Artificial fish swarm algorithm (AFSA) is one of the state-of-the-art swarm intelligence techniques, which is widely utilized for optimization purposes. Triaxial accelerometer error coefficients are relatively unstable with the environmental disturbances and aging of the instrument. Therefore, identifying triaxial accelerometer error coefficients accurately and being with lower costs are of great importance to improve the overall performance of triaxial accelerometer-based strapdown inertial navigation system (SINS). In this study, a novel artificial fish swarm algorithm (NAFSA) that eliminated the demerits (lack of using artificial fishes’ previous experiences, lack of existing balance between exploration and exploitation, and high computational cost) of AFSA is introduced at first. In NAFSA, functional behaviors and overall procedure of AFSA have been improved with some parameters variations. Second, a hybrid accelerometer error coefficients identification algorithm has been proposed based on NAFSA and Monte Carlo simulation (MCS) approaches. This combination leads to maximum utilization of the involved approaches for triaxial accelerometer error coefficients identification. Furthermore, the NAFSA-identified coefficients are testified with 24-position verification experiment and triaxial accelerometer-based SINS navigation experiment. The priorities of MCS-NAFSA are compared with that of conventional calibration method and optimal AFSA. Finally, both experiments results demonstrate high efficiency of MCS-NAFSA on triaxial accelerometer error coefficients identification.
Artificial fish swarm algorithm (AFSA) is one of the state-of-the-art swarm intelligence approaches, which was proposed by Li et al. [
Nevertheless, the standard AFSA (SAFSA) has not been extensively considered by researchers, due to its complexity in comparison with other swarm intelligence algorithms in this domain. Particularly, concerning particle swarm optimization (PSO), the results of SAFSA are not better than those of PSO [
Triaxial accelerometer is widely utilized in military and civilian fields [
At present, the optimal AFSA (OAFSA) with improvement on
Monte Carlo simulation (MCS) is a broad class of computational algorithms that relies on repeated random sampling to obtain numerical results [
In conclusion, the advantages of the MCS-NAFSA for accelerometer error parameters identification are (1) that the algorithm’s computational complexity is decreased to release the high computational cost, (2) that the algorithm’s convergence rate is improved by adopting AFs’ previous experiences during AFs optimization process, (3) that no external reference information is introduced during the identification process, and (4) that the high workload and costs in conventional calibration method are reduced greatly.
The rest of this paper is organized as follows. In Section
AFSA is one of the swarm intelligence methodologies and evolutionary optimization techniques and its framework is based on functions that are inspired by social behaviors of fishes in the nature. Generally, fishes move to the areas that have more food by AFs individual or swarm search. AF model is depicted by prey, swarm, free move, and following behaviors. AFs food consistency degree in a specific area is the AFSA objective function. Finally, the AFs approach the maximum food density point.
The state of AF
In nature, fishes search for food or position with more food. Normally, we choose the position
Afterward, the food consistency degree
One of the features of fishes as a swarm is that they always try to move along other swarm members, which causes fish swarm to scatter, and the generality of the swarm is kept. In AFSA, in order to maintain the swarm generality, AFs try to move to the center position of the searching space in each of the iterations. The central position of swarm is expressed as
In (
If
During the AFs moving process, when the fishes find food, neighbor fishes follow them to search for food. When the current AF in position
If AF
AFs would move freely when they are not successful in finding food. In AFSA, the AFs would move a random
The free move behavior is necessary to maintain the diversity of AFs swarm when the AFs search for a better position in problem space.
According to the characteristics of SAFSA, there are some demerits for its application to the recalibration of accelerometer error parameters [
According to the AFSA for accelerometer error coefficients identification, the mentioned drawbacks in Section
The SAFSA has fixed value on both
After dozens of iterations, the objective or indicator function starts to present divergence tendency because the AFs fall into the local extreme by unsuccessful prey behavior [
When OAFSA is used for accelerometer error parameters identification, only the second drawback of SAFSA is eliminated, but the other two drawbacks are not avoided during the optimization process. Moreover, by inducing the secondary initialization method, the lowest indicator function point selection at the first stage is artificially aided. The reload process of AF parameters and accelerometer error coefficients are completed manually. This means the method is nonautonomous during the optimization process. Therefore, there are also some shortcomings when OAFSA is applied to accelerometer error parameters identification.
