Cusp Modeling for 5-Axis Surface Machining Paul J. Gray

pjgray@mecheng1.uwaterloo.ca Fathy Ismail

fismail@mecheng1.uwaterloo.ca Sanjeev Bedi

sbedi@mecheng1.uwaterloo.ca Department of Mechanical Engineering

University of Waterloo Waterloo, Ontario, Canada N2L 3G1

001 (519) 888-4567

ABSTRACT Gouge checking and gouge measuring are essential parts of 5-axis tool-positioning strategies. To evaluate the quality of a tool path (i.e. the remaining material, commonly referred to as cusps), machining tests must usually be performed. If the tests prove to be unsatisfactory, new tool paths must be generated and run. In this paper, a new approach for modeling cusps is presented which makes use of recent advances in vector algebraic computation of swept volumes. The method uses the computer’s graphics card to compute the cusp geometry from the swept volumes. The algorithm was verified with a machining test by measuring the cusp height between two adjacent passes with a Coordinate Measuring Machine and comparing it to the modeled cusp height.

Categories and Subject Descriptors J.6 [Computer Applications]: Computer-Aided Engineering - Computer-Aided Design (CAD).

General Terms Algorithms, Measurement, Verification.

machining, modeling the surface that will be generated by a tool path is more difficult than for 3-axis. The simplicity of 3-axis machining allows for the development of simplified algorithms to compute and visualize the surface resulting from a tool path. Unfortunately, these simplifications cannot easily be generalized to 5-axis tool motion. Gouge checking and measuring for 5-axis tool paths has been a widely researched area [13]. However, gouge checking only measures the error of a tool path and does not measure its quality. To determine the quality, researchers usually perform a machining test to measure cusp heights (the material remaining on the workpiece between adjacent passes of the tool). If the test proves to be unsatisfactory, the researcher must compute new tool paths by adding more tool passes and perform a new machining test. This iterative process can be time consuming and expensive. If the resulting machined surface could be modeled before running a machining test, the speed of this iterative process could be improved. The approach presented in this paper generates a triangulated model of the machined surface from the swept volume of a 5-axis tool path using the computer’s graphics hardware.

ackground information and a literature review are presented in

B Keywords Swept volumes, Z-buffer, 5-Axis machining, Cusp modeling, Surface fitting.

1. INTRODUCTION Surface machining is an important part of the Tool and Die industry. Though the bulk of surface machining is still performed in 3-axis with ballnose tools, the use of 5-axis machines is growing. The additional rotation axes of a 5-axis machine has permitted the development of optimized tool positioning strategies that bring more of the tool's cutting surface closer to the design surface. This allows wider strips, which satisfy surface finish criteria, to be machined with each pass of the tool. Ultimately the goal is to reduce the total tool path length enough to make significant gains over 3-axis machining [2]. Naturally in 5-axis

the next section covering NC verification. The computation of swept volumes is also discussed since it is the basis upon which the cusp modeling algorithm is constructed. A brief look at computer graphics and their application to NC verification and evaluation is presented in the following subsections. In Section 2 the graphic-assisted cusp modeling algorithm is presented. Subsection 2.3 presents a simulation example and Section 3 presents the results of a machining experiment designed to verify the cusp modeling algorithm. Finally, conclusions and a discussion of future improvements are presented.

1.1 Solid Modeling Solid modeling has been recognized for sometime as a promising contender for NC verification software. However, even with todays computing performance and memory, achieving accurate results is slow. Performing a Boolean operation for each tool position in the tool path with a model of the workpiece is an expensive endeavour for surface machining which typically contains upwards of thousands of tool positions for any reasonably sized part [13]. For this reason solid modeling remains more useful for CAD applications than for NC verification

1.2 Polygon Clipping Like solid modeling, polygon clipping is also a view independent strategy. In polygon clipping, the tool path is interpolated in small linear steps between tool positions. The surface swept between the

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steps is clipped to the polygons of the model surface and the model surface is updated with the new polygons. Usually, an Octree hierarchical data structure is used that recursively subdivides the model space and keeps track of what objects are in each subdivision. Intersections are found by traversing the tree nodes where the tool is occupying space. Roy and Xu [11] implemented such a data structure for 3-axis tool path simulation. Generally, computation times for such approaches are significantly higher than for graphics-assisted ray clipping algorithms and their data structures are difficult to implement. Ultimately accuracy is usually sacrificed for computation speed.

1.3 Z-Map Ray Casting The most common visualization method is the simple z-map. Z-maps are graphics-assisted ray casting consisting of parallel rays that are grown from the model surface along the viewing direction (Figure 1) and are intersected with numerous static instances of the cutting tool. This is easy to implement and fast to compute because they are implemented with the depth buffer of the graphics hardware of the computer. The only disadvantage is that it is view dependent. To view the final image in different orientations, the tool path must be re-simulated from the beginning. The popularity of this approach can be attributed to its simplicity and its speed of computation since it can be implemented in hardware using the computer's depth buffer for the ray intersections.

