CN101739488A - Method for modeling paint deposition rate model adapting to complex free curved surface spraying - Google Patents

Abstract

The invention discloses a method for modeling a paint deposition rate model adapting to complex free curved surface spraying, which comprises the following steps: fitting spraying test data to form the paint deposition rate model of a planar workpiece by adopting a genetic algorithm based on the hypothesis that the paint deposition rate model in a single spraying torch on the planar workpiece is a specific function relation of a spraying point coordinate position, then establishing a judgment criteria of paint accessibility, and constructing the paint deposition rate model adapting to complex free curved surface by a curvature circle method and solving an included angle between a tangential plane of a spraying impact point of a spraying area and an attached imaginary plane of the maximum paint thickness. The method is helpful for improving the paint film thickness controlling precision of the free curved surface spraying, and provides specific operable theoretical and algorithm basis for realizing off-line programming and simulation of automatic spraying of a robot.

Description

The modeling method that adapts to the paint deposition rate model of complex free curved surface spraying

Technical field

The present invention relates to a kind of modeling method that adapts to the paint deposition rate model of complex free curved surface spraying in the Control During Paint Spraying by Robot off-line programing system that is applied in.

Background technology

Spraying makes the typical manufacture method of paint industry as the surface because its adaptability good to coating process has obtained using widely at numerous areas such as automobile, furniture, household electrical appliances, aviations.Adopt spraying robot to carry out that the automation paint operation can be improved the quality of products and stability in the spray painting operation, reduce the consumption of paint vehicle and energy, the workman is in poisonous environment all the time and causes acute or slow poisoning when having avoided manually spraying paint, and has improved labor productivity.Therefore, spraying robot more and more obtains people's attention in Application in Manufacturing Industry.

Early stage spraying robot is " teaching reproduction " type.By operating personnel's " teaching " spray gun operation overall process, robot controller memory teaching operation order is reappeared this operational motion in operation earlier, and its shortcoming is the teaching experience that spraying effect depends primarily on the individual.Along with improve day by day to the attention degree of environmental protection and labour protection various countries, also in order further to improve product quality and production efficiency, people begin to try hard to break away from the mode of production of traditional online " teaching " type, seek the method for spraying robot off-line programing, expectation utilizes computing machine to seek out the spray gun movement locus that can produce even film thickness, best spraying effect automatically.Therefore, utilize the spraying robot off-line programing be exactly to finally generate one can control robot the travelling speed, spray gun of spray gun of terminal clamping apart from the continuous walking route of the orientation relationship of height, spray gun and the Workpiece painting point of workpiece, promptly generate the spray gun track automatically, and carry out before the above-mentioned planning, a primary and underlying issue is how to set up paint deposition rate model.

At present,, mainly contain both at home and abroad: 1. infinite range model the research of spraying robot paint deposition rate model.Exist as AntonioJ K: " Optimal Trajectory Planning Problems for spray coating (being applied to the optimization trajectory planning problem in the field of spraying) " (IEEE International Conference on Robotics and Automation, USA:Atlanta, 1993, p2570-2577) (institute of electrical and electronic engineers: IEEE robot and robotization international conference, the U.S.: Atlanta, 1993, the 2570-2577 page or leaf) the Cauchy's distributed model that belongs to the infinite range model of Ti Chuing, this model is only thought when spray gun when the distance of any is tending towards infinity on surface of the work, the paint film growth rate functional value of this point just is zero, because they only adapt to the situation of spray gun perpendicular to surface of the work, so use less.2. limited range model.Such as KLEIN A at " CAD-based off-lineprogramming of painting robots (based on the off-line programing system of spraying robot CAD navigation) " (Robotica, 1987,5, p267-271) (robot, 1987 the 5th phases, the 267-271 page or leaf) the limited range model described in, the difference of it and infinite range model is: paint spray torch forms the paint film of similar round platform in spraying area, the form that the paint film deposition rate model adopts piecewise function to represent, only otherwise those surface of the work points within spray gun subtended angle scope, its paint film growth rate all is zero.The limited range model more tallies with the actual situation than infinite range model, thereby the paint film deposition function of setting up thus is more accurate.But also there is shortcoming in this model, because the paint film growth rate function of limited range model can only obtain by the method for numerical evaluation the integration of time, thereby relevant cost function is just smooth not as what adopt the resulting cost function of infinite range model.Like this, adopt the spray gun optimal trajectory planning problem of limited range model just to need more computing time.3. β distributed model.Arikan, M.A.Sahir Balkan etc. are at " Process modeling; simulation; and paint thicknessmeasurement for robotic spray painting (being applied to modeling, emulation and the thickness measure of the process modeling of spraying robot spraying) " (Journal of Robotic Systems, 2000,17 (9): 479-494) (robot system journal, 2000 17 the 9th phases of volume, the 479-494 page or leaf) described β distributed model thinks that the film thickness obedience β that forms distributes in circular spraying area.Because this model provides moulding parameter beta easily, so this model is considered to a kind of model preferably.4, oval two β model.People such as Zhang Yonggui are at " NEW MODEL FOR AIR SPRAY GUN OF ROBOTIC SPRAY-PAINTING " (mechanical engineering journal .2006 42 volume o. 11ths, the 226-233 page or leaf) the two β models of described ellipse, be to be the hypothesis of cone for circular, spray torch in spatiality based on the paint map sheet that is deposited on flat work pieces, be not inconsistent with applicable cases in the actual production and the model that proposes at infinite range model, limited range model and β distributed model.This model is thought in actual production, forms oval-shaped paint film distributed areas after the map sheet of painting perpendicular to the plane of spray gun axis is deposited on workpiece.The model and the reality that propose are more identical, but mainly remain with flat work pieces as spraying research object, and its model only is applicable to this spraying coating process form of aerial spraying.

Make a general survey of above-mentioned modeling method, all be based on the test that the relative workpiece of spray gun keeps motionless spraying flat work pieces, behind the measurement plane workpiece oil film thickness distributed data,, obtain the formula that embodies of this model by fitting technique.A fundamental purpose that adopts spraying robot to carry out the automation paint operation is exactly the uniform quality coating of thickness of surface attachment deposition that will realize workpiece to be sprayed.Yet above model all is to be the research modeling object with the flat work pieces, and actual to wait to be coated with its profile of industrial products rich and varied, ever-changing, mostly is the surface with free form surface feature.Because often there is bigger variation in its surface curvature, plane paint thickness distributed model is applied to the free curved surface spraying off-line programing obviously can brings than mistake.And the research that they launch is being assumed to be prerequisite or only being only applicable to this process form of aerial spraying of cone with the space distribution state of paint spray torch mostly.This obviously also can bring error to the spraying robot off-line programing.

Summary of the invention

Therefore, the purpose of this invention is to provide a kind of modeling method that can adapt to the paint deposition rate model of complex free curved surface Workpiece painting off-line programing, provide theoretical foundation and method to instruct for off-line programing, robot automation spray paint to process, realize the accuracy and the homogeneity of spraying thickness of workpiece quality index.

The technical solution used in the present invention is:

A kind of modeling method that adapts to the paint deposition rate model of complex free curved surface spraying, this method comprises the steps:

(1) selected flat work pieces paint deposition rate model function adopts genetic algorithm to simulate the paint deposition rate model of this flat work pieces by the flat work pieces spray test;

(2) set up the judgment criterion of painting accessibility;

(3) utilize circle of curvature method, ask section and maximum paint thickness to adhere to angle between the imaginary plane, free curve surface work pieces actual (real) thickness formula is set up in calculating, the flat work pieces paint deposition rate model that simulates is extended to the paint deposition rate model that adapts to the complex free curved surface workpiece.

The described approximating method that adopts genetic algorithm to simulate the paint deposition rate model of this flat work pieces by the flat work pieces spray test is: the parametric variable to be asked behind the selected flat work pieces paint deposition rate model function in the clear and definite pattern function, based on test figure, ask parametric variable to adopt real number coding method to waiting in the model, the span of each variable with reference to the experimental measurement data to determine in certain limit, in resulting each parameter area, produce an initial population of forming by real number at random, the scale of colony is taken as 200 * 8, whole colony is divided into 8 sub-populations with scale, the number of each sub-population is 200, evolutionary generation is 400, generation gap is 0.8, crossing-over rate is 1, aberration rate is 0.2, the insertion rate is 0.9, mobility is 0.2, sub-population migration algebraically is 20, select operator to adopt random ergodic sampling (sus) method, crossover operator adopts the discrete recombination operator, mutation operator is selected real-valued population mutation operator, genetic algorithm to asking parameter in global optimizing, principle according to the survival of the fittest is evolved, and calculates the parametric variable to be asked of flat work pieces paint deposition rate model function through the interative computation in given generation.