In order to solve the abovementioned demerits of SAFSA and the OAFSA when they are applied in accelerometer error coefficients identification, a novel artificial fish swarm algorithm is proposed. At first, the
Suppose that there are
The above equation generates a random CF in range
Next, the NAFSA behaviors will be discussed.
The individual behavior is made up of prey and free move behaviors. The AF
Because the position
In NAFSA, each AF moves towards a better position by individual behavior. But when it failed, it will perform random behavior within its
To keep the swarm characteristics of all the fishes and make the moving of AF to the best position are the two main targets in group behavior. The center position of AF swarm is obtained by (
Therefore, the AFs in a worse position would move towards center position by comparing with center position. When the position is better than center position, it will move towards best AF swarm position. Therefore, all the AFs will reach the best position by performing a group behavior. Consequently, in NAFSA, the best position searched by fish swarm would be adopted to accelerate the convergence rate with all AFs’ movement. So the group behavior is used to maintain the fishes swarm characteristics and avoid reducing in swarm diversity.
In group behavior, the center position AF may have better food density (indicator function) than best AF position. AF moves towards center position by (
The above equation is executed only when
In this section, the triaxial accelerometer static error model will be demonstrated at first. And then the optimization indicator function derivation process is given in detail. Finally, the triaxial accelerometer error coefficients identification procedures with NAFSA will be provided specifically.
The purpose of the error coefficients identification is to identify accelerometer error parameters quickly and accurately. There are various identification methodologies for triaxial accelerometer error parameters [
The NAFSA is terminated in one of the three conditions. The first is when the preset maximum iteration times are reached. The second condition is when the optimization indicator function during optimization process is below predefined threshold. The third condition is that when performing the next iteration, the deviation of current iteration result and the next iteration result is within an acceptable range. Therefore, the optimization indicator is a key factor for the termination condition of NAFSA optimization process. The following part will present an optimization indicator function for triaxial accelerometer error parameters identification based on NAFSA.
Theoretically, when the static triaxial accelerometer is at arbitrary space position, the accelerometer measured linear accelerations would satisfy the following equation:
In (
Actually, because of the errors caused by accelerometer itself, the calculated linear accelerations are different from theoretical values. Therefore, the linear acceleration mode square error (MSE) is adopted to represent the deviation, which is derived from (
Our target by using the NAFSA is to identify the error coefficients of accelerometer precisely and make the linear acceleration MSE as stable as possible. So the standard deviation function is used to evaluate the discrete degree of the accelerometer error coefficients:
In (
In this subsection, the implementation procedures of accelerometer error coefficients identification will be explained. Two main steps are conducted to illustrate the NAFSA optimization process. At the beginning, the variation characteristics of the 15 error coefficients in triaxial accelerometer are discussed and a clustering process is described on different triaxial accelerometer error parameters. After that, the specific MCS-NAFSA accelerometer error coefficients identification procedures are presented step by step.
In 15 triaxial accelerometer error coefficients’ identification, different error coefficients have different influences on the linear acceleration MSE and also the NAFSA requires all the AFs to have similar characteristics during optimization process. Therefore, the clustering on accelerometer error coefficients is a necessity before the accelerometer error coefficients identified by NAFSA.
