International Journal of Graphics Vol. 1, No. 1, November, November, 2010 2010
Study of Distance Computation between Objects Represented by Discrete Boundary Model M. S. Uddina*, K. Yamazakib School of Mechanical and Manufacturing Engineering, University of New South Wales, Sydney, NSW 2052, Australia b Department of Mechanical and Aerospace Engineering, University of California Davis, CA 95616, USA a,*
[email protected] ( *corresponding author) b
[email protected] a,*
Abstract Distance computation has become an effective means for identifying the proximity between objects in the applications of robotic path planning, collision detections in machine workspace (e.g. virtual NC program debugging). This paper presents an algorithm for computing the shortest distance between two objects represented by discrete boundary model (DBM). By using the DBM, the solid model of the object is discritized into three-dimensional (3D) points with a certain spacing distance (i.e. the mesh size). While two objects represented by the set of digitized points are placed in a 3D space, the proposed algorithm finds the pair of the closest points, one on the boundary surfaces of each object, and hence, calculates the shortest distance between objects. As a pre-processing step, bounding volume hierarchy (BVH) trees around the DBM models of objects are generated. Then, as the run time (i.e. distance query) step, a simple recursive search routine with the aid of pre-existing technique on BVH trees is applied to find the pair of the closest points on objects. Case studies are carried out on simple sphere models to verify the correctness of distance computation by the proposed algorithm. Computational results obtained from case studies demonstrate the capability of the proposed algorithm to calculate the shortest distance between objects with a reasonable error margin. Keywords: Discrete Boundary Model (DBM), Shortest Distance Calculation, Bounding Volume Hierarchy (BVH), Calculation Error, Computational Performance
1. Introduction The distance computation between objects has been used extensively in many applications such as path planning in robotics and computer graphics animations. By representing objects in terms of a mathematical model, a point on each object is searched such that the distance between the points is minimized. The technique of the distance computation is also adopted to determine the instance of collision between objects in many dynamic simulation environments. Recently some commercially available CAD/CAM software has incorporated such technique in an aim to determine the state (i.e. binary answer- True or False) of possible interference or collision between different objects in the machining workspace of the machine tools. Especially, during on-machine physical NC program debugging simulation, the value of the shortest distance will give the proximity or closeness between two objects such as between the tool and workpiece in the machining workspace of the machine tools. By comparing the shortest distance with an allowance distance, a machining operator can determine the NC block in NC programs, which is responsible for the possible interference
29
International Journal of Graphics Vol. 1, No. 1, November, November, 2010 2010
between objects. This, in turn, leads to a reduction of on-machine NC program debugging time and an increase of manufacturing productivity. The essence of the algorithm for such distance computation relies on the model representation of objects involved in the simulation environments. There are two major types of object’s model used for simulating the interactions between objects in the dynamics environments. One is polyhedral objects which may include CSG (Constitutive Solid Geometry) models and curved surfaces. The other is non-polyhedral objects; this can be classified into two groups – (1) without topological information, for example- set of polygons, (2) with topological information, for example- convex polyhedral model and nonconvex polyhedral models. As a non-polyhedral model, point is an important primitive to represent an object. Any kind of geometrical shape of objects without any topological information can be defined and represented by the set of points. A single point sample can be easily defined in 3D space and be stored in the computer memory. And when the boundary surface of the objects is represented by the sets of point samples, they create a point cloud of objects. Although such point cloud representation of objects consumes lots of memory storage in the computer, the data structure of the input model is very simple and a single point represents a bit of the computer memory, which enables computer’s processing engine to handle and process very quickly an abundant number of point sets. In addition, because of the evolution of advanced semiconductor memory chip technology, the computer’s memory capacity gets bigger as well as cheaper and the high speed processing engine becomes available in the market. Hence, the computational techniques for the distance calculation dealing with objects represented by set of points can be easily realized to achieve more realistic results in the dynamic simulations. FPGA (Field Programmable Gate Array) based semiconductor circuit hardware is an example potential system where those computational techniques can be practically implemented [1]. With this motivation, this paper presents an algorithm for the shortest distance computation between two objects represented by points. In order to convert the solid model of objects to a set of points, a discrete boundary model (DBM) where the boundary surface of the object is discritized into 3D points. In DBM, 3D scanning line grids with a certain spacing distance (i.e. mesh size) in three directions are created and the intersection between these scanning lines and the boundary surface of object generates the discrete points in 3D space. The shortest distance computation by the proposed algorithm involves two processing steps. As a preprocessing step, at first, bounding volume hierarchy (BVH) models around objects are created. Then, by using the simple search routine, the algorithm starts searching into BVH models of objects hierarchically to find the pair of the closest points, one on each BVH model. Finally the distance between the pair of the closest points gives the shortest distance between two objects. Case studies are presented to verify the correctness of the shortest distance computation by the proposed algorithm. Results obtained from case studies show the computational capability of the algorithm with a reasonable error margin. The rest of the paper is structured into the following sections. Section 2 describes briefly the previous works related to the distance calculation and collision detection. The basic idea of the discrete boundary model to represent the boundary surface of objects is illustrated in Section 3. Section 4 explains the proposed algorithm for the shortest distance calculation between two objects represented by discrete boundary points. Case studies to verify the correctness and computational performance of the proposed algorithm and a brief discussion on the computational results are highlighted in Section 5. Finally, Section 6 concludes the paper with a summary and future works of the current research study.
