Int J Adv Manuf Technol https://doi.org/10.1007/s00170-017-1263-4
ORIGINAL ARTICLE
Cutter partition-based tool orientation optimization for gouge avoidance in five-axis machining Xiyan Li 1 & Chen-Han Lee 1,2 & Pengcheng Hu 1 & Yang Zhang 1 & Fangzhao Yang 1
Received: 2 June 2017 / Accepted: 24 October 2017 # Springer-Verlag London Ltd. 2017
Abstract This paper presents a cutter partition-based tool orientation optimization algorithm that can identify and eliminate both local and rear gouging in five-axis machining. This algorithm is developed to generate smooth gouge-free tool paths for arbitrary free-form surfaces machined by general Automatically Programmed Tools (APT) cutters. The algorithm involves an iterative process of cutter projection to compute the gouge-free cutter positions. During each iteration, the cutter partition method is used to classify the cutting portion of the cutter, which in turn determines the gouging condition. Based on the classification, the algorithm applies an optimal gouge-avoidance strategy to minimize the size of tool orientation and position changes. The proposed algorithm can handle the combination of local and rear gouging (either separately or simultaneously), guarantee front edge cutting (avoiding damage to the cutter and surface), ensure a safety clearance between the rear edge of the cutter and the machined surface, and achieve minimal interruption to tool orientation and position from gouge avoidance. Simulation and cutting experiments confirm the effectiveness of the proposed algorithm.
Keywords Gouge avoidance . Five-axis machining . Cutter projection . Cutter partition . Tool orientation optimization
* Chen-Han Lee
[email protected]
1
Huazhong University of Science and Technology, Wuhan, China
2
Wuhan Institute of Technology, Wuhan, China
1 Introduction There are known advantages of five-axis machining owing to the two rotary degrees of freedom of the tool axes. The tool orientations can be allowed to change to ensure optimal tool accessibility and material removal rate (wider machining strip width) [1]. Five-axis machining can also guarantee an effective cutting profile and good machining quality [2]. A side effect of two additional rotary degrees of freedom is the higher possibility of cutter interference with the workpiece. In general, there are two types of cutter interference: gouging and collision. Gouging means that the cutter cuts into the workpiece deeper than the expected part geometry shape. It happens between the cutting portion of the cutter and the workpiece, and the consequence is overcutting the material [3]. The rest of interference is considered collision. When collision happens, it means that a portion of the cutter is making an unintended contact with the workpiece or check geometry and the consequence is severe, usually involving equipment damages. The conditions of unintended contact include the contact with the noncutting portion of the cutter (such as the upper shank) and the contact while the cutter is making a rapid movement. The scope of this paper is limited to solving gouging. Though not as severe as collision, gouging affects the tool life, machining accuracy, and surface quality [4]. As shown in Fig. 1, gouging includes local gouging, rear gouging, and the combination of local and rear gouging. Local gouging (Fig. 1a), which occurred in the vicinity of the cutter contact (CC) point at the front edge of the cutter, is usually due to the mismatch in curvatures between the tool swept surface and the part surface at the CC point [5]. Rear gouging (Fig. 1b) is caused by material removal with rear edge of the cutter. In the region of the high curvature, the combination of the local and rear gouging may occur, as shown in Fig. 1c.
Int J Adv Manuf Technol Fig. 1 The categories of gouging
Tool axis
Tool axis
Tool axis
Cutter
Cutter
Cutter Part surface
Feed direction
Part surface
Feed direction
Feed direction
(b) Rear gouging
(a) Local gouging
The classification of local and rear gouging is based on the location of material removal being at the front (near CC point) or rear of the cutter. The definition of front/rear is based on the feed direction of the cutter, as shown in Fig. 2. Front portion means the side of the cutter facing the feed direction while rear means the rest of the cutter outside of the front portion. Extensive works on the elimination of interference by either lifting or tilting the tool (change the lead and tilt angle or the inclination and screw angle, as shown in Fig. 3) in fiveaxis machining have been reviewed [6, 7]. There are numerous methods in previous works, including the surface properties analysis-based method [5, 8, 9], the convex hull-based method [4], the C-space-based method [1, 3, 10–14], the visibility and accessibility-based method [2, 15–17], the bounding volume and space partition method [18, 19], the distance calculation (vector)-based method [20–24], RBM&AIM [25–28], the radial projection [29, 30], the graphic-assisted method [31–33], the sweep plane method [34, 35], the bisection search method [36–38], the angleadjustment method [39], and offset CL-surface method [10, 40–42]. Junet al. [1] presented the C-space method to detect the interference and smooth the tool orientation in five-axis machining considering the local and rear gouging and global collision. This algorithm is limited to ball-end cutters and does not guarantee front edge cutting. Wang and Tang [2] presented an algorithm that can identify the set of valid orientations by the construction of discretized
Part surface
(c) Combination of local and rear gouging
visibility map (VMap) and automatically generate a five-axis machining tool path under various constraints. However, it does not provide a solution when the gouge-free VMap is empty. In addition, there are potential problems of drastic swing of tool orientation and cutting with the rear of the cutter. The study in Lee [3] involved a three-stage approach: the first stage to find a feasible orientation, the second to update the tool orientation, and the final stage to identify all candidate points and to update the tool orientation if potential gouging exists. This algorithm is effective only if the surface can be circular-approximated in the second stage, and the performance of rear gouging avoidance depends on the number of discrete checking points in the third stage. Rao and Sarma [5] investigated local gouging with an exact method in five-axis machining of C2 sculptured surfaces (which may be joined by C1 or C0 seams) using flat-end tools. The tool orientation can be changed to eliminate the local gouging while ensuring faster material removal rates and increased machining accuracy. However, rear gouging is not taken into consideration and this algorithm is only suitable for the flat-end cutter. Lee and Chang [4] employed a two-phase approach using the convex hull of the control mesh in the quick feasibility checking module to achieve tool interference checking and feasible tool orientation planning for fiveaxis machining. However, this approach does not allow a general tool and the gouging category is not identified to guarantee front edge cutting.
