Essay: Fibre optic gyroscope

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CHAPTER 1
INTRODUCTION
1.1 OVERVIEW
The second half of the twenty th century is marked not only by social changes; but also intensive and thorough investigations in fundamental and applied science. Two aspects are to be of the paramount importance in the field of navigation 1) creation of global satellite navigation systems, such as Transit, GPS and GLONASS; 2) the advent of a new generation of inertial navigation sensors -laser and fiber optical gyros (LG and FOG). These optical devices, making use of fundamental properties of electromagnetic waves, provide a basis for a new line of investigations in inertial navigation and make it possible to give up the fast rotating rotor. This situation opens a possibility to use new progressive technology in mass production. Exceptional properties of optical gyros (high accuracy, wide dynamic range, non-sensitivity to linear acceleration, et al.) stimulated the evolution of various highly accurate strap down inertial navigation systems (SINS). Integration of SINS based on optical, gyros with Satellite Navigation Systems, such as GPS and GLONASS, allows new positive properties necessary for dual usage in military and commercial technology.
There is an increasing demand for accurate, yet low-cost and highly reliable guidance, control, and navigation systems for measuring the direction and altitude of an object. Gyroscope is the core component for providing this information. Although different type of gyroscopes are available, Fiber Optic Gyroscope (FOG)[4] is a proven technology for measuring angular velocity of an object .It has the advantages of low reaction time, wide dynamic range, high accuracy and reliability. The basic operational principle of FOG is that the optical path difference induced by counter propagating beams in a rotating reference frame is proportional to the absolute rotation rate (Sagnac effect).
The advantages of a fiber gyroscope over a mechanical gyroscope are that there are no moving parts, no warm-up time is required, and there is no sensitivity to acceleration. They also promise to be low in cost. The required accuracy for a gyroscope depends on the application. Short-f1ight missile and automobile applications can probably be satisfied with an accuracy or 10 degree per hour. To appreciate this term, consider that the earth rotates at 15 degree per hour, and the hour hand on a clock rotates at 30 degree per hour. Both of these are considered to be relatively large rotation rates from a gyroscope\’s point of view. Gyroscope is the most important and critical component of the inertial navigation system (INS).
In the strap down inertial navigation system (SINS), Fiber optic gyroscope (FOG) has been used for measuring the rotation angle. Recently, FOG is being widely used for military and defense applications, due to its significant advantages such as small size, low cost, light weight, no moving parts, large dynamic range, low power consumption, and possible batch fabrication. The performance of FOG degrades due to the variation in environmental factors such as temperature, vibration, and pressure. Among different types of error in the FOG signal, random drift error leads to decrease the FOG performance over a period of time. The precision of the FOG sensor depends on the bias drift and noise in the measurement. FOG sensor has mainly two types of errors
1. Deterministic error
2. Stochastic error
Deterministic errors are due to the scale factor, bias and misalignment which can be eliminated by suitable calibration techniques in the laboratory environment. However, stochastic errors are due to the environmental temperature changes, electronic components and, other electronic equipment interfaced with it, it is difficult to eliminate these errors by calibration. Thus stochastic models are required to characterize these errors and signal processing techniques are required to suppress these errors.
1.2GYROSCOPE HISTORY
The advent of the fiber optic gyroscope (FOG) dates back to the mid-1970s when Vali and Short hill demonstrated the first fiber optic rotation sensor. This breakthrough followed the pioneering efforts of R. B. Brown from the Navy Laboratory in 1968, who proposed a coil of optical fiber as a rotation sensor. Fringes were demonstrated in an optical fiber ring interferometer in 1975 using low-loss, single mode fiber. During the years to follow, a number of researchers and developers worldwide made the FOG concept become a reality. A number of universities and industrial laboratories such as McDonnell Douglas, Northrop-Grumman (Litton), Honeywell, Northrop, Singer, Lear Siegler, Martin Marietta, and others have investigated the FOG. Gyroscope bias errors of 0.01°/hr were being achieved in the laboratory by the early 1980s.
