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Identification of workpiece location on rotary tables to minimize tracking errors in five-axis machining
Accepted Manuscript Identification of workpiece location on rotary tables to minimize tracking errors in fiveaxes machining Jixiang Yang, Deniz Aslan, Yusuf Altintas PII:
S0890-6955(17)30163-3
DOI:
10.1016/j.ijmachtools.2017.11.009
Reference:
MTM 3309
To appear in:
International Journal of Machine Tools and Manufacture
Received Date: 1 October 2017 Revised Date:
2 November 2017
Accepted Date: 10 November 2017
Please cite this article as: J. Yang, D. Aslan, Y. Altintas, Identification of workpiece location on rotary tables to minimize tracking errors in five-axes machining, International Journal of Machine Tools and Manufacture (2017), doi: 10.1016/j.ijmachtools.2017.11.009. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Identification of Workpiece Location on Rotary Tables to Minimize Tracking Errors in Five-Axes Machining
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Jixiang Yang*, Deniz Aslan, Yusuf Altintas Manufacturing Automation Laboratory, Department of Mechanical Engineering, The University of British Columbia, Vancouver, BC, Canada, V6T1Z4
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Abstract
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Five-axis CNC machine tools are widely used in machining parts with free form surfaces. This paper presents optimal placement of parts on the five-axis machine tool tables to minimize the tracking errors of the rotary servo drives. The cutting forces along the tool path are first simulated at the workpiece coordinate system in Computer-Aided-Manufacture (CAM) environment. The cutting torques transmitted to the rotary and translational drives are predicted using the location of the part on the table and kinematic configuration of the machine tool. The optimal location of the part on the table is identified by minimizing the forces transmitted to the rotary drives as torque disturbances. The proposed model has been experimentally validated on a five-axis machine with tilt-table configuration. It has been shown that the tracking and contouring errors can be significantly reduced with the proposed strategy, which can be used by process planners in digital simulation environment.
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Keywords: Cutting load, workpiece setup, tracking errors, five-axis, CNC machining
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Corresponding author. E-mail address: [email protected] (J.Yang).
Preprint submitted to International Journal of Machine Tools and Manufacture
November 2, 2017
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Identification of Workpiece Location on Rotary Tables to Minimize Tracking Errors in Five-Axes Machining
Abstract
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Five-axis CNC machine tools are widely used in machining parts with free form surfaces. This paper presents optimal placement of parts on the five-axis machine tool tables to minimize the tracking errors of the rotary servo drives. The cutting forces along the tool path are first simulated at the workpiece coordinate system in Computer-Aided-Manufacture (CAM) environment. The cutting torques transmitted to the rotary and translational drives are predicted using the location of the part on the table and kinematic configuration of the machine tool. The optimal location of the part on the table is identified by minimizing the forces transmitted to the rotary drives as torque disturbances. The proposed model has been experimentally validated on a five-axis machine with tilt-table configuration. It has been shown that the tracking and contouring errors can be significantly reduced with the proposed strategy, which can be used by process planners in digital simulation environment.
Nomenclature
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Keywords: Cutting load, workpiece setup, tracking errors, five-axis, CNC machining
Cutting forces acting on the workpiece Cutting loads transmitted to five axes Tool tip position in the workpiece coordinate system Tool orientation in the workpiece coordinate system Axes displacements in the machine coordinate system Tool pose Jacobian matrix Workpiece position relative to the work-table center Open loop transfer function of feed drive Position control loop of a servo drive Disturbance torque on feed drive Reference and actual position Axis tracking error Tool tip position and tool orientation contouring errors, respectively
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F = [Fx , Fy , Fz , nx , ny , nz ]T τ = [fx , fy , fz , τa , τc ]T P = [Px , Py , Pz ]T O = [Oi , Oj , Ok ]T q = [x, y, z, θa , θc ]T J wp = [wx , wy , wz ]T Gp Gc Td Pr , Pa e εp , εo
Preprint submitted to International Journal of Machine Tools and Manufacture
November 2, 2017
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1. Introduction
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Five-axis CNC machine tools are widely used in manufacturing parts with free-form surfaces, such as turbine blades, impellers, dies and molds. Traditionally, NC programs are planned to avoid collision, saturation of spindle and feed drives, excessive tool deflections while avoiding chatter and tool breakage. This paper presents the optimization of workpiece setup location to minimize tracking errors of rotary table contributed by the cutting forces transmitted to the rotary feed drives of the five-axis machine tools.
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Tool path is determined by the workpiece geometry and machine tool’s kinematic configuration. The geometry based optimization of NC programs is carried out at the ComputerAided-Manufacture (CAM) stage, where the cutter location (CL) data is generated before postprocessing it to run on a specific CNC Machine tool. The tool center position (TCP) and tool axis orientation (TAO) are planned to improve material removal rates [1], avoid gauging [2] and collision [3], and prevent kinematical singularities [4; 5] along the tool path. The physics based optimization algorithms are used to identify the feeds and spindle speeds by considering the physical constraints of the machine and process. The process based optimization methods consider chatter limit [6], maximum cutting forces to avoid tool breakage and excessive tool deflections [7–9], and feed profiling to avoid saturation of feed drives’ jerk and acceleration limits [10; 11].
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Part machining accuracy does not only depend on the tool deflections, but also on the kinematic and tracking errors of active servo drives along the tool path. The kinematic errors are minimized by reducing the axes motion through optimal selection of workpiece set-up on the machine tool table through Newton-Raphson [12] or particle swarm optimization [13] based search algorithms. Shaw et al. [14] developed a genetic algorithm to optimize setup location of a workpiece on a tilt-table type five-axis milling machine by minimizing three translational motions (x,y,z) during machining. Pessoles et al. [15] used a brute-force search method to achieve an optimal workpiece orientation that will minimize the machine’s translational distance. The workpiece location on the machine table also influences the collision of tool holder with the part and fixture. Hu et al. [16] presented a heuristic-based solution to search a collision free workpiece placement on a given five-axis machine tool. Cai et al. [17] considered collision free placement of prismatic parts by examining five-axis machine configurations. The workpiece setup highly affects the machine’s kinematic behavior, which dominates the overall processing time and energy consumption. Campatelli et al. [18] reduced the total energy consumption by adjusting the workpiece orientation about the Z-axis of the work-table. Xu et al. [19] considered both the positions and orientations of workpiece setup to minimize the energy consumption without modifying the tool path. Hu et al. [20] optimized the workpiece set-up and tools’ tilt and yaw angles by reducing the maximum angular acceleration of the rotary axes of the machine. Contouring errors left on the part surface are contributed by the projection of tracking errors on the curved tool path [21]. Servo tracking errors are dependent on the bandwidth of both position and external load disturbance transfer functions [22]. While modeling and compensation of servo contouring errors have been studied in the past, the optimal placement of workpiece on the machine to minimize the contouring errors induced by cutting loads has not seemed to be studied in the past. This paper presents the effect of workpiece placement location on the transmission of disturbance forces to feed drive motors, which lead to considerable rotary track2
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z
ing errors and five-axis contouring errors.It c is shown that while the cutting forces are directly transmitted to the translational drives, the workpiece location significantly alters the cutting Workpiece torque received by the rotary drives. Thepaper a mathematical model to select an 1 presents 2 optimum placement of workpiece while considering the working space of the five-axis machine tool. W p1 Table
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The paper is organized as follows: First, the transmission of cutting forces from the toola workpiece contact zone to machine tool feed drives as a function of workpiece location is derived in Section 2. The optimization of workpiece locationxis presented in Section 3. The proposed algorithms are validatedywith simulations and cutting tests in Section 4, where the effects on cutting forces on the tracking accuracy of rotary axes and contouring accuracy of five-axis motion are illustrated. The paper is concluded in Section 5. 2. Transmission of cutting forces from the workpiece to servo drives
Tool
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A five-axis machine tool with a tilting table architecture in Fig. 1 is used to illustrate the proposed model.
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Workpiece w p1
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Fig. 1. Illustration of workpiece setup optimization of five-axis CNC machining The cutting forces acting on the cutter-workpiece contact zone are measured in workpiece coordinate system as F = [Fx , Fy , Fz , nx , ny , nz ]T , where [Fx , Fy , Fz ]T are forces acting in linear directions (x, y, z) in the workpiece coordinate system and [nx , ny , nz ]T are moments around the three directions. These cutting forces are transmitted to machine tool servo drives as τ = [fx , fy , fz , τp , τs ]T , where [fx , fy , fz ]T are forces on the linear axes and [τp , τs ]T are torque on the the primary and secondary rotary axes in the kinematics chain counted from the workpiece to the cutting tool. The virtual position and orientation displacements caused by the cutting force vector in the workpiece coordinate system F are defined as δX = [δPx , δPy , δPz , δOi , δOj , δOk ]T , and the virtual translation and rotational displacements caused by drive load vector τ are δq = [δx, δy, δz, δθp , δθs ]T . The displacements at the drives can be transformed to workpiece coordinates using the kinematics of the five-axis machine as follows: 3
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δX = J · δq
←
∂P ∂O J= ; , ∂q ∂q
(1)
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where J is the Jacobian and P = [Px , Py , Pz ]T and O = [Oi , Oj , Ok ]T are the tool tip position and tool orientation related to the workpiece coordinate system. The virtual work of the entire machine tool in the two coordinate systems must be the same:
F T · δX − τ T · δq = F T J − τ T δq = 0.
