Accepted Manuscript

Comparing the kinematic efficiency of five-axis machine toolconfigurations through nonlinearity errors

O. Remus Tutunea-Fatan, Md Shafayet H. Bhuiya

PII: S0010-4485(11)00113-8DOI: 10.1016/j.cad.2011.05.003Reference: JCAD 1779

To appear in: Computer-Aided Design

Received date: 28 October 2010Accepted date: 8 May 2011

Please cite this article as: Remus Tutunea-Fatan O, Bhuiya MSH. Comparing the kinematicefficiency of five-axis machine tool configurations through nonlinearity errors.Computer-Aided Design (2011), doi:10.1016/j.cad.2011.05.003

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Comparing the kinematic efficiency of five-axis machine tool configurations

through nonlinearity errors

O. Remus Tutunea-Fatan1, Md Shafayet H. Bhuiya Department of Mechanical and Materials Engineering

The University of Western Ontario London, Ontario Canada N6A 5B9

Five-axis CNC machines represent a particular class of machine tools characterized by superior

versatility. Little attempts were made in the past to compare directly their performances through a

common indicator. In this sense, the present study proposes nonlinearity error as a valuable

method to quantify the kinematic efficiency of a particular five-axis configuration. Nonlinearity

error is defined as the maximum deviation of the cutter-location point from the reference plane

generated by the initial and final orientations of the tool during linearly-interpolated motions of

the cutter along the intended tool path. The proposed concept has demonstrated that nonlinearity

error occurs approximately around the middle of the linearly-interpolated interval and therefore

has validated the current post-processing practice of halfway cutter-location point insertion. The

employment of nonlinearity error in evaluation of the kinematic efficiency of vertical spindle-

rotating five-axis machine tools revealed that for an identical machining task, configurations

involving the vertical rotational axis tend to move more than those involving only horizontal

rotational axes.

Keywords: five-axis machine tools; vertical spindle-rotating configuration; kinematic efficiency; nonlinearity error

1 Corresponding author. Tel.: +1-519-661-2111, ext. 88289; fax: +1-519-661-3020; E-mail: rtutunea@eng.uwo.ca

  

1. Introduction

Five-axis CNC machining represents one of the most effective material removal

technologies used in manufacturing of a wide range of moulds, turbine blades,

automotive and aerospace parts whose geometries are typically defined by complex

surfaces [1]. Over the past decade, five-axis machine tools have already proved their

superiority compared to their three-axis counterparts. The most quoted advantages of

five-axis machines are their increased productivity and accuracy that enable significant

reductions of the manual polishing time that in some cases could amount up to 66-75% of

the total machining time [2]. However, their increased versatility is often achieved at the

expense of more complex tool path generation methods and higher initial investment

costs.

Both advantages and drawbacks of five-axis machine tools are a consequence of

their complex kinematic configurations that typically incorporates three translational and

two rotational joints [3]. At least from a theoretical standpoint, other combinations are

also possible, but it seems that this particular combination – that is practically an

extension of the traditional three-axis configuration – meets to a satisfactory level the

needs of the machining process. The most common design solution involves reciprocally

orthogonal axes for the five joints involved, although some other options are available,

typically involving nutating axes [4]. Since both tool posture and its relative motion with

respect of the workpiece are practically influenced by the configuration of the five-axis

machine tools, a large variety of research papers have investigated their kinematic

behaviour from different perspectives.

  

Among them, a first category of studies was mainly concerned with their

taxonomy and notable efforts were made in this direction. For instance, Ishizawa et al. [5]

were among the first to propose a classification scheme for the five-axis machines,

accompanied by a detailed outline of the essential differences between their basic

structural configurations. A few years later, Sakamoto and Inasaki [6] performed critical

assessments of the possible arrangements of the translational slides within the kinematic

chain of the five-axis machine. Other comprehensive classifications and structural

synthesis schemes were proposed by Bohez [3] and Chen [7].

A second important research direction was concerned with determination of the

errors introduced by the machine tool kinematics. This particular type of errors influences

tool posture along the tool path. One of the earliest studies in this direction was proposed

by Kiridena and Ferreira [8] who analyzed the influence of positional joint accuracy on

cutting tool position and orientation. Later studies have proposed models that were able

to predict various other types of errors generated by the kinematics of the five-axis,

including but not limited to geometric, thermal and link inaccuracies [9-12]. The vast

majority of these researchers have also addressed the need for appropriate corrective

measures that were intended to specifically compensate for this type of error.

A particular class of errors introduced by the kinematic chain of the five-axis

machines are the nonlinearity errors that are practically generated by the superposition

between translational and rotational motions. They determine the relative position and

orientation of the tool with respect of the workpiece as it travels along the intended tool

path. The existence of these errors has been acknowledged since early days of five-axis

machining [13], but extremely limited attempts were made to compensate them [14,15].

  

The most common method employed by the industrial practice in order to reduce their

amount is based on the initial suggestions made by Takeuchi and Watanabe [16] and Cho

et al. [13]. According to this approach, denoted by the term tool path linearization,

additional cutter-contact (CL) points are inserted by the post-processor typically midway

between the points originally determined by the Computer-Aided Manufacturing (CAM)

software during tool path generation algorithm [4,10,17].

The third category of research studies investigating the kinematics of five-axis

machine tools was concerned with the development of appropriate kinematic models

primarily intended for post-processing or tool motion analysis purposes. While only few

authors have proposed generalized solutions [4,18,19], a large variety of particular cases

are discussed by the majority of papers dealing with five-axis machining topics, since

conversion of the tool posture data from workpiece coordinate system into machine

coordinate system represents a mandatory step of tool path generation algorithm [20].

