Optimal Tool Path Planning for Compound ... - Semantic Scholar
Proceedings of the 2004 IEEE International Conference on Robotics & Automation New Orleans, LA • April 2004
Optimal Tool Path Planning for Compound Surfaces in Spray Forming Processes Weihua Sheng
Jindong Tan
Heping Chen, Ning Xi
Yifan Chen
ECE Dept. ECE Dept. Scientific Research Labs ECE Dept. Kettering University Michigan State University Michigan Technological University Ford Motor Company Flint, MI 48504 East Lansing, MI, 48824 Houghton, MI, 49931 Dearborn, MI, 48121 Email: [email protected]
Abstract— Spray forming is an emerging manufacturing process. The automated tool planning for this process is a nontrivial problem, especially for geometry-complicated parts consisting of multiple freeform surfaces. Existing tool planning approaches are not able to deal with this kind of compound surfaces. This paper proposes a tool path planning approach which considers the tool motion performance and the thickness uniformity. There are two steps in this approach. The first step partitions the part surface into flat patches based on its topology and normal directions. The second step determines the tool movement patterns and the sweeping directions for each flat patch. Based on that, optimal tool paths can be calculated. Experimental tests are carried out on automotive body parts and the results validate the proposed approach.
I. I NTRODUCTION A. Motivation Spray forming is an emerging preforming process used in manufacturing industry to fabricate products such as body parts of automobiles and airplanes. The process of spray forming can be briefly stated as follows: first, glass fibers are sprayed on a mold using a chopper gun; at the same time, the binders are applied; then a consolidation process is implemented to melt the binders so that the glass fibers are combined; last, the finish part is obtained by removing the mold. Spray forming has many advantages over traditional part manufacturing processes: (i) light weight of finish parts; (ii) low material cost; (iii) fit for high volume production; (iv) flexibility. The spray forming process is usually carried out by moving mechanisms such as robotic manipulators with specific spraying tools mounted on their end-effectors. Compared to the full automation in the spraying process, the automation in the spraying tool planning is far from satisfying. Currently, tool planning is done manually through a teaching method, which heavily relies on an operator’s experience and knowledge. It usually requires an operator to use a trial-and-error approach to find a good tool plan. It is even harder for an operator to figure out an optimal plan when some performance measures are considered. For example, in Ford Motor Company’s Aston Martin plant, it takes an experienced operator about 8 weeks to design a tool trajectory in order to spray a door panel. As a matter of fact, the spray forming can be considered as a special type of surface manufacturing, which is a process
0-7803-8232-3/04/$17.00 ©2004 IEEE
to add material to or remove material from the surfaces of a part. Therefore the automated tool planning for spray forming faces the same difficulty as the tool planning in many other surface manufacturing applications, such as spray painting, spray coating, rapid tooling, spray cleaning, etc. An automated tool planning system for the spray forming can be easily extended to other surface manufacturing applications. B. Previous work There are some reports on automated tool planning for specific surface manufacturing applications. Some of the work considers the coverage problem, which only requires that any point on the surface be covered or touched by the tool. Huang [1] presented an optimal line-sweep-based decomposition method to cover a surface by minimizing the number of turns a tool has to make. Mizugaki [2] developed a path planning method for a polishing robot. Takeuchi et al. [3] discussed polishing path generations using the CAD model of a surface. The thickness problem arises from spray painting applications and it is more complicated than the coverage problem. Asakawa et al. [4] developed a teachingless path generation method to paint a car bumper using the parametric surfaces. However, the resulted paint uniformity is not satisfying and no report on how to find the spray overlap percentage and the gun speed is available. Antonio et al. [5] developed a framework for optimal trajectory planning to deal with the optimal paint thickness problem. But in their method, the paint gun path must be given a priori. Compared to spray painting, spray forming has received even less attention. Penin et al. [6] developed an automatic path planning method to spray glass fibers on a panel with cement. The spray width and spraying speed are determined using spray rules. However, the material thickness constraints are not considered. Recently, Luo et al. developed the software and hardware for a rapid tooling system [7]. The software includes a tool path generation algorithm using zigzag patterns. Yang et al. [8] studied different scanning strategies for tool path planning in rapid tooling. An equidistant scanning algorithm was proposed and implemented. Hensinger et al. [9] developed a motion planning strategy to generate tool motion control instructions for Laser Engineered Net Shape (LENS). They used simple rastering patterns to generate tool paths for each layer. Neither
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ROBCADTM
CAD Model
Robot Motion Performance Evaluation Report
Automated Tool Model
Tool
Tool Trajectory
Planning System Constraints Verification Optimization Criteria
Fig. 2. Fig. 1.
Deposition Quality Evaluation Report
The automated tool planning system for spray forming.
A car inner hood.
of the above three approaches deals with free-form surfaces consisting of multiple patches or considers the thickness problem. To summarize, several problems need to be solved before an automated tool planning system can be adopted for factoryfloor use: (i) Most of the existing tool planning approaches can only handle path planning on 2D plane. Automated tool planning for free-form surfaces in 3D space is still an open problem. (ii) Most of the existing approaches can only deal with parts that have simple geometry while real-world parts usually have complicated shape or topology. A typical example is a car inner hood as shown in Figure 1. The complexity of the part geometry requires that the surfaces be partitioned. (iii) Little existing research work on automated tool planning is concerned about the motion performance of the moving mechanism, especially in the designing of tool paths. However, the motion performance could greatly affect the manufacturing efficiency and part quality. The ultimate objective of this research is to develop a systematic, mathematical-sound, automated tool planning system for spray forming in automotive parts manufacturing. In this paper, the major concern is the optimal path planning for multiple freeform surfaces. There are two steps: surface partition and path planning. The surface partition simplifies the tool planning problem by generating flat patches that have favorable shapes. The path planning designs tool paths that lead to high tool motion performance and good deposition uniformity. The paper is organized as follows: in Section 2, the general framework of tool planning is introduced; Section 3 discusses the surface partition problem; Section 4 solves the optimal path planning problem for multiple flat patches. Implementation and experimental testing are described in Section 5. Conclusions are provided in Section 6. II. G ENERAL FRAMEWORK Figure 2 depicts the proposed automated tool planning system for spray forming. There are four major inputs to this planning system: part CAD model, tool model, task constraints and optimization criteria. The planning system generates a tool plan which can be validated by a simulation module and a deposition quality verification module.
