Gear manufacturing is a fundamental technology used in machine production. With an increase in automotive demand world-wide, a highly efficient method for gear manufacturing is required. Gear is used not only in automobiles but in all kinds of mechanical products including aircrafts, ships, and power generating equipment, and the production quantity of the gear is enormous. Power skiving is rapidly spreading as an efficient method for the manufacturing of internal high-precision gears. Power skiving is a machining method used to continuously remove chips for shaping the gear grooves while rotating the workpiece and cutting tool at high speed during synchronization [1] . It is well known that the power skiving process is significantly faster than traditional gear shaping methods and is more flexible than a costly broaching procedure. A determination of the proper machining parameters, such as the cutter shape/position and rotation speed of the cutter and workpiece, is critical for efficient gear manufacturing.
To analyze the mechanical phenomena during power skiving, a geometric simulation of the power skiving process, particularly a visualization of the resulting machining shape and a prediction of the chip geometry generated during the cutting process, is important. For example, the cutting force during power skiving can be estimated by analyzing the thickness of the chip [2, 3, 4, 5] . In this paper, we propose a novel method for simulating the power skiving process using solid modeling. Using this method, shape changes to the workpiece during the machining process can be precisely analyzed. It is possible to generate a solid model of a non-deformed chip by executing the intersection shape computation between the workpiece and the cutter swept volume.
Tapoglou proposed a 3D CAD based approach for geometrically simulating the power skiving [6] . In the geometric simulation of power skiving, the set operation which
Gear manufacturing is a fundamental technology used in machine production. With an increase in automotive demand world-wide, a highly efficient method for gear manufacturing is required. Gear is used not only in automobiles but in all kinds of mechanical products including aircrafts, ships, and power generating equipment, and the production quantity of the gear is enormous. Power skiving is rapidly spreading as an efficient method for the manufacturing of internal high-precision gears. Power skiving is a machining method used to continuously remove chips for shaping the gear grooves while rotating the workpiece and cutting tool at high speed during synchronization [1] . It is well known that the power skiving process is significantly faster than traditional gear shaping methods and is more flexible than a costly broaching procedure. A determination of the proper machining parameters, such as the cutter shape/position and rotation speed of the cutter and workpiece, is critical for efficient gear manufacturing.
To analyze the mechanical phenomena during power skiving, a geometric simulation of the power skiving process, particularly a visualization of the resulting machining shape and a prediction of the chip geometry generated during the cutting process, is important. For example, the cutting force during power skiving can be estimated by analyzing the thickness of the chip [2, 3, 4, 5] . In this paper, we propose a novel method for simulating the power skiving process using solid modeling. Using this method, shape changes to the workpiece during the machining process can be precisely analyzed. It is possible to generate a solid model of a non-deformed chip by executing the intersection shape computation between the workpiece and the cutter swept volume.
Tapoglou proposed a 3D CAD based approach for geometrically simulating the power skiving [6] . In the geometric simulation of power skiving, the set operation which
Gear manufacturing is a fundamental technology used in machine production. With an increase in automotive demand world-wide, a highly efficient method for gear manufacturing is required. Gear is used not only in automobiles but in all kinds of mechanical products including aircrafts, ships, and power generating equipment, and the production quantity of the gear is enormous. Power skiving is rapidly spreading as an efficient method for the manufacturing of internal high-precision gears. Power skiving is a machining method used to continuously remove chips for shaping the gear grooves while rotating the workpiece and cutting tool at high speed during synchronization [1] . It is well known that the power skiving process is significantly faster than traditional gear shaping methods and is more flexible than a costly broaching procedure. A determination of the proper machining parameters, such as the cutter shape/position and rotation speed of the cutter and workpiece, is critical for efficient gear manufacturing.