Fig. 2 Definition of the cutting portion
Part surface
Feed direction
(a) Front edge cutting
Feed direction
Part surface
(b) Rear edge cutting
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Feed direction Surface normal Lead angle ∅ Tilt angle Inclination angle Screw angle Cutter contact point Tool orientation
Cutter
∅
Fig. 3 The definition of the tool orientation with different angles
Gouge detection and avoidance in five-axis NC machining of sculptured surfaces with a filleted-end cutter is proposed in [8]. An optimal cutter orientation is first determined by matching the instantaneous cutting profile of the cutter and the machined surface as close as possible to avoid local gouging. Rear gouge detection is identified and handled afterward. This algorithm cannot solve the combination of local and rear gouging, cannot guarantee front edge cutting, and may cause sudden change of the tool orientation. You and Chu [20] presented a systematic scheme for the verification of tool paths in five-axis machining of sculptured surfaces based on the subdivision of the part surface into sampling grid points. Since the interference detection is conducted only at discrete sample points, it is possible that the cutter invades the surface between the sample points. Jensen et al. [21] proposed algorithms for detection and correction of the local tool gouging by calculating the shortest distance between the cutter bottom and the part surface. Zhang et al. [22] subdivide the cutter bottom into a number of circles, then checked the intersection between the circles and the designed surface to detect gouging, and adjusted accordingly. Rear gouging cannot be identified and eliminated in these two algorithms. Kiswanto et al. [23] presented a method to eliminate gouging during faceted-model tool-path generation in five-axis milling based on a tool lifting. As shown in Fig. 4, lifting the tool to barely touch the gouged object at a high curvature region can result in a back motion of the cutting tool (Fig. 4 in blue) that drastically alters the location of the cutter, causing unexpected problems. Hansen and Arbab [24] presented the projection method to generate gouging-free tool paths in fixed three-axis machining. The problem of positioning a tool to a complex parametric surface and performing global interference checking is simplified to the problem of positioning a tool to facet models. This method ensures gouge-free between the cutter and the surface. This paper only covers three-axis machining.
Tool lifting
Feed direction Surface
Fig. 4 Gouging treatment in a high curvature surface
Rolling ball method (RBM) and arc intersect method (AIM) are two powerful tool orientation techniques for local and rear gouging avoidance. Gray et al. [25] considered the approximate curvature of the surface in the vicinity of the CC point. Local gouging checking is automatically built-in based on the approximate curvature. A graphics hardware-assisted approach to five-axis surface machining is further presented [26]. These methods have tolerancing issues due to the approximation. In AIM, the same shadow checking area and grid points in RBM are used, the surface under the tool shadow is rotated to intersect the surface [27]. This algorithm is an areabased method where the smallest resulting angle (equivalent to the largest tilt angle) is selected as the gouge-free orientation. Hosseinkhani et al. [28] introduced a penetrationelimination method (PEM) for five-axis machining and developed a quantitative definition for the gouging concept. Table 1 (yes/no)
Comparison of algorithms for gouge detection and avoidance
References Identify and Cut remove rear with gouging front edge
Solve Smooth Satisfy a gouging tool general combination orientations cutter
[1] [2, 11] [3, 8, 36] [5]
Yes No Yes No
No No No No
No Yes No No
Yes Yes No No
No Yes No No
[4, 10, 12, 15, 25–28] [9, 20] [21, 22, 24] [23] [39]
No
No
Yes
No
No
No No
No No
No Yes
No No
Yes Yes
Yes No
No Yes
Yes Yes
No No
No No
Int J Adv Manuf Technol
Input one original cutter position
Step1: Execute the cutter projection to compute an intermediate position
Step5: Tilt the tool backward with the clearance angle to obtain a new position , make
Step2: Carry out the cutter-partition classification
Step3: Is front edge cutting
N
Step6: Activate the tool orientation optimization loop
Y Y
Step4: Is the first iteration
Step7: Tilt the tool forward with the clearance angle
N
Step8: Perform Step1 to obtain a position ,make
Output a new cutter position Fig. 5 The flow chart of the proposed algorithm
However, these methods [27, 28] cannot adapt to the general APT cutter and ensure front edge cutting. Li and Feng [36] employed the bisection search method to distinguish and avoid local and rear gouging of flat-end mills in five-axis surface machining. However, the algorithm is only applicable to low curvature surfaces. Li and Jerard [39] presented an algorithm to detect and eliminate different types of gouging problems. Tool-position correction, including heelclearance checking for gouge avoidance is carried out. The tool is adjusted to contact the surface at the CC point while lifting the heel of the tool to avoid gouging. If the tool position cannot be adjusted by tilting, the tool is lifted along the toolaxis direction. Only the flat-end cutter is used and retract movements may occur in this algorithm.
