Lock-In Amplifier Technology in Laser Gyroscope North Finder of Constant Rate Biasing

Abstract

This paper presents a new type of strapdown north finder. A simplified configuration is proposed, including three parts: a pendulous force-feedback accelerometer, a ring laser gyroscope, and a single-axis rotating platform. The dynamic northfinding scheme of continuous rotation is adopted to eliminate the lock-in region of the laser gyroscope and to modulate the weak attitude signals; meanwhile the complex leveling process is avoided in this proposed configuration. To suppress the drift noises of the accelerometer and gyroscope, two digital lock-in amplifiers are used to extract the weak attitude signals. Simulation results show that the north finder can determine the heading angle in 72 s, when the rotating rate is 10^∘/s with the maximum heading error less than 1.0′; meanwhile the horizontal attitude angles as well as the constant drift noises of the inertial components could also be obtained.

1. Introduction

The equipment mounted in the body of the land-based vehicle would allow the directional reference with respect to geographical true north. This equipment is often referred to as north finder, which plays an important role in both civilian and military fields [1–6]. According to the principles of north finding, there are two kinds of north finders: non-gyro-north finder and gyroscope north finder.

In the non-gyro-north finder, magnetic compass and astronomical navigator are the two major categories. Magnetic compass enables the vehicle’s attitude to be determined with respect to the Earth’s magnetic field; the astronomical navigator fixed in the body of a vehicle could provide measurements of the remote star’s bearing and elevation with respect to the body frame relying on the optical sensors. But these two instruments are highly sensitive to the external environment, such as magnetic field, weather, and sometimes cannot be used any more.

In view of the limitations of both the aforementioned techniques, the gyroscope north finder has the characteristic of autonomy, which is not impacted by external environment; therefore more attentions have been paid to these greatest inventions. With the development of the inertial components, the gyroscope north finder is mainly divided into two categories: one is pendulous north finder as the representative of mechanical gyroscope, which is based on the principle of the gyrocompass with high accuracy but long alignment time. The basic principle of this equipment is the indication of true north by establishing the equilibrium between the effect of its pendulosity and the angular momentum of the rotation base carrying the mechanical gyroscope; the other is strapdown north finder using the optical gyroscope as a sensor [1, 2]. Usually, the strapdown north finder keeps the sensitive axis of the optical gyroscope in the horizontal plane accomplished by using the outputs of two orthogonal accelerations to drive the motor, respectively. Two measurements of the horizontal components of the Earth’s rate are sensed by optical gyroscope in two separations 90° apart. So the heading angle can be calculated from the ratio of these two measurements. The strapdown north finder locates the orientation with a little lower accuracy to the pendulous north finder but greatly decreases the time.

Through analyzing the dynamic output characteristics of inertial measurement unit in the presence of continuous rotation at a constant rate, this paper designs a new type of strapdown north finder with a simplified configuration and also can avoid the complex leveling process of rotating platform. Digital lock-in amplifier technology is adopted to extract the weak attitude signals submerged in the drift noise of inertial components effectively [7–10]. Three attitude angles are obtained in a short time with certain accuracy, as well as the estimations of the constant drifts of inertial components.

2. The Dynamic Output Characteristics of the Inertial Components at Constant Rate Rotation

2.1. The Measurement Principle of Laser Gyroscope of Constant Rate Biasing

The ring laser gyroscope (RLG) based on the Sagnac effect is an angular rate sensor, being an important inertial component currently. The RLG relies upon the detection of the frequency difference between the two counter-propagating beams of light in the optical resonant cavity, which is also proportional to the rotational motion about an axis perpendicular to the plane containing the light path in inertial space. It appears that the application of RLG to the sensing of angular rate can offer lots of advantages over the mechanical gyroscope, such as rapid response, wide dynamic detecting range, good linearity, little dynamic error, high precision, and high reliability and non-𝑔-dependent bias term. Therefore, RLG gets more and more attentions and now is considered as the ideal inertial component to replace the well-established mechanical gyroscope.

At very low angular rate, due to a variety of nonuniformity factors in the optical resonant cavity of RLG, the two counter propagating beams almost have the same frequency or “lock together,” so there is no output signal, which is usually called the lock-in phenomenon [11, 12], as shown in Figure 1. This phenomenon of frequency synchronization introduces a dead zone of the RLG measurement signal, so how to overcome the lock-in region becomes one of the key questions gradually. Although mechanical dither technology is widely used in RLG, it cannot eliminate the lock-in region completely, the components of which still cause measurement error in the mechanical dither.

Figure 1 

The RLG input/output characteristic.

Fortunately, the lock-in region could be eliminated completely by employing a so-called constant rate biasing technology which is superior to mechanical dither in principle [13]. RLG is installed on a platform with an appropriate rotating rate, which can make a projection rate on the sensitive axis of RLG, so the output signal can keep away from measurement error induced by the lock-in region. Theoretically, the faster the platform rotates, the smaller the laser measurement error is, but a faster rate will reduce the stability of the rotating platform and gyroscope sampling points in a whole rotation cycle, which will affect the calculation accuracy of the attitude solution. Usually, to keep the measurement of RLG mounted on the rotating platform away from the lock-in region, the biasing rate (𝜔_𝐵) provided by the platform should be ten to twenty times that of the lock-in rate (𝜔_𝐿).

2.2. The Dynamic Output Characteristics of Inertial Measurement Unit at Constant Rate Rotation

The Cartesian coordinate frames used in this paper are defined as follows [14, 15].

The inertial frame (𝑖-frame) sets its origin (𝑜) at the center of the Earth and axes (𝑜𝑥_𝑖,  𝑜𝑦_𝑖,  and  𝑜𝑧_𝑖) which are invariant in direction with respect to the fixed planets in space. In 𝑖-frame, axes  𝑜𝑥_𝑖  and  𝑜𝑦_𝑖  are parallel to the equator plane, with axis  𝑜𝑥_𝑖  coinciding with the spring equinox and axis  𝑜𝑧_𝑖  coinciding with the Earth’s polar axis, respectively.