The development of the FOG has flourished during the past 30 years. It has evolved from a laboratory experiment to the production floors, and thus into practical applications such as in navigation, guidance, and control of aircraft, missiles, automobiles, robots, and spacecraft. FOGs, which have replaced the RLG in a number of applications requiring 1.0°/hr performance accuracy, represent the prevalent solution in numerous applications of navigation, guidance, and stabilization in the better than 0.1°/hr regime (missiles, attitude heading and reference systems [AHRS], robotics, satellites, etc.). A great deal of effort has been made in the development of navigation-grade gyroscopes for aircraft and space applications with bias drift less than 0.01°/hr and scale factor of less than 10 parts per million (ppm). FOGs are currently used in the navigation system of aircraft such as the Boeing 777.
1.3FIBER OPTIC GYROSCOPE
Fibre optic gyroscope works on the principle of Sagnac effect.
Fig 1.1: Simple Fiber Optic Gyroscope.
The schematic of fibre optic gyro is as shown in fig, here the source is Super Luminescent Diode (SLD), and the receiver is photo diode (PD). the light wave generated by SLD is passing through optical fibre to fibre polarizer to maintain the polarization of light wave in single direction, the polarized light is given to 3-db coupler to split the light wave in to two equal intensity light waves, the splitted light waves are passing through closed loop optical fibre which is in circular shape. Hence light is travelled in opposite direction and the two light waves are collected at photo diode.
When GYRO there is no rotation the two light waves have the same path length, hence there is no phase difference in the received light waves. We will get zero rotation rates. If we rotate the gyro in clock wise direction the light wave in the clockwise direction will go further than the light travelled in the anti clock wise direction. This will cause a very small difference in the time of the received light beams as shown in fig 1.2. This can be detected and translated into a measurement of how far the sensor has turned.
Parameter Units Test data
Range o/s ±150
Bias over temperature range o/h max 50
Start up time Sec max 1
Operating temperature oc -30 – +60
Table 1: Specifications of High accuracy Fiber Optic Gyro
Fig 1.2: Path difference sensing in FOG.
1.4 KALMAN FILTER
Kalman filter is an optimal estimator that infers parameters of interest from indirect, inaccurate and uncertain observations[18]. The process of finding the “best estimate” from noisy data amounts to “filtering out” the noise.
Fig 1.3: Kalman filter circuit
KF uses state space representation of input and various other parameters. Process noise covariance matrix (Q_k) and measurement noise covariance matrix (R_k) are the two parameters that effect the output. There are several variants of the KF based on the way the values of Q_k and R_kare estimated. In the basic KF, fixed values of Q_k and R_k are considered to filter the signal by assuming that the process noise and measurement noise characteristics are statistically known.
The effective de-noising can be achieved by appropriate initialization of values of measurement noise covarianceR_k, process noise covariance Q_k and error covariancep_0.
The de noising process using KF has two stages
Prediction
Correction
In the first stage KF predicts the current state and error covariance based on the previous state estimate. The second stage corrects the previous predicted values using the present measured value. This procedure is repeated for each time step with the state of previous time step as initial value. Therefore the Kalman filter is called Recursive filter.
1.5 LITERATURE REVIEW:
Gyros
The word gyroscope was first coined by a French scientist, Leon Foucault, in 1852. It is derived from the Greek words “gyro,” meaning revolution, and “skopien,” meaning to view. The gyroscope, commonly called a GYRO[4], has existed since the first electron was sent spinning on its axis. Electrons spin and show all the characteristics of a gyro; so does the Earth, which spins about its polar axis at over 1000 miles per hour at the Equator. The Earth\’s rotation about its axis provides the stabilizing effect that keeps the North Pole pointed within one degree of Polaris (the North Star).
Characterization of Fiber Optics Gyro and noise compensation using DWT
This paper presents quantification of different types of random errors present in the Fiber Optics Gyroscope (FOG) measured data using Allan Variance analysis and denoising of the measured data using Discrete Wavelet Transform (DWT)[5]. Allan Variance analysis is performed before and after denoising the measured data. The experimental result shows that after denoising the angle random walk is reduced and therefore sensitivity of FOG is increased.
Optimization approach to adapt Kalman filters for the real-time application of accelerometer and gyroscope signals’ filtering[2]
A problem of accelerometer and gyroscope signals’ filtering is discussed in the paper. Triple-axis accelerometer and three single-axis gyroscopes are the elements of strap down system measuring head. Effective noise filtration impacts on measured signal reliability and the computation precision of moving object position and orientation. The investigations were carried out to apply Kalman filter in a real-time application of acceleration and angular rate signals filtering. The filter parameter adjusting is the most important task of the investigation, because of unknown accuracy of the measuring head and unavailability of precisely known model of the system and the measurement. Results of calculations presented in the paper describe relation between filter parameters and two assumed criterions of filtering quality: output signal noise level and filter response rate. The aim of investigation was to achieve and find values of the parameters which make Kalman filter useful in the real-time application of acceleration and angular rate signals filtering.
