# Motion Trajectory Planning and Design of Material Spraying Service Robot

Authors: Gang Wang; Hongyuan Wen; Jun Feng; Jun Zhou
Journal: Advances in Materials Science and Engineering
 (2022)

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## Abstract

The application of robots reduces repetitive and dangerous tasks for humans, especially the spraying robot, because many spraying materials are corrosive, toxic, and harmful. This paper designs the motion trajectory planning of the material spraying service robot. With the increasing demand for technology, there are strict requirements for the uniformity and thickness of spraying. In view of this, this paper proposes an algorithm for modeling kinematics using a joint screw and optimizes the modeling algorithm using particle swarm optimization. This makes the industrial spraying robot more intelligent and more capable of completing high-standard tasks. The experimental results in this paper show that when the spraying radii on the large-curvature cone surface are 44.5 and 49.5 mm, respectively, the coating distribution in the intersection area of the curved surfaces can be well controlled, and the optimized algorithm can better plan the path of the spraying robot.


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## Body

## 1. Introduction

A spraying robot is a special kind of industrial robot. As an advanced automatic painting production and processing equipment, its advantage lies in the automation of painting production. Its most notable feature is that it can avoid the harm of toxic particles to workers during manual spraying and can realize offline programming, so spraying robots are widely used in various spraying production occasions. Robot kinematics plays a key role in the design of robots or the implementation of control systems and is an important topic in the field of robotics.Today, almost all electromechanical products require surface coating in the production process. Especially in recent years, with the rapid development of aerospace, aviation, navigation, automobile, ship, and other industries, people have higher requirements for product functions, surface appearance, and other aspects. At the same time, the quality of the product appearance coating directly affects the appearance of the product surface and product performance, such as automobile surface paint film and aircraft surface special performance coating. Therefore, higher requirements are put forward for product surface coating technology.The innovations of this paper are as follows. (1) A method of analyzing the inverse kinematics subproblem from two angles of position change and direction change is proposed, which is convenient for solving the inverse kinematics of some serial manipulators. (2) A combined inverse kinematics algorithm is designed, which makes rational use of the analytical solution of approximate structure and avoids the problem of initial value selection in the nonlinear least-squares numerical iteration method. (3) An improved ISODATA algorithm is designed, which makes full use of the initial cluster information provided by hierarchical clustering and has a better clustering effect.


## 2. Related Work

Because of its wide application and great development potential, material spraying robots have been deeply studied by scholars from various disciplines. Especially with the advent of the age of intelligence and information, the development of intelligent robots has become a research hotspot, and there is also much research on spraying robots. Nakao et al. [1] studied the path planning and walking control algorithm of greenhouse pesticide spraying robots. He considered the effectiveness of path planning and travel control from simulation and experimental results obtained using a solid model of a greenhouse tunnel [1]. Chao et al. [2] introduced a measurement path planning algorithm based on the spraying robot system and laser displacement sensor technology for the problem of spraying the inner wall of the intake pipe. Based on the existing methods, they focused on the prescan measurement methods of different types of cross-section curves. Algorithm simulation and model reconstruction show that the research solves the collision avoidance problem of the scanning measurement of the inner wall of the intake pipe [2]. Nigam et al. [3] researched and explored a new metallization technique without any chemical hazards and with a high deposition rate, resulting in fairly reasonable coating densities, which clearly explains the strength and density of the copper coating deposition [3]. Kim et al. [4] proposed an automatic fine dust tracking system to accurately identify and track the location of dust generation. They conducted tracking experiments with a small excavator and a small water-jet robot. The results show sufficient tracking performance for water-jet machines [4]. Aiming at the problem that manual segmentation can be tedious and error-prone, Krishnan et al. [5] proposed a new segmentation method that combines the theory of mixed dynamical systems and Bayesian nonparametric statistics to automatically segment presentations [5]. Wang et al. [6] found that when a multilegged robot moves, the accuracy of the robot’s motion is affected by many interdependent noise factors. They proposed a parameter optimization method to improve the accuracy of motion trajectories. An important research goal is to discover parameter values that can be selected to make robot motion more precise without necessarily improving noise-generating factors. Their experimental results on quadruped robots show that their method is useful and will help further improve parameter optimization and design of multilegged robots [6]. At present, the spraying robots developed in China mostly adopt 3R skew nonspherical wrist, and their structural parameters are basically the same as those of similar foreign products, lacking innovation. There is still a big gap in the current spraying robot, so it is necessary to study it in this paper.


