A fast and global search method for grasping pose optimization in manufacturing

Abstract

To use the electromagnetic chuck to precisely absorb industrial parts in manufacturing, this paper presents a hybrid algorithm for grasping pose optimization, especially for the part with a large surface area and irregular shape. The hybrid algorithm is based on the Gaussian distribution sampling and the hybrid particle swarm optimization (PSO). The Gaussian distribution sampling based on the geometric center point is used to initialize the population, and the dynamic Alpha-stable mutation enhances the global optimization capability of the hybrid algorithm. Compared with other algorithms, the experimental results show that ours achieves the best results on the dataset presented in this work. Moreover, the time cost of the hybrid algorithm is near a fifth of the conventional PSO in the discovery of optimal grasping pose. In summary, the proposed algorithm satisfies the real-time requirements in industrial production and still has the highest success rate, which has been deployed on the actual production line of SANY Group.

Keywords

Particle swarm optimization Gaussian distribution alpha-stable distribution grasping pose

1 Introduction

Now industrial 4.0 is widely used in China and the global market size will reach to 214 billion USD in 2023. The reform of production line is one of the important contents in industrial 4.0 [1, 2]. This work focuses on the reform of the manufacture of heavy vehicles. At present, most steel parts in heavy industry are absorbed by the splicing electromagnetic chuck. However, many chucks are still carefully operated by hand [3, 4]. Especially for the part with an irregular shape, it is difficult to find the optimal grasping pose which is Pareto Optimality. The improper grasping pose may lead to the absorption of non-target parts, which has a serious impact on the downstream production. Moreover, the shape of the part is changeable and the area of the part is between 0.5 and 10 square meters, which brings more difficulties to calculate the optimal grasping pose.

In the manufacturing of heavy vehicles, steel parts with different shapes are cut on the whole homogeneous plate by the high-energy laser [5, 6]. Then the part embedded in the steel plate is absorbed by the electromagnetic chuck. After the part is placed in the placement area, it will be transported to the next scene by the plate chain for further processing. Figure 1 shows the real scene of part grabbing in the production of heavy vehicles. The upper area of the figure is a 45-degree view. The lower part of the figure is a top view.

Fig. 1

The scene of part grabbing in heavy vehicles production.

The methods of the pose search can be divided into traditional fixed point search and meta-heuristic search [7]. In the traditional method, some potential points on the surface of the part are selected. Then, the center point on the surface of the chuck coincides with the selected point on the part. After that, the part is fixed, and the electromagnetic chuck will be rotated at a certain angle in a limited range. The coverage area between the magnetic block on the chuck and the part can be calculated so as to determine the opening or closing of the magnetic block. Further, the overall magnetic force which comes from the opened magnetic blocks will be known. Therefore, the whole process can be summarized as the search for the optimal grasping pose, which includes the grasping point on the part and the rotated angle of the electromagnetic chuck. The pose search of certain fixed points can meet the low time cost, but the success rate of grasping is not high. Traditional meta-heuristic algorithms such as GA, PSO, etc [8 –12] search the entire pose space by simulating natural phenomena. These algorithms have improved the grasping success rate in pose search, however, the time cost is too large.

The optimal pose maximizes the coverage area of parts and electromagnetic chuck, which also maximizes the magnetic force. The detailed definition of the optimal pose can be found in Section 3.1. The search for the optimal pose search is time-consuming, but many industrial scenarios put forward stringent restrictions on the time cost and success rate for the pose search. The pose search of certain fixed points can meet the low time cost, but the success rate of grasping is not high. Therefore, it is necessary to find a method that satisfies the time cost and still has a higher grasping success rate.

Based on the above requirements, this work proposed a new hybrid PSO algorithm. The hybrid PSO algorithm which combines the Alpha-stable distribution mutation and dynamic parameters is used to find the optimal pose iteratively. In addition, the Gaussian distribution sampling is used to generate the initial coordinates of the population, and the mean value of the distribution is the coordinate of the geometric center point on the part.

The main contributions of this work include:

To speed up the process of search, the Gaussian distribution sampling based on the geometric center point was used to generate the initial population in PSO. The experiments confirmed that the initialization is convenient to accelerate convergence.

Aiming at the requirement of a high success rate in the scene of industrial production, a hybrid PSO algorithm was used. With the PSO algorithm is easy to fall into the local optimal solution, this paper introduced the Alpha-stable distribution and the SIGMOD function to the mutation in PSO, which has a balance on the exploitation and exploration tendency. The experiments showed that the method achieves the best result.