Research on filter method and model of MEMS gyro static drift[7]
The MEMS gyro in robot is researched. The noise feature of MEMS gyro is analyzed .Based on the AR model, the statistic parameters are reckoned. Then ,the Kalman filter method is used to denoise the signal of MEMS gyro. The simulation results show that the method of modeling and Kalman filter can achieve good performance on the denoise of MEMS gyro.
A novel adaptive filter mechanism for improving the measurement accuracy of the fiber optic gyroscope in the maneuvering case[10]
Although much work on modelling the drift of fiber optic gyroscope (FOG) in the quiescent case has been done, little attention is paid to eliminate the influence of the uncertain gyroscope drift in the maneuvering case. In this paper, a novel adaptive mechanism based on double transitive factors is highlighted, which is implemented in two stages. In the first stage one adaptive factor is investigated, and then the productions of the stage, the suboptimal estimated state vector and its suboptimal covariance matrix are passed to the second stage, where the other adaptive factor is derived, and then the modified covariance matrix of measurement noise is passed to the first stage; consequently the optimal estimation may be achieved based on the recursive mechanism. As demonstrated in the testing of a specific type of FOG, the proposed method, compared with the other scheme in this paper, greatly reduces the influence of insufficient kinematic model and stochastic error, thus improving the measurement accuracy of FOG in the maneuvering case
Error and Noise Analysis in an IMU using Kalman Filter [S.A.Quadri and Othman Sidek]
Kalman filtering is a well-established methodology used in various multi-sensor data fusion applications. In our experiment, we first obtain measurements from the accelerometer and gyroscope and fuse them using Kalman filter in an inertial measurement unit (IMU).[9] We estimate Kalman filter output and estimation error. The affect of process noise and measurement noise on estimation error is tested. It is explored that the measurement noise has significant role to increase estimation error in the data fusion process.
Kalman Filtering [DAN SIMON][18]
The Kalman filter is a tool that can estimate the variables of a wide range of processes. In mathematical terms we would say that a Kalman filter estimates the states of a linear system. The Kalman filter not only works well in practice, but it is theoretically attractive because it can be shown that of all possible filters, it is the one that minimizes the variance of the estimation error. Kalman filters are often implemented in embedded control systems because in order to control a process, you first need an accurate estimate of the process variables.
This article will tell you the basic concepts that you need to know to design and implement a Kalman filter. I will introduce the Kalman filter algorithm and we’ll look at the use of this filter to solve a vehicle navigation problem. In order to control the position of an automated vehicle, we first must have a reliable estimate of the vehicle’s present position. Kalman filtering provides a tool for obtaining that reliable estimate.
An Integrated MEMS Gyroscope Array with Higher Accuracy Output[13]
In this paper, an integrated MEMS gyroscope array method composed of two levels of optimal filtering was designed to improve the accuracy of gyroscopes. In the first level filtering, several identical gyroscopes were combined through Kalman filtering into a single effective device, whose performance could surpass that of any individual sensor. The key of the performance improving lies in the optimal estimation of the random noise sources such as rate random walk and angular random walk for compensating the measurement values. Especially, the cross correlation between the noises from different gyroscopes of the same type was used to establish the system noise covariance matrix and the measurement noise covariance matrix for Kalman filtering to improve the performance further. Secondly, an integrated Kalman filter with six states was designed to further improve the accuracy with the aid of external sensors such as magnetometers and accelerometers in attitude determination. Experiments showed that three gyroscopes with a bias drift of 35 degree per hour could be combined into a virtual gyroscope with a drift of 1.07 degree per hour through the first-level filter, and the bias drift was reduced to 0.53 degree per hour after the second-level filtering. It proved that the proposed integrated MEMS gyroscope array is capable of improving the accuracy of the MEMS gyroscopes, which provides the possibility of using these low cost MEMS sensors in high-accuracy application areas.
