IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 15, NO. 1, JANUARY 2007
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Optimal Robot Path for Minimizing Thermal Variations in a Spray Deposition Process Paul D. A. Jones, Stephen R. Duncan, Member, IEEE, Tim Rayment, and Patrick S. Grant
Abstract—This paper describes a method for determining the optimal robot path that minimizes thermal variations over a surface during a spray deposition process, where the deposited material is hotter than the surface. An analytical expression is formed for the amplitude of the thermal modes of the surface temperature profile. This expression is then used to determine the optimal robot velocity, scan angle, and start position. Experimental results from a metal spray deposition process are used to confirm the analysis. Index Terms—Distributed parameter systems, materials processing, optimal control, robot programming, spatially distributed systems, spraying, temperature control.
I. INTRODUCTION
T
HE DESIGN of the optimal robot path for spray processes where a robot is used to coat a substrate with a material has been extensively studied and examples include spray-glazing [1], generic spraying [2], and paint spraying [3], [4]. However, this body of work concentrates primarily on achieving uniform mass coverage of the substrate. In this paper, a thermal spraying process is considered, where a robot moves an arc spray gun over a substrate [5] and the aim is to determine the robot path that minimizes the temperature variations over the surface of the deposited metal, rather than minimizing variations in the deposited mass. Although the design of the robot path for minimizing thermal variations has much in common with the design of a path for achieving uniform deposition of mass, optimizing the temperature variations is more complex, primarily because it is necessary to account for the flow of heat within the surface of the deposited metal, which is an example of mobile control [6], [7]. The robot path can be described by the position and orienta. The path can, tion of the robot focus point for therefore, be described by the seven-dimensional (7-D) vector (1) where , , and represent the cartesian location of , , and denote anthe focus of the robot and gular rotations about the -, -, and -axes, respectively. All of Manuscript received September 21, 2004; revised December 12, 2005. Manuscript received in final form June 30, 2006. Recommended by Associate Editor D. Gorinevsky. This work was supported in part by the United Kingdom Engineering and Physical Research Council under Grant GR/M87832 and by the Ford Motor Company. P. D. A. Jones, S. R. Duncan, and P.S. Grant are with the Department of Engineering Science, Oxford University, Oxford OX1 3PJ, U.K. (e-mail: stephen.
[email protected]). T. Rayment is with Instron Ltd., High Wycombe HP11 2TJ, U.K. (e-mail:
[email protected]). Digital Object Identifier 10.1109/TCST.2006.883196
the elements of are continuous, therefore, an infinite number of distinct paths exist, but it is usually not possible to perform a continuous optimization of a general robot path over a general three-dimensional (3-D) substrate because this approach is computationally infeasible for the 7-D search space. As a result, most solutions have concentrated on lower-order optimizations. Much of the work in the area of robot path planning for spray processes has been motivated by paint spraying. Klein [8] describes a method for achieving even paint coverage, assuming a model of paint distribution in the spray cone that is flat in the center and sinusoidal at the edges. The method involves manual iteration of the path based on numerical simulations of coating thickness over a part. This method was improved by Hertling et al. [9] who proposed coupling the thickness simulation with data from thickness sensing of the paint coverage of the sprayed part for the purpose of improving the simulation model and aiding the design of automatic path planning strategies. Antonio [10] describes a different approach to finding the optimal path for minimizing thickness variation of the paint coating over the part. Antonio expresses the problem as a constrained variational optimization problem, where the constraints are the limits on the reach, velocity, and acceleration of the robot and position constraints that avoid collision with the part being sprayed. The search space is 7-D with a six-dimensional (6-D) vector describing position and orientation of the robot and the final dimension as time. The problem can be solved using standard nonlinear programming techniques, but due to limitations of processing power, Antonio only presents results for a 3-D search space, where the robot is constrained to move in the – plane and spray onto a flat substrate. Further work from Antonio and coworkers [3], [11], describes a suboptimal, or locally optimal, solution for the time profile along a specified spatial path, which reduces the order of the search space to one-dimension. Offline programming of robots for even thickness spray glazing of ceramic parts is described in a paper by Bidanda et al. [1]. Their path planning technique involves parameterizing the surface to be sprayed using a mesh of two families of intersecting curves, each family being orthogonal to the other. Each individual curve is represented by a polynomial fit of the surface being sprayed in either or , the parameters for the two orthogonal directions. The path plan is then described by a set of constant or constant curves which are offset by half the width of the glaze spray cone, in order to achieve even coverage with minimal wastage (it is assumed that the distribution of glaze inside the spray cone is constant). This technique is repeated for each surface patch of a part, but can produce poor results when many patches are present, due to overspray onto adjacent patches.
