Int J Adv Manuf Technol (2017) 92:3615–3625 DOI 10.1007/s00170-017-0403-1
ORIGINAL ARTICLE
A tool orientation smoothing method based on machine rotary axes for five-axis machining with ball end cutters Rufeng Xu 1 & Xiang Cheng 1 & Guangming Zheng 1 & Zhitong Chen 2
Received: 25 October 2016 / Accepted: 3 April 2017 / Published online: 30 April 2017 # Springer-Verlag London 2017
Abstract Avoiding significant fluctuations of tool orientation is an important problem in five-axis machining. The dramatic change of tool orientation may greatly increase the angular acceleration of machine rotary axes, produce larger nonlinear machining errors and gouging, reduce the feed rate of machine rotary axes, etc. In this paper, we propose a tool orientation smoothing method based on machine rotary axes for five-axis machining with ball end cutters to reduce fluctuations of tool orientation in the machine coordinate system (MCS). The core idea of the proposed method is to directly smooth machine rotary angles in the machine coordinate system for the smooth variation of tool orientation. First of all, we establish the relationship between the design variables of tool position and machine rotary angles. Then, we define an objective function of tool orientation smoothing based on machine rotary angles. In order to solve the above objective function, we also develop a simplified algorithm to obtain the minimum sum of squares of compound angular accelerations. Finally, a blade surface is used as a test example, and tool paths are generated by the Sturz and proposed methods, respectively. Comparison and analysis results show that the proposed method can improve the kinematics performance of five-axis machine tool, as well as surface machining quality and efficiency.
* Rufeng Xu
[email protected]
1
School of Mechanical Engineering, Shandong University of Technology, Zibo 255000, China
2
School of Mechanical Engineering and Automation, Beihang University, Beijing 100191, China
Keywords Tool orientation smoothing . Five-axis machining . Machine rotary axes . Sculptured surfaces . Ball end cutters
1 Introduction Compared with three-axis machine tools, five-axis machine tools have two additional rotary axes. It becomes more flexible and has more advantages, such as to cut more complex shaped parts, to enhance machining efficiency, and to improve machining quality. Five-axis machine tools are thus widely applied in modern manufacturing industries. Five-axis numerical control (NC) programming method is a core and key problem in five-axis machining, which has an important influence on the machining quality and efficiency. For example, when the tool moves along the specified trajectory, if tool orientations at two adjacent cutter contact (CC) points change dramatically, the actual motion of machine rotary axes may go beyond limits of angular velocity and acceleration, as well as serious gouging may occur. To avoid the abrupt change of tool orientation, we need to further optimize and smooth tool orientations. Therefore, tool orientation control plays a very important role in the machining quality and efficiency during five-axis machining of sculptured surfaces. Its main objective is to specify the parameters (such as tilt and inclination angles of the tool) that define the tool orientation at each CC point, so that the machining time is minimized and the machined surface is gouge-free, within tolerance, and of uniform quality across the whole surface [1]. In order to obtain the smooth tool path and tool orientation, many scholars have done some studies and made a lot of progress in the field of tool orientation control during five-axis machining of sculptured surfaces. For the determination of tool orientation in five-axis machining, Vickers and Quan [2] proposed a simple five-axis
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tool positioning method called the Sturz method, which is currently still applied in finish machining. For low curvature surfaces, much better tool paths can be generated by this method. While for some complex surfaces with high curvature variation, tool paths generated by this method can become very worse, such as abrupt changes of tool orientation. The main reason of the above problem is that the tool axis orientation largely depends on local surface properties, such as the surface normal, the principal curvature, etc. Therefore, the Sturz method needs to be further improved especially in the control of tool orientation. Lee [3] proposed a method of admissible tool orientation control for gouging avoidance in five-axis machining by considering both local and global surface shapes. Rao et al. [4] presented a principal axis method in which the tool inclination angle is determined in terms of the principal curvature directions of the design and tool surface at each CC point. Warkentin et al. [5] proposed a multi-point machining method to make the tool contact with the design surface at more than one contact points. Gray et al. [6] presented an arc-intersect method (AIM) to directly position the tool to contact the surface and generate gouge-free tool positions. Hsueh et al. [7] proposed a two-stage cutting tool collision check method to determine the collision-free ball end cutter orientation automatically. Barakchi Fard et al. [8] examined the effect of tool tilt angle on machining strip width in the determination of optimal tool orientation and feed direction in five-axis flat end milling. Lin et al. [9] proposed an efficient algorithm to generate tool posture collision-free area by sampling and interpolation methods for the whole free-form surface during five-axis computer numerical control (CNC) finishing period. Du et al. [10] presented a new method of automatic detection and elimination of cutter gouging when using the fillet end milling cutter to produce a complex surface on the five-axis machine tool. However, all about tool positioning methods mainly prevent the local gouging and increase the machining strip width under the specified scallop height. To avoid the local gouging for a ball end tool, the tool radius should not exceed the minimum concave radius of principle curvature on the surface. Hence, the variation of tool orientation along a tool path becomes a very important problem in five-axis NC programming. Abrupt changes of tool orientation between successive CC points can incur excessive angular velocity and angular accelerations of the machine rotary axes. Since machining errors are directly proportional to variation of rotation angles [11], more nonlinear machining errors are produced. Additionally, non-smooth variation of tool orientation can also increase machining times and decrease surface quality by leaving tool marks on the machined surface [12, 13].