This work collated the real production data of more than 1,000 kinds of the part in heavy industry. Moreover, the simulation data which shape is randomly generated was used to verify the universality of the hybrid algorithm. Finally, a total of 2,000 kinds of simulated data and more than 1,000 kinds of real data were used in the validation.

The remainder of the paper is organized as follows: Section 2 presents a brief overview of the related algorithms. Section 3 presents the description of the problem and the hybrid algorithm. Section 4 presents the experimental process and results. Section 5 concludes the paper with a summary of the contributions made and future work.

2 Related work

The search problem of the optimal grasping pose on the part in heavy industry can be approximated to the problem of maximum area coverage [13 –16]. This is a classic NP-hard problem involving computer science, computational complexity theory, and operations research. The problem of maximum area coverage has always attracted the study of many scholars. It has also been widely used in industrial applications, especially in the deployment of wireless network sensors and the planning of plant sites.

Meta-heuristic is an advanced program or heuristic algorithm in computer science and mathematical optimization [17, 18]. It is used to search, generate or select a heuristic result. And this result can provide a good solution for an optimization problem, especially for the situation where the information of the problem is incompleted or the computing power is limited. While the Particle Swarm Optimization (PSO) is a very powerful and simple optimization tool in the field of meta-heuristic [8 –10].

The PSO is originally an evolutionary computing technology developed by J. Kennedy and RC Eberhart et al in 1995 [19]. It was derived from the simulation of a simple social model. Through the observation of the social behavior of animals, it was found that the social sharing of information in the group provides an evolutionary trend, which was used as the basis for the development of the PSO algorithm. The initial version of PSO was formed by adding speed matching of neighbors, multi-dimensional search, and acceleration based on distance. Later, the inertial weight w was introduced to better control the exploitation and exploration trend, which formed a standard version [20, 21].

As an important optimization tool, the PSO algorithm has been successfully used in system identification, neural network training, and other fields [22 –25]. Compared with other meta-heuristic algorithms, the PSO has the advantage of quick convergence, especially suitable for the real-time requirement of the industrial field [26, 27]. Another widely used optimization algorithm is the genetic algorithm (GA). The genetic algorithm [11, 12] simulates the process of DNA evolution which includes coding, selection, crossover, and mutation, so as to gradually find the optimal solution. However, the disadvantage of the genetic algorithm is that the convergence time is too long.

Mihaela Cardei and David MacCallum defined the optimization of the largest disjoint coverage area in wireless networks in 2002 [8], and they introduced the PSO algorithm to optimize the solution. The theoretical analysis and experimental results proved that the meta-heuristic algorithm is more cost-effective than the rule-based approaches. Although the PSO algorithm provides the possibility of global search, it does not guarantee convergence to the global optimum. For a function with multiple local extreme points, it is easy to fall into the local extreme point, and the correct result may not be obtained. In the next few years, to improve the ability of the PSO algorithm to jump out of local extreme points, Hou et al proposed an improved PSO algorithm in 2004 [28]. The group diversity was enhanced by introducing a mutation operator in the genetic algorithm. The mutation probability of the current optimal particle was determined by the variance of the population’s fitness and the fitness value of the current optimal solution during operation. The mutation operator enhances the ability of the PSO algorithm to jump out of the local optimal solution. The experiments also showed that this algorithm is better than the conventional PSO in finding the optimal value of the function.

One of the emerging trends of recent years is to integrate the Lévy flight with PSO. Recently, H. Hakli and H. Uguz [29] combined the Lévy flight and mutation in the PSO algorithm to make some particles in the swarm have the ability to jump long distances. The experimental results showed that the Lévy mutation prevents the PSO algorithm from falling into convergence prematurely. At the same time, quite a few studies had shown that the movement patterns of many animals can be described by the Lévy flight. The Lévy flight is a non-Gaussian randomization process, and its random step size comes from the Lévy stable distribution [30, 31]. The Lévy stable distribution is a special case of the alpha-stable distribution [32, 33]. By adjusting the stability parameter α and the polarization parameter β in the alpha-stable distribution, the distribution can be changed. So the probability of a large stride generated can also be adjusted during mutation. Wu Deng and Huimin Zhao [34] used the alpha-stable distribution to generate random step size, which made the particles more capable of breaking through the local optimum. Compared with the previous method, they achieved the best result at the time. However, the stability parameter α and the polarization parameter β are fixed in their alpha-stable distribution. It is well known that the exploration trend should be increased in the early stage of the iteration for the group optimization algorithm, which means that the random step size should be increased during mutation and the algorithm tends to search globally. In the later stages of the iteration, the exploitation trend should be increased, which means that the random step size should be reduced during mutation, and the algorithm tends to local search. Based on the above understanding, further work by Shixin Cheng [35] dynamically set the parameters in the alpha-stable distribution and achieved better results. However, this method has not been tested in real scenes, and its practicability needs to be further tested.