Offset CL-surface method, also known as C-space method, was first proposed by Sato and Takahashi [40] and then by Choi, Kim, and Jerard [10] and other authors [41, 42]. This class of methods can only apply to fixed-axis or variable-axis with ball-end cutters. That is, they will not work for five-axis or four-axis machining with an arbitrary non-ball-end APT cutter, such as a toroidal cutter. In addition, the offset CLsurface is not analytical and it requires numerical computation based on subdivision of the original part surface, either a facet model [10] or G-buffer model [38]. Hence, these methods have inherited aliasing problems. According to the above analysis, most of those gougechecking and avoidance algorithms have at least one of the following limitations shown in Table 1:
Lift Lift Feed direction
Feed direction Projected cutter
New CC
Projected cutter
Local gouging
Part surface
(a) Front edge cutting and local gouging Fig. 6 The cutting portion and gouge category
Projection direction
Rear New CC gouging
(b) Rear edge cutting and rear gouging
Tilt
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Tilt backward
Fig. 7 Tool orientation adjustment with the clearance angle
Clearance angle
Tilt forward
Clearance angle Feed direction
Safety distance
Feed direction
Close
Part surface
(a) Checking for the new cutter position (1) Some algorithms cannot identify and remove both the local and the rear gouging. (2) Some algorithms cannot effectively handle the combination of local and rear gouging. (3) Some algorithms may produce drastic change of tool orientation to avoid gouging. (4) Some algorithms cannot guarantee cutting with the front cutting portion of the cutter. These may cause cutting with the rear end of the cutter, causing cutter damage and/or cut marks on the machined surface. (5) Some algorithms are limited to certain types of cutters or surfaces.
In this paper, we introduce a robust cutter partition-based tool orientation optimization for gouge avoidance in five-axis machining. The algorithm identifies and eliminates both local and rear gouging, guarantees cutting with front edge of the cutter, ensures the smooth variation of the tool orientations, and applies to arbitrary free-form surfaces machined by a general APT cutter. In the next section, the overview of the cutter partitionbased tool orientation optimization algorithm for gouge avoidance is presented. Section 3 describes the details of the algorithm, including the cutter projection along tool-axis direction, the cutter partition-based gouge classification, and tool
Part surface
(b) Remedy for the new cutter position
orientation optimization for gouge avoidance. In Section 4, implementation and experiments with blades as test cases are presented. Conclusions and future work are provided in Section 5.
2 Overview of the algorithm The input of the proposed gouge avoidance algorithm is predetermined CC curves generated using the iso-parametric method. The CC curves are sampled by CC points and each CC point is given a pre-optimized tool orientation. The initial tool orientation guarantees that the CC point is at the front of the cutter to ensure front cutting. The pre-optimization also ensures that the variation of the tool orientation is smooth (no abrupt changes to tool orientation). There is no constraint on the shape of the cutter, other than it is an APT cutter. To accomplish gouge avoidance for each initial CC point, the algorithm employs an iterative process to handle local and rear gouging and possible combination of them. During the first iteration, we apply the cutter projection along the toolaxis direction to clear the gouge, if there is any. This is the most effective method to handle local gouging–it does not alter the pre-optimized tool orientation and the change of CC point is minimal, since the amount of local gouge is usually minimal.