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Jun et al. [14] proposed a method of optimizing and smoothing the tool orientation control for five-axis machining by configuration space search method. By considering the adjacent part geometry and the alternative feasible tool orientations in the C-space, they globally optimize tool orientation to minimize the dramatic change of tool orientation. Ho et al. [15] proposed a tool orientation interpolation method to avoid abrupt changes of tool axes. In this method, the vector interpolation algorithm is used to obtain tool orientations at CC points between adjacent critical specified points in the workpiece coordinate system (WCS). Wang and Tang [16] presented an approach to handle the drastic change of tool orientation by considering the angular velocity limit of the tool axis design angles, i.e., tilt and inclination angles of the tool. Xu et al. [17] proposed a tool positioning algorithm based on smooth tool paths to avoid the local abrupt change of tool paths, whose key idea is to optimize tool position by considering the smoothness and continuity of a whole tool path together. Li et al. [18] presented a new method for generating five-axis tool paths with smooth tool motion and high efficiency based on the accessibility map (A-map) of the cutter at a point on the part surface. Farouki and Li [19] proposed a method to minimize the variation of tool orientation in fiveaxis machining with a ball end tool by keeping the constant cut speed. In the method, the tool axis is specified by a fixed vector in an orthonormal frame that is rotationminimizing with respect to the surface normal. Sun et al. [20] presented a cutter orientation adjustment method to obtain an optimized tool path which makes best use of the kinematic characteristics of angular feed for five-axis machining. He et al. [21] presented a method to reduce the fluctuation of machining strip width by modifying tool position, which also considers tilt angle smoothness along the tool path. Zhang et al. [22] presented an optimal tool path generation model for a ball end tool which strives to globally optimize a tool path with various objectives and constraints. The above methods mainly focus on how to optimize the tool orientation in WCS. Farouki et al. [23] proposed an approach to determine the rotary axis inputs to fiveaxis machine by minimizing variations of relative tool/workpiece orientation under the constraint of a fixed cutting speed with a ball end tool. Srijuntongsiri and Makhanov [24] proposed a new numerical algorithm to reduce the kinematic errors of a five-axis tool path by minimizing the variation of the rotation angles. In this method, they insert the additional cutter location (CL) points between key CL points and rotate/translate the part surface relative to workpiece into an optimal position. Some of the above methods have also started to consider the effect of the machine rotary axes.
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In order to obtain the smooth variation of tool orientation in the machine coordinate system, we establish an objective function of tool orientation smoothing based on the machine rotary angles in MCS, where the design variables are arbitrary two angles of the machine rotary axes in terms of configuration of given five-axis machines. To solve the objective function, we also develop a simplified algorithm to implement tool orientation smoothing by optimizing the two machine rotary angles directly, which can obtain the minimization of the sum of compound accelerations. The rest of this paper is organized as follows. The relationship between tool orientation and design variables of tool position is established in Section 2. The relationship between tool orientation and machine rotary angles is also established in Section 3. The relationship between the design variables of tool position and machine rotary angles is then established in Section 4. On this basis, a tool orientation smoothing method based on machine rotary axes is developed in Section 5. At last, a test example is provided in Section 6. Conclusions are reached in Section 7.
