Trajectory Optimization of Glue Spraying on Athletic Footwear Sole Based on Multivariable Spray Torch Model

Adopt robot arm for automatic spraying of footwear soles to solve the problems of low efficiency of manual spraying and substandard quality of spraying glue. Established a spray torch model with the pressure of the spraying swath <inline-formula> <tex-math notation="LaTeX">$p_{f}$ </tex-math></inline-formula>, the atomizing pressure <inline-formula> <tex-math notation="LaTeX">$p_{w}$ </tex-math></inline-formula>, spraying flow <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula>, spraying distance <inline-formula> <tex-math notation="LaTeX">$h$ </tex-math></inline-formula>, and spraying angle <inline-formula> <tex-math notation="LaTeX">$\theta $ </tex-math></inline-formula> as the variables, a visual trajectory offset function was established by introducing the sidewall <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> of the sole and the normal vector <inline-formula> <tex-math notation="LaTeX">$\overline n$ </tex-math></inline-formula> of the trajectory point, and the B-spline curve was used for interpolation to the offset trajectory. The improved non-dominated sorting genetic algorithm (INSGA-Ï) was adopted to solve the spraying trajectory with time, energy, and smoothness as the objectives. A normalized weight function was established that treated the three objectives with equal importance to optimize the solutions. Furthermore, a simulation to optimize the glue spray trajectory and glue spraying experiments were conducted on the sidewall of a shoe sole with a size of 41. As revealed in the results, provided that energy and smoothness are ensured, the proposed model can spray glue on the sidewall of the footwear sole uniformly within 7.27 s, no glue overflow and leakage phenomenon, and the spraying effect is uniform, peeling strength is more than 2.5Kgf/cm, greater than the process requirements of the index, and solved the bottleneck of shoe sole spraying.