MEMS Gyro Denoising Based on Second Generation Wavelet Transform
In order to improve the precision of MEMS gyro, the MEMS gyro denoising method based on second generation wavelet transform is investigated in this paper. After introducing the principles and establishment of the second generation wavelet transform, a wavelet thresholding denoising method based on the lifting scheme wavelet transform is used to remove the random drift of three axis MEMS gyro. The denoisng method we used showed good performance in speed and a little better precision compared to conventional wavelet thresholding schemes[14]. The experimental results and analysis indicate that the lifting denoising can effectively reduce the random drift error of MEMS gyro and can be used in real time system.
Modeling Random Gyro Drift Rate by Data Dependent Systems
A new modeling procedure is presented in which a mathematical model for random gyro drift rate is obtained directly from the original observations without requiring any theoretical conjectures or preprocessing of the data. The adequate model so obtained confirms the presence of a random walk component causing the “steady state” drift and a stochastic component of the nature of correlated noise. It also provides a quantitative measure of the relative contribution of each component, useful in determining the quality of the gyro in guidance over extended periods.
System on chip implementation of 1-D Wavelet transform based denoising of Fiber Optic Gyroscope signal on FPGA
This paper presents system on chip (SoC) implementation of 1-dimensional discrete wavelet transform (DWT) for real time denoising of fiber optic gyroscope signal (FOG) in Field Programmable Gate Array (FPGA). The Hardware DWT [14] IP is designed by using Distributed arithmetic filters. The design is carried out in Xilinx System generator for DSP and is integrated with Microblaze embedded system for SoC implementation. Simulation was performed on the Virtex5-FX70T -1136 platform for both static and dynamic gyro signal datasets. The results show that the algorithmic (floating point) results match with hardware (fixed point) results. The resource utilization is 60% in Virtex-5FX70T and the maximum frequency of operation is 180 MHz.
1.6 PROBLEM DEFINITION
In this project I studied about the various noises present in the fiber optic gyroscope signal and in order to reduce the stochastic errors like quantization noise, angle random walk, bias instability, rate random walk with the help of kalman filter. The kalman filter simulink model was developed and simulated for FOG based applications .The algorithms like discrete wavelet transform and kalman filter are successfully denoise the Fog signal in steady state condition. These algorithms fail while denoising the dynamic condition FOG signal. The proposed algorithm is efficiently denoise the FOG signal in both static and dynamic condition .the proposed algorithm is hybridization of kalman filter with adaptive moving average (AMA) technique and named it as adaptive moving average dual gain kalman filter (AMADGKF).
CHAPTER 2
BASIC CONFIGURATION OF FIBER GYROS
2.1 PRINCIPLE OF OPERATION
Sagnac first demonstrated the optical gyroscope principle in 1913. Optical gyroscopes implemented so far use Sagnac effect, which states that an optical path difference induced by counter propagating beams in a rotating reference frame is proportional to the absolute rotation. The induced optical path difference can be measured in two ways (i) by making a frequency measurement in a laser resonator or in a resonator fiber optic gyroscope (RFOG) (ii) by a phase measurement of two interfering beams as in an Interferometric fiber optic gyroscope (IFOG).
Fig2.1 Sagnac Interferometer [a] Bulk Optics [b] Fiber Optics and
[C] Simplified interferometer
The basic configuration for Sagnac interferometer in fiber optic form is shown in Fig. 2. A simplified interferometer that illustrates the principle of operation is shown in Fig. 2. (c). When system is at rest the light propagating in clockwise (CW) and counter clockwise (CCW) directions traverse identical paths and so there is no phase difference between them. When the system rotates at a angular velocity ‘Ω’, the light beam rotating with the ring has an optical path longer than the counter-rotating beam by a distance (LR/C)Ω .
The Sagnac phase shift due to rotation rate, Ω, is expressed as
Where ΔΦR is the Phase shift due to rotation rate, A= R2, the area enclosed by the path of radius R, L is length of the coil L=2RN, N being the number of turns of fiber loop and c is the speed of light,  is source wavelength. It is important to note a constant angular velocity yields a constant phase difference. We can thus think of a FOG as a rate gyro i.e., the output is proportional to rate.
As with any interferometer where two waves overlap in space, the response obtained is co sinusoidal. Thus, the optical intensity can be written as
P=Po (1+cos (ΦR))
The main limitations of the basic configuration are: Poor sensitivity for small rotation rates, direction ambiguity, limited dynamic range and non linearity.