## 3. Kinematics of the Spraying Robot

Spraying robots’ research and application have been conducted in foreign countries for more than 30 years. All the major foreign robot manufacturers have developed mature spraying robots and supporting equipment, and they can provide perfect solutions for specific application backgrounds. At present, the spraying products of ABB, KUKA, An Chuan, and other foreign manufacturers account for the vast majority of the global share.Smooth things can always be more liked by people, so spraying operations widely exist in life. It is an important step to spray the material evenly and smoothly on the surface of the object, from the small toy paint surface to the large space vehicle. For the traditional oh-um map operation, most of the workers do manual spraying, and through hand-held spraying equipment, in a static-free environment, empirical spraying is performed, and the effect is often poor or inefficient. In addition, due to the particularity of spraying materials, small particles or toxic substances are often volatilized, which seriously endangers the safety of workers. Therefore, in view of safety considerations, the development of spraying robots is getting better, and replacing manual operations with mechanical automation equipment has become the development trend in the spraying industry [7, 8].


### 3.1. The Wrist Joint Design of the Spraying Robot

Robot kinematics plays a key role in the design of robot mechanisms or the implementation of a control system, and it is an important topic in the field of robot technology. To the relationship between the position and posture of the robot end effector and the joint variables, it is necessary to establish a kinematic equation of the robot after the coordinates and parameters of the robot link are determined. In the context of the machine revolution, spraying robots were also quickly applied. The first large-scale use of small automatic spraying machines in the market, as shown in Figure1(a), due to its relatively fixed position, cannot control the spraying direction of materials, so it cannot carry out complex material spraying work. With the improvement of artificial intelligence and the level of industrialization of human society, spraying robots (Figure 1(b)) have emerged and gradually replaced automatic spraying machines. Up to now, the spraying robot has basically replaced manual spraying on the small car production lines in developed countries in Europe and America and achieved a high degree of automation.Figure 1
Automatic spraying equipment. (a) Automatic sprayer. (b) Spraying robot.
(a)(b)Generally, the mechanical body of the spraying robot includes two parts: the manipulator responsible for positioning and the wrist joint responsible for posture determination. Most operators of mainstream spraying robots adopt a 3R joint design, including base and arm, which is basically consistent with other types of multi-degree-of-freedom industrial robots. Generally, the wrist joint is also designed with the 3R joint to obtain higher flexibility. Considering that the painting robot requires high cleanliness around the body when it works, foreign manufacturers generally design the wrist joint of the painting robot as a hollow structure, and all kinds of pipelines pass through it and directly connect to the tail of the high-speed rotating cup. At the same time, considering the problem of bending and knotting the internal pipeline of the wrist, the oblique aspherical structure (the joint axes do not intersect at one point and are not orthogonal) is generally adopted [9]. Figure 2 shows Fancu’s latest P-250iB spraying robot, which adopts a 3R oblique aspherical wrist joint, and the angle between adjacent joints is designed to be 70°, which can reduce the problem of pipeline bending to a certain extent. The specific design parameters are shown in Tables 1 and 2.Figure 2
Fancu P-250iB spraying robot.Table 1
Connecting rod parameters of spraying robot.
iai−1αi−1diθiConnecting rod parameter value100°0θ1 (0°)d2 = 201.5 mm2d2−90°0θ2 (0°)L1 = 550.0 mm3L10°L2θ3 (−90°)L2 = 542.0 mm40−90°0θ4 (0°)d1 = 165.0 mm5090°0θ5 (0°)t0−90°d10°Table 2
Joint parameters of spraying robot.
Joint rotationAxis direction vectorReference point on the axisSpinner coordinate value1[0 0 1]T[0 0 0]T[0 0 1 0 0 0]T2[−1 0 0]T[0 0 0.340]T[−1 0 0 0 −0.340 0]T3[−1 0 0]T[0 0 1.640]T[−1 0 0 0 −1.640 0]T4[0 1 0]T[0 0 1.640]T[0 1 0 −1.640 0 0]T5[0 0.819 −0.574]T[0 1.509 1.640]T[0 0.819 −0.574 −2.209 0 0]T6[0 0.819 0.574]T[0 1.639 1.640]T[0 0.819 0.574 −0.404 0 0]T7[1 0 0]T[0 0 1.640]T[1 0 0 0 1.64 0]T