30
International Journal of Graphics Vol. 1, No. 1, November, November, 2010 2010
2. Related works As mentioned earlier, collision detection technique is related to the distance computation between objects. In this regard, previously many collision detection algorithms were applied to determine the state of interference between objects. There is considerable amount of research work on the minimum distance computation on the convex objects. For example, Lumelsky [2] presents an efficient algorithm for pairs of line segments. Chen et al. [3] presented an analytical technique to determine the minimum distance between two algebraic surfaces. By considering solid bodies of objects, Gilbert et al. [4], Bobrow [5], Cameron [6], and Lin [7] describe the algorithms that find the distance between two convex polyhedral objects. Each of these algorithms iteratively finds pairs of points, one on each object, such that the distance between the points monotonically converges to a minimum. However, the mechanism of the above algorithms completely depends on the properties of the convex objects, and it appears difficult to extend them directly to the non-convex objects. Further, Turnbull et al. [8] describes algorithm for computing distance between NURBS-defined convex objects. In order to compute the distance between non-convex objects, each object is broken into convex components and one of the above algorithms is applied to determine the distance between the components [9]. The smallest distance between any pair of convex components gives the distance between two non-convex objects. Since collision detection is closely related to the distance computation and underlying techniques rely on object representation, two related approaches for efficient collision detection as well as the minimum distance computation are hierarchical models and bounding representations. As for hierarchical model, an object is described at various levels of detail. The collision detection algorithm uses the different levels of detail to reduce the number of components that are examined. Similarly, the bounding representations approximately model an object with simple primitives such as triangles. Efficient algorithms, such as described by Baraff [10], determine if collision has occurred between the bounding representations and only then the components of the original model are examined. By using the sphere bounding volume hierarchy representation, Quinlan [11] described an efficient algorithm for the minimum distance computation between non-convex objects. He also incorporated a percentage of relative error in the distance search routine to further improve the efficiency of the algorithm. In this regard, using sphere tree, Palmer and Grimsdale [12] determined the state of collision for the purpose of computer animation. Chang et al. [13] presented an efficient algorithm for collision detection by using a dual OBBsphere bounding volume hierarchy, where sphere bounding volume is used to determine preliminary state of collision and OBB is used for the final confirmation of state of collision occurred between two objects. In this paper, an algorithm for the shortest distance calculation between objects represented by discrete boundary points is introduced. Unlike other distance calculation (or collision detection) algorithms described above, the proposed algorithm finds the pair of the closed points on the boundary surfaces using simple sphere bounding volume hierarchy (BVH) technique without any computational complexity. As the boundary surfaces of the objects are represented by discrete points with a certain mesh size, the accuracy of shortest distance calculation depends on the mesh size and distribution of points on the boundary surfaces of objects. Please note that the authors may not be aware of all other similar algorithms for distance calculation; however, the essence of the algorithm presented in this paper solely relies on finding the shortest distance between two objects when only the
31
International Journal of Graphics Vol. 1, No. 1, November, November, 2010 2010
boundary surface of objects is represented by discrete points. Hence, this, the authors strongly believe, eventually makes the current research study more distinct from other related works in literature.