The judgment criterion of described foundation paint accessibility is:

(1) spray gun sprays paint and moves to when certain is put on the free form surface, according to the spraying coating process requirement, it is that what is called is just sprayed that spray gun rifle body keeps vertical with surface of the work, this spraying point on the free form surface is called as positive specking at this moment, judge whether the neighbor point on the free form surface and the line of spray gun end central point in a certain scope of this positive specking pass within the deposition map sheet in the corresponding flat work pieces spray test of positive specking, when this condition satisfies, neighbor point will be subjected to the spray gun spraying influence of positive specking top, promptly so-called spraying influences a little, change judgment criterion (2) over to, otherwise quit a program;

(2) section of the spraying influence point in the positive a certain nearby sphere of specking and maximum paint thickness are adhered to the angle between the imaginary plane, judge this angle whether interval [0,90), otherwise it is excessive to be sprayed the curvature corner that influences a little on curved surface, spray gun ejection paint is unreachable, quits a program; It was the perpendicular plane of line between this spray gun spraying influence point and terminal central point of spray gun and the spraying influence point that wherein maximum paint thickness is adhered to imaginary plane.

The described circle of curvature method of utilizing, ask section and maximum paint thickness to adhere to angle between the imaginary plane, free curve surface work pieces actual (real) thickness formula is set up in calculating, the flat work pieces paint deposition rate model that simulates is extended to the paint deposition rate model that adapts to the complex free curved surface workpiece, and concrete steps are as follows:

(1) set up paint deposition rate model on this spraying influence point and the imaginary plane parallel with flat work pieces, the expression formula of the paint deposition rate model on the imaginary parallel plane is:

$q_B=q_F{\left(\frac{h}{l}\right)}^2{\left(\frac{2h_1\mathrm{<figure-callout\; id="1"\; label="l\; h"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">l</figure-callout>}}{h_{}^{}}\right)}^2---\left(1\right)$

Q wherein _F, q _BBe respectively paint particles be traveling in flat work pieces in the flat work pieces spray test along a certain rectilinear direction on, cross the paint deposition rate (μ m/s) on free curved surface spraying influence point and the imaginary plane parallel, h, h with flat work pieces ₁Be respectively the height of spray gun centre distance flat work pieces, positive specking, l, l _ABBe respectively the length between length between spray gun center and the spraying influence point and positive specking and the spraying influence;

(2) set up spraying influence point and adhere to paint deposition rate model on the imaginary plane direction in maximum paint thickness, the expression formula that maximum paint thickness is adhered to the paint deposition rate model on the imaginary plane direction is:

$q_B^{\#}=q_B/\cos \mathrm{\&theta;}=q_F{\left(\frac{h}{l}\right)}^2{\left(\frac{2h_1\mathrm{<figure-callout\; id="1"\; label="l\; h"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">l</figure-callout>}}{h_{}^{}}\right)}^3---\left(2\right)$

Q wherein ^# _BFor spraying influence point adheres to paint deposition rate (μ m/s) on the imaginary plane direction in maximum paint thickness, θ adheres to angle between the imaginary plane for the imaginary plane parallel with flat work pieces and maximum paint thickness;

(3) set up the paint deposition rate model of spraying influence point on the section, the paint deposition rate model on this section is the paint deposition rate model of free curved surface spraying, and its expression formula is:

$q_B^{*}=q_B^{\#}\cos \mathrm{\&alpha;}=q_F{\left(\frac{h}{l}\right)}^2\frac{{\mathrm{GO}}_B^2-({\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">l</figure-callout>}}^2+R_B^2)}{l{R}_B}\frac{4{\mathrm{<figure-callout\; id="1"\; label="h"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">h</figure-callout>}}_1^3{l}^3}{{({\mathrm{<figure-callout\; id="1"\; label="h"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">h</figure-callout>}}_1^2+l^2-l_{\mathrm{AB}}^2)}^3}---\left(3\right)$

Q wherein _B ^*For GB direction plane BD goes up the paint deposition rate (μ m/s) that B is ordered; R _BBe the curvature radius of a ball value of free form surface at spraying influence point, GO _BBe the length of spray gun center to the spraying influence point center of curvature, some O _BSphere center position for spraying influence point place curvature ball, to be free form surface adhere to angle between the imaginary plane in the maximum paint thickness of the section of spraying influence point and this point to angle α, more than four determine that according to the circle of curvature method point G is present position, spray gun center.

The paint deposition rate model of the adaptation complex free curved surface spraying that described modeling method generates is applied to the free curve surface work pieces solid model be carried out the simulation calculation of paint thickness in the spraying robot off-line programming software.

The present invention has very strong practicality, and the modeling method that the robot with complex free curved surface feature workpiece is sprayed automatically the off-line programing paint deposition rate model can be provided, and can improve the control accuracy of spraying, has guaranteed the homogeneity of spraying workpiece.

Description of drawings

Fig. 1 is to be that the spray gun spray square at center deposits to free form surface f (x, y, working condition chart z) with the A point.

Fig. 2 is the single oval map sheet that the present invention selects for use.

Fig. 3 is the schematic three dimensional views of right cylinder side spraying special case.

Fig. 4 is a right cylinder side spraying special case two-dimension projection.

Fig. 5 is the spray robot Test Drawing that sprays paint.

Fig. 6 is the back workpiece that sprays paint.

Fig. 7 is genetic algorithm fit Plane mould oil film thickness distribution surface figure.

Fig. 8 is a genetic algorithm fit Plane mould oil film thickness error profile surface chart.

Fig. 9 is the interior section x=-40 oil film thickness profile diagram of single map sheet on the flat work pieces.

Figure 10 is the interior section y=20 oil film thickness profile diagram of single map sheet on the flat work pieces.

Figure 11 is right cylinder fixed point spraying Theoretical Calculation oil film thickness distribution surface figure.

Figure 12 is a right cylinder fixed point spray oil film thickness distribution error surface chart.

Figure 13 is the spraying exemplar of right cylinder side one-way trip spray test.

Figure 14 is the sprayed three-dimensional synoptic diagram of right cylinder side one-way trip spray test.

Figure 15 is the comparison diagram of right cylinder side one-way trip spray oil film thickness measured value and theoretical value.

Figure 16 imports bumper entity workpiece in the off-line programing system of exploitation.

Figure 17 is a spraying bumper exemplar.

Figure 18 is the film thickness distribution plan that calculates in the off-line programing system of exploitation.

Figure 19 is the film thickness distribution plan of actual spraying.

Embodiment

Below in conjunction with the drawings and specific embodiments the present invention is described in further detail.

The modeling method of the paint deposition rate model of adaptation complex free curved surface spraying of the present invention comprises the steps:

1, selectes the mathematical model of plane paint deposition rate

According to concrete spraying coating process method, the selected plane paint thickness mathematical model that is suitable for this spraying coating process method is determined the parametric variable to be asked in the pattern function.

2, with the flat work pieces is the spraying object, relative workpiece transfixion fixed point one short time of spraying of spray gun, obtain paint thickness distribution in the single map sheet of flat work pieces, utilize genetic algorithm to carry out the match of film thickness curved surface, ask parametric variable to adopt real number coding method to waiting in the model, the span of each variable is determined in certain limit with reference to experimental measurement data and experience, in resulting each parameter area, produce an initial population of forming by real number at random, genetic algorithm to asking parameter in global optimizing, principle according to the survival of the fittest is evolved, calculate the parametric variable to be asked of flat work pieces paint deposition rate model function through the interative computation in given generation, thereby obtain the formula that embodies of selected plane paint deposition rate model.Wherein the scale of colony is taken as 200 * 8, and whole colony is divided into 8 sub-populations with scale, and the number of each sub-population is 200, evolutionary generation is 400, generation gap is 0.8, and crossing-over rate is 1, and aberration rate is 0.2, the insertion rate is 0.9, mobility is 0.2, and sub-population migration algebraically is 20, selects operator to adopt random ergodic sampling (sus) method, crossover operator adopts the discrete recombination operator, and mutation operator is selected real-valued population mutation operator.