Figure 1 Z-map and surface normal vectors

Wang and Wang [15] implemented such an approach to compare a polyhedral approximation of the swept volume of the tool path with the model surface. Van Hook [3] uses the depth buffer to perform Boolean subtractions between discrete tool positions and the model surface instead of using the swept volumes. However, Van Hook uses lists of dexels for the operation, which are not supported by hardware. Kim et al. [4] implemented a z-map method without graphics hardware acceleration. The computation speed of their approach could probably be much improved with the use of the computer's depth buffer.

1.4 Offset Surface A more accurate approach than the z-map would be to cast rays that extend along the surface normals (Figure 1) and intersect them with the tool [13]. It can also be thought of as generating an offset surface and intersecting it with the tool path. Chappel [17] developed this strategy and the equations for a randomly oriented cylindrical tool to check for gouging. Unfortunately this approach cannot be hardware accelerated since the rays are cast in a variety of different directions. This makes it difficult to determine which

of the rays grown from the surface will intersect each tool instance leading to computationally expensive search strategies. The high computation cost of this strategy often leads to the use of sparse ray densities in order to maintain reasonable computation times. Most implementations only use discrete tool positions to generate the machined surface. To generate the surface between tool positions usually the software generates several linearly interpolated instances of the tool between each tool position to generate a simple swept volume. The approach proposed in this paper takes advantage of the latest developments in the generation of swept volumes for NC simulation discussed in the next section.

1.5 Swept Volumes Swept volumes is a useful tool for computing the surface left by a cutting tool as it traverses a 5-axis tool path. Wang et al. [14] developed a method to compute the boundary points for the swept volume of a 7-parameter APT tool moving in 5-axis motion using the Sweep-Envelope Differential Equation method (SEDE) for NC verification. Their work is an implementation of SEDE originally developed by Blackmore et al. [1]. Until the developments by Sheltami et al.[12], Roth et al.[10], and Mann et al. [7], swept volumes computations have been mired with complex differential equations that require numerical methods to solve. A new approach developed by Roth et al. [9][10] eliminates the use of the complicated SEDE equations and replaces it by directly computing points on the swept volume using vector algebra. Though the algorithm can only compute discrete points on the swept volume, it is still useful as the points can easily be connected to create a triangulated definition of the swept volume for NC verification. The main benefit is that it is simple to compute and any number of points can be computed for the swept volume. Also, Mann and Bedi [7] have shown that the algorithm can be generalized to any surface of revolution. Grazing points are discrete points that the surface of the cutting tool will leave behind as it moves through space. To compute the grazing points Roth et al. discretize a toroidal cutting tool into a finite number of cross-sections through its axis of revolution. The sections are called pseudo-inserts (Figure 2). A grazing point is computed for each pseudo-insert of a given tool position. The algorithm works as follows:

1. The direction vector ( dr

) that connects the pseudo-insert centre of the current tool position to the next tool position for each pseudo-insert is computed. The tool is not considered to be rotating about its axis as it would when it is actually cutting metal.

2. The vector that is perpendicular to the pseudo-insert plane is ( nr ).

3. A grazing point ( pqmr ) is given as the intersection of the

cross-product of dr

and nr with the pseudo-insert’s circular edge (Eqn. 1).

Similar to Sheltami et al.'s work [12] Roth et al. divide the grazing curves (the collection of grazing points for each tool position) into two regions: the front and the back halves. The dividing points are called demarcation points which are at the pseudo-inserts whose planes are perpendicular to the instantaneous motion vector of the current tool’s torus centre point ( f

r in Figure 2). Separate front

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and back swept volume surface are generated by triangulating the grazing points of adjacent tool positions. This algorithm is simple to implement and fast to compute. The accuracy is mainly dependent on the number of pseudo-inserts the tool is discretized with. An important point to note is that the algorithm assumes linearly interpolated motion of the tool axis between two subsequent tool positions. For small forward step distances and small orientation changes in the tool axis, this assumption will be valid.