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Sheng et al. [4] use a method to reduce the search space for a spray painting process. The method involves splitting the complex geometry of the surface being sprayed into a set of flat patches which can be sprayed individually in turn. This method reduces the 7-D problem into a set of 3-D problems. This method is effective for the spray painting process because the aim is an even mass coverage, but this method would not be effective for the sprayforming process, because spraying small areas of the part sequentially would set up large temperature variations over the surface. The problem of determining the path of a scanning heat source that minimizes the thermal variation has received less attention. Demetriou and Doumanidis and coworkers have cast the path planning problem in terms of a quadratic optimization problem, where the path is expressed in terms of a series of discrete locations and the optimal path defines the movement of the robot between these locations. This leads to a path plan for a one-dimensional (1-D) problem [7], [12], which was extended to the two-dimensional (2-D) case in [13]. By contrast, this paper takes a different approach, where a continuous robot path consisting of a series of straight line passes over the surface is parameterized in terms of the robot velocity, the orientation of the scans, and the start position. As with the approaches described in [12] and [13], the robot path is chosen to minimize the mean square deviation of surface temperature. However, in the approach presented in this paper, an analytical solution for the variance is obtained, which can be optimized directly without the need to solve a Riccati equation. In fact, the approach described here is more closely related to previous work by Doumanidis and coworkers [14]–[16], where a state space model of the process is used to determine the optimal path for a plasma-arc cutting tool. The benefits of using the optimal path are demonstrated in this paper using results from trials on the sprayforming process and it is shown that the optimal path significantly reduces the temperature variance over the surface compared to the standard “standard” faster robot path that is commonly used in spray processes [3], [9]. This paper is laid out as follows. The sprayforming process and the parameterized robot path are described in Section II and a dynamic model of the thermal flows within the sprayed material is developed in Section III, which leads to an analytic expression for the temperature variance. Section IV shows how the path that minimizes the temperature variance can be obtained and Section V describes the results from using the path in practice. Section VI concludes the paper.
II. DESCRIPTION OF PROCESS The sprayforming process deposits molten metal from electric arc spray guns onto a ceramic substrate to build up a metal shell. The molten metal is produced in the guns by direct current arcing between two oppositely charged wires that are made from the metal that is being sprayed. The arcing causes the wire tips to melt and a high-pressure inert gas stream strips molten metal from the arc, atomizing it into a stream of droplets. The gas stream carries the droplets to the surface of the ceramic where they are deposited to build up a thick metal shell. As
Fig. 1. Diagram of sprayforming process.
shown in Fig. 1, four-arc spray guns are mounted on an industrial six-axis robot, which moves the guns over the surface in a predetermined, repetitive manner, referred to as the “path plan.” The guns act not only as source of material, but also as a source of heat because the molten droplets transfer their heat to the metal shell as they cool and solidify. A key feature of the process is that the temperature of the surface needs to be controlled throughout spraying in order to ensure that the metal droplets undergo prescribed phase transformations that offset the natural contraction of the metal as it cools. A feedback system has been implemented [5], which uses a thermal imaging camera to measure the temperature variations over the sprayed surface and then adjusts the spray rate in order to regulate the surface temperature. Experience with using this controller showed that large temperature variations were induced when the spray guns follow suboptimal paths and it was then necessary for the feedback controller to use large control actions to remove these variations. In order to avoid introducing large temperature variations, this paper describes how to determine the path that the robot should follow in order to minimize the temperature variations introduced by the spray guns as they are moved over the surface. This has two main purposes. • Open loop—In the absence of any feedback control, utilizing the optimal spray path will provide the optimal thermal profile and as a result, minimize the stresses that develop within the deposited metal. • Closed loop—Even if online feedback control is applied to minimize temperature variations, using the optimal path plan will minimize the thermal gradients introduced by
JONES et al.: OPTIMAL ROBOT PATH FOR MINIMIZING THERMAL VARIATIONS IN A SPRAY DEPOSITION PROCESS
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In this paper, the analysis is limited to the case where and 45 90 , although the cases where and/or 45 can be readily analyzed using the same approach. III. MODELING THE HEAT FLOW WITHIN THE SURFACE The evolution of the temperature , at a given point on the sprayed surface can be described by the heat diffusion partial differential equation [5] (3)
Fig. 2. Schematic of the robot path over a rectangular substrate of dimension L by L , parameterized in terms of scan velocity v , scan angle , and scan start point x .