Thinking about the different error coefficients’ influences on linear acceleration MSE and based on our previous experiences, the accelerometer scale factors have the highest impacts on linear acceleration MSE, followed by the biases, and then the accelerometer axis misalignment errors, and the last parameters are accelerometer quadratic nonlinear coefficients. So accelerometer error coefficients are divided into four different categories. They are 3 accelerometer scale factors
Through the discussion in Section
Begin for each AF initialize AFs parameters, category two, category three and category four end for each AF Perform end for each AF Perform end Update for each AF initialize AFs parameters, category one, category three and category four end for each AF Perform end for each AF Perform end Update for each AF initialize AFs parameters, category one, category two and category four end for each AF Perform end for each AF Perform end Update for each AF initialize AFs parameters, category one, category two and category three end for each AF Perform end for each AF Perform end Update Until terminate condition is meet End
In the first phase, accelerometer scale factors
In the second phase, accelerometer biases
In the third phase, accelerometer axis misalignment errors
In the last phase, accelerometer quadratic nonlinear coefficients
Finally, repeat the above four phases until the accelerometer error coefficients meet the termination conditions, when optimization indicator reaches
In this section, the triaxial accelerometer error coefficients simulation experiment is conducted by MCS-NAFSA. Before the simulation experiment, the AFs parameters and the nonoptimized triaxial accelerometer error parameters at each phase should be preset. Subsequently, the simulation process on the triaxial accelerometer error parameters is shown by MCS-NAFSA method.
Section
NAFSA triaxial accelerometer preset AFs parameters.
Accelerometer parameter types | NAFA AFs parameters | |||
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Maximum iteration times |
Contraction factors |
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(5.0000, 2.0000, 0.1000) | 50 | 60 | [0.000001, 0.999999] |
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(0.0050, 0.0020, 0.0001) | 50 | 60 | [0.000001, 0.999999] |
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(0.0005, 0.0002, 0.00001, 0.000005, 0.000002, 0.000001) | 50 | 60 | [0.000001, 0.999999] |
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(0.0050, 0.0020, 0.0001) | 50 | 60 | [0.000001, 0.999999] |
Meanwhile, in Section
Triaxial accelerometer error coefficients preset values.
Parameters types | Preset parameters |
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It is worth noting that, in each iteration, the dimensions of vector
After all the parameters preset procedure (i.e., the initialization process) is completed, the AFs start to execute the NAFSA optimization procedure. In order to increase the triaxial accelerometer error parameters’ credibility degree during NAFSA optimization process, 100-time MCS is conducted after the single NAFSA to bring in the random factors. Figure
Triaxial accelerometer scale factors: MCS-NAFSA curves.
Triaxial accelerometer biases: MCS-NAFSA curves.
Triaxial accelerometer axis misalignment errors: MCS-NAFSA curves.
Triaxial accelerometer quadratic nonlinear coefficients: MCS-NAFSA curves.
From Figures
Triaxial accelerometer error parameters MCS-NAFSA identification results.
Parameters | Preset value | MCS-NAFSA identification result | Relative error | Standard deviation |
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Through the comparison of preset parameters and the NAFSA identification results in Table
Furthermore, the standard deviation of the estimates in the last column of Table
At the same time, the variation tendency of indicator functions among the SAFSA, the OAFSA, and the NAFSA, when identifying the triaxial accelerometer scale factors, is demonstrated in Figure
Indicator functions tendency of AFSA, OAFSA, and NAFSA.
In Figure
To validate the feasibility and priorities of the proposed NAFSA on triaxial accelerometer error parameters optimization, the static 24-position verification experiment and the accelerometer-based SINS static navigation experiment are conducted, respectively. Additionally, for comparison, the experiments with conventional calibration and Section
In both experiments, the triaxial accelerometer-based SINS is developed by the Institute of Inertial Navigation and Measurement & Control Technology at the Harbin Engineering University. The main performance indicators of triaxial accelerometer are demonstrated in Table
Accelerometer performance indicators.
Parameter items | Performance indicators |
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Accelerometer dynamic range | ±300 g |
Accelerometer bias |
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Accelerometer linearity | 30 ppm |
Accelerometer scale factor | 10 ppm |
Triaxial accelerometer-based SINS.
After the triaxial accelerometer error coefficients are identified by adopting the proposed MCS-NAFSA, a static 24-position verification experiment is conducted to testify the precision of the triaxial accelerometer error parameters. Meanwhile, the triaxial accelerometer error parameters identified by the OAFSA are also testified in this experiment to reveal the advantages of the proposed MCS-NAFSA. In 24-position high-precision turntable verification experiment, when
Triaxial accelerometer measurement values with MCS-OAFSA identified parameters.