3. Discrete Boundary Model (DBM) As mentioned earlier, this study uses the advantage of a discrete boundary model to represent the boundary surfaces of objects [14]. Figure 1 shows the basic idea of the DBM. In the DBM, the boundary surface of an object is finitely discritized into three-dimensional (3D) points with a certain space resolution. YZ scanning line ZX scanning line
Geometric model Discrete points XY scanning line
∆Z
∆X
Z Y
∆Y
Spacing distance between X directional scanning lines
X Figure 1. Concept of discrete boundary model (DBM) To create such discrete points, at first, scanning line grids with a certain spacing distance on three orthogonal planes are generated. When the object is placed in the space of 3D scanning lines grids, the scanning lines intersect the boundary surfaces of the solid model, creating discrete points on the boundary surface of model. In Figure 1, ∆X, ∆Y, and ∆Z are respectively the spacing distance between scanning lines along X, Y and Z directions. The accuracy of model representation depends on the spacing distance between scanning lines. In this paper, the spacing distance between scanning lines can be defined as the mesh size in two-dimensional (2D) space. Since DBM is a kind of digital model defined by the set of points, it can be stored in computer by 3D memory addressing, and is very much suitable for the repetitive point-by-point data manipulation and calculation. To this end, by using the properties of the DBM, the main focus of the algorithm introduced in this paper is to find the pair of the closest discrete points, one on the boundary surfaces of each object, which eventually gives the shortest distance between objects.
4.0 Proposed algorithm for the shortest distance calculation 4.1. Overview The main idea of the algorithm presented in this paper is that when two DBM models are placed in a 3D space, the algorithm finds the pair of the closest points on the models, and then calculates the shortest distance between two models. Figure 2 shows an example of the shortest distance calculation between two DBM models.
32
International Journal of Graphics Vol. 1, No. 1, November, November, 2010 2010
SD = Shortest distance SD Model B
The pair of the closest points Model A Figure 2. Example of the shortest distance calculation between two DBM models As can be seen, each DBM model contains huge number of points on its boundary surfaces. If one performs a point-by-point distance checking between two models (Model A and Model B), it will take huge computational time to find the pair of the closest points on models. In such case, the computational complexity becomes O (nm), where n and m are the number of points on each model. In order to avoid such computational complexity, and hence to speed up the searching for the pair of the closest points, this paper proposes an algorithm for the shortest distance calculation between objects using the advantage of the bounding volume hierarchy (BVH) algorithm. In the next section, the details of the algorithm are explained. 4.2. Main algorithm Proposed in the paper is a two-step algorithm for the shortest distance calculation. This includes: (1) Preprocessing step – to build the bounding volume hierarchy (BVH) tree around the object’s DBM model (2) Distance query step – to search for the pair of the closest points, and hence the shortest distance by traversing BVH trees of two models The details of these two steps are briefly explained in the following subsections. 4.2.1. Preprocessing step: In this step, before computing the shortest distance between models, the underlying DBM model of the object is used to build a bounding volume hierarchy (BVH) tree. Here, a sphere is selected as a bounding volume (BV) because it is the simplest geometric model and can be specified with its center location and radius. To calculate the distance between two spheres one needs only seven additions, three multiplications, and one square root [11]. Other BVs such as OBB (orientated bounding box) or ellipsoid may better approximate the object; however the simplicity of the sphere makes it the preferred bounding volume. The following are the salient steps required to build BVH tree around object’s DBM model.
