An Improved Gyrocompass Alignment Method for Large Azimuth Misalignment

Due to the impact of the nonlinear factor caused by large azimuthmisalignment, the conventional gyrocompass alignment method is hard to favorablymeet the requirement of alignment speed under the condition of large azimuthmisalignment of INS. In order to solve this problem, an improved gyrocompass alignmentmethod is presented in this paper.The improvedmethod is designed based on the nonlinearmodel for large azimuthmisalignment and performed by opening the azimuth loop.The influence of the nonlinear factor on gyrocompass alignment will be reduced when opening the azimuth loop. Simulation and experimental results show that the initial alignment can be efficiently accomplished through using the improved method in the case of existing large azimuth misalignment, and in the same conditions, the alignment speed of the improved method is faster than that of the conventional one.


Introduction
The initial alignment of inertial navigation systems (INS) is an important process performed prior to normal navigation procedure [1].It is well known that the initial alignment result of the system is of fundamental importance to the following navigation accuracy [2].Therefore, many researchers have investigated this topic, mainly concentrated on gyrocompass alignment and optimal estimation techniques.In contrast to the optimal estimation techniques, the former method does not need precise mathematical and noise model [3].With many years' development, gyrocompass alignment method based on classical control theory is very mature now.In 1961, Cannon firstly presented gyrocompass alignment method for platform INS [4].After that, gyrocompass alignment is described extensively in the literatures [5,6], including alignment technique and error analysis.In recent years, with the development of strapdown INS, gyrocompass alignment is applied to strapdown INS [1,[7][8][9].In all the previous works, gyrocompass alignment is usually designed based on the small angle assumption (i.e., less than 5 degrees), and under this situation, the alignment system can then be approximated as linear model in the case of small azimuth misalignment.However, under the condition of large azimuth misalignment, such approximation is invalid, and the alignment system will then be influenced by the nonlinear factor.Then the conventional method is not effective to properly accomplish initial alignment in the case of existing large azimuth misalignment.
Therefore, in this paper a new improved gyrocompass alignment method, which is applicable to the INS that causes large azimuth misalignment, is established based on the nonlinear model.So far, many works are attempted to model large azimuth misalignment, and several models have been provided, such as the nonlinear psi-angle model [10,11], the rotation vector error and quaternion error models [12], and the nonlinear phi-angle model [13].In this work, the nonlinear psi-angle model is adopted.The improved method proposed in this paper is performed by opening the azimuth loop, and through using this scheme, the nonlinear factors can be regarded as constant inputs, which contain azimuth misalignment information.The estimation of the azimuth misalignment is implemented by using the horizontal velocity signals to estimate those nonlinear factors.At this time, the impact of the nonlinear factor caused by large azimuth misalignment on gyrocompass alignment can be reduced.The remainder of this paper is organized as follows: Section 2 describes the conventional gyrocompass alignment method and then analyzes the existing problem of this method in detail.Section 3 details the improved gyrocompass alignment method, which includes the establishment, operation, and implementation of the improved method.Simulation and experimental results that validate the proposed approach are presented and discussed in Section 4. Finally, conclusions are given in Section 5.