### 3.2. Trajectory Planning of Spraying Robot

Trajectory planning in robot joint space is to interpolate the positions and postures of given path points, such as the starting and ending points, under the condition of satisfying a series of constraints to generate the motion trajectory. The trajectory planning in the joint space can be said to be intuitive, on the one hand, because the joint motion can be controlled directly according to the change of the joint angle to achieve the purpose of driving the robot to move, or it can be said that it is not intuitive. On the other hand, it is difficult to guarantee the trajectory of the robot end effector because of its joint angle interpolation, which leads to the nonintuitive trajectory of the robot end effector. Therefore, trajectory planning in joint space is generally used for trajectory planning that does not require high motion trajectories or has a short motion distance and can ignore the influence of motion trajectories, such as point-to-point motion. In the joint space, three polynomial interpolation methods commonly used in path planning methods are high-order polynomial interpolation, parabolic interpolation, and spline curve interpolation [10, 11].For the movement from the position and attitude of the starting point to the position and attitude of the end point in a given time, using the method of solving the inverse kinematics of the robot introduced in the previous chapter, the problem of solving each angle of the corresponding start and end time can be solved. The change in joint angle over time is described by a functionq (t) with respect to time. For the convenience of description, the problem can be abstracted as follows: at the starting point, that is, when t0 = 0, the joint angle value is q0, and at the end point, time tf, the joint angle value is qf. In order to ensure the smoothness of the motion of the joint, the speed limit of the starting and the end points is added to the joint function q (t). At this point, the problem becomes finding a solution to the joint angle function q (t) that satisfies the four constraints while being smooth.At present, industrial robot systems mainly use teaching reproduction, robot language, and offline programming, among which teaching reproduction is the most widely used method. Reproduction is to reproduce according to the needs of the task and record some key points on the trajectory. If the trajectory of the robot’s end effector is strictly consistent with the motion requirements, enough points must be obtained. The robot’s motion trajectory is generally circular or straight, and the complex motion trajectory can be the general, straight line and arc fitting. The main spraying trajectories of the five-degree-of-freedom spraying robot studied in this paper are also spatial straight lines and spatial curves. The following describes the use of interpolation algorithm of spatial straight line based on parabola transition and interpolation algorithm of spatial arc based on the local coordinate system and the trajectory planning in Cartesian space.


### 3.3. Motion Trajectory Algorithm

The exponential product formulation is a method of modeling kinematics using joint spinners, which has a clear geometrical meaning. From the perspective of geometric analysis, some characteristics of rigid body motion can be used to eliminate the coupled joints so that the complex inverse kinematics problem is divided into simple subproblems, simplifying the inverse kinematics solution process [12].In general, the elimination of joint variables can be based on several principles shown in Figure3: (a) rotation does not change the position of the point on the axis; (b) rotation does not change the distance from a point outside the axis to the axis; (c) rotation does not change the vector along the axis; and (d) translation does not change the attitude.Figure 3
Joint screw modeling method.
(a)(b)(c)(d)Principle (a) can be described by a homogeneous expression as(1)eξθp¯=p¯.Principle (b) can be described by a homogeneous expression as(2)eξθP¯−R¯=P¯−R¯.Principle (c) can be described by a homogeneous expression as(3)eξθu¯=u¯,or a normal expression,(4)eω^θu=u,eω^θ=uT.Principle (d) can be described by a homogeneous expression as(5)eξθRP01=RP101.At present, according to the above basic principles, predecessors have deduced most of the inverse kinematics subproblems [13], which can be summarized in 11 cases, as shown in Table 3 [14] (R means rotating joint, T means moving joint). Most of the inverse kinematic problems of robots with nonredundant degrees of freedom can be solved by reducing them to these subproblems.Table 3
Inverse kinematics subproblem classification.
SubproblemOrderSymbolic representation11SubProb-R21SubProb-T32SubProb-RR42SubProb-RT52SubProb-TT63SubProb-RRR73SubProb-R RT83SubProb-RTR93SubProb-RTT103SubProb-TRT113SubProb-TTTAs the robot used in this paper only has rotary joints, this paper only discusses subproblems 1, 3, and 6. Among them, subproblem 1 is generally obtained by simplifying more than two series joints, and subproblem 3 is generally obtained by simplifying more than three series joints. Subproblem 6 is generally obtained by simplifying more than four series joints. It should be emphasized that, for the case where the four axes of rotation intersect at one point, if there is no joint linkage restriction, there is a redundant degree of freedom, and the number of inverse kinematics solutions is infinite. Figure4 can illustrate the problem more intuitively. In the case of the intersection of the transfer trajectory (an inverse kinematics solution can be obtained), the rotation radius around the axis ξ2 can be selected in countless ways, and there are countless types of joint angles (inverse kinematics solutions) that can be obtained.Figure 4
Schematic diagram of redundant degrees of freedom case.For the inverse kinematics subproblem, there are two approaches to deal with it. One is to analyze from the point of position change [15], that is, to simplify the inverse kinematics problem using the two principles (a) and (b) mentioned above. The other is to analyze from the perspective of the direction change of the vector, that is, to use the (c) principle to simplify the inverse kinematics problem. The analysis method of the position change angle is as follows.The exponential product formula for the 2R structure is given:(6)T02θ=eθ1ζ1eθ2ζ2T020.IfT02θ=Tgiven, it can be transformed into(7)eθ1ζ1eθ2ζ2=TgivenT020−1.From the perspective of the position change of the point, take any pointP on ξ2, and multiply P on both sides of the formula to have(8)eθ1ζ1eθ2ζ2P=TgivenT020−1P.According to principle (a), it can be simplified to(9)eθ1ζ1P=TgivenT020−1P=Q.At this time, subproblem 1 is the situation shown in Figure5(a), the known joint screw coordinates,(10)ξ=ω;R×ω∈se3,P,Q∈RR3.Figure 5
Schematic diagram of subproblem 1.
(a)(b)P and Q are two points outside the joint axis. If point P rotates around the joint axis to reach Q, find the rotation angle θ. It is described by the following expression:(11)eξθP=Q.First, take a pointR on the axis of rotation and define(12)U=P−R,v=Q−R.Because of principle (a),(13)eξθP=Q,eξθR=R.So,(14)eξθu=v.As shown in Figure5(b), define u,,v, as the projections of u and v on the plane perpendicular to the axis of rotation ξ; then, we have(15)u,=u−ωωTu,v′=v−ωωTv.It can get(16)ωTu=ωTv′u′=v′.θ can be obtained from the vector u,,v,. If u,≠0, then(17)u′∙v′=u′v′cosθ,u′×v,=ωu,v,sinθ.That is,(18)θ=atan2ωTu′×v′,u′Tv′.Ifu′=0, then there are infinitely many solutions.