To analyze the mechanical phenomena during power skiving, a geometric simulation of the power skiving process, particularly a visualization of the resulting machining shape and a prediction of the chip geometry generated during the cutting process, is important. For example, the cutting force during power skiving can be estimated by analyzing the thickness of the chip [2, 3, 4, 5] . In this paper, we propose a novel method for simulating the power skiving process using solid modeling. Using this method, shape changes to the workpiece during the machining process can be precisely analyzed. It is possible to generate a solid model of a non-deformed chip by executing the intersection shape computation between the workpiece and the cutter swept volume.
Tapoglou proposed a 3D CAD based approach for geometrically simulating the power skiving [6] . In the geometric simulation of power skiving, the set operation which 
Gear manufacturing is a fundamental technology used in machine production. With an increase in automotive demand world-wide, a highly efficient method for gear manufacturing is required. Gear is used not only in automobiles but in all kinds of mechanical products including aircrafts, ships, and power generating equipment, and the production quantity of the gear is enormous. Power skiving is rapidly spreading as an efficient method for the manufacturing of internal high-precision gears. Power skiving is a machining method used to continuously remove chips for shaping the gear grooves while rotating the workpiece and cutting tool at high speed during synchronization [1] . It is well known that the power skiving process is significantly faster than traditional gear shaping methods and is more flexible than a costly broaching procedure. A determination of the proper machining parameters, such as the cutter shape/position and rotation speed of the cutter and workpiece, is critical for efficient gear manufacturing.
To analyze the mechanical phenomena during power skiving, a geometric simulation of the power skiving process, particularly a visualization of the resulting machining shape and a prediction of the chip geometry generated during the cutting process, is important. For example, the cutting force during power skiving can be estimated by analyzing the thickness of the chip [2, 3, 4, 5] . In this paper, we propose a novel method for simulating the power skiving process using solid modeling. Using this method, shape changes to the workpiece during the machining process can be precisely analyzed. It is possible to generate a solid model of a non-deformed chip by executing the intersection shape computation between the workpiece and the cutter swept volume.
Tapoglou proposed a 3D CAD based approach for geometrically simulating the power skiving [6] . In the geometric simulation of power skiving, the set operation which 48th SME North American Manufacturing Research Conference, NAMRC 48 (Cancelled due to subtracts the swept shape of the moving cutter from the solid model of the workpiece is repeated. In the usual 3D CAD technology, it is necessary to reconstruct the graph structure representing the connection between faces, edges, and vertices of the solid model on the basis of the result of the intersecting computation between the objects in the set operation. This process is very complicated, and sometimes fails due to inevitable computation errors. Differing from this method, we apply triple-dexel solid modeling in a simulation. Since this modeling method does not use graph structure in the representation, our method is quite robust in the set operation.
In triple-dexel solid modeling, the object shape can be represented as a set of parallel segments defined for each grid point in the coordinate planes (XY-, YZ-, or ZX-plane). The geometric cutting simulation can be simplified as a set of Boolean computations between overlapping segments defined for the same grid point. This computation can be accelerated using the parallel processing function of the geometric processing unit (GPU), which is now installed on most personal computers. In the next section, some related studies on a simulation of the power skiving process is briefly introduced. In Section 3, the kinematics of the power skiving process is described. Our geometric simulation technology using the triple-dexel representation is given in Sections 4 and 5. Some geometric simulation results are illustrated in Section 6. Finally, Section 7 provides some concluding remarks. 
Power skiving is a gear manufacturing method, especially for internal gears. In this method, it was necessary to synchronize the cutting tool and the gear with high precision, therefore it was not popular at the beginning. Technological problems of the power skiving and their analysis methods are discussed in [7] . First simulation model of power skiving process was proposed by Spath and Huhsam [8] . Guo et al. studied the cutting mechanism with power skiving [9] . They extended their studies on the design of the cutting tool for machining involute gears [10] . Tapoglou developed a cutting simulation method of power skiving using 3D CAD system (Autodesk inventor) [6] . Our research work has some similarities to his work, except that we use triple-dexel solid modelling for representing and processing workpiece shape and use the parallel processing function of GPU for accelerating the simulation.