3, adapt to the foundation of complex free curved surface spraying paint deposition rate model

As shown in Figure 1, (x, y z) are the free form surface of workpiece to f, carve t at a time, and spray gun G moves to the top h that free form surface A is ordered ₁The place, h ₁Be that already known processes requires distance, and the normal direction that the direction of spray gun and A are ordered is opposite.Purpose is will find the solution B point in a certain nearby sphere that A is ordered on this curved surface to be subjected to the accompanying paint thickness regularity of distribution of spray gun influence above the A point.

Derivation is as follows:

As shown in Figure 1, this moment, A, B were the point on the curved surface, connected GA, GB, supposed ∠ AGB=θ, crossed B and made plane BC perpendicular to GB.Cross the B point and make plane BM perpendicular to straight line GA and hand over GA, know that easily the angle between plane MB and the plane BC equals θ (getting acute angle) in the M point.Make tangent plane to a surface BD after the B point and be cut in the B point, establishing the circle of curvature center of circle that B orders is O _B, radius is R _B(at three dimensions, the actual curvature ball that is interpreted as of the circle of curvature), (x, y z) express free form surface, and the angle between plane BC and the section BD is α by f; Along the GA direction, be starting point intercepting GE=h (h is that spray gun is sprayed paint in the experiment apart from the height of workpiece in the plane) with the G point, cross the E point and make plane EF and meet at the F point perpendicular to GA and with GB.GA=h ₁, establish GB=l.According to step 2, be research object with flat work pieces EF earlier, plane EF is carried out spray test, ask for the paint thickness of arbitrfary point distribution in the single map sheet on the EF plane, distribution function based on plane paint thickness mathematical model, spray gun carries out spray test apart from level altitude of plane EF, utilize genetic algorithm to set up fitness function according to the principle of least square, ask for the parameter of selected model, obtain the formula that embodies of this areal model, then the paint thickness of arbitrfary point just can be obtained according to the coordinate position relation in the last whole map sheet of plane EF.Then, the process conditions that flat work pieces is tested immobilize and are applied to complex free curved surface continuation test.Existing issue just is converted into known flat work pieces paint thickness sedimentation model, when carving on the t spray gun spraying free form surface A point at a time, asks for B and orders the distributed model of the oil film thickness that deposited.Known conditions is spray gun coordinate G (x _G, y _G, z _G), A point coordinate A (x _A, y _A, z _A), B point coordinate B (x _B, y _B, z _B), spray gun apart from the Equation f of flat work pieces height h and free form surface (x, y, z).

Then (1) sets up the judgment criterion of paint accessibility

(i) whether the intersection point F that judges line GB that any B point wait to investigate on the curved surface and G are ordered and plane EF within the ellipse map sheet of plane, when this condition satisfies, can guarantee that the B point is sprayed by the A point to influence just to carry out oil film thickness and analyzes, otherwise quit a program;

(ii) judge 0≤α＜90 °, otherwise corner is too big, spray gun ejection paint is unreachable, also quits a program.The programming of asking at α angle can be with reference to formula (16).

Satisfy above two criterions, enter next step.

(2) calculation is derived

According to technological requirement, the spray gun axis is right against the A point during spraying A point, and is opposite with A point direction of normal.

By GE=h, ∠ AGB=θ, then EF=h * tan θ;

Technological parameter and plane are consistent during the spraying free form surface, do not consider gravity effect.If paint is straightaway from spray gun ejection back paint particles, paint flux (μ m/s) along unit interval of a certain direction is a steady state value, be conservation along the volume of a certain direction paint flow simultaneously through the space, therefore the thickness that when the GB direction is flowed through the B point of the F point of imaginary plane EF and plane MB respectively, adheres to of the paint that comes out from the G point, be inversely proportional to the view field's area that projects this plane respectively, and area and G point are the relations that becomes square direct ratio to the distance of corresponding flat, therefore, promptly have:

$\frac{q_F}{q_B}=\frac{S_B}{S_F}={\left(\frac{h_2}{h}\right)}^2---\left(4\right)$

Q wherein _F, q _BBe respectively the paint thickness (μ m) that GB direction plane EF, plane MB are ordered at F, B; S _B, S _FPaint the paint area that on corresponding flat, adheres to for straight line GB direction; H, h ₂Be respectively the height of spray gun apart from plane EF, plane MB.

Therefore:

$q_B=q_F{\left(\frac{h}{h_2}\right)}^2$

And

h ₂＝lcosθ

Then:

$q_B=q_F{\left(\frac{h}{l}\right)}^2\frac{1}{{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&theta;}}---\left(5\right)$

We know that when spray gun was in over against workpiece point, the paint thickness that this point is obtained was maximum, promptly belong to and just spray the position, and the angle between plane MB and the plane BC equals θ simultaneously, and then the paint thickness that B is ordered on the BC plane is:

$q_B^{\#}=q_B/\cos \mathrm{\&theta;}=q_F{\left(\frac{h}{l}\right)}^2\frac{1}{{\mathrm{<figure-callout\; id="3"\; label="cos"\; filenames="H2009101146141E00062.png"\; state="\{\{state\}\}">cos</figure-callout>}}^3\mathrm{\&theta;}}---\left(6\right)$

Therefore, the paint thickness of the actual acquisition of B point should be:

$q_B^{*}=q_B^{\#}\cos \mathrm{\&alpha;}=q_F{\left(\frac{h}{l}\right)}^2\frac{\cos \mathrm{\&alpha;}}{{\mathrm{<figure-callout\; id="3"\; label="cos"\; filenames="H2009101146141E00062.png"\; state="\{\{state\}\}">cos</figure-callout>}}^3\mathrm{\&theta;}}---\left(7\right)$

In formula (7), q _F, q ^# _BBe respectively the GB direction plane EF paint thickness that B is ordered on F point and BC plane (μ m), q _B ^*For GB direction plane BD goes up the paint thickness (μ m) that B is ordered.q _FAs previously mentioned, in match can obtain according to coordinate position relation behind the paint deposition rate model of plane; H, l, θ can obtain according to known conditions, become known quantity, and have only the α angle to be amount to be asked.As shown in Figure 1, the α angle between section BD and the plane BC is actually section and maximum paint thickness and adheres to angle between the imaginary plane, and its midplane BC is that maximum paint thickness is adhered to imaginary plane.Be useful for the spraying operation of complex free curved surface off-line programing and trajectory planning for Control During Paint Spraying by Robot, under existing known conditions, utilize the curvature circule method, according to the corresponding geometry relation of curvature with circle of curvature radius, ask section BD and maximum paint thickness to adhere to angle α angle between the imaginary plane BC, α obtains at the angle, and then whole problem solves.

(3) the curvature circule method asks section and maximum paint thickness to adhere to angle α between the imaginary plane

Connect GO _B, at triangle Δ GBO _BIn, ∠ GBO _B=pi/2+pi/2-α=π-α.Concern computing formula according to curvature and curvature and the circle of curvature, have

$k_B=\frac{f_B^{\&prime;\&prime;}}{{(1+f_B^{\&prime;})}^{3/2}}$

So radius of curvature R at B point place _BCan try to achieve

$R_B=\frac{1}{k_B}=\frac{{(1+f_B^{\&prime;})}^{3/2}}{\left|f_B^{\&prime;\&prime;}\right|}---\left(8\right)$

K wherein _BBe the curvature value of curved surface at B point place, f _B', f _B" be respectively single order, the second derivative value of curved surface at B point place.

For circle O _BHave and (for ease of calculating, be assumed to be two-dimensional state, at this moment f _B'=y _B', f _B"=y _B")

${(x-x_{o_B})}^2+{(y-y_{o_B})}^2=R_B^2---\left(2\right)$

Y wherein _B', y _B" (x is y) in single order, the second derivative value at B point place to be respectively two-dimensional curve f.