(2)
=⇒
τ = J TF .
By combining Eq. (1), Eq. (3) as
∂θs
∂Py ∂x ∂Py ∂y ∂Py ∂z ∂Py ∂θp ∂Py ∂θs
∂Pz ∂x ∂Pz ∂y ∂Pz ∂z ∂Pz ∂θp ∂Pz ∂θs
∂Oi ∂x ∂Oi ∂y ∂Oi ∂z ∂Oi ∂θp ∂Oi ∂θs
∂Oj ∂x ∂Oj ∂y ∂Oj ∂z ∂Oj ∂θp ∂Oj ∂θs
(3)
∂Ok Fx ∂x ∂Ok Fy ∂y ∂Ok Fz ∂z . ∂Ok nx ∂θp ny ∂Ok nz ∂θs
(4)
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∂Px ∂x fx ∂Px fy ∂y ∂Px f τ = = ∂z z τp x ∂P ∂θp τs ∂Px
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τ T = F TJ
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Hence, the cutting forces at the tool tip are transmitted to the drives as:
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In summary, once the cutting forces acting on the cutter of a five-axis machine tool are known, their components acting on each axis can be determined by the differentials of the tool tip position and the tool orientation in terms of axis displacements. The differential kinematics model in Eq. (4) is determined by the machine topology, and derived for the A-C type tabletilting five-axis machine shown in Fig. 2 used in this paper. The translational X-axis carries the rotary table, and the Y-axis carries the Z-axis and the spindle. The rotary axis (C) is mounted on the tilting axis (A), and the workpiece is clamped on the rotary table. [Fx , Fy , Fz ]T are cutting forces acting on the tool in linear directions (x, y, z) of the workpiece coordinate system. [fx , fy , fz ]T and [τa , τc ]T are cutting force and torque components acting on linear axes and rotary axes, respectively, and they are parallel to five-axis motion commands [x, y, z, θa .θc ]T . wp = [wx , wy , wz ]T is the workpiece location related to the table center, which is the origin of the machine coordinate system. It must be pointed out that aside from location offsets of the workpiece setup, there can also be orientation variance of the workpiece related to the base frame, which will change the five drives’ motion commands and affect the cutting force transmitted to individual drives. In general, the workpiece orientation adjustment is much more complex than the location adjustment and need a specially designed fixture, which causes extra cost of CNC machining in practice. Hence, only the location of the workpiece setup is optimized in this paper. The forward kinematics transformation from five-axis motion commands [x, y, z, θa , θc ]T in the machine coordinate system to the tool tip position P = [Px , Py , Pz ]T and the tool orientation 4
a
b
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Y Z
wp
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Fy
z, f z
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Fx
Setup
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a , a
c , c
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A
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Fig. 2. Kinematics analysis of a table-tilting five-axis machine: (a) solid model; (b) kinematics chain
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O = [Oi , Oj , Ok ]T in the workpiece coordinate system can be obtained by using Homogeneous Transformation matrices as:
(5)
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Px = Cc (x + wox ) + Ca Sc (y + woy ) + Sa Sc (z + woz ) − Sa Sc Lz − wox , Py = −Sc (x + wox ) + Ca Cc (y + woy ) + Sa Cc (z + woz ) − Sa Cc Lz − woy , P = −S (y + w ) + C (z + w − L ) + L − w , z a oy a oz z z oz Oi = Sa Sc , Oj = Sa Cc , Ok = Ca ,
where Sa = sin(θa ), Ca = cos(θa ) , Sc = sin(θc ) and Cc = cos(θc ). Lz = −29.92mm is the offset between A-axis and C-axis in the z direction when motions of all axes equal to 0. The displacement of each axis is evaluated from the inverse kinematics solved from Eq. (5) as: θa = arccos(Ok ), θc = arctan(Oi , Oj ), x = Cc (Px + wox ) − Sc (Py + woy ) − wox , y = Ca Sc (Px + wox ) + Ca Cc (Py + woy ) − Sa (Pz + woz ) + Sa Lz − woy , z = S S (P + w ) + S C (P + w ) + C (P + w ) − C L + L − w . a c x ox a c y oy a z oz a z z oz 5
(6)
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The Jacobian function, which is defined as the differential of the forward kinematics function in Eq. (5) related to the five axes’ displacements, is given as:
Ca Sc Ca Cc −Sa 0 0 0
Sa Sc Sa Cc Ca 0 0 0
−Sa Sc (y + wy ) + Ca Sc (z + wz − Lz ) −Sa Cc (y + wy ) + Ca Cc (z + wz − Lz ) −Ca (y + wy ) − Sa (z + wz − Lz ) Ca Sc Ca Cc −Sa
−Sc (x + wx ) + Ca Cc (y + wy ) + Sa Cc (z + wz − Lz ) −Cc (x + wx ) − Ca Sc (y + wy ) − Sa Sc (z + wz − Lz ) 0 . Sa Cc −Sa Sc 0
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Cc −Sc 0 J = 0 0 0
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fx Cc fy fz = −Sc τ 0 a τc
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Since the contact zone between the cutter and the workpiece is small in contour milling, torque magnitudes [nx , ny , nz ]T acting on the cutter can be neglected. Therefore, only the force components [Fx , Fy , Fz ]T are considered here. Combining Eqs. (3)-(4) and (7), the cutting loads acting on individual drives of the five-axis machine in Fig. 2 are
Ca Sc Ca Cc −Sa
Sa Sc Sa Cc Ca
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−Sa Sc (y + wy ) + Ca Sc (z + wz − Lz ) −Sa Cc (y + wy ) + Ca Cc (z + wz − Lz ) −Ca (y + wy ) − Sa (z + wz − Lz )
−Sc (x + wx ) + Ca Cc (y + wy ) + Sa Cc (z + wz − Lz ) T Fx −Cc (x + wx ) − Ca Sc (y + wy ) − Sa Sc (z + wz − Lz ) Fy Fz 0
.
(8)
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From Eq. (8), the cutting loads acting on the translational axes are not affected by the workpiece clamping locations wp = [wx , wy , wz ]T . Hence, only the cutting loads acting on the rotary axes need to be considered to find the optimal location of workpiece
−Sa Sc (y + wy ) + Ca Sc (z + wz − Lz ) τa = −Sa Cc (y + wy ) + Ca Cc (z + wz − Lz ) τc −Ca (y + wy ) − Sa (z + wz − Lz )
T Fx −Sc (x + wx ) + Ca Cc (y + wy ) + Sa Cc (z + wz − Lz ) −Cc (x + wx ) − Ca Sc (y + wy ) − Sa Sc (z + wz − Lz ) Fy . 0 Fz
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Although the cutting load transmission model is derived for the A-C type table-tilting five-axis machine here, generalized cutting load transmission model for various five-axis topologies can be derived based on the five-axis generalized kinematics model introduced in [23; 24].
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3. Optimal Placement of Workpiece The objective is to place the workpiece on the allowable zone of the table which leads to minimum disturbance loads transmitted to the drives. The optimization process is designed as follows. The cutting forces acting on the tool-workpiece contact zone are predicted using - MACHpror Virtual Machining System developed at our laboratory [25]. The mechanics of machining behind - MACHpror can be found in past publications [8; 26]. Cutting forces are transmitted to the drives using the kinematics model of the machine as described in previous section. An objective function is designed to minimize the transmitted forces to the drives by searching the optimal location of the workpiece on the machine tool table as follows. The cutting loads transmitted to the rotary drives as torques (τa and τc ) can be expressed from Eq. (9) as a function of workpiece location as:
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wx + x τa 0 −Fx Sa Sc − Fy Sa Cc − Fz Ca Fx Ca Sc + Fy Ca Cc − Fz Sa wy + y = τc −Fx Sc − Fy Cc Fx Ca Cc − Fy Ca Sc Fx Sa Cc − Fy Sa Sc wz + z − Lz , wx = T m wy + T n , wz
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where
0 −Fx Sa Sc − Fy Sa Cc − Fz Ca Fx Ca Sc + Fy Ca Cc − Fz Sa Tm = , −Fx Sc − Fy Cc Fx Ca Cc − Fy Ca Sc Fx Sa Cc − Fy Sa Sc x 0 −Fx Sa Sc − Fy Sa Cc − Fz Ca Fx Ca Sc + Fy Ca Cc − Fz Sa y . Tn = −Fx Sc − Fy Cc Fx Ca Cc − Fy Ca Sc Fx Sa Cc − Fy Sa Sc z − Lz
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It can be observed from Eq.(10) that the transmitted torques to rotary drives are dependent on cutting forces (Tm , Tn ) and workpiece location on the table [wx , wy , wz ]T .