A CAD model contains the geometric information of a part and a tessellated part model can be expressed as Mc = ∪ni=1 Ti , where Ti represents a triangle and n is the total number of triangles. A tool is the deposition equipment mounted on the end-effector of a moving mechanism, such as a robotic manipulator. A general deposition tool model in spray forming can be represented by a spray cone [10]. Task constraints include thickness requirements, kinematics constraints of the moving mechanism, etc. Optimization criteria can be time, motion performance of the moving mechanism and wastage, etc. The automated tool planning problem for spray forming can be formulated as follows: Given the CAD model Mc of the part surface, tool model Mt , task constraints Ω, and optimization criteria O, find a mapping F : (Mc , Mt , Ω, O) → Γ. Γ is the overall tool trajectory which specifies a series of positions and associated velocities of the tool tip. In practice, a constant standoff distance h is usually used and the tool tip is assumed to point toward the surface along the reverse of the local normal direction [5]. Based on a divide-and-conquer philosophy, the overall strategy of the automated tool planning is as follows. The compound surface is first partitioned into easy-to-solve patches; second, for each patch, a path is obtained; third, the associated tool velocities are calculated for each path; finally, the trajectories for each patch is integrated to obtain an overall trajectory. This paper mainly addresses the first two steps. III. S URFACE PARTITION The first step in the tool planning is surface partition. It is observed that the geometry of most automotive parts is complicated in that (i) there are holes within the outer contour, (ii) the surface normals may not be consistent and neighboring surface areas may form certain angle where they meet. These two facts introduce difficulties in planning tool trajectory that achieves high motion performance and minimum thickness error. By partitioning the surfaces the tool planning problem can be simplified. Based on the above two facts, a two-step partition process is proposed. First, the compound surface is partitioned so that each patch does not contain any hole and has simple, regular shape, or topology. Second, each patch is, if necessary, further partitioned based on the normals so that the final patches are relatively flat.
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A. Topology-based partition To analyze the topology of most of the automotive parts, first, the compound is projected to a plane whose Pn surface P n normal is ~na = i=1 Ai~ni /( i=1 Ai ), and ~ni is the normal of triangle Ti and Ai is its area; second, the outer contour and all the inner contours (holes) are identified; third, an approximation process is applied to detect the critical points on the contours and then connect the critical points sequentially to get the approximated contours, where the critical points are where the contour has local maximum 2D curvature. Therefore, the problem of partitioning the compound surfaces is converted to the problem of partitioning a 2D polygon into multiple subpolygons. The criteria used is the tool motion performance, which is highly related to the geometry of the subpolygons. The following measures can characterize a subpolygon: (i) Regularity: It is easier to control the tool to move along the paths in rectangle-like area. Sharp angles always bring difficulties in automated tool planning and motion control due to the small area at the corner [8]. Essentially, the regularity requirement implies a favor on right or obtuse interior angles. (ii) Convexity: It is easier to plan paths and control the robot on convex subpolygons than on concave ones since concavity implies more changes of the tool moving direction. Therefore it is necessary to avoid those interior angles that are greater than π. (iii) The number of turns a tool makes: Fewer turns usually imply less travelling time for the tool since making turns requires a tool to slow down and speed up again [1]. The number of turns a tool has to make to finish a subpolygon can be represented by the minimum altitude, ALTmin , of the subpolygon [1]. (iv) The length of common borders: The common border is where two subpolygons meet. It is usually hard to achieve thickness uniformity along the common borders which receive material from paths on two different patches [10]. Hence it is desirable to reduce the total length of common borders. A measure called shape fairness is defined as follows: The shape fairness of a subpolygon, Fs , is defined as Fs = ωs1 h1 (RC) + ωs2 h2 (ALTmin ) + ωs3 h3 (Lcb ). Here RC is the regularity and convexity measure, which is defined as a function of the interior angles of the subpolygons; Lcb is the length of common borders; ωsi (i = 1, 2, 3) are weights and the hi (·) (i = 1, 2, 3) functions are defined such that when the each measure achieves its optimality, hi (·) achieves its maximum. A good partition of the polygon means that the overall shape fairness approaches maximum. Based on the above discussions, the topology-based surface partition problem can be formulated as the following geometric problem: Given an n-edge polygon with k polygonal holes, each hole has ni edges (i = 1, 2, ..., k), partition the polygonPinto m subpolygons so m that the overall shape fairness Fs = i=1 Fsi is maximized. Here we propose a new decomposition-based approach to solve the surface partition problem. It first decomposes the polygon into multiple small cells. For example, for the polygon
(a)
(b)
(c) Fig. 3. (a) A polygon with two holes. (b) The polygon decomposed into cells. (c) The corresponding CAG.
in Figure 3(a), its decomposition is shown in Figure 3(b). Then the cells are combined to form different subpolygons, which can be done through a subgraph search [11] on the Cell Adjacency Graph (CAG) as shown in Figure 3(c). For each subpolygon, its shape fairness is assigned to it as a weight. Therefore, the geometric partition problem can be converted into a weighted set partition problem [12] as follows: max z =
Ns X
cj xj
(1)
j=1
subject to :
Ns X
aij xj = 1, i = 1, 2, ..., Nc
j=1
with xj = 0 or 1, cj = Fsj , j = 1, 2, ..., Ns .
(2)
Here Nc is the number of cells and Ns is the total number of subpolygons. cj is the cost of jth subpolygon, which is the corresponding shape fairness Fsj . aij = 1 means cell i belongs to jth subpolygon, otherwise not. Variable xj = 1 means the jth subpolygon is selected in the final partition, otherwise not. The weighted set-partitioning problem is N P -hard [12]. Usually a sub-optimal solution can be obtained, which implies a partition of the polygon with a near maximum overall shape fairness. B. Normal-based partition The patches obtained through topology-based partition may exhibit big curvatures. For these patches, it is very difficult to obtain well-behaved trajectories that minimize the thickness error. Therefore a further step is needed to partition the patches into flat patches, or patches that satisfy the flatness constraint: its maximum deviation angle (MDA) is less than a certain threshold βth .
