CN102500498B

CN102500498B - Optimization method for spray gun track of spraying robot on irregular polyhedron

Info

Publication number: CN102500498B
Application number: CN201110355488.6A
Authority: CN
Inventors: 陈伟; 曾庆军; 汤养; 李春华; 章飞; 王彪
Original assignee: Jiangsu University of Science and Technology
Current assignee: Jiangsu Newblue Intelligent Equipment Co ltd
Priority date: 2011-11-11
Filing date: 2011-11-11
Publication date: 2014-06-04
Anticipated expiration: 2031-11-11
Also published as: CN102500498A

Abstract

A spray gun trajectory optimization method for a spraying robot on an irregular polyhedron. Triangulation is performed on each face of the polyhedron according to the CAD data of the polyhedron, the normal vector of each triangular face is calculated, and several adjacent triangular faces are connected according to the topological structure. For a larger piece, build a "cuboid model" and generate spray paths on each face of the polyhedron. Taking the variance of the actual coating thickness and the ideal coating thickness as the objective function, the optimal value of the width of the overlapping area of the coating is obtained by using the golden section method on each surface of the irregular polyhedron. In order to improve the spraying efficiency, the undirected connection graph is used to represent the optimal combination problem of the spray gun trajectory on the surface, and the improved particle swarm optimization algorithm is used to solve it. Thereby, the spraying quality is guaranteed, and the spraying efficiency is improved at the same time.

Description

Optimization method of spray gun trajectory of spraying robot on irregular polyhedron

技术领域 technical field

本发明涉及喷涂机器人自动喷涂过程中，针对多面体工件进行喷涂作业时的喷枪轨迹优化方法。The invention relates to a spray gun trajectory optimization method for a polyhedron workpiece during the automatic spraying process of a spraying robot.

背景技术 Background technique

喷涂机器人是一种非常重要的先进涂装生产装备，在国内外广泛应用于汽车等产品的涂装生产线。对于诸如汽车、电器及家具等产品，其表面的喷涂效果对质量有相当大的影响。在自动喷涂操作中，喷涂机器人的机械手围绕待涂工件表面来回移动，适当的轨迹和其它过程参数的选择都能使生产成本得到节约。喷涂机器人的喷涂效果与物体表面形状密切相关。对于诸如汽车、电器及家具等产品，其表面的喷涂效果对质量有相当大的影响。实际生产中，喷涂机器人喷涂作业的优化目标主要有两个：一是工件表面的涂层尽量均匀；二是喷涂时间尽量短。然而，这两个优化目标(效果和效率)通常是相互制约的。如何在保证喷涂效果的前提下，尽量提高喷涂效率是喷涂机器人轨迹优化问题中的难点之一。Spraying robot is a very important advanced coating production equipment, which is widely used in the coating production line of automobiles and other products at home and abroad. For products such as automobiles, electrical appliances and furniture, the spraying effect on the surface has a considerable impact on the quality. In the automatic spraying operation, the manipulator of the spraying robot moves back and forth around the surface of the workpiece to be coated, and the selection of appropriate trajectory and other process parameters can save production costs. The spraying effect of the spraying robot is closely related to the surface shape of the object. For products such as automobiles, electrical appliances and furniture, the spraying effect on the surface has a considerable impact on the quality. In actual production, there are two main objectives for optimizing the spraying operation of the spraying robot: one is that the coating on the surface of the workpiece is as uniform as possible; the other is that the spraying time is as short as possible. However, these two optimization goals (effectiveness and efficiency) are usually mutually restrictive. How to improve the spraying efficiency as much as possible on the premise of ensuring the spraying effect is one of the difficulties in the trajectory optimization of the spraying robot.