The problems of poor sensitivity and ambiguity in the determination of direction are usually overcome by the application of the differential phase modulation (also known as dynamic phase bias). A phase modulator is used for this purpose and it is placed in an asymmetric position in the sensing fiber loop so that the two counter propagating waves pass through the modulator at different time instants before that interfere with each other. This creates a phase difference between the two counter propagating beams. If the phase modulator is located at one end of the gyro-sensing loop, the time difference corresponds to the transit time of light through the length of fiber in the sensing loop.
The Open loop FOG uses may in the former, a lock-in amplifier is used to measure the photo detector output at the fundamental modulating frequency and the rotation rate is directly computed from this measurement. Additionally, the amplitude of the second and the higher harmonics may also be measured to compensate for the variations in the returning optical power and changes in the amplitude of the differential phase modulation.
The basic configuration of single axis all fiber gyro is shown in fig.2.2.
A basic FOG configuration is shown in figure. Light from a broadband source, such as a super luminescent diode (SLD), is projected into a 3-dB fiber optic coupler that splits the light into two waves. After traversing the coupler, the two light waves propagate equally in opposite directions around the fiber optic coil. The light waves interfere upon return to coupler and project a fringe pattern onto a photodetector.
Fig2.2 Basic configuration of Fiber Optical Gyro (FOG)
In accordance with any two-wave interferometer, the intensity on the photo detector, which represents a mixture of the two light waves, varies as cosine of Sagnac phase with its maximum value at zero as shown in Figure This intensity is expressed as
Where, I0 is the mean value of the intensity.
The detected intensity is used to calculate the rotation rate. In the case of no rotation = 0, the light waves will combine in phase, which results in maximum intensity.
Figure2.3: Optical intensity versus phase difference between interfering waves
In the presence of a rotation, the light waves travel different path lengths and mix slightly out of phase. The intensity is reduced due to the degree of destructive interference. The cosine function, which is symmetrical about zero, has its minimum slope there. For small rotation rates, it is impossible to determine the direction of rotation (CW or CCW) from the symmetrical aspect of above figure, where the slope is near zero. Furthermore, the gyroscope operating in this mode has minimum sensitivity near zero. Incorporating a dithering phase modulator with drive modulation capability a symmetrically in the loop (near one end of the coil) provides a means to introduce a nonreciprocal phase shift to bias the gyroscope to its maximum sensitivity point. This corrective measure solves both the low-sensitivity problem and the issue of ambiguous direction of rotation at low rotation rates.
2.2Open-Loop Configuration Of Fiber Gyros
A major problem of the basic configurations is the output nonlinearity for small = 0, which hinders high sensitivity measurements of small rotation angles without sign ambiguity. This limitation is overcome by transforming the baseband cosine-dependence into a sinusoidal function, for example, by translating the output signal from baseband to a carrier at angular frequency w. Although different solutions have been proposed and demonstrated, the optical phase modulation technique is nowadays commonly used. A phase modulator is inserted in the fiber coil, close to a coupler output, so that a different phase delay is cumulated by the counter propagating waves.
The all-fiber version phase modulator is constructed by winding and cementing a few fiber turns on a short, hollow piezoceramic tube (PZT). By applying to the PZT a modulating voltage, a radial elastic stress and a consequent optical path length variation due to the elasto-optic effect are generated.
Fig 2.4: Open-loop FOG configuration
When a phase modulator is used, the expression for the intensity on the photo detector is,
Where, ∆Φs is the Sagnac phase shift
Where,
λ = wavelength of source
C=velocity of light
L=length of the fiber coil
D=diameter of the fiber coil
Ω= rotation rate
2.3 Closed Loop Fiber Optic Gyroscope
In the open loop configuration, the lock-in output signal V is given by V=V0sinφS and
V0 is the fringe amplitude. A first problem of this signal is the intrinsic nonlinearity and limited dynamic range of the sinusoidal function, which may represent a restriction in some applications. A second issue is related to the insufficient accuracy and stability of either the fringe amplitude and the scale factor which multiplies the rotation rate. The presence of only the analog output is also considered a third drawback of this configuration.
A closed-loop scheme has been proposed with different implementations for solving most of the above mentioned problems. The basic idea consists in using a feedback effect which cancel the Sagnac phase shift by adding a controlled phase delay, thus directly proportional to the rotation rate to be detected. This solution however was not the most appropriate in terms of maintaining reciprocity. Alternatively, the frequency variation is simulated by a phase ramp modulation, which has to be superimposed and synchronized to the biasing phase modulation.