Our simulation technology is related to the NC geometric milling simulation. The main purpose of a geometric milling simulation is to visualize the change in workpiece shape during the milling process and detect possible collisions between a cutter holder and the workpiece. A milling operation is geometrically equivalent to a Boolean subtraction of the swept volume of a cutter moving along a cutter path from a geometric model representing the stock shape. The majority of systems used realize a milling simulation based on this idea. One of the major differences among such systems is the representation scheme of the workpiece shape. CSG, Z-map, dexel, voxel, octree, and vector representations have been used in prior studies [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] . Applying the parallel processing capability of a GPU to accelerate the simulation has also been studied [15, 17] . Fig. 1 illustrates the configuration of a cutter and workpiece during power skiving. In the figure, some parameters controlling the kinematics of the cutter/workpiece in the machining process are specified, where ωc (rpm) and ωw (rpm) represent the rotational speed of the cutting tool and workpiece, respectively. The feed rate of the cutting tool per rotation of the workpiece is f (mm/rev). The cutting tool is shifted rx (mm) in the X-axis direction from the center of the workpiece. The tool is further tilted about the X-axis by θx (degrees). For our implementation, Fig. 2 shows the outline of the cutting edge. The rake face is flat and its normal vector is perpendicular to the X-axis of the cutter coordinate frame. 
The sequence of points De on the cutting edge is expressed as follows:
(1)
During our simulation, the workpiece model is considered to be fixed in the world coordinate frame, and only the cutting edge is moved, engraving the workpiece surface. The position Ce(t) of the sequence of points on the cutting edge at time t can be obtained through the following coordinate transformation.
where Rc is a matrix representing the rotation of the cutter about the Z-axis of the world coordinate frame. This transformation corresponds to the rotation parameter ωc at time t. Points on the cutting edges are then rotated by θx about the X-axis. Matrix Rx represents this rotation. Points on the cutter edge are further translated by rx in the X-axis direction. The points are finally rotated about the Z-axis of the world coordinate frame. This operation corresponds to the rotation parameter ωw at time t. To cancel the workpiece motion in the space, a rotating operation using Rw is applied in the reverse direction. This formula does not evaluate the tool feed. The handling of the tool feed will be described later.
The left side of Fig. 3 illustrates a trajectory of the cutting edge moved during a certain range of time with a time interval of 5.0 × 0.1 -6 (min). The right side of the figure shows a closeup of a part specified by the small square on the left side. Because the workpiece model is fixed and the cutter edge is only moved in our simulation, it can be seen that the points on the cutting edge approaches the workpiece from above, and engraves the workpiece surface and escapes downward. In this computation, the configuration parameters are set to ωc = 1,311.00 rpm, ωw = 764.75 rpm, rx = 11.326 mm, and θx = 20 degree.
In this section, our geometric simulation method using the triple-dexel representation model is described. 
A cutting operation is geometrically equivalent to a Boolean subtraction of the swept volume of the moving cutter from a solid model representing the workpiece shape. During power skiving, the cutting edge moves along a spatial trajectory curve changing its position and orientation. In the cutting simulation, a series of thin swept volumes of the cutting edge during motion are computed for each short time interval, and are subtracted from the workpiece model to obtain the resulting cutting shape. Fig. 4(a) illustrates the changes in position and posture of the cutting edge at every small interval t. We obtain a thin swept volume of the cutting edge by placing two successive cutting edges Ce(t) and Ce(t + t), and by filling the space between them with small polygons, as shown in Fig. 5 . Fig. 4(b) shows a shaded display of a series of thin swept volumes corresponding to the cutting edges given in (a).