the motion of the spray guns over the surface, which reduces the level of control authority required to regulate the process. To reduce the search space and computational effort required for the optimization of the sprayforming process, this paper analyzes the specific case of flat, rectangular substrates of dimenby , sprayed using paths consisting of straight line sion robot motions at a constant speed of from one edge of a rectangular substrate to another. The motion of the robot is constrained to be in the – plane (so that there is no movement in the direction) with the orientation of the robot wrist held constant so that the robot focus remains normal to the substrate surface. The robot path can, therefore, be described by the 3-D vector (2) The substrate is positioned such that its edges are coincident with the - and -axes and the robot motion is always at an angle to the -axis and 90 to the -axis, where is of referred to as the scan angle. The start point for the path is at an from the origin in the direction. When the robot offset of reaches an edge of the rectangle, its velocity normal to the edge is reversed so that it scans back across the rectangle, as shown in Fig. 2. In order to program the robot movements, it is necessary for the path to consist of a finite number of moves and the ideal path (from the programming point of view) is for the robot to return to the start point following a finite number of “reflections” at the edges. Under these circumstances this closed path, or “scan,” can be repeated either until a new path is required or until spraying is complete. Because the path consists of reflections at the edges of the sprayform, it is referred to as a “mirrorbox” path and a closed-mirrorbox path contains a finite number of reflection points on the edges of the rectangle. In the mathematics literature, the choice of angle that results in a closed path is an example of a “billiard ball” problem [17] and a single scan of the mirrorbox path consists of reflections from the top and bottom edges of the rectangle and reflections from the sides.
where is the specific heat of the sprayed material, is its denis the thickness of sity, is the thermal conductivity, and the material, which is taken to be the same over the surface. is the heat transfer coefficient between the sprayed material and the air flowing over the top surface while is the temperature is the heat transfer coefficient between of the air. Similarly, denotes the temthe sprayed material and the ceramic and perature of the ceramic. The ceramic is preheated to match the average temperature of the sprayed metal in order to minimize the thermal flow between the metal and the ceramic and given the thermal mass of the ceramic, it is assumed that remains fixed during spraying. In the process that forms the basis of this paper, there is a strong air stream blowing across the surface being sprayed. The primary purpose of this air stream is to remove dust and fumes from the surface, but it has the additional effect of providing a strong forced convection from the surface. Although there is also heat loss due to radiation from the surface, when an energy balance was calculated, it showed that convection from the top surface of the sprayed material was the dominant method of heat loss. For this reason, the model used in the analysis presented here uses convection as the main source of heat loss. For the more general case, it would be possible to include the effect of radiation, but it would then be necessary to linearize the model around a given operating temperature, as the optimization of the path is based on a linear model. is the 2-D second spatial derivative The operator (4) The term denotes the heat flux from the spray. This can be considered as a fixed spatial “footprint,” , whose location on the sprayed surface is determined by the path followed by the robot, so that if at time , the robot is positioned at , , then point (5) It was assumed that this fixed footprint was truncated as the spray moved over the edge of the surface. This will not be exactly accurate as the flow pattern of the droplets will be disrupted as the spray moves across the edge. For this reason, the derived path will not be optimal near the edges, as can be seen in the results presented in Section V. In practice, the temperature variations introduced at the edges were removed by the feedback control system. The shape of the spatial footprint was determined experimentally [18], but the footprint can be approximated by a 2-D Gaussian function with circular symmetry. The assumption that
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the footprint has a Gaussian shape has been used by a number of authors [14], [19]–[21], although the analysis presented here is applicable to other footprints, provided that their shape is smooth. If the orientation of the robot and the offset from the surface remains fixed as the spray moves over the surface, then the shape of this footprint remains the same and the time depenarises solely from the changing location of dence in the robot , . Although there will be some convection from the edges of the sprayed material, the thickness of the sprayed material is very much less than the dimensions of the surface (so that and ), and the heat loss from the edges is small by comparison to the heat loss from the top surface. As a result, it is reasonable to use the Neumann boundary conditions, so that 0 for 0 and and 0 for 0 and . The eigenfunctions of the spatial operator that satisfy these boundary conditions on a rectangular surface, are
the orthogonality of the eigenfunctions, leads to a set of ordinary differential equations describing the time evolution of the amplitude of each mode [5] (10) where (11)
(12)
(13) with
(6) so that (7)
(14) for for for
, ,
or
,
(15)
which is derived from where with
are the eigenvalues associated with
,