Figure 2 Swept volumes cross-product method

( )pq = TGrazingPoint InsertRadius d n+ × ⊗rr r Eqn. 1

1.6 Cusp Modeling Cusp modeling has taken a backseat to gouge detection and measurement research. The weakness of most z-map approaches is that they are usually view dependent. The cusp modeling algorithm presented in this paper is designed to address the view-dependence problem. The geometric swept volumes computation by Roth et al. [10] discussed above is used instead of interpolations between discrete tool positions. Lee and Chang [5] developed a method to compute the cusp height between adjacent tool passes using differential geometry. The cutting shape of two adjacent tool positions is assumed to be the ellipse that is projected onto the plane that is perpendicular to the tool's forward direction of motion. A plane containing the two cutter contact points of the two tool positions and containing the surface normal at one of the cutter contact points is established. The two cutting shapes are projected onto this plane and their intersection is found. This is considered to be the cusp peak. The weakness of this method is the double projection of the cutting shape. The first projection of the cutting shape does not take into account any orientation change in the tool's axis. The second projection simply adds to this error. The planar ellipsoid cutting shape is not the true cross-sectional shape that is left behind by the cutting tool as it moves through space. This is only true for a linear tool path without any orientation changes in the tool axis. In simultaneous 5-axis surface machining it is possible that the sides of the tool that make up the outer edges of the ellipsoid will have moved in the following tool position far from the ellipsoid drawn at the current cutter contact point. For this reason swept volumes were chosen to model the cusp in the current work. The swept volumes generated by Roth et al. [10] do not show the actual surface that will remain after machining because grazing points generated in the tool path may actually be cut away by

other tool positions in the pass or by adjacent passes of the tool. A graphics-assisted algorithm to model the machined surface is proposed in the next section.

2. Graphics-Assisted NC Verification and Evaluation Using the computer's graphics hardware can simplify complicated geometric computations and speed up computation times. The sections below explain the operation of the depth buffer for computer graphics and how it is harnessed in this work to generate a cusp model of the machined surface.

2.1 Proposed Use of Computer Graphics for Cusp Modeling The idea is to render a triangulation of all the swept volumes for the whole tool path and, using the depth buffer, only render the grazing points that are closest to the model surface. This can be done using a depth test typically used to eliminate the rendering of hidden surfaces. It is computed on a per-pixel basis by comparing each object's depth buffer value for each pixel and only rendering the objects that are closest to the eye position. Instead, for cusp modeling, the depth test is set to render only the points that are furthest from the eye coordinate since the eye is positioned above the whole surface in the model's z-axis direction. To remove the view dependence, a backwards transformation of the rendered pixel coordinates from the image space to Euclidean space is performed. With the viewing direction being parallel to the model's z-axis, the transformation is reduced simply to a translation offset and a scaling factor. The view window's pixel coordinates and depth buffer correspond to the model's x, y and z axes respectively. The backwards-transformed coordinates are then triangulated and rendered to the screen. They can now be dynamically rotated and zoomed. What has been achieved is that the cusp, which is the intersection of swept volumes of adjacent passes of the tool, has been computed indirectly with the depth buffer by only rendering the points that are on the material side of the swept volumes. This eliminates the grazing points that are cut away by successive tool positions along the tool paths and by adjacent tool paths. All this is performed with the graphics hardware. The accuracy is controlled by the number of pseudo-inserts, the viewing volume and window sizes, and to some degree, the view direction.

2.2 Cusp Height Measurement The proximity of the machined surface model to the actual model surface can also be measured using the computer's graphics hardware. If the view direction is set along the surface normal at a point on the surface, then the distance between the machined surface and the point on the surface can be measured directly with the depth buffer. Measuring the surface in the vicinity of the cusp will give the cusp height. We propose to measure the maximum cusp height by bucketing the surface into a fine grid in the vicinity of the cusp and measuring the maximum distance between the model surface and the machined surface in each bucket using the depth buffer. The surface normal of the centre point of each bucket determines the bucket’s view direction. In our implementation we used functional parametric surfaces with parallel tool passes. Dividing a functional parametric surface into a grid is simple since the

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parametric space is a direct ratio to Euclidean space. Furthermore, the parallel tool passes simplified defining the cusp area on the surface. To measure the distance between the surface point and the machined surface for a bucket, the eye is positioned at the bucket’s centre point and offset along the surface normal by half the viewing volume depth. The viewing direction is towards the model surface. First the model surface is rendered in green. The depth buffer and the green colour buffers are read. Then the machined surface is rendered alone using the same viewing volume and direction. The depth buffer and colour buffers are read again. For each pixel that has the model surface and the machined surface rendered to it, the depth buffers are compared. The maximum depth from the depth buffer will be the largest distance between the surface model and the machined surface in the bucket region. The maximum measured depth is stored for the buckets and the process repeats itself for the entire bucketed surface region to compute the maximum cusp height. Naturally, the accuracy of this approach will depend on how small the buckets are. In the following section a simulation test was performed which demonstrates the results of the machined surface modeling and gives some computation times.