近年来，随着喷涂机器人的广泛应用，机器人喷涂已基本上能满足工业生产的需要。但由于制造工业的不断发展，出现了许多非规则多面体工件。由于多面体的结构复杂多变，采用一般的Bezier曲面和B样条曲面造型方法很难对非规则多面体上表面进行处理。目前，大多数喷涂机器人轨迹优化方法只适用于二维平面或曲面上的喷涂作业，不适用于三维多面体上的喷涂作业。对三维多面体进行喷涂作业的系统还不能进行轨迹优化，存在工作效率低、机器人位置和速度控制精度低、喷涂效果不理想等缺点，从而使得表面涂层不均匀，产品不能达到较高的工艺水平。因此非规则多面体工件上的喷枪轨迹优化设计是机器人轨迹规划问题中的又一个难点。公开号CN 101239346提供了一种复杂曲面上的喷涂机器人喷枪轨迹优化方法，可实现对复杂曲面的工件进行喷涂作业时的机器人自动喷涂，但该方法对于非规则多面体并不适用。公开号CN101367076A提供了一种非规则平面多边形的静电喷涂机器人变量喷涂方法，但该方法只能用于二维平面上，不能应用于三维多面体上。In recent years, with the widespread application of spraying robots, robot spraying has basically been able to meet the needs of industrial production. However, due to the continuous development of the manufacturing industry, many irregular polyhedron workpieces have appeared. Due to the complex and changeable structure of the polyhedron, it is difficult to deal with the upper surface of the irregular polyhedron by using the general Bezier surface and B-spline surface modeling methods. At present, most spraying robot trajectory optimization methods are only suitable for spraying operations on two-dimensional planes or curved surfaces, but not on three-dimensional polyhedrons. The system for spraying three-dimensional polyhedrons cannot perform trajectory optimization, and has disadvantages such as low work efficiency, low robot position and speed control accuracy, and unsatisfactory spraying effects, resulting in uneven surface coating and products that cannot reach a high level of technology . Therefore, the optimal design of the spray gun trajectory on the irregular polyhedron workpiece is another difficult point in the robot trajectory planning problem. Publication number CN 101239346 provides a spray gun trajectory optimization method for spraying robots on complex curved surfaces, which can realize automatic spraying by robots when spraying workpieces with complex curved surfaces, but this method is not suitable for irregular polyhedrons. Publication No. CN101367076A provides a variable spraying method of an electrostatic spraying robot for irregular plane polygons, but this method can only be used on a two-dimensional plane and cannot be applied on a three-dimensional polyhedron.

因此，在实际生产中喷涂非规则多面体工件时，产品外观质量不能得到进一步提升，而且不能实现复杂、多面上的全自动喷涂。机器人喷涂工件的主要部分后仍需人工进行喷涂，费时、费力、费料，且工人仍处于有害环境中。Therefore, when spraying irregular polyhedron workpieces in actual production, the appearance quality of the product cannot be further improved, and complex, multi-faceted automatic spraying cannot be realized. After the robot sprays the main part of the workpiece, it still needs to be sprayed manually, which is time-consuming, laborious, and material-consuming, and the workers are still in a harmful environment.

发明内容 Contents of the invention

本发明正是为了解决上述问题，目的在于提供一种专门的针对非规则多面体上的喷涂机器人喷枪轨迹优化方法，在保证喷涂效果和喷涂效率的前提下，以实现机器人对非规则多面体的自动喷涂，满足实际工业生产的需要。The present invention is just to solve the above problems, and the purpose is to provide a special method for optimizing the spray gun trajectory of the spraying robot on the irregular polyhedron, under the premise of ensuring the spraying effect and spraying efficiency, to realize the automatic spraying of the irregular polyhedron by the robot , to meet the needs of actual industrial production.

本发明解决其技术问题所采用的技术方案是：本发明非规则多面体上的喷涂机器人喷枪轨迹优化方法，将多面体工件的CAD数据输入GID软件，通过GID网格图形输出功能对多面体的每个面进行三角划分，计算每个三角面的法向量，按照相邻三角面之间拓扑结构连接生成若干个较大的片，建立“长方体模型”并生成多面体每个面上的喷涂路径；以实际涂层厚度与理想涂层厚度方差为目标函数，在非规则多面体的每个面上采用黄金分割法求解涂层重叠区域宽度的最优值；为提高喷涂效率，利用无方向的连接图表示曲面上的喷枪轨迹优化组合，并采用改进的粒子群算法进行求解。The technical scheme that the present invention solves its technical problem adopts is: the spraying robot spray gun trajectory optimization method on the irregular polyhedron of the present invention, the CAD data input GID software of polyhedron workpiece, each face of polyhedron by GID grid graphics output function Carry out triangulation, calculate the normal vector of each triangular face, generate several larger slices according to the topological structure connection between adjacent triangular faces, establish a "cuboid model" and generate the spraying path on each face of the polyhedron; The variance between the layer thickness and the ideal coating thickness is the objective function, and the golden section method is used to solve the optimal value of the width of the overlapping area of the coating on each surface of the irregular polyhedron; in order to improve the spraying efficiency, a non-directional connection graph is used to represent The trajectory optimization combination of the spray gun is solved by an improved particle swarm algorithm.

