Initial alignment of compass based on genetic algorithm-particle swarm optimization

Yi-feng Liang ,Peng-fei Jiang,Jiang-ning Xu ,Wen An,Miao Wu

College of Electrical Engineering,Naval Univ.of Engineering,Wuhan 430033,China

Keywords:Inertial alignment Genetic algorithm SINS Compass alignment

ABSTRACT The rapidity and accuracy of the initial alignment influence the performance of the strapdown inertial navigation system(SINS),compass alignment is one of the most important methods for initial alignment.The selection of the parameters of the compass alignment loop directly affects the result of alignment.Nevertheless,the optimal parameters of the compass loop of different SINS are also different.Traditionally,the alignment parameters are determined by experience and trial-and-error,thus it cannot ensure that the parameters are optimal.In this paper,the Genetic Algorithm-Particle Swarm Optimization(GA-PSO)algorithm is proposed to optimize the compass alignment parameters so as to improve the performance of the initial alignment of strapdown gyrocompass.The experiment results show ed that the GA-PSO algorithm canfind out the optimal parameters of the compass alignment circuit quickly and accurately and proved the effectiveness of the proposed method.?2020 China Ordnance Society.Production and hosting by Elsevier B.V.on behalf of KeAi Communications Co.This is an open access article under the CCBY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

1.Introduction

Because of its simple structure,small spatial volume,low cost,good environmental adaptability,strong autonomy and high concealment,SINS has become mainstream in the research and development of modern inertial navigation system.It has attracted wide attention and been applicated extensively in aviation,space flight,navigation and other fields[1-4].Initial alignment technology is one of the key technologies of strapdown inertial navigation system,and the results of initial alignment directly affect the accuracy of inertial navigation calculation[5,6].The initial alignment method based on compass loops is an important and usual approach to initial alignment of SINS[7].

The basic principle of inertial alignment of SINS under the condition of high azimuth misalignment was described in Refs.[8-10].In Ref.[11],Li Yao et al.studied the self alignment technology of the compass method under the swaying state of the SINS.In reference to the problem that ships are easily disturbed by wind waves and surges under mooring conditions,the alignment method based on time-varying parameters is studied in Ref.[9].The strapdown compass alignment method has the advantages of simple algorithm,small calculation and strong anti-interference ability,so it is suitable for the initial alignment of SINS.The value of compass parameter Tdi(damping oscillation period)can directly affect the accuracy of compass alignment[7].Different motion state,environment and constant error and random noise of inertial sensors affect the selection of Tditogether.The parameter Tdiis often determined by experience or repeated test in the traditional way,and it is difficult to obtain the optimal alignment parameter[12,13].

He H.Yet al.optimized the parameters of the strapdown compass by genetic algorithm(GA),and obtained good results[12].Zhu B proposed to optimize the parameters by PSO algorithm,and also achieved good performance[13].How ever,there are some defects in the GA and PSO algorithm.The computational complexity of the genetic algorithm and the hardness of the coding and genetic manipulation make it difficult to achieve it[14].The precocious phenomenon often occurs when the extreme value of the function calculated in the PSO algorithm,which may lead to great deviation in the solution the function extremum[15,16].GA algorithm adopt selection,crossover and mutation operator in a probabilistic way for function optimization and take directly the objective function as search information,so the global optimization ability,the evolution speed,the convergence precision of PSO algorithm can be improved by GA,so GA-PSO algorithm has obvious advantages compared to GA or PSO algorithm[17,18].In this paper,GA-PSO algorithm is utilized to optimize the alignment parameters Td1 and Td2.The experimental results show that,the optimal parameters of gyrocompass loop(Tdbest1,Tdbest2)can be searched automatically by GA-PSO algorithm compared with the traditional method,GA algorithm optimization method and PSO optimization,and it is a feasible and effective method in the initial alignment of strapdown gyrocompass.

2.Com pass alignment principle of SINS

The basic principle of the compass alignment method is to realize the convergence of the IMU(Inertial Measurement Unit)yaw angle by using the compass effect in the north and east circuits.In the process of compass alignment,the horizontal alignment is first carried out follow ed by the direction alignment when the horizontal attitude angle error converges.The basic principle of the initial alignment of the strapdown compass is shown in Figs.1-3.

Where e，i，n，b denote the earth frame,the inertial navigation frame,the local level navigation frame(East-North-Up)and SINS body frame(Right-Forth-Up)respectively;KE1,KE2,KE3denote the east compass loop parameters,KN1,KN2,KN3for the north,KU2,KU2,KU3for the yaw loop parameters;is the attitude matrix from b frame to n frame,a mathematical platform that contains errors;ωbiband fbdenote the angular rate of the carrier measured by gyroscope and specific force measured by accelerometer in the body frame;is the control angular speed applied to the mathematical platform~Cnb;ωieand Rerepresent the rotation speed and the radius of the earth;δvnEand δvnNrespectively indicate eastward velocity error and northward velocity error;δpEand δpNintermediate variables.