The Earth frame (𝑒-frame) sets origin (𝑜) at the center of the Earth and axes (𝑜𝑥_𝑒,  𝑜𝑦_𝑒,  and  𝑜𝑧_𝑒)   which are in respect to the Earth. In 𝑒-frame, axes  𝑜𝑥_𝑒  and  𝑜𝑦_𝑒  are parallel to the equator plane, with axis  𝑜𝑥_𝑒 along the 0° longitude, axis  𝑜𝑦_𝑒  along the 90° longitude, and axis 𝑜𝑧_𝑒coincident with the Earth’s polar axis, respectively. The  𝑒-frame rotates relative to the 𝑖-frame at a rate  𝜔_𝑖𝑒  about the axis  𝑜𝑧_𝑒(𝑜𝑧_𝑖).

The navigation frame (𝑛-frame), the reference frame for the attitude solution, is a local geographic one. Its origin (𝑜) is set at the location of the navigation system. And axes   (𝑜𝑥_𝑛,  𝑜𝑦_𝑛,  and  𝑜𝑧_𝑛) lie along the directions of the local east, the local north, and the vertical upturn, respectively.

The body frame (𝑏-frame) is attached to the base of the vehicle with its origin (𝑜) located at the center of vehicle. Axes (𝑜𝑥_𝑏,  𝑜𝑦_𝑏,  and  𝑜𝑧_𝑏) lie along the directions of the pitch (right), roll (front), and yaw (upturn) of the vehicle, respectively.

The gyroscopes frame (𝑔-frame) is defined by the axes (𝑜𝑥_𝑔, 𝑜𝑦_𝑔,  and  𝑜𝑧_𝑔)   which are parallel to the sensitive axes of the three gyroscopes, respectively.

The accelerometers frame (𝑎-frame) is defined by the axes (𝑜𝑥_𝑎,  𝑜𝑦_𝑎,  and  𝑜𝑧_𝑎) which are parallel to the sensitive axes of the three accelerometers, respectively.

Moreover, the coordinate transformation is defined as 𝑉²=𝐶²₁𝑉¹. 𝑉¹,  𝑉² represent vectors expressed in the 1-frame, 2-frame, respectively, and 𝐶²₁ is the direction cosine matrix transferring 𝑉¹ into 𝑉². The 𝜔³₁₂ represents the angular rate of the 2-frame with respect to the 1-frame expressed in the 3-frame. The positive rotations about each axis are taken to be in a clockwise direction looking along the axis from the origin.

The vehicle’s attitude matrix 𝐶^𝑏_𝑛 is the transform matrix from 𝑛-frame to 𝑏-frame by three successive rotations about heading axis, pitch axis, and roll axis in turn, as shown in Figure 2: 𝑜-𝑥_𝑛𝑦_𝑛𝑧−−−−−−−−−→_𝑛^{RotΨ,−𝑧𝑛}𝑜-𝑥₁𝑦₁𝑧−−−−−−−→₁^{Rot𝜃,𝑥1}𝑜-𝑥₂𝑦₂𝑧_{2Rot𝛾,𝑦2}−−−−−−−→𝑜-𝑥_𝑏𝑦_𝑏𝑧_𝑏. (1) Rotate through heading angle 𝜓 about reference −𝑧_𝑛-axis, 𝜓∈[0^∘,360^∘). Rotate through pitch angle 𝜃 about new 𝑥₁-axis, 𝜃∈[−90^∘,90^∘]. Rotate through roll angle 𝛾 about new 𝑦₂-axis, 𝛾∈[−90^∘,90^∘].

Figure 2 

The transformation from 𝑛-frame to 𝑏-frame.

where  𝜓,  𝜃,  and  𝛾 are referred to as the attitude angles of the vehicle. Usually, 𝜃 and 𝛾 represent the horizontal attitude angles.

Thus the direction cosine matrix 𝐶^𝑏_𝑛 could be expressed as follows: 𝐶^𝑏_𝑛=𝐶^𝑏₂𝐶²₁𝐶¹_𝑛, (2) where 𝐶^𝑏₂=cos𝛾0−sin𝛾010sin𝛾0cos𝛾,𝐶²₁=,𝐶1000cos𝜃sin𝜃0−sin𝜃cos𝜃_1𝑛=.cos𝜓−sin𝜓0sin𝜓cos𝜓0001 (3)

Substituting and rearranging the above equation yield 𝐶^𝑏_𝑛=𝐶₁₁𝐶₁₂𝐶₁₃𝐶₂₁𝐶₂₂𝐶₂₃𝐶₃₁𝐶₃₂𝐶₃₃=.cos𝛾cos𝜓+sin𝛾sin𝜃sin𝜓−cos𝛾sin𝜓+sin𝛾sin𝜃cos𝜓−sin𝛾cos𝜃cos𝜃sin𝜓cos𝜃cos𝜓sin𝜃sin𝛾cos𝜓−cos𝛾sin𝜃sin𝜓−sin𝛾sin𝜓−cos𝛾sin𝜃cos𝜓cos𝛾cos𝜃 (4)

The purpose of attitude solution is to figure out these three attitude angles [14].

