KINEMATIC ANALYSIS OF UNIVERSAL SPLIT HEADS FOR MILLING MACHINES
D. Castagnetti, E. Dragoni, G. Scirè Mammano
Dipartimento di Scienze e Metodi dell’Ingegneria – Università di Modena e Reggio Emilia
Via Fogliani 1 - 42100 Reggio Emilia (Italy)
E-mail: [email protected], [email protected], [email protected]
Introduction. Most milling machines are equipped with split heads aimed at orienting the tool at any angle with respect
to the workpiece. Typical examples are given in Fig. 1. The general geometry of these heads can be reduced to the
linkage shown in Fig. 2. Each particular architecture is achieved by choosing angle θ between the member axis and by
selecting the orientation of the tool with respect to the final member. The head features two degrees of freedom,
embodied by the joint rotations α and β, through which the configuration can be changed to adjust the orientation of
the tool. The kinematic analysis of the head can be posed in two ways: a) direct analysis if the values of the joint
rotations, α and β, are given and the orientation of the tool axis with respect to the machine frame is sought; b) inverse
analysis if the desired orientation of the tool axis is given and the joint rotations, α and β, are sought that enforce that
orientation. This paper provides the analytical solution of both direct and inverse problems. The work falls within a
research program carried out with the partnership of a renowned machine tool manufacturer of Reggio Emilia.
Analysis.
The kinematic analysis of the head is focused on
the angular orientation of the tool axis. The actual
position of the tool tip is disregarded. This is because
the position of the tip can always be adjusted by overall
translations of the head (applied by the milling
machine) which leave unaffected the achieved
orientation.
Fig. 2 shows the schematic of the head used for the
analysis. The linkage comprises two members (ma, mb)
and two revolute joints (A, B) that rotate by angles α
and β about coplanar axes a and b, respectively. The
two revolute axis form an angle θ ranging from 0 to
90°, according to the selected architecture.
Four right-handed, orthogonal reference systems,
instrumental to the mathematical development are
established. The first reference system, xoyozo, is fixed
to the frame of the machine, with origin nearby joint B.
Axis yo is aligned with revolute axis b and points away
from joint A. The second reference system, xbybzb, is
fixed to member mb, with the same origin as xoyozo. For
β=0, references xbybzb and xoyozo coincide. In any other
situation, reference xbybzb is rotated by an angle β about
axis yb≡yo. The third reference system, xaoyaozao, is also
fixed to member mb and has origin at the intersection
point of revolute axes a and b. Axis yao is aligned with
revolute axis a and points away from member ma. Axis
xao is parallel to axis xb. The fourth reference system,
xayaza, is fixed to member ma with the same origin as
xaoyaozao. For α=0, references xayaza and xaoyaozao
coincide. In any other situation, reference xayaza is
rotated by an angle α about axis ya≡yao.
The orientation of the tool with respect to member
ma is provided by the direction cosines (uax, uay, uaz) of
the tool axis with respect to reference system xayaza.
Those direction cosines are conveniently collected in a
T
tool direction vector {ua } = ⎡uax uay uaz ⎤
⎣
⎦
za0
ma
uaz
Figure 1b
za
y0≡yb
b
x0
β
xa
A
zb
B
θ
Uay
Figure 1a
z0
mb
ya0≡ya≡ya’
xa0
a
{u}
uax
Figure 2
α
xb
If {uo} is the tool direction vector referred to the fixed
coordinate system xoyozo, the relationship between {ua}
and {uo} is given by
{uo } = [T ] a ⋅ {ua }
o
(1)
where [T ] o is the matrix that transforms system xayaza
a
into system xoyozo. By splitting the transformation into
three subsequent transformations through the four
reference systems defined above [1], matrix [T ] o can
a
be cast as:
[T ] a = [T ] b ⋅ [T ] a ⋅ [T ] a
o
o
b
ao
o
=
(2)
⎡ cos α ⋅ cos β − sin α ⋅ sin β ⋅ cosθ sin β ⋅ sin θ sin α ⋅ cos β + cos α ⋅ sin β ⋅ cosθ ⎤
⎥
=⎢
sin α ⋅ sin θ
cosθ
− cos α ⋅ sin θ
⎢
⎥
⎢⎣− cos α ⋅ sin β − sin α ⋅ cos β ⋅ cosθ cos β ⋅ sin θ − sin α ⋅ sin β + cos α ⋅ cos β ⋅ cosθ ⎦⎥
Equations (1) and (2) provide the solution of the
kinematic analysis of the head. Once the architecture of
the head is given (θ and {ua} are known), (1) and (2)
contain the tool vector {uo} and joint rotations α and β
as the only variables.
Direct analysis is performed by assigning α and β
in (2) and calculating {uo} through (1). Inverse analysis
is performed by assigning {uo} in (1) and solving for α
and β the resulting system of transcendent equations.
Whatever the architecture of the head, direct
analysis is straightforward and can be solved in closed
form. By contrast, exact solution of the inverse
problem is not possible for the general case. However,
closed-form solutions exist for the two important cases
corresponding to {θ = 45°; {ua}=[0, -sin45, -cos45]T}
(Fig.1a) and to {θ = 90°; {ua}=[1, 0, 0]T} (Fig.1b).
From the second of equations (1) two values of α
are retrieved that verify the relationships α2=2π–α1
(Fig.1a) and α2=π–α1 (Fig.1b). When entered into the
first and the third equation, these solutions provide
explicit conditions for sinβ and cosβ from which a
single value of β is calculated. As a general rule, two
pairs of α and β are found that satisfy the problem,
showing that the same tool orientation can be achieved
with two distinct layouts of the head.
Results and Discussion
A map of the work region explored by the tool for a
configuration of the head defined by θ = 45° and
{ua}=[0, -sin45, -cos45]T is provided in Fig.3a. Line
OP represents the tool vector for α=β=0. When
connected to point O of the figure, each point on the
surface identifies a feasible orientation of the tool axis.
The vertical circles correspond to the tool trajectory at
constant values of α. The other circles correspond to
constant values of β. The symmetry axis of the surfaces
corresponds to axis yo of the main reference system.
If the same value θ = 45° is associated with a
different tool vector, the overall pattern of Fig.3b is
produced, describing a different coverage of the work
region. With other choices of the tool vector, the
surface of Fig.3a remains unchanged, apart from a hole
appearing on its left side. Similar effects are observed
if the value of θ is changed in combination with the
value of {ua} used for Fig.3a.
In Fig.4a a map of the work region achieved by the
tool is provided for a configuration of the head
corresponding to θ = 90° and {ua}=[1, 0, 0]T. In this
case, complete exploration of the space is achieved. If
the same value θ = 90° is combined with a different
tool vector, the symmetrically holed surface of Fig.4b
is produced. Again, a similar effect is consequent upon
associating values of θ other than 90° to the same tool
vector as in Fig.4a.
O
P
Figure 3a
Figure 3b
Figure 4a
Figure 4b
Conclusions
The paper deals with the kinematic analysis of
universal split heads for milling machines formed by
two members linked by in-plane revolute pairs. An
exact solution is given that supplies the orientation of
the tool axis with respect to the machine frame for any
head architecture and any rotation of the revolute
joints. Closed form solutions of the inverse problem are
also highlighted that provide the joint rotations needed
to achieve any feasible tool orientation for two popular
head architectures
References
[1] L. Sciavicco, B. Sciliano, “Robotica Industriale” McGraw Hill