Again because the tangent line and the curvature radius of a circle O at B point curve place _BB is perpendicular, so have

$y_B^{\&prime;}=-\frac{1}{k_{O_{B}B}}=-\frac{x_B-x_{O_B}}{y_B-y_{O_B}}---\left(10\right)$

(9) formula, the cancellation of (10) formula simultaneous

, can get

${(y-y_{O_B})}^2=\frac{{\mathrm{<figure-callout\; id="2"\; label="R\; B"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">R</figure-callout>}}_B^{}}{1+y_B^{\&prime;2}}=\frac{{(1+y_B^{\&prime;2})}^2}{y_B^{\&prime;\&prime;2}}---\left(11\right)$

Again because as y " _B＞0 o'clock curve is a concave arc, $y-y_{O_B}<0;$ As y " _B＜0 o'clock curve is a convex arc, $y-y_{O_B}>0.$ Hence one can see that y " _BWith

Contrary sign, so after the following formula both sides open radical sign, have

$y-y_{O_B}=-\frac{1+y_B^{\&prime;2}}{y_B^{\&prime;\&prime;}}---\left(12\right)$

And

$x-x_{O_B}=-y_B^{\&prime;}(y-y_{O_B})=\frac{y_B^{\&prime;}(1+y_B^{\&prime;2})}{y_B^{\&prime;\&prime;}}---\left(13\right)$

So O _BCentral coordinate of circle have:

Be GO _BLength can also can try to achieve by range formula.GB=l, OB _B=R _B, triangle Δ GBO _BIn, have according to the cosine law

$\mathrm{\&alpha;}=\mathrm{\&pi;}-\mathrm{<figure-callout\; id="2"\; label="arccos\; l"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">arccos</figure-callout>}\frac{l^{}}{}\frac{{}^2+R_B^2-\mathrm{<figure-callout\; id="2"\; label="G\; O\; B"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">G</figure-callout>}{O}_B^{}{}_2^{}}{2l{R}_B}---\left(4\right)$

Formula (14) back substitution to formula (7), can be obtained the actual acquisition of B point paint thickness:

$q_B^{*}=q_F{\left(\frac{h}{l}\right)}^2\frac{{\mathrm{GO}}_B^2-({\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">l</figure-callout>}}^2+R_B^2)}{l{R}_B}\frac{4{\mathrm{<figure-callout\; id="1"\; label="h"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">h</figure-callout>}}_1^3{l}^3}{{({\mathrm{<figure-callout\; id="1"\; label="h"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">h</figure-callout>}}_1^2+l^2-l_{\mathrm{AB}}^2)}^3}---\left(15\right)$

Can try to achieve by (16), (17) formula for cos α, cos θ

$\cos \mathrm{\&alpha;}=-\frac{{\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">l</figure-callout>}}^2+R_B^2-\mathrm{<figure-callout\; id="2"\; label="G\; O\; B"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">G</figure-callout>}{O}_B^{}{}_2^{}}{2l{R}_B}---\left(16\right)$

$\cos \mathrm{\&theta;}=\frac{{\mathrm{<figure-callout\; id="1"\; label="h"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">h</figure-callout>}}_1^2+l^2-{\mathrm{<figure-callout\; id="2"\; label="l\; AB"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">l</figure-callout>}}_{\mathrm{AB}}^{}}{2h_{1}l}---\left(17\right)$

By formula (7) and formula (15) as can be seen:

(1) the free curved surface spraying paint deposition rate model is towards complex free curved surface, and when α=0 °, (7) formula is exactly a plane paint sedimentation model situation.Therefore, this model has been contained workpiece preferably and has been plane and two kinds of forms of free form surface, and is more realistic than areal model, has adaptability and applicability preferably.

(2) but when free form surface be the time by concrete continuous second order derived function formal representation, circle of curvature radius R _BWith central coordinate of circle O _BCan draw by asking the curvature that function is ordered at B, so the establishment of program is easy to realize.

(3) in the CAD/CAM system of exploitation, what the workpiece entity extensively adopted is the Discrete Surfaces expression-form of triangle grid model.What therefore, realistic meaning arranged more is that the free form surface paint deposition rate model is applied in such system.Triangle mesh curved surface is a kind of burst linear surface, does not have continuous method resultant curvature.The discrete curvature Calculation Method is more, and is wherein commonly used with the Taubin method.Taubin has done the derivation of complete differential geometry of surfaces characteristic, obtain the discrete approximation (being that the normal curvature of summit on certain adjoint point direction is approximate only relevant with vertex scheme arrow, summit and adjoint point coordinate) of curvature tensor on the grid surface, thereby the weighted average calculation that can vow with the triangular plate method earlier goes out the method arrow on each summit, find the solution the eigenwert and the proper vector of the symmetric matrix that constitutes by each curvature tensor weighting then with the Householder method, eigenwert is exactly a principal curvatures, and proper vector then is a principal direction.Thereby, after the relevant information of the method arrow, apex coordinate and the discrete curvature that have obtained the triangle gridding dough sheet, just these numerical value can be calculated R with the form assignment of vector matrix _BAnd GO _BThereby, can carry out concrete Model Calculation to formula (15).

Concrete implementation step in off-line programing up-to-date style (15) is described as follows:

1, finds the solution the intersection point F coordinate of straight line GB and plane EF;

2, judge intersection point F whether within the single map sheet in plane, when this condition satisfies, can guarantee that the B point is influenced by the spraying of A point just to carry out oil film thickness to analyze, otherwise quit a program;

3, judge whether straight line GB satisfies 0≤α＜90 ° at vertical plane that B is ordered and the angle α of free form surface between the section that B is ordered, otherwise corner is too big, spray gun ejection paint is unreachable, also quits a program.The programming of asking at α angle can be with reference to formula (16);

4, calculate circle of curvature central coordinate of circle and the place circle of curvature radius of free form surface on the B point;

5, according to the flat work pieces test match single paint map sheet paint deposition rate model that sprays paint;

6, formula (15) is write code, with q _F, h, h ₁, l, l _AB, GO _B, R _BWait each variable assignments to calculate substitution formula (15) and specifically deposited the paint thickness value.

7, this functions of modules finishes, and enters subordinate's computing module and trajectory planning module.

Embodiment 1

Following mask body is that example is verified validity of the present invention and practicality with the fixed point aerial spraying test of right cylinder side.

1, the theoretical preparation

In actual production, consider that aerial spraying map sheet shape might be circular and oval two kinds of situations, the present invention produces actual oval two β distributed models as shown in Figure 2 selected meeting more:

$q(x,y)=q_{\max }{(1-\frac{x^2}{a^2})}^{{\mathrm{\&beta;}}_1-1}{[1-\frac{y^2}{b^2(1-x^2/a^2)}]}^{{\mathrm{\&beta;}}_2-1}---\left(18\right)$

Wherein-a≤x≤a-b (1-x ²/ a ²) ^1/2≤ y≤b (1-x ²/ a ²) ^1/2

As shown in Figure 2, the ellipse that is deposited on certain flat work pieces of diagram ellipse representation air gun spraying paints map sheet.After setting up coordinate system, (x y) is thickness distribution function in the oval map sheet, q to the q in (18) formula _MaxBe the paint thickness of spray gun central axis at the workpiece subpoint, the thickness that this point obtains is maximum in the whole map sheet.A, b are respectively ellipse long and short shaft length.β ₁, β ₂It is respectively the distribution parameter on the long and short axle.Work as β ₁=1 and β ₂≠ 1 or β ₂=1 and β ₁≠ 1 o'clock (is β ₁And β ₂Have one to equal 1, but can not be 1 simultaneously), paint map sheet shape is exactly circular; In addition, paint map sheet shape is exactly oval.This model is to be that ellipse cone is a research object with the air gun model, has contained two kinds of spray gun model case of ellipse awl and circular cone, has more rationality than single β model.

With the flat work pieces is the spraying object, and paint thickness distribution in the single map sheet of unit interval inner plane workpiece is obtained in the spraying of the relative workpiece transfixion fixed point of spray gun, utilizes genetic algorithm to carry out the match of film thickness curved surface, asks for a, b, β ₁, β ₂, q _MaxValue, obtain (18) formula embody formula after, can obtain the further concrete expression formula of (15) formula

$q_B^{*}=q_{\max }{\left(\frac{h}{l}\right)}^2\frac{{\mathrm{GO}}_B^2-({\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">l</figure-callout>}}^2+R_B^2)}{l{R}_B}\frac{4{\mathrm{<figure-callout\; id="1"\; label="h"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">h</figure-callout>}}_1^3{l}^3}{{({\mathrm{<figure-callout\; id="1"\; label="h"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">h</figure-callout>}}_1^2+l^2-l_{\mathrm{AB}}^2)}^3}{(1-\frac{x_F^2}{a^2})}^{{\mathrm{\&beta;}}_1-1}{[1-\frac{x_F^2}{b^2(1-{\mathrm{<figure-callout\; id="2"\; label="y\; F"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">y</figure-callout>}}_F^{}/a^2)}]}^{{\mathrm{\&beta;}}_2-1}---\left(19\right)$

Because concrete function expression the unknown of free form surface in (19) formula, cause when checking is found the solution B point coordinate and corresponding curvature specifically to obtain, therefore select for use entity workpiece to carry out spray test with a certain special form face.Consider the operability of test and the convenience of data sampling, the curvature of selected workpiece shape face should have certain representativeness simultaneously, because the arbitrfary point on the right cylinder side surface all equates with respect to right cylinder central shaft curvature of a curve and circle of curvature radius, has therefore selected the right cylinder side as test spraying object here.To calculate and checking in order being beneficial to, at first should to carry out the paint deposition rate of right cylinder side fixed point spraying and derive.