In conventional B-rep solid modeling usually used in the 3D CAD system, a three dimensional object is represented as a collection of closed surfaces connected together. A Boolean subtraction operation must handle both the intersection curve computation between the model surfaces and the topological reconstruction by trimming and reconnecting the surfaces into a new closed model. These operations are computationally expensive, and they tend to be unstable owing to the existence of unavoidable floating-point computation errors. In our geometric simulation system, a triple-dexel model is used to represent the workpiece shape. Dexel modeling is known as a discrete method for representing a 3D shape [22] . With this method, the object shape is represented by a bundle of Z-axis-aligned segments defined for each grid point of a square mesh in the XY-plane. Because no topological information is used in the model representation, a Boolean operation in the dexel modeling is much more robust than the operation used in B-rep modeling. With dexel modeling, nearvertical surfaces have inevitable large shape errors caused by a finite grid resolution. The triple-dexel model was proposed to overcome this nonuniformity of the shape errors and to realize an accurate shape representation [23] . In this representation, the 3D shape is not only defined by a Z-axis-aligned dexel model but also by an X-axis-aligned dexel model based on a square mesh in the YZ-plane and a Y-axis-aligned dexel model based on a mesh in the ZX-plane (see Fig. 6(a) ). In the triple-dexel representation, the simulation space can be divided into a set of cubes defined by properly positioned X-, Y-, and Z-axis-aligned dexels, as shown in Fig. 6(b) . We define a regular spatial grid with a cubic cell structure in an axis-aligned rectangular box that holds the workpiece shape. The grid is projected onto the XY, YZ, and ZX-planes to define an axis-aligned square mesh in each plane (see Fig. 7 ). Lines starting from the grid points in each plane, perpendicular to the plane, correspond to the grid lines organizing the spatial grid structure in the box. Here, X-, Y-, and Z-axis-aligned dexels are defined using these meshes in the YZ, ZX, and XY-planes, respectively. End points of the dexels are sampled as the intersection points between the grid lines and the surface of the object, therefore the space between the adjacent grid lines becomes the worst value of the model accuracy as the same as the voxel model. The representation accuracy of the triple-dexel model is generally much higher than the voxel model of the same resolution, because the endpoint of each dexel records the exact position. Fig. 8 illustrates one cubic cell in the spatial grid structure. Because this cell is located in the middle of the workpiece surface, four dexel segments of the cell have intersection points with the surface. Small black points in the figure correspond to the intersection points (= end points of dexels). Small polygons representing a part of the workpiece surface are obtained in the cell by properly connecting the intersection points. The marching cube algorithm [24] is used for generating the polygons in our current implementation.
In our power skiving simulation software, the cutting result shape is obtained by successively subtracting thin swept volumes of the cutting edge from the workpiece model. Before starting the simulation, the initial workpiece shape model is converted to its equivalent triple-dexel model. The following describes our simulation algorithm.
• Repeat the following process for each thin swept volume of the cutting edge moving along the trajectory curve.
Step 1: Convert the cutter swept volume to a model in the triple-dexel representation.
Step 2: Subtract dexels of the cutter swept volume (red dexels in Fig. 9 ) from other dexels of the workpiece model (gray dexels in the same figure). In the following explanation, only X-axis-aligned dexels are used. In the actual processing, the same operations are repeated for other Y-and Z-axis-aligned dexels. As shown in Fig. 2 , the cutting edge has a convex shape. When the time interval t of two successive cutting edges Ce(t) and Ce(t + t) is sufficiently small, a thin swept volume obtained by filling a space between Ce(t) and Ce(t + t) also has a convex shape. For each line of the X-axis-aligned dexel model, the intersection points between the line and all surface elements mentioned above are calculated.
Because the swept volume has a convex shape, up to two intersection points are obtained. A segment connecting the two points becomes a dexel on the line. This operation is iterated for all X-axis-aligned lines, and a dexel model of the cutter swept volume is obtained. By properly using the axis-aligned bounding box of the swept volume, the number of potential Xaxis-aligned lines intersecting the volume can be limited. After obtaining the X-axis-aligned dexel model of the swept volume, this shape is subtracted from the dexel model of the workpiece in Step 2. This process can be simplified to a subtracting operation of a segment from another segment on the same line.