Conventional Gyrocompass Alignment Method
In literatures [8,9], a fourth-order gyrocompass alignment system was used for INS alignment.It is a higher-order system which has a good performance in the case of small azimuth misalignment.In this paper, the conventional gyrocompass alignment method analyzed here is the fourth-order system (as the representative of the similar methodologies), and the form of this system is simplified.The error diagram of reduced conventional gyrocompass alignment is shown in Figure 1.
In Figure 1, ∇  is the north accelerometer error.  and   are the east and up gyro errors.  and   represent the east level and azimuth misalignments, respectively. represents the acceleration due to gravity, Ω represents the earth rate, and  represents the local latitude. 1 ,  2 ,   , and   are the control gains.
According to the Mason gain formula, we can get the characteristic equation of conventional gyrocompass alignment from Figure 1: In order to ensure the stability of the alignment system, four eigenvalues of (1) are set as  1,2 = −  and  3,4 = −  ± √1 −  2   , where  = 0.707 and   is the natural frequency which is adjustable.Then (1) can be rewritten as Comparing ( 1) with ( 2), the values of the control gains under the small angle can be obtained: In the knowledge of a classical control theory, it is obvious that the alignment speed of gyrocompass alignment is determined by the eigenvalues.Worth noting is that the four eigenvalues of the conventional gyrocompass alignment system can be given any value desired by appropriate choice of the control gains.In other words, the alignment speed of conventional gyrocompass alignment is determined by choice of the control gains  1 ,  2 ,   , and   .However, if the azimuth misalignment is large, the scale factor Ω cos  in Figure 1 will become the nonlinear factor Ω cos  ⋅ sin   /  .The characteristic equation of conventional gyrocompass alignment becomes where (  ) denotes the nonlinear factor   / sin   .Then only if the values of the control gains are set as can the four eigenvalues of conventional gyrocompass alignment system then be set as   gains cannot be set as (7), (8), and (9).In fact, under the large azimuth misalignment, the values of the control gains are still set as (3), (4), and (5).Then four eigenvalues of conventional gyrocompass alignment cannot be set as the desired values.In this case, the conventional method is hard to favorably meet the requirement of alignment speed.The particular reason of this problem also can be expressed by the following equation: where    is the theoretical value calculated by (9) and   is the actual value obtained by (5).From (10), it is clear that the actual value   adopted by the alignment system is smaller than the theoretical value    , due to the fact that (  ) ≥ 1 (domain:   ∈ [−180 ∘ , 180 ∘ ]).The functional relationship between (  ) and |  | is shown in Figure 2.
It is obvious from Figure 2 that the value of (  ) is increasing along with the growth of |  |; then the difference between   and    will increase together with the growth of |  |.That means the difference between the actual and desired eigenvalues increases along with the growth of |  |; however, the performance of the alignment system will be poor.

Improved Gyrocompass Alignment Method
Considering the poor performance of conventional gyrocompass alignment under the condition of large azimuth misalignment, an improved gyrocompass alignment method is proposed in this paper to improve the performance of gyrocompass alignment.It is clear that the four eigenvalues of conventional gyrocompass alignment cannot be set as the desired values in the case of large azimuth misalignment, because the uncertain nonlinear factor is included in the characteristic equation.So the key to design the improved method is to reduce the impact of the nonlinear factor on characteristic equation.This problem is solved by opening the azimuth loop; at this time, the influence of the nonlinear factor on characteristic equation disappeared, and the estimation of the azimuth misalignment is implemented by using the horizontal velocity signals to estimate the nonlinear factors.