2 Relationship between tool orientation and design variables of tool position 2.1 Tool position calculation in the local coordinate system As shown in Fig. 1, the part surface S: r(u, v), is machined with a ball end cutter. Let Pcc(ucc,vcc) be an arbitrary point on the surface, and ncc the unit normal vector at point Pcc. Suppose OL is the origin of the local coordinate system, rOL ¼ rPcc þ rnCC , where rOL and rPcc are position vectors of point OL and Pcc, respectively. Let vectors e1 and e3 represent the unit tangent vector of the feed direction and the unit normal vector at point Pcc, respectively. We may have the vector e 2 = e 3 × e 1 . A local coordinate system OLXLYLZL can be built at point OL, where e1, e2, and e3
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are unit vectors in the directions of the positive XL-, YL-, and Z L -axes, respectively. A tool coordinate system OTXTYTZT is also built on the tool, where the origin OT is located at the center of the ball. In the initial state, the direction of XT-, YT-, and ZT-axes coincides with one of XL-, YL-, and ZL-axes, respectively. Here, the tool still has two degrees of freedom: one is the inclination angle θ around YLaxis and the other is the tilt angle ψ around XL-axis. The two angles, that is, the design variables of tool position, may be adjusted to obtain the optimal tool orientation. In the local coordinate system, vectors of tool position and tool orientation can be expressed as T TLCS axis ¼ R ðX L ; ψÞR ðY L ; θÞð0; 0; 1Þ
ð1Þ
T TLCS pos ¼ R ðX L ; ψÞR ðY L ; θÞð0; 0; −r Þ
ð2Þ
w h e r e RðY L ; θÞ ¼ ð cosθ 0 sinθ 010 −sinθ0cosθÞ, π πR ðX L ; ψÞ ¼ ð 1 0 0 0cosψ−sinψ 0sinψcosψÞ, θ∈ − 2 ; 2 , andψ∈ − π2 ; π2 . 2.2 Tool position calculation in the workpiece coordinate system Let axis components of local coordinate system OLXLYLZL be e1 = (x1, y1, z1)T, e2 = (x2, y2, z2)T, and e3 = (x3, y3, z3)T, respectively. In the workpiece coordinate system OWXWYWZW, vectors of tool position and tool orientation can then be expressed as LCS TWCS axis ¼ ðe1 e2 e3 ÞTaxis
TWCS pos
¼ ð e1
e2
e3 ÞTLCS pos
ð3Þ ð4Þ
Substituting Eq. (1) into Eq. (3) yields 0
TWCS axis
1 x1 sinθ−x2 cosθsinψ þ x3 cosθcosψ ¼ @ y1 sinθ−y2 cosθsinψ þ y3 cosθcosψ A z1 sinθ−z2 cosθsinψ þ z3 cosθcosψ
ð5Þ
3 Relationship between tool orientation and machine rotary angles
Fig. 1 Ball end tool positioning
From Eq. (5), one can see that tool orientation at certain CC point can be determined by the design variables of tool position and components of the local coordinate axes. In order to execute cutter location data (i.e., tool position and tool orientation) on a five-axis machine tool, we need to convert them into the G-code data according to different types of five-axis machine. And the machine rotary angles will denote tool
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orientation in the G-code file. Therefore, we will discuss how tool orientation is determined by machine rotary angles in the machine coordinate system as follows. 3.1 Types of generalized five-axis machines The five-axis machine tool is generally classified as the following: the dual heads, the head and table, and the dual tables. However, in accordance with different machine rotary axes, the five-axis machine tool is also classified as follows: (1) AB type, (2) AC type, (3) BA type, and (4) BC type. For any above type, in terms of the different location of rotary axis (such as the spindle head and the table), it can further be classified into three basic types. For example, the five-axis machine tool of AB type can further be classified as the following: AB type (both A- and B-axes are located on the spindle head), AB′ type (A-axis is located on the spindle head and B-axis located on the table), and A′B′ (both A- and B-axes are located on the table), where the notation “′” denotes that the rotary axis is located at the table. In general, the direction of each axis of the WCS coincides with one of the MCS. Therefore, a generalized tool orientation formula in the machine coordinate system can be given by
TWCS axis
0 1 0 1 i 0 ¼ @ j A ¼ Rðβ 2 ÞRðβ1 Þ@ 0 A k 1
ð6Þ
where β1 denotes either of machine rotary angles A about Xaxis and B about Y-axis; β2 denotes any angle among the machine rotary angles A about X-axis, B about Y-axis, and C about Z-axis, and are different from the angle β1; R denotes the rotational transformation matrix. For the above four types of five-axis machines, the relationship between machine rotary angles and tool orientation will be detailed as follows.