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A block-diagram of position control loop of a single-axis system is shown in Fig.3, where Pr and Pa are the reference and actual position, respectively. Gc is the transfer function of position controller, Gp is the transfer function of worm gear system, and Td is the equivalent disturbance torque acted on feed drive’s motor shaft. The tracking error of the position control loop is expressed as:
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Gp 1 Pr + Td . 1 + Gp Gc 1 + Gp Gc
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e=
Pr
e
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Gc
Td
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Fig. 3. Block-diagram of position control loop of single-axis system The transfer function shows that the tracking errors of the servo drives are directly proportional to disturbance loads on the drives. Therefore, the minimization of disturbance cutting torques transmitted to the drives would lead to minimized tracking errors of servo drives, and the contouring errors contributed by the projection of tracking errors on the curved tool path can also be reduced. From the cutting force decomposition model in Eq. (10), the disturbance 7
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min
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toques transmitted to rotary drives are affected by the workpiece placement. Hence, in order to minimize tracking errors of the rotary table and improve the contouring accuracy of five-axis CNC machining, the maximum magnitudes of the cutting forces are simulated along the tool path at Ml segment length intervals, and the optimal objective is to minimize the transmitted cutting loads to A and C axes as:
(|τai Mli | + |τci Mli |) ,
i=1
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where N is the number of tool path segments. If required, the optimal objectivecan also focus PN on reducing the transmitted cutting loads on individual axis of A or C, i.e. min |τ Ml | i i=1 ai P N or min i=1 |τci Mli | .
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The optimal variables are the workpiece location wp = [wx , wy , wz ]T related to the table center. The relation between the optimal objective (Eq. (13)) and the optimal variables is connected through the cutting torque calculation model in Eq. (10). Since the working space of table-tilting machine is limited, the workpiece should normally be placed near the center of the work-table. Thus, there should be a physical constraint on the location of workpiece setup in order to guarantee the accessibility of five-axis machine tool,
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(xw , yw , zw ) ∈ ([xwl , xwu ], [ywl , ywu ], [zwl , zwu ]) ,
where [xwl , xwu ] represents the lower and upper bounds of xw , and so forth for yw and zw . Eq. (14) includes the domain of the optimal variables.
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A brute-force search algorithm is used to find the optimal location of workpiece on the table by minimizing Eq. (13). Since the adjustment in Z-direction is tightly restricted by the height of workpiece block, the optimization is constrained to the adjustment of workpiece location in xy plane.
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The optimization algorithm is implemented as follows: (1) Cutting forces Fi = [Fxi , Fyi , Fzi ]T at discrete tool path locations [Pi , Oi ] (i = 1, 2, ..., N ) are simulated by the in-house developed MACHpror virtual machining software [25], which is used in industry. The discrete tool path segment interval Mli is selected by considering the tool path curvatures. (2) The feasible workpiece placement zone on the machine tool’s work-table is determined, and divided into m×n grids as shown in Fig. 4. The workpiece placement location at each grid is obtained as wp(j,k) = [wx(j,k) , wy(j,k) , wz(j,k) ]T , with j = 1, 2, ..., m and k = 1, 2, ..., n. (3) Substituting the workpiece placement wp(j,k) of each grid into the inverse kinematics model in Eq. (6), corresponding axes displacements qi(j,k) = [xi(j,k) , yi(j,k) , zi(j,k) , θai(j,k) , θci(j,k) ]T related to cutting locations [Pi , Oi ] are obtained. Then Tmi(j,k) , Tni(j,k) and the accumulated cutting P τami(j,k) Mli are evaluated from Eqs. (10)-(11). loads transmitted to the rotary axes N i=1 τ cmi(j,k) Mli 8
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(4) By using the objective in Eq. (13), the optimal workpiece location wp = [wx , wy , wz ]T that leads to minimized accumulated cutting load transmitted to the motor shafts of A and C-rotary axes are obtained.
Feasible setup zone
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Work-table
Fig. 4. Illustration of the grids division in feasible setup zone used for brute-force search algorithm
4. Simulation and Experiments
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The proposed workpiece setup optimization algorithm has been experimentally validated on Quaser UX-600 5-axis machining center as shown in Fig. 5(a). Kistler 9123C rotary dynamometer has been used to measure the cutting forces acting on the part. Torque and tracking errors of all servo drives have been collected in real time at 10kHz from Heidenhein CNC via TNC Ethernet connection.
Fig. 5. The A-C type table-tilting five-axis machine used in experiments: (a) machine tool platform; (b) rotary feed drive mechanism A workpiece with curved tool paths has been used in the simulations and experiments (Fig. 6(a)). The time stamped position commands of all five drives, which are extracted from the CNC in the workpiece coordinate system, are shown in Fig. 6(b). The workpiece material is Aluminum 7050 and cutting tool is a 16 mm diameter 2-fluted ball end mill with a regular pitch and 30 degrees helix angle. The spindle speed has been set to 1000 rev/min in the experiments. 9
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Fig. 6. Testing workpiece used in simulation and experiments: (a) solid model; (b) motion commands in the workpiece coordinate system
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The virtual machining system - MACHpror [25] has been used to simulate the cutting forces along the tool path. MACHpror receives the raw workpiece geometry in the form of an STL file and NC tool path in standard APT format, and uses cutter-workpiece engagement and process mechanics to predict the cutting forces Fi = [Fxi , Fyi , Fzi ]T at discrete (Ml = 2mm) tool path intervals [Pi , Oi ] (i = 1, 2, ..., N ). The simulated minimum-maximum force amplitudes have been validated experimentally in Fig. 7 along the tool path. Hence, the cutting forces simulated by MACHpror have acceptable accuracy to be used in the proposed optimal workpiece placement algorithm ahead of actual machining of parts.
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The brute force search algorithm introduced in Section 3 is used to estimate the optimal location of workpiece which leads to minimum cutting torque felt by the rotary drives. The size of the work-table is la = 380mm, lb = 510mm as illustrated in Fig. 4. The feasible workpiece setup zone on the work-table has been chosen as lm = 240mm, ln = 360mm, which is divided into 100 × 100 grids to conduct the brute-force search algorithm. The height of workpiece coordinate system related to the work-table surface in z-direction is fixed to wz = 60mm when searching the optimal set-up in the xy plane. The objective function given in Eq. (13) is used, where integrated cutting force components transmitted to the motor shafts of A- and C-axes are minimized. The optimized workpiece location wmin = [wmin,x , wmin,y ]T in xy plane has been obtained as wmin = [−34.5mm, −25.5mm]T . For clear comparison, the workpiece location wmax = [wmax,x , wmax,y ]T corresponding to the largest accumulated cutting load transmitted to the rotary axes in the feasible setup zone has also been found as wmax = [−180mm, 120mm]T . The transmitted cutting torque on two rotary drives have been simulated for the two workpiece locations as shown in Fig. 8. It can be seen that for the same tool path and cutting conditions, the optimized workpiece location yields to significantly reduced load transmission to the rotary drives. It should be noticed that load transmission reduction on rotary axes would also be ob10
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Fig. 7. Accuracy verification of cutting force simulation by MACHpror
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served comparing the optimized location with other locations in the feasible setup zone, because the optimized location is corresponding to the minimized transmitted cutting loads on A and C axes (Eq. (13)).
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First, the model of force transmission as a disturbance torque to feed drive motor has been experimentally validated. The force is transmitted to the translational drive through nutballscrew and to the rotary drive through worm gear (as shown in Fig. 5(b)). The disturbance transfer function between the cutting force on the table and torque received by the A and C rotary servo motors have been identified through impulse modal tests on the five-axis machine. The inertial acceleration and friction components of the servo motor torque are subtracted from the drive torque measurements to isolate the cutting components in identifying the disturbance transfer functions. An impulse force Fhi (ω) is given to each individual rotary drive and converted as an equivalent applied (input) torque τni (ω), where the response torque from the motor is τmi (ω) [27], with i = a, c representing A- and C-axis. The experimentally measured FRFs are identified by a modal curve fitting technique [22] which leads to the following transfer function (Φdi (s))
τmi (s) = τni (s) · Φdi (s), where Φdi (s) =
τmi (s) X αki + βki s = , (i = a, c), 2 2 τni (s) s + 2ζ ki ωnki s + ωnki k
(15)
where αki and βki , ζki , ωnki are the residues, damping and natural frequency of mode k, respectively. The specific parameters of the measured rotary drives are given in Table 1. Taking the workpiece location at [−180mm, 120mm]T as an example, the measured cutting forces have been passed through the disturbance transfer functions of the rotary drives to 11
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Fig. 8. Simulated cutting load components on rotary axes with optimal objective of P (|τ Ml min N ai i | + |τci Mli |) i=1
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Table 1. Modal parameters for rotary drives’ disturbance FRFs Mode
ωn (Hz)
ζ(%)
αk
βk
A-axis
1 2 3 4
19.9 50 63.86 81.31
0.69 0.47 0.22 0.16
−8053 −123072 105425 14992
−26.2 263.9 −132 −115
C-axis
1 2 3 4 5
45.3 52.6 66.3 75.2 98.9
0.14 0.086 0.035 0.022 0.052
−11820 70505 30915 9365 −1126
−244 453.6 16.6 6.44 −65.5
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estimate the torque felt by the servo motors. The predicted and measured torques at the motor agree at the cutting zone of the tool path as shown in Fig. 9.