Theoretical values (m/s2) | Calculated values (m/s2) | Measurement errors (mg) | |||||||
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Triaxial accelerometer measurement values with MCS-NAFSA identified parameters.
Theoretical values (m/s2) | Calculated values (m/s2) | Measurement errors (mg) | |||||||
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In Tables
Triaxial accelerometer measurement errors statistical results.
Mean value (mg) | Standard error (mg) | |||||
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MCS-OAFSA |
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MCS-NAFSA |
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In Figure
Finally, the statistical results in mean values and standard deviations of the triaxial accelerometer measurement errors calculated by MCS-OAFSA and MCS-NAFSA were demonstrated in Table
In this subsection, a static navigation experiment is carried out by the triaxial accelerometer-based SINS. At the beginning, we install the triaxial accelerometer-based SINS on the marble benchmark to eliminate external disturbances on system positioning precision. Next, start up the SINS and the navigation information (attitude, velocity, position, etc.) which are shown in the monitor. Meanwhile, store the measurement values of triaxial accelerometer and triaxial FOGs and the navigation information for later data processing. After that, make the triaxial accelerometer error parameters identification with OAFSA and the proposed NAFSA with the stored triaxial accelerometer data, respectively. Finally, conduct the navigation mechanization with the OAFSA and NAFSA optimized parameters, respectively. The positioning error calculation formula is given by [
Figure
The comparison of positioning errors with three methods.
The corresponding numerical results of static positioning errors with the three different methods are shown in Table
Static positioning results of three different methods.
Methods | Conventional calibration | OAFSA identification | NAFSA identification |
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24-hour position error (nmile) | 4.4895 | 4.4988 | 4.2550 |
Therefore, we can summarize that the proposed NAFSA has advantages in workload and costs compared to the conventional calibration method. Moreover, the NAFSA has better performances in long-term navigation precision and has been more acceptable for actual engineering applications with lower computation complexity and faster convergence rate.
After the triaxial accelerometer-based SINS operated for a period of time, the triaxial accelerometer would be vulnerable by the working environmental disturbances, such as gravitational field, magnetic field, and thermal field. These exterior disturbances could influence the triaxial accelerometer error parameters’ stability directly or indirectly. Even though some measures are taken to eliminate these effects, high-precision navigation application is far from enough.
The research work in this paper is based on one of the swarm intelligence algorithms, artificial fish swarm algorithm, mainly on its optimization and improvement algorithm for triaxial accelerometer error parameters identification. The proposed NAFSA has the advantages of lower computational complexity and higher convergence rate than the OAFSA during optimization process. And it also has lesser workload and costs requirements than the conventional triaxial accelerometer error parameters calibration method. Furthermore, the proposed method could implement shorter recalibration interval time with higher precision in some harness application environment.
Both the 24-position triaxial accelerometer verification experiment and the triaxial accelerometer-based SINS navigation experiment results show that when the triaxial accelerometer-based SINS is in navigation condition, the proposed NAFSA on triaxial accelerometer error parameters identification could implement the SINS navigation process rapidly and accurately. Moreover, the NAFSA-identified triaxial accelerometer error parameters have better environment adaptive ability, which means higher positioning accuracy and better tracking performance. Therefore, the proposed NAFSA has better ability than the conventional calibration method and the OAFSA in triaxial accelerometer error parameters identification applications.
However, the AFSA on triaxial accelerometer error parameters identification is only in exploration phase and all the navigation experiments are based on the stored data. So our work for next stage is to realize the algorithm in real-time navigation.
The authors declare that there is no conflict of interests regarding the publication of this paper.
Lianwu Guan is sponsored by the China Scholarship Council (CSC) for his joint Ph.D. research training program at the Queen’s University, Kingston, ON, Canada. This research work is supported by the Ministry of Science and Technology of China (no. 2014DFR10010) and sponsored by the National Natural Science Foundation of China (51309059). The authors also would like to thank Dr. Jonathan Wylie for helping to edit the language of the paper and anonymous reviewers for improving this paper.