Fig. 3 Coordinate systems in five-axis machine tool with rotary axes B and A′
3.2 Relationship between tool orientation and machine rotary angles For the simplicity and convenience of presentation, a five-axis machine of BA′ type is chosen as an example, as shown in Fig. 2. The coordinate systems in the five-axis machine tool of BA′ type are illustrated in Fig. 3. Therefore, the formula between tool orientation and machine rotary angles B and A can be given by
TWCS axis
0 1 0 ¼ RðX ; AÞRðY ; BÞ@ 0 A 1
ð7Þ
where RðX ; AÞ ¼ ð 1 0 0 0cosA−sinA 0sinAcosAÞ, R ðY ; BÞ ¼ ð cosB 0 sinB 010 −sinB0cosBÞ. Rearranging Eq. (7), we can have T TWCS axis ¼ ðsinB; −cosBsinA; cosBcosAÞ
ð8Þ
By using the above same method, the relationship between tool orientation and machine rotary angles for the other three types of five-axis machining tool are derived. Furthermore, the formula between tool orientation and machine rotary angles A and B for five-axis machine of AB type can be given by T TWCS axis ¼ ðcosAsinB; −sinA; cosAcosBÞ
Fig. 2 Schematic of five-axis machine tool with rotary axes B and A′
ð9Þ
Similarly, the formula between tool orientation and machine rotary angles A and C for five-axis machine of AC type can be given by
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T TWCS axis ¼ ðsinAsinC; −sinAcosC; cosAÞ
ð10Þ
Similarly, the formula between tool orientation and machine rotary angles B and C for five-axis machine of BC type can be given by T TWCS axis ¼ ðsinBcosC; sinBsinC; cosBÞ
ð11Þ
Substitute Eq. (11) into Eq. (5), the relationship equation between the design variables of tool position and machine rotary angles can be given by 0
1 0 1 x1 sinθ−x2 cosθsinψ þ x3 cosθcosψ sinBcosC @ sinBsinC A ¼ @ y1 sinθ−y2 cosθsinψ þ y3 cosθcosψ A ð15Þ cosB z1 sinθ−z2 cosθsinψ þ z3 cosθcosψ
5 Tool orientation smoothing method based on machine rotary axes 4 Relationship between the design variables of tool position and machine rotary angles Section 2 states that tool orientation can be determined by the design variables of tool position in the workpiece coordinate system, and Section 3 also states that tool orientation can be determined by the machine rotary angles in the machine coordinate system. Hence, we can find the relationship between the design variables of tool position and machine rotary angles as follows. (1) Five-axis BA type machine Substitute Eq. (8) into Eq. (5), the relationship equation between the design variables of tool position and machine rotary angles can be given by 0
1 0 1 x1 sinθ−x2 cosθsinψ þ x3 cosθcosψ cosAsinB @ −sinA A ¼ @ y1 sinθ−y2 cosθsinψ þ y3 cosθcosψ A ð12Þ cosAcosB z1 sinθ−z2 cosθsinψ þ z3 cosθcosψ
In this section, one first defines an objective function of tool orientation smoothing based on machine rotary angles to smooth the tool orientation in five-axis machining. To solve the above objective function, we then develop a novel simplified tool orientation smoothing algorithm by using the above formulas to implement the smooth variation of tool orientations in the machine coordinate system. 5.1 Objective function of tool orientation smoothing We first introduce a concept of a compound angular acceleration in MCS. The compound angular acceleration at certain CC point Pi on the design surface can be defined as α P i ðβ 1 ; β 2 Þ ¼
¼d ¼2
(2) Five-axis AB type machine Substituting Eq. (9) into Eq. (5), the relationship equation between the design variables of tool position and machine rotary angles can be given by 0
1
0
1
x1 sinθ−x2 cosθsinψ þ x3 cosθcosψ sinB @ −cosBsinA A ¼ @ y1 sinθ−y2 cosθsinψ þ y3 cosθcosψ A ð13Þ cosBcosA z1 sinθ−z2 cosθsinψ þ z3 cosθcosψ
(3) Five-axis AC type machine Substitute Eq. (10) into Eq. (5), the relationship equation between the design variables of tool position and machine rotary angles can be given by 0
1 0 1 x1 sinθ−x2 cosθsinψ þ x3 cosθcosψ sinAsinC @ −sinAcosC A ¼ @ y1 sinθ−y2 cosθsinψ þ y3 cosθcosψ A ð14Þ cosA z1 sinθ−z2 cosθsinψ þ z3 cosθcosψ
(4) Five-axis BC type machine
d2 TPi ðβ1 ; β 2 Þ dt 2
þ
∂TPi ðβ1 ; β 2 Þ ∂TPi ðβ 1 ; β2 Þ ω1 þ ω2 =dt ∂β 1 ∂β 2
∂ 2 T P i ðβ 1 ; β 2 Þ ∂2 TPi ðβ1 ; β 2 Þ 2 ω1 ω2 þ ω1 ∂β 1 β 2 ∂β 1 2
∂2 TPi ðβ1 ; β 2 Þ 2 ∂TPi ðβ 1 ; β 2 Þ ∂TPi ðβ 1 ; β2 Þ ω2 þ α1 þ α2 ∂β 1 ∂β2 ∂β 2 2