33
International Journal of Graphics Vol. 1, No. 1, November, November, 2010 2010
For the given set f points in DBM model, calculate maximum and minimum of points, which creates coordinates of four extreme boundary points of an axis aligned bounding box (1) Calculate the center of the box, which will give the center of sphere BV (2) Calculate the radius of the sphere such that it covers all the points (3) Calculate the longest axis along which points have maximum variation (4) Divide the point-set into two groups along the longest axis (5) Recursively do the above operations (1~4) for each group of points until each group contains a single point Root node Sphere BV
Children nodes
Leaf node Figure 3. Concept of building a sphere BVH tree around a DBM model Figure 3 shows a simple concept of building sphere BVH tree around DBM model. Most methods for building BVH tree fall into two categories: bottom-up and top-down. Bottom-up method begins with a bounding volume for each primitive point in model and merge volume into larger volumes until the tree is complete. On the other, top-down method begins with a group of all the points in model and recursively subdivide until all leaf nodes are invisible. In this paper, a top-down strategy is applied to create the sphere BVH tree. At first, all the points in the model are enclosed by a sphere that is called root sphere or node. Then the point-set is divided into two groups by using the principal component analysis and calculate the sphere that covers grouped point sets (the basic procedure to divide and to calculate the sphere BV is described in the above five (5) steps). Then, by recursively partitioning point sets and sphere
34
International Journal of Graphics Vol. 1, No. 1, November, November, 2010 2010
covering at different node levels, a hierarchical sphere BV tree is built. It is seen that the root sphere contains all descendent spheres including children spheres, while leaf node contains the points on the model (See Fig. 3). By applying the above procedure as a preprocessing step, the sphere bounding volume hierarchy (BVH) tree for each model in their local coordinate frames is generated. Before computing the distance between two objects at a specific position and orientation, one must augment each sphere BVH tree with the corresponding transformation matrix describing the position and orientation of the object with respect to some global coordinate frame. Before a node is used in the search routine, the node’s sphere is mapped through the tree’s transformation matrix. The mapping is done only when a node is used because the search routine only needs to examine a small fraction of the total nodes of a tree. Mapping all the nodes before the search would cause lower efficiency in computation. 4.2.2. Distance query step This step describes the details of the algorithm to find the pair of the closest points on models and hence the shortest distance between two models. The algorithm proceeds on by simultaneously traversing sphere BVH trees of two models. The search routine of the algorithm is based on building a bounding volume test tree (BVTT) comprised of BVH trees of two models [15]. This BVTT will represent the hierarchy of distance query tests performed during a query. Figure 4 shows an example of BVTT derived from two BVH trees. Each node in the BVTT corresponds to a single distance test between a pair of BVs. For the distance query algorithm presented in this paper, a BV test (i.e. a node in the BVTT) often leads to an additional BV tests (i.e. its child nodes in the BVTT). The root node of the BVTT is the BV test between the roots of the BVH trees. The leaf nodes of the BVTT are either a test between two BVH leaf nodes or else a test between a pair of nodes which are far away each other, which prunes their sub-tree from the BVTT. The detailed description of the algorithm based on the above BVTT to find the closest pair of points is explained as follows. By using the BVTT as described earlier, the algorithm finds the pair of closest points, one on each object such that the distance between the points is less than or equal to the distance between any other pair. Initially the distance d is set to infinity or the distance between two arbitrary points, one on each object. The key to the algorithm is to show the two objects are a distance d apart without examining all possible pairs of points on the objects. The search routine finds pairs of leaf nodes that are less than a distance d apart. On the BVTT, the search examines pairs of nodes in the depth-first manner [16] starting with the root nodes of the two BVH trees. If the distance between the node’s spheres is greater or equal to the current value of d then, from the structure of the BVH trees, it is known that the distance between the two sets of descendent leaf spheres is greater or equal to d and can thus be ignored. If the two nodes are less than d apart, then further examination is carried out on the children of the nodes. This way the process continues until leaf nodes which contain the points on boundary surface of object are reached.
35
International Journal of Graphics Vol. 1, No. 1, November, November, 2010 2010
BVH tree of Model A
BVH tree of Model B X A
A + B A
C A
Y B
Z C
AX
BX
BY
CX
BZ
CY
CZ
Simultaneous BVTT for model A and B Figure 4. Example of BVTT resulted from BVH trees of two models During carrying out the search routine on the BVTT to find the pair of closest points, there are three cases to be considered. Case 1: If both nodes are from the interior of the BVH tree, one of the nodes is split into its two children. Then recursively search the two pairs of consisting of a child and the node not split. Deciding which node to split is based on the heuristic of splitting the node with the larger associated sphere. Case 2: In the case of one interior node and one leaf node, the interior node is split. The order of the two subsequent searches is the same as above. Case 3: In the case where two leaf nodes reach, the distance between the points labeled by leaf nodes is calculated. If the distance between the points is less than d, then a new minimum is reached. If the distance is zero, i.e. the two objects intersect, then the distance between two objects is zero and the search need not continue. Otherwise, d is set to the new distance and continue the search. Figure 5 shows a simple example of distance query tests between two BVH trees in 2D space showing the sequence of finding the pair of closest points. By following the above procedure, the pair of the closest points, one on each object, can be searched, which eventually gives the shortest distance between two objects. During the whole searching process, there are two types of search; searching between BV pairs (i.e. between children nodes) and searching between point pairs (i.e. between leaf nodes). So, the number of BV
36
International Journal of Graphics Vol. 1, No. 1, November, November, 2010 2010
pair and point pair tests required to calculate the shortest distance between two objects will give an overall impression of the computational performance of the algorithm.