The analog solution, based on an analog phase ramp (also indicated as serrodyne modulation) in addition to the sinusoidal biasing modulation, does not represent a very efficient solution. A great improvement is obtained with the all-digital approach based on a square wave biasing modulation and on a digital phase ramp for closed-loop processing.
Essentially, a digital feedback loop is added to the open-loop structure previously reported in Figure. The lock-in amplifier output is sampled and quantized yielding the error signal, which is maintained closed to zero by the digital feedback. The sampling frequency corresponds to the inverse of the radiation transit time τ,for the required synchronisation of the ramp and the biasing signal
Fig:2.5 closed loop fiber optic gyroscope
Starting from the error signal, the controller drives the phase modulator so that it generates phase steps of amplitude equal to the Sagnac phase shift and duration τ. The digital to analog converter automatically creates the ramp reset, by means of its overflow. The reset step corresponds to a phase variation of 2π radian, in order to get always the correct Sagnac phase shift. In this scheme the rotation rate is directly obtained, in a digital format, from the error signal. Another advantage of this configuration, with respect to the analog solution, is the phase stability during the signal recovering.
CHAPTER 3
INERTIAL SENSOR ERRORS
The performance characteristics of inertial sensors (either gyroscopes or accelerometers) are affected by a variety of errors. Most errors can be categorized into sensor bias, scale factor, axes misalignment, and noise. In the following section, these errors will be discussed briefly.
3.1 BIAS
The bias for gyro/accelerometer is the average over a specified time of accelerometer/gyro output measured at specified operating conditions that have no correlation with input acceleration or rotation. The gyro bias is typically expressed in degree per hour (°/h) or radian per second (rad/s) and the accelerometer bias is expressed in meter per Second Square [m/s2 or g]. Bias generally consists of two parts: a deterministic part called bias offset and a random part. The bias offset, which refers to the offset in the measurement provided by the inertial sensor, is deterministic in nature and can be determined by calibration. The random part is called bias drift, which refers to the rate at which the error in an inertial sensor accumulates with time. The bias drift and the sensor output uncertainty are random in nature and they should be modeled as a stochastic process. Bias errors can be reduced from the reference values, but the specific amount is range and type dependent.
In addition to the above, there are another two characteristics used to describe the sensor bias. The first is the bias asymmetry (for gyro or accelerometer), which is the difference between the bias for positive and negative inputs, typically expressed in degree per hour (°/h) or meter per second square [m/s2, g]. The second is the bias instability (for gyro or accelerometer), which is the random variation in the bias as computed over specified finite sample time and averaging time intervals. This non-stationary (evolutionary) process is characterized by a 1/f power spectral density. It is typically expressed in degree per hour (°/h) or meter per second square [m/s2, g], respectively.
3.2 SCALE FACTOR
Scale factor is the ratio of a change in the input intended to be measured. Scale factor is generally evaluated as the slope of the straight line that can be fit by the method of least squares to input-output data. The scale factor error is deterministic in nature and can be determined by calibration. Scale factor asymmetry (for gyro or accelerometer) is the difference between the scale factor measured with positive input and that measured with negative input, specified as a fraction of the scale factor measured over the input range. Scale factor asymmetry implies that the slope of the input-output function is discontinuous at zero input. It must be distinguished from other nonlinearities.
Scale factor stability, which is the capability of the inertial sensor to accurately sense angular velocity (or acceleration) at different angular rates (or at different accelerations), can also be used to describe scale factor. Scale factor stability is presumed to mean the variation of scale factor with temperature and its repeatability, which is expressed as part per million (ppm). Deviations from the theoretical scale are due to system imperfections.
3.3 MISALIGNMENT
Axes misalignment is the error resulting from the imperfection of mounting the sensors. It usually results in a non-orthogonality of the axes defining the INS body frame. As a result, each axis is affected by the measurements of the other two axes in the body frame. Axes misalignment can, in general, be compensated or modeled in the INS error equation.
3.4 SENSORS GENERAL MEASUREMENT MODEL
3.4.1 Gyros measurement model:
Gyroscope is an angular rate sensor providing either angular rate in case of rate sensing type or attitude in the case of the rate integrating type. The following model represents most of the errors contained in a single gyroscope measurement of the angular rate:
〖 I〗_ω= ω+b_ω+S_ω+e (3.1)
Where I_ωthe vector of measurements in (deg/hr) is, ω is the vector of true angular velocities (the theoretically desired measurement) (deg/hr), b_ω is the vector of gyroscope instrument bias (deg/hr), S_ω is the gyro scale factor e is the gyro sensor noise (deg/hr). These errors are, in principle, minimized by estimation techniques.