I. INTRODUCTION
Glue spraying on the sole is vital for producing athletic footwear and directly determines the product's quality [1]. Currently, this work is mainly completed by the workforce, which cannot guarantee the quality and appearance of the spraying. Moreover, the glue also harms workers' health [2]. In recent years, some companies have introduced glue-spraying robot arms, glue-spraying processes, gluespraying trajectory planning, and spray torches, all of which impact the spray quality [3]. The glue spray area on the sole contains the sidewall and midsole. The latter has almost no requirement for spraying precision, but the former has a high The associate editor coordinating the review of this manuscript and approving it for publication was Mohammad Alshabi . precision requirement for the spraying trajectory. Among the trajectory planning for spraying glue on the sidewall of the athletic footwear sole, the spray torch is the foundation for trajectory planning for the spraying robot arm. Only with a precise spray torch model, the evenness and boundary of the glue film can be predicted reliably to plan to spray trajectory and control film thickness [4], [5]. The sole of athletic footwear is made by injection molding PU(polyurethanes) and may deform to some extent after demolding. Yet, the sidewall is an irregular spatially curved face. Therefore, it is necessary to establish a precise glue distribution model and plan a trajectory for the robot arm to ensure uniform and precise glue spraying [6].
The establishment of glue gun model affects the cumulative uniformity of glue film and the trajectory planning of manipulator [7], [8], The existing spray torch models can predict the glue film distribution precisely with specific glue spraying parameters; whereas if the specific glue spraying parameters or the glue film distribution changes, the precision will be out of control and even invalid. Therefore, existing models lack universality and generalization ability [9], [10]. There are Aerosol models the volume-of-fluid method and large eddy simulation [11], multi-scale spray model of mixed Euler-Lagrange coatings [12] to solve the problem of spray effect of glue gun and poor coating quality, and they are widely used in automobile surface spraying [13], daily furniture spraying [14] and trajectory planning [15]. However, the spraying surface of automobile surface and ceramic furniture mainly has large curved surface area, clear boundary, small curvature and low precision of boundary spraying glue characteristics. As a special spraying object, the sole has the characteristics of small spraying area, large curvature fluctuation and higher precision requirement of the boundary spraying, and these characteristics resulting in more complex spraying model. In the field of sole glue spraying, a glue gun model is established based on the external input pressure parameters of the glue gun [6], but did not taken the flow rate of glue and the characteristic information of sole glue spraying area as the input of model building, lead to the phenomenon of glue overflow and leakage. For sports soles with high quality requirements of spraying glue, it is difficult to meet the technological requirements directly by using the existing torch model. Combining with the technological requirements of spraying glue, the torch model suitable for sole spraying glue is established in order to fill the research field of fine spraying glue in shoe industry.
Considering the spraying process and external influence factors, the spraying effect is relevant to the spraying process, the spray torch model, and the spraying trajectory and is related to the control of the robot arm for glue spraying. The model of robot arm glue spraying and glue gun installation in Figure 1, wherein the angle between the spraying axis of the glue gun and the Z-axis of the local coordinates O(X,Y,Z) of the end of the robot arm is 45 • . In terms of handling and assembly, most studies on robot arm trajectory planning [16], joint space optimization [17], and dynamic performance focused on point-to-point control [18], neglecting precision trajectory control. While the precision control of motion trajectory is emphasized in the fields of welding [19], polishing [20] and spraying glue [21]. The spraying area of the sole side wall is an irregular free-form surface in space, which has the curvature of the spraying area and large fluctuations in the area and normal direction, leading to difficulties in controlling the trajectory accuracy of the sole sidewall. Existing studies on shoe spray trajectory are mainly based on the overall bias of 3D visual trajectory [6] or the arc length of adjacent trajectory points as the bias [22], without combining the width of the spray area and normal vector information, leading to theoretical defects in trajectory planning. Combined with the motion characteristics of the robot arm, for the side wall area with large curvature, the motion velocity of the robot arm should be reduced to ensure trajectory precision. Meanwhile, the spray flow should also be changed to avoid the accumulation of glue [23]. Therefore, in this paper, under comprehensive consideration of time, energy and smoothness, the trajectory optimization is carried out on the sole side wall spraying glue, so as to solve the influence of manipulator jitter on trajectory accuracy, spraying efficiency and spraying effect during the biased trajectory operation.
The sidewall has different widths (large in the part near the heel and small near the toe cap) and different curvatures, making precision spray control of the robot arm difficult. In this study, the authors established a variable spray torch model of a glue gun based on the glue spraying parameters, which offsets the visual trajectory based on the sidewall widths and optimizes the offset trajectory with time, energy, and smoothness. The structure and main contents of this paper are as follows: (1) Comprehensive analysis of the relationship between adhesive film thickness and peel strength, combined the pressure of the spraying swath p f ,, the atomizing pressure p w , spraying flow q, spraying distance h, and spraying angle θ, to established the torch model with spray width L.
(2) Combining the visual trajectory points and the normal vector of the spraying area, and introducing the spraying width variable L to establish the bias function of the spraying trajectory.
(3) The k-order B-spline curve is used to interpolate the robot arm joint trajectory for the biased spraying trajectory. Constraint speed v max , acceleration a max and jert c max , and INSGA− algorithm is used to optimize the spraying trajectory in terms of robot arm motion time, energy consumption and trajectory smoothness; a normalized weight function is established for Pareto solution set finding and compared with single objective optimization and linear weighted optimization.
(4) Combined with the experimental platform to carry out glue spraying experiments of shoe soles and test the peeling strength of soles and uppers.

A. THE RELATIONSHIP BETWEEN THICKNESS AND PEELING STRENGTH OF GLUE FILM
Peeling strength is the main basis for assessing the glue spraying effect. According to the peeling strength test methods (GB / T3903.3) and relevant literature [24], [25], the peeling strength of the glue bonded to the sole and upper of footwear increases first and then decreases with increasing film thickness; when the film thickness is 0.2 mm, the peeling strength reaches the maximum and the bonding effect is also the best. The reasons are as follows: if the amount of glue used is little, the film thickness formed will be over small, causing 'point contact' and low peeling strength; if the amount is moderate, a continuous film with moderate thickness and high peeling strength will be formed; if the amount is over large, the film will be over thick so that the solvent beneath the film will VOLUME 11, 2023  be hard to volatilize and even produce bubbles, reducing the peeling strength of the film.