Assuming that the sampling time is Ts(Ts＜＜Td),the control law of the compass alignment loop can be discretized into the first order difference.According to Figs.1 and 2,the control laws of level alignment loop can be formulated as follows:

Fig.1.East alignment loop of strapdown gyrocompass.

Fig.2.North alignment loop of strapdown gyrocompass.

Fig.3.Azimuth alignment loop of strapdown gyrocompass.

The control rules of azimuth alignment loop can be written as:

The typical values of the control parameters in alignment loops are[5]:

Where σi(i=1，2)denote attenuation coefficient which determined by the precision,the rapidity and the interference strength of the alignment,ωsis S chuler period,ξ is damping ratio,usually ξ=denote the frequency of damping oscillation,then ωd=2π/Tdand the relation between σiand Tdican be expressed asTherefore,the value of Tdiwill directly affect the value of the control parameters of the compass alignment loop,which has an important influence on the final result of the alignment.Different SINS(different characteristics of IMU)are different in the optimal value of Tdi.Therefore,it is of great significance to determine the optimal value of Tdiintelligently for improving the performance of the compass alignment of the SINS.

3.GA-PSO algorithm based com pass alignment

3.1.Fundamental principle of GA-PSO algorithm

The particle swarm optimization(PSO)algorithm was proposed by American scholars Eberhart and Kennedy in 1995.The PSO algorithm updates particle speed and position by tracking individual optimal particle pmbestand group optimal particle gmbestduring operation,the speed and position updating formulas are:

Where d=1，2，…，K;i=1，2，…，N respectively are the dimension of search space and population size;r1,r2are random numbers in the range of(0,1);c1,c2are learning factors which indicates respectively the ability to learn from themselves and other particles,usually between(0,2).ω is constant inertia weight used to adjust the diversity of particles;particle velocity v∈［vmin，vmax］;m is the algebra of current population,indicate the current position and velocity of the particle;espectively represents the position of current individual and group optimal particle.

Genetic algorithm(GA)is an evolutionary algorithm,its basic principle is to follow the evolution law of“survival of the fittest in natural selection,survival of the fittest”in biology.Genetic algorithm is to encode the problem parameters into chromosomes,and then select,crossover and mutation iteratively to exchange information of chromosomes in the population,and finally generate the chromosomes meeting the optimization target.The purpose of mutation operation is to maintain diversity of population by which selects an individual from the population randomly and selects one point of the individual to mutate to produce a better individuals.The method of mutation operation for the j gene aijof individual i is:

Where amaxis the upper bound of gene aij,aminis the lower bound;r2is a radom number,g is the number of iterations,Gmaxis the largest number of evolutionary times,r is a random number of the[0,1]interval.

Crossover operation refers to randomly selecting two individuals from the population,and inheriting excellent characteristics of parent string to the sub string by crossover of two chromosomes,thus generating new substrings(new excellent individuals).The cross operation method of the k chromosome akand the l chromosome alat bit j is the method for the cross operation.

When the PSO algorithm calculates the extreme value of the function,the precocious phenomenon often occurs,which leads to the extreme deviation of the extreme value of the solution function.How ever,the genetic algorithm for function optimization using selection,crossover and mutation operator,directly to the objective function as the search for information,to the probability of a way,thus enhances the particle swarm algorithm global optimization ability,accelerate the speed of evolution algorithm,improves the precision of convergence.

3.2.Procedure of the proposed algorithm

When GA-PSO algorithm is used to optimize the parameters of the strapdown compass alignment,the particles i｝,and particle velocitiesThe flow of the GA-PSO algorithm used to optimize the strapdown compass alignment parameters as shown in Fig.4.

3.3.Establishment of fitness function

The performance of initial alignment mainly depends on the accuracy,rapidity and stability.The convergence speed and accuracy of heading angle are usually difficult to guarantee simultaneously in compass alignment.Therefore,the convergence speed and accuracy of heading angle are important references for constructing fitness function.Based on the alignment of the middle time,the whole alignment process is divided into two parts:alignment,starting time to the middle part of alignment,the alignment of the first half,alignment of the middle to the end of alignment.The convergence precision of heading angle by heading angle error,the overshoot is commonly used to measure the transition(alignment)process stability,fast response rise time alignment,so the second half of the alignment error of heading angle and azimuth alignment process overshoot and rise time as a constraint condition to construct the cost function.In order to ensure or further improve the accuracy of horizontal alignment,the angular error of the horizontal attitude is also considered as one of the best indexes for constructing the cost function.Therefore,the cost function is defined as:

Fig.4.Flow chart of strapdown gyrocompass initial alignment based on GA-PSO algorithm.

In the function above,tbis the starting time of alignment,teis the ending time,tmis the middle time,tm=（tb+te）/2;|e（t）|=|eE（t）|+|eN（t）|+|eU（t）|,where eE（t）,eN（t）,eU（t）are the pitch angle error,roll angle error and heading angle error respectively;tuis the rise time in the horizontal alignment;α1,α2,α3and α4are weight coefficients;In order to search for the optimal parameters,the objective of the optimization is to obtain the minimum value of the cost function,so the adaptive value function is defined as F=1/J.