Population initialization is another important step in the population-based optimization algorithm, which has an important influence on the speed of convergence and the quality of the result [36, 37]. For example, Hajira Jabeen [38] proposed an improved method of initial population generation which is based on opposite particle. The experimental results on multiple problems showed that the population initialization based on this strategy is better than the traditional random initialization. In addition, Borhan Kazimipour [7] studied the impact of various population initialization techniques on optimization performance. This research showed that initialization plays a more important role in evolutionary algorithms especially for the problem with medium or low dimensions. Generally, if the prior information exists, it can be used to generate the initial population. Otherwise, the random initialization is usually used.

This paper uses the hybrid PSO algorithm which combines the α-stable mutation with dynamic change in parameters. The parameters in α-stable mutation are dynamically changed, and the parameters α and β are changed through the sigmoid function. At the same time, to further speed up the discovery of optimal grasping pose, the Gaussian distribution sampling based on the geometric center point of the part is used to initialize the population in PSO. Compared with other current typical algorithms, it is shown that the algorithm proposed in this work has stronger performance in both the optimization score and the convergence speed, which also indirectly shows that the algorithm has better practicability.

3 The hybrid algorithm for grasping pose optimization

3.1 Problem statement

The framework of the production system is shown in Fig. 2. Firstly, the parts with different shape in the plate is used as the input of the algorithm. Then, it is needed to output a grasping pose for the corresponding part after analysis. Finally, The grasping pose will be sent to the robot by PLC, and the robot will move the electromagnetic chuck to the corresponding coordinate and angle. As shown in Fig. 11, it is necessary to maximize the coverage area between the electromagnetic chuck and the part, so as to maximize the electromagnetic suction. In this problem, the solution represents the grasping pose which is denoted as a vector [x, y, angle].

Fig. 2

The framework of the production system.

Objective function: Equation 1 gives the definition of the objective function. To maximize the electromagnetic suction, the objective function in search is set to maximize the ratio of force generated by electromagnetic chuck at the specified pose to the gravity of the part. $\begin{matrix} max : score = \sum_{i = 1}^{N} I_{i} \cdot (\frac{S_{r} \cdot F_{i}}{S_{a} \cdot G}) \end{matrix}$ (1)

$s . t . : I_{i} = {\begin{matrix} 1 & if \frac{S_{r}}{S_{a}} = 1 \\ 0 & otherwise \end{matrix}$ (2) Here N represents the number of magnetic blocks in the electromagnetic chuck. I_i is the indication function for the ith magnetic block in the chuck. F_i is the rated force of the corresponding magnetic block, and the force will be generated only when the magnetic block is opened. G represents the gravity of the part. The final score is the ratio of force generated by the chuck to the gravity of the part. The higher the score, the better the search result.

Related constraints: Equation 2 gives the related constraints of the objective function. When the magnetic block is opened, I_i is 1. Otherwise, the magnetic block is closed and I_i is 0. S_r is the coverage area between the magnetic block and the part. S_a is the area of the corresponding magnetic block. It is agreed that the corresponding magnetic block will be opened, only when the magnetic block is fully covered by the part to prevent the non-target part is sucked up.

The next subsections introduce a hybrid search algorithm of grasping pose optimization. This algorithm makes it possible for heavy parts to be grasped safely and quickly. Several specific aspects of the hybrid algorithm, including the Gaussian distribution sampling based on the geometric center point, Alpha-stable distribution mutation and SIGMOD function, are presented in the following.