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No motion, caused by static friction
No motion, caused by static friction
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Fig. 9. Calculated torques delivered to the servo compared with measured motor torques from the CNC in experiments (fixing location [−180mm, 120mm]T )
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The cutting loads transmitted to the translational drives are at the same level regardless of workpiece locations since the forces act parallel to the linear ball screw drives. However, the cutting forces are transmitted to the rotary drives with different amplitudes as a function of workpiece locations, see Fig. 10.
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The tracking errors of the rotary drives are proportional to the transmitted cutting torque (see Eq. (12)). The tracking errors of all five drives have been measured from the CNC during the tests and given in Fig. 11. Similar to the transmitted forces to the drives, the tracking errors of translational axes are at the same level regardless of setup location. However, the optimal placement of the workpiece, which gives the minimum transmitted force at the rotary drives, significantly reduces the tracking errors of rotary drives. As presented in Table 2, the maximum and mean tracking errors of the A-axis ea are reduced by 63.96% and 64.22% at the optimized workpiece location in comparison to un-optimized set-up, and the improvement is about 88.76% and 85.62% on the C-axis. The contouring error, which is defined as the path error between the actual and the reference trajectories [28], are passed as form errors on the machined part. The generalized five-axis contouring errors have been calculated from the axis tracking errors and tool path as demonstrated by Yang et al. [21]. The contouring errors predicted from the measured tracking errors of the machine are shown in Fig. 12, which show significant reduction in tool tip and tool orientation contouring errors along the tested tool path when the workpiece location is optimally selected. When the optimal and un-optimal set-ups are compared, the maximum and mean absolute values of the tool tip position contouring errors εp are reduced by 80.65% and 68.29% . The 13
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Fig. 10. Measured cutting torques on each axis in experiments
Table 2. Tracking errors of rotary axes before and after workpiece setup optimization in experiments max|ea | Before optimization After optimization 1−
After optimization Before optimization
× 100%
mean|ea |
max|ec |
mean|ec |
193.73[µrad] 82.29[µrad] 69.81[µrad] 29.44[µrad]
1055.9[µrad] 359.0[µrad] 118.68[µrad] 51.63[µrad]
63.96%
88.76%
64.22%
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Fig. 11. Tracking errors of each axis in experiments
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maximum and mean absolute values of the tool orientation contouring errors εo are also reduced by 84.04% and 79.71%, see Table 3.
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Fig. 12. Contouring errors of five-axis motion in experiments Table 3. Tool tip position and tool orientation contouring errors before and after workpiece setup optimization in experiments
After optimization Before optimization
× 100%
mean|εp |
max|εo |
mean|εo |
120.76[µm] 23.36[µm]
37.92[µm] 12.02[µm]
1.8605[mrad] 0.2970[mrad]
0.6253[mrad] 0.1269[mrad]
80.65%
68.29%
84.04%
79.71%
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5. Conclusion
The cutting forces are transmitted to the feed drives of machine tools as disturbance loads. The cutting forces are transmitted to the translational feed drives directly regardless of workpiece location on the table. However, due to moment arm effects of the cutting forces, the location of workpiece on the table significantly affects the amplitude of the torque transmitted to the rotary drives. This paper demonstrates that the optimal but physically allowable selection of workpiece location on the table can reduce the transmitted cutting load to the rotary drives significantly. Since the tracking errors of the servo drives are directly proportional to the disturbance cutting load on the drives, the contouring accuracy of the CNC can also be significantly improved, i.e. 68% in the sample case shown in the paper. The proposed optimal part placement strategy can 16
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be calculated in virtual environment ahead of costly physical trials, hence it can be used by process planners in industry. Acknowledgement
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This research is supported by the National Natural Science Foundation of China (NSFC) under Grant Nos. 51505167 and 51535004, the NSERC CANRIMT2 Strategic Research Network partners, and the International Postdoctoral Exchange Fellowship supported by China Postdoctoral Council. Industrial Technology Research Institute (ITRI) of Taiwan donated the CNC Machining Center. References
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[1] M. Fard and H. Feng. Effective determination of feed direction and tool orientation in five-axis flat-end milling. ASME Journal of Manufacturing Science and Engineering, 132(6):061011, 2010. [2] J. Chiou and Y. Lee. Optimal tool orientation for five-axis tool-end machining by swept envelope approach. ASME Journal of Manufacturing Science and Engineering, 127(4):810– 818, 2005.
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[3] K. Morishige, Y. Takeuchi, and K. Kase. Tool path generation using c-space for 5-axis control machining. ASME Journal of Manufacturing Science and Engineering, 121(1):144– 149, 1999. [4] A. Affouard, E. Duc, C. Lartigue, J. Langeron, and P. Bourdet. Avoiding 5-axis singularities using tool path deformation. International Journal of Machine Tools and Manufacture, 44(4):415–425, 2004.
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[5] Z. Lin, J. Fu, H. Shen, G. Xu, and Y. Sun. Improving machined surface texture in avoiding five-axis singularity with the acceptable-texture orientation region concept. International Journal of Machine Tools and Manufacture, 108:1–12, 2016.
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[6] Y. Altintas and E. Budak. Analytical prediction of stability lobes in milling. CIRP AnnalsManufacturing Technology, 44(1):357–362, 1995. [7] H. Erdim, I. Lazoglu, and B. Ozturk. Feedrate scheduling strategies for free-form surfaces. International Journal of Machine Tools and Manufacture, 46(7):747–757, 2006. [8] W. Ferry and Y. Altintas. Virtual five-axis flank milling of jet engine impellers - part ii: feed rate optimization of five-axis flank milling. ASME Journal of Manufacturing Science and Engineering, 130(1):011013, 2008. [9] K. Erkorkmaz, S. Layegh, I. Lazoglu, and H. Erdim. Feedrate optimization for freeform milling considering constraints from the feed drive system and process mechanics. CIRP Annals-Manufacturing Technology, 62(1):395–398, 2013.
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[10] J. Dong and J. Stori. Optimal feed-rate scheduling for high-speed contouring. ASME Journal of Manufacturing Science and Engineering, 129(1):63–76, 2007.
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[11] S. Mansour and R. Seethaler. Feedrate optimization for computer numerically controlled machine tools using modeled and measured process constraints. ASME Journal of Manufacturing Science and Engineering, 139(1):011012, 2017. [12] W. Anotaipaiboon, S. S Makhanov, and E. Bohez. Optimal setup for five-axis machining. International Journal of Machine Tools and Manufacture, 46(9):964–977, 2006.
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[13] Z. Lin, J. Fu, H. Shen, and W. Gan. On the workpiece setup optimization for five-axis machining with rtcp function. The International Journal of Advanced Manufacturing Technology, 74(1-4):187–197, 2014.
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[14] D. Shaw and G. Ou. Reducing x, y and z axes movement of a 5-axis ac type milling machine by changing the location of the work-piece. Computer-Aided Design, 40(10):1033– 1039, 2008. [15] X. Pessoles, Y. Landon, S. Segonds, and W. Rubio. Optimisation of workpiece setup for continuous five-axis milling: application to a five-axis bc type machining centre. The International Journal of Advanced Manufacturing Technology, 65(1-4):67–79, 2013. [16] P. Hu, K. Tang, and C. Lee. Global obstacle avoidance and minimum workpiece setups in five-axis machining. Computer-Aided Design, 45(10):1222–1237, 2013.
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[17] N. Cai, L. Wang, and H. Feng. Adaptive setup planning of prismatic parts for machine tools with varying configurations. International Journal of Production Research, 46(3):571–594, 2008.
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[18] G. Campatelli, A. Scippa, L. Lorenzini, and R. Sato. Optimal workpiece orientation to reduce the energy consumption of a milling process. International Journal of Precision Engineering and Manufacturing-Green Technology, 2(1):5–13, 2015.
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[19] K. Xu and K. Tang. Optimal workpiece setup for time-efficient and energy-saving five-axis machining of freeform surfaces. ASME Journal of Manufacturing Science and Engineering, 139(5):051003, 2017. [20] P. Hu and K. Tang. Improving the dynamics of five-axis machining through optimization of workpiece setup and tool orientations. Computer-Aided Design, 43(12):1693–1706, 2011. [21] J. Yang and Y. Altintas. A generalized on-line estimation and control of five-axis contouring errors of cnc machine tools. International Journal of Machine Tools and Manufacture, 88:9– 23, 2015. [22] Y. Altintas. Manufacturing automation: metal cutting mechanics, machine tool vibrations, and CNC design. Cambridge university press, 2012. [23] J. Yang, T. Huang, M. Yang, H. Ding, and H. Zhang. Generalized cutting loads decomposition model of five-axis serial machine tools based on the screw theory. The International Journal of Advanced Manufacturing Technology, 91(1-4):399–410, 2017. 18
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[24] J. Yang and Y. Altintas. Generalized kinematics of five-axis serial machines with nonsingular tool path generation. International Journal of Machine Tools and Manufacture, 75:119–132, 2013.
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[25] MACHpror , advanced virtual machining system, manufacturing automation laboratory, the university of british columbia, canada. http://www.malinc.com/products/machpro/. [26] S. Merdol and Y. Altintas. Virtual simulation and optimization of milling applications part ii: optimization and feedrate scheduling. ASME Journal of Manufacturing Science and Engineering, 130(5):051005, 2008.