所述生成喷枪路径的方法如下：首先沿垂直于“长方体模型”右侧方向作若干个距离为l的切平面，即得到切平面与曲面片的若干段相交线；然后再在相交线上平均地作出距离为d的一组点，d即为两条喷涂路径之间的距离，该距离的大小可人为设定或通过优化计算得出；最后将这些点沿“长方体模型”右侧方向连接起来，从而生成喷枪空间路径，其中l取R/2～R，R为喷枪喷涂半径。The method for generating the spray gun path is as follows: first make several tangent planes with a distance of l along the direction perpendicular to the right side of the "cuboid model", that is, to obtain some intersection lines of the tangent planes and curved surface pieces; Make a group of points with a distance of d, d is the distance between two spraying paths, the size of the distance can be set artificially or calculated through optimization; finally connect these points along the right side of the "cuboid model" Up, thus generating the spray gun space path, where l takes R/2~R, and R is the spraying radius of the spray gun.

所述利用无方向的连接图表示曲面上的喷枪轨迹优化组合的方法如下：用一个无方向的连接图G(V，E，R，ω：E→Z⁺)表示喷枪轨迹优化组合，其中V表示顶点集，E表示边集，R表示E的任意一个子集，ω表示边的权即实际喷枪轨迹的长度，在无方向的连接图G中求出一条经过所有边且只经过一次的具有最短距离的回路；M＝{d_ij}是由图G中不在同一条边上的顶点i和顶点j之间的最短轨迹所组成的集合，i，j＝1，2，...，n，n为多面体面的个数。The method for expressing the optimal combination of spray gun trajectories on a curved surface using a non-directional connection graph is as follows: use a non-directional connection graph G (V, E, R, ω: E→Z ⁺ ) to represent the optimal combination of spray gun trajectories, where V Represents the vertex set, E represents the edge set, R represents any subset of E, ω represents the weight of the edge, that is, the length of the actual spray gun trajectory, in the undirected connection graph G, find a path that passes through all edges and only once. The circuit of the shortest distance; M={d _ij } is the set consisting of the shortest trajectories between vertices i and j that are not on the same edge in graph G, i, j=1, 2, ..., n , n is the number of polyhedron faces.

采用改进的粒子群算法进行求解的方法如下：Using the improved particle swarm optimization algorithm to solve the problem is as follows:

Step1初始化：初始化粒子位置

i＝1，2，...，m；初始化每个粒子的速度

i＝1，2，...，m；选择速度最大阈值ε和最大迭代次数N_max，迭代次数k＝0；Step1 initialization: Initialize the particle position

i=1,2,...,m; initialize the velocity of each particle

i=1, 2,..., m; select the maximum speed threshold ε and the maximum number of iterations N _max , and the number of iterations k=0;

Step2测量每个粒子的适应值

表示为

令

Step2 measures the fitness value of each particle

Expressed as

make

Step3迭代次数k←k+1；更新速度更新位置

Step3 The number of iterations k←k+1; update speed update location

Step4测量z_i的适应值，表示为

取更新

和

Step4 measures the fitness value of _zi , expressed as

Pick renew

and

Step5若

且k＜N_max，则跳转到Step3；若k≥N_max，循环停止，输出计算结果。Step5 if

And k<N _max , jump to Step 3; if k≥N _max , stop the loop and output the calculation result.

本发明具有很强的实用性，能够实现对非规则多面体工件的机器人自动喷涂，可提高喷涂机器人工作效率以及产品的品质。The invention has strong practicability, can realize robot automatic spraying on irregular polyhedral workpieces, and can improve the working efficiency of the spraying robot and the quality of products.

附图说明Description of drawings

图1长方体模型示意图；Fig. 1 schematic diagram of cuboid model;

图2多面体任意一个面上的喷枪空间路径示意图；Fig. 2 schematic diagram of the spray gun space path on any face of the polyhedron;

图3平面上的喷涂示意图；The schematic diagram of spraying on the plane of Fig. 3;

图4无方向的连接图。Figure 4 Connection diagram without direction.

具体实施方式 Detailed ways

下面结合附图和实施方式对本发明进一步说明。本发明实施步骤由多面体上喷涂机器人喷枪空间路径的生成、多面体每个面上的喷枪轨迹优化、喷枪轨迹优化组合三大部分组成，具体实施方式如下。The present invention will be further described below in conjunction with the accompanying drawings and embodiments. The implementation steps of the present invention are composed of three parts: the generation of the space path of the spray gun of the spraying robot on the polyhedron, the optimization of the trajectory of the spray gun on each face of the polyhedron, and the optimization combination of the trajectory of the spray gun. The specific implementation is as follows.