Iso-parametric CC curve
Tool axis
Drive point
Part surface
Cutter
Fig. 8 The cutter projection along the tool-axis direction
Part surface
Fig. 9 Computation of an intermediate gouge-free cutter position
Int J Adv Manuf Technol Fig. 10 Cutting portion definition of APT cutter
Z
Y
Torus section
Upper cone section
X CC region
Lower cone section
X
(a)
(b) Cutting portions of APT cutter
General APT cutter
However, if the gouging is at the rear, the above projection method leaves the contact at the rear that must be avoided. As the remedy, we develop cutter partition method to classify the cutting portion of the cutter based on the projection result (details will be provided in Section 3). When the rear cutting is detected, we keep the original CC point (before the projection) and perform the tool orientation optimization instead. In the tool orientation optimization process, the lead angle (the forward-leaning angle) of the tool orientation is optimized using the combination of linear and bisection search methods while keeping the tilt angle unchanged, until cutting with the front edge of the cutter is secured. As a result, the tool is rotated around the original CC point towards the feed direction, until the rear of the tool is cleared by a specified clearance angle from the part surface. In the first iteration, either local or rear gouging is handled accordingly. However, the clearance of rear gouging may still leave residual local gouging and local gouging avoidance may not handle rear gouging properly. To be robust, the iteration is repeated again to ensure there is no local or rear gouging. The flow chart in Fig. 5 shows the procedure of the proposed algorithm, starting with the input original cutter position Pori (CCori, CLori, TAori), that contains the original CC point, CL point, and tool-axis (orientation). Step 1: Take Pori (CCori, CLori, TAori) as the initial position, execute the cutter projection along tool axis to obtain an intermediate gouge-free cutter position, Pint (CCint, CLint, Tnew). Identify the cutting condition
Step 2:
Step 3: Step 4:
Step 5:
Step 6:
with the displacement vector, VCL = CLint − CLori. Details will be presented in Section 3.1. Carry out the cutter partition-based classification to identify the cutting portion (front or rear portion) of the new cutter position and the corresponding gouging category (local or rear gouging). Details will be presented in Section 3.2. If the front edge cutting (local gouging for the original cutter position) is detected, shown in Fig. 6a, turn to Step 4; else turn to Step 5. If it is the first iteration, turn to Step 5; else turn to Step 6. The reason for performing Step 5 (see below) is to make sure that the clearance between the heel (rear) of the cutter and the workpiece is at least the size of a pre-determined clearance angle. Tilt the tool with the clearance angle γ = arctan(dC/D) towards the reversal of the feed direction shown in Fig. 7a to obtain a new position, Pori. dC is the safe clearance and D is the diameter of the cutter. Make Pnew = Pint and turn to Step 1. The backward-tilted position, Pori, will be checked again in the next iteration to make sure there is a rear clearance. The position before backward tilt, Pnew = Pint, is saved for output in the second iteration, if it is proven to be safe. When the rear edge cutting (of the original position) is detected, shown in Fig. 6b, the tool orientation optimization loop will be activated, to be described in Section 3.3. The tool is repeatedly tilted around the original CC point CCori until a new contact
Fig. 11 The cutter partition
α Part surface Rear cutting portion
(a) Parameter definition
Front cutting portion
(b) Top view of the cutter
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Z Y Cutter
Part surface
O
X
Fig. 14 Elimination of the rear gouging by rotating the tool axis
Fig. 12 Definition of the position checking vector
location in the front cutting portion is achieved, and turn to Step 7. Step 7: To ensure a safe clearance from the workpiece at the rear of the cutter, this step tilts the tool forward, towards the feed direction shown in Fig. 7b, with the clearance angle. The previous step ensures no rear gouging while this step adds a safe clearance. Turn to Step 8. Step 8: After the previous two steps take care of rear gouging and clearance, this final step performs the projection again (similar to Step 1) to take care of possible residual front gouging (if there is any) before output the final position, Pnew = Pint. This step makes sure that the combination of rear and front gouging is handled properly.
Begin
The cutter partition-based tool orientation optimization for gouge avoidance mainly includes three algorithms: the cutter projection along tool-axis direction, the cutter partition-based gouge classification, and the tool orientation optimization. Details will be presented in Section 3.
3 Details of the algorithm In the master flowchart described in the previous section, the Steps 1, 2, and 6 are more complex and require additional explanation. The following subsections provide more details. In Section 3.1, the cutter projection along tool-axis direction for local gouge clearance in Step 1 is explained. In Section 3.2, cutter-partition classification to distinguish between local and rear gouging in Step 2 is developed. Section 3.3 focus on the tool orientation optimization for rear gouge handling.
Input the CC point, the tool axis , the feed direction , the CL point
Calculate the cutter partition vector Compute the cutter position checking vector and parameter
Identify the cutting portion and the gouging category ∅
End Fig. 13 The flowchart of the cutting portion detection
Fig. 15 Definition of the tool axis in the local coordinate system
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displacement vector is zero, |VCL| = 0, and the cutting condition is contact.