Despite the relatively large number of research papers focused on kinematics of

the five-axis machines, very limited attempts were made to compare their structures

through a common performance indicator. One of the earliest comparison measures

proposed for this purpose was the maximum linear movement area required to machine a

square workpiece [6,7]. Later, Bohez [3] indicated a minimal set of criteria to be used in

selecting a five-axis machine structure that would be suitable for a certain machining

task. Nonlinearity of tool motion caused by the two mutually orthogonal rotary axes

represents one of the important sources of error in five-axis machining and it also

translates into inefficient machine tool movements. Nonlinearity errors are typically

associated with current offline tool path generation methods based on linearly-

  

interpolated motions [14,21]. Being generated by the inherent kinematics of the five-axis

machine tool, these machining errors can be reduced but not completely eliminated

through enhanced NC post-processing techniques.

As a result, the present study will employ nonlinearity error as a measure of the

kinematic efficiency associated with a particular five-axis machine configuration.

Kinematic efficiency translates into a reduced amount of motions on translational and

rotational axes that in turn translates into energetically efficient machining operations.

Detailed methodology on its calculation, along with its specific values for all feasible

configurations of vertical spindle-rotating five-axis machine with reciprocally orthogonal

and intersecting rotational axes will be provided in the following sections. The two

rotational axes of the analyzed five-axis machine tool configurations are always assumed

at the end of the kinematic chain, since this represents the most common constructive

solution. Only the linear interpolation scheme will be assumed throughout the study,

since it is extensively used in industrial practice and it allows a clearer illustration of the

investigated phenomenon.

2. The concept of nonlinearity error

2.1 Cutter location curve

Generation of the sculptured surfaces through five-axis end milling is generally

accomplished through sequential motions of the cutter along the intended tool paths.

Determination of the tool paths on a given design surface constitutes the object of the

path generation algorithms that are implemented within CAM software. The trajectory of

the tool path and the feed rate at which it is being travelled by the cutter depends on the

machining strategy selected by the user. Based on the specific constraints aiming a

  

simultaneous increase of the overall effectiveness of the machining process, CAM

software determines optimized tool orientations for each cutter contact (CC) point of the

analyzed tool path. On the other hand, the main function of the numerical controller

installed on the five-axis machine tool is to coordinate its motions in order to ensure the

contact of the tool with the intended tool path and to preserve the optimal tool

orientations as established during path planning phase.

The tool posture along the tool path Ω is defined by its position vector PCL and its

tool axis orientation unit vector Tk̂ along ZT axis (Figure 1):

)ˆ,( TCL kP=Ω (1)

However, because tool postures that are embedded into CL data are expressed by the

CAM software in workpiece coordinate system (WCS), their conversion into joint

movements of the machine tool is mandatory. This conversion of the tool posture from

machine independent (CL data) into machine dependent format (G-code) represents the

task of the post-processor that is a mandatory element of the information flow in five-axis

machining. In order to attain a certain tool posture, five-axis machine has to move its

joints according to its particular kinematic structure. Due to the serial kinematic structure

of the five-axis machines, they can be assimilated with a manipulator attempting to reach

a desired posture for its end-effector – represented by the milling cutter, in this case. As a

result, the amount of motion required for each of the five joints within its kinematic chain

can be determined through inverse kinematic analysis.

  

One of the available options to perform this type of analysis involves the use of a

generalized kinematic model for a five-axis CNC machine like the one proposed in [18].

According to this model, the generalized homogeneous coordinate transformation matrix

[T]WT that is required to convert the position of a point from tool coordinate system

(TCS) into WCS can be expressed as:

⎥⎦

⎤⎢⎣

⎡=

10[R]

[T]WT

b (2)

where [R] is a generalized rotational matrix and b is a generalized position vector, both

dependent upon the particular kinematic configuration of the five-axis machine tool

(Figure 2). Machine kinematics is essentially a consequence of the physical structure of

the machine tool (Figure 3). The vertical spindle-rotating (SR) machine depicted in Figs.

2 and 3 is able to perform three translational motions (sX, sY and sZ) and two rotational

motions (A and C). The most significant point of the kinematic structure is denoted by the

term “pivot point” [22] and is placed at the intersection of the two rotary axes of the

machine (point OP in Figure 2). This intersection becomes defined only when the only

nonzero component of b5 – connecting the origins of the two rotational axes – becomes

collinear with axis of rotation of the primary rotational joint.

The generalized position vector b in Equation (2) is dependent on the machine

control coordinates (MCC) that represent the amount of translational and rotational

motions to be performed by the joints in order to achieve a desired configuration:

)),(( jiMCCf bb λ= with 5..1∈i , 6..0∈j and ]1,0[∈λ (3)

  

where λ is the interpolation parameter and bj are the vectors required to position

significant coordinate systems (joints) located along the kinematic chain of the machine

tool (Figure 2). Alternatively, the general coordinate transformation matrix can also be

expressed as:

⎥⎦

⎤⎢⎣

⎡=

1000

ˆˆˆ[T] CLTTTW

TPkji (4)

By equating the expression of the fourth column of [T]WT in Equation (1) and (3), the

general expression of position vector associated with CL point PCL for a five-axis

machine becomes:

=)(CL λP )),(( jiMCCf bλ (5)

which means that trajectory of the CL point is simultaneously affected by machine tool

kinematics and type of interpolation function used to calculate intermediate cutter

postures along the tool path. The generalized expression of the CL curve presented in

Equation (5) can be individualized for various machine configurations by applying

indices that are specific to the previously mentioned generic kinematic model.