In 𝑏-frame, axes  𝑜𝑥_𝑏,  𝑜𝑦_𝑏  are parallel to the plane of the rotating platform with axis 𝑜𝑧_𝑏 along the rotation axis of the platform. An inertial measurement unit (IMU) generally consists of three mutually perpendicular gyroscopes and accelerometers to measure the angular rates and linear accelerations, respectively. The IMU is mounted on the rotating platform, and the axes  𝑜𝑥_𝑎,  𝑜𝑦_𝑎  in 𝑎-frame are parallel to the plane of the rotating platform. The axis  𝑜𝑧_𝑎  lies along the direction of the rotation axis  𝑜𝑧_𝑏,   with axis  𝑜𝑥_𝑎  coinciding with axis 𝑜𝑥_𝑏 before rotating. In order to provide the three orthogonal gyroscopes with the same biasing rate, the angular rate of the rotating platform should satisfy the following: 𝜔^𝑔_𝑥²+𝜔^𝑔_𝑦²+𝜔^𝑔_𝑧²=𝜔^𝑟_𝑧²,𝜔^𝑔_𝑥=𝜔^𝑔_𝑦=𝜔^𝑔_𝑧. (5)

Then the following is obtained: 𝜔^𝑔_𝑥=𝜔^𝑔_𝑦=𝜔^𝑔_𝑧=𝜔_bias=1√3𝜔^r_𝑧,𝜒₁=𝜒₂=𝜒₃1=arctan√3=54.73^∘. (6)

So this configuration has each of the three sensitive axes of mutually perpendicular gyroscopes at 54.73° to the rotation axis 𝑜𝑧_𝑏(𝑜𝑧_𝑎) and is equally spaced at 120° to each other in the plane of the rotating platform [16]. This system is rotated away from axis 𝑜𝑧_𝑏(𝑜𝑧_𝑎), so the three gyroscopes are provided 1/√3 of the rotating rate  𝜔₀. Meanwhile ensuring that axes  𝑜𝑥_𝑎,  𝑜𝑥_𝑔, and 𝑜𝑧_𝑏(𝑜𝑧_𝑎) are in the same plane, the orthogonal instrument cluster arrangement structure relationship between 𝑏-frame coordinate, 𝑔-frame coordinate, and 𝑎-frame coordinate is as shown in Figure 3.

Figure 3 

The schematic diagram of inertial measurement unit.

Fixed to the rotating platform and the zero-angle position of the rotating platform lying along the vehicle longitudinal axis, the IMU provides measurement of angular rates and linear accelerations about three orthogonal axes. In the presence of continuous rotation at constant rate, IMU dynamic output can be represented by [14, 15] 𝑓^𝑎=𝐶^𝑎_𝑏𝐶^𝑏_𝑛𝑓^𝑛+𝑛^𝑎(𝑡)+𝑏^𝑎𝜔^𝑔_𝑖𝑔=𝐶^𝑔_𝑎𝐶^𝑎_𝑏𝐶^𝑏_𝑛𝜔^𝑛_𝑖𝑒+𝜔^𝑛_𝑒𝑛+𝐶^𝑎_𝑏𝜔^𝑏_𝑛𝑏+𝜔^𝑎_𝑏𝑎+𝜔^𝑔_𝑎𝑔+𝑛^𝑔(𝑡)+𝑏^𝑔, (7) where  𝑓^𝑎 is the measured linear acceleration in 𝑎-frame; 𝜔^𝑔_𝑖𝑔 is the measured angular rate in 𝑔-frame.  𝑛^𝑎(𝑡),  𝑏^𝑎  are the random drift noise and constant drift noise of the accelerator, respectively. 𝑛^𝑔(𝑡), 𝑏^𝑔 are the random drift noise and constant drift noise of the gyroscope respectively. It is assumed that the vehicle is stationary relative to the Earth, so 𝜔^𝑛_𝑒𝑛 and 𝜔^𝑛_𝑛𝑏 are zero. Also there is no relative movement between 𝑎-frame and 𝑔-frame, which means  𝜔^𝑔_𝑎𝑔=0. Equation (7) could be simplified as 𝑓^𝑎=𝐶^𝑠_𝑏𝐶^𝑏_𝑛𝑓^𝑛+𝑛^𝑎(𝑡)+𝑏^𝑎,𝜔^𝑔_𝑖𝑔=𝐶^𝑔_𝑎𝐶^𝑎_𝑏𝐶^𝑏_𝑛𝜔^𝑛_𝑖𝑒+𝜔^𝑎_𝑏𝑎+𝑛^𝑔(𝑡)+𝑏^𝑔. (8)

Ignoring scale-factor error and installation error of the inertial component and substituting the above coefficients, (8) can be further written as (9) and (11).

As discussed above, in the presence of continuous and unidirectional rotation with constant rate, the specific forces acting on the vehicle, measured by three mutually perpendicular accelerometers, are given by 𝐹^𝑎_𝑥𝐹^𝑎_𝑦𝐹^𝑎_𝑧=cos𝜔₀𝑡sin𝜔₀𝑡0−sin𝜔₀𝑡cos𝜔₀𝑡0𝐶001¹¹𝐶₁₂𝐶₁₃𝐶₂₁𝐶₂₂𝐶₂₃𝐶₃₁𝐶₃₂𝐶₃₃00𝑔+𝑛^𝑎_𝑥(𝑡)𝑛^𝑎_𝑦(𝑡)𝑛^𝑎_𝑧(𝑡)+𝑏^𝑎_𝑥𝑏^𝑎_𝑦𝑏^𝑎_𝑧. (9)

This equation can be rearranged and expressed as follows: 𝐹^𝑎_𝑥𝐹^𝑎_𝑦𝐹^𝑎_𝑧=𝐶₁₃𝑔cos𝜔₀𝑡−𝐶₂₃𝑔sin𝜔₀𝑡+𝑛^𝑎_𝑥(𝑡)+𝑏^𝑎_𝑥𝐶₁₃𝑔sin𝜔₀𝑡+𝐶₂₃𝑔cos𝜔₀𝑡+𝑛^𝑎_𝑦(𝑡)+𝑏^𝑎_𝑦𝐶₃₃𝑔+𝑛^𝑎_𝑧(𝑡)+𝑏^𝑎_𝑧. (10)

As could be seen from (10), the outputs of the three accelerometers involve 𝐶₁₃, 𝐶₂₃, and 𝐶₃₃ of the attitude matrix 𝐶^𝑏_𝑛 in (4), which are only associated with the horizontal attitude angles 𝛾,  𝜃. It will be seen that the specific forces  𝐹^𝑎_𝑥  and  𝐹^𝑎_𝑦  are expressed in the form of sine and cosine waveforms with the same amplitudes but different phases. Similarly, ignoring the random drift noise, the output of the accelerometer  𝐹^𝑎_𝑧  is approximated as a constant value.