B. DETERMINING THE INDEPENDENT VARIABLES OF THE SPRAY TORCH MODEL
Air spraying is a glue spraying process in which compressed air is used to get the glue atomized into tiny particles and deposited on the sprayed surface, forming a continuous glue film. The filming process of glue is complex and involves the flow, atomization, volatilization, deposition, and other physical processes of the glue fluid. Therefore, the thickness of the glue film is affected by many factors, which are divided into 4 categories, as shown in Table 1.
The glue spraying device and the external environment are unchanged for the same set of robot arm for glue spraying or the same batch of glue spraying operations. Among the glue spraying parameters, the position of the pin valve is not changed once it is established, the spraying swath pressure, the atomizing pressure, and the spraying flow are adjusted in the spraying operation, and the change in the glue feeding pressure is reflected in the spraying flow. Among the path parameters, the spraying distance and spraying velocity are adjusted according to the spraying process; and in most glue spraying operations, the axis of the glue gun is vertical to the sprayed surface or has a fixed angle. Therefore, the pressure of the spraying swath p f , the atomizing pressure p w , spraying flow q, spraying distance h, and spraying angle θ were determined as the variables of the spray torch model. The trajectory extracted by the visual system was offset based on the spray torch model in Figure 2. Further, the spraying distance h was adjusted to ensure that the film thickness was equal to the width of the sidewall L of the sole. The pressure of the spraying swath p f and the atomizing pressure p w were adjusted by the glue feeder, the spraying flow q by the glue gun, and the spraying distance h and spraying angle θ by the robot arm.

C. ESTABLISHING THE MULTI-VARIABLE SPRAY TORCH MODEL
The Nordson LS-373 is designed for high-precision cold adhesive spray applications with a maximum adhesive pressure of 3.04Mpa, a maximum atmospheric pressure of 1.52Mpa, and a 10ms opening and closing time. The innovative nozzle design can provide a variety of spray requirements from narrow to wide, widely used in footwear, packaging, carton and other industries. A NORDSON LS-373 glue gun was used to experiment with vertical glue spraying on the  sprayed surface in Figure 3. Consequently, a linear glue film was obtained by straight spraying, while an elliptical glue film was obtained by spraying to a fixed point within 0.3∼0.5 s (after turning on the glue gun and before turning off the glue gun).
As can be seen in the experiment, the glue was sprayed evenly, forming an elliptical film on the sprayed surface. As presented in Figure 3, a coordinate system o(x,y,z) was established taking the center o of the ellipse as the origin, the major axis 2a of the ellipse as the x-axis, the minor axis 2b as the y-axis, and the axial direction of the spraying as the z-axis. After the glue gun was turned on, the glue would accumulate in the sprayed area, and the film thicknesses on the cross-sections of xy and yz had similar curve shapes. Therefore, assuming that the growth rate curves of the glue film in the cross sections of xy and yz were both in β distribution [26], namely, the film accumulation rate was relevant to the β distribution value (range: (0, 1)), the accumulation rate was inversely proportional to the distance from the accumulation point to the center along the spraying axis (z-axis). The following displays the relationships between the major and minor axes (2a and 2b) of the elliptical coverage on the sprayed surface and the spraying distance h: where α x is the spray cone angle on the x-axis of the glue gun, • ; α y is the spray cone angle on the y-axis of the glue gun, • . The said two spray cone angles were affected by the structure of the glue gun, the flowing principle of glue, the pressure of the spraying swath p f , and the atomizing pressure p w . The following exhibits the relationships between the two spray cone angles and the pressure of the spraying swath p f , the atomizing pressure p w , and the spraying flow q, respectively: where K x α is the x-axis coefficient of the glue gun, while K y α the y-axis coefficient; p f is the pressure of the spraying swath, MPa; p w is the atomizing pressure, MPa; q is the glue spraying flow, mL·min −1 ; a 1 and b 1 are the combined index variables of p f and p w ; a 2 and b 2 are the index variables of the spraying flow q. Figure 4 shows the changing curves of the two spray cone angles (α x and α y ) with the change of the spray flow q at different spraying swath pressure p f and atomizing pressure p w in the above experiment. Based on the experimental data, the following solutions were obtained: When the p f , p w , and q changed, the α x and α y were predicted as per equation (2), and the spraying distance h was calculated based on equation (1).
If the sidewall of the sole is greatly slanted, vertical spraying may cause interference between the glue gun and the sole or may be invalid for the inner corner of the sole. Therefore, the glue gun was adjusted to the extent that the angle between its spraying axis and the vertical spraying axis was θ to ensure that the sprayed glue film has a width identical to the sidewall of the sole. The slanted spraying principle is shown in Figure 5.
where θ is the spraying angle, • . Based on equations (1), (2), and (3), the spray torch model for spraying glue on the sidewall of the shoe sole was obtained as follows:

III. THE TRAJECTORY OFFSET OF SPRAYING
It was assumed that the trajectory point extracted by the visual system was p g (x g , y g , z g ), (g = 1, 2, · · · , G), corresponding to the width L = (L 1 , L 2 , · · · , L g ) of the sidewall and the normal vector n(a g , b g , c g ) of the point p g . Calculation principle of glue gun spraying trajectory in FIGURE 6. Then, the spraying distance h = (h 1 , h 2 , · · · , h g ) was obtained based on equation (4); and the offset trajectory point p ′ g (x ′ g , y ′ g , z ′ g ) corresponding to the trajectory point p g (x g , y g , z g ) was obtained VOLUME 11, 2023 as follows: where ).
Due to the variations in the curvature and height of the sidewall, the trajectory points p g (x g , y g , z g ) of the sidewall extracted by the visual system were unevenly distributed, and the height to the sole varied. The offset trajectory points p ′ g (x ′ g , y ′ g , z ′ g ) obtained using equation (5) were also unevenly distributed in space and also had varying heights to the sole. In this part of the study, the 3D spatial trajectory of the sidewall of a shoe sole was extracted by a visual system and then offset, obtaining the offset spray trajectory on the sidewall of the sole in Figure 7.

IV. MULTI-OBJECTIVE TRAJECTORY OPTIMIZATION A. THE INTERPOLATING FOR ROBOT ARM'S JOINT TRAJECTORY
According to equation (5) and Figure 7, the offset spraying trajectory points p ′ g (x ′ g , y ′ g , z ′ g ) on the sidewall of the footwear sole were unevenly distributed in space, where the trajectory points on the toe cap are dense, causing unstable motion and vibration of the robot arm and affecting the spraying evenness and precision. The interpolation method was used to plan the joint trajectory of the robot arm to ensure a smooth joint motion trajectory. In the polynomial curve interpolation, if one point changes, the whole curve will be changed; whereas, the B-spline curve only has an impact on the value of the next point. Therefore, a k-order B-spline curve was used to interpolate the joint trajectory of the robot arm. The B-spline curve with n + 1 control points is expressed as: where, d i (i = 0, 1, · · · , n) is the i-th control point of the B-spline curve; N i,k (t) is the primary function of the k-order B-spline curve, namely: where, [t 0 , t 1 , · · · , t n+k+1 ] is the node vector of the B-spline curve.
The node vector was calculated with the accumulated chord length method [27]: 79426 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.   where, l i is the chord length between the control points d i and d i−1 .
The D-H parametric model of the robot arm was used to solve the offset trajectory based on inverse kinematics, obtaining the joint angle with time sequence in the robot arm's joint space J B m = ( J β m , t m ), J = 1, 2, · · · , 6; m = 0, 1, · · · , G − 1}. In this formula, J is the joint sequence of the robot arm; m is the number of joints of the robot arm (each joint has G joint angles); J β m is the angle of the robot arm's joint J at the joint angle point m; t m is the time when the robot arm's joint J was at the joint angle point m. To ensure that the joint motion curve of the robot arm passed by a given joint angle J β m , B-spline interpolation was carried out based on the vector quantity J d j of the control point on the robot arm's joint interpolation curve of the robot arm as obtained by reverse calculation of the B-spline curve, taking the J β m as a point on the B-spline curve. The following displays the G equations satisfying the condition for B-spline interpolation: Based on the simultaneous equation (9), the vector quantity J d j of the control point on the robot arm's joint interpolation curve was calculated. Then, the joint interpolation curve p(t) was obtained by forward calculation of the B-spline curve. Further, the derivative of the curve p(t) was calculated for the first, second, and third times, obtaining the velocity v(t), acceleration a(t), and jerk c(t) of the robot arm's joint motion, respectively:

B. MOTION CONSTRAINTS AND MULTI-OBJECTIVE SOLUTION
The efficiency and precision of the spraying of glue on the side wall of the shoe sole can be ensured by controlling the time, energy consumption, and smoothness of the motion of the robot arm. In this study, the said spraying efficiency and precision problems were converted into multi-objective optimization of the joint motion of the robot arm, involving VOLUME 11, 2023 79427 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. velocity, acceleration, and jerk. The objective function is provided below: where, S 1 is the robot arm's motion time to measure the execution velocity of the robot arm's execution velocity; S 2 the robot arm joint's average acceleration for measuring the energy consumption; and the average jerk of the joint to measure the smoothness of the joint motion.
where, t m = t m+1 − t m is the time interval between two adjacent nodes in the trajectory, s.
Since the B-spline curve has a convex hull, the extreme points of the joint's velocity v(t), acceleration a(t), and jerk c(t) were at the control point J d j . The motion constraints of the robot arm were converted into the constraints at the control point J d j of the B-spline curve: (13) where, v max , a max and c max are the motion constraints of the robot arm, as listed in Table 2.
Based on equation (13), the improved nondominated sorting genetic algorithm (INSGA-II) was used to optimize the three objectives, with the process shown in Figure 8. The first step was to enter the motion range of the robot arm joint's motion range [θ min , θ max ], velocity constraint v max , acceleration constraint a max , and jerk constraint c max . Then, a contrast experiment was conducted to understand the impacts of crossover rate P c , mutation rate P m , crossover distribution index η c , and mutation distribution index η m on the multi-objective search performance of multi-objective search, provided that the population size S pop was 150. As a result, the crossover rate P c and mutation rate P m had less impact on the multi-objective search performance than the crossover distribution index η c , and mutation distribution index η m ; when P c = 0.75, P m = 0.1, η c = 20, and η m = 10, the multi-objective optimization converged well. In the genetic operation, a simulated binary crossover operator and a polynomial mutation operator were used, and many non-inferior solutions (i.e., many groups of Pareto solution sets) were obtained in the iteration process. Then, the following objective normalization weight function was established to optimize the Pareto solution sets: where, ω 1 is the weight factor of time; ω 2 is the weight factor of energy; ω 3 is the weight factor of smoothness; ψ 1 is the normalized coefficient of time; ψ 2 is the normalized coefficient of energy; ψ 3 is the normalized coefficient of smoothness.