When fixed point spraying right cylinder side, as shown in Figure 3, the A point be on the right cylinder side spray gun over against the spraying point, the B point is point to be asked.For the ease of understanding, with right cylinder along central axis to vertical paper direction projection, obtain Fig. 4 equatorial projection figure, to show contrast.Wherein the right cylinder radius is R, GE=h, ∠ AGB=θ, GA=h ₁, this moment O _AOn the right cylinder central axis, for cutting the round center of circle: O in right cylinder side, A point place _BBe the circle of curvature center of circle that B is ordered, the right cylinder radius is R, easily knows R=R _BThe angle on the BC plane that section BD is vertical with GB is similarly α, and with reference to the solution procedure of free form surface, its solution procedure is as follows:

The actual paint thickness q that adheres on the B point section ^* _BSubstitution formula (7) has

$q_B^{*}=q_B^{\#}\cos \mathrm{\&alpha;}=q_F{\left(\frac{h}{l}\right)}^2\frac{\cos \mathrm{\&alpha;}}{{\mathrm{<figure-callout\; id="3"\; label="cos"\; filenames="H2009101146141E00062.png"\; state="\{\{state\}\}">cos</figure-callout>}}^3\mathrm{\&theta;}}$

At right-angle triangle Δ GO _AO _BIn, because GO _A=GA+R _B, easily know

${\mathrm{GO}}_B=\sqrt{{\mathrm{<figure-callout\; id="2"\; label="GO\; A"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">GO</figure-callout>}}_A^{}+O_{\mathrm{<figure-callout\; id="2"\; label="A\; O\; B"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">A</figure-callout>}}{O}_B^{}{}_2^{}}=\sqrt{{(\mathrm{GA}+A{O}_A)}^2+O_{\mathrm{<figure-callout\; id="2"\; label="A\; O\; B"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">A</figure-callout>}}{O}_B^{}{}_2^{}}=\sqrt{{({\mathrm{<figure-callout\; id="1"\; label="h"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">h</figure-callout>}}_1+R_B)}^2+{\mathrm{\&rho;}}^2}---\left(20\right)$

Establish line segment O in the following formula _AO _BLength equal ρ, ρ can try to achieve h by A, B two point coordinate ₁, R _BFor oneself knows condition, so GO _BCan in the hope of; Δ GBO _BIn, GB=l, BO _B=R _B, can try to achieve cos α by formula (16).

Therefore according to formula (16), have

$\cos a=\frac{{({\mathrm{<figure-callout\; id="1"\; label="h"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">h</figure-callout>}}_1+R_B)}^2+{\mathrm{\&rho;}}^2-l^2-{\mathrm{<figure-callout\; id="2"\; label="R\; B"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">R</figure-callout>}}_B^{}}{2l{R}_B}---\left(21\right)$

Simultaneously in triangle Δ GAB, because h ₁, l, l _ABKnown, can try to achieve cos θ by formula (17).

So

$q_B^{*}=q_F{\left(\frac{h}{l}\right)}^2\frac{{({\mathrm{<figure-callout\; id="1"\; label="h"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">h</figure-callout>}}_1+R_B)}^2+{\mathrm{\&rho;}}^2-l^2-{\mathrm{<figure-callout\; id="2"\; label="R\; B"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">R</figure-callout>}}_B^{}}{2l{R}_B}\frac{8{\mathrm{<figure-callout\; id="1"\; label="h"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">h</figure-callout>}}_1^3{l}^3}{{({\mathrm{<figure-callout\; id="1"\; label="h"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">h</figure-callout>}}_1^2+l^2-l_{\mathrm{AB}}^2)}^3}---\left(22\right)$

With following formula substitution (18) formula, after the simplification, have

$q_B^{*}=q_{\max }{(1-\frac{x_F^2}{a^2})}^{{\mathrm{\&beta;}}_1-1}{[1-\frac{y_F^2}{b^2(1-{\mathrm{<figure-callout\; id="2"\; label="x\; F"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">x</figure-callout>}}_F^{}/a^2)}]}^{{\mathrm{\&beta;}}_2-1}\frac{4h^2{\mathrm{<figure-callout\; id="1"\; label="h"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">h</figure-callout>}}_1^3({({\mathrm{<figure-callout\; id="1"\; label="h"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">h</figure-callout>}}_1+R_B)}^2+{\mathrm{\&rho;}}^2-l^2-R_B^2)}{R_B{({\mathrm{<figure-callout\; id="1"\; label="h"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">h</figure-callout>}}_1^2+l^2-l_{\mathrm{AB}}^2)}^3}---\left(23\right)$

(wherein $-a\&le;x_F\&le;a-b{(1-{\mathrm{<figure-callout\; id="2"\; label="x\; F"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">x</figure-callout>}}_F^{}/a^2)}^{1/2}\&le;y_F\&le;b{(1-{\mathrm{<figure-callout\; id="2"\; label="x\; F"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">x</figure-callout>}}_F^{}/a^2)}^{1/2}$ )

Like this, just derived the paint deposition rate of right cylinder side spraying special case, carried out the example preparation for entering the verification experimental verification stage.

2, verification experimental verification

In order to verify the correctness of free curved surface spraying paint deposition rate model, carried out two parts test.It is motionless with respect to flat work pieces at first to carry out spray gun, and rifle is apart from the test of spraying paint (test one) of d=200mm, Switch of spray pistol rifle time t=2.50s; Keep technological parameter constant, it is motionless with respect to cylindrical workpiece to carry out spray gun then, rifle is apart from the side spray test (test two) of d=220mm, before formally carrying out side spray test two, note the major axis position of map sheet on the flat work pieces, central axis or side bus after making cylindrical workpiece place are parallel with oval map sheet major axis, and guarantee the centre position of the high bus of spray gun center behind the right cylinder side, it is the A point among Fig. 3 or Fig. 4, spray gun just moves up and down along vertical when adjusting, to obtain best test spraying effect.Cylinder dimensions: Φ 275mm * 400mm, Switch of spray pistol rifle time t=2.13s; After waiting to paint bone dry, obtain the film thickness distributed intelligence in plane spraying area, spray zone, cylinder side.Fig. 5 is the spray robot Test Drawing that sprays paint, and Fig. 6 is the back workpiece that sprays paint, and wherein Fig. 6 top is the plane exemplar, and the bottom is a cylinder side spray exemplar.

With spray gun center line and flat work pieces intersection point is initial point, transverse is the x axle, minor axis is the y axle, set up coordinate system, measurement plane EF goes up the physical dimension and the film thickness of spraying area, film thickness adopts German byko-test 7500 electromagnetic type film thickness testers, the distributed data of the thickness that obtains spraying paint, totally 603 groups of data.As space is limited, only provide the part measurement data.See Table 1.