The Establishment of the Improved Gyrocompass Alignment
Method.In this paper, , , , , and  denote the inertial frame, the earth fixed coordinate frame, the sensor body frame, the navigation frame, and the computed navigation frame, respectively.In this work, we choose the local level geographic coordinate frame as the navigation frame.Under the large azimuth misalignment, the direction cosine matrix (DCM) C   can be described as follows [13,14]: where   is the north level misalignment;   and   (level misalignments) are small, and   (azimuth misalignment, the departure of computed north from actual north) is large.For a quasistationary initial alignment, the average value for velocity will be zero [15].Then under quasistationary situations, the navigation equations for strapdown INS attitude and velocity can be represented, respectively, as where C   represents the DCM, relating the transformation from  frame to the  frame;    is the angular velocity vector of  frame with respect to  frame resolved in  frame; [×] denotes a skew-symmetric matrix operator; V  is the computed velocity of strapdown INS; f is the specific force from the accelerometer output; g  is the gravity vector resolved in  frame.The angular velocity    is derived by where ω  is the angular velocity vector measured by gyros;    is the angular velocity vector of  frame with respect to  frame resolved in  frame.
These angular velocity vectors ω  ,    and accelerometer output f can be described, respectively, as where    is the true angular velocity vector,   is the gyro error vector, g  is the gravity vector resolved in  frame, a  is the disturbing motion vector under quasistationary situations, and ∇  is the accelerometer error vector.
Substituting (15c) in ( 13), we obtain Then, from ( 16), the horizontal components of V  can be described, respectively, as where   and   represent the east and north horizontal velocities, respectively; (⋅)  and (⋅)  represent the  and  elements of column matrix (⋅) separately, whose entries from top to bottom are the , , and  elements.From ( 11), (12), and ( 14), the misalignment equation of strapdown INS under large azimuth misalignment can be represented as [11,13] where  is the misalignment vector and The leveling alignment of the improved method is performed by introducing control angular velocity    into the calculation of angular velocity    ; then we have where is the control angular velocity vector.From (19), the horizontal components of  can be, respectively, represented as Because the azimuth misalignment's rate of change is small [10], the approximation ψ  = 0 is admitted; namely,   is a constant value [15].
The azimuth information is obtained by structuring azimuth estimation functions.The block diagram of    and azimuth estimation functions is shown in Figure 3.
In Figure 3,   and   represent the control gains; () and () represent the control networks used for reducing the influence of disturbing motions; (  ) and (  ) represent the functions of   .From (17), (20), (21), and Figure 3, the error diagrams of improved gyrocompass alignment can be obtained as in Figures 4 and 5.
It can be seen from Figures 4 and 5 that the nonlinear factors sin   Ω cos  and (1−cos   )Ω cos  both are caused by the large azimuth misalignment and become the constant inputs of the system through opening the azimuth loop.So  the nonlinear factors will not be included in the characteristic equation.In addition, these constant inputs contain azimuth misalignment information, and the estimation of the azimuth information can be obtained by (  ) and (  ).

The Operation of the Improved Gyrocompass Alignment
Method.In this section, the control gains   and   and control networks () and () are provided.Then (  ) and (  ) can be obtained, and the azimuth estimation is performed by these two functions.
Firstly, we provide the values of the control gains   and   for improved gyrocompass alignment; the detailed deduction for these control gains can be found in the following.
According to Mason gain formula, we can get the characteristic equation of north loop from Figure 4: In the same way, from Figure 5, the characteristic equation of east loop can be obtained as follows: It can be seen from ( 22) and ( 23) that the north and east loops have the same characteristic equation.In order to ensure the stability of alignment system, we set two eigenvalues of ( 22) and (23) as Then the characteristic equation can be rewritten as Comparing ( 22) with (25), we can get Secondly, in order to simplify the subsequent analysis, we assume that the inertial sensor errors are basically constant drifts, and the vehicle in which the INS is mounted is totally stopped.Under this assumption, the disturbing motion vector a  is equal to zero.At this time, the control networks can be set as () = () = 1, because no disturbing motion is introduced into the alignment system.The design procedure of azimuth estimation is described in the following, and it can be divided into two steps.The first step: the analysis of the relationships between functions (  ) and (  ) and azimuth misalignment   is made with the aid of the Laplace transformation.The second step: based on the former analysis, the azimuth estimation equations are provided.

Relationships between Functions 𝑆(𝜓 𝑈
) and (  ) and Azimuth Misalignment   .The response to east level misalignment   and function (  ) can be written in Laplace form directly from Figure 4 and can be seen as follows (according to the Mason gain formula): where {⋅}() represents the Laplace transformation of {⋅}.
Similarly, from Figure 5, it yields (29) Then we have where (⋅) ss represents the steady-state value of (⋅).
We regard sin   , cos   ,  ss , and  ss as the unknown values; (30) can then be easily solved.Then, the relationships between functions (  ) and (  ) and azimuth misalignment   can be obtained: The detailed derivations of (31) can be found in Appendix A.