## 3.1. The Wrist Joint Design of the Spraying Robot

Robot kinematics plays a key role in the design of robot mechanisms or the implementation of a control system, and it is an important topic in the field of robot technology. To the relationship between the position and posture of the robot end effector and the joint variables, it is necessary to establish a kinematic equation of the robot after the coordinates and parameters of the robot link are determined. In the context of the machine revolution, spraying robots were also quickly applied. The first large-scale use of small automatic spraying machines in the market, as shown in Figure1(a), due to its relatively fixed position, cannot control the spraying direction of materials, so it cannot carry out complex material spraying work. With the improvement of artificial intelligence and the level of industrialization of human society, spraying robots (Figure 1(b)) have emerged and gradually replaced automatic spraying machines. Up to now, the spraying robot has basically replaced manual spraying on the small car production lines in developed countries in Europe and America and achieved a high degree of automation.Figure 1
Automatic spraying equipment. (a) Automatic sprayer. (b) Spraying robot.
(a)(b)Generally, the mechanical body of the spraying robot includes two parts: the manipulator responsible for positioning and the wrist joint responsible for posture determination. Most operators of mainstream spraying robots adopt a 3R joint design, including base and arm, which is basically consistent with other types of multi-degree-of-freedom industrial robots. Generally, the wrist joint is also designed with the 3R joint to obtain higher flexibility. Considering that the painting robot requires high cleanliness around the body when it works, foreign manufacturers generally design the wrist joint of the painting robot as a hollow structure, and all kinds of pipelines pass through it and directly connect to the tail of the high-speed rotating cup. At the same time, considering the problem of bending and knotting the internal pipeline of the wrist, the oblique aspherical structure (the joint axes do not intersect at one point and are not orthogonal) is generally adopted [9]. Figure 2 shows Fancu’s latest P-250iB spraying robot, which adopts a 3R oblique aspherical wrist joint, and the angle between adjacent joints is designed to be 70°, which can reduce the problem of pipeline bending to a certain extent. The specific design parameters are shown in Tables 1 and 2.Figure 2
Fancu P-250iB spraying robot.Table 1
Connecting rod parameters of spraying robot.
iai−1αi−1diθiConnecting rod parameter value100°0θ1 (0°)d2 = 201.5 mm2d2−90°0θ2 (0°)L1 = 550.0 mm3L10°L2θ3 (−90°)L2 = 542.0 mm40−90°0θ4 (0°)d1 = 165.0 mm5090°0θ5 (0°)t0−90°d10°Table 2
Joint parameters of spraying robot.
Joint rotationAxis direction vectorReference point on the axisSpinner coordinate value1[0 0 1]T[0 0 0]T[0 0 1 0 0 0]T2[−1 0 0]T[0 0 0.340]T[−1 0 0 0 −0.340 0]T3[−1 0 0]T[0 0 1.640]T[−1 0 0 0 −1.640 0]T4[0 1 0]T[0 0 1.640]T[0 1 0 −1.640 0 0]T5[0 0.819 −0.574]T[0 1.509 1.640]T[0 0.819 −0.574 −2.209 0 0]T6[0 0.819 0.574]T[0 1.639 1.640]T[0 0.819 0.574 −0.404 0 0]T7[1 0 0]T[0 0 1.640]T[1 0 0 0 1.64 0]T