3.5 SENSOR ERRORS ELIMINATION/MINIMIZATION TECHNIQUES
In order to minimize the effect from bias and scale factor, the sensor errors elimination methods are necessary. The estimation and differencing are introduced only for the bias elimination process in accelerometer measurements; however, the principle for the gyro is the same. The calibration method is more generally used in inertial sensor bias and scale factor elimination.
3.5.1Calibration
Calibration is defined as the process of comparing instrument outputs with known reference information and determining coefficients that force the output to agree with the reference information over a range of output values. The calibration parameters to be determined can change according to the specific technology in an inertial measurement unit (IMU). The calibration usually takes place in a lab environment in which the inertial system is mounted on a level table with each sensitive axis pointing alternatively up and down (six positions).
For the gyros, placing the sensor in static mode with the axis being calibrated pointing vertically upward and using the average of 10 to15 minutes’ measurements,
I_(ω_up )= b_ω+(1+S_ω)ω_e sin⁡∅(3.2)
Whereb_ω,S_ω, ∅ and ω_e represent bias, scale factor, latitude of the gyro location and Earth’s rotation rate, respectively. Rotate the sensor 180° such that the same axis is pointing vertically downward and get the average measurement,
I_(ω_down )= b_ω-(1+S_ω)ω_e sin⁡∅ (3.3)
The bias and scale factor of the gyro under calibration can be calculated from these measurements as,
b_ω= ((I_(ω_up )+I_(ω_down ))/2)(3.4)
S_ω= ((I_(ω_up )-I_(ω_down )-〖2ω〗_e sin⁡∅)/(〖2ω〗_e sin⁡∅ ))(3.5)
As described above, the noise is ignored during the calibration process. However, the value of the bias and scale factor is still contaminated by residual noise after calibration. As a result, further techniques are needed to estimate noise in navigation algorithm.
3.6 IMPACT OF SENSOR BIAS
An uncompensated gyro bias in the X or Y gyro error will introduce an angle error proportional to time t,
δθ= ∫▒b_ω dt= b_ω t (3.6)
This small angle will cause misalignment of the INS, and therefore the acceleration vector will be projected wrong. This, in turn, will introduce acceleration in one of the horizontal channels with a value a=gsin⁡δθ≈gδθ ≈gb_ω t.
V= ∫▒〖b_ω gt〗 dt=(1/2) b_ω gt^2 (3.7)
P= ∫▒V dt=∫▒〖(1/2) b_ω gt^2 〗 dt=(1/6) b_ω gt^3 (3.8)
Therefore, a gyro bias introduces second order errors in velocity and third order errors in the position. If there is a 0.2 deg/h level axis gyro drift, the error in position due to gyro drift is 0.0016m after 10 seconds and 1600m after 1000seconds (assume g=10m/s2). That means gyro drift is a significant error source.
3.7 IMPACT OF SCALE FACTOR
According to Equation (3.1), if only the scale factor error is considered, then
I_ω=S_Tω+S_Eω(3.9)
Where ST is the true scale facto and SE is the scale factor error. Clearly, the scale factor error worked as a gyro bias, and the value is proportional to the angular rate. Based on the previous analysis, gyro drift is a significant error source and a gyro bias introduces second order errors in velocity and third order errors in the position.
Errors Characteristic Procedures to Remove/Minimize
Deterministic Random Calibration Compensation Stochastic
Modeling
Bias Offset
Bias Drift
Scale Factor
Misalignment
Noise
TABLE 2: Summary of Inertial Sensor Errors
3.8 NOISES
Noise is an additional signal resulting from the sensor itself or other electronic equipment that interferes with the output signals trying to measure. Noise is in general non-systematic and therefore cannot be removed from the data using deterministic models. It can only be modeled by stochastic process.
Noise is the unwanted component that effects the output of the system. There exist several noises that effect the output of the FOG. Those noises are:
Thermal noise
Shot noise
Quantization noise
Rate ramp noise
Rate random walk noise
Bias instability noise (1/f noise)
Angular random walk noise & Non reciprocal effect
3.8.1Thermal noise:
Thermal noise is the electronic noise generated by the thermal agitation of the charge carriers (usually the electrons) inside an electrical conductor at equilibrium, which happens regardless of any applied voltage. The generic, statistical physical derivation of this noise is called the fluctuation-dissipation theorem, where generalized impedance or generalized susceptibility is used to characterize the medium.