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A growth algorithm is proposed to find flat patches. First, the triangle with the biggest area of the remaining triangles is chosen as the seed triangle, and it is taken as the initial flat patch, then its neighboring triangles are tested by the flatness constraint. Any neighboring triangle that satisfies the constraint is added and a new patch is formed. This process is repeated until no more triangles can be added. Then in the remaining triangles, the triangle with the biggest area is chosen and the process repeats until all the triangles are pulled into one of the flat patches. After the two-step surface partition, flat patches are generated. The whole compound surface can be described by a Flat Patch Adjacency Graph (FPAG) which models each flat patch as a node and the adjacency relationship between two patches as a corresponding edge. IV. T OOL PATH PLANNING Movement patterns are usually used to design the tool path. The two widely-used movement patterns are zigzag and spiral [13]. Zigzag patterns lead to simple tool movement but the non-isotopic nature introduces difficulty in maintaining thickness uniformity near the patch borders. Spiral patterns have isotopic nature but may not be able to avoid disconnected path segments for certain patch shapes [14], as shown in Figure 4(a). In most of the previous work, only one movement pattern can be pre-selected. In this work, the automated tool planner considers both patterns and uses whatever pattern is appropriate.
Disconnected Segment Nominal Spacing Distance
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and the neighboring relationship with other patches. Here the critical points are the points with local maximum curvature and they can be found using the parallel algorithm developed in [15]. There are multiple choices to design the paths. As shown in Figure 4(a), 4(b) and 4(c), it can be zigzag or spiral. If zigzag pattern is used, there are different sweeping directions. For simplicity, we only consider the sweeping directions parallel to border segments. For each border segment, the line connecting the two ends of the border segments can roughly characterize the direction of the border segment. We call it the characteristic line of the border segment. For any two border segments, the characteristic angle φ is the angle between the two corresponding characteristic lines. We have the following definition. The parallel index, τ , of border segment b, is the number of border segments that are “nearly parallel” to b in that patch, including b itself. Here “nearly parallel” means that the characteristic angle φ between the two border segments is smaller than a certain threshold φth . Based on the definition of parallel index, if a patch has nb border segments,Pthe total number of different sweeping nb directions is nb − i=1 (1 − τ1i ) since one set of “nearly parallel” border segments contributes only one sweeping direction. Considering the spiral the total number of candidate Pnpath, b (1 − τ1i ), paths is nc = 1 + nb − i=1 To guide the design of good paths, it is desirable to quantitatively evaluate the fairness of a path regarding to the tool motion performance. Ideal tool paths should have fewer turns, fewer disconnected segments which may cause the spray gun to be turned on and off, longer strokes and equal spacing distance between strokes. The path fairness is defined below. The path fairness, F , is defined as F = 1 e−[ω1 nt +ω2 ns +ω3 lm +ω4 σd ] . Here ω1 to ω4 are the weights; nt is the number of turns; ns is the number of disconnected path segments; lm is the minimum segment length and σd = dmax − dmin is the variation of path spacing distance d. Obviously, this fairness measure is directly related to the choice of tool movement pattern and the sweeping directions of the zigzag pattern. Based on this definition, in the extreme case, a single long stroke path has a fairness near 1.
dmax
B. Determination of movement patterns and sweeping directions dmin
(c) Fig. 4. (a) Spiral pattern with nt = 16, ns = 1. (b) Zigzag pattern with nt = 18, ns =0. (c) Zigzag pattern with nt = 17, ns = 1.
A. Evaluation of path fairness For each flat patch, its contour is partitioned into multiple slightly-curved border segments based on the critical points
To determine the tool movement patterns and sweeping directions for all the patches, we also have to consider the effect of path sweeping direction on the deposition thickness near the common border segment between two patches. It has been found out that it is much easier to maintain small thickness error if both paths are parallel to the common border segment [10]. If either path is not parallel to the common border segment, the thickness error will be much bigger. The above observation implies that the paths in adjacent patches should “coordinate” to maximize the border parallelity while maintaining overall path fairness. Figure 5(a) displays one scenario with movement patterns and sweeping directions
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Here the first component is the reward for the parallelity between zigzag paths and the second component is the reward for the parallelity between the zigzag and spiral paths. Obviously there is a trade-off between the path fairness and the path parallelity. To achieve the optimal overall performance, the following polynomial integer programming (PIP) problem can be formulated:
Common Border Segment Hole
Sweeping Directions
max z(xij ) = ωp1 Fo + ωp2 Re with : xij = 0 or 1, i ∈ (1, ...Nr ), j ∈ (1, ...Nn ) Nr X Nn X xij subject to : =1 (3) τ i=1 j=1 ij
Border Segment
(a)
(b)
Fig. 5. (a) Movement patterns and sweeping directions. (b) The Flat Patch Adjacency Graph enhanced by virtual nodes and virtual edges. The dotted circles and lines indicate virtual nodes and virtual edges.
indicated for each patch. This problem can be modelled as a polynomial integer programming (PIP) problem [16]. As shown in Figure 5(b), the solid lines and circles represent the Flat Patch Adjacency Graph. An edge uniquely identifies the common border segment between the two patches. For those border segments not shared with other patches, virtual nodes and virtual edges are created so that each border segment is uniquely identified by an actual or virtual edge. On the enhanced FPAG, the total number of nodes Nn is the sum of the number of real nodes nr and the number of virtual nodes nv , Nn = Nr + Nv . For patch Pi , if the movement pattern and sweeping directions are known, a path can be calculated based on a nominal spacing distance dn [17]. The fairness of the zigzag path whose sweeping direction is parallel to the border segment identified by eij is denoted as Fij (j ∈ (1, 2, ...Nn ), i 6= j) and the fairness of the spiral path which is parallel to all the border segments of patch Pi is denoted as Fii . τij is the parallel index of the common border segment identified by edge eij in Pi when i 6= j. τii = 1 for all i ∈ (1, 2, ..., Nr ). Let xij be the binary variable that represents which candidate path is selected. xij = 1 (i 6= j) means that the zigzag path parallel to the border segment identified by eij is selected, otherwise not. xii = 1 means that the spiral path is selected, otherwise not. We have the following expression to represent the overall path fairness: X X xij Fo = · Fij τij i∈(1,...,Nr ) j∈(1,...Nn )
To encourage achieving parallelity along the common border segments, a positive reward aij is introduced when two paths in adjacent patches are parallel to the common border segment identified by eij . The following expression represents the overall reward due to the parallelity along the borders. X X Re = xij · xji · aij +
xij1 = ... = xijτij , ∀i ∈ (1, 2...Nr )
Equation(3) means only one candidate path can be selected and Equation(4) represents the “nearly parallel” constraint among border segments on patch Pi . The solution to the above PIP problem gives, for each patch, the tool movement patterns and the sweeping directions. Once the tool movement patterns and sweeping directions are known for each patch, we design the path for each patch with the goal of minimizing the thickness uniformity. First, an optimal spacing distance dn is calculated which minimizes the thickness error on a plane [17]. Second, depending on the path approaching situation at the common border segments, three different cases, i.e., parallel-parallel (PA-PA), parallelperpendicular (PA-PE) and perpendicular-perpendicular (PEPE) are optimized to obtain the optimal path-to-border distance h [10]. Once dn and h are known, the overall path can be determined on the flat patch. V. I MPLEMENTATION AND T ESTING R ESULTS The algorithms are implemented in C++ and several automotive body parts from the Ford Motor company are tested. Here are the results for the car inner hood, which is shown in Figure 1. The first step is the surface partition. The weights of the shape fairness are set as ωs1 = 1/3, ωs2 = 1/3, ωs3 = 1/3. The solution to the weighted set partitioning problem gives the partition shown in Figure 6(a). To verify that the overall shape fairness z reaches optimality, two other different partitions, as shown in Figure 6(b) and Figure 6(c) are manually generated. The corresponding overall shape fairness z is calculated and compared with the optimal solution. The comparison results is shown in Table I. From Figure 6(a), 6(b) and 6(c), it is clear that the patches in the first partition have higher regularity RC and also have shorter common border length Lcb . TABLE I T HE COMPARISON OF OVERALL SHAPE FAIRNESS Case Optimal partition Manual partition 1 Manual partition 2
i∈(1,...,Nr ) j∈(1,...Nr ),j6=i Nr X i=1
[xii (
Nr X
(4)
xji · aij )].