4.Experiment

We installed a navigation-level optical fiber strapdown inertial navigation system to collect the output data of its gyroscopes and accelerometers on HNT2G69MF1C1 dual-axis multi-function test turntable,which is mainly used for speed,position,swing and other functions.It has the characteristics of stable performance,reliability and simple operation.The acquisition time is 600 s,and the acquisition frequency is 10 Hz.The experimental data acquisition system is shown in Fig.5,the main performance indicators of fiber IMU listed in Table 1.

In the experiment,the true attitude angles of the IMU are[pitch,roll,yaw]=[0°,0°,36.8°].Optimize the parameters of the compass by GA-PSO algorithm and verify the validity of the initial compass alignment optimized by the proposed algorithm.In the experiment,the parameters of the GA-PSO algorithm are as follow s:c1=2.0,c2=1.5;vmin=-1.5,vmax=1.5;The population size is 20,and the population evolution termination algebra is 50,The span of the compass parameters is Tdi∈［1，300］;in the fitness function F=1/J,weight coefficients α1=0.05,α2=1.0,α3=0.10,α4=0.15;the crossover possibility is pc=0.7,the mutation possibility is pm=0.3.

The following four experiments were carried out to validate the performance of the proposed algorithms:

Alignment 1:Traditional compass alignment with the common used alignment parameters Td1=20 s,Td2=100 s;

Alignment 2:Compass alignment by the GA based optimal compass alignment method proposed in Ref.[12].The used parameters are Td1=174.4941 s,Td2=106.5374 s;

Alignment 3:Compass alignment by the PSO based optimal compass alignment method proposed in Ref.[13].The used parameters are Td1=173.0240 s,Td2=87.2268 s;

Alignment 4:Compass alignment by the GA-PSO based optimal compass alignment method, the used parameters are Td1=114.2278 s,Td2=50.4647 s.

The results of these alignments are shown in Figs.6-8.In Figs.6-8,Traditional method’means the results of experiment 1 that the parameters determined by traditional method,by experience or‘trial and error’;‘GA algorithm’means the parameters optimized by GA algorithm in Ref.[12];‘PSO algorithm’means the parameters optimized by PSO algorithm in Ref.[13];‘the proposed algorithm’means the alignment parameters optimized by proposed GA-PSO.

Fig.5.IMU on the turntable.

Table 1 Performance of inertial devices in IMU.

Fig.6.Estimates curve of roll angle.

Fig.7.Estimates curve of pitch angle.

Fig.8.Estimates curve of yaw angle.

From the three diagrams,we can see that the red curve is the attitude angle of three directions of compass parameter alignment optimized by genetic particle swarm optimization,and the blue line is the attitude angle of compass alignment obtained by selecting empirical parameters 100 s and 20 s.Obviously,the convergence time of the three attitude angles of the optimal alignment is about 100 s,while the convergence time of the traditional parameter horizontal attitude angle is about 250 s,and the convergence time of the course is about 300 s,so the whole alignment time can be reduced by about 70%.The convergence time of GA-PSO is earlier than that proposed in Refs.[12,13].Therefore,the compass parameters obtained by the optimization algorithm proposed in this chapter can greatly shorten the alignment time,and there is almost no difference between the azimuth alignment time and the horizontal attitude alignment time,which proves the effectiveness of the proposed optimization method.

It is clearly shown that the alignment performance by the proposed method is much better than that of traditional compass alignment in this contribution.The attitude convergence speed by scheme 4 is much faster than that by scheme 1.That is to say,the proposed method can achieve the same endpoint with shorter time and the dynamic performance of the convergence process is much better.Besides,it is notable that the proposed method is to achieve an optimal compromise between accuracy and rapidity of compass alignment so as to meet certain alignment requirements.The main superiority of the proposed method lies in the accuracy,rapidity and feasibility in compass parameters optimization.The experimental results verified that the proposed method can search out the optimal compass alignment parameters and achieve an optimal compass alignment of SINS.

5.Results and discussion

Genetic particle swarm optimization is a hybrid intelligent algorithm combining the advantages of genetic algorithm and particle swarm optimization,which avoids the difficulty of coding and slow search speed of genetic algorithms and the shortcoming of PSO algorithm which is easy to fall into the local optimal.It has obvious advantages in parameter optimization.In this paper,the GA-PSO is proposed to optimize the parameters of the compass alignment of the SINS.The optimized parameters of the compass alignment are verified by the simulation data and the measured data,and compared with the parameters optimized by other methods,the GA-PSO,proved the effectiveness and advance of GAPSO in compass alignment parameter optimization of SINS.This method also provides a reference for other kinds of parameter optimization problems.

Acknowledgment

Research supported by the National Natural Science Foundation of China(Nos.4157069,41404002 and 61503404),the National Key Research and Development Program (2016YFB0501700,2016YFB0501701).