Table 1 EF flat work pieces spraying area paint thickness DATA REASONING (unit: μ m)

The data that obtain according to flat work pieces fixed point spray test, for formula (18), based on the MATLAB environment adopt genetic algorithm (Genetic algorithm) tool box with the least square of measured value and calculated value with set up the match that fitness function carries out the film thickness curved surface, ask for a, b, β ₁, β ₂, q _MaxValue.5 unknown variables are arranged in the thickness model, variable is adopted real number coding method, each individuality is one 5 dimensional vector, for determining of the span of each variable, determines a, b and q with reference to the experimental measurement data _MaxConstant interval, β ₁, β ₂Constant interval can determine that then the span of transverse a is [100,180] in the formula (18) with reference to the β distribution curve that pertinent literature provides, the span of ellipse short shaft b is [20,60], maximum film thickness q _MaxSpan be [40,120], profile exponent β ₁, β ₂Constant interval be [2,25].The scale of colony is 200 * 8, whole colony is divided into 8 sub-populations with scale again, the number of each sub-population is 200, and evolutionary generation is 400, and generation gap is 0.8, crossing-over rate is 1, aberration rate is 0.2, and the insertion rate is 0.9, and mobility is 0.2, sub-population migration algebraically is 20, try to achieve and all produce an initial population of forming by real number in each parameter area at random.Wherein select operator to adopt random ergodic sampling (sus) method, crossover operator adopts the discrete recombination operator, and mutation operator is selected real-valued population mutation operator.Genetic algorithm, is evolved according to the principle of the survival of the fittest in global optimizing to asking parameter, at last can be through the interative computation in 400 generations in the hope of a=142.112mm, and b=43.032mm, β ₁=4.1987, β ₂=6.1692, q _Max=81.12 μ m.In order to compare,, draw genetic algorithm match paint deposition map sheet model curved surface, as shown in Figure 7 with above parameter substitution formula (18) with test one data.Draw oil film thickness error curved surface with the difference of theoretical value and measured value, as shown in Figure 8.Simultaneously, get x=-40 and y=20 tests as section, on section, get a little every a determining deviation, oil film thickness to these points carries out Theoretical Calculation and actual measurement, result such as Fig. 9 and shown in Figure 10, correlation curve from figure as can be seen, the comparatively accurate match of GA two β models of plane paint deposition rate.Like this can be in the hope of the formula that embodies of this kind process conditions following formula (18).

So have for (23) formula

$q_B^{*}=81.12\&times;\frac{2.13}{2.50}\&times;{(1-\frac{{\mathrm{<figure-callout\; id="2"\; label="x\; F"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">x</figure-callout>}}_F^{}}{142.112^2})}^{4.1987-1}\&times;{[1-\frac{{\mathrm{<figure-callout\; id="2"\; label="y\; F"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">y</figure-callout>}}_F^{}}{43.032^2(1-{\mathrm{<figure-callout\; id="2"\; label="x\; F"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">x</figure-callout>}}_F^{}/142.112^2)}]}^{6.3092-1}$

$\&times;\frac{4\&times;{200}^2\&times;{220}^3({(220+137.5)}^2+{\mathrm{\&rho;}}^2-l^2-R_B^2)}{137.5\&times;{({220}^2+l^2-l_{\mathrm{AB}}^2)}^3}$

Have after the arrangement

$q_B^{*}=8.56353\&times;{10}^{11}\&times;{(1-\frac{{\mathrm{<figure-callout\; id="2"\; label="x\; F"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">x</figure-callout>}}_F^{}}{142.112^2})}^{3.1987}\&times;{[1-\frac{{\mathrm{<figure-callout\; id="2"\; label="y\; F"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">y</figure-callout>}}_F^{}}{43.032^2(1-{\mathrm{<figure-callout\; id="2"\; label="x\; F"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">x</figure-callout>}}_F^{}/142.{112}^2)}]}^{5.3092}\&times;\frac{108900+{\mathrm{\&rho;}}^2-l^2}{{({220}^2+l^2-l_{\mathrm{AB}}^2)}^3}---\left(24\right)$

After right cylinder side spray test two is finished, set up coordinate system, true origin fixes on spray gun center G, and the x axle is parallel to the map sheet major axis, and the y axle is parallel to minor axis, and coordinate system meets the right-hand rule.Write down the A point position of spray gun, at this moment A point coordinate (X over against the right cylinder side _A, Y _A, Z _A) be (0,0,220).With the right cylinder side be launched into right cylinder on the tangent plane of A point bus, in the spraying area of whole plane, choose coordinate points B ' (X every 5mm along x axle, y axle _{B '}, Y _{B '}, Z _{B '}), carry out Theoretical Calculation and actual measurement, oil film thickness distribution surface and error curved surface such as Figure 11, shown in Figure 12, its calculated value and measured value are basic coincideing, maximum error is 3.343 μ m, and error mean is 0.4207 μ m, and square error is 0.5998 μ m.

Simultaneously, in order to understand whole computation process, provide the some B ' (X that launches x=-45 section on the plane _{B '}, Y _{B '}, Z _{B '}), as the verification experimental verification sample, totally 13 sample points carry out Theoretical Calculation and actual measurement, list in table 2 with them.Every symbology meaning can be with reference to Fig. 3 or Fig. 4 in the table.

Table 2 right cylinder side spraying special case checking paint deposition rate

Sequence number	Right cylinder launches plane B ' coordinate (X B′，Y B′，Z B′)	Right cylinder side corresponding B coordinate (X B，Y B，Z B)	F respective coordinates (X on the EF flat work pieces F，Y F，Z F)	GB length l (mm)	The length l of AB AB(mm)	??O AO BLength ρ (mm)	Calculated value (um)	Measured value (um)
									??1	??(-45，-30，220)	??(-45，-29.76，223.26)	??(-40.31，-26.66，200)	??229.69	??54.0503	??45	??2.394	??2.2
??2	??(-45，-25，220)	??(-45，-24.86，222.27)	??(-40.49，-22.37，200)	??228.13	??51.4614	??45	??6.663	??6.4
									??3	??(-45，-20，220)	??(-45，-19.93，221.45)	??(-40.64，-18.00，200)	??226.85	??49.2371	??45	??13.877	??13.6
??4	??(-45，-15，220)	??(-45，-14.97，220.82)	??(-40.76，-13.56，200)	??225.85	??47.4318	??45	??23.359	??23.6
									??5	??(-45，-10，220)	??(-45，-9.99，220.36)	??(-40.84，-9.07，200)	??225.13	??46.0972	??45	??33.135	??33.1
??6	??(-45，-5，220)	??(-45，-5.00，220.09)	??(-40.89，-4.54，200)	??224.70	??45.2769	??45	??40.545	??40.3
									??7	??(-45，0，220)	??(-45，0，220)	??(-40.91，0，200)	??224.56	??45	??45	??43.313	??43.1
??8	??(-45，5，220)	??(-45，5.00，220.09)	??(-40.89，4.54，200)	??224.70	??45.28	??45	??40.545	??41.3

Sequence number	Right cylinder launches plane B ' coordinate (X B′，Y B′，Z B′)	Right cylinder side corresponding B coordinate (X B，Y B，Z B)	F respective coordinates (X on the EF flat work pieces F，Y F，Z F)	GB length l (mm)	The length l of AB AB(mm)	??O AO BLength ρ (mm)	Calculated value (um)	Measured value (um)
									??9	??(-45，10，220)	??(-45，9.99，220.36)	??(-40.84，9.07，200)	??225.133	??46.10	??45	??33.135	??35.2
??10	??(-45，15，220)	??(-45，14.97，220.82)	??(-40.76，13.56，200)	??225.8527	??47.43	??45	??23.359	??23.6
									??11	??(-45，20，220)	??(-45，19.93，221.45)	??(-40.64，18.00，200)	??226.8549	??49.24	??45	??13.877	??13.7
??12	??(-45，25，220)	??(-45，24.86，222.27)	??(-40.49，22.37，200)	??228.1349	??51.46	??45	??6.663	??6.3
									??13	??(-45，30，220)	??(-45，29.76，223.26)	??(-40.31，26.66，200)	??229.6862	??54.05	??45	??2.394	??2.1

In the table two, when theory is calculated, for formula (18), by B ' (X _{B '}, Y _{B '}, Z _{B '}), A point coordinate (0,0,220) and known coordinate position relation adopts following transformational relation can try to achieve B point coordinate (X _B, T _B, Z _B), F point coordinate (X _F, T _F, Z _F), the length l of GB, the length l of AB _ABAnd O _AO _BLength ρ.