5. Case study - simple examples In order to verify the correctness of the shortest distance calculation and to study the computational performance of the proposed algorithm, several case studies are carried out. For simplicity to implement the algorithm, sphere models as input objects are considered. Sphere model always offers universal geometric shape and it is very simple to theoretically calculate the shortest distance between two sphere models. The algorithm is implemented on Microsoft Visual C++ 6.0 platform with OpenGL graphics tool using a desktop PC of 2 GHz processing engine and 1 GB Random Access Memory capacity. Root node
Model 2 d
Ignore searching inside child node
d Model 1
d d
Closer child BV pair and continue searching their child node Pair of the closest points
dmin
Figure 5. Example of distance query tests between sphere BVH trees of two objects in 2D space At first, a sphere solid model of 50 mm diameter is converted to discrete points on its surface by using the DBM processing engine as described in Section 3. The mesh size (i.e. the spacing distance between scanning lines in 2D space) considered is of 0.5 mm, 1.0 mm, 1.5 mm and 2 mm, which gives four sets of DBM data of the sphere model. The mesh size defines the accuracy of the model’s surface representation in terms of discrete points. The number of points in the sphere model for the mesh size of 0.5 mm, 1.0 mm, 1.5 mm, and 2 mm are 47119, 11780, 5233, and 2904 respectively. For two sphere DBM models of the given mesh size, the shortest distance is calculated where both sphere modes are placed at different positions allowing a certain distance between them. For simplicity, no rotations are applied to the models. This way four case studies are carried out; one for each different mesh size of sphere model. In each case study, the percentage of error between the theoretical and the calculated (obtained by our proposed algorithm) shortest distances is calculated. In order to justify the computational performance
37
International Journal of Graphics Vol. 1, No. 1, November, November, 2010 2010
of the algorithm, the number of BV pair and point pair tests and the total distance query time (in milliseconds) required to calculate the shortest distance are computed. For each case study, to check effect of separation distance between models on calculation error, the shortest distance is calculated while keeping one model is fixed at a position and the other is placed at eight different positions. This gives us total eight trials of the distance calculation. For all case studies with four different mesh sizes of sphere DBM models, the percentage of calculation error is estimated by taking an average of all eight trials of the distance calculation.
Figure 6. Percentage of calculation error with the mesh size of models As Figure 6 shows, the percentage of calculation error increases with an increase in the mesh size of model. This can be due to the coarse arrangement of the discrete points on the boundary surface of models for larger mesh size. The calculation error increases sharply when the mesh size ranges from 1 mm to 1.5 mm and it remains constant for the mesh size less than 1 mm. When the mesh size further gets small, it is likely to observe a constant level of calculation error. This means that the mesh size ranging from 0.5 mm to 1 mm would be optimized one to maintain an acceptable calculation error in distance calculation between objects in a dynamic simulation environment. Further, for each case study, the calculated shortest distance is found to be always larger than the theoretical one. Note that the boundary surface of model is defined by discrete points located exactly at the grid points of 3D scanning lines (as described in Sections 3 and 4) and the present algorithm always finds two such closest discrete points to calculate the shortest distance. Also, points on the theoretical boundary surface of a model at any location may not correspond to the discrete points on DBM model. The above would be a possible fact behind why the calculated shortest distance is larger than the theoretical one.