j=1,j6=i
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Overall Shape Fairness z 6.2 5.4 5.7
(a)
tool paths that can achieve good motion performance for the moving mechanism and optimal deposition uniformity on a compound surface. Solving this problem requires the part surfaces be partitioned and the tool movement patterns and sweeping directions be carefully determined. Two measures, the shape fairness and the path fairness are defined to guide the surface partition and path design. Two integer programming problems are formulated and solved. Experimental results are provided, which validate the proposed approaches. Our future work will focus on the tool velocity planning and trajectory integration. This tool planning system can be easily extended to other similar surface manufacturing applications.
(b)
R EFERENCES
(c) Fig. 6. (a) Optimal partition by solving weighted set-partitioning problem. (b) Manually generated partition: case 1. (c) Manually generated partition: case 2.
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Fig. 7. (a) Movement patterns and sweeping directions: optimal. (b) Movement patterns and sweeping directions: non-optimal.
The second step is the designing of movement patterns and sweeping directions. The first partition (Figure 6(a) is used. The weights for overall path fairness Fo and the reward Re , are ωp1 = 0.5, ωp2 = 0.5. The movement pattern and sweeping directions are shown in Figure 7(a). The overall path fairness is 0.92. For comparison, the manually-picked movement patterns and sweeping directions as shown in Figure 7(b) has a overall path fairness of 0.83. This observation verifies that the solution to the PIP problem gives the optimal movement pattern and sweeping directions. As can be seen from Figure 7(a), there is no PA-PE or PE-PE cases regarding the path directions along the common borders.
[1] W.H.Huang. Optimal line-sweep-based decomposition for coverage algorithms. In Proceedings of the 2001 IEEE International Conference on Robotics and Automation, 2001. [2] Y. Mizugaki, M. Sakamoto, and K. Kamijo. Fractal path application in a metal mold polishing robot system. In Fifth International Conference on Advanced Robotics, pages 431 – 436, Pisa, Italy, June 1991. [3] Y. Takeuchi, D. Ge, and N. Asakawa. Automated polishing process with a human-like dexterous robot. In IEEE, pages 950–956, Barcelona, Spain, Oct. 1993. [4] N. Asakawa and Y. Takeuchi. Teachingless spray-painting of sculptured surface by an industrial robot. In IEEE International Conference on Robotics and Automation, pages 1875 –1879, Albuquerque, New Mexico, April 1997. [5] J. K. Antonio, R. Ramabhadran, and T. L. Ling. A framework for optimal trajectory planning for automated spray coating. International Journal of Robotics and Automation, 12(4):124–134, 1997. [6] L. F. Penin, C. Balaguer, J. M. Pastor, F. J. Rodriguez, A. Barrientos, and R. Aracil. Robotized spraying of prefabricated panels. IEEE Robotics and Automation Magazine, 5(3):18 –29, 1998. [7] R. C. Luo and J. H. Tzou. Investigation of a linear 2-d planar motor based rapid tooling. In IEEE International Conference on Robotics and Automation, pages 1471–1476, Washington, DC, USA, May 2002. [8] Y.Yang, H.T.Loh, F.Y.H.Fuh, and Y.G.Wang. Equidistant path generation for improving scanning efficiency in layered manufacturing. Rapid Prototyping Journal, 8(1):30–37, 2002. [9] D.M.Hensinger, A.L.Ames, and J.L.Kuhlmann. Motion planning for a direct metal deposition rapid prototyping system. In Proceedings of the 2000 International Conference on Robotics and Automation, 2000. [10] H. Chen, N. Xi, Z. Wei, Y. Chen, and J. Dahl. Trajectory integration for a surface with multiple patches. In Proceeding of the IEEE International Conference on Robotics and Automation, 2003. [11] 2002. Personal communication with D.Israel. [12] E.Balas. Combinatorial Optimization, chapter 7, pages 151–210. 1979. [13] K.Ramaswami. Process planning for shape deposition manufacturing. PhD thesis, Stanford University, 1997. [14] J.H.Kao and F.B.Prinz. Optimal motion planning for deposition in layered manufacturing. In Proceedings of 1998 ASME Design Engineering Technical Conferences, Atlanta, Georgia, USA, Sept 1998. [15] C.H.The and R.T.Chin. On the detection of dominant points on digital curves. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11(8), 1989. [16] S.Arora, D.Karger, and M.Karpinski. Polynomial time approximation schemes for dense instances of np-hard problems. Journal of comput. System. Sci., 1998. [17] H. Chen, W. Sheng, N. Xi, M. Song, and Y. Chen. Automated robot trajectory planning for spray painting of free-form surfaces in automotive manufacturing. In IEEE International Conference on Robotics and Automation, volume 1, pages 450 –455, 2002.