3.2 The Gaussian distribution sampling based on the geometric center point

In the conventional PSO, the initial population is randomly generated, which leads to the PSO cost too much time in finding a better individual. From the experimental results, most of the optimal grasping points are concentrated near the geometric center point, especially for the part with regular shape. For example, the workpiece in the upper left area of Fig. 5. If the angle of the chuck is 0, the relationship between x, y and objective function near the geometric center point area can be drawn in Fig. 3. It is clear that there is an obvious peak value of objective function near the geometric center point. So the x and y of the peak can be recorded. Then, through increasing the angle by 10 degrees at each time, the x and y of new peak can be recorded again. The angle is between 0 to degree 360 and all the peak coordinates are drawn in Fig. 4. It can be seen that a large number of peak coordinates are gathered near the geometric center point of the part.

Fig. 3

The relationship between x, y and objective function near the geometric center point area.

Fig. 4

The peak coordinates with only the angle change

Fig. 5

The examples of four representative shapes. Except for the part in the upper left area, other parts are irregular in shape, which bring more difficulties to the search.

So the Gaussian distribution sampling which mean is the coordinate of the geometric center is used to generate the initial population. In addition, if starting from the geometric center point and then spreading around the part, the search also has a good symmetry advantage in space, which will be good for reducing the convergence time. The subsequent ablation study also proved the advantages of the sampling.

3.3 The hybrid PSO algorithm

The PSO algorithm is based on population and fitness function. Each individual in the population corresponds to a possible solution of the problem, and contains two attributes which are position and velocity. The fitness value of individual position is the value of the objective function, and represents the quality of a possible solution. The PSO algorithm will gradually find the optimal solution by updating individuals. In iteration, each individual updates itself by tracking two extrema. One is the individual extremum which is called pbest and found by the particle itself. Another is the global extreme value which is called gbest and found by all particles in the population. The following two equations are used for each update: $\begin{matrix} v = & w * v + & c 1 * rand () * (pbest - position) + \\ c 2 * rand () * (gbest - position) \end{matrix}$ (3)

$\begin{matrix} position = position + v \end{matrix}$ (4) v represents the update speed of the individual. position represents the position of individual and w is the inertial weight. The first term w * v is called the part of inertia keeping. c1 and c2 are learning factors. The rand () represents a random number between 0 to 1. The second term in Equation (3) represents the part of self-cognition in individual updating, and the third term represents the part of group cognition.

Equations (3) and (4) respectively represent the velocity update and position update of the individual. It can be learnt from the formulas that the individuals in the population will approach to the individuals with higher fitness value.

Algorithm 1 shows the pseudo-code of the hybrid PSO search. The input parameters include the number of particles in the population P, N; the number of iterations, M; and the position with the largest target value that each particle can find, namely pbest; the position with the largest target value in the population, namely gbest. The final output is gbest.

Algorithm 1 The pseudo-code of the hybrid PSO.
Input: The population P with size N; The number of iteration M; The position with the largest target value in population, gbest; the position with the largest target value about each particle, pbest;
Output:gbest;
1: The gaussian distribution sampling based on geometric center point.
2: for each i ∈ [1, N] do
3: [x, y] = GaussianSampling () ; // the variance is one
4: angle = random () ;
5: pbest [i] = [x, y, angle] ;
6: end for
7: gbest = the maximum target value in pbest;
8:
9: for each j ∈ [1, M] do
10: P = Alpha-StableMutation() // preform alpha stable
mutation
11: for each i ∈ [1, N] do
12: Update the position of particle i by Equations (3)(4);
13: Calculate the target value under the new position by Equations (1)(2);
14: if the target value > pbest[i] then
15: // update pbest
16: pbest[i] = the target value
17: end if
18: end for
19: if the maximum value in pbest > gbestthen
20: // update gbest
21: gbest = the maximum target value in pbest
22: end if
23: end for
24: The gbest is output.

In the first step, the coordinate of the particle in population is generated by the Gaussian distribution sampling and the values of pbest and gbest are recorded. In consideration of the genetic algorithm is prone to jump out of local optimum, so this paper introduces α-stable mutation into PSO. Generally, the performance of PSO heavily depends on the balance between exploration and exploitation. A common belief is that PSO should start with exploration (global search) and then gradually step into exploitation (local search). So the dynamic parameter should also be used in PSO. The detailed pseudo-code of the dynamic mutation operation can be found in Algorithm 2.