V. EXPERIMENTAL VERIFICATION AND ANALYSIS A. THE SIMULATION OF TRAJECTORY OPTIMIZATION
Based on the offset trajectory shown in Figure 7, a 7-order B-spline curve was used to interpolate the robot arm's joint trajectory. Then, the multi-objective optimization was simulated, obtaining the Pareto solution set of the spraying trajectory as provided in Figure. 9. Black is the optimization results; red is the projection of optimization results on time and acceleration planes; green is the projection of optimization results on acceleration and jerk surfaces; blue is the projection of optimization results on time and jerk surfaces. The minimum time is at point A and the maximum is at point D, while the energy and smoothness are on the contrary. The time optimality closer to point A is superior, while the energy optimality and smoothness optimality closer to point D is better. The multi-objective optimization results at points A, B, C, and D shown in Figure 9 were extracted to compare with the single-objective optimization and linear weight optimization results obtained by the genetic algorithm of the BP neural network. The comparison results are listed in Table 3. The single-objective optimization was to optimize the time, energy, and smoothness separately. The linear weight optimization was carried out under the same smoothness weight factor condition to compare the impacts of time and energy on the objective, where, as shown, the weight factors of time, energy, and smoothness at point B+ were [0.4, 0.4, 0.3], while those at point C+ were [0.5, 0.2, 0.3]. The results of the single-objective optimization were superior to those of the other two methods, indicating that the single-objective optimization ensured the absolute advantage in optimizing the single objective. Further, the performances at points B and B+, and points C and C+ were compared. The results demonstrate that when the energy performances were equivalent, both the time and smoothness indices of multi-objective optimization increased, while the linear weight optimization discarded the performance of some optimization objectives. The results at points A, B, C, and D reveal that the time index at point A was optimal, while the energy and smoothness were inferior. The time index at point D was poor, while the energy and smoothness were optimum. The indices at points B and C were between points A and D. These results 79428 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
suggest that the INSGA-II algorithm can weaken at least one objective function while improving the other one or more objective functions.
However, provided that time, energy, and smoothness were equally important, namely ω 1 = 1, ω 2 = 1, and ω 3 = 1, the time, energy, and smoothness in the Pareto solution set were normalized. Given that ψ 1 = 30, ψ 2 = 18, and ψ 3 = 32 according to Figure 9, formula (14) was used to optimize the normalized weight of the solution set. Consequently, the simulation and optimization results obtained for the spraying trajectory on the side wall of the footwear with a size of 41 were at point E in Figure 9, namely, the optimal solution was S 1 = 7.27s, S 2 = 12.09 • · s −2 , and S 3 = 13.82 • · s −3 . The positions, velocities, accelerations, and jerks of the robot arm joints are presented in FIGURE 10. The range of motion of the robot arm depends on joints 1∼3, while the attitude of motion is determined by joints 4∼6. Restricted by the sidewall space of the footwear sole, joints 1∼3 moved within 20 • ; limited by the spraying attitude of the glue gun, joint 4 always moved no more than 10 • around 90 • , while joint 5 moved within 20 • ; since the sidewall should be sprayed for a round, the joint 6 moved within -150 • ∼200 • . The velocity curves of joints 1∼5 were gentle, without mutation at the beginning and ending positions; for a quick response to the change in the spraying attitude, the fluctuation in motion velocity of joint 6 reached 100 • ·s −1 . For the robot arm joints, the acceleration curve was smoother than the jerk curve. For rapid adjustment of the velocity to respond to the change in the sprayed area on the sidewall, the jerk curve of joint 1 underwent a larger fluctuation than the acceleration curve. To adapt to the change in the height of the sidewall, the acceleration and jerk curves of joints 2 and 3 were gentle and the curves of the two joints were in opposite directions. Additionally, to quickly adjust the velocity to adapt to the spraying attitude in different areas of the sidewall, fluctuations in the jerk curves of joint 4∼6 were greater than the acceleration curves.

B. THE EXPERIMENT AND ANALYSIS FOR GLUE SPRAYING AND PEELING STRENGTH
As shown in Figure 11(a), an experimental table was set up, with ESTUN robot arm ER10-900, Keba CP505 controller, and NORDSON LS-373 glue gun as the main devices. Then, the INSGA-II-based multi-objective optimization method was built into the robot arm controller, and the athletic footwear soles (size: 41) with sidewalls of varying curvatures were used to do the glue spraying experiment. The spraying results are illustrated in Figure 11(b). As can be seen, the toe cap and heel with difficulty in spraying were evenly sprayed with glue, forming clear film boundaries. It took not more than 7.27s to spray glue on the sidewall of a single sole. Over an experimental verification of the glue spraying on the sidewalls of footwear soles with different models and sizes, the results reveal that the multi-objective optimization-based spraying trajectory had a better spraying effect and its spraying efficiency satisfies the requirement for the production cycle, 12s. However, the accumulation of adhesive film is less  in the side wall area of the sole, and more in the shoe head and heel area, it exist an obvious uneven phenomenon of spraying adhesive. The glue rate of the glue gun is unchanged, and the reason for the uneven spraying of glue mainly comes from the VOLUME 11, 2023 fluctuation of the speed of the robot arm. The curvature of the side wall of the sole is small, and the speed of the manipulator is large, which leads to less accumulation of the film; In the shoe head and heel area, the curvature changes greatly, and the running speed of the manipulator is small, which leads to the accumulation of more adhesive film. 79430 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.    Then, the sprayed sole was pressed on the upper of the footwear with the shoemaking process, forming a qualified product. After that, a peeling strength experiment was conducted on the finished product. The sample for this experiment (namely the bonding area of the sidewall and upper of the footwear) was obtained by cutting down the upper area along the upper cutting path of the finished product and removing the sole area along the sole cutting path in Figure 12. Then, 30 test points were selected on the sample at an interval of 1∼2 cm and a tensile testing machine was used to peel the sample. The obtained results are shown in Figure 13. As observed, the peeling strength at each point of the left and right shoes was higher than 2.5 Kgf/cm, the index required by the process, which suggests that the film strength obtained by spraying glue on the sidewall along the multi-objective optimization-based spraying trajectory met the application requirement. The maximum peeling strength of the right sole was 5.31 Kgf/cm and the minimum peeling strength was 2.86 Kgf/cm. The maximum peeling strength of the left sole was 4.75 Kgf/cm and the minimum peeling strength was 2.76 Kgf/cm. The peeling strength between each point of the left and right soles fluctuated, and the maximum difference was 2.45Kgf /cm, both of which were greater than the technical requirement index of 2.5Kgf/cm, indicating that the adhesive film strength obtained by spraying the soles side wall with multi-objective optimization spraying trajectory met the application requirements. Combined with the spraying effect in Figure 11 (b), the impact on peeling strength mainly comes from the inconsistent film thickness caused by the uneven spraying of glue. Combined with Table 1, the factors affecting the thickness of the film mainly come from the spraying speed of the path parameter.