2.2 Definition of nonlinearity error

In linearly interpolated five-axis machining, inherent machine tool kinematics

prevents a continuous and permanent contact between cutter and intended tool path. As a

result, this type of motion translates into a sequence of discrete CC points along the tool

path (Figure 4) between which the tool moves according to instantaneous MCC values

determined by a linear interpolation law:

  

1)1()( +⋅+⋅−= mm MCCMCCMCC λλλ (6)

While the distance between successive CC points – similar to CCm and CCm+1 – can be

dictated by variety of constraints, the most common technique employed in practice to

establish the length of the forward step is based on the chordal deviation [23]. When

travelling along each of the linearly interpolated segments, the machine tool continuously

adjusts tool postures between its intended initial (Ωm) and final (Ωm+1) values according

particular kinematic poses of the machine tool that are determined by instantaneous MCC

values (Figure 5). As a result, simultaneous motions of all five-axis joints induce a

nonlinear trajectory of the CL point, while machine’s pivot point follows a linear

trajectory. The kinematics of the vertical SR five-axis machine tool imposes a motion of

the CL point that is laterally away and not contained within the bilinear surface Γ

determined by CLm, CLm+1, mPO and 1PO+m

points.

For general five-axis motions, the trajectory of the CL is dictated by a

simultaneous superposition of the translational and rotational movements performed by

the five joints of the machine. The approach used in this study to determine the

nonlinearity error resembles the idea of decoupling translational from rotational motions

of the tool [24]. According to this, a general five-axis motion can be seen as an overlap of

two elementary movements: a planar motion contained within ΠT plane and a rotational

motion about a fixed point 1PO+m

that determines the ΠR plane (Figure 6). Since the linear

motion of the CL point between CLm and im 1CL + is characterized by a zero nonlinearity

errror, it becomes clear that this type of error is introduced by the two rotational motions

  

of the machine tool that are responsible for motion of the CL point between im 1CL + and

fm 1CL + . The two end positions of the tool determined by mm POCL and

1P1 OCL++ mfm

segments are noncoplanar, implying that the two motion planes involved ΠT and ΠR are

different. When ΠT and ΠR coincide, it means that five-axis motion has been reduced to

a particular case of four-axis motion. Intuitively, since the length of the translational

motion is identical regardless of machine tool structure, it means that the length of the

general motion is essentially set by the rotational motion. A longer rotational motion

means that the CL point will have to deviate more from the surface Γ in order to end up

in the same location as one that was subjected to a shorter rotational motion caused by a

different kinematics. Essentially, this implies that a direct proportionality relationship

exists between the amount of maximum deviation δ, the length of the CL curve, and the

length of the rotational motions required to perform a general five-axis tool motion.

By taking into consideration only the rotational motions performed along each of

the linearly interpolated segments of the tool motion (Figure 7), it may be observed that

CL point traces between CLi and CLf a sphere curve that belongs to the spherical surface

of radius L centered in OP. The nonlinearity error of this motion εnonlinear is defined as the

maximum deviation δ from the reference plane ΠR determined by the two end tool

orientation vectors is given by:

⎭⎬⎫

⎩⎨⎧ ×=⋅== +ΠΠ

∈1

TCL

]1,0[nonlinear

ˆˆand|ˆ))((|)()max(RRR mm kknnP λλδδε

λ (7)

  

Three important observations are to be made regarding the linearly-interpolated

trajectory of the CL point between its initial and final positions:

(1) For a family of five-axis motions, characterized by identical initial and final tool

postures, the larger εnonlinear is, the more the machine will have to move its

rotational joints, and this decreases the kinematic efficiency of the analyzed five-

axis machine. As a result, it can be inferred that small nonlinearity errors are

generally desirable in five-axis machining at least from an energy consumption

perspective.

(2) The absence of kinematic constraints would allow a more direct, shorter and

hence more energetically-efficient motion of the CL point between its initial and

final positions. Such calculations could be based on quaternion-based approaches

that are common in computer graphics [25] and have been recently extended to

kinematic analysis of five-axis machine tools [26]. One typical example of

shortest path motion is represented by the dash-dotted geodesic shown in Figure

7. This “in-plane” motion happens in this case along one of the great circles of the

sphere and is accompanied by a zero nonlinearity error. However, such motions

cannot be accommodated in practice due to the constraints set by machine tool

kinematics that makes the five-axis “in-plane” tool motion virtually impossible.

(3) The superposition of the interpolated translational and rotational motions forces

the CL point laterally away from bilinear surface Γ. This departure, expressed

through deviation δ, has a major – but until presently little accounted for – effect

on the length of tool motions in five-axis machining. In common industrial

  

practice, tool postures along the tool path are established within CAM systems

that have no information on the machine tool kinematics that will be used [20]. It

is important to emphasise that this decouple of the translational from rotational

motions is performed solely in the context of qualitative and quantitative

evaluation of deviation δ, to be later used to compare qualitatively the kinematic

efficiency of five-axis machines. An alternate approach involving an interative

numerical evaluation of the distance from instantaneous position of CL to surface

Γ based on Newton-Raphson method is also possible, but this procedure would be

more computationally expensive and thus not preferred. Furthermore, a pure

numerical approach would practically minimize most of the insight provided via a

more geometrically intuitive procedure like the one adopted in this study.

It is perhaps important to note here that while nonlinearity error has an important

impact on kinematic performances of five-axis machines when analyzed in the context of

linearly-interpolated motions, some of the particular instances of multi-axis machining

might be affected by it to lesser degree. For example, the nonlinearity error associated

with 3 ½ ½-axis machining operations [27] is expected to be relatively small, since the

cutter is expected to execute a quasi-three-axis motion along the tool path. Moreover,

depending on the local configuration of the machined surface, the rather fortuitous

nonlinear motion of the tool might be able to actually reduce the overall machining

errors, despite of the inherent inefficiency of simultaneous five-axis movements.

  

3. Determination of nonlinearity error

3.1 Deviation from reference plane

According to Equations (2-4) and previously proposed generalized formula for

position vector b of a SR five-axis machine [18], the trajectory of the CL point is given

by:

65

4

0CL ][R][R][R bbbP ⋅⋅+⋅+=∑

=SSS SPP

ii (8)

where [SPR ] and [

SSR ] represent the rotational matrices associated with primary and

secondary rotational axis of the machine, respectively. The position vectors bi ( 6..0∈i )

in Equation (8) are used to locate spatial position of the characteristic coordinate systems.