## 3.2. Trajectory Planning of Spraying Robot

Trajectory planning in robot joint space is to interpolate the positions and postures of given path points, such as the starting and ending points, under the condition of satisfying a series of constraints to generate the motion trajectory. The trajectory planning in the joint space can be said to be intuitive, on the one hand, because the joint motion can be controlled directly according to the change of the joint angle to achieve the purpose of driving the robot to move, or it can be said that it is not intuitive. On the other hand, it is difficult to guarantee the trajectory of the robot end effector because of its joint angle interpolation, which leads to the nonintuitive trajectory of the robot end effector. Therefore, trajectory planning in joint space is generally used for trajectory planning that does not require high motion trajectories or has a short motion distance and can ignore the influence of motion trajectories, such as point-to-point motion. In the joint space, three polynomial interpolation methods commonly used in path planning methods are high-order polynomial interpolation, parabolic interpolation, and spline curve interpolation [10, 11].For the movement from the position and attitude of the starting point to the position and attitude of the end point in a given time, using the method of solving the inverse kinematics of the robot introduced in the previous chapter, the problem of solving each angle of the corresponding start and end time can be solved. The change in joint angle over time is described by a functionq (t) with respect to time. For the convenience of description, the problem can be abstracted as follows: at the starting point, that is, when t0 = 0, the joint angle value is q0, and at the end point, time tf, the joint angle value is qf. In order to ensure the smoothness of the motion of the joint, the speed limit of the starting and the end points is added to the joint function q (t). At this point, the problem becomes finding a solution to the joint angle function q (t) that satisfies the four constraints while being smooth.At present, industrial robot systems mainly use teaching reproduction, robot language, and offline programming, among which teaching reproduction is the most widely used method. Reproduction is to reproduce according to the needs of the task and record some key points on the trajectory. If the trajectory of the robot’s end effector is strictly consistent with the motion requirements, enough points must be obtained. The robot’s motion trajectory is generally circular or straight, and the complex motion trajectory can be the general, straight line and arc fitting. The main spraying trajectories of the five-degree-of-freedom spraying robot studied in this paper are also spatial straight lines and spatial curves. The following describes the use of interpolation algorithm of spatial straight line based on parabola transition and interpolation algorithm of spatial arc based on the local coordinate system and the trajectory planning in Cartesian space.


## 3.3. Motion Trajectory Algorithm

The exponential product formulation is a method of modeling kinematics using joint spinners, which has a clear geometrical meaning. From the perspective of geometric analysis, some characteristics of rigid body motion can be used to eliminate the coupled joints so that the complex inverse kinematics problem is divided into simple subproblems, simplifying the inverse kinematics solution process [12].In general, the elimination of joint variables can be based on several principles shown in Figure3: (a) rotation does not change the position of the point on the axis; (b) rotation does not change the distance from a point outside the axis to the axis; (c) rotation does not change the vector along the axis; and (d) translation does not change the attitude.Figure 3
Joint screw modeling method.
(a)(b)(c)(d)Principle (a) can be described by a homogeneous expression as(1)eξθp¯=p¯.Principle (b) can be described by a homogeneous expression as(2)eξθP¯−R¯=P¯−R¯.Principle (c) can be described by a homogeneous expression as(3)eξθu¯=u¯,or a normal expression,(4)eω^θu=u,eω^θ=uT.Principle (d) can be described by a homogeneous expression as(5)eξθRP01=RP101.At present, according to the above basic principles, predecessors have deduced most of the inverse kinematics subproblems [13], which can be summarized in 11 cases, as shown in Table 3 [14] (R means rotating joint, T means moving joint). Most of the inverse kinematic problems of robots with nonredundant degrees of freedom can be solved by reducing them to these subproblems.Table 3
Inverse kinematics subproblem classification.
SubproblemOrderSymbolic representation11SubProb-R21SubProb-T32SubProb-RR42SubProb-RT52SubProb-TT63SubProb-RRR73SubProb-R RT83SubProb-RTR93SubProb-RTT103SubProb-TRT113SubProb-TTTAs the robot used in this paper only has rotary joints, this paper only discusses subproblems 1, 3, and 6. Among them, subproblem 1 is generally obtained by simplifying more than two series joints, and subproblem 3 is generally obtained by simplifying more than three series joints. Subproblem 6 is generally obtained by simplifying more than four series joints. It should be emphasized that, for the case where the four axes of rotation intersect at one point, if there is no joint linkage restriction, there is a redundant degree of freedom, and the number of inverse kinematics solutions is infinite. Figure4 can illustrate the problem more intuitively. In the case of the intersection of the transfer trajectory (an inverse kinematics solution can be obtained), the rotation radius around the axis ξ2 can be selected in countless ways, and there are countless types of joint angles (inverse kinematics solutions) that can be obtained.Figure 4
Schematic diagram of redundant degrees of freedom case.For the inverse kinematics subproblem, there are two approaches to deal with it. One is to analyze from the point of position change [15], that is, to simplify the inverse kinematics problem using the two principles (a) and (b) mentioned above. The other is to analyze from the perspective of the direction change of the vector, that is, to use the (c) principle to simplify the inverse kinematics problem. The analysis method of the position change angle is as follows.The exponential product formula for the 2R structure is given:(6)T02θ=eθ1ζ1eθ2ζ2T020.IfT02θ=Tgiven, it can be transformed into(7)eθ1ζ1eθ2ζ2=TgivenT020−1.From the perspective of the position change of the point, take any pointP on ξ2, and multiply P on both sides of the formula to have(8)eθ1ζ1eθ2ζ2P=TgivenT020−1P.According to principle (a), it can be simplified to(9)eθ1ζ1P=TgivenT020−1P=Q.At this time, subproblem 1 is the situation shown in Figure5(a), the known joint screw coordinates,(10)ξ=ω;R×ω∈se3,P,Q∈RR3.Figure 5
Schematic diagram of subproblem 1.
(a)(b)P and Q are two points outside the joint axis. If point P rotates around the joint axis to reach Q, find the rotation angle θ. It is described by the following expression:(11)eξθP=Q.First, take a pointR on the axis of rotation and define(12)U=P−R,v=Q−R.Because of principle (a),(13)eξθP=Q,eξθR=R.So,(14)eξθu=v.As shown in Figure5(b), define u,,v, as the projections of u and v on the plane perpendicular to the axis of rotation ξ; then, we have(15)u,=u−ωωTu,v′=v−ωωTv.It can get(16)ωTu=ωTv′u′=v′.θ can be obtained from the vector u,,v,. If u,≠0, then(17)u′∙v′=u′v′cosθ,u′×v,=ωu,v,sinθ.That is,(18)θ=atan2ωTu′×v′,u′Tv′.Ifu′=0, then there are infinitely many solutions.