ð16Þ where Pi denotes an arbitrary point on the CC curve, β1 and β2 any two machine rotary angles at point Pi, T the function of tool orientation whose variables are machine rotary anglesβ1 and β2 at Pi, t the time, ω1 and ω2 the corresponding angular velocities of β1 and β2, ω1 ¼ dβdt1 , ω2 ¼ dβdt2 , and α1 and α2 the corresponding angular accelerations of β1 and β2, α1 ¼
d2 β 1 dt 2 ,
α2 ¼ ddtβ22 . The above equation illustrates that the compound angular acceleration at an arbitrary point relates to the machine rotary angles, the angular velocity, and the acceleration of each machine rotary axis. In the practice, if we do not know machine rotary angles at the successive CC points, then the current compound angular acceleration is very difficult to solve directly and analytically. In general, we first need to evaluate tool orientations at adjacent CC points by using existing tool 2
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positioning algorithms [2–10], such as the Sturz method [2], an arc-intersect method [6], a two-stage cutting tool collision check method [7], a new method of automatic detection and elimination of cutter gouging [10], etc. Then, we also compute the machine rotary angles at adjacent CC points by using Eqs. (12)–(15). Next, the compound angular acceleration at the specified CC point will be solved from Eq. (16). On the basis, an objective function of tool orientation smoothing based on machine rotary angles may be defined as N
Γ ¼ ∑ kα P i ð β 1 ; β 2 Þ k2
ð17Þ
i¼1
From Eq. (17), we observe that Γ is the function of all angles β1 and β2 at CC points Pi. In order to obtain the smooth tool orientation, we may have the following optimization problem of minimum sum of squares of the compound angular acceleration at certain CC point Pi. 8 > β 1 ∈β 1limit > > > > > β 2 ∈β 2limit > > < N 2 ω 1 ∈ω1limit minΓ ¼ ∑ kαPi ðβ 1 ; β 2 Þk s:t: > i¼1 > ω2 ∈ω2limit > > > > α ∈α > > : 1 1limit α2 ∈α2limit
þ
∂2 TPi ðB; AÞ ∂2 TPi ðB; AÞ 2 ω1 ω1 ω2 þ ∂B∂A ∂B2
∂2 TPi ðB; AÞ 2 ∂TPi ðB; AÞ ∂TPi ðB; AÞ ω2 þ α1 þ α2 ∂B ∂A ∂A2
∂TPi ðB; AÞ ¼ ðcosB; sinBsinA; −sinBcosAÞT ∂B ∂TPi ðB; AÞ ¼ ð0; −cosBcosA; −cosBsinAÞT ∂A
ð20Þ ð21Þ
∂2 TPi ðB; AÞ ¼ ð0; sinBcosA; sinBsinAÞT ∂B∂A
ð22Þ
∂2 TPi ðB; AÞ ¼ ð−sinB; cosBsinA; −cosBcosAÞT ∂B2
ð23Þ
∂2 TPi ðB; AÞ ¼ ð0; cosBsinA; −cosBcosAÞT ∂A2
ð24Þ
Substitute Eqs. (20)–(24) into Eq. (19) yields αPi ðB; AÞ 0
1 α1 cosB 2 2 ω þ ω2 cosBsinA þ α1 sinBsinA−α2 cosBcosA A ¼ @ 2ω1 ω2 sinBcosA þ 1 2ω1 ω2 sinBsinA− ω1 2 þ ω2 2 cosBcosA−α1 sinBcosA−α2 cosBsinA
ð18Þ
where N denotes the number of CC points on a tool path, β 1limit and β 2limit sets of admissible range of angle β1 and β2, ω1limit and ω2limit sets of admissible angular velocity of angle β1 and β2, and α1limit and α2limit sets of admissible angular acceleration of angle β1 and β2. Note that β1limit , β 2limit , ω1limit , ω2limit , α1limit , and α2limit depend on the kinematics constraints of the given five-axis machine tool. Assume that a five-axis machine of BA′ type is used as the specified machine tool now. Then β1 and β2 denote machine rotary angles B and A, respectively. According to the specifications of the given five-axis machine tool, we can easily obtain the sets β1limit and β 2limit , i.e., the admissible range of angle B and A, ω1limit and ω2limit , i.e., admissible angular velocity of angle B and A, and α1limit and α2limit , i.e., admissible angular acceleration of angle B and A. For the five-axis machine of BA′ type, let β1limit be a close interval [−90°, 90°], β2limit a close interval [−180°, 180°], ω1limit and ω2limit a close interval [−200°/s, 200°/s], and α1limit and α2limit a close interval [−1000°/s2, 1000°/s2]. Substituting B and A into Eq. (16), then the equation can be rewritten as
αPi ðB; AÞ ¼ 2
By differentiating Eq. (8) with respect to B and A, respectively, we can obtain
ð19Þ
ð25Þ For the given CC point Pi on the design surface, if angles B and A are determined, then the compound angular acceleration can be completely dependent on the variables ω1, ω2, α1, and α2 at point Pi. Hence, for the given point Pi, a function of (ω1, ω2, α1, α2), namely squares of the compound angular acceleration at certain CC point Pi, can be defined as F ðω1 ; ω2 ; α1 ; α2 Þ ¼ kαPi ðβ1 ; β 2 Þk2
ð26Þ
Rewriting Eq. (26) through Eq. (25) gives F ðω1 ; ω2 ; α1 ; α2 Þ ¼ α1 2 þ 4ω1 2 ω2 2 þ ω1 4 þ ω2 4 −2ω1 2 ω2 2 þ α2 2 cos2 B þ α1 ω1 2 þ α1 ω2 2 ‐2ω1 ω2 α2 sin2B
ð27Þ