3.2 Cutter partition gouge classification
Surface Fig. 16 The linear search method for finding the optimal tool axis
3.1 Cutter projection for gouge detection From the pre-generated CC path, every drive point is produced and the cutter projection along tool axis direction is executed to compute the new gouge-free cutter position. The displacement vector from the original cutter position to the intermediate gouge-free cutter position is used to identify the cutting and gouging conditions. The principle of the projection method in is shown in Fig. 8. The drive point is the original CC position on the part surface from iso-parametric curve on the part surface. The cutter projection along any pre-determined direction has been comprehensively implemented in [43]. This method can be used to calculate the intermediate gouge-free cutter position for each original cutter position on tool path. The displacement vector from the original to the new cutter position is shown in Fig. 9. CCori is the CC point of the original position, and CLori is the corresponding CL point. Similarly, CCint is the intermediate gouge-free CC point obtained as a result of the cutter projection, and CLint is the corresponding CL point. TA is the unit vector of the tool axis, VCL is the displacement vector determined by CLori and CLint, and VCC is the vector decided by CCori and CCint. If VCL is non-zero and have the same direction as TA, a gouging occurs with the depth of |VCL|. If VCL and TA have the reversed direction, undercut appears. In the ideal condition, the Fig. 17 The flow chart of the linear search method
Since the rear edge cutting will lead to damage to both the machined surface and the cutter, we need to identify rear gouge or rear cutting and avoid them. Details of the cutter partition-based gouge classification to identify rear gouge/ cutting will be described below. As shown in Fig. 10, the initial CC point in our algorithm is always confined to the toroidal section of the general APT cutter, to satisfy the optimal cutting condition. The goal is that after the tool-orientation optimization process, the new CC point is also at or near the original CC point on the toroidal section. As shown in Fig. 11a, TA is the unit vector of the tool axis, FD is the unit vector of the feed direction, and the unit vector VTF can be defined as Eq. (1). V TF ¼
T A FD jT A FD j
ð1Þ
As shown in Fig. 11b, VP is the reference vector, which is used to classify the position of the new gouge-free contact point. From the top view of the cutter, the bottom portion of the cutter is divided into the front cutting portion if the angle satisfy [−α, α] interval and the rear cutting portion if the angle satisfy [α, 2π − α] interval. The partition angle α can affect the smoothness of the tool orientations, is optimally defined as 5π/12. V P ¼ V TF T A
ð2Þ
The cutting and gouging category can be identified based on the cutter partition. As shown in Fig. 12, the XOY plane is End
Begin
Increase the lead angle forward by step and obtain the intermediate tool axis
Output the optimal lead angle N
Y
Is front edge cutting Execute the cutter projection to compute an intermediate cutter position
Carry out the cutter-partition classification N
Is front edge cutting
Carry out the cutter-partition classification Execute the cutter projection
Y
Increase the lead angle backward by step and obtain the intermediate tool axis
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Begin Front cutting portion
Input the minimum lead angle Increase the lead angle forward by step and obtain the maximum lead angle
Rear cutting portion
Execute the cutter projection to compute an intermediate cutter position
Fig. 18 The bisection search method for finding the optimal lead angle
determined with the coordinate original O (CL point), the X axis (X = VP), and the Yaxis (Y = VTF). The vector VCC begins from the original point O (CL point) and ends at the new gouge-free contact point CC. The projection of the vector VCC is the position checking vector VL in Eq. (3). The cutting portion and gouging category can be distinguished from the angle α of the position checking vector VL and the partition
Carry out the cutter-partition classification Is front edge cutting Execute the binary search method to find the optimal lead angle
Begin
End Fig. 20 The flow chart of the compound linear/bisection search method
Input the lead angle
and
vector VP. The variable ρ in Eq. (4) is denoted as the cutter position parameter. If ρ ≥ cosα, the new cutter contact point is on the front cutting portion and the gouging category for the original cutter contact point is local gouging. If ρ < cosα, the cutter contact point is on the rear cutting portion and the gouging category for the original cutter contact point is rear gouging.
Compute the intermediate tool axis from
Execute the cutter projection to compute an intermediate cutter position
VL ¼ Carry out the cutter-partition classification
V CC −ðT A ∙VCC ÞT A jV CC −ðT A ∙VCC ÞT A j
ρ ¼ V L ∙V P Y
Is front edge cutting
N
N Output the optimal lead angle
End Fig. 19 The flow chart of the bisection search method
ð4Þ
The comprehensive procedure of the cutter partition-based cutting gouge classification is shown in Fig. 13.