## 4. Airbrush Trajectory Optimization for Curvature Composite Surfaces

For the aesthetics, rust prevention, insulation, and specific functions of the product surface, it is necessary to apply one or more layers of coatings to these curved surfaces. However, the surface features of these workpieces to be sprayed are often complex, which not only increases the difficulty of effective spraying by spraying robots, but also brings greater challenges to uniform spraying. For this purpose, a series of large-curvature curved surfaces and small-curvature curved surfaces (including plane ones) are formed; that is, the complex free-form surface is an arbitrary combination of a series of large-curvature surfaces and small-curvature surfaces, which are called curvature combination surfaces here. When spraying a curved surface with a combination of curvatures, especially when the surface coating uniformity is required to be high, the quality of the coating cannot be guaranteed only by the experience of workers or by random spraying [16, 17]. In order to reduce the complexity of the problem, based on the curvature characteristics of the surface, the curvature combined surface is divided into a large-curvature patch group and a small curvature patch group and then divided into three forms: small curvature patch and small curvature patch, small curvature patch and large-curvature patch, and large-curvature patch and large-curvature patch, which are discussed, respectively, according to their different intersecting combinations. Of course, some areas will become horns due to excessive curvature, which needs to be discussed separately. For the convenience of specific analysis, the common large-curvature cylindrical sheet, conical sheet, and spherical sheet represent the large-curvature surface sheet for analysis and discussion.


### 4.1. Optimization Methods for Small Curvature Complex and Large Curvature Surfaces

For the surface with small curvature, its own shape features are relatively single, and the surface area is larger and flatter. However, when the formed small-curvature surface patches are irregular, contain holes, and are in different curvature patches, in order to avoid the problem of paint wasting and prolonging the spraying cycle caused by the deposition of paint at the holes, it cannot be sprayed arbitrarily [18, 19]. The method of geometric topology decomposition is used to decompose such patches into subpatches with simple topology (without holes); that is, on these subpatches, a spray gun space path that can be sprayed with a simple back-and-forth movement path can be planned.For the large-curvature surface, in the previous chapter, based on the mesh CAD model and the triangle patch algorithm, the curved combined surface is modeled, and the definition of a large-curvature surface and the method of extracting large and small curvature areas are given. In order to improve the spraying effect of spraying large-curvature surfaces, the least-squares surface fitting algorithm is used to reshape the large-curvature surfaces into simple, regular, and easy-to-spray quadric surfaces. In this way, the problems of difficult spraying and poor spraying effect on complex and large-curvature surfaces are converted into spraying problems of quadratic surfaces that are simple and easier to spray. Here, the large-curvature surface patches are embodied as large-curvature cylinders, cones, and spherical patches for analysis and discussion.