The primary rotational axis is placed closer to the machine bed (ground) within

the kinematic chain of the machine tool. Depending on the rotational axis of the joints

installed on the five-axis machine tool, [PS] and [SS] can be determined by equating them

with classical expressions of the rotational matrices about an axis of the coordinate

system:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−=

)cos()sin(0)sin()cos(0

001]R[

AAAAA (9’)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−=

)cos(0)sin(010

)sin(0)cos(]R[

BB

BB

B (9’’)

  

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −=

1000)cos()sin(0)sin()cos(

]R[ CCCC

C (9’’’)

Depending on the combination of rotary axes installed on the machine tool, four main

types of vertical SR five-axis machines exist: CA, CB, AB, BA. The first letter of the pair

indicates the primary rotational axis. The other two possible combinations (AC and BC)

were previously shown as infeasible for this type of machine tool due to the symmetry of

the tool about its own axis. For the purpose of this work, CA/CB or AB/BA will be treated

as distinct configurations based on the home position of the axes as set by the

manufacturer.

Without losing the generality of the solution when analyzing the trajectory of the

CL point during nonlinearity error-generating motion shown in Figure 7, a simplified

machine tool configuration will be assumed: bi = 0 ( 5..0∈i ) and ( ) [ ]L−= 00T6b

This structure accurately describes the most common types of vertical SR machines

encountered in practice with intersecting rotary axes that were analytically proved to be

more effective than those with non-intersecting axes [18]. As a result, combination of

Equations (8) and (9) allows determination of parametric expression for CL point

rotational only curve RCLP , value that can be used subsequently to calculate the

instantaneous value of the deviation δ from the ΠR plane according to Equation (7). Both

formulas are summarized in Table 1 for all four types of vertical SR machines. All

intermediate values for rotational angles A, B, C are to be determined based on the linear

interpolation law presented in Equation (5) and their end values Af, Bf, Cf are to be

calculated for λ = 1.

  

3.2 Interpolation parameter characteristic to nonlinearity error

An analysis of the formulas presented in Table 1 reveals that nonlinearity error

defined by Equation (6) is in fact a function of three variables, namely: interpolation

parameter λ, primary final fSP and secondary final

fSS rotational angles of the machine

tool. As a result, a generalized form of the nonlinearity error would be:

)],,,([max]1,0[nonlinear ff SSSS SPSPδε

λ∈= (10)

Since this study assumes linear interpolation only as defined by Equation (5),

intermediate rotational angles can be determined based on the intended rotational speeds

SPv and SSv :

})1({)( fSSPS PPvPS

=⋅= λλ (11’)

})1({)( fSSSS SSvS

S=⋅= λλ (11’’)

Based on this assumption, the Equation (9) can be transformed as:

)],,([max

]1,0[nonlinear SS SP vvλδελ∈

= (12)

Determination of εnonlinear involves a classical one-variable maximization problem,

according to which for each known pair of primary and secondary angles, a certain value

of interpolation parameter λmax will allow maximization of the deviation value δ.

However, no closed form solutions were found in this case since the first derivative test

0dd =λδ yields inevitably transcendental equations. As a result, a numerical subroutine

  

based on Brent’s method that is built in Matlab was used to calculate λmax for various

combinations of fSP and

fSS angles. A graphical representation of the values obtained

for λmax is depicted in Figure 8. No difference in results was observed between CA and

CB or between AB and BA configurations, respectively. To obtain meaningful results that

are also unaffected by the periodicity of the trigonometric functions involved in δ

expressions presented in Table 1, the range of the two rotational angles analyzed was

varied only within 0° to 90° interval. Furthermore, to avoid singularities introduced by

null denominators, only rotational angles larger than 1° were considered in a first

iteration. The extreme values of λmax for all four types of vertical SR machines analyzed

are synthesised in Table 2, along with the specific combination of rotational angles that

generated them. According to the tabulated data presented, the nonlinearity error is

produced for interpolation parameter varying between 0.4523 and 0.5477 when C axis is

involved and between 0.5 and 0.57735 when C axis is not involved in kinematic

configuration of the five-axis machine tool.

For most general five-axis motions programmed along the tool path and subjected

to various constraints, the range of primary and secondary final angle previously

analyzed is too broad. In order to provide more insightful information on λmax that is

characteristic to small rotational motions, typically restricted to less than 1° values [28],

a second iteration was used to establish its value in these cases. In order to solve for

interpolation parameter values, Taylor’s expansion was used to approximate

trigonometric functions since no closed form solutions can be used to determine the

maximum value of δ. Once again, a clear distinction is to be made between the vertical

  

SR configurations involving or not C axis: while for CA and CB machines Taylor’s first

order terms yield λmax = 0.5, for AB and BA types up to fifth order terms were needed to

calculate the characteristic value of the interpolation parameter that is associated with

nonlinearity error. As a consequence, some minimal corrections are to be made on λmax

values for AB and BA configurations as reported during the first iteration (Table 1): λmax

= 0.57741 (AB type) and λmax = 0.58223 (BA type) when both axes rotate with extremely

small amounts (A, B → 0).

Summing up the results acquired so far, it may be concluded for general five-axis

linearly-interpolated motions, nonlinearity error will be obtained for interpolation

parameter λmax ranging between 0.4523 and 0.5477 for CA and CB machines and between

0.5 and 0.58223 for AB and BA machines, respectively. For all analyzed cases, both

primary final and secondary final rotational angles can take any values between 0 and

90°, but without attaining the boundaries in order to avoid singularities: °<< 900fSP

and °<< 900fSS . Although possible, rotational angles larger than 90° are considered

outside of the scope of the current work, since the largest five-axis motions used in

practical surface generation applications will rarely exceed the 90° upper bound.