VI. C ONCLUSIONS This paper addresses a challenging problem in automated tool path planning for spray forming. That is, how to design
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Optimal Tool Path Planning for Compound Surfaces in Spray Forming Processes Weihua Sheng
Jindong Tan
Heping Chen, Ning Xi
Yifan Chen
ECE Dept. ECE Dept. Scientific Research Labs ECE Dept. Kettering University Michigan State University Michigan Technological University Ford Motor Company Flint, MI 48504 East Lansing, MI, 48824 Houghton, MI, 49931 Dearborn, MI, 48121 Email: [email protected]
Abstract— Spray forming is an emerging manufacturing process. The automated tool planning for this process is a nontrivial problem, especially for geometry-complicated parts consisting of multiple freeform surfaces. Existing tool planning approaches are not able to deal with this kind of compound surfaces. This paper proposes a tool path planning approach which considers the tool motion performance and the thickness uniformity. There are two steps in this approach. The first step partitions the part surface into flat patches based on its topology and normal directions. The second step determines the tool movement patterns and the sweeping directions for each flat patch. Based on that, optimal tool paths can be calculated. Experimental tests are carried out on automotive body parts and the results validate the proposed approach.
I. I NTRODUCTION A. Motivation Spray forming is an emerging preforming process used in manufacturing industry to fabricate products such as body parts of automobiles and airplanes. The process of spray forming can be briefly stated as follows: first, glass fibers are sprayed on a mold using a chopper gun; at the same time, the binders are applied; then a consolidation process is implemented to melt the binders so that the glass fibers are combined; last, the finish part is obtained by removing the mold. Spray forming has many advantages over traditional part manufacturing processes: (i) light weight of finish parts; (ii) low material cost; (iii) fit for high volume production; (iv) flexibility. The spray forming process is usually carried out by moving mechanisms such as robotic manipulators with specific spraying tools mounted on their end-effectors. Compared to the full automation in the spraying process, the automation in the spraying tool planning is far from satisfying. Currently, tool planning is done manually through a teaching method, which heavily relies on an operator’s experience and knowledge. It usually requires an operator to use a trial-and-error approach to find a good tool plan. It is even harder for an operator to figure out an optimal plan when some performance measures are considered. For example, in Ford Motor Company’s Aston Martin plant, it takes an experienced operator about 8 weeks to design a tool trajectory in order to spray a door panel. As a matter of fact, the spray forming can be considered as a special type of surface manufacturing, which is a process
0-7803-8232-3/04/$17.00 ©2004 IEEE
to add material to or remove material from the surfaces of a part. Therefore the automated tool planning for spray forming faces the same difficulty as the tool planning in many other surface manufacturing applications, such as spray painting, spray coating, rapid tooling, spray cleaning, etc. An automated tool planning system for the spray forming can be easily extended to other surface manufacturing applications. B. Previous work There are some reports on automated tool planning for specific surface manufacturing applications. Some of the work considers the coverage problem, which only requires that any point on the surface be covered or touched by the tool. Huang [1] presented an optimal line-sweep-based decomposition method to cover a surface by minimizing the number of turns a tool has to make. Mizugaki [2] developed a path planning method for a polishing robot. Takeuchi et al. [3] discussed polishing path generations using the CAD model of a surface. The thickness problem arises from spray painting applications and it is more complicated than the coverage problem. Asakawa et al. [4] developed a teachingless path generation method to paint a car bumper using the parametric surfaces. However, the resulted paint uniformity is not satisfying and no report on how to find the spray overlap percentage and the gun speed is available. Antonio et al. [5] developed a framework for optimal trajectory planning to deal with the optimal paint thickness problem. But in their method, the paint gun path must be given a priori. Compared to spray painting, spray forming has received even less attention. Penin et al. [6] developed an automatic path planning method to spray glass fibers on a panel with cement. The spray width and spraying speed are determined using spray rules. However, the material thickness constraints are not considered. Recently, Luo et al. developed the software and hardware for a rapid tooling system [7]. The software includes a tool path generation algorithm using zigzag patterns. Yang et al. [8] studied different scanning strategies for tool path planning in rapid tooling. An equidistant scanning algorithm was proposed and implemented. Hensinger et al. [9] developed a motion planning strategy to generate tool motion control instructions for Laser Engineered Net Shape (LENS). They used simple rastering patterns to generate tool paths for each layer. Neither
45
ROBCADTM
CAD Model
Robot Motion Performance Evaluation Report
Automated Tool Model
Tool
Tool Trajectory
Planning System Constraints Verification Optimization Criteria
Fig. 2. Fig. 1.
Deposition Quality Evaluation Report
The automated tool planning system for spray forming.
A car inner hood.
of the above three approaches deals with free-form surfaces consisting of multiple patches or considers the thickness problem. To summarize, several problems need to be solved before an automated tool planning system can be adopted for factoryfloor use: (i) Most of the existing tool planning approaches can only handle path planning on 2D plane. Automated tool planning for free-form surfaces in 3D space is still an open problem. (ii) Most of the existing approaches can only deal with parts that have simple geometry while real-world parts usually have complicated shape or topology. A typical example is a car inner hood as shown in Figure 1. The complexity of the part geometry requires that the surfaces be partitioned. (iii) Little existing research work on automated tool planning is concerned about the motion performance of the moving mechanism, especially in the designing of tool paths. However, the motion performance could greatly affect the manufacturing efficiency and part quality. The ultimate objective of this research is to develop a systematic, mathematical-sound, automated tool planning system for spray forming in automotive parts manufacturing. In this paper, the major concern is the optimal path planning for multiple freeform surfaces. There are two steps: surface partition and path planning. The surface partition simplifies the tool planning problem by generating flat patches that have favorable shapes. The path planning designs tool paths that lead to high tool motion performance and good deposition uniformity. The paper is organized as follows: in Section 2, the general framework of tool planning is introduced; Section 3 discusses the surface partition problem; Section 4 solves the optimal path planning problem for multiple flat patches. Implementation and experimental testing are described in Section 5. Conclusions are provided in Section 6. II. G ENERAL FRAMEWORK Figure 2 depicts the proposed automated tool planning system for spray forming. There are four major inputs to this planning system: part CAD model, tool model, task constraints and optimization criteria. The planning system generates a tool plan which can be validated by a simulation module and a deposition quality verification module.