### 4.2. Simulation Experiment

Set ideal coating thicknessqd = 50 μm, maximum allowable error of coating thickness ∆qd = 10 μm, spray radius range [41 mm, 50 mm], spray gun cone angle φ=45°, and spray flow rate Q = 200 ml/s. The spraying time of each point on the spraying track is ∆t=0.5s, the utilization coefficient of the paint is 0.822, the variation range of the allowable spraying height h is [99.0 mm, 120.5 mm], and the variation range of the distance between adjacent spray gun tracks on the optimized plate is [50.0 mm, 60.5 mm]. Moreover, the solution for the optimization objective function is run and calculated in the software MATLABR2013a environment.Let the radius of the cylinder beρ=50mm, the radius of the bottom surface of the cone R = 50 mm, and the height H = 50 mm. The large-curvature cylindrical sheet and the large-curvature conical sheet are, respectively, ¼ of the surface size of the right cylinder and the right cone, both are tangent to the small curvature surface at the junction, namely, γ=90°, the length of the intersection line is L = 50 mm, and the spray radius on the plane is R = 50 mm.For the intersection of the small curvature surface and the large-curvature cylinder patch, the optimized spray gun speeds on the respective surfaces are 396.4 and 392.1 mm/s and are constant on the respective patches. After spraying for 1 s, the optimized valueh1 = 24.8 mm and β=23.7° was obtained by calculation so that the variation range of coating thickness at the interface of curved surfaces was [49.1 μm, 50.9 μm], as shown in Figure 6.Figure 6
Coating thickness distribution at the intersection of the large-curvature cylindrical sheet and the approximately planar sheet.For the intersection of the small curvature surface and the large-curvature cone, set the lengthL2 = 50 mm when the spray gun moves from the maximum radius to the minimum radius on the cone surface, and adopt the method of segmental optimization of the intersecting straight generatrix trajectory. Here, the trajectory is evenly divided into three segments and optimized, respectively, and the spraying radius from large to small can be obtained. The spray gun rates are 390.3, 395.6, 403.8, and 394.7 mm/s on the small curvature surface, they are all constant on the respective patches, h1 = 23.6 mm, and β=23.3° is obtained by optimization. Figure 7 shows the coating distribution in the intersection area of the curved surfaces when the spraying radii on the large-curvature cone surface are 44.5 and 49.5 mm, respectively.Figure 7
Coating thickness distribution at the intersection of the large-curvature conical surface and the approximately flat sheet.To sum up, the coating thicknesses at the junction of curved surfaces with different curvatures meet the requirements of the coating deviation range, which also shows the effectiveness of the established spray gun model and the feasibility of the solution for the uneven coating at the junction of different curvature surfaces and more concretely discusses the spraying problem facing the curvature composite surface.


## 4.1. Optimization Methods for Small Curvature Complex and Large Curvature Surfaces

For the surface with small curvature, its own shape features are relatively single, and the surface area is larger and flatter. However, when the formed small-curvature surface patches are irregular, contain holes, and are in different curvature patches, in order to avoid the problem of paint wasting and prolonging the spraying cycle caused by the deposition of paint at the holes, it cannot be sprayed arbitrarily [18, 19]. The method of geometric topology decomposition is used to decompose such patches into subpatches with simple topology (without holes); that is, on these subpatches, a spray gun space path that can be sprayed with a simple back-and-forth movement path can be planned.For the large-curvature surface, in the previous chapter, based on the mesh CAD model and the triangle patch algorithm, the curved combined surface is modeled, and the definition of a large-curvature surface and the method of extracting large and small curvature areas are given. In order to improve the spraying effect of spraying large-curvature surfaces, the least-squares surface fitting algorithm is used to reshape the large-curvature surfaces into simple, regular, and easy-to-spray quadric surfaces. In this way, the problems of difficult spraying and poor spraying effect on complex and large-curvature surfaces are converted into spraying problems of quadratic surfaces that are simple and easier to spray. Here, the large-curvature surface patches are embodied as large-curvature cylinders, cones, and spherical patches for analysis and discussion.