Y
−
ð3Þ
Fig. 21 Parameter definition on tool-path curve
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C surface
R5 R30
R5
R40
(a) Part 1 Airfoil surface of the blade part
(b) Part 2 Airfoil surface of the blade part
tool orientation before the gouge-avoidance optimization is also presented at the end of this subsection. The tool orientation optimization, adjusting the lead angle of the cutter while keeping the tilt angle fixed, is carried out to ensure machining with the front edge of the cutter. Since drastic change of tool orientation causes discontinuous tool path, we need to find the smallest lead-forward angle that clears the rear gouging. The effect of tool orientation optimization is shown in Fig. 14, the cutter contact point moves from the rear cutting portion to the front cutting portion. The rear gouging is eliminated when the tool tilt around the original CC point CCori by increasing the lead angle, θ. The relationship between the adjusted new tool axis orientation and the lead angle is given in Eq. (5). As shown in Fig. 15, XL is the unit vector along the feed direction, and ZL is the unit vector of the surface normal. XL, YL = ZL × XL, and ZL construct the local coordinate system where the tool axis is defined. To determine the tool orientation, we start with ZL,first rotate it around YL with the lead angle θ and then rotate it around XL with the tilt angle, ∅, to obtain the tool orientation. 2 3 1 0 0 T Anew ¼ 4 0 cosð∅Þ sinð∅Þ 5 ð5Þ 0 −sinð∅Þ cosð∅Þ 2
(c) Part 3 Fig. 22 Three parts with compound surfaces
3.3 Tool orientation optimization When the rear gouge/cutting is detected in the previous section, we need to adjust and optimize the tool orientation and reposition the cutter onto the part surface to find the new gouge-free position with contact point on the front cutting portion. The details of the tool orientation optimization are presented in this subsection. In addition, the pre-optimization of the
Fig. 23 Gouge-free tool paths and simulation verification of the part 1
cosðθ þ ΔθÞ ∙4 0 sinðθ þ ΔθÞ
0 1 0
3 −sinðθ þ ΔθÞ 5∙Z L 0 cosðθ þ ΔθÞ
Three iterative search methods, i.e., linear search, bisection search, and compound linear/bisection method, are proposed to find the smallest value of the lead angle that ensures front edge cutting. We first explain the linear search method. As shown in Figs. 16 and 17, we first forward rotate the tool axis by gradually increasing the lead angle with 1°, execute the cutter projection to obtain the new contact, identify the cutting portion and gouging category, and repeat until the intermediate
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Fig. 24 Gouge-free tool paths and simulation verification of the part 2
cutter position is on the front cutting portion of the cutter. Then we rotate the tool axis backward by decreasing the lead angle with 0.02° to find the critical value of the rotated angle which happened to guarantee front edge cutting. The bisection search method is shown in Fig. 19, to find the optimal lead angle θopt between θmin (the original tool axis) and θmax (a given value that is always bigger than θmin). The variable lg = ρ − cosα is defined to identify the front (positive value) and rear (negative) cutting (Fig. 18). The parameter θavg = (θmax − θmin)/2 is used to update the iteration (Fig. 19). The compound linear/bisection method is developed to improve the efficiency for finding the smallest lead angle to guarantee front edge cutting. The minimum lead angle θmin
Approach
Fig. 25 Simulation and machining results without and with gouge avoidance
Departure
corresponds the original tool axis. The lead angle is linearly searched by the step Δθ = 3° to find the first lead angle θmax to clear the rear gouge. As shown in Fig. 20, the bisection search method is then employed to determine the optimal lead angle θopt between the current and previous θmax, to find the smallest lead angle that ensures front edge cutting. Among the three methods, we choose the compound linear/ bisection method, as it is the fastest one. It is worth mentioning that the pre-optimization of tool orientations is developed before gouge avoidance to ensure smooth tool orientation. The tool orientation optimization loop for rear gouge avoidance is carried out to find the smallest value of the lead angle to ensure front edge cutting.
Tool path
(a) Tool path generated without gouge avoidance Airfoil surface
(b) Tool path generated with gouge avoidance Airfoil surface
Gouge
(c) Machining result without gouge avoidance
(d) Machining result with gouge avoidance
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Feed direction
Feed direction
Feed direction Front cutting portion
Front cutting portion
Front cutting portion
Tool path Approach Tool orientation Departure (b) Cutting with the front (a) Cutting with the front edge of the cutter for part 2 of the cutter for part 1
(c) Cutting with the front edge of the cutter for part 3
Fig. 26 Front edge cutting
Tool path Tool orientation
Approach Departure
Smooth tool orientations
Tool path Tool orientation (a) Display of the tool paths and the tool orientations
seconddifferential differential of theof tool The The second theorientations tool orientations
0.2
0.02
0.15
0.01
0.1
0
0.05
-0.01
0
-0.02 37 44 51 131 138 145 226 233 314 321 328 409 416 497 504 511 592 599 680 687 694 775 782 789 870 877 958 965 972
0.25
0.03
1 91 181 271 361 451 541 631 721 811 901 991 1081 1171 1261 1351 1441 1531 1621 1711 1801 1891 1981 2071 2161 2251 2341 2431 2521 2611 2701 2791 2881
first differential of the tool orientations The firstThedifferential of the tool orientations
(b) Display of the details of the tool orientations
No No optimization optimization
PrePre-optimization optimization
Both pre-and-loop optimization
-0.03
Both pre-and-loop optimization