(8) Including convection from the edges of the sprayed material effectively changes the boundary conditions for the model, which can be handled by modifying the eigenfunctions in (7). It is conceptually straightforward to deal with this case, although the analysis does become more complex. For the particular problem being considered here, the results showed that a good prediction of the optimal path was obtained by assuming that the heat loss from the edges was negligible. The temperature profile of the surface can be expressed as a basis function expansion in terms of the spatial eigenfunctions (9) where denotes the time varying amplitude associated with each spatial “mode.” Strictly, the true solution for is obtained as the number of eigenfunctions in the summation of (9) approaches infinity, but this would create an infinite dimensional model that could not be used in practice. Because the footprint of the heating source is smooth, it will be shown below the amplitude of modes for large values of and will negligible, so it is reasonable to truncate the summation to and terms, respectively. The choice of suitable values for and will be discussed below. Multiplying the partial differential equation in (3) throughout , then integrating over and by , substituting for using (6) and exploiting
(16) in (11) is time varying due to the increase in thickStrictly, , as spraying proceeds. However, the rate at which maness terial thickness increases tends to be slow compared to the rate will be taken as a constant. of heat flow and as a result, If necessary, the optimization can be performed for a range of as increases. values for Because is a constant, in (13) will be 0 and 0. This means that the zero except when term will only affect , which corresponds to the average temperature over the surface. Since the aim is to minimize deviterm will not ations of temperature from the average, the affect the variance of the temperature and for this reason, it will be omitted from the subsequent analysis. at time , then using If the robot is positioned at (5) the expression for in (12) becomes
(17) where the time dependence of and has been omitted to simplify the notation. Using the derivation in Appendix A, the reduces to expression for (18) where is obtained from the expansion of the combined spatial footprint of the guns when it is positioned in the center does of the surface, so that the region where
JONES et al.: OPTIMAL ROBOT PATH FOR MINIMIZING THERMAL VARIATIONS IN A SPRAY DEPOSITION PROCESS
not extend beyond the edges of the surface. The guns will be and positioned at the center of the surface when , so that
If the robot path starts at the first sweep over the surface
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, then during
(26) (27)
(19)
Substituting these positions into the expression for (18)
in (28)
The footprint of the heat source is taken to have a Gaussian shape, which is a smooth function in the sense that its higher order spatial derivatives are finite, and as shown in Appendix B, as and increase. For for smooth footprints this reason, the heat source has no effect on the higher order in modes, which justifies truncating the expansion for (9), provided that and are chosen such that 0 for and . Strictly, it is not possible for a spray footprint that it is localized in space to have a finite bandwidth in the spatial frequency domain, but for smooth functions, such as the Gaussian shape considered here, the response will be approximately bandlimited in the sense there exists a constant , for and . such that A. Analysis of Mirrorbox Paths The robot follows a mirrorbox path in the – plane as shown , in Fig. 2. If the path starts at point is a rational number, the path will pass then provided that through the same point after having made a finite number of reflections at the edges. This gives a closed path consisting a finite number of moves that can be programmed into the robot and the sequence executed repeatedly during spraying. The number of reflections is related to the scan angle , and if denotes the number of reflections on the top or bottom edge of the rectangle, while denotes the reflections on either of the side edges, then
, the robot reaches the top edge and the At time -component of the velocity is reversed, so that . Substituting this value into (28) gives
(29) (30) A similar argument applies when the robot reverses the -component of its velocity at the right-hand edge of the surface. This is a consequence of the Neumann boundary conditions for the heat diffusion equation, which means that the spatial eigen, consist of cosine functions. As a result, functions is an even function with respect to both and , so in (18) holds as the sign of and this expression for switches when the heat source reverses direction at the edge of the surface. in Using a trignometric identity, the expression for (30) can be rearranged to give (31) where
(20)
(32) (33)
(21)
The total length of the closed path, denoted by , is (22) and if the velocity of the robot is , then the time to complete a single scan , is (23)
(34) Applying Laplace transforms to the ordinary differential equations describing the time evolution of the amplitude of each mode (10), gives (35) where and function from
and are the Laplace transforms of , respectively, so that , the transfer to , is given by
The velocity of the arc spray guns can be split into its components in the and directions
(36)
(24) (25)
is the sum of two sinusoids From (31) it can be seen that , is also a at different frequencies, so each of the modes
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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 15, NO. 1, JANUARY 2007
combination of sinusoids at the same frequencies, but with magnitude and phase shifts determined by the frequency response of . The expression for the modal amplitudes is, therefore
(37) (44)
where (38) (39) with (40)
If the mean variance is taken over a complete scan, then and from (71) and (72) in Appendix C, will be an integer and . Using orthogonality, the multiple of both integrals in (44) become (45)
(41) (46) IV. DERIVATION OF THE OPTIMIZATION CRITERION
(47)
