An effective method is proposed to estimate the parameters of a dynamic grain flow model (DGFM). To this end, an improved artificial bee colony (IABC) algorithm is used to estimate unknown parameters of DGFM with minimizing a given objective function. A comparative study of the performance of the IABC algorithm and the other ABC variants on several benchmark functions is carried out, and the results present a significant improvement in performance over the other ABC variants. The practical application performance of the IABC is compared to that of the nonlinear least squares (NLS), particle swarm optimization (PSO), and genetic algorithm (GA). The compared results demonstrate that IABC algorithm is more accurate and effective for the parameter estimation of DGFM than the other algorithms.
Grain flow sensor is an important part of yield monitoring system in precision agriculture. There are mainly four types of grain flow sensor: impacttype [
Work schematic of impact grain flow sensor.
Grain flow model is used to characterize the relationship between the grain flow and the impact force, which contains linear models and nonlinear models. Linear models have been utilized due to being easy for model building [
Recently, many artificial intelligent techniques have emerged as useful tool to solve complex problems, and these techniques have been utilized to parameter estimation of nonlinear models. In [
ABC has been proven to be a powerful and efficient tool for realworld optimization problems [
The paper is organized as follows. In Section
As mentioned above, it is a trend to model grain flow using nonlinear models. DGFM is proposed to give an accurate estimation of grain flow [
Total grain mass on a paddle is obtained as follows:
Horizontal velocity and vertical velocity at the end of the release stage are given by
The impact force caused by grain can be expressed as
DGFM is represented by (
In (
As discussed above, the DGFM consists of four unknown parameters. Unknown parameter estimation problem can be transformed into an optimization problem by minimizing a root mean square error (RMSE). For this purpose, the impact force is measured and computed by grain flow sensor and DGFM, respectively. Unknown parameters are estimated by the proposed objective function defined as follows:
The ABC algorithm simulating the foraging process of honey bees was proposed to solve complex optimization problems [
If the current food position is
Each onlooker bee chooses and follows the employed bees based on the quality of food sources. The quality of food sources denoted as
If a food source fails to get better after predetermined value “limit,” then that food source needs to be abandoned. The employed bees become scouts later. A new food source is randomly generated in space using (
This section presents the detailed description of the proposed improved artificial bee colony (IABC) algorithm. There are two improvements on basic ABC algorithm: modifying the search method of the onlooker bees and proposing a novel probability model.
In ABC, production of new food source (see in (
In IABC, the search method of MABC is modified to further improve the exploitation. The new method used to calculate a new food source is shown in
Compared with (
It is obvious that (
In (
From the above analysis, two equations are modified ((
A set of six benchmark functions is applied to verify the performance of IABC in the simulation experiments. The benchmark functions are summarized in Table
Detail for the 6 benchmark functions used in experiments.
Functions  Dimension  Search range  Optimum value 

Rosenbrock  2 and 3 

0 
Schaffer  2 and 3 

0 
Rastrigin  30 and 60 

0 
Griewank  30 and 60 

0 
Sphere  30 and 60 

0 
Ackley  30 and 60 

0 
Three ABC variants, GABC [
Effect of
Maximum number of generations  Dimension 

Functions  

Rastrigin  Griewank  Sphere  Ackley  
1000  30  100  Mean  0 



Std  0 




1000  30  200  Mean  0 



Std  0 




1000  30  300  Mean  0 



Std  0 




1000  30  400  Mean 




Std 




Given the above, the population size, the maximum number of generations, and the
The final results comparison of IABC, MABC, GABC, EABC, and ABC.
Functions  Dimension  ABC  GABC  MABC  EABC  IABC  

Mean  Std  Mean  Std  Mean  Std  Mean  Std  Mean  Std  
Rosenbrock  2 










3 











Schaffer  2 










3 











Rastrigin  30 










60 











Griewank  30 










60 











Sphere  30 










60 











Ackley  30 










60 










Table
Figure
Convergence performance of ABC, GABC, MABC, and IABC on six benchmark functions.
To make it more accurate for DGFM to calculate grain flow, it is very necessary to use a more efficient optimization algorithm to solve parameters estimation problem of DGFM. In view of the excellent performance of the proposed IABC algorithm in this paper, the IABC algorithm is used for parameters estimation of DGFM. A schematic diagram of the approach is illustrated in Figure
Schematic diagram of parameter estimation of DGFM.
Impact force and grain flow signals were obtained from a test rig illustrated in Figure
Machine geometric parameters of test rig.










0.176  0.151  0.074  0.367  0.326  0.021  70 
Schematic of test rig for DGFM.
To estimate the theoretical parameters of DGFM, the population size, the maximum number of generations, and the
To verify the superiority of IABC algorithm in parameter estimation of DGFM, the IABC algorithm is compared with nonlinear least squares (NLS), particle swarm optimization (PSO), and genetic algorithm (GA). In all experiments in this section, the population size and the maximum number of generations used in each algorithm were chosen to be the same as the IABC algorithm, respectively. The other specific parameters of algorithms are given below.
For PSO algorithm, the learning rates
Table
Comparison of estimated performance for NLS, PSO, GA, and IABC.
Sprocket rotational speed (r/min)  Parameters  Search range  Algorithm  

NLS  PSO  GA  IABC  
320 

0~1  0.186  0.213  0.192  0.216 

0~1  0.054  0.069  0.059  0.068  

750~2000  1081.9  1123.6  1098.3  1123.3  

0~5  1.984  2.112  2.031  2.118  
RMSE  0.013 




Std  0.032 






360 

0~1  0.181  0.214  0.195  0.219 

0~1  0.056  0.066  0.058  0.067  

750~2000  1081.1  1123.8  1099.9  1123.5.  

0~5  1.989  2.114  2.036  2.119  
RMSE  0.012 




Std  0.009 






400 

0~1  0.187  0.216  0.193  0.212 

0~1  0.057  0.067  0.053  0.065  

750~2000  1081.6  1123.1  1099.5  11.23.4  

0~5  1.981  2.111  2.038  2.117  
RMSE  0.014 




Std  0.065 



To make a better comparison between IABC and PSO, search range of the theoretical parameters was enlarged. The comparison results are shown in Table
Comparison of estimated performance for IABC and PSO.
Algorithm  Parameters  Search range  Sprocket rotational speed (r/min)  

320  360  400  
IABC 

−1~2 




−1~2 


 

100~3000 


 

−5~10 


 
RMSE 




Std 






PSO 

−1~2 




−1~2 


 

100~3000 


 

−5~10 


 
RMSE 




Std 



Comparison of convergence curves of IABC and PSO at sprocket rotational speed of 360 r/min.
This paper presented a method of estimating parameters of DGFM using IABC algorithm. In the proposed algorithm, the research method of original ABC was modified to improve its convergence rate. But the research method resulted in diversity loss, so a probability model was proposed to maintain diversity of population. With testing against a set of six benchmark functions, the IABC algorithm had a better performance than the other ABC variants. Then, the proposed algorithm was used to estimate parameters of DGFM. By comparing IABC algorithm with NLS algorithm, GA algorithm, and PSO algorithm, the proposed algorithm had higher accuracy solutions and a faster convergence rate, which is more suitable for parameter estimation of DGFM.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This project is fully supported by the Scientific Research Foundation of the Education Department of Liaoning Province, China (no. 2017LNQN22); the Young Teachers Foundation of University of Science and Technology Liaoning, China (no. 2017QN04); and the Nature Science Foundation of Liaoning Province, China (no. 2015020128).