Especially for primary rotational axis, larger than 90° (and up to 360°) working angles

are also possible, but their upper-range values are generally used in context of

positioning/indexing motions solely.

Interestingly, these results confirm that in case of linearly interpolated motions

performed on vertical SR five-axis machine tools, the current empirical post-processing

practice of inserting an additional CL point at the mid distance between two other pre-

  

determined ones in order to limit machining errors [4,10,13,16,17] is in fact correct and it

is now supported by computational evidence. The middle CL point is the point associated

with largest nonlinearity error. By passing through an additional mid CL point, the tool is

practically constrained to deviate less from the bilinear surface Γ, but this comes on the

expense of additional motions to be performed by the machine. For CA and CB machines,

mid CL point insertion is almost an accurate procedure since λmax ≅ 0.5 when Cf and Af or

Bf angles have small values. By contrast, AB and BA machines are characterized by a

larger but still acceptable approximation, since λmax approaches 0.58 when both Af and Bf

angles are small. For all four vertical SR machine types, once λmax has been determined

for a certain general tool motion, the nonlinearity error can be calculated with tabulated

formulas presented in Table 1. The required interpolated and final rotational angles are

generated from known rotational parametric speeds as introduced in Equation (11).

4. Kinematic efficiency

Based on the concept of nonlinearity error defined above, the kinematic efficiency

of vertical SR five-axis machine tools will be compared. For this purpose, two main

scenarios will be considered: i) both primary and secondary final angular values are the

same; and ii) all four machines move to the same final tool orientation. According to the

definition of the nonlinearity error proposed, the CL deviation from the reference plane is

only generated by the combination of the two rotational motions presented in Figure 7.

The underlying assumption of this analysis is that the kinematic efficiency of the machine

tool is directly proportional with the amount of nonlinearity error.

  

4.1 Identical final rotational motions

The goal of this assessment is to predict the most kinematically efficient vertical

SR five-axis machine in the event of identical motions performed by the two rotary axes

of the five-axis machine tool. To simplify the comparison, the initial tool orientation will

be assumed the same.

Figure 9 depicts representative CL trajectories for all four types of vertical SR

machines investigated when they all started to move from an initial vertical position

characterized by 0==ii SS SP . The CL curves are plotted for equal primary final and

secondary final angles: °== 24ff SS SP and a tool length L = 212.5 mm. The graph

shows clearly that in the event of identical primary final and identical secondary final

rotational motions applied to the tool, AB and BA configurations move more than CA and

CB types. This remark can be validated consistently for all possible combinations of fSP

and fSS angles (Figure 10) by analyzing the magnitude of the direct 3D angle θ between

iTk̂ and fTk̂ vectors (Figure 7):

fSCBCA S=,θ

(13’)

))cos()(cos(cos 1

, ff SSBAAB SP ⋅= −θ (13’’)

It is relatively straghtforward to show that BAABCBCA θθθ =≤, for all ]90,0[, °∈ff SS SP ,

meaning that AB and BA machine types tend to travel longer and equal distances

compared to CA and CB that travel less and in different amounts.

  

A second important remark may be made regarding the amount of nonlinearity

error generated during rotational motions characterized by identical fSP and

fSS angles.

By analyzing the plot of εnonlinear calculated from Equations (10-12) and Table 1 (Figure

10), it can be inferred that AB and BA machines are consistently more kinematically

efficient than CA and CB types. No difference in nonlinearity error amounts was noticed

between AB and BA or CA and CB machines, respectively. By combining both

observations, it can be concluded that when vertical SR five-axis machines are

constrained to move identically their primary and secondary rotary axes, respectively, the

AB and BA machines are capable to travel longer distances and their motions are

characterized by smaller nonlinearity errors compared to CA and CB configurations.

4.2 Identical final orientation

The scope of this analysis is to provide a direct comparison of the kinematic efficiency

associated with the four possible types of vertical SR five-axis machines, in the event that

they are all constrained to move between identical initial and final orientations, which is

equivalent to an identical machining task.

As outlined in Section 4.1, when identical fSP and

fSS rotational motions are

used, the final orientation of the tool is generally different for each machine type. As a

result, for the purpose of the current comparison, the CA configuration was chosen as

reference, while the other three machine types were constrained to move to match its

final orientation starting from an initial vertical position defined by 0==ii SS SP . To

satisfy this constraint, individual values were calculated for primary final and secondary

final angles involved in each transformation according to the transformation formulas

  

presented in Table 3. In case of multiple solutions available for the same final orientation,

the combination that enabled the shortest travel path between initial and final orientation

was selected.

Figure 11 presents variation of CL trajectories for all investigated machine types

when CA reference configuration was moved to a final tool orientation as determined by

°= 80fA and °= 20fC angles. The viewing angle of the graph was chosen in such a

way to emphasise that nonlinearity errors associated with each machine type have

completely different values in this case. This finding is further reinforced by a general

plot of the nonlinearity error over the entire analyzed range: ]90,0[, °∈ff CA . The chart,

presented in Figure 12, reveals that depending on the final tool orientation, different

vertical SR five-axis machine types will introduce various amounts of nonlinearity errors.

However, unlike the situation when machines move identically on both their rotary axes,

no absolute worst or best configuration exists. Furthermore, the overall aspect of the

εnonlinear variation is more irregular than in the previously analyzed case.

Two main zones are identifiable on the graph and they are delimited by a vertical

plane located at Cf = 45°. When this threshold value is crossed, different machine

configurations become responsible for generation of the largest/smallest nonlinearity

error. For example, when Cf < 45°, the smallest nonlinearity error generated for a

particular combination of Cf and Af angles is associated with AB machine, while the

largest nonlinearity error is associated with CB type. By contrast, when Cf > 45° the

smallest nonlinearity error is generated by BA and the largest by CA machine. At Cf =

  

45°, the nonlinearity errors introduced by AB and BA or CA and CB configurations

become equal, respectively.