A CAD model contains the geometric information of a part and a tessellated part model can be expressed as Mc = ∪ni=1 Ti , where Ti represents a triangle and n is the total number of triangles. A tool is the deposition equipment mounted on the end-effector of a moving mechanism, such as a robotic manipulator. A general deposition tool model in spray forming can be represented by a spray cone [10]. Task constraints include thickness requirements, kinematics constraints of the moving mechanism, etc. Optimization criteria can be time, motion performance of the moving mechanism and wastage, etc. The automated tool planning problem for spray forming can be formulated as follows: Given the CAD model Mc of the part surface, tool model Mt , task constraints Ω, and optimization criteria O, find a mapping F : (Mc , Mt , Ω, O) → Γ. Γ is the overall tool trajectory which specifies a series of positions and associated velocities of the tool tip. In practice, a constant standoff distance h is usually used and the tool tip is assumed to point toward the surface along the reverse of the local normal direction [5]. Based on a divide-and-conquer philosophy, the overall strategy of the automated tool planning is as follows. The compound surface is first partitioned into easy-to-solve patches; second, for each patch, a path is obtained; third, the associated tool velocities are calculated for each path; finally, the trajectories for each patch is integrated to obtain an overall trajectory. This paper mainly addresses the first two steps. III. S URFACE PARTITION The first step in the tool planning is surface partition. It is observed that the geometry of most automotive parts is complicated in that (i) there are holes within the outer contour, (ii) the surface normals may not be consistent and neighboring surface areas may form certain angle where they meet. These two facts introduce difficulties in planning tool trajectory that achieves high motion performance and minimum thickness error. By partitioning the surfaces the tool planning problem can be simplified. Based on the above two facts, a two-step partition process is proposed. First, the compound surface is partitioned so that each patch does not contain any hole and has simple, regular shape, or topology. Second, each patch is, if necessary, further partitioned based on the normals so that the final patches are relatively flat.
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A. Topology-based partition To analyze the topology of most of the automotive parts, first, the compound is projected to a plane whose Pn surface P n normal is ~na = i=1 Ai~ni /( i=1 Ai ), and ~ni is the normal of triangle Ti and Ai is its area; second, the outer contour and all the inner contours (holes) are identified; third, an approximation process is applied to detect the critical points on the contours and then connect the critical points sequentially to get the approximated contours, where the critical points are where the contour has local maximum 2D curvature. Therefore, the problem of partitioning the compound surfaces is converted to the problem of partitioning a 2D polygon into multiple subpolygons. The criteria used is the tool motion performance, which is highly related to the geometry of the subpolygons. The following measures can characterize a subpolygon: (i) Regularity: It is easier to control the tool to move along the paths in rectangle-like area. Sharp angles always bring difficulties in automated tool planning and motion control due to the small area at the corner [8]. Essentially, the regularity requirement implies a favor on right or obtuse interior angles. (ii) Convexity: It is easier to plan paths and control the robot on convex subpolygons than on concave ones since concavity implies more changes of the tool moving direction. Therefore it is necessary to avoid those interior angles that are greater than π. (iii) The number of turns a tool makes: Fewer turns usually imply less travelling time for the tool since making turns requires a tool to slow down and speed up again [1]. The number of turns a tool has to make to finish a subpolygon can be represented by the minimum altitude, ALTmin , of the subpolygon [1]. (iv) The length of common borders: The common border is where two subpolygons meet. It is usually hard to achieve thickness uniformity along the common borders which receive material from paths on two different patches [10]. Hence it is desirable to reduce the total length of common borders. A measure called shape fairness is defined as follows: The shape fairness of a subpolygon, Fs , is defined as Fs = ωs1 h1 (RC) + ωs2 h2 (ALTmin ) + ωs3 h3 (Lcb ). Here RC is the regularity and convexity measure, which is defined as a function of the interior angles of the subpolygons; Lcb is the length of common borders; ωsi (i = 1, 2, 3) are weights and the hi (·) (i = 1, 2, 3) functions are defined such that when the each measure achieves its optimality, hi (·) achieves its maximum. A good partition of the polygon means that the overall shape fairness approaches maximum. Based on the above discussions, the topology-based surface partition problem can be formulated as the following geometric problem: Given an n-edge polygon with k polygonal holes, each hole has ni edges (i = 1, 2, ..., k), partition the polygonPinto m subpolygons so m that the overall shape fairness Fs = i=1 Fsi is maximized. Here we propose a new decomposition-based approach to solve the surface partition problem. It first decomposes the polygon into multiple small cells. For example, for the polygon
(a)
(b)
(c) Fig. 3. (a) A polygon with two holes. (b) The polygon decomposed into cells. (c) The corresponding CAG.
in Figure 3(a), its decomposition is shown in Figure 3(b). Then the cells are combined to form different subpolygons, which can be done through a subgraph search [11] on the Cell Adjacency Graph (CAG) as shown in Figure 3(c). For each subpolygon, its shape fairness is assigned to it as a weight. Therefore, the geometric partition problem can be converted into a weighted set partition problem [12] as follows: max z =
Ns X
cj xj
(1)
j=1
subject to :
Ns X
aij xj = 1, i = 1, 2, ..., Nc
j=1
with xj = 0 or 1, cj = Fsj , j = 1, 2, ..., Ns .
(2)
Here Nc is the number of cells and Ns is the total number of subpolygons. cj is the cost of jth subpolygon, which is the corresponding shape fairness Fsj . aij = 1 means cell i belongs to jth subpolygon, otherwise not. Variable xj = 1 means the jth subpolygon is selected in the final partition, otherwise not. The weighted set-partitioning problem is N P -hard [12]. Usually a sub-optimal solution can be obtained, which implies a partition of the polygon with a near maximum overall shape fairness. B. Normal-based partition The patches obtained through topology-based partition may exhibit big curvatures. For these patches, it is very difficult to obtain well-behaved trajectories that minimize the thickness error. Therefore a further step is needed to partition the patches into flat patches, or patches that satisfy the flatness constraint: its maximum deviation angle (MDA) is less than a certain threshold βth .