Meanwhile, the angular rates sensed by three orthogonal gyroscopes sensitive axes are given by 𝐺^𝑔_𝑥𝐺^𝑔_𝑦𝐺^𝑔_𝑧=2√601√3−1√61√21√3−1√6−1√21√3×cos𝜔₀𝑡sin𝜔₀𝑡0−sin𝜔₀𝑡cos𝜔₀𝑡0𝐶001¹¹𝐶₁₂𝐶₁₃𝐶₂₁𝐶₂₂𝐶₂₃𝐶₃₁𝐶₃₂𝐶₃₃×0𝜔^𝑛_𝑖𝑒𝑁𝜔^𝑛_𝑖𝑒𝑈−00𝜔₀+𝑛^𝑔_𝑥(𝑡)𝑛^𝑔_𝑦(𝑡)𝑛^𝑔_𝑧(𝑡)+𝑏^𝑔_𝑥𝑏^𝑔_𝑦𝑏^𝑔_𝑧. (11)

The above equation can be rearranged and expressed as follows: 𝐺^𝑔_𝑥𝐺^𝑔_𝑦𝐺^𝑔_𝑧=2𝑅√6cos𝜔₀𝑡+2𝑀√6sin𝜔₀𝑡+𝑁−𝜔₀√3+𝑛^𝑔_𝑥(𝑡)+𝑏^𝑔_𝑥√3𝑀−𝑅√6cos𝜔₀𝑡−𝑀+√3𝑅√6sin𝜔₀𝑡+𝑁−𝜔₀√3+𝑛^𝑔_𝑦(𝑡)+𝑏^𝑔_𝑦−√3𝑀+𝑅√6cos𝜔₀𝑡−𝑀−√3𝑅√6sin𝜔₀𝑡+𝑁−𝜔₀√3+𝑛^𝑔_𝑧(𝑡)+𝑏^𝑔_𝑧, (12)where  𝑅=𝐶₁₂𝜔^𝑛_𝑖𝑒𝑁+𝐶₁₃𝜔^𝑛_𝑖𝑒𝑈,  𝑀=𝐶₂₂𝜔^𝑛_𝑖𝑒𝑁+𝐶₂₃𝜔^𝑛_𝑖𝑒𝑈,  and  𝑁=𝐶₃₂𝜔^𝑛_𝑖𝑒𝑁+𝐶₃₃𝜔^𝑛_𝑖𝑒𝑈.

According to (12), the output of a triad of gyroscopes is related to both the heading angle  𝜓  and horizontal attitude angles  𝛾,  𝜃,  which comprise sinusoidal oscillation superimposed on the constant parameter  (𝑁−𝜔₀)/√3. Clearly, it biases the measurement signals of the RLG away from the lock-in region, reducing the measurement errors.

As mentioned above, the rotating rate modulates the stationary attitude signals as sinusoidal signals, which facilitates the subsequent process of attitude solution. According to the characteristic of the output signals, the acceleration outputs  𝐹^𝑎_𝑥,  𝐹^𝑎_𝑦  only contain the horizontal attitude signals, and the gyroscope outputs  𝐺^𝑔_𝑥,  𝐺^𝑔_𝑦,  and  𝐺^𝑔_𝑧 contain all the attitude signals. Since the outputs of the inertial components have similar propagation rule, the attitude angles could be determined from each combination outputs of 𝐹^𝑎_𝑥- 𝐺^𝑔_𝑥,  𝐹^𝑎_𝑥- 𝐺^𝑔_𝑦,   𝐹^𝑎_𝑥- 𝐺^𝑔_𝑧,   𝐹^𝑎_𝑦- 𝐺^𝑔_𝑥,   𝐹^𝑎_𝑦- 𝐺^𝑔_𝑦, and 𝐹^𝑎_𝑦- 𝐺^𝑔_𝑧, which are six simplified configurations in total. The solution of horizontal attitude angles is the premise to calculate the heading angle.

3. The Proposed Algorithm for North Finding

3.1. The Principle of the Lock-In Amplifier

Lock-in amplifier (LIA) was firstly used to suppress 1/f noise and constant drift induced by the thermal effects of the circuits and to extract weak signal from those with strong noises [7–10]. It is also widely employed in resource exploration and precision measurement field, and better results are obtained especially for the sine waves and square waves. According to the characteristic of the measurement signal  𝑥(𝑡), the lock-in amplifier generates two orthogonal reference signals  𝑟_𝐼(𝑡) and 𝑟_𝑄(𝑡), the frequencies of which are consistent with 𝑥(𝑡). These two reference signals have the same amplitude but 90° phase difference. The spectrum of the measurement signal 𝑥(𝑡) could be changed by multiplication and accumulation operations of two reference signals 𝑟_𝐼(𝑡) and 𝑟_𝑄(𝑡) in the phase sensitive devise, and then the drift noise in the signal is removed through a low-pass filter consequently. Since the output signal of LIA is sensitive to the amplitude and phase of the measurement signal  𝑥(𝑡), both amplitude discriminating and phase discriminating are accomplished to extract the feature parameters (amplitude and phase) of the measurement signal, as shown in Figure 4.

Figure 4 

Lock-in amplifier structure figure.