The Azimuth Estimation
The detailed derivations of (34) can be found in Appendix B. At this time, the azimuth estimation ψ can be obtained from (34) and is shown as follows: It is obvious from the previous discussion that the azimuth estimation equations consist of (32)-( 35).Finally, we take the disturbing motions into consideration, because the alignment is often performed under quasistationary conditions.The disturbing motions can be generally considered to be sinusoidal [4,16].Then, in order to reduce the influence of the disturbing motions on azimuth estimation, low pass filters need to be added to the system; namely, the control networks () and () should have the capability of restraining disturbance.The design law of the control networks () and () is provided as follows.
(a) The magnification of () and () in low frequency must be equivalent to 1; that is, the following equation should be satisfied: The reason for this requirement is that the relationships between functions (  ) and (  ) and azimuth misalignment   under quasistationary conditions also can be represented by (31) as the requirement is met.
(b) In high frequency, for the purpose of reducing the influence of disturbing motions, they need be capable of restraining disturbance.
Then, under quasistationary conditions, the azimuth estimation equations also consist of (32)-( 35).In this work, the control networks () and () are set as At this time, the denominators of {(  )}() and {(  )}() can be set as It is obvious from the azimuth estimation equations that the performance of the azimuth alignment is determined by ( 32) and (33).Since both (32) and (33) consist of (  ) and (  ), the four eigenvalues of {sin ψ }() and {cos ψ }() could be set as  1,2 = −  and  3,4 = −  ± √1 −  2   , without the influence of the nonlinear factors.Therefore, the impact of the nonlinear factor caused by large azimuth misalignment on gyrocompass alignment is reduced, and the performance of the improved gyrocompass alignment will be better than the conventional one.

The Implementation of the Improved Gyrocompass Alignment Method.
In this section, the implementation of the proposed method in real time operating device is discussed, and meanwhile a brief summary of this method is provided.An inertial navigation system implements the proposed alignment method using a cluster of accelerometers to sense the specific force vector components f , a triad of gyros to measure the angular velocity ω  , and a digital signal processor (DSP) to perform the alignment algorithm.The direct expression of this implementation can be seen in Figure 6.According to Figure 6, the specific implementation steps of the improved gyrocompass alignment in real time operation device are described as follows.
(a) At the beginning, a preliminary alignment often called coarse alignment is performed, and after that, a rough DCM C   is obtained.The specific coarse alignment method can be found in [15]; the level misalignments are small and azimuth misalignment is usually large.
(b) Secondly, by utilizing the specific force f and DCM C   , the derivatives of the east and north horizontal velocities V  and V  can be obtained.Then, according to the calculation method shown in Figure 3, the control velocity    and azimuth functions (  ) and (  ) can be acquired.Finally, by spending a period of time, an accurate azimuth estimation ψ can be obtained, and, with the compensation of azimuth misalignment using ψ , an accurate DCM C   can be obtained.

Simulations and Experiments
In this section, to evaluate the performance of the improved gyrocompass alignment method under the condition of large azimuth misalignment, simulations and experiments are carried out.

Simulation Results and Analysis.
Both the conventional and improved gyrocompass alignment methods are performed during the simulations simultaneously, and the simulations are conducted under the conditions of, respectively, choosing different azimuth misalignments; namely,   = {50 ∘ , 100 ∘ , 145 ∘ , −50 ∘ , −100 ∘ , −145 ∘ }.In these simulations, gyro and accelerometer outputs are generated by the strapdown INS simulator; we assume that the vehicle is in disturbing motions caused by sea waves.The initial conditions are presented as follows.
The constant gyro errors: The constant accelerometer errors: .
The acceleration due to gravity:  = 9.78049 m/s 2 .
The natural frequency of conventional and improved methods:   = 0.02 rad/s.
Under the disturbing motions, the vehicle undertakes angular and lineal movements.In angular movement, the pitch   , the roll   , and the yaw   are controlled as In the lineal movement, the vehicle lineal movement velocities are taken as But, under the condition of large azimuth misalignment, the performance of conventional method will be poor along with the growth of |  |.The reason is that the difference between the actual and desired eigenvalues is increasing along with the growth of |  |.
Figure 9 shows convergence time for different azimuth misalignments with a converged azimuth estimation error of less than 1 degree.It is clear that large azimuth misalignment needs more time to converge under the conventional method.However, for the improved method, the change of the convergence time is not obvious.That means the performance of the improved method is not affected by the large azimuth misalignment.The reason is that the eigenvalues of improved method can be set as the desired values, without the influence of the nonlinear factors caused by the large azimuth misalignment.Moreover, it is obvious that the alignment speed of the improved method is faster than that of the conventional one.