The robot path should minimize the deviation of the surface temperature about the average temperature and also be short enough to be converted into a manageable sequence of lines of code in the robot programming language. The aim of the overall optimization problem is, therefore, to minimize a combination of the thermal variance and the path length. The deviation from , is given by , where the average temperature is the amplitude of the 0 state, , so the deviation of the temperature profile from the mean is obtained by removing the 0,0 term from the summations in (20), giving (42) where . At any time , the mean square deviation, or variance, of the thermal profile over the surface is
except for the case when duces to
0 when the second integral re-
(48) which from (33), occurs when (49) Under these circumstances (50) and the mean variance over a scan becomes
(51) (43) with defined in (15). Using the expression for in (37), the mean thermal variance over period is given by
where from (40) (52)
(53)
The thermal optimization criterion is to minimize , the mean of the thermal variance of the surface over a complete scan,
JONES et al.: OPTIMAL ROBOT PATH FOR MINIMIZING THERMAL VARIATIONS IN A SPRAY DEPOSITION PROCESS
while at the same time, minimizing the path length, . Combining the two criteria gives a cost function for an optimal robot path
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between the amplitude of and scan velocity. As a result, to minimize the temperature variations, the scan velocity should be as fast as possible.
(54) where is a relative weighting factor for the two criteria. When choosing a regular scanning path, there are 3 degrees-of-freedom (DOF) for adjusting , the scan start point , the scan velocity , and scan angle . A. Optimization of the Start Point The scan start point does not affect the path length because it has no effect on the number of edge points, just their positioning. term in (51), for modes However, it does affect via the 0, because from (34), depends on . From (72) where 0 when in Appendix C, (55) for some integers and . If then this implies that
and for
share no common factors (56)
, and hence, , is minimized by choosing an to minimize for the values of given in (56). From (34), it can be shown that 0 when (57) or (58) Equations (56) and (58) show that it is not possible to set 0 for all values of . However, the term , which from (11) depends upon . is scaled by Since depends upon , decreases rapidly with increasing . This roll off means that it is more important to of the lowest value of than any eliminate the effect on to minimize is other. Therefore, the optimal value of (59) It is possible that the magnitude of could get larger as in. creases, which might offset the roll off due to the term Under these circumstances, a higher value of may be optimal. deHowever, this is unlikely to occur in practice as pends upon and as mentioned above, the spatial footprint of the spray is usually smooth, which means that the magnitude also tends to decrease with increasing . This is the of case for the Gaussian footprint consider here, where decreases increases and the choice of in monotonically to zero as (59) is optimal. B. Optimization of Scan Velocity The robot or scan velocity , has no effect on the path length. Therefore, its effect on is through the term. From (51)–(53), it can be seen that increasing the velocity , reduces the magnitude of and there is an approximately inverse relationship
C. Optimization of Scan Angle There are an infinite set of paths described by including unbounded paths that never repeat, but because of the need to generate repeatable robot paths of finite length, in this application it is sufficient to consider only the values of that give closed paths. From (20) and (21), these are the values of given by (60) where and are integers. The scan angle , affects both the path length and the average thermal variance. From (22), the path length increases as both and increase, but the effect of scan angle on is more subtle. From (51) it can be seen that is a summation over each mode . In general, for smooth thermal footprints, the magnitude of will decrease as and increase. This means that the contribution to decreases with increasing and , due to the roll off of both and as and become large. However, a large peak in 0 due to the the contribution to occurs when term becoming maximized. As was shown in (56), this high excitation occurs for modes where and are integer multiples of and , respectively. As was seen in the optimization of the start point, one of these contribution peaks from the highly excited modes can be removed by judicious choice of , but the rest will remain. The scan angle is, therefore, chosen such that 0 for each of the highly excited modes so that their effect on is minimal. This process can also be thought of as choosing the scan angle so that the modes that get highly excited are outside the spatial bandwidth of the . The procecombined heating and cooling footprint dure for minimizing is a tradeoff between choosing a scan angle corresponding to large values of and , and minimizing the path length which requires small values of and . Because the cost function is nonconvex, the optimum scan angle is found searching over all possible values for and in the range . This is not as computationally expensive as might be expected, because it is only necessary to search over a finite number of discrete integer values for and . with for the case Fig. 3 shows the variation of , , 0.3 m, 0.3 m, where ms , gs , m s , s , , and is a 2-D Gaussian function with circular symmetry and a standard de. The plot shows that there are some “bad” viation of angles with large values of . These bad angles are due to either long robot paths or high average thermal variance. One , which results in such bad angle is given by . This angle is bad because the highly excited modes are within the spatial bandwidth of the heating and cooling footprint, the effect is compounded by the fact that the start point has not been chosen such that it cancels out the fun-
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Fig. 3. Plot of E against (; ) for
50 and 50.