In Figure 4, the single frequency measurement signal  𝑥(𝑡) without drift noise can be represented by 𝑉_𝑠sin(𝜔₀𝑡+𝜂). The two square waves reference signals are  𝑟_𝐼(𝑡),  𝑟_𝑄(𝑡)   with the same amplitude 𝑉_𝑟  and 90° phase-difference. After the signal sampling with integral multiple of rotation periods, In-phase (𝐼) and Quadrature (𝑄) components are calculated: 1𝐼=𝑡^𝑡₀𝑥(𝑡)⋅𝑟_𝐼(𝑡)=2𝜋𝑉_𝑠𝑉_𝑟1cos𝜂,𝑄=𝑡^𝑡₀𝑥(𝑡)⋅𝑟_𝑄(𝑡)=2𝜋𝑉_𝑠𝑉_𝑟sin𝜂. (13)

The following are obtained: 𝑉_𝑠=𝜋2𝑉_𝑟𝐼²+𝑄²,𝑄𝜂=arctan𝐼. (14)

As mentioned above, the feature parameters (amplitude  𝑉_𝑠 and phase  𝜂) of the sinusoidal measurement signal could be identified by the lock-in amplifier [10].

3.2. The Attitude Solution Algorithm Proposed for the North Finder

As a result of the preponderance of trigonometric terms in (10) and (12), the accelerometer signals only contain the horizontal attitude signals. The amplitude and phase of the measurement signal could be extracted by the lock-in amplifier, allowing two horizontal attitude angles to be obtained by resolving the accelerometer measurements. Similarly, the heading angle could be obtained by resolving gyroscope measurement signal through another lock-in amplifier consequently according to (13) and (14).

In practice, the rotation axis should be kept perpendicular to the vehicle base so that the rolling back and forth motion of accelerometer in the gravitational field could be avoided effectively, and the accelerometer works in linear measurement region.

For the sake of easy installation, a simplified configuration of north finder system is proposed. The system mainly consists of 𝑜𝑥_𝑎-axis accelerometer, 𝑜𝑥_𝑔-axis laser gyroscope, and a single-axis rotating platform. 𝐹^𝑎_𝑥  and  𝐺^𝑔_𝑥  are the 𝑜𝑥_𝑎-axis accelerometer output signal and 𝑜𝑥_𝑔-axis gyroscope output signal, respectively, which are also used as the measurement signals of the lock-in amplifiers.

In terms of the components 𝐶₁₃ and 𝐶₂₃ of the unknown matrix  𝐶^𝑏_𝑛, the amplitude  𝑉_𝑠  and the phase  𝜂 of the 𝑜𝑥_𝑎-axis accelerometer signal 𝐹^𝑎_𝑥 could be written as 𝐹^𝑎_𝑥=𝐶₁₃𝑔cos𝜔₀𝑡−𝐶₂₃𝑔sin𝜔₀𝑡+𝑛^𝑎_𝑥(𝑡)+𝑏^𝑎_𝑥=𝐶²₁₃+𝐶²₂₃𝑔sin𝜔₀𝐶𝑡−arctan¹³𝐶₂₃+𝑛^𝑎_𝑥(𝑡)+𝑏^𝑎_𝑥. (15)

Due to the presence of the random drift noise, the amplitude and phase calculated by the lock-in amplifier have some deviations from the true values, shown as 𝑉_𝑆=𝐶²₁₃+𝐶²₂₃𝐶𝑔,𝜂=−arctan¹³𝐶₂₃. (16)

Combining (4), (14), and (16), the amplitude and phase of the signal  𝐹^𝑎_𝑥  from the accelerometer could be calculated, and then 𝐶₁₃  and  𝐶₂₃ are determined. Since the inverse-sine function is monotonic in the main interval [−90^∘,90^∘], the horizontal attitude angles 𝜃 and 𝛾 could be determined solely from 𝐶₁₃  and  𝐶₂₃, as shown in 𝜃=𝜃_main=sin⁻¹𝐶₂₃,𝛾=𝛾_main=sin⁻¹−𝐶₁₃,cos𝜃 (17) where 𝜃_main and 𝛾_main are the main values of the 𝜃 and 𝛾 in the inverse-sine function.

Similarly, The output of the 𝑜𝑥_𝑔-axis laser gyroscope signal  𝐺^𝑔_𝑥  is expressed as 𝐺^𝑔_𝑥=2𝑅√6cos𝜔₀𝑡+2𝑀√6sin𝜔₀𝑡+𝑁−𝜔₀√3+𝑏^𝑔_𝑥+𝑛^𝑔_𝑥(𝑡)=2𝑀²+𝑅²√6sin𝜔₀𝑅𝑡+arctan𝑀+𝑁−𝜔₀√3+𝑏^𝑔_𝑥+𝑛^𝑔_𝑥(𝑡), (18) where 𝑅=𝐶₁₂𝜔^𝑛_𝑖𝑒𝑁+𝐶₁₃𝜔^𝑛_𝑖𝑒𝑈,  𝑀=𝐶₂₂𝜔^𝑛_𝑖𝑒𝑁+𝐶₂₃𝜔^𝑛_𝑖𝑒𝑈,  and 𝑁=𝐶₃₂𝜔^𝑛_𝑖𝑒𝑁+𝐶₃₃𝜔^𝑛_𝑖𝑒𝑈.