38
International Journal of Graphics Vol. 1, No. 1, November, November, 2010 2010
Figure 7. Computation time with the mesh size of models
Figure 8. Computation time with the number of points in a sphere model Figure 7 shows the computation time required to calculate the shortest distance with respect to the mesh size of models. The computation time increases with the decrease of the mesh size. This is due to the fact that the smaller mesh size increases the total number of points in the model, which eventually results in a larger number of leaf nodes in BVH tree of the model. During traversing BVH trees of two models, the algorithm needs to perform larger number of distance query tests in terms of number of BV pair and point pair tests required to find the pair of the closest points, which give the shortest distance between models. Figure 8 shows the trend of computation time with respect to the number of points in the model. In this graph, the number of points means the total number of points in a single sphere model because both spheres used in distance query tests have the same number of total points. The
39
International Journal of Graphics Vol. 1, No. 1, November, November, 2010 2010
computation time shows a linear increasing trend which is more obvious and can be attributed to the similar fact as for Fig. 7.
Figure 9. Number of search with the shortest distance between models
Figure 10. Percentage of calculation error with the shortest distance between models Figure 9 shows the computation results in terms of number of search with respect to shortest distance (i.e. separation) between models for the case where the mesh size of the models is 0.5 mm. The number of search includes the number of BV and point pair tests required to find the pair of the closest points. It is seen that, for each distance query test, the number of BV pair tests is always larger than that of point pair tests. This means that the algorithm requires performing more searches between internal children nodes (i.e. BVs) compared to those between leaf nodes (i.e. points). The number of BV pair tests increases little sharply when models are placed close to each other and increases very slowly with the
40
International Journal of Graphics Vol. 1, No. 1, November, November, 2010 2010
increase of distance between them. On the other hand, at the close proximity between models, the number of point pair tests is larger and it decreases when models are placed far distance away from each other. Figure 10 shows the percentage of calculation error with respect to the shortest distance between models for the case where the mesh size of the sphere model is 0.5 mm. The calculation error remains almost constant except for its larger value when models are very close to each other. Further, when two models are placed at a distance of 20 mm or more, the proposed algorithm is able to calculate the shortest distance with a calculation error of less than 0.1%, indicating its effectiveness and correctness in distance calculation.
6. Conclusion and future work This paper presents an algorithm for the shortest distance calculation between objects represented by discrete boundary model (DBM). By utilizing the advantage of bounding volume hierarchy technique, the algorithm finds the pair of the closest points on the boundary surfaces of objects. And, the distance between the closest points gives the shortest distance between two objects. Several case studies are carried out on sphere models with different mesh sizes and separation distances to verify the correctness of distance calculation by the proposed algorithm. Results from case studies validate the correctness of the shortest distance calculation by the proposed algorithm with a reasonable calculation error. Some computational results in terms of the number of searches and computation time are discussed. Due to the advancement of computer memory and data processing technology, the algorithm involving the repetitive calculations between discrete points presented in this paper can be easier to implement. FPGA (Field Programmable Gate Array) based semiconductor circuit hardware is one of the potential technologies where such repetitive calculations can be performed within a shorter period of time. By integrating such FPGA based hardware technology into the CNC system, a machining operator can easily debug NC programs within a shorter period of time. This will tremendously reduce machining lead time, and increase manufacturing productivity as well. Hence, further extension in terms of the application of the proposed algorithm to FPGA based hardware integrated into CNC system is left for the future work of the current research study.
Acknowledgments The authors wish to sincerely thank the Mori Seiki Co. for providing its generous financial support to carry out the research work presented in the paper.
7. References [1] [2] [3] [4] [5]
N. Atay, J. W. Lockwood, and B. Bayazit, “A collision detection chip on reconfigurable hardware”, Technical Report WUCSE-2005, Washington University in St. Louis, St. Louis MO, 2005. D. Baraff, “Curved surfaces and coherence for non-penetrating rigid body simulation”, Computer Graphics, 24(4), 1990, pp. 19-28. J. E. Bobrow, “Optimal robot path planning using the minimum-time criterion”, IEEE Journal of Robotics and Automation, 4(4), 1988, pp. 443-450. J. W. Chang, W. Wang, and M. S. Kim, “Efficient collision detection using a dual OBB-sphere bounding volume hierarchy”, Computer Aided Design, 42(1), 2009, pp. 50-57. S. Cameron, “Enhancing GJK: computing minimum and penetration distances between convex polyhedral” In: Proc of Int. conf. on Robotics and Automation, Albuquerque, NM USA, 1997, pp. 3112-3117.