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[27] Y. Altintas and D. Aslan. Integration of virtual and on-line machining process control and monitoring. CIRP Annals-Manufacturing Technology, 66(1):349–352, 2017.
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[28] B. Sencer, Y. Altintas, and E. Croft. Modeling and control of contouring errors for fiveaxis machine tools - part i: modeling. ASME Journal of Manufacturing Science and Engineering, 131(3):031006, 2009.
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Highlights
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1. A workpiece setup optimization algorithm is proposed to minimize tracking errors of rotary table in five-axis machining. 2. The cutting torques transmitted to rotary drives are predicted using the location of the workpiece on the table and the kinematic configuration of the machine tool.
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3. The optimal location of the workpiece on the table is identified by minimizing the forces transmitted to rotary drives as torque disturbances.
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4. The proposed algorithm has been experimentally validated on a five-axis machine, where rotary tracking errors and five-axis contouring errors are significantly reduced.
S0890-6955(17)30163-3
DOI:
10.1016/j.ijmachtools.2017.11.009
Reference:
MTM 3309
To appear in:
International Journal of Machine Tools and Manufacture
Received Date: 1 October 2017 Revised Date:
2 November 2017
Accepted Date: 10 November 2017
Please cite this article as: J. Yang, D. Aslan, Y. Altintas, Identification of workpiece location on rotary tables to minimize tracking errors in five-axes machining, International Journal of Machine Tools and Manufacture (2017), doi: 10.1016/j.ijmachtools.2017.11.009. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Identification of Workpiece Location on Rotary Tables to Minimize Tracking Errors in Five-Axes Machining
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Jixiang Yang*, Deniz Aslan, Yusuf Altintas Manufacturing Automation Laboratory, Department of Mechanical Engineering, The University of British Columbia, Vancouver, BC, Canada, V6T1Z4
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Abstract
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Five-axis CNC machine tools are widely used in machining parts with free form surfaces. This paper presents optimal placement of parts on the five-axis machine tool tables to minimize the tracking errors of the rotary servo drives. The cutting forces along the tool path are first simulated at the workpiece coordinate system in Computer-Aided-Manufacture (CAM) environment. The cutting torques transmitted to the rotary and translational drives are predicted using the location of the part on the table and kinematic configuration of the machine tool. The optimal location of the part on the table is identified by minimizing the forces transmitted to the rotary drives as torque disturbances. The proposed model has been experimentally validated on a five-axis machine with tilt-table configuration. It has been shown that the tracking and contouring errors can be significantly reduced with the proposed strategy, which can be used by process planners in digital simulation environment.
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Keywords: Cutting load, workpiece setup, tracking errors, five-axis, CNC machining
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Corresponding author. E-mail address: [email protected] (J.Yang).
Preprint submitted to International Journal of Machine Tools and Manufacture
November 2, 2017
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Identification of Workpiece Location on Rotary Tables to Minimize Tracking Errors in Five-Axes Machining
Abstract
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Five-axis CNC machine tools are widely used in machining parts with free form surfaces. This paper presents optimal placement of parts on the five-axis machine tool tables to minimize the tracking errors of the rotary servo drives. The cutting forces along the tool path are first simulated at the workpiece coordinate system in Computer-Aided-Manufacture (CAM) environment. The cutting torques transmitted to the rotary and translational drives are predicted using the location of the part on the table and kinematic configuration of the machine tool. The optimal location of the part on the table is identified by minimizing the forces transmitted to the rotary drives as torque disturbances. The proposed model has been experimentally validated on a five-axis machine with tilt-table configuration. It has been shown that the tracking and contouring errors can be significantly reduced with the proposed strategy, which can be used by process planners in digital simulation environment.
Nomenclature
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Keywords: Cutting load, workpiece setup, tracking errors, five-axis, CNC machining
Cutting forces acting on the workpiece Cutting loads transmitted to five axes Tool tip position in the workpiece coordinate system Tool orientation in the workpiece coordinate system Axes displacements in the machine coordinate system Tool pose Jacobian matrix Workpiece position relative to the work-table center Open loop transfer function of feed drive Position control loop of a servo drive Disturbance torque on feed drive Reference and actual position Axis tracking error Tool tip position and tool orientation contouring errors, respectively
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F = [Fx , Fy , Fz , nx , ny , nz ]T τ = [fx , fy , fz , τa , τc ]T P = [Px , Py , Pz ]T O = [Oi , Oj , Ok ]T q = [x, y, z, θa , θc ]T J wp = [wx , wy , wz ]T Gp Gc Td Pr , Pa e εp , εo
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November 2, 2017
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1. Introduction
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Five-axis CNC machine tools are widely used in manufacturing parts with free-form surfaces, such as turbine blades, impellers, dies and molds. Traditionally, NC programs are planned to avoid collision, saturation of spindle and feed drives, excessive tool deflections while avoiding chatter and tool breakage. This paper presents the optimization of workpiece setup location to minimize tracking errors of rotary table contributed by the cutting forces transmitted to the rotary feed drives of the five-axis machine tools.
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Tool path is determined by the workpiece geometry and machine tool’s kinematic configuration. The geometry based optimization of NC programs is carried out at the ComputerAided-Manufacture (CAM) stage, where the cutter location (CL) data is generated before postprocessing it to run on a specific CNC Machine tool. The tool center position (TCP) and tool axis orientation (TAO) are planned to improve material removal rates [1], avoid gauging [2] and collision [3], and prevent kinematical singularities [4; 5] along the tool path. The physics based optimization algorithms are used to identify the feeds and spindle speeds by considering the physical constraints of the machine and process. The process based optimization methods consider chatter limit [6], maximum cutting forces to avoid tool breakage and excessive tool deflections [7–9], and feed profiling to avoid saturation of feed drives’ jerk and acceleration limits [10; 11].
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Part machining accuracy does not only depend on the tool deflections, but also on the kinematic and tracking errors of active servo drives along the tool path. The kinematic errors are minimized by reducing the axes motion through optimal selection of workpiece set-up on the machine tool table through Newton-Raphson [12] or particle swarm optimization [13] based search algorithms. Shaw et al. [14] developed a genetic algorithm to optimize setup location of a workpiece on a tilt-table type five-axis milling machine by minimizing three translational motions (x,y,z) during machining. Pessoles et al. [15] used a brute-force search method to achieve an optimal workpiece orientation that will minimize the machine’s translational distance. The workpiece location on the machine table also influences the collision of tool holder with the part and fixture. Hu et al. [16] presented a heuristic-based solution to search a collision free workpiece placement on a given five-axis machine tool. Cai et al. [17] considered collision free placement of prismatic parts by examining five-axis machine configurations. The workpiece setup highly affects the machine’s kinematic behavior, which dominates the overall processing time and energy consumption. Campatelli et al. [18] reduced the total energy consumption by adjusting the workpiece orientation about the Z-axis of the work-table. Xu et al. [19] considered both the positions and orientations of workpiece setup to minimize the energy consumption without modifying the tool path. Hu et al. [20] optimized the workpiece set-up and tools’ tilt and yaw angles by reducing the maximum angular acceleration of the rotary axes of the machine. Contouring errors left on the part surface are contributed by the projection of tracking errors on the curved tool path [21]. Servo tracking errors are dependent on the bandwidth of both position and external load disturbance transfer functions [22]. While modeling and compensation of servo contouring errors have been studied in the past, the optimal placement of workpiece on the machine to minimize the contouring errors induced by cutting loads has not seemed to be studied in the past. This paper presents the effect of workpiece placement location on the transmission of disturbance forces to feed drive motors, which lead to considerable rotary track2
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ing errors and five-axis contouring errors.It c is shown that while the cutting forces are directly transmitted to the translational drives, the workpiece location significantly alters the cutting Workpiece torque received by the rotary drives. Thepaper a mathematical model to select an 1 presents 2 optimum placement of workpiece while considering the working space of the five-axis machine tool. W p1 Table
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The paper is organized as follows: First, the transmission of cutting forces from the toola workpiece contact zone to machine tool feed drives as a function of workpiece location is derived in Section 2. The optimization of workpiece locationxis presented in Section 3. The proposed algorithms are validatedywith simulations and cutting tests in Section 4, where the effects on cutting forces on the tracking accuracy of rotary axes and contouring accuracy of five-axis motion are illustrated. The paper is concluded in Section 5. 2. Transmission of cutting forces from the workpiece to servo drives
Tool
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A five-axis machine tool with a tilting table architecture in Fig. 1 is used to illustrate the proposed model.