Algorithm 2α-StableMutation
Input: The population P with size N; mr (Mutation rate); The position with the largest target value in population, gbest; α (the stable parameter); β (the skewness parameter)
Output: The final population, P′
1: if rand() < mrthen 2: // The mutation is Performed
3: for each i ∈ [1, N] do R=rand()
4: if R <0.4 then
5: $P_{i}^{'}$ =P_i+(gbest-P_i)*α-StableRand(α,β,1,0)
6: end if R >0.6 then
7: $P_{i}^{'}$ =gbest+(P_i-gbest)*α-StableRand(α,β,1,0)
8: end else
9: md=0.5(P_i+gbest*)
10: $P_{i}^{'}$ =md+abs(gbest-P_i)*α-StableRand(α,β,1,0)
11: end if
12: end for
13: end if

Then the Equations (3) (4) (1) (2) are repeated to update and calculate for each particle, and whether to update the corresponding pbest and gbest are determined. Finally, gbest is output.

3.3.1 Alpha-stable distribution mutation

Alpha-stable distribution is determined by four parameters which are α, β, γ and δ. α is the characteristic exponent in the range of 0-2 which describes the tail of the distribution. The smaller value of α, the more samples deviate from the median. With the increase of α, the tail of the distribution will become shallower and the impact strength of the distribution will be decreased. So the number of sample which deviates from the median will also be decreased.

β is the skewness parameter which determines the degree of symmetry about the distribution. When δ is 0, the distribution is symmetric which is called Symmetric α-Stable (SaS) distribution. If δ > 0 the distribution is right-skewed or left-skewed (δ < 0). γ is a scale parameter, which is a measure of the deviation of a distribution sample from its mean. The meaning of γ is similar to the variance in the Gaussian distribution. δ is the Location parameter.

Alpha-stable distribution can be evolved into three different distributions which are the Gaussian distribution, the Cauchy distribution, and the Lévy distribution by adjusting α and β. Through the change of α and β, this paper used a dynamic mutation operator based on Alpha-stable distribution.

The mutation operators are classified into three types. When mutation operation is performed, the details of three types of mutation are formulated from line 5, line 7, and line 9 in Algorithm 2. A random number R between 0-1 is generated by the roulette algorithm. The first operator has a 40% probability of mutation centered on the current individual, The second operator has also a 40% probability of mutation centered on the optimal particle gbest, which is used to guide the individuals of swarm to move in particle swarm optimization. The third operator has a 20% probability of mutation centered on the median of P_i and gbest.

In the initial stage of the search, smaller α and larger β should be used to generate random numbers which have stronger randomness. In the later stage of the search, it is needed to increase α and decrease β which makes the α-stable distribution closes to Gaussian distribution and the generated random numbers are relatively concentrated. Therefore, the sigmoid function is used to dynamically set α and β in the work.

3.3.2 Dynamic parameter settings based on sigmod function

The improved SIGMOD function is shown in Equation 5. y stands for α or β, and x is the number of iterations. L_i and L_f is the initial value and final value of y. x₀ = r₁ * maxIter, maxIter is the maximum number of iterations and r₁ is a constant; while k= r₂*(L_f-L_i)/maxIter, and r₂ is also a constant.

$\begin{matrix} y (x) = \frac{\max (L_{i}, L_{f}) - \min (L_{i}, L_{f})}{1 + \exp (- k (x - x_{0}))} + \min (L_{i}, L_{f}) \end{matrix}$ (5)

It is very difficult to harmonize r₁ and r₂ with respect to α, β to obtain the best optimization results. Therefore, a simple method is adopted to obtain satisfactory settings. Firstly, two parameters (α, β) are sorted by their contribution to the results, and the understanding of their contribution comes from empirical testing; Then, the two coefficients (r₁, r₂) of α are linearly adjusted and the best settings are recorded; finally, the two coefficients (r₁, r₂) of β are independent adjusted and the best settings are recorded again. The relevant settings are recorded in Table 1.

Table 1

The best settings of α and β

parameter	L _i	L _f	r ₁	r ₂
α	0.07	2	0.5	15
β	0.5	0.1	0.4	30

The curve of α and β are shown in Fig. 6. It is clear that α is small in the early stage of the search. So the variables generated by alpha-stable distribution have stronger randomness. The new individuals generated by mutation operator are far away from the central positions, which prevents the premature convergence in PSO. After that, α is changed to a larger value and finally is closed to 2. The randomness of the variables generated by alpha-stable distribution is weakened, which makes a subtle difference between the new individuals generated by mutation operator and the original individuals. The PSO algorithm is similar to local search in the later evolution period.