B point coordinate (X _B, Y _B, Z _B):

$X_B=X_{B^{\&prime;}}=;Y_B=R_B\&times;\frac{360\&times;Y_{B^{\&prime;}}}{2\mathrm{\&pi;}{R}_B};Z_B=R_B\&times;(1-\cos \frac{360\&times;Y_{B^{\&prime;}}}{2\mathrm{\&pi;}{R}_B})+Z_{B^{\&prime;}}---\left(25\right)$

F point coordinate (X _F, Y _F, Z _F):

$X_F=\frac{200X_B}{Z_B};Y_F=\frac{200Y_B}{Z_B};Z_F=200---\left(26\right)$

The length l of GB:

$l=\sqrt{{\mathrm{<figure-callout\; id="2"\; label="X\; B"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">X</figure-callout>}}_B^{}+{\mathrm{<figure-callout\; id="2"\; label="Y\; B"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">Y</figure-callout>}}_B^{}+Z_B^2}---\left(27\right)$

The length l of AB _AB:

$l_{\mathrm{AB}}=\sqrt{{\mathrm{<figure-callout\; id="2"\; label="X\; B"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">X</figure-callout>}}_B^{}+{\mathrm{<figure-callout\; id="2"\; label="Y\; B"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">Y</figure-callout>}}_B^{}+{(Z_B-220)}^2}---\left(28\right)$

O _AO _BLength ρ:

ρ＝|X _B|????(29)

From Figure 11, Figure 12 and table 2 as can be seen, using the fixed point spraying of right cylinder side is approximate consistent as the theoretical value that special case calculates on computers with actual spraying measurement result, the Theoretical Calculation process is reliable, the test point maximum error is less than 4 μ m, and overall square error is less than 0.6 μ m.Since in the process of the test interference and the measurement point of various factors can not accurately locate, the measuring error of surveying instrument, brought certain error thus, this error in permissible range, thereby verified the correctness of free form surface aerial spraying paint deposition rate model.

Embodiment 2

For validity and the practicality of institute's established model for spray gun continuous motion operating mode further is discussed, under test two conditions, choose same specification right cylinder side, keep process conditions constant, spray gun is done rectilinear motion along the central axis of the relative cylindrical workpiece of right cylinder bus with the speed of 100mm/s, and the spray gun center continues the one-way trip spray test is carried out in the right cylinder side over against the right cylinder central axis.After the cylindrical workpiece one-way trip spraying as shown in figure 13, in order to keep and the consistance of testing one, two, copy test two, set up coordinate system, true origin fixes on spray gun center G, with the right cylinder side be launched into right cylinder on the tangent plane of A point bus, the y axle is parallel to the highest cylindrical bus, be designated as the x axle in the direction perpendicular to the y axle, coordinate system meets the right-hand rule, right cylinder is carried out film thickness detect.Theoretically, in the paint range coverage, arbitrfary point on the periphery is from entering spraying area up to leaving this zone, all in the deposition of accepting from the coating cloud particle of spray gun, and amount of paint differences that receive in the different moment because in the residing position of spraying area difference just.Because spray gun does linear uniform motion, therefore the paint thickness that is deposited on the difference of same bus of right cylinder side of paint equates, and is consistent cut the paint shape of curved surface profile of circle cross section perpendicular to the difference of fortune rifle direction.Therefore, in order to eliminate measuring error, measure to section along a plurality of y on the plane launching, same y measures the identical data of a plurality of x values and asks on average as measurement data on section, can obtain measured value.Measured value has reflected that paint film is at the paint film average thickness that cuts each point on the circle cross section perpendicular to the spray gun moving direction.

For the paint thickness of periphery arbitrfary point B, in the spraying of spray gun fixed point, know that by preceding surface analysis its oil film thickness can derive out by the oil film thickness that corresponding F on the flat work pieces is ordered.At spray gun certain t constantly with speed v motion, the single oval map sheet that is deposited on flat work pieces crosses along spray gun direction of motion that F point work is parallel to the straight line of y axle and the border of oval map sheet meets at F as shown in figure 14 ₁, F _N+12 points, because the relativity of motion, the paint thickness that the F point is deposited in the motion of spray gun one-way trip on the flat work pieces should equal the F point with the paint thickness deposition of counter motion through single map sheet zone process, equals line segment F on the numerical value ₁F _N+1Between the thickness product of paint section divide sum.Connect GF ₁, GF _N+1Cross the bus that B orders with right cylinder and meet at B respectively ₁, B _N+12 points, because the curvature of the arbitrfary point on the same bus is identical, in like manner, during the one-way trip spraying, the paint thickness of periphery arbitrfary point B equals to be deposited on line segment B in the single map sheet zone ₁B _N+1Between the thickness product of paint section divide sum.Set integral accuracy n, with F ₁F _N+1Be divided into the n equal portions, at this moment F ₁F _N+1On n+1 point arranged, F _iBe F ₁F _N+1On i branch, B ₁B _N+1The corresponding n equal portions that also are divided into have on it and F ₁F _N+1Go up corresponding n+1 point, B _iBe B ₁B _N+1Go up and F _iI corresponding branch.Utilize formula (23), calculate line segment B ₁B _N+1The oil film thickness of last n+1 point according to preceding surface analysis, is exactly the oil film thickness integrated value corresponding to the arbitrfary point B on this bus with this n+1 oil film thickness addition of putting as can be known, obtains formula (30):

$q_{B_i}=\underset{0}{\overset{T}{\&Integral;}}{q}_u{(1-\frac{x_F^2}{a^2})}^{{\mathrm{\&beta;}}_1-1}{\{1-\frac{{[2b{(1-{\mathrm{<figure-callout\; id="2"\; label="x\; F"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">x</figure-callout>}}_F^{}/a^2)}^{1/2}-\mathrm{vt}]}^2}{b^2(1-{\mathrm{<figure-callout\; id="2"\; label="x\; F"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">x</figure-callout>}}_F^{}/a^2)}]}^{{\mathrm{\&beta;}}_2-1}\mathrm{dt}$ (30)

$=\underset{i=1}{\overset{n+1}{\mathrm{\&Sigma;}}}{q}_u\&times;{(1-\frac{x_F^2}{a^2})}^{{\mathrm{\&beta;}}_1-1}\&times;{\{1-\frac{{[2b{(1-{\mathrm{<figure-callout\; id="2"\; label="x\; F"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">x</figure-callout>}}_F^{}/a^2)}^{1/2}-{\mathrm{vt}}_i]}^2}{b^2(1-{\mathrm{<figure-callout\; id="2"\; label="x\; F"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">x</figure-callout>}}_F^{}/a^2)}\}}^{{\mathrm{\&beta;}}_2-1}\&times;\frac{T}{n}$

Q wherein _uFor testing a maximum oil film thickness q _MaxWith the ratio of test one fixed point spray time, the μ m/s of unit.T is that the F point passes through line segment F in the single map sheet zone along the opposite direction of spray gun speed ₁F _N+1The time that needs, t _iFor the F point along paint map sheet in line segment F ₁F _N+1From F ₁To F _iThe time of being experienced.For T and t _iCalculate according to formula (31), (32) formula.

$T=2b{(1-{\mathrm{<figure-callout\; id="2"\; label="x\; s"\; filenames="H2009101146141E00062.png,H2009101146141E00081.png"\; state="\{\{state\}\}">x</figure-callout>}}_s^{}/a^2)}^{1/2}/v---\left(31\right)$

$t_i=\frac{T}{n}(i-1),i=\mathrm{1,2,3}...n+1---\left(32\right)$

Therefore, in computing machine, utilize formula (30) programming, consider the precision that numerical integration is calculated, x _FEach corresponding time integral upper limit T is equally divided into 10 ⁴Equal portions utilize trapezoidal method to carry out numerical integration and calculate as calculating step-length, obtain the simulation calculation result.Then, the measured value and the computer theory result of calculation of spraying back film thickness are compared, as shown in figure 15, measurement result and simulation result are near consistent, the maximum error of measurement result and simulation result is 0.6579 μ m, the thickness square error is 0.2207 μ m, and average error is 0.1683 μ m.Therefore at the uniform velocity spray operating mode for free form surface, above institute established model is effectively, can obtain a rational paint thickness calculated value.

In the actual spraying process, paint is after the ejection of spray gun nozzle is deposited on workpiece, because spray pattern middle part paint film is thicker, the edge is thinner, spray pattern overlaps mutually before and after must making during spraying, just the point in the part zone in the middle of two adjacent strokes of spray gun will be by spraying twice, and their film thickness is this twice spraying result's stack.In the teach programming system, the overlapping width of spraying swath between spray gun path and adjacent two paths established a capital really according to experience and carry out, distribute if adopt model of the present invention then can simulate film thickness on computers, seek out rational spray gun motion path and overlapping width, obtain uniform film thickness on the surface of the work thereby make.Therefore further illustrate validity and the practicality that the free form surface paint deposition rate model is set up.

