Mathematical Problems in Engineering

Volume 2016, Article ID 9369047, 7 pages

http://dx.doi.org/10.1155/2016/9369047

## Prediction Model of Coating Growth Rate for Varied Dip-Angle Spraying Based on Gaussian Sum Model

^{1}School of Mechanical Engineering, Yancheng Institute of Technology, Yancheng, Jiangsu 224051, China^{2}College of Electrical and Information Engineering, Lanzhou University of Technology, Lanzhou, Gansu 730050, China

Received 27 September 2016; Accepted 15 November 2016

Academic Editor: Alessandro Gasparetto

Copyright © 2016 Yong Zeng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In automatic spraying of spray painting robot, in order to solve the problems of coating growth rate modeling for varied dip-angle spraying technology, a prediction mode of coating growth rate using the Gaussian sum model is proposed. Based on the Gaussian sum model, a theoretical model for coating growth rate with varied dip-angle spraying is established by using the theory of differential geometry. The coating thickness of the sample points in the distribution range of the coating was obtained by making the experiment of varied dip-angle spraying. Based on the theoretical model, the nonlinear least square method is used to fit the coating thickness of the sample points and the parameter values of the theoretical model are calculated. By analyzing the variation law of the parameters with the spray dip-angle, the prediction model of coating growth rate for varied dip-angle spraying is established. Experiments have shown that the prediction model has good fitting precision; it can satisfy the real-time requirement with varied dip-angle spraying trajectory planning in the offline programming system.

#### 1. Introduction

Trajectory planning of spray painting robot for complex free surface has been a hot research topic both at home and abroad. One of the key difficulties in the research of spraying trajectory planning is to establish a coating growth rate model with high accuracy and wide applicability [1–3]. At present, there are two main kinds of models for coating growth rate; a class of Gaussian distribution model [4, 5] and Cauchy distribution model [6] were proposed by Antonio and Freund, both of them belong to an infinite range model, and this kind of model is only suitable for the spray gun perpendicular to the surface of the workpiece. There is also limited range models, for example, piecewise function model [7], *β* distribution model [8], analytical deposition model [9], parabolic model [10], ellipse dual-*β* model [11], and Gaussian sum model [12]. The mathematical expressions of these models are usually derived by means of mathematical analysis. At the same time, based on the mathematical expressions, the final model of coating growth rate is approximated by the fitting of the experimental data. These models can be widely used in the case of complicated surface shape if these models meet the error requirement. In these models, the Gaussian sum model has a wide range of application; it can be applied to many kinds of spray gun and its fitting precision is higher.

In this paper, a prediction model of coating growth rate for varied dip-angle spraying based on the Gaussian sum model is established by the theory study and spraying experiment. The model can be used to generate the coating growth rate model under a certain spray dip-angle quickly and accurately.

#### 2. Theoretical Modeling of the Coating Growth Rate for Varied Dip-Angle Spraying

Spraying process is the coating produced by the spray gun that atomize to the surface of the workpiece; the process is rather complicated. Under the action of the spray gun and the pressure vessel, the paint can realize the atomization and it is sprayed from the spray gun nozzle; the paint is formed in a conical or ellipse-conical shape in space. In this paper, the coating growth rate model is established for varied dip-angle spraying based on a kind of spray gun, the paint space distribution of the spray gun is conical, and the torch of the spray gun is a circular cross section and isotropic. Under the assumptions that the spray distance , the spray flow , and the cone angle *φ* are constant when the spray gun is still sprayed, a mathematical model is used to describe the distribution of spray coating on a flat plate. The Gaussian sum model has the advantage of high fitting precision and not affected by the shape of the coating distribution, so it is assumed that the coating on the flat plate is distributed by the Gaussian sum model; its expression iswhere , , and are the unknown parameters, ; is the radius of spray painting. When tends to infinity, the function can approximate any form of coating distribution [13], but its complexity has increased dramatically, so generally take .

In consideration of the dip-angle spraying, assume that the angle between the axis of the spray gun and the plane normal direction is , as the spray dip-angle. Using the area amplification theorem of differential geometry, assuming that the small round regions are and by spraying, respectively, in the reference planes P1 and P2 which are perpendicular to the spray direction of the spray gun, as shown in Figure 1(a), the areas of and are and respectively, and the corresponding coating thicknesses are and . is the distance from any point on the plane to the nozzle along the spray gun axis. Therefore,So the coating thicknesses of and are satisfied:Suppose there are two plane regions at the same cone angle, respectively, which are and , as shown in Figure 1(b), where is perpendicular to the spray direction; the angle between and the spray direction is , so the coating thickness of can be expressed asCombining formulas (2)–(4), the theoretical model of coating growth rate which takes into account the spray dip-angle as a variable can be expressed aswhere and are the horizontal and vertical coordinates of any point within the coating range.