Two important comments are to be made regarding the practical insights related

to this comparison:

(1) For the most common five-axis machining operations, that are generally

characterized by small angular motions ( °≤<° 5,0 ff CA ), only a small difference

in nonlinearity errors will be noticed between AB and BA configurations, such that

ABnonlinearε BAnonlinearε .

(2) Although from a strictly mathematical standpoint, the tool orientation determined

by 0== ii AC for CA machine is identical with tool orientation determined by

0== ii BC for CB machine, the latter option occurs rarely in practice because of

the computational singularities associated with angular conversions. As a result,

most inverse kinematic algorithms tend to convert a 0== ii AC tool orientation

(CA case) into one that is determined by °−= 90iC , °= 0iB (CB case) based on

the formulas presented in Table 3. Therefore, for most practical tasks, no

difference will be noticed between the cutter motions along tool path that are

characteristic to CA or CB five-axis machines (CBCA nonlinearnonlinear εε = ). However,

differences are possible, depending on the angular conversion formulas used by

the post-processor.

By coupling the nonlinearity variations depicted in Figure 12 with these two

observations, it may be concluded that the kinematic efficiency of the four studied

  

machine types can be ordered – for practical purposes – according to the following

relationship:

ABnonlinearε

CBCABA nonlinearnonlinearnonlinear εεε =< (14)

5. Case Study

In order to further emphasise the practical aspects related to the aforementioned

findings, all four vertical SR machine configurations were used to perform tool path

tracking simulations on the sculptured surface patch of approximately 110x130 mm

shown in Figure 13. For this purpose, the linearly-interpolated motions of a 5 mm

diameter flat-end cutter travelling between 11 discrete CC points located along the tool

path were tracked in terms of CL point position and associated joint movements. The 11

CC points used were selected in such a way to allow a clear visualization of the CL point

trajectory for each of the linearly-interpolated tool motions. The tool orientation at each

of the CC points was determined based on curvature matching and gouging avoidance

constraints that are common to five-axis machining operations. The distance between CL

and pivot point was assumed as 212.5 mm.

As implied by the ordering of the nonlinearity errors for motions with identical

initial and final orientations and small angular motions introduced by Equation (14), the

length of the CL trajectory for the investigated five-axis machine types follows an

identical trend (Table 4). In terms of MCCs, while all machine types perform identical

translational motions along the tool path, significant differences can be observed in terms

of rotational angles when compared through a common indicator related to the length of

total rotational movement defined as follows:

  

∑ ∫= ⎟⎟

⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛= ++

10

0

1

0

22

dd

dd

d1,1,

m

SSR

mmmmSP

I λλλ

(15)

where )(1,λ

+mmSP and )(1,λ

+mmSS represent the instantaneous amounts of primary and

secondary rotational angles, respectively for linearly-interpolated motion defined by CCm

and CCm+1 end points. From the data presented in Table 4, it can be inferred that in order

to trace the same tool path, CA machine has to perform approximately

3.1 times longer rotational motions than AB machine. This observation has important

consequences on the amount of energy consumed by vertical SR five-axis machine tools

while performing an identical machining task.

6. Conclusions

The study proposes a new method to evaluate the nonlinearity method introduced

by the kinematics of the vertical SR five-axis machine tools. Although most modern five-

axis post-processors generally acknowledge the existence of nonlinearity errors through

tool path linearization process, little attempt was made to quantify their magnitude and/or

variation along the tool path. In this work, evaluation of the nonlinearity errors during

linearly-interpolated motions of the cutter along the intended tool path is primarily based

on the separation of the translational and rotational movements of the tool. The maximum

deviation of the CL point of the tool with respect of the reference plane generated by the

initial and final orientation of the tool are then used to quantify the magnitude of

nonlinearity errors.

The technique used to calculate nonlinearity errors demonstrated that common

industrial practice used to limit their amount through halfway CL point insertion is in fact

  

correct and present study provides computational evidence required to support this

decision. For all analyzed five-axis machines, nonlinearity error occurs at interpolation

parameters approximately equal to 0.5, with small variations caused by the machine type

and the amplitude of the rotational only motion performed. Nonlinearity errors also

constitute an useful tool that can be used to compare the kinematic efficiency of the

vertical SR five-axis machines. Determination of these errors in context of identical final

rotational motions or of identical tool orientations revealed that, while tracing an identical

tool path, AB and BA machines generally tend to move their rotary axes less than CA and

CB types. This conclusion has equally important implications on tool path generation in

five-axis and on selection of a particular five-axis kinematic configuration that is

energetically efficient for a specific machining task.

Although all considerations made throughout the study specifically refer to

spindle-rotating machines, the procedure can be extended to the other two main types of

machines: table-rotating or hybrid, as long as the same framework is used to calculate the

nonlinearity error. This is a direct consequence of the equivalence between rotational

only motions performed in a scenario of decoupled translational and rotational

movements of the cutter. Future research efforts will be directed towards definition of

nonlinearity errors in context of more complex interpolation schemes, followed by their

integration in general five-axis tool path planning algorithms.

References

[1] Lasemi A, Xue D, Gu P. Recent development in CNC machining of freeform surfaces: A state-of-the-art review. Computer-Aided Design 2010. 42(7):641-654.

  

[2] Makhanov SS, Anotaipaiboon W. Advanced Numerical Methods to Optimize Cutting Operations of Five Axis Milling Machines. Springer 2007.

[3] Bohez ELJ. Five-axis milling machine tool kinematic chain design and analysis, International Journal of Machine Tools and Manufacture 2002; 42(4):505-520.

[4] She CH, Chang CC. Development of a five-axis postprocessor system with a nutating head. Journal of Materials Processing Technology 2007; 187:60-64.