Fig. 4. Example of a poor path plan: (9; 2) = 77:47 , E = 0:065532.
damental highly excited mode. The mirrorbox path for this scan angle is shown in Fig. 4. is shown An expansion of Fig. 3 in the range in Fig. 5. This plot shows the optimal angle for this range to , which gives . The be mirrorbox robot path for this scan angle is shown in Fig. 6. V. EXPERIMENTAL RESULTS Three experiments were performed under the same spraying conditions but with different path plans. The spray guns were set at a distance of 160 mm from the surface of the sprayform and the robot moved at a constant velocity of 0.2 ms . The guns each deposited mass at 1.8 gs onto a square ceramic of dimensions 300 300 mm. The guns followed a fixed path plan which covered an area of 380 380 mm. The variations in the thermal profile were recorded by taking an image using the thermal imaging camera, every quarter of the way through each repeat of the path plan. From the recorded thermal images, the thermal variance of the surface was calculated.
Fig. 5. Plot of E against (; ) for 50 and 50 in the range 70 75 . The optimum occurs at (13; 4) = 72:89 .
Fig. 6. Optimal robot path plan: (13; 4) = 72:89 , E = 2:79
2 10
.
Three different path plans were compared. 1) Suboptimal mirrorbox—A path plan with a “poor” 380 mm pattern, launch angle of 77.47 for the 380 which excites low order spatial modes associated with the thermal footprint of the guns. 2) Optimal mirrorbox—A path plan with an optimized 380 mm pattern launch angle of 75.07 for the 380 size that avoids the exciting the low-order spatial modes associated with the thermal footprint of the guns. 3) Raster pattern of size 380 380 mm, where the guns scan across the sprayform in a direction parallel to one edge. When the guns reach the edge of the spray pattern, they are moved a short distance parallel to the other edge and then scan back across the sprayform parallel to the original track, but in the opposite direction. This is repeated until the guns reach the edge of the spray pattern when the path is reversed. This spray path is commonly used in spraying operations [3].
JONES et al.: OPTIMAL ROBOT PATH FOR MINIMIZING THERMAL VARIATIONS IN A SPRAY DEPOSITION PROCESS
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Fig. 7. Thermal image for the suboptimal mirrorbox pattern.
Fig. 9. Thermal image for raster pattern.
Fig. 8. Thermal image for the optimal mirrorbox pattern.
Fig. 10. Plot of thermal variance for images from the suboptimal mirrorbox path, optimal mirrorbox path, and raster path.
An image taken at an equivalent time for each path plan was chosen as a typical result. The image for the suboptimal mirrorbox pattern is shown in Fig. 7, while Figs. 8 and 9 show the corresponding images for the optimal mirrorbox pattern and the raster path plan, respectively. The image sequences were analyzed to determine their thermal variance . Fig. 10 shows a plot of thermal variance over the same period in the thermal image sequence for each of the path plans. It is clear from Fig. 10 that although the raster pattern displays low variance at some stages during a scan, namely half way through and at the end, it has a high thermal variance at the intermediate stages of a quarter and three quarters of the way through a scan. This causes it to have a higher average thermal variance than the optimal mirrorbox path that displays a low thermal variance throughout its scan. As expected the suboptimal mirrorbox path has a higher thermal variance on average than its optimal equivalent.