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Fig. 1. Illustration of workpiece setup optimization of five-axis CNC machining The cutting forces acting on the cutter-workpiece contact zone are measured in workpiece coordinate system as F = [Fx , Fy , Fz , nx , ny , nz ]T , where [Fx , Fy , Fz ]T are forces acting in linear directions (x, y, z) in the workpiece coordinate system and [nx , ny , nz ]T are moments around the three directions. These cutting forces are transmitted to machine tool servo drives as τ = [fx , fy , fz , τp , τs ]T , where [fx , fy , fz ]T are forces on the linear axes and [τp , τs ]T are torque on the the primary and secondary rotary axes in the kinematics chain counted from the workpiece to the cutting tool. The virtual position and orientation displacements caused by the cutting force vector in the workpiece coordinate system F are defined as δX = [δPx , δPy , δPz , δOi , δOj , δOk ]T , and the virtual translation and rotational displacements caused by drive load vector τ are δq = [δx, δy, δz, δθp , δθs ]T . The displacements at the drives can be transformed to workpiece coordinates using the kinematics of the five-axis machine as follows: 3
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δX = J · δq
←
∂P ∂O J= ; , ∂q ∂q
(1)
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where J is the Jacobian and P = [Px , Py , Pz ]T and O = [Oi , Oj , Ok ]T are the tool tip position and tool orientation related to the workpiece coordinate system. The virtual work of the entire machine tool in the two coordinate systems must be the same:
F T · δX − τ T · δq = F T J − τ T δq = 0.
(2)
=⇒
τ = J TF .
By combining Eq. (1), Eq. (3) as
∂θs
∂Py ∂x ∂Py ∂y ∂Py ∂z ∂Py ∂θp ∂Py ∂θs
∂Pz ∂x ∂Pz ∂y ∂Pz ∂z ∂Pz ∂θp ∂Pz ∂θs
∂Oi ∂x ∂Oi ∂y ∂Oi ∂z ∂Oi ∂θp ∂Oi ∂θs
∂Oj ∂x ∂Oj ∂y ∂Oj ∂z ∂Oj ∂θp ∂Oj ∂θs
(3)
∂Ok Fx ∂x ∂Ok Fy ∂y ∂Ok Fz ∂z . ∂Ok nx ∂θp ny ∂Ok nz ∂θs
(4)
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∂Px ∂x fx ∂Px fy ∂y ∂Px f τ = = ∂z z τp x ∂P ∂θp τs ∂Px
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τ T = F TJ
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Hence, the cutting forces at the tool tip are transmitted to the drives as:
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In summary, once the cutting forces acting on the cutter of a five-axis machine tool are known, their components acting on each axis can be determined by the differentials of the tool tip position and the tool orientation in terms of axis displacements. The differential kinematics model in Eq. (4) is determined by the machine topology, and derived for the A-C type tabletilting five-axis machine shown in Fig. 2 used in this paper. The translational X-axis carries the rotary table, and the Y-axis carries the Z-axis and the spindle. The rotary axis (C) is mounted on the tilting axis (A), and the workpiece is clamped on the rotary table. [Fx , Fy , Fz ]T are cutting forces acting on the tool in linear directions (x, y, z) of the workpiece coordinate system. [fx , fy , fz ]T and [τa , τc ]T are cutting force and torque components acting on linear axes and rotary axes, respectively, and they are parallel to five-axis motion commands [x, y, z, θa .θc ]T . wp = [wx , wy , wz ]T is the workpiece location related to the table center, which is the origin of the machine coordinate system. It must be pointed out that aside from location offsets of the workpiece setup, there can also be orientation variance of the workpiece related to the base frame, which will change the five drives’ motion commands and affect the cutting force transmitted to individual drives. In general, the workpiece orientation adjustment is much more complex than the location adjustment and need a specially designed fixture, which causes extra cost of CNC machining in practice. Hence, only the location of the workpiece setup is optimized in this paper. The forward kinematics transformation from five-axis motion commands [x, y, z, θa , θc ]T in the machine coordinate system to the tool tip position P = [Px , Py , Pz ]T and the tool orientation 4
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O = [Oi , Oj , Ok ]T in the workpiece coordinate system can be obtained by using Homogeneous Transformation matrices as:
(5)
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Px = Cc (x + wox ) + Ca Sc (y + woy ) + Sa Sc (z + woz ) − Sa Sc Lz − wox , Py = −Sc (x + wox ) + Ca Cc (y + woy ) + Sa Cc (z + woz ) − Sa Cc Lz − woy , P = −S (y + w ) + C (z + w − L ) + L − w , z a oy a oz z z oz Oi = Sa Sc , Oj = Sa Cc , Ok = Ca ,
where Sa = sin(θa ), Ca = cos(θa ) , Sc = sin(θc ) and Cc = cos(θc ). Lz = −29.92mm is the offset between A-axis and C-axis in the z direction when motions of all axes equal to 0. The displacement of each axis is evaluated from the inverse kinematics solved from Eq. (5) as: θa = arccos(Ok ), θc = arctan(Oi , Oj ), x = Cc (Px + wox ) − Sc (Py + woy ) − wox , y = Ca Sc (Px + wox ) + Ca Cc (Py + woy ) − Sa (Pz + woz ) + Sa Lz − woy , z = S S (P + w ) + S C (P + w ) + C (P + w ) − C L + L − w . a c x ox a c y oy a z oz a z z oz 5
(6)
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The Jacobian function, which is defined as the differential of the forward kinematics function in Eq. (5) related to the five axes’ displacements, is given as:
Ca Sc Ca Cc −Sa 0 0 0
Sa Sc Sa Cc Ca 0 0 0
−Sa Sc (y + wy ) + Ca Sc (z + wz − Lz ) −Sa Cc (y + wy ) + Ca Cc (z + wz − Lz ) −Ca (y + wy ) − Sa (z + wz − Lz ) Ca Sc Ca Cc −Sa
−Sc (x + wx ) + Ca Cc (y + wy ) + Sa Cc (z + wz − Lz ) −Cc (x + wx ) − Ca Sc (y + wy ) − Sa Sc (z + wz − Lz ) 0 . Sa Cc −Sa Sc 0
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Cc −Sc 0 J = 0 0 0
(7)
fx Cc fy fz = −Sc τ 0 a τc
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Since the contact zone between the cutter and the workpiece is small in contour milling, torque magnitudes [nx , ny , nz ]T acting on the cutter can be neglected. Therefore, only the force components [Fx , Fy , Fz ]T are considered here. Combining Eqs. (3)-(4) and (7), the cutting loads acting on individual drives of the five-axis machine in Fig. 2 are
Ca Sc Ca Cc −Sa
Sa Sc Sa Cc Ca
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−Sa Sc (y + wy ) + Ca Sc (z + wz − Lz ) −Sa Cc (y + wy ) + Ca Cc (z + wz − Lz ) −Ca (y + wy ) − Sa (z + wz − Lz )
−Sc (x + wx ) + Ca Cc (y + wy ) + Sa Cc (z + wz − Lz ) T Fx −Cc (x + wx ) − Ca Sc (y + wy ) − Sa Sc (z + wz − Lz ) Fy Fz 0
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(8)
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From Eq. (8), the cutting loads acting on the translational axes are not affected by the workpiece clamping locations wp = [wx , wy , wz ]T . Hence, only the cutting loads acting on the rotary axes need to be considered to find the optimal location of workpiece
−Sa Sc (y + wy ) + Ca Sc (z + wz − Lz ) τa = −Sa Cc (y + wy ) + Ca Cc (z + wz − Lz ) τc −Ca (y + wy ) − Sa (z + wz − Lz )
T Fx −Sc (x + wx ) + Ca Cc (y + wy ) + Sa Cc (z + wz − Lz ) −Cc (x + wx ) − Ca Sc (y + wy ) − Sa Sc (z + wz − Lz ) Fy . 0 Fz
(9)
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Although the cutting load transmission model is derived for the A-C type table-tilting five-axis machine here, generalized cutting load transmission model for various five-axis topologies can be derived based on the five-axis generalized kinematics model introduced in [23; 24].
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3. Optimal Placement of Workpiece The objective is to place the workpiece on the allowable zone of the table which leads to minimum disturbance loads transmitted to the drives. The optimization process is designed as follows. The cutting forces acting on the tool-workpiece contact zone are predicted using - MACHpror Virtual Machining System developed at our laboratory [25]. The mechanics of machining behind - MACHpror can be found in past publications [8; 26]. Cutting forces are transmitted to the drives using the kinematics model of the machine as described in previous section. An objective function is designed to minimize the transmitted forces to the drives by searching the optimal location of the workpiece on the machine tool table as follows. The cutting loads transmitted to the rotary drives as torques (τa and τc ) can be expressed from Eq. (9) as a function of workpiece location as:
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wx + x τa 0 −Fx Sa Sc − Fy Sa Cc − Fz Ca Fx Ca Sc + Fy Ca Cc − Fz Sa wy + y = τc −Fx Sc − Fy Cc Fx Ca Cc − Fy Ca Sc Fx Sa Cc − Fy Sa Sc wz + z − Lz , wx = T m wy + T n , wz
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where
0 −Fx Sa Sc − Fy Sa Cc − Fz Ca Fx Ca Sc + Fy Ca Cc − Fz Sa Tm = , −Fx Sc − Fy Cc Fx Ca Cc − Fy Ca Sc Fx Sa Cc − Fy Sa Sc x 0 −Fx Sa Sc − Fy Sa Cc − Fz Ca Fx Ca Sc + Fy Ca Cc − Fz Sa y . Tn = −Fx Sc − Fy Cc Fx Ca Cc − Fy Ca Sc Fx Sa Cc − Fy Sa Sc z − Lz
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(10)
(11)
It can be observed from Eq.(10) that the transmitted torques to rotary drives are dependent on cutting forces (Tm , Tn ) and workpiece location on the table [wx , wy , wz ]T .