Experimental Results and Analysis.
The experiments were implemented in the lab, as the true yaw angle of the strapdown INS was known, so different   could be exactly set.We fixed the strapdown INS on the SGT-3 threeaxis turntable to implement the alignment experiments.The strapdown INS and the turntable can be seen in Figure 10.At the start of each experiment, the turntable turned to a static position for 20 minutes, that is, pitch angle 0 ∘ , roll angle 0 ∘ , and yaw angle 135 ∘ .Two alignment methods were performed under the condition of, respectively, setting initial yaw angle as 185 ∘ and 235 ∘ ; namely,   = {50 ∘ , 100 ∘ }. Figure 11 provides the comparison of the yaw angle using the two alignment methods under the condition of different   .
As the results above show, the yaw angle can converge to 135 ∘ through using the two alignment methods.However, for the conventional method, the convergence time is lengthened with the growth of the azimuth misalignment.Compared with the conventional method, the convergence time of improved method is hardly changed.The experimental results are consistent with the simulation results.The experiments demonstrate that the performance of the improved method is better than that of the conventional one.

Conclusions
In this paper, an improved gyrocompass alignment method has been proposed for large azimuth misalignment of INS.The improved new method is performed by opening the azimuth loop, and the estimation of the azimuth misalignment is implemented by using the horizontal velocity signals to estimate the nonlinear factors.Through using this method, the impact of the nonlinear factor caused by large azimuth misalignment on gyrocompass alignment is reduced, and the performance of gyrocompass alignment is improved.The alignment speed of the improved method is faster than the conventional one in the same conditions.The performance of the improved method is well validated by simulations and experiments.

A. The Derivations of (31)
In this section, (31) are obtained in the following.
According to (30), their matrix form can be written as where Multiply the first and third rows of (A.1) by   /2Ω cos , and multiply the second and fourth rows of (A.1) by  2  /Ω cos ; we can obtain where

Figure 3 :
Figure 3: Block diagram of    and azimuth estimation functions.

Figure 6 :
Figure 6: Flow chart of improved gyrocompass alignment in real time operating device.
(c) On the one hand, the control velocity    and angular velocity ω  are used to update the DCM C   used for the next calculation step.(d) On the other hand, the azimuth functions (  ) and (  ) are used to compute sin ψ and cos ψ .And, after the final process which is achieved by utilizing (34)-(35), the azimuth estimation ψ is obtained.(e) Then, steps (b), (c), and (d) are performed repeatedly.

𝑡) m/s 2 ,
Figures 7 and 8 provide the azimuth estimation errors of conventional and improved methods under the conditions

Figure 11 :
Figure 11: Comparison of the yaw angle using the two alignment methods.
1,2 = −  and  3,4 = −  ± √1 −  2   (desired values).Since   cannot be obtained, (  ) is an uncertain factor.Thus, the values of the control Equations.It is clear that sin   and cos   can be supplied exactly by (31) if the accelerometer errors and gyro errors (sensor errors) are known.Since the sensor errors are uncertain, sin   and cos   are calculated by the following equations: , the errors between computed values (sin ψ and cos ψ ) and theoretical values (sin   and cos   ) are only determined by sensor errors, and these errors are allowable.Worth noting is that (32) and (33) do not preserve the unit-norm property of the trigonometric function; that is, (sin ψ ) 2 + (cos ψ ) where sin ψ and cos ψ are the computed values of sin   and cos   , respectively.When the improved alignment system is stable The derivations of (34) are carried out based on the principle of least squares fitting, setting the Euclidean minimum norm  between sin ψ , cos ψ and sin ψ , cos ψ (the normalized data) as the normalization index.Obviously, the parameters sin ψ and cos ψ that made  minimum are equal to the ones that make  minimum.Respectively, the partial derivatives of the parameters sin ψ and cos ψ are calculated, and, by making them equal to zero, we can get First, define  = [(sin ψ − sin ψ ) 2 + (cos ψ − cos ψ ) 2 ]