VI. CONCLUSION AND FUTURE WORK A. CONCLUSION
Based on the establishment of the glue gun torch model and the optimization of the glue spraying trajectory, this paper effectively solves the problems of poor glue spraying accuracy and substandard glue spraying quality of the traditional robotic arm, and lays the foundation for the transformation of the shoe industry to the direction of automated intelligent manufacturing.
(1) A spray torch model was established, with the spray swath pressure p f , the atomizing pressure p w , the spray flow q, the spray distance h and the spray angle as variables, to provide the theoretical basis for precision assurance of glue spraying.
(2) A visual trajectory offset function was created based on the spray torch model, the width L of the sidewall of the footwear sole, and the normal vector n of the trajectory point, to provide the theoretical calculation basis for the offset of the trajectory, and solve the phenomenon of overflowing and leakage of spraying area in the traditional spraying process.
(3) To ensure the proper allocation of robot arm runtime, energy and smoothness in multi-objective optimization, the INSGA-II was used to conduct multi-objective simulation for the offset trajectory on the sidewall of the footwear sole, and a normalized weight function was used to select the optimal objectives.
(4) The multi-objective optimization method was built in the robot arm controller to perform the glue spray experiment on the sidewall of the footwear sole, with the spraying time, not more than 7.27s. As a result, the peeling strength obtained was greater than 2.5 Kgf/cm, the index required by the process. Over an experimental verification of the glue spraying on the sidewalls of footwear soles with different models and sizes, the results demonstrate that the multiobjective optimization-based spraying trajectory had better spraying effects; namely, the formed film boundary was clear, and both the spraying efficiency and peeling strength satisfied the process requirements.

B. FUTURE WORK
(1) In this experiment, the relevant glue spraying tests were carried out for traditional soles. Later, the glue spraying experiments should be carried out for special soles to further improve the torch model.
(2) For the uneven thickness of the adhesive film, the function relationship between the glue gun and the spraying speed of the robot arm is established to dynamically adjust the amount of glue output.
(3) The thickness of the adhesive film of the spraying affects the peel strength. The 3D data of the soles after spraying were collected to verify the spray accuracy and the thickness of the film. She is currently the Laboratory Director of the college of Mechanical Engineering, Donghua University. She is a master supervisor. Her research interests include textile intelligent equipment and machine vision. She has won the second prize in National Science and Technology Progress Award as a participant.

SUN YIZE was born in Sichuan, China, in 1958.
He is currently a Professor with the College of Mechanical Engineering, Donghua University. His research interests include complex mechanical systems and their intelligent measurement and control technology, high-end textile equipment technology, and systems. His major academic achievements are presided over more than 30 projects, such as the National Key Research and Development Program, National Science and Technology Support Program, National Development and Reform Commission Major Special Project for Textile Machinery, and National Natural Science Foundation. He has won a two-second prize in the National Science and Technology Progress Award and three prizes at provincial and ministerial levels as the first finisher. He has published more than 100 SCI papers. It has more than 70 authorized invention patents. He is also a Discipline Leader of Mechanical and Electronic Engineering, an Innovation Team Leader of the Ministry of Education, an Excellent Innovation Team Leader of the Ministry of Education, and an Academic Leader of China Textile, leading talents of Shanghai. He received the State Council Special Allowance, the Baosteel Excellent Teacher Award, and other honors.