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A block-diagram of position control loop of a single-axis system is shown in Fig.3, where Pr and Pa are the reference and actual position, respectively. Gc is the transfer function of position controller, Gp is the transfer function of worm gear system, and Td is the equivalent disturbance torque acted on feed drive’s motor shaft. The tracking error of the position control loop is expressed as:
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Fig. 3. Block-diagram of position control loop of single-axis system The transfer function shows that the tracking errors of the servo drives are directly proportional to disturbance loads on the drives. Therefore, the minimization of disturbance cutting torques transmitted to the drives would lead to minimized tracking errors of servo drives, and the contouring errors contributed by the projection of tracking errors on the curved tool path can also be reduced. From the cutting force decomposition model in Eq. (10), the disturbance 7
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min
N X
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toques transmitted to rotary drives are affected by the workpiece placement. Hence, in order to minimize tracking errors of the rotary table and improve the contouring accuracy of five-axis CNC machining, the maximum magnitudes of the cutting forces are simulated along the tool path at Ml segment length intervals, and the optimal objective is to minimize the transmitted cutting loads to A and C axes as:
(|τai Mli | + |τci Mli |) ,
i=1
(13)
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where N is the number of tool path segments. If required, the optimal objectivecan also focus PN on reducing the transmitted cutting loads on individual axis of A or C, i.e. min |τ Ml | i i=1 ai P N or min i=1 |τci Mli | .
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The optimal variables are the workpiece location wp = [wx , wy , wz ]T related to the table center. The relation between the optimal objective (Eq. (13)) and the optimal variables is connected through the cutting torque calculation model in Eq. (10). Since the working space of table-tilting machine is limited, the workpiece should normally be placed near the center of the work-table. Thus, there should be a physical constraint on the location of workpiece setup in order to guarantee the accessibility of five-axis machine tool,
(14)
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(xw , yw , zw ) ∈ ([xwl , xwu ], [ywl , ywu ], [zwl , zwu ]) ,
where [xwl , xwu ] represents the lower and upper bounds of xw , and so forth for yw and zw . Eq. (14) includes the domain of the optimal variables.
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A brute-force search algorithm is used to find the optimal location of workpiece on the table by minimizing Eq. (13). Since the adjustment in Z-direction is tightly restricted by the height of workpiece block, the optimization is constrained to the adjustment of workpiece location in xy plane.
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The optimization algorithm is implemented as follows: (1) Cutting forces Fi = [Fxi , Fyi , Fzi ]T at discrete tool path locations [Pi , Oi ] (i = 1, 2, ..., N ) are simulated by the in-house developed MACHpror virtual machining software [25], which is used in industry. The discrete tool path segment interval Mli is selected by considering the tool path curvatures. (2) The feasible workpiece placement zone on the machine tool’s work-table is determined, and divided into m×n grids as shown in Fig. 4. The workpiece placement location at each grid is obtained as wp(j,k) = [wx(j,k) , wy(j,k) , wz(j,k) ]T , with j = 1, 2, ..., m and k = 1, 2, ..., n. (3) Substituting the workpiece placement wp(j,k) of each grid into the inverse kinematics model in Eq. (6), corresponding axes displacements qi(j,k) = [xi(j,k) , yi(j,k) , zi(j,k) , θai(j,k) , θci(j,k) ]T related to cutting locations [Pi , Oi ] are obtained. Then Tmi(j,k) , Tni(j,k) and the accumulated cutting P τami(j,k) Mli are evaluated from Eqs. (10)-(11). loads transmitted to the rotary axes N i=1 τ cmi(j,k) Mli 8
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(4) By using the objective in Eq. (13), the optimal workpiece location wp = [wx , wy , wz ]T that leads to minimized accumulated cutting load transmitted to the motor shafts of A and C-rotary axes are obtained.
Feasible setup zone
ln
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la
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Work-table
Fig. 4. Illustration of the grids division in feasible setup zone used for brute-force search algorithm
4. Simulation and Experiments
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The proposed workpiece setup optimization algorithm has been experimentally validated on Quaser UX-600 5-axis machining center as shown in Fig. 5(a). Kistler 9123C rotary dynamometer has been used to measure the cutting forces acting on the part. Torque and tracking errors of all servo drives have been collected in real time at 10kHz from Heidenhein CNC via TNC Ethernet connection.
Fig. 5. The A-C type table-tilting five-axis machine used in experiments: (a) machine tool platform; (b) rotary feed drive mechanism A workpiece with curved tool paths has been used in the simulations and experiments (Fig. 6(a)). The time stamped position commands of all five drives, which are extracted from the CNC in the workpiece coordinate system, are shown in Fig. 6(b). The workpiece material is Aluminum 7050 and cutting tool is a 16 mm diameter 2-fluted ball end mill with a regular pitch and 30 degrees helix angle. The spindle speed has been set to 1000 rev/min in the experiments. 9
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Fig. 6. Testing workpiece used in simulation and experiments: (a) solid model; (b) motion commands in the workpiece coordinate system
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The virtual machining system - MACHpror [25] has been used to simulate the cutting forces along the tool path. MACHpror receives the raw workpiece geometry in the form of an STL file and NC tool path in standard APT format, and uses cutter-workpiece engagement and process mechanics to predict the cutting forces Fi = [Fxi , Fyi , Fzi ]T at discrete (Ml = 2mm) tool path intervals [Pi , Oi ] (i = 1, 2, ..., N ). The simulated minimum-maximum force amplitudes have been validated experimentally in Fig. 7 along the tool path. Hence, the cutting forces simulated by MACHpror have acceptable accuracy to be used in the proposed optimal workpiece placement algorithm ahead of actual machining of parts.
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The brute force search algorithm introduced in Section 3 is used to estimate the optimal location of workpiece which leads to minimum cutting torque felt by the rotary drives. The size of the work-table is la = 380mm, lb = 510mm as illustrated in Fig. 4. The feasible workpiece setup zone on the work-table has been chosen as lm = 240mm, ln = 360mm, which is divided into 100 × 100 grids to conduct the brute-force search algorithm. The height of workpiece coordinate system related to the work-table surface in z-direction is fixed to wz = 60mm when searching the optimal set-up in the xy plane. The objective function given in Eq. (13) is used, where integrated cutting force components transmitted to the motor shafts of A- and C-axes are minimized. The optimized workpiece location wmin = [wmin,x , wmin,y ]T in xy plane has been obtained as wmin = [−34.5mm, −25.5mm]T . For clear comparison, the workpiece location wmax = [wmax,x , wmax,y ]T corresponding to the largest accumulated cutting load transmitted to the rotary axes in the feasible setup zone has also been found as wmax = [−180mm, 120mm]T . The transmitted cutting torque on two rotary drives have been simulated for the two workpiece locations as shown in Fig. 8. It can be seen that for the same tool path and cutting conditions, the optimized workpiece location yields to significantly reduced load transmission to the rotary drives. It should be noticed that load transmission reduction on rotary axes would also be ob10
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Fig. 7. Accuracy verification of cutting force simulation by MACHpror
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served comparing the optimized location with other locations in the feasible setup zone, because the optimized location is corresponding to the minimized transmitted cutting loads on A and C axes (Eq. (13)).
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First, the model of force transmission as a disturbance torque to feed drive motor has been experimentally validated. The force is transmitted to the translational drive through nutballscrew and to the rotary drive through worm gear (as shown in Fig. 5(b)). The disturbance transfer function between the cutting force on the table and torque received by the A and C rotary servo motors have been identified through impulse modal tests on the five-axis machine. The inertial acceleration and friction components of the servo motor torque are subtracted from the drive torque measurements to isolate the cutting components in identifying the disturbance transfer functions. An impulse force Fhi (ω) is given to each individual rotary drive and converted as an equivalent applied (input) torque τni (ω), where the response torque from the motor is τmi (ω) [27], with i = a, c representing A- and C-axis. The experimentally measured FRFs are identified by a modal curve fitting technique [22] which leads to the following transfer function (Φdi (s))
τmi (s) = τni (s) · Φdi (s), where Φdi (s) =
τmi (s) X αki + βki s = , (i = a, c), 2 2 τni (s) s + 2ζ ki ωnki s + ωnki k
(15)
where αki and βki , ζki , ωnki are the residues, damping and natural frequency of mode k, respectively. The specific parameters of the measured rotary drives are given in Table 1. Taking the workpiece location at [−180mm, 120mm]T as an example, the measured cutting forces have been passed through the disturbance transfer functions of the rotary drives to 11
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Fig. 8. Simulated cutting load components on rotary axes with optimal objective of P (|τ Ml min N ai i | + |τci Mli |) i=1
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Table 1. Modal parameters for rotary drives’ disturbance FRFs Mode
ωn (Hz)
ζ(%)
αk
βk
A-axis
1 2 3 4
19.9 50 63.86 81.31
0.69 0.47 0.22 0.16
−8053 −123072 105425 14992
−26.2 263.9 −132 −115
C-axis
1 2 3 4 5
45.3 52.6 66.3 75.2 98.9
0.14 0.086 0.035 0.022 0.052
−11820 70505 30915 9365 −1126
−244 453.6 16.6 6.44 −65.5
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estimate the torque felt by the servo motors. The predicted and measured torques at the motor agree at the cutting zone of the tool path as shown in Fig. 9.