1多面体上喷涂机器人喷枪空间路径的生成1 Spatial Path Generation of Spraying Robot Spray Gun on Polyhedron

多面体上喷涂机器人喷枪空间路径具体步骤如下：The specific steps of spraying robot spray gun space path on polyhedron are as follows:

(1)由工件CAD模型确定非规则多面体，在GID7.2软件中输入非规则多面体CAD图形后，通过GID网格图形输出功能，得到非规则多面体的三角网格图形(允许误差2mm，每一个网格称为三角面或三角片)。多面体的每个面进行三角网格划分后可以用数学表达式表示为：(1) The irregular polyhedron is determined by the CAD model of the workpiece. After inputting the CAD graphics of the irregular polyhedron in the GID7.2 software, the triangular mesh graphic of the irregular polyhedron is obtained through the GID grid graphic output function (allowable error 2mm, each The mesh is called a triangular face or triangular sheet). After each face of the polyhedron is divided into triangular meshes, it can be expressed mathematically as:

M＝{T_i：i＝1，...，M} (1)M={T _i : i=1, . . . , M} (1)

这里T_i是三角网格中的第i个三角面(片)，M是三角网格中三角面的总个数。Here T _i is the ith triangular face (piece) in the triangular mesh, and M is the total number of triangular faces in the triangular mesh.

(2)计算每个三角面的法向量，按照相邻三角面之间拓扑结构连接生成若干个较大的片。假设沿平面上的优化喷涂轨迹进行喷涂后，平均涂层厚度为

整个平面上某一点的最大涂层厚度为

某一点的最小涂层厚度为

再设每个面上的法向量与每个面投影平面的法向量的最大夹角为β_th(只考虑二者的法向量指向每个面的同侧)，则多面体任意一个面上任一点s上的涂层厚度q_s可能的范围为：(2) Calculate the normal vector of each triangular face, and generate several larger slices according to the topological connection between adjacent triangular faces. Assuming that after spraying along the optimized spraying trajectory on the plane, the average coating thickness is

The maximum coating thickness at a point on the entire plane is

The minimum coating thickness at a point is

Then set the maximum angle between the normal vector on each surface and the normal vector of each surface projection plane to be β _th (only consider that the normal vector of the two points to the same side of each surface), then any point s on any surface of the polyhedron The possible range of coating thickness q _s on is:

${\overset{&OverBar; &OverBar;}{q q}}_{min min} cos cos (({β β}_{th the th})) \leq \leq {q q}_{s the s} \leq \leq {\overset{&OverBar; &OverBar;}{q q}}_{max max} - - - - - - ((22))$

如果面上的任意一点s上的涂层厚度满足：If the coating thickness at any point s on the surface satisfies:

$| | {q q}_{s the s} - - {\overset{&OverBar; &OverBar;}{q q}}_{d d} | | \leq \leq {q q}_{w w} - - - - - - ((33))$

其中，q_w为允许最大涂层厚度偏差。那么，Among them, q _w is the allowable maximum coating thickness deviation. So,

${\overset{&OverBar; &OverBar;}{q q}}_{max max} - - {q q}_{d d} \leq \leq {q q}_{w w} - - - - - - ((44))$

${q q}_{d d} - - {\overset{&OverBar; &OverBar;}{q q}}_{min min} cos cos (({β β}_{th the th})) \leq \leq {q q}_{w w} - - - - - - ((55))$

如果式(4)始终成立，则可以根据式(5)求出β_th。这也就是说，对于多面体上的任意一个面，如果面上的法向量与其投影平面的最大夹角β满足β≤β_th，则该面上某一点的涂层厚度就能够满足式(3)。在求出β_th后，即可生成多面体上任意一个面的每一片。各个三角面连接成一块整体片的步骤如下：If formula (4) always holds true, then β _th can be calculated according to formula (5). That is to say, for any surface on the polyhedron, if the maximum angle β between the normal vector on the surface and its projection plane satisfies β≤β _th , then the coating thickness at a certain point on the surface can satisfy the formula (3) . After calculating _βth , each slice of any face on the polyhedron can be generated. The steps of connecting each triangular face into a whole piece are as follows:

①指定任意一个三角面为初始三角面。① Designate any triangular face as the initial triangular face.