And  𝜔^𝑛_𝑖𝑒𝐸=0,  𝜔^𝑛_𝑖𝑒𝑁=𝜔^𝑛_𝑖𝑒⋅cos𝐿,  and 𝜔^𝑛_𝑖𝑒𝑈=𝜔^𝑛_𝑖𝑒⋅sin𝐿  are the Earth’s rate components which could be calculated from the known local latitude  𝐿. Thus, parameters  𝑀,  𝑅  could be expressed as 𝑀=𝐶₂₂⋅𝜔^𝑛_𝑖𝑒𝑁+𝐶₂₃⋅𝜔^𝑛_𝑖𝑒𝑈=cos𝜃cos𝜓⋅𝜔^𝑛_𝑖𝑒𝑁+sin𝜃⋅𝜔^𝑛_𝑖𝑒𝑈,𝑅=𝐶₁₂⋅𝜔^𝑛_𝑖𝑒𝑁+𝐶₁₃⋅𝜔^𝑛_𝑖𝑒𝑈=−cos𝛾sin𝜓+sin𝛾sin𝜃cos𝜓⋅𝜔^𝑛_𝑖𝑒𝑁−sin𝛾cos𝜃⋅𝜔^𝑛_𝑖𝑒𝑈. (19)

The main value of heading angle 𝜓_main could be computed as follows: cos𝜓=𝑀−sin𝜃⋅𝜔^𝑛_𝑖𝑒𝑈cos𝜃⋅𝜔^𝑛_𝑖𝑒𝑁,sin𝜓=sin𝛾tan𝜃⋅𝑀−sin𝜃⋅𝜔^𝑛_𝑖𝑒𝑈−sin𝛾cos𝜃⋅𝜔^𝑛_𝑖𝑒𝑈−𝑅cos𝛾⋅𝜔^𝑛_𝑖𝑒𝑁,𝜓_main=arctansin𝜓cos𝜓,where𝜓_main∈−90^∘,90^∘. (20)

As the heading angle  𝜓  lies in the interval [0^∘,360^∘), it could be calculated through 𝜓_main in (20), finishing the whole process of north finding. The truth table of the heading angle 𝜓 is shown in Table 1.

Table 1 

The truth value table of the heading angle 𝜓.

It is shown that the range of heading angle 𝜓 is 𝜓∈[0^∘,360^∘), and the range of main value 𝜓_main is 𝜓_main∈(−90^∘,90^∘). There are totally eight cases for the heading angle  𝜓, according to the signs of  cos𝜓,  sin𝜓.

3.3. Determination of the Inertial Components Constant Drift Noise

As shown in Figure 5, the outputs of the accelerometer 𝐹^𝑎_𝑥 and gyroscope 𝐺^𝑎_𝑥 are sinusoidal oscillatory motions with equal frequency summed by a constant parameter. The constant drift noises of the inertial components are estimated by averaging the data in an entire rotation period, which can eliminate the influence of sinusoidal oscillation. The estimations of the constant drift noises of the inertial components are mainly affected by the intensity of random noise and the stability of the rotating rate. Combining (10), (12) and ignoring the random drift noise, the expressions for constant drift noises of accelerometer and gyroscope are given as 𝑏^𝑎_𝑥≈1𝑡^𝑡₀𝐶₁₃𝑔cos𝜔₀𝑡−𝐶₂₃𝑔sin𝜔₀𝑡+𝑛^𝑎_𝑥(𝑡=1)𝑡^𝑡₀𝐹^𝑎_𝑥,𝑏^𝑔_𝑥≈1𝑡^𝑡₀2𝑀√6sin𝜔₀𝑡+2𝑅√6cos𝜔₀𝑡+𝑛^𝑔_𝑥(𝑡=1)𝑡^𝑡₀𝐺^𝑔_𝑥−𝑁−𝜔₀√3,𝑁=𝐶₃₂𝜔^𝑛_𝑖𝑒𝑁+𝐶₃₃𝜔^𝑛_𝑖𝑒𝑈=−sin𝛾sin𝜓⋅𝜔^𝑛_𝑖𝑒𝑁−cos𝛾sin𝜃cos𝜓⋅𝜔^𝑛_𝑖𝑒𝑁+cos𝛾sin𝜃⋅𝜔^𝑛_𝑖𝑒𝑈, (21) where  𝑡=𝑛𝑇₀,  𝑛∈𝑁⁺. 𝑇₀ is the time period for rotating one circle (360°), and 𝑡 is the time for 𝑛 circles of the platform rotation.

Figure 5 

The structure diagram of the north finder.

4. Configuration and Experiment of the North Finder

4.1. Configuration of the North Finder System

All of the components of the north finder system are fixed on the base of the vehicle, and the plane of rotating platform is parallel to the plane of the vehicle base. Make sure that the sensitive axis of the accelerometer, the sensitive axis of the laser gyroscope, and the rotation axis lie in the same plane, and the sensitive axis of the gyroscope forms an angle  𝜒₁ with the rotation-axis for the equipment configuration considered above. The skewed sensor configuration ensures that the rotating platform provides the gyroscope with a biasing frequency to overcome the lock-in phenomenon. The zero-angle position of the rotating platform is usually arranged to be aligned with the longitudinal axis of the vehicle; then the heading angle could be determined by combining the outputs from the inertial components and angle encoders [14]. The structure diagram of the north finder configuration system is shown in Figure 5.

Since the rotating platform continuously rotates at a constant rate, a conductive ring is required to connect the signal line and power line. An accurate controller of rotating rate with the compensation for friction torque is critical to the accurate estimation of heading angle [17]. To guarantee stability of rotating rate, a bipolar PWM control mode is employed in the servo system, and Phase Locked Loop (PLL) technology is also used, which comprises four parts: the control algorithm, a frequency discriminator, a low-pass filter, and power driver [18–20]. The angle between rotating platform and the base is measured by an angle encoder with arc-second accuracy, the output of which would be feed back via torque motor to rotate the platform at the reference rate provided. The rotating platform can be also oriented at the zero-angle position at initial time automatically.