(c) Comparison of the smoothness of the tool orientations Fig. 27 The display and analysis of tool orientations for part 1
No optimization
No optimization
PrePre-optimization optimization
Both pre-and-loop optimization
Both pre-and-loop optimization
(d) Comparison of the cutting vibration
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Since the tool orientation changed during rear gouge avoidance, the variation of the tool orientations is quantified to guarantee the final tool orientation smooth. As shown in Fig. 21, Ci(i ∈ [0, n]) denotes a series of CC points, TAi represents the corresponding tool orientations, and li, i + 1 is the distance between the two adjacent CC points Ci and Ci + 1. To minimize the rate of change of the entire tool orientations, the objective function Eq. (6) is defined. The preoptimization of tool orientations by adjusting the lead and tilt angle is carried out. The first differential γTi of the tool orientations in Eq. (7) is used to evaluate the smoothness of tool orientations. The second differential γTi′ of the tool orientations in Eq. (8) is calculated to identify the cutting vibration. ðT Aiþ1 −T Ai Þ2 2l i;iþ1 i¼0
n−1
ð6Þ
E¼ ∑
T Aiþ1 −T Ai l i;iþ1 γ Tiþ1 −γ Ti ¼ l i;iþ1 þ l iþ1;iþ2 =2
γ Ti ¼
ð7Þ
0
ð8Þ
γ Ti
4 Implementation and experiments To validate the effectiveness of the proposed cutter partition-based tool orientation optimization for gouge avoidance in five-axis tool path computation, experiments
are carried out on three parts with compound surfaces. In the experiments, gouge-free tool paths with smooth changing of tool orientations are generated. All kinds of cutters are used to cut the part surface. The simulation and experiment results indicate that the method is effective in removing gouging, ensuring front edge cutting, and guaranteeing smooth tool orientations. The proposed algorithm is implemented in C++ with Visual Studio 2010, and incorporated as the NC Blade software system, which was developed for automatic planning and programming for five-axis machining. Three parts with complex compound surfaces, the concave and the convex surfaces, are shown in Fig. 22. For the part 1 in Fig. 22a, four circles with the radius 5, 30, 40, and 5 mm and tensile length 80 mm form the C surface. The part 2 in Fig. 22b and the part 3 in Fig. 22c are blade models applied in the practical industry. Gouge-free tool paths of the above three parts are generated with the proposed algorithm. Gouge-free tool paths are generated in our developed CAM software (NC Blade). The simulation results in Figs. 23 and 24 reflect that gouging on the concave region of the part surface is totally eliminated with the effective gouge avoidance algorithm. The cutter partition-based tool orientation optimization for gouge avoidance is then practically tested by machining the part 3, i.e., five-axis airfoil surface finishing of the compressor blade. The part 3 have been machined on a five-axis milling machine (C.B.Ferrari A176 five-axis table-head machining
Smooth tool orientations
Approach Departure
Tool path Tool orientation
(a) Display of the tool paths and the tool orientations
(b) Display of the details of the tool orientations
The differential first differentialof of the the tool The first toolorientations orientations 4.5
The second differential of of the orientations The second differential thetool tool orientations 30
4
25
3.5
20
3 2.5
15
2
10
1.5
1
5
0.5 5 35 73 140 232 283 354 452 547 567 600 740 782 853 878 948 1017 1128 1150 1184 1233 1267 1348 1391 1406 1536 1577 1622 1627 1640
No Pre Both pre-and-loop No optimization Pre-optimization Both pre-and-loop optimization optimization optimization optimization (c) Comparison of the smoothness of the tool orientations Fig. 28 The display and analysis of tool orientations for part 2
-5
1 50 99 148 197 246 295 344 393 442 491 540 589 638 687 736 785 834 883 932 981 1030 1079 1128 1177 1226 1275 1324 1373 1422 1471 1520 1569 1618
0
0
Pre No Both pre-and-loop No optimization Pre-optimization Both pre-and-loop optimization optimization optimization optimization (d) Comparison of the cutting vibration
Int J Adv Manuf Technol
Smooth tool orientations
Approach Tool path Departure Tool orientation (a) Display of the tool path and the tool orientations
Tool path Tool orientation (b) Display of the details of the tool orientations
The first differentialof ofthe the tool The first differential toolorientations orientations
The second differential of of the orientations The second differential thetool tool orientations
2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0
8 6 4
2 1 76 151 226 301 376 451 526 601 676 751 826 901 976 1051 1126 1201 1276 1351 1426 1501 1576 1651 1726 1801 1876 1951 2026 2101 2176 2251 2326 2401
0 -2 -4 -6 14 40 104 125 147 211 218 249 262 322 329 364 398 437 444 456 485 548 555 567 596 660 667 697 708 766 773 802 818 881
-8
No optimization
No optimization
Preoptimization
Pre-optimization
Both pre-and-loop optimization
-10
Both pre-and-loop optimization
No No optimization optimization
PrePre-optimization optimization
Both pre-and-loop optimization
Both pre-and-loop optimization
(d) Comparison of the cutting vibration
(c) Comparison of smoothness of the tool orientations Fig. 29 The display and analysis of the tool orientation for part 3
center) with the test part are steel (10705BA) material with dimension 298 mm × 100 mm × 58 mm. The machining tolerance is set to be 0.05 mm. D20R1.6 torus-shaped milling cutter is used. The machining result in Fig. 25c shows that gouging
Clearance angle =0
(a1) Rear edge of the cutter close to the Part1 surface Clearance angle =0.2
(b1) Rear edge of the cutter not close to the Part1 surface
seriously affects machining quality, while Fig. 25d shows that the proposed gouge avoidance algorithm can remove gouging. The proposed tool-orientation optimization algorithm ensures cutting the surface with the front cutting portion. The