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A growth algorithm is proposed to find flat patches. First, the triangle with the biggest area of the remaining triangles is chosen as the seed triangle, and it is taken as the initial flat patch, then its neighboring triangles are tested by the flatness constraint. Any neighboring triangle that satisfies the constraint is added and a new patch is formed. This process is repeated until no more triangles can be added. Then in the remaining triangles, the triangle with the biggest area is chosen and the process repeats until all the triangles are pulled into one of the flat patches. After the two-step surface partition, flat patches are generated. The whole compound surface can be described by a Flat Patch Adjacency Graph (FPAG) which models each flat patch as a node and the adjacency relationship between two patches as a corresponding edge. IV. T OOL PATH PLANNING Movement patterns are usually used to design the tool path. The two widely-used movement patterns are zigzag and spiral [13]. Zigzag patterns lead to simple tool movement but the non-isotopic nature introduces difficulty in maintaining thickness uniformity near the patch borders. Spiral patterns have isotopic nature but may not be able to avoid disconnected path segments for certain patch shapes [14], as shown in Figure 4(a). In most of the previous work, only one movement pattern can be pre-selected. In this work, the automated tool planner considers both patterns and uses whatever pattern is appropriate.
Disconnected Segment Nominal Spacing Distance
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“Nearly Parallel” Border Segments
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and the neighboring relationship with other patches. Here the critical points are the points with local maximum curvature and they can be found using the parallel algorithm developed in [15]. There are multiple choices to design the paths. As shown in Figure 4(a), 4(b) and 4(c), it can be zigzag or spiral. If zigzag pattern is used, there are different sweeping directions. For simplicity, we only consider the sweeping directions parallel to border segments. For each border segment, the line connecting the two ends of the border segments can roughly characterize the direction of the border segment. We call it the characteristic line of the border segment. For any two border segments, the characteristic angle φ is the angle between the two corresponding characteristic lines. We have the following definition. The parallel index, τ , of border segment b, is the number of border segments that are “nearly parallel” to b in that patch, including b itself. Here “nearly parallel” means that the characteristic angle φ between the two border segments is smaller than a certain threshold φth . Based on the definition of parallel index, if a patch has nb border segments,Pthe total number of different sweeping nb directions is nb − i=1 (1 − τ1i ) since one set of “nearly parallel” border segments contributes only one sweeping direction. Considering the spiral the total number of candidate Pnpath, b (1 − τ1i ), paths is nc = 1 + nb − i=1 To guide the design of good paths, it is desirable to quantitatively evaluate the fairness of a path regarding to the tool motion performance. Ideal tool paths should have fewer turns, fewer disconnected segments which may cause the spray gun to be turned on and off, longer strokes and equal spacing distance between strokes. The path fairness is defined below. The path fairness, F , is defined as F = 1 e−[ω1 nt +ω2 ns +ω3 lm +ω4 σd ] . Here ω1 to ω4 are the weights; nt is the number of turns; ns is the number of disconnected path segments; lm is the minimum segment length and σd = dmax − dmin is the variation of path spacing distance d. Obviously, this fairness measure is directly related to the choice of tool movement pattern and the sweeping directions of the zigzag pattern. Based on this definition, in the extreme case, a single long stroke path has a fairness near 1.
dmax
B. Determination of movement patterns and sweeping directions dmin
(c) Fig. 4. (a) Spiral pattern with nt = 16, ns = 1. (b) Zigzag pattern with nt = 18, ns =0. (c) Zigzag pattern with nt = 17, ns = 1.
A. Evaluation of path fairness For each flat patch, its contour is partitioned into multiple slightly-curved border segments based on the critical points
To determine the tool movement patterns and sweeping directions for all the patches, we also have to consider the effect of path sweeping direction on the deposition thickness near the common border segment between two patches. It has been found out that it is much easier to maintain small thickness error if both paths are parallel to the common border segment [10]. If either path is not parallel to the common border segment, the thickness error will be much bigger. The above observation implies that the paths in adjacent patches should “coordinate” to maximize the border parallelity while maintaining overall path fairness. Figure 5(a) displays one scenario with movement patterns and sweeping directions
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Here the first component is the reward for the parallelity between zigzag paths and the second component is the reward for the parallelity between the zigzag and spiral paths. Obviously there is a trade-off between the path fairness and the path parallelity. To achieve the optimal overall performance, the following polynomial integer programming (PIP) problem can be formulated:
Common Border Segment Hole
Sweeping Directions
max z(xij ) = ωp1 Fo + ωp2 Re with : xij = 0 or 1, i ∈ (1, ...Nr ), j ∈ (1, ...Nn ) Nr X Nn X xij subject to : =1 (3) τ i=1 j=1 ij
Border Segment
(a)
(b)
Fig. 5. (a) Movement patterns and sweeping directions. (b) The Flat Patch Adjacency Graph enhanced by virtual nodes and virtual edges. The dotted circles and lines indicate virtual nodes and virtual edges.
indicated for each patch. This problem can be modelled as a polynomial integer programming (PIP) problem [16]. As shown in Figure 5(b), the solid lines and circles represent the Flat Patch Adjacency Graph. An edge uniquely identifies the common border segment between the two patches. For those border segments not shared with other patches, virtual nodes and virtual edges are created so that each border segment is uniquely identified by an actual or virtual edge. On the enhanced FPAG, the total number of nodes Nn is the sum of the number of real nodes nr and the number of virtual nodes nv , Nn = Nr + Nv . For patch Pi , if the movement pattern and sweeping directions are known, a path can be calculated based on a nominal spacing distance dn [17]. The fairness of the zigzag path whose sweeping direction is parallel to the border segment identified by eij is denoted as Fij (j ∈ (1, 2, ...Nn ), i 6= j) and the fairness of the spiral path which is parallel to all the border segments of patch Pi is denoted as Fii . τij is the parallel index of the common border segment identified by edge eij in Pi when i 6= j. τii = 1 for all i ∈ (1, 2, ..., Nr ). Let xij be the binary variable that represents which candidate path is selected. xij = 1 (i 6= j) means that the zigzag path parallel to the border segment identified by eij is selected, otherwise not. xii = 1 means that the spiral path is selected, otherwise not. We have the following expression to represent the overall path fairness: X X xij Fo = · Fij τij i∈(1,...,Nr ) j∈(1,...Nn )
To encourage achieving parallelity along the common border segments, a positive reward aij is introduced when two paths in adjacent patches are parallel to the common border segment identified by eij . The following expression represents the overall reward due to the parallelity along the borders. X X Re = xij · xji · aij +
xij1 = ... = xijτij , ∀i ∈ (1, 2...Nr )
Equation(3) means only one candidate path can be selected and Equation(4) represents the “nearly parallel” constraint among border segments on patch Pi . The solution to the above PIP problem gives, for each patch, the tool movement patterns and the sweeping directions. Once the tool movement patterns and sweeping directions are known for each patch, we design the path for each patch with the goal of minimizing the thickness uniformity. First, an optimal spacing distance dn is calculated which minimizes the thickness error on a plane [17]. Second, depending on the path approaching situation at the common border segments, three different cases, i.e., parallel-parallel (PA-PA), parallelperpendicular (PA-PE) and perpendicular-perpendicular (PEPE) are optimized to obtain the optimal path-to-border distance h [10]. Once dn and h are known, the overall path can be determined on the flat patch. V. I MPLEMENTATION AND T ESTING R ESULTS The algorithms are implemented in C++ and several automotive body parts from the Ford Motor company are tested. Here are the results for the car inner hood, which is shown in Figure 1. The first step is the surface partition. The weights of the shape fairness are set as ωs1 = 1/3, ωs2 = 1/3, ωs3 = 1/3. The solution to the weighted set partitioning problem gives the partition shown in Figure 6(a). To verify that the overall shape fairness z reaches optimality, two other different partitions, as shown in Figure 6(b) and Figure 6(c) are manually generated. The corresponding overall shape fairness z is calculated and compared with the optimal solution. The comparison results is shown in Table I. From Figure 6(a), 6(b) and 6(c), it is clear that the patches in the first partition have higher regularity RC and also have shorter common border length Lcb . TABLE I T HE COMPARISON OF OVERALL SHAPE FAIRNESS Case Optimal partition Manual partition 1 Manual partition 2
i∈(1,...,Nr ) j∈(1,...Nr ),j6=i Nr X i=1
[xii (
Nr X
(4)
xji · aij )].