Combining the given machine’s constraints in Eqs. (18) and (27), the minimum value of the above function F is zero if and only if ω1 = ω2 = α1 = α2 = 0, namely the compound angular acceleration is zero. In other words, the machine rotary axes B and A are fixed, and only the linear axes X, Y, and Z can be moved. Similarly, for all CC points, the minimum value of the objective function of tool orientation smoothing in Eq. (18) is also zero with the given constraints. However, if changes of the machine rotary angles B and A are nonzero, in order to obtain the minimum value of the object function in Eqs. (18) or (26), then the angular velocity and acceleration needs to be reduced as much as possible.
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In order to obtain the minimum sum of squares of compound angular accelerations at all CC points, as mentioned above, we first need to solve the initial machine rotary angles β1 and β2 at all CC points by using the tool positioning algorithm, then calculate the compound accelerations, and finally judge if the sum of squares of compound angular accelerations is minimum. The above process needs to be iterated until the algorithm converges. However, due to the time-consuming and difficulty to solve the compound angular acceleration analytically and directly, we develop a simplified algorithm to obtain the minimum sum of squares of compound angular accelerations at all CC points, with which one can obtain the smoother angular variation, and lower angular velocity and acceleration of each machine rotary axis. For the convenience, we will still choose a BA ′ type five-axis machine as an example, and the proposed algorithm is detailed as follows. 5.2 Tool orientation smoothing method based on machine rotary axes The core idea of the tool orientation smoothing method based on machine rotary axes is to directly smooth machine rotary angles in the machine coordinate system for the smooth variation of tool orientation. The proposed method mainly consists of three steps, as shown in Fig. 4. The detailed process of the proposed method is described as follows. Let the initial inclination and tilt angles be θ and ψ, respectively. Let h be the step length tolerance, N the number of sampled CC point, and M the total actual number of CC points on each CC curve. The first stage is to obtain initial tool orientation at each sampled CC point by using the Sturz method [2] or the other collision-free tool positioning methods [3–10], such as an arc-intersect method [6], a two-stage cutting tool collision check method [7], a new method of automatic detection and elimination of cutter gouging [10], etc. For simplicity and convenience, we will here employ the simple Sturz method to implement the proposed method. For a given design surface, we first need to generate sampled CC point CCi on the CC curve in terms of N by using the constant parameter discretizing method. For each sampled point CCi, we can then calculate tool position Tpos and tool orientation Taxis in terms of specified angles θ and ψ by using the Sturz method. The second stage is to fit all of machine rotary angles by the cubic spline interpolation function. On the basis of the above obtained tool orientations, we may compute machine rotary angles A and B by using Eq. (12) at all sampled CC points. Then, we need to fit two sets of the above rotary angles A and B into the cubic spline curves by the cubic spline interpolation function, respectively. The third stage is to compute new tool orientations for all of given CC points. For the specified CC curve, we first generate a series of CC points CCi in terms of M by
Fig. 4 Flow chart of tool orientation smoothing method based on machine rotary angles
using the constant parameter discretizing method or h by using constant chord height discretizing method. For an arbitrary point CCi on the CC curve, we can then calculate values of angles A and B at point CCi by using the cubic spline interpolating algorithm and obtain new tool orientation at point CCi by Eq. (12). Finally, one can repeat the above process until all of new tool orientations on the CC curve are calculated. It should be noted that cutter center point does not need to be calculated again for the ball end tool although tool orientation may change at each CC point.