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Fig. 9. Calculated torques delivered to the servo compared with measured motor torques from the CNC in experiments (fixing location [−180mm, 120mm]T )
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The cutting loads transmitted to the translational drives are at the same level regardless of workpiece locations since the forces act parallel to the linear ball screw drives. However, the cutting forces are transmitted to the rotary drives with different amplitudes as a function of workpiece locations, see Fig. 10.
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The tracking errors of the rotary drives are proportional to the transmitted cutting torque (see Eq. (12)). The tracking errors of all five drives have been measured from the CNC during the tests and given in Fig. 11. Similar to the transmitted forces to the drives, the tracking errors of translational axes are at the same level regardless of setup location. However, the optimal placement of the workpiece, which gives the minimum transmitted force at the rotary drives, significantly reduces the tracking errors of rotary drives. As presented in Table 2, the maximum and mean tracking errors of the A-axis ea are reduced by 63.96% and 64.22% at the optimized workpiece location in comparison to un-optimized set-up, and the improvement is about 88.76% and 85.62% on the C-axis. The contouring error, which is defined as the path error between the actual and the reference trajectories [28], are passed as form errors on the machined part. The generalized five-axis contouring errors have been calculated from the axis tracking errors and tool path as demonstrated by Yang et al. [21]. The contouring errors predicted from the measured tracking errors of the machine are shown in Fig. 12, which show significant reduction in tool tip and tool orientation contouring errors along the tested tool path when the workpiece location is optimally selected. When the optimal and un-optimal set-ups are compared, the maximum and mean absolute values of the tool tip position contouring errors εp are reduced by 80.65% and 68.29% . The 13
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Fig. 10. Measured cutting torques on each axis in experiments
Table 2. Tracking errors of rotary axes before and after workpiece setup optimization in experiments max|ea | Before optimization After optimization 1−
After optimization Before optimization
× 100%
mean|ea |
max|ec |
mean|ec |
193.73[µrad] 82.29[µrad] 69.81[µrad] 29.44[µrad]
1055.9[µrad] 359.0[µrad] 118.68[µrad] 51.63[µrad]
63.96%
88.76%
64.22%
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Fig. 11. Tracking errors of each axis in experiments
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maximum and mean absolute values of the tool orientation contouring errors εo are also reduced by 84.04% and 79.71%, see Table 3.
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Fig. 12. Contouring errors of five-axis motion in experiments Table 3. Tool tip position and tool orientation contouring errors before and after workpiece setup optimization in experiments
After optimization Before optimization
× 100%
mean|εp |
max|εo |
mean|εo |
120.76[µm] 23.36[µm]
37.92[µm] 12.02[µm]
1.8605[mrad] 0.2970[mrad]
0.6253[mrad] 0.1269[mrad]
80.65%
68.29%
84.04%
79.71%
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5. Conclusion
The cutting forces are transmitted to the feed drives of machine tools as disturbance loads. The cutting forces are transmitted to the translational feed drives directly regardless of workpiece location on the table. However, due to moment arm effects of the cutting forces, the location of workpiece on the table significantly affects the amplitude of the torque transmitted to the rotary drives. This paper demonstrates that the optimal but physically allowable selection of workpiece location on the table can reduce the transmitted cutting load to the rotary drives significantly. Since the tracking errors of the servo drives are directly proportional to the disturbance cutting load on the drives, the contouring accuracy of the CNC can also be significantly improved, i.e. 68% in the sample case shown in the paper. The proposed optimal part placement strategy can 16
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be calculated in virtual environment ahead of costly physical trials, hence it can be used by process planners in industry. Acknowledgement
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This research is supported by the National Natural Science Foundation of China (NSFC) under Grant Nos. 51505167 and 51535004, the NSERC CANRIMT2 Strategic Research Network partners, and the International Postdoctoral Exchange Fellowship supported by China Postdoctoral Council. Industrial Technology Research Institute (ITRI) of Taiwan donated the CNC Machining Center. References
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[1] M. Fard and H. Feng. Effective determination of feed direction and tool orientation in five-axis flat-end milling. ASME Journal of Manufacturing Science and Engineering, 132(6):061011, 2010. [2] J. Chiou and Y. Lee. Optimal tool orientation for five-axis tool-end machining by swept envelope approach. ASME Journal of Manufacturing Science and Engineering, 127(4):810– 818, 2005.
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[3] K. Morishige, Y. Takeuchi, and K. Kase. Tool path generation using c-space for 5-axis control machining. ASME Journal of Manufacturing Science and Engineering, 121(1):144– 149, 1999. [4] A. Affouard, E. Duc, C. Lartigue, J. Langeron, and P. Bourdet. Avoiding 5-axis singularities using tool path deformation. International Journal of Machine Tools and Manufacture, 44(4):415–425, 2004.
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[6] Y. Altintas and E. Budak. Analytical prediction of stability lobes in milling. CIRP AnnalsManufacturing Technology, 44(1):357–362, 1995. [7] H. Erdim, I. Lazoglu, and B. Ozturk. Feedrate scheduling strategies for free-form surfaces. International Journal of Machine Tools and Manufacture, 46(7):747–757, 2006. [8] W. Ferry and Y. Altintas. Virtual five-axis flank milling of jet engine impellers - part ii: feed rate optimization of five-axis flank milling. ASME Journal of Manufacturing Science and Engineering, 130(1):011013, 2008. [9] K. Erkorkmaz, S. Layegh, I. Lazoglu, and H. Erdim. Feedrate optimization for freeform milling considering constraints from the feed drive system and process mechanics. CIRP Annals-Manufacturing Technology, 62(1):395–398, 2013.
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[10] J. Dong and J. Stori. Optimal feed-rate scheduling for high-speed contouring. ASME Journal of Manufacturing Science and Engineering, 129(1):63–76, 2007.
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[11] S. Mansour and R. Seethaler. Feedrate optimization for computer numerically controlled machine tools using modeled and measured process constraints. ASME Journal of Manufacturing Science and Engineering, 139(1):011012, 2017. [12] W. Anotaipaiboon, S. S Makhanov, and E. Bohez. Optimal setup for five-axis machining. International Journal of Machine Tools and Manufacture, 46(9):964–977, 2006.
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[13] Z. Lin, J. Fu, H. Shen, and W. Gan. On the workpiece setup optimization for five-axis machining with rtcp function. The International Journal of Advanced Manufacturing Technology, 74(1-4):187–197, 2014.
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[14] D. Shaw and G. Ou. Reducing x, y and z axes movement of a 5-axis ac type milling machine by changing the location of the work-piece. Computer-Aided Design, 40(10):1033– 1039, 2008. [15] X. Pessoles, Y. Landon, S. Segonds, and W. Rubio. Optimisation of workpiece setup for continuous five-axis milling: application to a five-axis bc type machining centre. The International Journal of Advanced Manufacturing Technology, 65(1-4):67–79, 2013. [16] P. Hu, K. Tang, and C. Lee. Global obstacle avoidance and minimum workpiece setups in five-axis machining. Computer-Aided Design, 45(10):1222–1237, 2013.
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[19] K. Xu and K. Tang. Optimal workpiece setup for time-efficient and energy-saving five-axis machining of freeform surfaces. ASME Journal of Manufacturing Science and Engineering, 139(5):051003, 2017. [20] P. Hu and K. Tang. Improving the dynamics of five-axis machining through optimization of workpiece setup and tool orientations. Computer-Aided Design, 43(12):1693–1706, 2011. [21] J. Yang and Y. Altintas. A generalized on-line estimation and control of five-axis contouring errors of cnc machine tools. International Journal of Machine Tools and Manufacture, 88:9– 23, 2015. [22] Y. Altintas. Manufacturing automation: metal cutting mechanics, machine tool vibrations, and CNC design. Cambridge university press, 2012. [23] J. Yang, T. Huang, M. Yang, H. Ding, and H. Zhang. Generalized cutting loads decomposition model of five-axis serial machine tools based on the screw theory. The International Journal of Advanced Manufacturing Technology, 91(1-4):399–410, 2017. 18
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[24] J. Yang and Y. Altintas. Generalized kinematics of five-axis serial machines with nonsingular tool path generation. International Journal of Machine Tools and Manufacture, 75:119–132, 2013.
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Highlights
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1. A workpiece setup optimization algorithm is proposed to minimize tracking errors of rotary table in five-axis machining. 2. The cutting torques transmitted to rotary drives are predicted using the location of the workpiece on the table and the kinematic configuration of the machine tool.
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3. The optimal location of the workpiece on the table is identified by minimizing the forces transmitted to rotary drives as torque disturbances.
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4. The proposed algorithm has been experimentally validated on a five-axis machine, where rotary tracking errors and five-axis contouring errors are significantly reduced.