During the computation process mentioned above, a number of intersection point calculations and Boolean subtracting computations of dexels on the same line are executed. These computations on a line are independent of those on the other lines. Thus, an intersection point calculation and a dexel-wise Boolean computation can be conducted in a parallel manner. To implement parallel computation software, the compute unified device architecture (CUDA) is used [25] . Current GPUs are designed to have thousands of streaming processors (SPs) on a single chip. Using CUDA, programmers can utilize a GPU as a general-purpose parallel processor in which each SP executes a unit computation (or thread). In our simulation software, each thread is assigned to the intersection point calculations and Boolean subtracting computations of dexels on a single line, as shown in Fig. 9 . Fig. 10 Results of geometric simulation of power skiving. Fig. 10(a) shows the result of a subtraction operation of the swept volume from the workpiece model with X-axis-aligned dexels. After the subtracting operation, the result of the tripledexel model is polyhedralized using the marching cubes algorithm. The resulting model is given in Fig. 10(b) .
Using the geometric simulation technology mentioned above, we implemented the following two applications. 1. Geometric modeling of the resulting gear shape. 2. Geometric modeling of non-deformed chip shape. In this section, the computation method used in these applications is described.
During the first application, the simulation algorithm described in Section 4 is used to compute the resulting gear shape. In our simulation algorithm, the effect of the cutter feed is not considered, and therefore, the final gear shape cannot be obtained even if this simulation process is executed over a long period. To overcome this limitation, we modified our algorithm such that the simulation process is repeatedly executed while moving the swept volume of the cutting edge in the Z minus direction by the feed amount each time. The following shows our algorithm for visualizing the resulting gear shape.
• Repeat the following process until the resulting gear shape is obtained. Here, i is a counter representing the number of repetitions thus far.
Step 1: For a predetermined time period (from ts to te), compute the positions of the cutting edge per short time interval (t) using Equation (2) . Simulation times ts and te are set such that the cutting edge approaches from the outside of the workpiece, cuts the workpiece, and is sufficiently far from the workpiece after the cutting.
Step 2: The obtained cutting edge position Ce(t) is further translated in the Z-axis minus direction by f i, where f is the feed rate of the cutting tool per rotation of the workpiece.
Step 3: By using the translated cutting edges, the cutter swept volume is computed. The volume is then subtracted from the workpiece model. Fig. 11 illustrates the workpiece shape changing process using our algorithm. The figure on the far right shows the resulting gear shape obtained through our simulation. 
To analyze the cutting force during the power skiving, a non-deformed chip shape generated through the cutting process is necessary. In the power skiving, this shape is obtained as an intersection volume between the workpiece shape and the swept volume of the cutting edge (see Fig. 12 ). We realize this computation by using only the subtraction operation. The following is the algorithm for computing the non-deformed chip shape.
Step 1: The geometric model of the workpiece shape in the middle of the machining is calculated using the resulting shape visualization algorithm mentioned above. Obtained tripledexel model is saved as model S0.
Step 2: Consider a series of swept volumes of the cutting edge finally used in the workpiece shape calculation in Step 1.
This volume is further shifted by the feed amount f in the Z minus direction.
Step 3: A new triple-dexel model S1 is obtained by subtracting the swept volumes obtained in Step 2 from the workpiece model.
Step 4: Subtract model S1 obtained in Step3 from model S0 saved in Step1. The resulting shape S0 -S1 corresponds to the non-deformed chip shape.
During the cutting force analysis, the thickness of the chip shape must be known. The thickness is measured at the position where the cutting edge contacts the chip. We convert the chip shape model in the triple-dexel representation into its equivalent polyhedral model using the marching cube algorithm. For each point p on the cutting edge, a ray is extended in the cross product direction of the tangential direction t and the moving direction f at p (Fig. 13) . The intersection points between the ray and polygons of the chip shape model are obtained. The distance d between the two intersection points nearest p corresponds to the thickness of the chip at point p along the cutting edge. 