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6 Test example To verify the validation and effectiveness of the proposed method, an aircraft engine blade surface (as shown in Fig. 5) is machined on a five-axis machine tool with the swing spindle of B-axis and rotary table of A-axis. First of all, we will generate one tool path on the blade surface by using both the Sturz method and proposed method, respectively. Then, we will compare and analyze the angles, angular velocity, and angular acceleration of each machine rotary axes generated by two different methods. 6.1 Tool path generation Suppose the inclination angle θ is 10°, the tilt angle ψ is 2°, the machining parametric ranges are u = 0.5, and v∈[0.0, 1.0], respectively, the number of CC points is 101, and the feed direction is along the parameter v. The number of sampled CC points is 10, the diameter of the ball end tool is 0.25 in., and the interval time between two adjacent CC points is 0.05 s. One tool path and tool axes at CC points on the leading surface of the blade are generated by the Sturz method and the proposed method, respectively, as shown in Figs. 6 and 7. Comparing tool axes in Fig. 6 with those in Fig. 7, we can observe that variation of tool axes in the workpiece coordinate system with the proposed method is slightly larger than that with the Sturz method. 6.2 Comparison and analysis 6.2.1 The angle of machine rotary axes with two different methods Angle A at each CC point can be obtained by the Sturz and proposed methods, respectively, as shown in Fig. 8. From the figure, it can be observed that angle A generated by the Sturz and proposed method varies from −95° to −55°, while change of angle A generated by the proposed method is smoother than that generated by the Sturz method.
Fig. 5 An aircraft engine blade surface
Fig. 6 One tool path and tool axes at CC points generated by the Sturz method
Angle B at each CC point can be obtained by the Sturz and proposed methods, respectively, as shown in Fig. 9. From the figure, it can be observed that angle B generated by the Sturz and proposed method varies from −35° to −25°, while change of angle B generated by the proposed method is smoother than that generated by the Sturz method. 6.2.2 The angular velocity curves of rotary axes with two different methods Angular velocity of angle A generated by two methods can be calculated by the values of angle A at adjacent CC points, as shown in Fig. 10. From the figure, it can be observed that angular velocity of angle A generated by the Sturz method greatly varies from about −60°/s to 120°/s, while one generated by the proposed method varies from about −40°/s to 65°/ s. Thus, the maximum angular velocity of angle A generated by the proposed method is reduced by nearly half. In addition, the change of angular velocity of angle A generated by the proposed method is smoother than that generated by the Sturz method.
Fig. 7 One tool path and tool axes at CC points generated by the proposed method
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Fig. 8 Variation curves of angle A with two different methods
Fig. 10 Variation curves of angular velocity of angle A with two different methods
Angular velocity of angle B generated by two methods can be calculated by the values of angle B at adjacent CC points, as shown in Fig. 11. From the figure, it can be observed that angular velocity of B generated by the Sturz method greatly varies from about −190°/s to 60°/s, while one generated by the proposed method varies from about −75°/s to 30°/s. Thus, the maximum angular velocity of angle B generated by the proposed method is reduced by over half. In addition, the change of angular velocity of angle B generated by the proposed method is smoother than that generated by the Sturz method.