VI. CONCLUSION Performing a general robot path optimization over a 3-D shape is computationally infeasible due to the size of the search space. Instead a restricted optimization can be performed that optimizes mirrorbox paths over flat rectangular substrates. The parameters for the optimization are the robot start point, the robot velocity and the robot scan angle. The optimization criterion is a combination of a path length criterion and a mean thermal variance criterion. The optimal start point is , which eliminates the effect of the given by fundamental highly excited mode. The optimal robot velocity is the maximum velocity that the robot can achieve. The optimal scan angle is a tradeoff between the thermal constraint that requires all of the highly excited modes to be outside the spatial bandwidth of the combined heating and cooling footprint and the path length constraint which requires the path length to be as short as possible. Once the optimal parameters for the mirrorbox path have been selected by evaluating the cost
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function over the search space, the robot program is generated automatically by a software program and uploaded to the robot for the commencement of spraying.
for some finite value , then by expanding of the eigenfunctions, such that
in terms
(65) APPENDIX then using the expression for the eigenfunctions in (6)
A) Derivation of (18): Applying the change of variables , to the integral in (17) gives
(66) (67) (61) In order for the left-hand side of this expression to remain finite as , 0. A similar result applies for the th spatial derivative in the direction. and are Integer multiples of : C) Proof that and in (32) and (33), together From the expressions for with and
and using the trigonometric identity, , gives
(68) (69) From (68) and (20)–(22)
(62) where
(70) Canceling terms gives (71) and using the same approach for
gives
(63) (72) If the spatial range of the heating and cooling source is lim0, for and for , then ited, so that the limits of the integrations in (62) can be truncated. Also, if is an even function with respect to both and , then only the integrand in the first integral is even. The other three integrands are odd functions and will, therefore, integrate to zero, , , , and provided that . This will not be the case when the robot is and , close to the edges of the surface, but if this error will be small and will be ignored in the rest of the analysis. 0 as increase for smooth B) Proof that : If the spatial response is smooth, in the sense that its th spatial derivative (in both and direction) is continuous, such that (64)
From (71) and (72), it can be shown that and are because , , , and are intealways integer multiples of gers. REFERENCES [1] B. Bidanda, V. Narayanan, and J. Rubinovitz, “Computer-aided-design-based interactive off-line programming of spray-glazing robots,” Int. J. Comput. Integr. Manuf., vol. 6, pp. 357–365, 1993. [2] A. Hansbo and P. Nylen, “Models for the simulation of spray deposition and robot motion optimization in thermal spraying of rotating objects,” Surface Coatings Technol., vol. 122, pp. 191–201, 1999. [3] J. Antonio, R. Ramabhadran, and T.-L. Ling, “A framework for optimal trajectory planning for automated spray coating,” Int. J. Robot. Autom., vol. 12, no. 4, pp. 124–134, 1997. [4] W. Sheng, N. Xi, M. Song, Y. Chen, and P. MacNeville, “Automated CAD-guided robot path planning for spray painting of compund surfaces,” in Proc. IEEE/RSJ Int. Conf. Intell. Robot. Syst., 2000, pp. 1918–1923. [5] P. Jones, S. Duncan, T. Rayment, and P. Grant, “Control of temperature profile for a spray deposition process,” IEEE Trans. Contr. Syst. Technol., vol. 11, no. 5, pp. 656–667, Sep. 2003.
JONES et al.: OPTIMAL ROBOT PATH FOR MINIMIZING THERMAL VARIATIONS IN A SPRAY DEPOSITION PROCESS
[6] A. Butkovskiy and L. Pustylnikov, Mobile Control of Distributed Parameter Systems. Chichester, NY: Ellis Horwood Ltd, 1987. [7] M. Demetriou, A. Paskaleva, O. Vayena, and H. Doumanids, “Scanning actuator guidance scheme in a 1-D thermal manufacturing process,” IEEE Trans. Contr. Syst. Technol., vol. 11, no. 5, pp. 757–764, Sep. 2003. [8] A. Klein, “CAD-based off-line programming of painting robots,” Robotica, vol. 5, pp. 267–271, 1987. [9] P. Hertling, L. Hog, R. Larsen, J. Perram, and H. Petersen, “Task curve planning for painting robots—Part 1: Process modeling and calibration,” IEEE Trans. Robot. Autom., vol. 12, no. 3, pp. 324–330, Apr. 1996. [10] J. Antonio, “Optimal trajectory planning for spray coating,” in Proc. IEEE Int. Conf. Robot. Autom., 1994, pp. 2570–2577. [11] R. Ramabhadran and