②寻找到离初始三角面中心点距离小于喷枪喷涂半径的所有三角面。②Find all the triangular faces whose distance from the center point of the initial triangular face is less than the spraying radius of the spray gun.

③计算第②步中找到的所有三角面的法向量与初始三角面法向量的夹角，若夹角小于β_th，则将该三角面与初始三角面连接。③Calculate the angle between the normal vectors of all the triangles found in step ② and the normal vector of the initial triangle, if the angle is less than β _th , connect the triangle to the initial triangle.

④寻找尚未连接成片的三角面作为新的初始三角面，重复②、③步，直到所有三角面都连接成片。④Find the triangular faces that have not been connected into pieces as the new initial triangular faces, and repeat steps ② and ③ until all the triangular faces are connected into pieces.

多面体任意一个面分片后，其中的某一片可以表示为：After any face of the polyhedron is sliced, one of the slices can be expressed as:

式中S_i表示第i片，

和是第j个三角面和第k个三角面的法向量，D(T_j，T_k)表示第i个三角面和第k个三角面中心点的距离。由此，多面体任意一个面将被分成一片或者若干片。In the formula, S _i represents the i-th piece,

and is the normal vector of the j-th triangle and the k-th triangle, and D(T _j , T _k ) represents the distance between the i-th triangle and the center point of the k-th triangle. Thus, any face of the polyhedron will be divided into one or several pieces.

(3)在每一片上建立“长方体模型”，并生成每一片上的喷枪路径。附图1所示的是在某一片上建立的“长方体模型”。“长方体模型”是一个恰好包含了整个片的长方体，它主要具有以下两个性质：(i)其前端方向是与整个片的法向量方向相反；(ii)其各个面的长方形的面积都尽可能的最小。为了生成喷枪路径，首先沿垂直于“长方体模型”右侧方向作若干个距离为l的切平面(l通常取

R为喷枪喷涂半径)，即可得到切平面与曲面片的若干段相交线；然后再在相交线上平均地作出距离为d的一组点；最后将这些点沿“长方体模型”右侧方向连接起来，从而生成喷枪空间路径(如附图2所示)。(3) Establish a "cuboid model" on each piece, and generate the spray gun path on each piece. What shown in accompanying drawing 1 is " cuboid model " that builds up on a certain piece. The "cuboid model" is a cuboid that just contains the whole piece, and it mainly has the following two properties: (i) the direction of its front end is opposite to the direction of the normal vector of the whole piece; (ii) the area of the rectangle on each surface is as large as possible. the smallest possible. In order to generate the spray gun path, first make several tangent planes with a distance of l along the direction perpendicular to the right side of the "cuboid model" (l is usually taken as

R is the spraying radius of the spray gun), then you can get several intersection lines between the tangent plane and the curved surface sheet; then make a group of points with a distance of d on average on the intersection line; finally put these points along the right direction of the "cuboid model Connected to generate the spray gun space path (as shown in Figure 2).

2多面体每个面上的喷枪轨迹优化2 Spray gun trajectory optimization on each face of the polyhedron

附图3所示的是平面上的喷涂过程。图中R表示喷涂半径，v表示喷枪移动速度，d表示为两个喷涂行程的涂层重叠区域宽度，x表示喷涂半径内某一点s到第一条路径的距离，s′为s点在路径上的投影，O点为喷枪中心投影点，则点s的涂层厚度为：Shown in accompanying drawing 3 is the spraying process on the plane. In the figure, R represents the spraying radius, v represents the moving speed of the spray gun, d represents the width of the coating overlapping area of two spraying strokes, x represents the distance from a point s within the spraying radius to the first path, and s′ represents the distance between point s and the path The projection on the point O is the projection point of the spray gun center, then the coating thickness at point s is:

${q q}_{s the s} ((x x)) = = \{\begin{matrix} {q q}_{11} ((x x)) & 00 \leq \leq x x \leq \leq R R - - d d \\ {q q}_{11} ((x x)) + + {q q}_{22} ((x x)) & R R - - d d < < x x \leq \leq R R \\ {q q}_{22} ((x x)) & R R < < x x \leq \leq 22 R R - - d d \end{matrix} - - - - - - ((77))$

q₁(x)和q₂(x)分别表示两条相邻路径上喷涂时s点的涂层厚度，q₁(x)和q₂(x)计算公式为：q ₁ (x) and q ₂ (x) respectively represent the coating thickness at point s when spraying on two adjacent paths, and the calculation formulas of q ₁ (x) and q ₂ (x) are:

$q_{1} (x) = 2 {&Integral;}_{0}^{t_{1}} f (r_{1}) dt,$ 0≤x≤R； $q_{2} (x) = 2 {&Integral;}_{0}^{t_{2}} f (r_{2}) dt,$ R-d≤x≤2R-d (8) $q_{1} (x) = 2 {&Integral;}_{0}^{t_{1}} f (r_{1}) dt,$ 0≤x≤R; $q_{2} (x) = 2 {&Integral;}_{0}^{t_{2}} f (r_{2}) dt,$ Rd≤x≤2R-d (8)

其中， $t_{1} = \sqrt{R^{2} - x^{2}} / v;$ $t_{2} = \sqrt{R^{2} - {(2 R - d - x)}^{2}} / v$ in, $t_{1} = \sqrt{R^{2} - x^{2}} / v;$ $t_{2} = \sqrt{R^{2} - {(2 R - d - x)}^{2}} / v$

${r r}_{11} = = \sqrt{{((vt vt))}^{22} + + {x x}^{22}};;$ ${r r}_{22} = = \sqrt{{((vt vt))}^{22} {((22 R R - - d d - - x x))}^{22}}$

t₁和t₂分别表示两条相邻喷涂路径上喷枪在s点喷涂时间的一半；r₁和r₂分别表示s点到两条相邻喷涂路径上的喷枪中心投影点的距离；t为喷枪从点O运动到点s′的时间。由(8)式可得：t ₁ and t ₂ represent half of the spraying time of the spray gun at point s on two adjacent spraying paths; r ₁ and r ₂ represent the distances from point s to the projection point of the spray gun center on two adjacent spraying paths; t is The time for the spray gun to move from point O to point s'. From formula (8), we can get:

${q q}_{s the s} ((x x,, d d,, v v)) = = \frac{11}{v v} J J ((x x,, d d)) - - - - - - ((99))$

其中J为x和d的函数。为了使工件表面涂层厚度尽可能均匀，取s点的实际涂层厚度与理想涂层厚度之间的方差为优化目标函数：where J is a function of x and d. In order to make the surface coating thickness of the workpiece as uniform as possible, the variance between the actual coating thickness and the ideal coating thickness at point s is taken as the optimization objective function:

$\underset{d d &Element; &Element; [[00,, R R]],, v v}{min min} {E E.}_{11} ((d d,, v v)) = = {&Integral; &Integral;}_{00}^{22 R R - - d d} {(({q q}_{d d} - - {q q}_{s the s} ((x x,, d d,, v v))))}^{22} dx dx - - - - - - ((1010))$

式中q_d为理想涂层厚度。由于最大涂层厚度q_max和最小涂层厚度q_min决定了工件表面上涂层厚度的均匀性，因此，q_max和q_min也需要进行优化：Where q _d is the ideal coating thickness. Since the maximum coating thickness q _max and the minimum coating thickness q _min determine the uniformity of the coating thickness on the workpiece surface, therefore, q _max and q _min also need to be optimized:

$\underset{d d &Element; &Element; [[00,, R R]],, v v}{min min} {E E.}_{22} ((d d,, v v)) = = {(({q q}_{max max} - - {q q}_{d d}))}^{22} + + {(({q q}_{d d} - - {q q}_{min min}))}^{22} - - - - - - ((1111))$

由式(9)、(10)、(11)可得：From formula (9), (10), (11) can get:

$\underset{d d &Element; &Element; [[00,, R R]],, v v}{min min} E E. ((d d,, v v)) = = \frac{11}{22 R R - - d d} {E E.}_{11} ((d d,, v v)) + + {E E.}_{22} ((d d,, v v)) - - - - - - ((1212))$

又由(7)式，最大涂层厚度和最小涂层厚度表达式可写为：Also by formula (7), the expressions of maximum coating thickness and minimum coating thickness can be written as:

${q q}_{max max} = = \frac{11}{v v} {J J}_{max max} ((d d)),,$ ${q q}_{min min} = = \frac{11}{v v} {J J}_{min min} ((d d)) - - - - - - ((1313))$

令

由式(8)、(12)、(13)可得：make

From formula (8), (12), (13) can get:

$v v = = \frac{\frac{11}{22 R R - - d d} {&Integral; &Integral;}_{00}^{22 R R - - d d} {J J}^{22} ((x x,, d d)) dx dx - - {J J}_{max max}^{22} ((d d)) - - {J J}_{min min}^{22} ((d d))}{{q q}_{d d} [[\frac{11}{22 R R - - d d} {&Integral; &Integral;}_{00}^{22 R R - - d d} J J ((x x,, d d)) dx dx + + {J J}_{max max} ((d d)) + + {J J}_{min min} ((d d))]]}$

由此看出，喷枪速率v可表示成d的函数，因此，E(d，v)的最小值只和d有关。可采用黄金分割法求出d优化值，从而可得到非规则多面体上每一面上的优化轨迹。It can be seen that the spray gun speed v can be expressed as a function of d, therefore, the minimum value of E(d, v) is only related to d. The optimal value of d can be obtained by using the golden section method, so that the optimal trajectory on each surface of the irregular polyhedron can be obtained.

3非规则多面体上的喷枪轨迹优化组合3 Optimal Combination of Spray Gun Trajectory on Irregular Polyhedron

为简化问题，将非规则多面体上每一面片上的轨迹看成是一条边。如图4所示，可用一个无方向的连接图G(V，E，R，ω：E→Z⁺)表示喷枪轨迹优化组合(Tool Trajectory OptimalIntegration，TTOI)问题，其中V表示顶点集，E表示边集，R表示E的任意一个子集，ω表示边的权(实际喷枪轨迹的长度)。TTOI问题就是在图G中求出一条经过所有边且只经过一次的具有最短距离的回路。设M＝{d_ij}(i，j＝1，2，...，n)是由图G中不在同一条边上的顶点i和顶点j之间的最短轨迹所组成的集合，且各顶点间的最短距离矩阵可使用Floyd算法算出。图2所示的是一个有5条边的连接图G，图中实线表示边，虚线表示从一个顶点到其他任意一个不在同一条边上的顶点的轨迹。To simplify the problem, the trajectory on each patch on the irregular polyhedron is regarded as an edge. As shown in Figure 4, an undirected connection graph G (V, E, R, ω: E→Z ⁺ ) can be used to represent the tool trajectory optimization combination (Tool Trajectory Optimal Integration, TTOI) problem, where V represents the vertex set, and E represents Edge set, R represents any subset of E, ω represents the weight of the edge (the length of the actual spray gun trajectory). The TTOI problem is to find a circuit with the shortest distance that passes through all edges and only once in graph G. Let M={d _ij }(i, j=1, 2,..., n) be a set composed of the shortest trajectories between vertices i and j that are not on the same edge in graph G, and each The shortest distance matrix between vertices can be calculated using Floyd's algorithm. Figure 2 shows a connected graph G with 5 edges. The solid line in the figure represents the edge, and the dashed line represents the trajectory from a vertex to any other vertex that is not on the same edge.

由于喷涂机器人喷枪轨迹组合问题自身的特点，应用粒子群算法与其他优化算法相比，易于实现，没有很多参数需要调整，且不需要梯度信息，是解决优化组合问题的有效工具。算法中，每个个体为一个粒子，每个粒子代表着一个潜在的解。设z_i＝(z_i1，z_i2，...，z_iD)为第i个粒子的D维位置矢量，根据适应度函数计算z_i当前的适应值，即可衡量粒子位置的优劣，而TTOI问题中可选取计算喷枪轨迹长度最小值为适应度函数。v_i＝(v_i1，v_i2，...，v_iD)为粒子i的飞行速度，即粒子移动的距离；p_i＝(p_i1，p_i2，...，p_iD)为粒子迄今为止搜索到的最优位置；p_g＝(p_g1，p_g2，...，p_gD)为整个粒子群迄今为止搜索到的最优位置。每次迭代中，粒子可根据下式更新速度和位置：Due to the characteristics of the spray gun trajectory combination problem of the spraying robot, compared with other optimization algorithms, the application of particle swarm optimization algorithm is easy to implement, there are not many parameters to adjust, and gradient information is not required. It is an effective tool to solve the optimal combination problem. In the algorithm, each individual is a particle, and each particle represents a potential solution. Let z _i =(z _i1 , z _i2 ,..., z _iD ) be the D-dimensional position vector of the i-th particle, and calculate the current fitness value of z _i according to the fitness function, which can measure the quality of the particle position, In the TTOI problem, the minimum value of the calculated spray gun trajectory length can be selected as the fitness function. v _i = (v _i1 , v _i2 , ..., v _iD ) is the flying speed of particle i, that is, the distance that the particle moves; p _i = (p _i1 , p _i2 , ..., p _iD ) is the particle i p _g = (p _g1 , p _g2 , . . . , p _gD ) is the optimal position searched so far by the entire particle swarm. In each iteration, the particle can update its speed and position according to the following formula:

${v v}_{id id}^{k k + + 11} = = {v v}_{id id}^{k k} + + {c c}_{11} {r r}_{11} (({p p}_{id id} - - {z z}_{id id}^{k k})) + + {c c}_{22} {r r}_{22} (({p p}_{gd gd} - - {z z}_{id id}^{k k}))$

${z z}_{id id}^{k k + + 11} = = {z z}_{id id}^{k k} + + {v v}_{id id}^{k k + + 11}$

其中，i＝1，2，...，m，d＝1，2...D，r₁和r₂为[0，1]之间的随机数，c₁和c₂为学习因子。Wherein, i=1, 2,..., m, d=1, 2...D, r ₁ and r ₂ are random numbers between [0, 1], c ₁ and c ₂ are learning factors.

由此，TTOI问题的粒子群算法步骤为：Therefore, the steps of particle swarm algorithm for TTOI problem are:

Step1初始化。初始化粒子位置

i＝1，2，...，m；初始化每个粒子的速度

i＝1，2，...，m；选择速度最大阈值ε和最大迭代次数N_max，迭代次数k＝0。Step1 initialization. Initialize particle position

i=1,2,...,m; initialize the velocity of each particle

i=1, 2, . . . , m; select the maximum speed threshold ε and the maximum number of iterations N _max , and the number of iterations k=0.

Step2测量每个粒子的适应值

表示为

令 Step2 measures the fitness value of each particle

Expressed as

make

Step3迭代次数k←k+1；更新速度更新位置 Step3 The number of iterations k←k+1; update speed update location

Step4测量z_i的适应值，表示为

取更新和

Step4 measures the fitness value of _zi , expressed as

Pick renew and

Step5若

Claims

1. A spraying robot spray gun trajectory optimization method on an irregular polyhedron is characterized in that the CAD data of a polyhedron workpiece is input to GID software, and each face of the polyhedron is triangulated by the GID grid graphic output function, and each triangle is calculated The normal vector of the face, according to the topological connection between adjacent triangular faces, generate several larger slices, establish a "cuboid model" and generate the spray gun path on each face of the polyhedron; take the variance of the actual coating thickness and the ideal coating thickness As the objective function, the golden section method is used on each surface of the irregular polyhedron to solve the optimal value of the width of the overlapping area of the coating; in order to improve the spraying efficiency, the optimal combination of spray gun trajectories on the curved surface is represented by a non-directional connection graph, and the The improved particle swarm algorithm solves; the method of the spray gun path on each face of the described generation polyhedron is as follows: first make several tangent planes with a distance of l along the direction perpendicular to the right side of the "cuboid model", that is, to obtain the tangent plane and curved surface Intersecting lines of several sections of the sheet; then averagely make a group of points with a distance of d on the intersecting lines; finally connect these points along the right side of the "cuboid model" to generate a spray gun space path, where l is R/ 2～R, R is the spraying radius of the spray gun, d is the distance between two spraying paths, the distance can be set artificially or obtained through optimization calculation.

2. the spraying robot spray gun trajectory optimization method on the irregular polyhedron according to claim 1, is characterized in that: the described method of the spray gun trajectory optimization combination on the curved surface that utilizes the connection diagram without direction to represent is as follows: use a directionless The connection graph G (V, E, R, ω: E→Z ⁺ ) represents the optimized combination of the spray gun trajectory, where V represents the vertex set, E represents the edge set, R represents any subset of E, and ω represents the weight of the edge, that is, the actual The length of the spray gun trajectory, in the undirected connection graph G, find a circuit with the shortest distance that passes through all edges and only once; M={di _j } is determined by the vertices i and A collection of shortest trajectories between vertices j, i,j=1,2,...,n, where n is the number of polyhedron faces.

3. the spraying robot spray gun trajectory optimization method on the irregular polyhedron according to claim 1, is characterized in that, adopts improved particle swarm algorithm to solve the method as follows:

Step1 initialization: Initialize the particle position

Initialize the velocity of each particle Select the maximum speed threshold ε and the maximum number of iterations N _max , and the number of iterations k=0;

Step2 measures the fitness value of each particle