Two orthogonal square waves with 90° out of phase are adopted as the reference signals. When the platform rotates at the instant that the output of angle encoder is an integer multiple of 90°, interrupts are triggered to switch to both polarities of the reference signals, respectively. The data from inertial components are not collected until a steady motor rate is reached. After data collection, the two horizontal attitude angles could be calculated from the accelerometer output  𝐹^𝑎_𝑥 by the LIA1; then the heading angle is determined from the gyroscope output 𝐺^𝑔_𝑥 by the LIA2, the principle diagram of which is shown in Figure 6.

Figure 6 

The principle diagram of the north finder.

4.2. Simulation Experiment

Under static condition, the vehicle’s attitudes [𝛾𝜃𝜓] are [12^∘−15^∘276^∘], and the skewed angle  𝜒₁ is 54.73°. Each sensor output is sampled at a rate of 100 Hz, with the rotating rate 𝜔₀=10°/s and the operation time 𝑡=72 s. According to the proposed principle, a simulation experiment is designed with the typical performance parameters for the inertial components shown in Table 2.

Table 2 

Typical performance parameters for the inertial components.

Ten groups of north finding experiments are executed, and data processing results are shown in Table 3.

Table 3 

Experimental data processing results of ten groups repeated north finding.

It can be seen that the accuracy of horizontal attitude angles  𝛾,  𝜃 is higher than that of heading angle 𝜓 from the above ten groups’ results. Simulation data results show that the maximum heading error is less than 1.0′ in 72 s, when the rotating rate is 𝜔₀=10^∘/s.

4.3. Error Analysis

4.3.1. Effect of Drift Noise on the Accuracy of the North Finding

Uniformity effects on drift noise can be very significant either optically or electronically, which is often modeled as additive Gaussian white noise and constant noise [11, 21]. The output of the inertial component is given by 𝑥(𝑡)=𝑉_𝑠sin𝜔₀𝑡+𝜂+𝑛(𝑡)+𝑏, (22) where 𝑛(𝑡) and 𝑏  represent the random drift noise and constant drift noise, respectively. Taking the wideband drift noise 𝑛(𝑡) into account, (22) can be rewritten as 𝑥(𝑡)=𝑉_𝑠sin𝜔₀𝑡+𝜂+^∞_𝑛=1𝑉_𝑛sin𝜔_𝑛𝑡+𝜂_𝑛+𝑏, (23) where  𝑉_𝑠 is the amplitude of the measurement signal, 𝑉_𝑛 and 𝜂_𝑛 are the amplitude and phase of the random drift noise  𝑛(𝑡) in each frequency, respectively.

One of the reference signals can be expanded into the Fourier series [22]: 𝑟(𝑡)=4𝑉_𝑟𝜋^∞_𝑛=1(−1)^𝑛+1)𝜔2𝑛−1cos(2𝑛−1₀𝑡. (24)

The product of 𝑥(𝑡) by 𝑟(𝑡) yields 𝑢_out(𝑡)(𝑡)==𝑥×𝑟2𝑉_𝑠𝑉_𝑟𝜋^∞_𝑛=1(−1)^𝑛+1)𝜔2𝑛−1sin(2𝑛−2₀+𝑡−𝜂2𝑉_𝑠𝑉_𝑟𝜋^∞_𝑛=1(−1)^𝑛+12𝑛−1sin2𝑛𝜔₀+𝑡+𝜂2𝑉_𝑠𝑉_𝑟𝜋^∞_𝑛=1(−1)^𝑛+1𝜔2𝑛−1sin_𝑛+(2𝑛−1)𝜔₀𝑡+𝜂_𝑛+2𝑉_𝑠𝑉_𝑟𝜋^∞_𝑛=1(−1)^𝑛+1𝜔2𝑛−1sin_𝑛−(2𝑛−1)𝜔₀𝑡+𝜂_𝑛+4𝑏𝑉_𝑟𝜋^∞_𝑛=1(−1)^𝑛+1)𝜔2𝑛−1cos(2𝑛−1₀𝑡. (25)

As shown in (25), the product  𝑢_out  consists of five components: only 𝑛=1 frequency difference has an output after the low-pass filter in the first term, which is a useful information to obtain the amplitude  𝑉_𝑠 and phase  𝜂 of the measurement signal  𝑥(𝑡). Other frequency components in the first term, the second term, and the third term can be filtered out by the low-pass filter. There is no frequency difference in the fifth term, allowing no output through the low-pass filter, so the output is not impacted by the constant drift noise  𝑏; most of the frequency components of wideband random drift noise 𝑛(𝑡) can be filtered out in the fourth term, but when the frequency of the drift noise and the corresponding harmonic frequency of the reference square wave have the relationship 𝜔_𝑛=(2𝑛−1)𝜔₀, a direct current output of a phase sensitive is produced, which influences the result to some extent. Therefore, in order to eliminate the effect of the harmonics, a band-pass filter with the center frequency 𝜔₀ is added to filter out the higher harmonic in the measurement signal channel, using the function of its narrowband characteristic.

4.3.2. Effect of Rate Stability on the Accuracy of the North Finding

Here, the drift noises of the inertial components are ignored, and only the deviation of rotating rate is considered [2]. Thus the constant rate  𝜔₀ has a random frequency component, which randomizes slightly the angular motion of the rotating platform [23]. Therefore, the output signal of the inertial measurement is given by 𝑥(𝑡)=𝑉_𝑠sin𝜔₀𝑡+𝜂+𝜃₀sin(𝛼𝑡)=𝑉_𝑠sin𝜔₀𝑡+𝜂cos𝜃₀𝜔sin𝛼𝑡+cos₀𝑡+𝜂sin𝜃₀sin𝛼𝑡. (26)