Fig. 6

The curve of α and β.

3.4 Overview

The overall flowchart of the hybrid algorithm for solving the optimal pose in the scene of part grabbing is shown in Fig. 7. Firstly, the Gaussian distribution sampling which is based on the geometric center point is used to generate the initial particles in PSO. Then, a dynamic mutation operator is added to the search of the PSO algorithm, and the parameters in the mutation operator are generated by using the alpha-stable distribution and the sigmoid function. Finally, the gbest which presents the optimal particle in population is output as the optimal grasping pose.

Fig. 7

The overall flow chart of the hybrid algorithm.

4 Experiment

4.1 Dataset and implementation details

The simulated dataset contains 2,000 kinds of parts which shape are randomly generated by assigning different sizes to common shapes such as squares, rectangles, triangles, diamonds, and circles. The use of simulated data helps to test the generalization of the algorithm. The real dataset is more than 1,000 kinds of parts and comes from the No.18 intelligent Workshop which belongs to the SANY Heavy Industry Group in Changsha. More data is available at https://github.com/tiantianhuanle/part-grabbing. The scale of the intelligent Workshop ranks the second in the world [39].

The examples of representative real parts are shown in Fig. 5. Different search methods are compared in the simulated dataset and real dataset. Moreover, four types of splicing electromagnetic chuck are designed so as to verify the influence of different shapes of the electromagnetic chuck. These chucks can be found in Fig. 2. The experiment results are shown in the following.

The experiment uses Python 3 for programming, and the CPU of the hardware platform is Intel Core i7 8565u@1.8GHz. The memory size is 16G and there is no GPU acceleration.

4.2 Ablation study

To further explore the respective roles of the Gaussian distribution sampling based on geometric center point and dynamic alpha-stable mutation in PSO search, the following ablation experiments are designed in this section.

4.2.1 The Gaussian distribution sampling

For the Gaussian distribution sampling, this section presents the average time cost of the hybrid algorithm from the beginning to the convergence under different variances and mean values. The convergence is defined as the optimal particle no longer changes in the last 5 iterations. As shown in Fig. 8, no matter how the variance δ changes, the average time cost for the convergence is the shortest when the mean of Gaussian distribution is the geometric center point (x, y). This fully illustrates the effectiveness of the Gaussian distribution sampling based on geometric center point for accelerating convergence. Under the comparative performance of five different variances, the hybrid algorithm achieves the best results when the variance is 1.0. So this is the reason for the variance of 1.0 is recommended in Algorithm 1.

Fig. 8

The average time cost of the hybrid algorithm from the beginning to the convergence under different variances and means. The algorithm is ran on all the data.

4.2.2 The dynamic mutation

To verify the usefulness of the dynamic mutation operator, the following four strategies are compared, which are the fixed α and β, only α changed, only β changed, and the both changed. The average score of these strategies are calculated on all datasets. The results are shown in Fig. 9. It can be seen that the highest average score is achieved when the α and β are both changed, while the fixed α and β is the worst. This shows that dynamic mutation is helpful to the optimization process. The average score of the strategy only α changed is higher than the strategy only β changed, which shows that the α plays a more important role in the dynamic mutation.

Fig. 9

The performances of four strategies on α-StableMutation.

4.3 Effect of different chucks

To get a good combined shape of splicing electromagnetic chuck, this work compares the performance of several different shapes of the chuck presented in Fig. 2. Table 2 shows the test results of different shapes of splicing electromagnetic chuck on the simulated dataset and real dataset. All chucks are run on the hybrid PSO algorithm for comparison.

Table 2
The test results of different shapes of splicing electromagnetic chuck on the simulated dataset and real dataset

results chuck 1 chuck 2 chuck 3 chuck 4

average score (real dataset) 3.8573 3.6848 3.3214 3.5471

average score (simulated dataset) 5.7470 5.5392 5.2608 5.6725

success rate 98.37% 97.86% 91.25% 94.41%

results	chuck 1	chuck 2	chuck 3	chuck 4
average score (real dataset)	3.8573	3.6848	3.3214	3.5471
average score (simulated dataset)	5.7470	5.5392	5.2608	5.6725
success rate	98.37%	97.86%	91.25%	94.41%

The above two lines in Table 2 show the average score on different datasets. The final line in the table is the grasping success rate. It can be learnt from Table 2 that the performance of the electromagnetic chuck 1 is better than the electromagnetic chuck 2, 3 and 4 in both the simulated dataset and real dataset. The magnetic blocks in electromagnetic chuck 1 are all circular, while the magnetic blocks in electromagnetic chuck 2, 3 and 4 are square or strip-shaped, which shows that the use of circular block in the grabbing scene may have more advantages than the square block.