Clearance angle =0
(a2) Rear edge of the cutter close to the Part2 surface Clearance angle =0.2
(b2) Rear edge of the cutter not close to the Part2 surface
Fig. 30 Display of the distance between the rear edge of the cutter and the part surface
Clearance angle =0
(a3) Rear edge of the cutter close to the Part3 surface Clearance angle =0.2
(b3) Rear edge of the cutter not close to the Part3 surface
Int J Adv Manuf Technol
Fig. 31 Definition of a general APT cutter with six parameters
tool orientations are distributed uniformly and front edge cutting is ensured (Fig. 26). Based on the pre-optimized tool orientation, the proposed the tool orientation optimization loop for rear gouge avoidance can guarantee smooth tool orientations. Tool paths with tool orientations of the part 1, part 2, and part 3 are shown in Figs. 27, 28, and 29. The cutter positions with rear gouging, which need to adjust the tool orientations, are selected to be
Flat-end cutter
analyzed. The curves of the first differential of tool orientations, which represents the smoothness of tool orientations, are shown in Figs. 27c, 28c, and 29c. The blue curve is fiercely fluctuated without the tool optimization, the red curve is smooth with the pre-optimization of tool orientations before gouge avoidance, and the green curve resembles the red curve with the preoptimization and then tool orientation optimization loop for gouge avoidance. The results indicate that the pre-optimized tool orientation before gouge avoidance is effective to ensure smooth tool orientations. The tool orientation optimization loop for gouge avoidance, finding smallest lead angle, can avoid drastic change of the tool orientations. However, the tiny fluctuation of the tool orientations is caused after adjusting tool orientations to guarantee gouge-free and front edge cutting. To further check the effect of the tiny fluctuation on the cutting vibration, the second differential of tool orientations is calculated and the curves are shown in Figs. 27d, 28d, and 29d. The green curves show that there is no drastic cutting vibration compared with the cutting vibration in red curve with preoptimization of tool orientations. Therefore, smooth tool orientations can be guaranteed with the pre-optimization and tool orientation optimization loop for gouge avoidance. As shown in Fig. 30, the rear edge of the cutter is not close to the machined part surface owing to the adjustment of tool orientation with the defined clearance angle when the front edge cutting occurs. The proposed algorithm is developed for a general APT cutter to generate the gouge-free tool path. As shown in
Ball-end cutter
Approach Tool path Departure (a) Tool path generated with the flat-end cutter Toroidal cutter
Approach Tool path Departure (b) Tool path generated with the ball-end cutter
Approach Tool path Departure (c) Tool path generated with the toroidal cutter
Approach Tool path Departure (d) Tool path generated with the APT cutter
Fig. 32 Gouge-free tool path generated with different types of cutter
APT cutter
Int J Adv Manuf Technol
Fig. 31, the cutter is defined with six parameters: the tip angleA1, the tapper angle A2, and the corner radius R, the diameter D, the total cutter height H, and the flute length L. Figure 32a shows the gouge-free tool path generated with the bottom-end cutter (D = 10 mm, H = 100 mm, and L = 20 mm). Figure 32b shows the gouge-free tool path generated with the ball-end cutter (R = 10 mm, D = 20 mm, H = 100 mm, and L = 20 mm). Figure 32c shows the gouge-free tool path generated with the toroidal cutter (R = 5 mm, D = 10 mm, H = 100 mm, and L = 20 mm). Figure 32d shows the gouge-free tool path generated with the general APT cutter (A1 = 5°, A2 = 5°, R = 5 mm, D = 10 mm, H = 100 mm, and L = 20 mm).
References 1.
2.
3.
4.
5.
5 Conclusions and future research 6.
This paper presents an effective cutter partition-based tool orientation optimization for gouge avoidance in five-axis machining. The cutter projection is used to obtain new gougefree cutter positions. The cutter-partition classification is developed to distinguish between local and rear gouging. The tool orientation optimization loop, after pre-optimization, is used to ensure front edge cutting and avoid sudden change of the tool orientations. A clearance angle is defined to ensure a safety clearance between the rear edge of the cutter and the machined surface. The accomplishments are as follows:
7.
8.
9.
10.
(1) This algorithm can identify and remove both the local and the rear gouging. (2) This algorithm can effectively handle the combination of local and rear gouging. (3) This algorithm minimizes the sudden change to smooth tool orientations due to gouge avoidance. (4) This algorithm can ensure machining with the front cutting portion of the cutter and a safety clearance between the rear edge of the cutter and the machined surface. (5) This algorithm is applicable to a general APT cutter and arbitrary free-form surfaces. We plan to further improve the current work in the following two aspects. The optimization of tilt angle will be considered to handle more general cutting conditions, including the side cutting (swarfing). The second enhancement will be the avoidance of cutter collision, in addition to gouging.
11.
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14.
15.
16. Acknowledgments We acknowledge the support of Dongfang Turbine Co., Ltd. for the machining experiments, and the help of Lixiong Gan, Fan Yang, and Changya Yan. 17. Funding information The authors gratefully acknowledge the support of the National Science and Technology Major Project of the Ministry of Science and Technology of China (2013ZX04007-041).
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