A geometric simulation system of power skiving using a triple-dexel representation was implemented. VisualStudio 2010 and CUDA 7.5 are used in the implementation. A 64-bit PC with an Intel Core i7 Processor (2.8 GHz), 32 GB memory, and an nVIDIA GeForce GTX-1050 GPU was used in the experiments. Fig. 14(a) shows the projection of the cutting edge shape to the XY-plane. During the simulation, parameters ωc = 1,394.0 rpm, ωw = 975.8 rpm, rx = 9.0 mm, and θx = 20 degree are used. The developed software aims at geometric simulation of gear shape obtained by power skiving, therefore specification of materials of tools and workpieces is not necessary in the simulation. Fig. 14(b) shows the resulting gear shape. In this study, to obtain highly accurate simulation results, only a limited part of the workpiece shape where the swept volume of the cutting edge can intersect is converted into a triple-dexel model. Tripledexels used to represent the workpiece shape were generated using three square meshes in the XY, YZ, and ZX-planes, which were obtained by projecting a spatial grid with a resolution of 826 × 994 × 2,369. The size of each grid is 0.01048 mm representing the worst value of the simulation accuracy. During this computation, a total of 1042.93 s is necessary. Fig. 15 Simulation result for initial cutting. (a) Workpiece shape before cutting, (b) cutting edges superimposed on the workpiece, (c) computed non-deformed chip shape, and (d) chip thickness analysis result along the cutting edges. Fig. 15, 16, and 17 show non-deformed chip shapes obtained by using our simulation software. Triple-dexel model based on a spatial grid with a resolution of 657 × 1,013 × 2,325 is used in the computation. The size of each grid is 0.00912 mm. Fig.  15 corresponds to the initial cutting operation (rx = 8.0 mm), and Fig.16 and Fig. 17 show the cutting results of the intermediate cutting (rx = 8.5 mm) and the finish cutting (rx = 9.0 mm). In each figure, (a) shows the workpiece shape before the cutting and (b) shows the cutting edges superimposed on the workpiece. (c) shows the obtained non-deformed chip shape viewing from front and back. They are very similar to those described in the papers [1] and [6] , and it is considered that the accurate result is obtained. (d) illustrates computed chip thickness along the cutting edges viewing from the back side of the chip. Red corresponds to a thickness of 0 mm, and blue corresponds to a maximum thickness, which is 0.206 mm for Fig. 15 (d) , 0.101 mm for Fig.16 (d) and 0.109mm for Fig. 17 (d).
In this paper, a novel geometric simulation method of the power skiving was proposed. The resulting shape of the power skiving is computed by subtracting the swept volume of the cutting edge from the solid workpiece model. To realize a robust computation in the subtraction operation, triple-dexel solid modeling is used to represent the workpiece shape. A thin swept volume of the cutting edge is defined as a polyhedral model. It is converted into an equivalent triple-dexel model, and is subtracted from the workpiece model in the triple-dexel representation. To accelerate the conversion and subtraction operations, the parallel processing function of the GPU is used. Using our simulation technology, two applications for assisting the power skiving were implemented. The first application is the resulting gear shape visualization software. In this software, the simulation process is repeatedly executed while moving the swept volume of the cutting edge in the Z minus direction by the feed amount each time. Another application is for computing the non-deformed chip shape. In this application, the geometric model of the workpiece shape in the middle of the machining is calculated. The intersection shape between a swept volume of the cutting edge and the workpiece shape corresponds to the non-deformed chip shape. An experimental simulation system was implemented, and some computational experiments were conducted.
In the proposed method, triple-dexel model is converted into an equivalent polyhedral model when the thickness of the chip shape is evaluated. Marching cube algorithm is used in the conversion. This method ignores very thin shapes in constructing the polyhedron. In order to solve this problem, we are currently developing a software that performs direct thickness evaluation using a triple-dexel model.