41
International Journal of Graphics Vol. 1, No. 1, November, November, 2010 2010
[6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
[16]
X.D. Chen, J. H. Yong, G. Q. Zheng, J. C. Paul, and J. G. Sun, “Computing minimum distance between two algebraic surfaces”, Computer Aided Design, 38, 2006, pp.1053-1061. Ding, K., A study on a new geometric modeling for off-line and on-line multi-axis machining simulation system. Thesis (PhD). University of California Davis. USA, 2004. E. Gilbert, D. E. Johnson, and S. Keerthi, “A fast procedure for computing the distance between complexobjects in three-dimensional space” IEEE Journal of Robotics and Automation, 1988, pp.193-203. D. E. Johnson and E. Cohen, “Bound coherence for minimum distance computations” In: Proc. of Int. Conf. on Robotics and Automation, Detroit MI USA. 15, 1999, pp. 1843-1848. E. Larsen, S. Gottschalk, M. C. Lin, and D. Manocha, “Fast proximity queries with swept sphere volumes” In: Proc. of Int. Conf. on Robotics and Automation, 2000, pp. 3719-3726. V. Lumelsky, “On fast computation of distance between line segments” Information Processing Letters, 21, 1985, pp. 55-61. M.C. Lin, Efficient Collision Detection for Animation and Robotics, Thesis (PhD), University of California Berkeley, 1997. I. Palmer and R. Grimsdale, “Collision detection for animation using sphere trees” Computer Graphics Forum, 14(2), 1995, pp.105-116. S. Quinlan, “Efficient distance computation between non-convex objects”, In: Proc. of IEEE Int. Conf. on Robotics and Automation, 1994, pp. 3324—3329. Y. Sato, M. Hirata, T. Maruyama, and Y. Arita, “Efficient collision detection using fast distance calculation algorithms for convex and non-convex objects”, In: Proc. of 1996 IEEE Int. Conf. on Robotics and Automation, Minnesota USA. 1996, pp. 771-778. C. Turnbull and S. Cameron, “Computing distances between NURBS-defined convex objects”, In: Proc. of IEEE Intl. Conf. on Robotics and Automation, Leuven Belgium. 1998, pp. 16-21.
Author M. S. Uddin is currently working as a Postdoctoral Research Fellow in the School of Mechanical and Manufacturing Engineering at University of New South Wales (UNSW) in Australia, where his research focuses on design, manufacturing, CAD/CAM integration, geometric modeling. Earlier he worked for a year as a Postdoctoral Research Fellow in the IMSMechatronics Lab at University of California Davis (UCDavis) in USA, mainly concentrating on algorithms and geometric modeling for CAM software development for manufacturing automation. Dr Uddin published a significant number of research papers in reputed international journals and conferences. For his outstanding research, he received the “Young Researcher Award” from the Mazak Foundation (Japan) in 2006. He received PhD and MEng degrees, both in Mechanical Engineering from Kyoto University (Japan) in 2007 and National University of Singapore in 2004 respectively. K. Yamazaki is currently a Professor and Director of the IMS-Mechatronics Lab in the School of Mechanical and Aerospace Engineering at University of California Davis (UCDavis) in USA. Professor Yamazaki is rigorously engaged in research on machine tools technology development, which attracted growing attention of machine tools industries across the world. Because of his outstanding and real-world focused research, he received a vast amount of financial support from Mori Seiki Co. and Sodick Co., the world-
42
International Journal of Graphics Vol. 1, No. 1, November, November, 2010 2010
leading machine tools companies. Professor Yamazaki published more than a couple of hundreds papers in the research area of machine tools technologies. He received PhD (1975), MS (1972), BS (1970) degrees from Keio University in Japan. During his doctoral study, he developed an innovative microprocessor-based controller for CNC machines. Professor Yamazaki holds highly prestigious Fellowship positions in a number of reputed professional societies such as ASME, SME and CIRP.
43
International Journal of Graphics Vol. 1, No. 1, November, November, 2010 2010
44
