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: [email protected]
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