Embodiment 3

On above experimental basis, below we utilize the free form surface paint deposition rate model to weave program, be applied in the spraying robot off-line programming software, with free curve surface work pieces---bumper is the simulation calculation that example carries out paint thickness, bumper part after the import system discretize as shown in figure 16, technological parameter according to table 3 sprays, and sprays the back workpiece as shown in figure 17, accessory size 1495mm * 395mm * 320mm.Sample every 100mm on the bumper surface, totally 55 sample points, off-line system can obtain theoretical value through simulation calculation after sample point generates corresponding normal vector automatically.The spraying Theoretical Calculation distribution value of sampled point and actual measurement distribution value are respectively as Figure 18, shown in Figure 19.

Table 3 spraying coating process condition

Room temperature	??23℃
		Paint	From dried metal colored paint
Viscosity	??16sec/NK-2
		Go out the lacquer amount	??120ml/min
Coating pressure	??0.15MPa
		The spraying air pressure	??0.20MPa
The map sheet air pressure	??0.15MPa
		The map sheet width	??130mm
The map sheet overlap distance	??160mm
		Fortune rifle speed	??750mm/sec

From Figure 18, Figure 19 as can be seen, spray gun sprays according to track shown in Figure 180, and oil film thickness is compared thin partially with central region in bumper corner.In the process of the test, owing to have the spraying dead angle area in the corner of bumper, promptly paint unreachable zone, Control During Paint Spraying by Robot is painted and can not all be deposited time this place, causes comparing with other zones in bumper corner paint thickness obviously thin partially.In the actual engineering, regular way is before the robot automation sprays this part to be carried out the pre-spray treatment in part, with homogeneity that reaches the whole work-piece paint thickness and the consistance of painting aberration.Whole work-piece is 1.937 μ m at the measured value of 55 sample point and the maximum error of calculated value, and error mean is 0.5717 μ m.As can be seen, measured value and calculated value are near consistent.Thereby illustrate that the free form surface paint deposition rate model can be finished the prediction to the paint thickness of free curve surface work pieces spraying, for such CAD/CAM system, can use equally.

Claims (5)

1. a modeling method that adapts to the paint deposition rate model of complex free curved surface spraying is characterized in that, this method comprises the steps:

(1) selected flat work pieces paint deposition rate model function adopts genetic algorithm to simulate the paint deposition rate model of this flat work pieces by the flat work pieces spray test;

(2) set up the judgment criterion of painting accessibility;

(3) utilize circle of curvature method, ask section and maximum paint thickness to adhere to angle between the imaginary plane, free curve surface work pieces actual (real) thickness formula is set up in calculating, the flat work pieces paint deposition rate model that simulates is extended to the paint deposition rate model that adapts to the complex free curved surface workpiece.

2. the modeling method of the paint deposition rate model of adaptation complex free curved surface spraying according to claim 1, it is characterized in that, the described approximating method that adopts genetic algorithm to simulate the paint deposition rate model of this flat work pieces by the flat work pieces spray test is: the parametric variable to be asked behind the selected flat work pieces paint deposition rate model function in the clear and definite pattern function, based on test figure, ask parametric variable to adopt real number coding method to waiting in the model, the span of each variable with reference to the experimental measurement data to determine in certain limit, in resulting each parameter area, produce an initial population of forming by real number at random, the scale of colony is taken as 200 * 8, whole colony is divided into 8 sub-populations with scale, the number of each sub-population is 200, evolutionary generation is 400, generation gap is 0.8, crossing-over rate is 1, aberration rate is 0.2, the insertion rate is 0.9, mobility is 0.2, sub-population migration algebraically is 20, select operator to adopt random ergodic sampling (sus) method, crossover operator adopts the discrete recombination operator, mutation operator is selected real-valued population mutation operator, genetic algorithm to asking parameter in global optimizing, principle according to the survival of the fittest is evolved, and calculates the parametric variable to be asked of flat work pieces paint deposition rate model function through the interative computation in given generation.

3. the modeling method of the paint deposition rate model of adaptation complex free curved surface spraying according to claim 1 is characterized in that, the judgment criterion of described foundation paint accessibility is:

(1) spray gun sprays paint and moves to when certain is put on the free form surface, according to the spraying coating process requirement, it is that what is called is just sprayed that spray gun rifle body keeps vertical with surface of the work, this spraying point on the free form surface is called as positive specking at this moment, judge whether the neighbor point on the free form surface and the line of spray gun end central point in a certain scope of this positive specking pass within the deposition map sheet in the corresponding flat work pieces spray test of positive specking, when this condition satisfies, neighbor point will be subjected to the spray gun spraying influence of positive specking top, promptly so-called spraying influences a little, change judgment criterion (2) over to, otherwise quit a program;

(2) section of the spraying influence point in the positive a certain nearby sphere of specking and maximum paint thickness are adhered to the angle between the imaginary plane, judge this angle whether interval [0,90), otherwise it is excessive to be sprayed the curvature corner that influences a little on curved surface, spray gun ejection paint is unreachable, quits a program; It was the perpendicular plane of line between this spray gun spraying influence point and terminal central point of spray gun and the spraying influence point that wherein maximum paint thickness is adhered to imaginary plane.

4. the modeling method of the paint deposition rate model of adaptation complex free curved surface spraying according to claim 1, it is characterized in that, described step is utilized circle of curvature method in (3), ask section and maximum paint thickness to adhere to angle between the imaginary plane, free curve surface work pieces actual (real) thickness formula is set up in calculating, the flat work pieces paint deposition rate model that simulates is extended to the paint deposition rate model that adapts to the complex free curved surface workpiece, and concrete steps are as follows:

(1) set up paint deposition rate model on this spraying influence point and the imaginary plane parallel with flat work pieces, the expression formula of the paint deposition rate model on the imaginary parallel plane is:

$q_B=q_F{\left(\frac{h}{l}\right)}^2{\left(\frac{{2h}_{1}l}{h_1^2+l^2-l_{\mathrm{AB}}^2}\right)}^2---\left(1\right)$

Q wherein _F, q _BBe respectively paint particles be traveling in flat work pieces in the flat work pieces spray test along a certain rectilinear direction on, cross the paint deposition rate (μ m/s) on free curved surface spraying influence point and the imaginary plane parallel, h, h with flat work pieces _lBe respectively the height of spray gun centre distance flat work pieces, positive specking, l, l _ABBe respectively the length between length between spray gun center and the spraying influence point and positive specking and the spraying influence;

(2) set up spraying influence point and adhere to paint deposition rate model on the imaginary plane direction in maximum paint thickness, the expression formula that maximum paint thickness is adhered to the paint deposition rate model on the imaginary plane direction is:

$q_B^{\#}=q_B/\cos \mathrm{\&theta;}=q_F{\left(\frac{h}{l}\right)}^2{\left(\frac{{2h}_{1}l}{h_1^2+l^2-l_{\mathrm{AB}}^2}\right)}^3---\left(2\right)$

Q wherein ^# _BFor spraying influence point adheres to paint deposition rate (μ m/s) on the imaginary plane direction in maximum paint thickness, θ adheres to angle between the imaginary plane for the imaginary plane parallel with flat work pieces and maximum paint thickness;

(3) set up the paint deposition rate model of spraying influence point on the section, the paint deposition rate model on this section is the paint deposition rate model of free curved surface spraying, and its expression formula is:

$q_B^{*}=q_B^{\#}\cos \mathrm{\&alpha;}=q_F{\left(\frac{h}{l}\right)}^2\frac{{\mathrm{GO}}_B^2-(l^2+R_B^2)}{{\mathrm{lR}}_B}\frac{{4h}_1^3{l}^3}{{(h_1^2+l^2-l_{\mathrm{AB}}^2)}^3}---\left(3\right)$

Q wherein _B ^*For GB direction plane BD goes up the paint deposition rate (μ m/s) that B is ordered; R _BBe the curvature radius of a ball value of free form surface at spraying influence point, GO _BBe the length of spray gun center to the spraying influence point center of curvature, some O _BSphere center position for spraying influence point place curvature ball, to be free form surface adhere to angle between the imaginary plane in the maximum paint thickness of the section of spraying influence point and this point to angle α, more than four determine that according to the circle of curvature method point G is present position, spray gun center.

5. as right 1 described a kind of modeling method that adapts to the paint deposition rate model of complex free curved surface spraying, it is characterized in that: the paint deposition rate model of the adaptation complex free curved surface spraying that described modeling method generates, be applied to the free curve surface work pieces solid model be carried out the simulation calculation of paint thickness in the spraying robot off-line programming software.

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