Expanding cos(𝜃₀sin𝛼𝑡) and sin(𝜃₀sin𝛼𝑡) into the form of the first Bessel function yields 𝑥(𝑡)=𝑉_𝑠𝐽₀sin𝜔₀𝑡+𝜂+𝑉_𝑠^∞_𝑚=1𝐽_𝑚sin𝜔₀+()+𝑚𝑎𝑡+𝜂−1^𝑚sin𝜔₀−𝑚𝑎𝑡+𝜂, (27) where 𝐽_𝑚 is the coefficient of the first Bessel function, 𝑚 is the order, and 𝜃₀ is the amplitude of the frequency error. As 𝜃₀ is very small, 𝐽₁=𝜃₀/2,  𝐽₂=𝐽₃=𝐽₄⋯≈0, and (27) could be rewritten as 𝑥(𝑡)=𝑉_𝑠sin𝜔₀𝑡+𝜂+𝑉_𝑠𝜃₀cos𝜔₀𝑡+𝜂. (28)

One of the reference signals can be expanded into the Fourier series [22]: 𝑟(𝑡)=4𝑉_𝑟𝜋^∞_𝑛=1(−1)^𝑛+1)𝜔2𝑛−1cos(2𝑛−1₀𝑡. (29)

According to (28) and (29), the product of  𝑥(𝑡) by 𝑟(𝑡) yields: 𝑢_out(𝑡)(𝑡)==𝑥×𝑟2𝑉_𝑠𝑉_𝑟𝜋^∞_𝑛=1(−1)^𝑛+1)𝜔2𝑛−1sin(2𝑛−2₀+𝑡−𝜂2𝑉_𝑠𝑉_𝑟𝜋^∞_𝑛=1(−1)^𝑛+12𝑛−1sin2𝑛𝜔₀+𝜃𝑡+𝜂₀𝑉_𝑠𝑉_𝑟𝜋^∞_𝑛=1(−1)^𝑛+1)𝜔2𝑛−1sin(2𝑛−2₀+𝜃𝑡−𝛼𝑡−𝜂₀𝑉_𝑠𝑉_𝑟𝜋^∞_𝑛=1(−1)^𝑛+12𝑛−1sin2𝑛𝜔₀−𝜃𝑡+𝛼𝑡+𝜂₀𝑉_𝑠𝑉_𝑟𝜋^∞_𝑛=1(−1)^𝑛+1)𝜔2𝑛−1sin(2𝑛−2₀−𝜃𝑡+𝛼𝑡−𝜂₀𝑉_𝑠𝑉_𝑟𝜋^∞_𝑛=1(−1)^𝑛+12𝑛−1sin2𝑛𝜔₀𝑡−𝛼𝑡+𝜂. (30)

As shown in (30), the product  𝑢_out  consists of six components: only 𝑛=1 frequency difference has an output after the low-pass filter in the first term, which is considered useful information to obtain the amplitude 𝑉_𝑠 and phase 𝜂 of the measurement signal 𝑥(𝑡). Other frequency components in the first term, the second term, the fourth term, and the fifth term can be filtered out by the low-pass filter. The errors of the result are from the frequency-difference of 𝛼=(2𝑛−2)𝜔₀ in the third term, the difference component of 𝛼=2𝑛𝜔₀  in the fifth term and in the sixth term, causing an error output through the lock-in amplifier. Since the amplitude  𝜃₀  of the error is small, its effect on the result is not significant.

4.3.3. Effect on North Finder Accuracy due to Installing Error

In the ideal condition, the gyroscope sensitive axis  𝑜𝑥_𝑔 is OA-axis with the skewed angle  𝜒₁ to the rotation axis  𝑜𝑧_𝑏(𝑜𝑧_𝑎). Two parameters of installing error  𝜕,  𝛽 represent the deviation of the matrix 𝐶^𝑔_𝑎 from the ideal condition, and the result of the gyroscope sensitive 𝑜𝑥^′_𝑔-axis is turned into 𝑂𝐶-axis. It is stressed that, although there is no effect on the output of the 𝑜𝑥_𝑎-axis accelerometer, the angular rate that the 𝑜𝑥^′_𝑔-axis gyroscope sensed at this time can be expressed as 𝐺^𝑔_𝑥=𝑅cos𝛽cos𝜒₁−𝜕+𝑀sin𝛽cos𝜔₀𝑡+𝑀cos𝛽cos𝜒₁−𝜕−𝑅sin𝛽sin𝜔₀𝑡+𝑁−𝜔₀cos𝛽cos𝜒₁−𝜕+𝑏^𝑔_𝑥+𝑛^𝑔_𝑥(𝑡). (31)

The horizontal attitude angle solutions will not be affected as the output of the 𝑜𝑥_𝑎-axis accelerometer is not changed by the errors. However, the heading angle solution will be affected. Under the conditions 𝜕=12^′′ and 𝛽=12^′′, 15′′ heading error will be induced, which illustrates that this type of installation error should be reduced as small as possible [24].

In addition to the installing error described in Figure 7, when the sensitive center of the accelerometer does not go through the center of the rotation axis sufficiently, a steady drift bias of acceleration is introduced as a result of size effect [14]. It does not affect the solution of attitude angles, which is similar to the constant drift noise of the accelerometer.

Figure 7 

Installing error of sensitive axis of the inertial component.

5. Conclusion

This paper focuses on a dynamic strapdown north finding scheme at the constant rate rotation. A simplified configuration of the north finder system consists of three major parts: a pendulous force-feedback accelerometer, an RLG, and a rotating platform, which reduces the cost of equipment. In order to suppress the drift noise of the inertial components, two lock-in amplifiers are employed to extract and amplify the vehicle’s attitude signals that are weak and slow-varying. The main advantage of the new scheme keeps the rotating platform away from the process of leveling, which is convenient for engineering applications. The vehicle’s attitude angles could be determined in a short time with certain accuracy, as well as the estimations of the constant drift noise of accelerometer and gyroscope.

The only remaining problem of using this scheme arises when the installation errors of gyroscope and accelerometer are significant. These installation errors cause the heading error during north finding, which indicates that this type of installation errors should be reduced as small as possible.