2.3 Cusp Surface Examples Figure 3 shows the example test surface. The cutting tool used is a 38.1mm outer diameter, 3-insert toroidal tool with 12.7mm diameter inserts. The swept volumes were generated using 100 pseudo-inserts. Using a 32 Mb NVidia GeForce 2 Go Mobile graphics card with a 32 bit depth buffer and a view depth of 100mm, the cusp model surface in Figure 4, generated with a 1000x1000 pixel view window, was computed in 3.345 seconds. Unfortunately it is difficult to see the cusps with a printed figure of the model. Computing the same simulation with a 300x300 pixel view window took only 0.301 seconds. The computation time increases almost linearly with the number of pixels of the viewing window. The main speed advantage of this algorithm comes from the fact that the swept volumes for the whole tool path are rendered only once to compute the machined surface instead of having to intersect each tool position individually.

Figure 3 Example surface

Cusps

Figure 4 Example cusp surface from 1000x1000 pixel view

An experimental evaluation of the graphics-assisted algorithm is presented in the following section.

3. Experimental Evaluation The experiment was designed to evaluate the graphics-assisted cusp modeling algorithm's ability to measure cusp heights. Because it is difficult to measure curved surfaces, a quadratic convex test surface curved in only one direction was used to simplify measuring. Figure 5 shows the surface and grazing curves for a single tool pass. The first pass was generated with a surface that is about 0.5mm smaller than the model surface to generate a asymmetric cusp. The second pass was computed with the correct sized model surface. The cutting tool was a 3-insert toroidal tool with an outer diameter of 38.1mm and an insert diameter of 12.7mm. The cusp size was computed with the graphics-assisted cusp modeling algorithm. To verify the simulations, the workpiece was then measured with a Mitutoyo BHN 305 coordinate measuring machine. Figure 6 shows a measured cross-section of the workpiece. The cusp height measured from the deeper tool pass (left-hand side pass of Figure 6) is 1.112mm.

Figure 5 Model surface and grazing curves

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Figure 6 CMM measured cross-section of tool passes

3.1 Cusp Height Simulation Results The cusp surface was computed using the graphics-assisted algorithm outlined earlier with a viewing window of 300×300 pixels and a view depth of 100mm. The swept volumes for the passes were generated using 100 pseudo-inserts. Figure 7 and Figure 8 show the cusp model results. The computation time was 0.301 seconds with the above mentioned computer system. The deviations of the cusp model at the start and end of the tool passes in Figure 7 and Figure 8 are due to the swept volume of the tool moving in the vertical direction. This is not really significant since the tool pass starts and ends outside of the workpiece and therefore, these tool positions are not actually cutting any material. They could be removed by clipping the triangles to the actual edge boundary of the workpiece.

Tool path start

Tool path end

Figure 7 Cusp model of test surface

Tool path start Tool path end

Figure 8 Cusp model of test surface

To validate the cusp height computation, the top-most cross-section line across the two passes was measured with the CMM. The data was then compared to the cusp height computed for the cross-section using the simulation. The relative height of the cusp measured from the lowest valley of the workpiece was compared to the simulation. The CMM measured cusp height of 1.112mm compares favourably with the simulated measurement of 1.1825mm. The results show a difference of 0.0705mm giving an error of 6.34%. The CMM was set to measure points along the cross-section in steps of 0.005mm. It is possible that the exact peak of the cusp fell between two consecutive measured points. This would mean that the CMM would measure a smaller cusp than actually exists.

4. Conclusions and Recommendations An algorithm was developed to visualize the cusp generated between adjacent 5-axis tool passes over the surface using swept volumes. The algorithm is fast and simple to compute. To evaluate the quality of a tool path the cusp height between adjacent passes of the tool was computed by comparing the machined cusp surface to the model surface using the computer's depth buffer. A machining experiment was designed to evaluate the graphics-assisted cusp modeling. The first of the two adjacent passes was designed to gouge the model surface by 0.5mm to generate an asymmetric cusp. The cusp height was estimated to be 1.1826mm using the cusp model simulation. The workpiece was measured using a CMM and the measured data was compared to the simulated data. The measured results show a cusp height of 1.112mm giving an error of 6.34%. The small difference could be attributed to the limitations in measuring surfaces using the CMM. The implementation for cusp height measuring should be expanded to compute the cusp height anywhere along a cusp for any number of cusps generated by the complete tool path. The algorithm would have to be adapted to allow the user to select which cusp and what region of the cusp is to be inspected. The swept volumes that generate the selected cusp would have to be identified and rendered alone. This would not be very difficult to do; a machined surface model for each set of adjacent passes of the tool could be computed and stored. The user could then select from the list which cusp to analyze and then identify what region of the cusp to analyze.

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5. Acknowledgements This research was funded by the Natural Sciences and Engineering Research Council of Canada, by Ontario Innovation Trust, and the Canadian Foundation for Innovation.

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