Compared with the electromagnetic chuck 3, the long strip block is reduced in the electromagnetic chuck 2 and the better results are available, which might mean that it is not suitable to use the long strip block in the scene. Compared with the electromagnetic chuck 3, the chuck 4 uses the square magnetic block with larger area and the better results are obtained. So the compared results reveal that it is maybe appropriate to use the large area block on the edge of the electromagnetic chuck.

4.4 Comparison with other methods

Several groups of comparative experiments on the simulated dataset and real dataset show the effectiveness of ours hybrid algorithm proposed in this paper. These algorithms in the experiment are the LMPSO in [35], the conventional PSO, the conventional GA, and the hybrid algorithm. The conventional PSO and conventional GA mean that the algorithm is used to search separately. The hybrid algorithm includes the Gaussian distribution sampling in subsection 3.2 and the hybrid PSO algorithm in subsection 3.3. The electromagnetic chuck used in the comparative experiments of different algorithms is the chuck 1 presented in Fig. 2, which has the best performance. Table 3 shows the input parameters of different algorithms. Therefore, all the following results are obtained under these settings.

Table 3
Hyper parameters of different algorithms

Algorithms Parameters value

PSO Population 500

Number of iterations 30

GA Population 500

Number of iterations 30

Mutation rate 1/5

LMPSO Population 500

Number of iterations 30

Mutation rate 1/5

α 3/2

β 3/10

Ours Population 500

Number of iterations 30

Mutation rate 1/5

delta 1

Algorithms	Parameters	value
PSO	Population	500
	Number of iterations	30
GA	Population	500
	Number of iterations	30
	Mutation rate	1/5
LMPSO	Population	500
	Number of iterations	30
	Mutation rate	1/5
	α	3/2
	β	3/10
Ours	Population	500
	Number of iterations	30
	Mutation rate	1/5
	delta	1

Figure 10 shows the average score of different methods in real dataset (left columns) and simulated dataset (right columns). The score is the ratio of magnetic force to gravity. The larger the score, the better the algorithm.

Fig. 10

The average score of different methods in real dataset and simulated dataset.

It can be seen from Fig. 10 that the hybrid algorithm has the highest average score of 5.747 in the simulated dataset, and its magnetic force is the largest, which also indicates the hybrid algorithm has the best ability to find the optimal grasping pose. The second best is the LMPSO algorithm and its score is 5.585. The average score of conventional GA is 5.38. The worst is the conventional PSO and its average score is 4.368. Howerver, considering the complexity and variety of the real parts, the average scores of all algorithms is reduced for the real dataset. Among them, the conventional PSO has the worst ability to find the grasping pose, and the average score is 2.9110. The average score of the conventional GA algorithm is 3.5772. The average score of the LMPSO algorithm is 3.6359. The average score of the hybrid search is 3.8573.

Most parts have a regular shape in simulation dataset while many parts are irregular shapes in the real dataset. As mentioned above, for the parts with regular shape, the optimal grasping poses are mostly concentrated near the geometric center point of the part, which contributes to the discovery of optimal pose. But for the parts with the irregular shape, the distribution of optimal grasping poses is more random, which brings more difficulty to search the optimal pose.

Compared with the conventional PSO algorithm and GA algorithm, the global optimization capability of the hybrid algorithm has been greatly enhanced due to the introduction of dynamic alpha-stable mutation. The best performance of the hybrid algorithm on the simulated dataset and real dataset also reflects the situation. At the same time, compared with the LMPSO algorithm, the hybrid algorithm can also achieve better performance on the average score, which also indirectly shows the effectiveness of the Gaussian distribution sampling based on the geometric center point.

Figure 11 shows the optimal grasping poses about the part with the irregular shape. These poses are given by different algorithms on the electromagnetic chuck 1 and the parts come from the real dataset. The Fig. 11(a) shows the optimal grasping poses given by conventional PSO. The grasping pose in in Fig. 11(b) is given by LMPSO. The Fig. 11(c) and Fig. 11(d) show the grasping poses found by the hybrid algorithm.

