds of degrees required by the traditional method, achieving a short setup time in RINS.

## Acknowledgments

The authors would like to thank Chinese academician Peide Feng for his professional suggestions and postgraduate Ariel Cho from Tufts University for her linguistic revise. This work was supported by the National Natural Science Foundation (NO.51574012), the State Key Laboratory of Geo-Information Engineering (NO. SKLGIE2015-M-2-3), and the Fund of BUAA for Graduate Innovation and Practice (NO.YCSJ-01-2015-08).

## References and links

**1. **P. G. Savage, “Blazing gyros: The evolution of strapdown inertial navigation technology for aircraft,” J. Guid. Control Dyn. **36**(3), 637–655 (2013). [CrossRef]

**2. **J. N. Chamoun and M. J. F. Digonnet, “Noise and bias error due to polarization coupling in a fiber optic gyroscope,” J. Lightwave Technol. **33**(13), 2839–2847 (2015). [CrossRef]

**3. **Z. Wang, H. Zhao, S. Qiu, and Q. Gao, “Stance-phase detection for ZUPT-aided foot-mounted pedestrian navigation system,” IEEE/ASME Trans. Mech. **20**(6), 3170–3181 (2015).

**4. **J. K. Stockton, K. Takase, and M. A. Kasevich, “Absolute geodetic rotation measurement using atom interferometry,” Phys. Rev. Lett. **107**(13), 133001 (2011). [CrossRef] [PubMed]

**5. **L. Feng, J. Wang, Y. Zhi, Y. Tang, Q. Wang, H. Li, and W. Wang, “Transmissive resonator optic gyro based on silica waveguide ring resonator,” Opt. Express **22**(22), 27565–27575 (2014). [CrossRef] [PubMed]

**6. **Y. Yan, H. Ma, and Z. Jin, “Reducing polarization-fluctuation induced drift in resonant fiber optic gyro by using single-polarization fiber,” Opt. Express **23**(3), 2002–2009 (2015). [CrossRef] [PubMed]

**7. **O. Gerberding, F. G. Cervantes, J. Melcher, J. R. Pratt, and J. M. Taylor, “Optomechanical reference accelerometer,” Metrologia **52**(5), 654–665 (2015). [CrossRef]

**8. **Z. Wang, W. Zhang, W. Huang, and F. Li, “Liquid-damped fiber laser accelerometer: Theory and experiment,” IEEE Sens. J. **15**(11), 6360–6365 (2015). [CrossRef]

**9. **Z. Wu, X. Hu, M. Wu, H. Mu, J. Cao, K. Zhang, and Z. Tuo, “An experimental evaluation of autonomous underwater vehicle localization on geomagnetic map,” Appl. Phys. Lett. **103**(10), 104102 (2013). [CrossRef]

**10. **J. Waldmann, R. I. G. Silva, and R. A. J. Chagas, “Observability analysis of inertial navigation errors from optical flow subspace constraint,” Inf. Sci. **327**, 300–326 (2016). [CrossRef]

**11. **B. Yuan, D. Liao, and S. Han, “Error compensation of an optical gyro INS by multi-axis rotation,” Meas. Sci. Technol. **23**(2), 025102 (2012). [CrossRef]

**12. **Z. Zheng, S. Han, and K. Zheng, “An eight-position self-calibration method for a dual-axis rotational Inertial Navigation System,” Sen. Actuat. A. Phys. **232**, 39–48 (2015).

**13. **W. Sun, D. Wang, L. Xu, and L. Xu, “MEMS-based rotary strapdown inertial navigation system,” Meas. **46**(8), 2585–2596 (2013). [CrossRef]

**14. **L. Wang, W. Wang, Q. Zhang, and P. Gao, “Self-calibration method based on navigation in high-precision inertial navigation system with fiber optic gyro,” Opt. Eng. **53**(6), 064103 (2014). [CrossRef]

**15. **Q. Ren, B. Wang, Z. Deng, and M. Fu, “A multi-position self-calibration method for dual-axis rotational inertial navigation system,” Sen. Actuat. A-Phys. **219**, 24–31 (2014).

**16. **Q. Zhang, L. Wang, Z. Liu, and P. Feng, “An accurate calibration method based on velocity in a rotational inertial navigation system,” Sensors (Basel) **15**(8), 18443–18458 (2015). [CrossRef] [PubMed]

**17. **X. Wang, J. Wu, T. Xu, and W. Wang, “Analysis and verification of rotation modulation effects on inertial navigation system based on MEMS sensors,” J. Navig. **66**(5), 751–772 (2013). [CrossRef]

**18. **F. P. Quintián, N. Calarco, A. Lutenberg, and J. Lipovetzky, “Performance of an optical encoder based on a nondiffractive beam implemented with a specific photodetection integrated circuit and a diffractive optical element,” Appl. Opt. **54**(25), 7640–7647 (2015). [CrossRef] [PubMed]

**19. **N. Al-Emadi, L. Ben-Brahim, and M. Benammar, “A new tracking technique for mechanical angle measurement,” Meas. **54**, 58–64 (2014). [CrossRef]

**20. **X. Wang, “Fast alignment and calibration algorithms for inertial navigation system,” Aerosp. Sci. Technol. **13**(4–5), 204–209 (2009). [CrossRef]

**21. **S. Gao, W. Wei, Y. Zhong, and Z. Feng, “Rapid alignment method based on local observability analysis for strapdown inertial navigation system,” Acta Astronaut. **94**(2), 790–798 (2014). [CrossRef]

**22. **P. M. G. Silson, “Coarse alignment of a ship’s strapdown inertial attitude reference system using velocity loci,” IEEE Trans. Instrum. Meas. **60**(6), 1930–1941 (2011). [CrossRef]

**23. **J. Ali, M. Ushaq, and S. Majeed, “Robust aspects in dependable filter design for alignment of a submarine inertial navigation system,” J. Mar. Sci. Technol. **17**(3), 340–348 (2012). [CrossRef]

**24. **P. Marland, “The NATO ships inertial navigation system (SINS),” J. Nav. Eng. **33**(3), 688–700 (1992).

**25. **F. Pei, L. Zhu, and J. Zhao, “Initial self-alignment for marine rotary SINS using novel adaptive kalman filter,” Math. Probl. Eng. **2015**, 320536 (2015).