[5] Ishizawa H, Hamada M, Tanaka F, Kishinami T. Form shaping function model of 5-axis machine tools – classification of 5-axis machine tools based on form shaping functions. In: Proceedings of the 5th Sapporo International Computer Graphics Symposium, 1991. Sapporo, Japan, 64-69.

[6] Sakamoto S, Inasaki I. Analysis of generating motion for five-axis machining centers. Transactions of NAMRI/SME 1993; 21:287-293.

[7] Chen FC. On the structural configuration synthesis and geometry of machining centres. Journal of Mechanical Engineering Science 2001. 215(6):641-652.

[8] Kiridena V, Ferreira PM. Mapping the effects of positioning errors on the volumetric accuracy of five-axis CNC machine tools. International Journal of Machine Tools and Manufacture, 1993; 33(3):417-437.

[9] Munlin M. Errors estimation and minimization for the 5-axis milling machine. In: Proceedings of the IEEE International Conference on Industrial Technology IEEE ICIT 2002; 1013-1018 vol.10122002.

[10] Bohez ELJ. Compensating for systematic errors in 5-axis NC machining. Computer-Aided Design 2002; 34(5):391-403.

[11] Bohez ELJ, Ariyajunya B, Sinlapeecheewa C, Shein TMM, Lap DT, and Belforte G, Systematic geometric rigid body error identification of 5-axis milling machines. Computer-Aided Design 2007; 39(4):229-244.

[12] Uddin MS, Ibaraki S, Matsubara A, and Matsushita T. Prediction and compensation of machining geometric errors of five-axis machining centers with kinematic errors. Precision Engineering 2009; 33(2):194-201.

[13] Cho HD, Jun YT, Yang MY. 5-Axis CNC Milling for Effective Machining of Sculptured Surfaces. International Journal of Production Research, 1993; 31(11):2559-2573.

[14] Liang H, Hong H, Svoboda J. A combined 3D linear and circular interpolation technique for multi-axis CNC machining. Journal of Manufacturing Science and Engineering-Transactions of the ASME 2002; 124(2):305-312.

  

[15] Ye T, Xiong CH. Geometric parameter optimization in multi-axis machining. Computer-Aided Design 2008. 40(8):879-890.

[16] Takeuchi Y, Watanabe T. Generation of 5-Axis Control Collision-Free Tool Path and Postprocessing for NC Data, CIRP Annals - Manufacturing Technology 1992 41 (1), 539-542.

[17] Ho MC, Hwang YR. Machine codes modification algorithm for five-axis machining. Journal of Materials Processing Technology 2003. 142(2):452-460.

[18] Tutunea-Fatan OR, Feng HY. Configuration analysis of five-axis machine tools using a generic kinematic model. International Journal of Machine Tools & Manufacture 2004. 44(11):1235-1243.

[19] Mann S, Bedi S, Israeli G, Zhou X. Machine models and tool motions for simulating five-axis machining. Computer-Aided Design 2010. 42(3):231-237.

[20] Makhanov S. Adaptable geometric patterns for five-axis machining: a survey. International Journal of Advanced Manufacturing Technology 2010. 47(9):1167-1208.

[21] Tutunea-Fatan OR, Feng HY. Determination of geometry-based errors for interpolated tool paths in five-axis surface machining. Journal of Manufacturing Science and Engineering-Transactions of the ASME 2005. 127(1):60-67.

[22] Apro K. Secrets of 5-Axis Machining. Industrial Press 2008.

[23] Faux ID, Pratt MJ, Computational Geometry for Design and Manufacture. Halsted Press 1979.

[24] Hsu YY, Wang SS, Mapping geometry errors of five-axis machine tools using decouple method. International Journal of Precision Technology 2007. 1(1):123-132.

[25] Shoemake K. Animating rotation with quaternion curves, In: Proceedings of the 12th annual conference on Computer graphics and interactive techniques SIGGRAPH 1995. 19(3):245–254.

[26] Guo RF, Li PN, Research on Kinematics Based on Dual Quaternion for Five-axis Milling Machine, In: Global Design to Gain a Competitive Edge, Springer 2008.

[27] Roman A, Bedi S, Ismail F, Tool path planning for 3½½-axis machining, International Journal of Manufacturing Research 2006. 1(2):248–265.

[28] Roth D, Gray P, Ismail F, Bedi S. Mechanistic modelling of 5-axis milling using an adaptive and local depth buffer. Computer-Aided Design 2007. 39(4):302-312.

  

List of Figures

Figure 1. Tool posture in five-axis machining

Figure 2. Kinematic configuration of vertical spindle-rotating machines

Figure 3. Physical configuration of vertical spindle-rotating machines: a) AB; b) BA and c) CA types

Figure 4. Discretized CC points along the intended tool path

Figure 5. General cutter motion in five-axis machining

Figure 7. Rotational only component of a general five-axis motion

Figure 8. Interpolation parameter characteristic to nonlinearity error

Figure 9. CL trajectories for °== 24ff SS SP

Figure 10. Nonlinearity error for identical final rotational motions

Figure 11. CL trajectories for Cf = 20° and Af = 80°

Figure 12. Nonlinearity error for identical final orientations set by CA reference type

Figure 13. Tool movements for vertical SR machine tool configurations

  

List of Tables

Table 1. Detailed expressions of rotational only CL curve and instantaneous deviation from reference plane for vertical SR five-axis machines

Table 2. Extreme values for interpolation parameter characteristic to nonlinearity error

Table 3. Angular conversion formulas relative to CA reference machine

Table 4. Kinematic efficiency of vertical SR five-axis machines

  