j=1,j6=i
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Overall Shape Fairness z 6.2 5.4 5.7
(a)
tool paths that can achieve good motion performance for the moving mechanism and optimal deposition uniformity on a compound surface. Solving this problem requires the part surfaces be partitioned and the tool movement patterns and sweeping directions be carefully determined. Two measures, the shape fairness and the path fairness are defined to guide the surface partition and path design. Two integer programming problems are formulated and solved. Experimental results are provided, which validate the proposed approaches. Our future work will focus on the tool velocity planning and trajectory integration. This tool planning system can be easily extended to other similar surface manufacturing applications.
(b)
R EFERENCES
(c) Fig. 6. (a) Optimal partition by solving weighted set-partitioning problem. (b) Manually generated partition: case 1. (c) Manually generated partition: case 2.
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Fig. 7. (a) Movement patterns and sweeping directions: optimal. (b) Movement patterns and sweeping directions: non-optimal.
The second step is the designing of movement patterns and sweeping directions. The first partition (Figure 6(a) is used. The weights for overall path fairness Fo and the reward Re , are ωp1 = 0.5, ωp2 = 0.5. The movement pattern and sweeping directions are shown in Figure 7(a). The overall path fairness is 0.92. For comparison, the manually-picked movement patterns and sweeping directions as shown in Figure 7(b) has a overall path fairness of 0.83. This observation verifies that the solution to the PIP problem gives the optimal movement pattern and sweeping directions. As can be seen from Figure 7(a), there is no PA-PE or PE-PE cases regarding the path directions along the common borders.
[1] W.H.Huang. Optimal line-sweep-based decomposition for coverage algorithms. In Proceedings of the 2001 IEEE International Conference on Robotics and Automation, 2001. [2] Y. Mizugaki, M. Sakamoto, and K. Kamijo. Fractal path application in a metal mold polishing robot system. In Fifth International Conference on Advanced Robotics, pages 431 – 436, Pisa, Italy, June 1991. [3] Y. Takeuchi, D. Ge, and N. Asakawa. Automated polishing process with a human-like dexterous robot. In IEEE, pages 950–956, Barcelona, Spain, Oct. 1993. [4] N. Asakawa and Y. Takeuchi. Teachingless spray-painting of sculptured surface by an industrial robot. In IEEE International Conference on Robotics and Automation, pages 1875 –1879, Albuquerque, New Mexico, April 1997. [5] J. K. Antonio, R. Ramabhadran, and T. L. Ling. A framework for optimal trajectory planning for automated spray coating. International Journal of Robotics and Automation, 12(4):124–134, 1997. [6] L. F. Penin, C. Balaguer, J. M. Pastor, F. J. Rodriguez, A. Barrientos, and R. Aracil. Robotized spraying of prefabricated panels. IEEE Robotics and Automation Magazine, 5(3):18 –29, 1998. [7] R. C. Luo and J. H. Tzou. Investigation of a linear 2-d planar motor based rapid tooling. In IEEE International Conference on Robotics and Automation, pages 1471–1476, Washington, DC, USA, May 2002. [8] Y.Yang, H.T.Loh, F.Y.H.Fuh, and Y.G.Wang. Equidistant path generation for improving scanning efficiency in layered manufacturing. Rapid Prototyping Journal, 8(1):30–37, 2002. [9] D.M.Hensinger, A.L.Ames, and J.L.Kuhlmann. Motion planning for a direct metal deposition rapid prototyping system. In Proceedings of the 2000 International Conference on Robotics and Automation, 2000. [10] H. Chen, N. Xi, Z. Wei, Y. Chen, and J. Dahl. Trajectory integration for a surface with multiple patches. In Proceeding of the IEEE International Conference on Robotics and Automation, 2003. [11] 2002. Personal communication with D.Israel. [12] E.Balas. Combinatorial Optimization, chapter 7, pages 151–210. 1979. [13] K.Ramaswami. Process planning for shape deposition manufacturing. PhD thesis, Stanford University, 1997. [14] J.H.Kao and F.B.Prinz. Optimal motion planning for deposition in layered manufacturing. In Proceedings of 1998 ASME Design Engineering Technical Conferences, Atlanta, Georgia, USA, Sept 1998. [15] C.H.The and R.T.Chin. On the detection of dominant points on digital curves. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11(8), 1989. [16] S.Arora, D.Karger, and M.Karpinski. Polynomial time approximation schemes for dense instances of np-hard problems. Journal of comput. System. Sci., 1998. [17] H. Chen, W. Sheng, N. Xi, M. Song, and Y. Chen. Automated robot trajectory planning for spray painting of free-form surfaces in automotive manufacturing. In IEEE International Conference on Robotics and Automation, volume 1, pages 450 –455, 2002.
VI. C ONCLUSIONS This paper addresses a challenging problem in automated tool path planning for spray forming. That is, how to design
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