Fig. 11

The example of the optimal grasping poses given by the different algorithms.

It is shown on Fig. 11(a) that only 4 small circular magnetic blocks completely covered the part under the optimal pose given by the conventional PSO algorithm. However, there are 6 small circular magnetic blocks completely covered the part under the optimal pose given by the hybrid algorithm. The same situation is shown in Fig. 11(b) and Fig. 11(d). There are 11 small circular magnetic blocks completely covered the part in the optimal pose given by the LMPSO algorithm. There are 10 circular magnetic blocks completely covered the part, but one magnetic block has a larger size. In general, the coverage area between the part and the electromagnetic chuck is larger and the suction force is also larger under the optimal pose given by the hybrid algorithm. Under these conditions, it means that the hybrid algorithm can find better grasping poses for the part with an irregular shape which is difficult to grasp.

To verify the grasping ability of these algorithms, the four algorithms were deployed on the real production line. Figure 12 shows the grasping success rate of four different algorithms for more than 1,000 kinds of parts on real production. It can be found that the hybrid algorithm still has the best success rate than other algorithms. But even under the optimal pose given by the hybrid algorithm, A small number of parts still cannot be grasped successfully, which shows that there may be some other physical indicators that also need to be considered in the grasping scene.

Fig. 12

The grasping success rate of different methods on real dataset.

4.5 Computational efficiency

The search based on population requires double cycles, which means that m iterations are performed on the population with n individuals. So the time complexity is O(n*m). The conventional PSO only includes one update of the particle position in each iteration, while each iteration of GA includes three operations of selection, crossover, and mutation. Therefore, the time cost of GA is higher than that of the conventional PSO algorithm. LMPSO adds the mutation operation on the basis of PSO, which has stronger global optimization ability and can converge faster than the conventional PSO. The operation of hybrid algorithm is similar to LMPSO. However, due to the special initialization and dynamic mutation operation, the hybrid algorithm has more powerful optimization ability and faster convergence than the LMPSO in time cost.

Figure 13 shows the average time cost of different algorithms from the beginning to the convergence. The smaller the value, the lower the time cost. It can be seen from the figure that the average time cost of the GA is too high, which is close to 30 minutes. The average time cost of the conventional PSO is about 20 minutes and the LMPSO is about 14 minutes. However, due to the Gaussian distribution sampling based on the geometric center point initializes the population in PSO, a better solution can be found in the initial stage and the average time cost of the hybrid algorithm is greatly reduced. Finally, the average time cost can be reduced by about 5 times compared with the conventional PSO.

Fig. 13

The Comparison of the average time cost of different algorithms for grasping pose optimization.

In short, the hybrid algorithm proposed in this work not only achieves the best results in the average score and grasping success rate, but also is the shortest in the average time cost. This fully illustrates the excellent performance of the hybrid algorithm under the problem of parts grabbing, especially for industrial manufacturing which strictly controls time cost.

5 Conclusions

Aiming at the special requirements of the success rate and time cost of the part grabbing, this paper puts forward a new hybrid algorithm for grasping pose optimization in the manufacturing of heavy vehicles. The hybrid algorithm has been deployed on the real industrial production line in No.18 intelligent Workshop belong to the SANY Heavy Industry Group in Changsha. The hybrid algorithm combines the Gaussian distribution sampling based on the geometric center point and dynamic Alpha-stable mutation in PSO. The Gaussian distribution sampling speeds up the process of search. The dynamic Alpha-stable mutation enhances the global optimization ability of the hybrid algorithm. The time cost of the hybrid algorithm is near a fifth of the conventional PSO in the discovery of optimal grasping pose. Moreover, the hybrid algorithm still has the highest grasping success rate.

In the future, more physical indicators for the part grabbing and combined shapes for the splicing electromagnetic chuck will be studied to further improve the success rate of the production. At the same time, the parallel algorithm will also be the key to further shorten the average time cost.

Footnotes

Acknowledgments

This work was supported in part by the National Key Research and Development Program of China (2018AAA0102200), the National Natural Science Foundation of China (61572507, 61532003, 61622212, 61902435), and Hunan Provincial Natural Science Foundation of China (2019JJ50808).

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