Table 1. Detailed expressions of rotational only CL curve and instantaneous deviation from

reference plane for vertical SR five-axis machines

Machine Type

Rotational only CL Curve Deviation from Reference Plane ΠR

CA ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡⋅−⋅

⋅=A

ACAC

Lcos

sincossinsin

RCLP [ ]ACACACACA

Lffff

f

sincossinsinsinsinsincossin

⋅⋅⋅−⋅⋅⋅⋅=δ

CB ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡⋅⋅

⋅=B

BCBC

Lcos

sinsinsincos

RCLP [ ]BCBCBCBCB

Lffff

f

sincossinsinsinsinsincossin

⋅⋅⋅−⋅⋅⋅⋅=δ

AB ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⋅⋅−⋅=

BABA

BL

coscoscossin

sin

RCLP [ ]

fff

fff

BBA

BABBBAL222 sincossin

cossinsinsincossin

+⋅

⋅⋅−⋅⋅⋅=δ

BA ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⋅−

⋅⋅=

ABA

ABL

coscossin

cossin

RCLP [ ]

fff

fff

ABA

AABABAL222 cossinsin

sincossincossinsin

⋅+

⋅⋅−⋅⋅⋅=δ

  

Table 2. Extreme values for interpolation parameter characteristic to nonlinearity error

Machine Type

Final Rotational Angles Interpolation Parameter

λmax fSP [deg.]

fSS [deg.] A B C A B

CA and CB

- - 1 90 - 0.4523 - - 1 - 90 - - 90 1 - 0.5477 - - 90 - 1

AB and BA

- 90 - 90 - 0.5 90 - - - 90 - 1 - 1 - 0.57735 1 - - - 1

Table 3. Angular conversion formulas relative to CA reference machine

Machine Type Final Rotational Angles

BA

)sin(cossin 1fffBA ACA ⋅= −

⎥⎥⎦

⎤

⎢⎢⎣

⎡ ⋅= −

fBA

fffBA A

ACB

cossinsin

sin 1

CB

)cot(tan 1ffCB CC −= −

⎥⎥⎦

⎤

⎢⎢⎣

⎡ ⋅= −

fCB

fffCB C

ACB

cossinsin

sin 1

AB

)sin(sinsin 1fffAB ACB ⋅= −

⎥⎥⎦

⎤

⎢⎢⎣

⎡ ⋅= −

fAB

fffAB B

ACA

cossincos

sin 1

  

Table 4. Kinematic efficiency of vertical SR five-axis machines

Machine Type CL Trajectory Length [mm] IR

AB 119.50 1.207

BA 119.73 1.230

CA, CB 127.5 3.806

Figure 1. Tool posture in five-axis machining

CC

Work piece CL

Tool path ZWCS

YWCS

XWCS

PCL

ZT

XT

YT

Tk̂

Figure 2. Kinematic configuration of vertical spindle-rotating machines

b1

O5

O6 OP

O4

Y

X O3

b0

Z

O1

b5

O6 OP

TCS

b6

O7

O5

TCS

A

C B

A

CA type BA type

TCS O7

O6 OP

b6

b5

O5

AB type

B A b5

sX

sY

sZ

O0

O2

O7

b2

b3

b4

b6

WCS

a) b) c)

Figure 3. Physical configuration of vertical spindle-rotating machines:

a) AB; b) BA and c) CA types

sX sY

sZ

C

A

sX sY

sZ

A B B

sX sY

sZ

A

Figure 4. Discretized CC points along the intended tool path

…

… CCm

CCm+1

design surface

intended tool path

Figure 5. General cutter motion in five-axis machining

CLm+1

1CLmP

mCLP

CL point trajectory

CLm

mTk̂

mPO

1POm

1Tˆ

mk

pivot point trajectory

Z

X

Y

Y

X

Figure 6. Resolving general tool motion into translational

and rotational only components

fm 1CL

CLm

mPO

1POm

T

R

im 1CL

1POm

Figure 7. Rotational only component of a general five-axis motion

Z

X

Y

L

nonlinear = δmax

OP

CLi

CLf

δ( )

R

RCLP

shortest path

motion

iTk̂

fTk̂

Figure 8. Interpolation parameter characteristic to nonlinearity error

020

4060

80100

020

4060

80

0.44

0.46

0.48

0.50

0.52

0.54

0.56

0.58

0.60

fSP [deg] fSS [deg]

max

CA and CB

AB and BA

Figure 9. CL trajectories for 24ff SS SP

-100

-80

-60

-40

-20

0

-40-20

020

4060

80100

-220

-210

-200

-190

-180

-170

X [mm]

Y [mm]

Z [mm]

CA

CB

AB

BA

a)

b)

Figure 10. Nonlinearity error for identical final rotational motions

020

4060

80100

0

50

100

0

20

40

60

80

100

120

020

4060

80100

020

4060

80100

0

20

40

60

80

100

120

fSS

nonlinear

CB and CB

AB and BA

fSP [deg]

fSS [deg]

AB and BA CB and CB nonlinear [mm]

fSP

Figure 11. CL trajectories for Cf = 20 and Af = 80

-120 -100 -80 -60 -40 -20 0 050

100150

200-220

-200

-180

-160

-140

-120

-100

-80

-60

-40

-20

CA

X [mm] Y [mm]

Z [mm]

BA

AB

CB

Figure 12. Nonlinearity error for identical final orientations set

by CA reference type

020

4060

80100

020

4060

80100

0

20

40

60

80

100

120

CA

BA

CB

AB

nonlinear

Cf [deg]

Af [deg]

Figure 13. Tool movements for vertical SR machine tool configurations

CA, CB

AB, BA

tool path

machined surface

CC point

CL point

Research Highlights

Nonlinearity error can be used to assess the kinematic efficiency of five-axis machines

In linear interpolation, maximum nonlinearity error occurs around mid-parametric point

AB and BA vertical 5-axis SR machines tend to move farther than CA and CB types

CA and CB tend to move more than AB and BA machines

Vertical 5-axis SR machines involving C-axis are less kinematically efficient