## 4.2. Simulation Experiment

Set ideal coating thicknessqd = 50 μm, maximum allowable error of coating thickness ∆qd = 10 μm, spray radius range [41 mm, 50 mm], spray gun cone angle φ=45°, and spray flow rate Q = 200 ml/s. The spraying time of each point on the spraying track is ∆t=0.5s, the utilization coefficient of the paint is 0.822, the variation range of the allowable spraying height h is [99.0 mm, 120.5 mm], and the variation range of the distance between adjacent spray gun tracks on the optimized plate is [50.0 mm, 60.5 mm]. Moreover, the solution for the optimization objective function is run and calculated in the software MATLABR2013a environment.Let the radius of the cylinder beρ=50mm, the radius of the bottom surface of the cone R = 50 mm, and the height H = 50 mm. The large-curvature cylindrical sheet and the large-curvature conical sheet are, respectively, ¼ of the surface size of the right cylinder and the right cone, both are tangent to the small curvature surface at the junction, namely, γ=90°, the length of the intersection line is L = 50 mm, and the spray radius on the plane is R = 50 mm.For the intersection of the small curvature surface and the large-curvature cylinder patch, the optimized spray gun speeds on the respective surfaces are 396.4 and 392.1 mm/s and are constant on the respective patches. After spraying for 1 s, the optimized valueh1 = 24.8 mm and β=23.7° was obtained by calculation so that the variation range of coating thickness at the interface of curved surfaces was [49.1 μm, 50.9 μm], as shown in Figure 6.Figure 6
Coating thickness distribution at the intersection of the large-curvature cylindrical sheet and the approximately planar sheet.For the intersection of the small curvature surface and the large-curvature cone, set the lengthL2 = 50 mm when the spray gun moves from the maximum radius to the minimum radius on the cone surface, and adopt the method of segmental optimization of the intersecting straight generatrix trajectory. Here, the trajectory is evenly divided into three segments and optimized, respectively, and the spraying radius from large to small can be obtained. The spray gun rates are 390.3, 395.6, 403.8, and 394.7 mm/s on the small curvature surface, they are all constant on the respective patches, h1 = 23.6 mm, and β=23.3° is obtained by optimization. Figure 7 shows the coating distribution in the intersection area of the curved surfaces when the spraying radii on the large-curvature cone surface are 44.5 and 49.5 mm, respectively.Figure 7
Coating thickness distribution at the intersection of the large-curvature conical surface and the approximately flat sheet.To sum up, the coating thicknesses at the junction of curved surfaces with different curvatures meet the requirements of the coating deviation range, which also shows the effectiveness of the established spray gun model and the feasibility of the solution for the uneven coating at the junction of different curvature surfaces and more concretely discusses the spraying problem facing the curvature composite surface.


## 5. Solving the ORPP Problem Based on the Improved Particle Swarm Algorithm

Particle swarm optimization (PSO) is an optimization computing technology, which is an intelligent algorithm proposed in 1995 for the analysis and research of the bird flock model-Boid model [20]. The PSO algorithm requires few parameters, has a fast convergence speed, and especially has a powerful global search ability. Since it was proposed, it has been widely used in various fields such as scheduling optimization and economic allocation, chemical engineering, data mining, biological engineering, nonlinear programming, robotics, system identification, and optical fiber communication because of its simple concept, easy implementation, fast convergence speed, few required parameters, and strong global search capabilities. In the algorithm model, a set of position and velocity vectors are set, which represent the feasible solution of the problem and the direction of motion in the search space to denote each particle. After repeated learning, the particle finally achieves the global optimal search.In this paper, MATLAB is used to run the algorithm program to solve the problem of spray gun trajectory combination and connection. Assuming that a surface is composed of patches with different curvatures, the connection graphG has five edges and 10 vertices. Therefore, in the algorithm, the population size is 10, and the maximum number of iterations is 200.The combined curvature surface shown in Figure8 is composed of a small curvature surface sheet 1, a large-curvature column (2 and 4), a cone (3), and a ball (5) curved surface sheet. After obtaining the optimal trajectory on each surface, the next step is to connect the trajectories on each surface to form a complete spray gun path. Here, the projection coordinates of the path inflection point when turning from each curved piece to connect different curved pieces are shown in Figure 9.Figure 8
A combined surface with large and small curvatures.Figure 9
The optimized shortest path connection graph.As shown in Figure10, when running this algorithm, with the continuous evolution process, the path length of the spray gun shows a rapidly decreasing trend. It can be seen from the optimization results in Figure 10 that the iterative effect of optimizing the particle swarm optimization algorithm is still good. As the number of iterations is updated, the optimization of the path is stepped. The iterative convergence of the spray gun path length planning is very fast, and the final distance path length converges at 410 mm, which also reflects the excellence of the PSO algorithm.Figure 10
Optimization results.


## 6. Conclusions

This paper studies the key technologies of the spraying robot trajectory planning system. First of all, it analyzes the research background of industrial robots, and not only understands the research status of spraying robots, but also explains its deficiencies. Then, the path planning of the spraying robot is analyzed, the research and design of the color pattern information clustering algorithm, the research and design of the obstacle avoidance path planning algorithm, the final program writing and simulation, and the method of clustering the color information of the model are studied. This paper first proposes an easy-to-implement combined hierarchical clustering algorithm, then designs an improved ISODATA algorithm for the shortcomings of the hierarchical clustering algorithm, and finally proves the effectiveness of the proposed algorithm. However, there are still some problems that need to be improved. For example, the spraying rate of the spraying robot has not been tested, so this will be the goal of follow-up research.



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*Source: 8923901-2022-08-08.xml*