s2, while one generated by the proposed method smoothly varies from about −200°/s2 to 300°/s2. Thus, the maximum angular acceleration of angle A generated by the proposed method is reduced by nearly three times. In addition, the change of angular acceleration of angle A generated by the proposed method is smoother than that generated by the Sturz method. Angular acceleration of angle B generated by two methods can be calculated by the values of angular velocity of B at adjacent CC points, as shown in Fig. 13. From the figure, it can be observed that angular acceleration of B generated by the Sturz method greatly varies from about −1800°/s2 to 1600°/s2 which exceed the given allowable limit, while one generated by the proposed method smoothly varies from about −200°/s2 to 200°/s2. Thus, the maximum angular acceleration of angle B generated by the proposed method is reduced by nearly nine times. In addition, the change of angular acceleration of angle B generated by the proposed method is smoother than that generated by the Sturz method.
6.2.3 The angular acceleration curves of rotary axes with two different methods Angular acceleration of angle A generated by two methods can be calculated by the values of angular velocity of A at adjacent CC points, as shown in Fig. 12. From the figure, it can be observed that Angular acceleration of B generated by the Sturz method greatly varies from about −800°/s2 to 900°/
Fig. 9 Variation curves of angle B with two different methods
Fig. 11 Variation curves of angular velocity of angle B with two different methods
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Fig. 12 Variation curves of angular acceleration of angle A with two different methods
6.3 Discussions Comparisons of tool paths generated from two different methods demonstrate that variation of tool axes in the workpiece coordinate system with the proposed method is slightly larger than that with the Sturz method. From comparison of the angle variation of machine rotary axes generated by the Sturz and proposed method, one can find that angle variation ranges of two methods are almost identical, while angle change of machine rotary axes with the proposed method is smoother than that with the Sturz method. From the comparison of the angular velocity of machine rotary axes generated by the Sturz and proposed methods, one can find that the maximum angular velocity of machine rotary angles A and B with the proposed method may be reduced by nearly half. Through comparison of the angular acceleration of machine rotary axes generated by two different methods, the maximum angular
Fig. 13 Variation curves of angular acceleration of angle B with two different methods
Int J Adv Manuf Technol (2017) 92:3615–3625
acceleration of machine rotary angles A and B with the proposed method may be greatly reduced by three and nine times, respectively. Therefore, the proposed method can greatly decrease the angular velocity and acceleration of machine rotary axes only through adjusting the two machine rotary angles in a very small range. We can further obtain the minimum sum of squares of compound angular accelerations at all CC points. In other words, the proposed method is an effective and efficient method to solve Eq. (18). Due to the decrease of the angular velocity and acceleration of machine rotary axes by using the proposed method, gouging and nonlinear machining error is greatly reduced, the case to exceed the machine angular velocity and acceleration limit is also avoided, and the motion of machine rotary axes becomes more stable and smoother. Therefore, the proposed method can improve the kinematics performance of five-axis machine tool, as well as surface machining quality and efficiency.
7 Conclusions In this paper, we propose a tool orientation smoothing method based on machine rotary axes for five-axis machining with ball end cutters to reduce fluctuations of tool orientation in the machine coordinate system. The core idea of the proposed method is to directly smooth machine rotary angles in the machine coordinate system for the smooth variation of tool orientation. We firstly establish the relationship between the design variables of tool position and machine rotary angles. And, we also define an objective function of tool orientation smoothing based on machine rotary angles. In order to solve the objective function, we develop a simplified algorithm to obtain the minimum sum of squares of compound angular accelerations. The test results show that the proposed method can greatly decrease the angular velocity and acceleration of machine rotary axes. Moreover, gouging and nonlinear machining error is significantly reduced, the case to exceed the machine angular velocity and acceleration limit is also avoided, and the motion of machine rotary axes becomes more stable and smoother. Therefore, the proposed method can improve the kinematics performance of five-axis machine tool, as well as surface machining quality and efficiency. However, since the tool orientation correction does not affect the location of tool center for ball end cutters, the proposed method is currently only suitable for ball end tools. For other types of tool, we still need to make a further research by considering the interference between the tool and the part surface and tool position correction.
Int J Adv Manuf Technol (2017) 92:3615–3625 Acknowledgements This work was partially supported by the National Natural Science Foundation of China (51105026, 51505264), the Promotive Research fund for Excellent Young and Middle-aged Scientists of Shandong Province, China (BS2013ZZ002), and Natural Science Foundation of Shandong Province, China (ZR2015EL023).
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