J. Antonio, “Planning spatial paths for automated spray coating applications,” in Proc. IEEE Int. Conf. Robot. Autom., 1996, pp. 1255–1260. [12] O. Vayena, M. Demetriou, and H. Doumanidis, “An LQR-based optimal actuator guidance in thermal processing of coatings,” in Proc. Amer. Contr. Conf., 2000, pp. 549–554. [13] M. Demetriou, O. Vayena, and H. Doumanidis, “Optimal actuator guidance scheme for a 2-D thermal processing,” in Proc. IEEE Mediterranean Conf. Contr. Autom., 2000, pp. C3–2. [14] N. Fourligkas and C. Doumanidis, “Temperature field regulation in thermal cutting for layered manufacturing,” J. Manuf. Sci. Eng., Trans. ASME, vol. 121, no. 3, pp. 440–447, 1999. [15] C. Doumanidis, “In-proceess control in thermal rapid prototyping,” IEEE Contr. Syst. Technol., vol. 17, no. 4, pp. 46–54, Jul. 1997. [16] ——, “Thermal manufacturing process control by lumped MIMO and distributed-parameter control,” J. Dyn. Syst. Meas. Contr., vol. 117, no. 4, pp. 625–632, 1995. [17] H. Steinhaus, Mathematical Snapshots, 3rd ed. New York: Dover, 1999. [18] Z. Djuric and P. Grant, “An inverse problem in modelling liquid metal spraying,” Appl. Math. Model., vol. 27, no. 5, pp. 379–396, 2003. [19] J. Hattel and N. Pryds, “A unified spray forming model for the prediction of billet shape geometry,” Acta Mater., vol. 52, no. 18, pp. 5275–5288, 2004. [20] P. Mathur, D. Apelian, and A. Lawley, “Analysis of the spray deposition process,” Acta Metall., vol. 37, no. 2, pp. 429–443, 1989. [21] H.-K. Seok, H. Lee, K. Oh, J.-C. Lee, H.-I. Lee, and H. Ra, “Formulation of rod-forming models and their application in spray forming,” Metall. Mater. Trans., A, vol. 31, no. 5, pp. 1479–1488, 2000. Paul D. A. Jones was born in Chester, U.K., in 1977. He received the M.Eng. and D.Phil. degrees in engineering science at the University of Oxford, U.K., in 2001 and 2004, respectively. Since 2000, he has been working as a Research Assistant at the Department of Engineering Science, University of Oxford. His research interests include the implementation of real-time control systems and finite element modeling of heat flow.
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Stephen R. Duncan (M’96) was born in Liverpool, U.K., in 1959. He received the M.A. degree in physics and theoretical physics from Cambridge University, U.K., in 1981, and the M.Sc. and Ph.D. degrees in control systems from Imperial College London, London, U.K., in 1985, and 1989, respectively. From 1989 to 1993, he was a Director of Greycon Ltd., London, U.K., and from 1993 to 1998 he was with the Control Systems Centre, UMIST, Manchester, U.K., where he was appointed Reader. Since 1998, he has been with the Department of Engineering Science at the University of Oxford, U.K., where he is a Reader. His current research interests include the design of control systems for industrial processes, particularly distributed parameter systems.
Tim Rayment was born in Oxford, U.K., in 1976. He received the M.A. and M.Sci. degrees in materials science and metallurgy from Cambridge University, Cambridge, U.K., in 1999, and the D.Phil degree in materials from Oxford University, Oxford, U.K., in 2003. He is currently a Software Engineer with Instron Ltd., High Wycombe, U.K., where he is developing advanced materials testing applications. Prior to joining Instron, he was an Industrial Research Fellow at Oxford University, where his research concentrated on industrial process control, thermal spraying, and process diagnostics.
Patrick S. Grant received the B.Eng. degree in metallurgy and materials science from Nottingham University, Nottingham, U.K., in 1987 and the D.Phil degree in materials from Oxford University, Oxford, U.K., in 1991. He was a Royal Society University Research Fellow and Reader in the Department of Materials at Oxford University and became Cookson Professor of Materials in 2004. He was Director of the Oxford Centre for Advanced Materials and Composites (1999–2004) that coordinates industrially related materials at Oxford University, and he is currently Director of Faraday Advance, a government and industry funded national partnership that links the science base with industry in the field of advanced materials. His research interests include advanced materials and processes for industrial structural and functional applications, especially in the aerospace and automotive sectors. He is the author of over 100 scientific publications and three granted and licensed patents. Prof. Grant is a member of the 2008 Research Assessment Exercise Panel for Materials and a member of the Defense and Aerospace National Advisory Committee for Materials and Structures.