Thermal noise in an ideal resistor is approximately white, meaning that the power spectral density is nearly constant throughout the frequency spectrum when limited to a finite bandwidth, thermal noise has a nearly Gaussian amplitude distribution.
Thermal noise arises from the thermal fluctuations in the electron density within a conductor. In a formulation due to H. Nyquist in 1928, an idealized resistor is assumed to contain a voltage generator causing a fluctuating emf at the terminals, the mean squared value of which is
〈V^2 〉=4RkT∆f (3.10)
Where R is the resistance of the conductor, k is Boltzmann\’s constant, T is the absolute temperature, and ∆f is the bandwidth of the measuring instrument
Shot Noise:
The concept of shot noise was first introduced in 1918 by Walter Schottky who studied fluctuations of current in vacuum tubes. Shot noise in electronic circuits consists of random fluctuations of the electric current in a DC current which originate due to fact that current actually consists of a flow of discrete charges (electrons). Because the electron has such a tiny charge, however, shot noise is of relative insignificance in many (but not all) cases of electrical conduction. For instance 1 ampere of current consists of about 6.24×1018 electrons per second; even though this number will randomly vary by several billion in any given second, such a fluctuation is minuscule compared to the current itself. In addition, shot noise is often less significant as compared with two other noise sources in electronic circuits, flicker noise and Johnson–Nyquist noise. However, shot noise is temperature and frequency independent, in contrast to Johnson–Nyquist noise, which is proportional to temperature, and flicker noise, with the spectral density decreasing with the frequency. Therefore at high frequencies and low temperatures shot noise may become the dominant source of noise.
I_shot=√(2qBI_0 )(3.11)
Where B= is the measuring bandwidth
Q= is the electron charge
Intensity changes due to source variation can be referenced out, but fluctuations due to shot noise cannot be reduced, as they arise from a random process. This is because of statistical fluctuations in photon flux at detector. The uncertainty in the actual phase signal is given by
δφ=(photon shot noise)/(fringe slope)(3.12)
Plugging this into the sensitivity of a fiber gyro, the ultimate rotation uncertainty is
δΩ=λc/2πLD δφ(3.13)
Quantization Noise:
Fig 3.1: Quantization noise
Quantization, in mathematics and digital signal processing, is the process of mapping a large set of input values to a (countable) smaller set – such as rounding values to some unit of precision. A device or algorithmic function that performs quantization is called a quantizer. The round-off error introduced by quantization is referred to as quantization error.
In analog-to-digital conversion, the difference between the actual analog value and quantized digital value is called quantization error or quantization distortion. This error is either due to rounding or truncation. The error signal is sometimes modelled as an additional random signal called quantization noise because of its stochastic behaviour. Quantization is involved to some degree in nearly all digital signals processing, as the process of representing a signal in digital form ordinarily involves rounding.
3.8.4 Rate Ramp Noise:
The rate ramp noise results from the external environmental factors such as the temperature variations. This noise may be seen as a deterministic effect but generally it is considered as a stochastic process with a certain distribution.
3.8.5Rate Random Walk Noise:
This noise is usually associated with the aging effects in oscillators it can be approximately attained with the integration of white noise and it statistical characteristic obeys Brownian motion.
Bias Instability Noise (1/F Noise):
This noise denotes the instability of the FOG and is the representation of polarization wander in the FOG. The simplest model for bias instability noise is a random constant. The observed spectrum of physical 1/f noise varies as 1/f but gives in consistent power spectrum. 1/f noise has zero mean but has infinite variance, so it is difficult to handle mathematically.
Angular Random Walk Noise:
This noise is assumed to Gaussian distributed and to have white power spectral with zero mean. It arises from the photon shot noise with Poisson density function, the white Gaussian thermal noise of the amplifier and the light source’s relative intensity noise (RIN) which is nearly white and Gaussian.
Nonreciprocal Effects
It is assumed that primary waves are reciprocal if they travel the same physical path. This is not always true. Reciprocity is broken in some situations:
Shupe Effect:
If fiber properties vary with time, CW and CCW waves see fiber properties at different time.
Kerr Effect:

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