Abstract— This paper describes the workspace and the 
inverse and direct kinematic analysis of the VERNE machine, 
a serial/parallel 5-axis machine tool designed by Fatronik for 
IRCCyN. This machine is composed of a three-degree-of-
freedom (DOF) parallel module and a two-DOF serial tilting 
table. The parallel module consists of a moving platform that 
is connected to a fixed base by three non-identical legs. This 
feature involves (i) a simultaneous combination of rotation 
and translation for the moving platform, which is balanced by 
the tilting table and (ii) workspace whose shape and volume 
vary as a function of the tool length. This paper summarizes 
results obtained in the context of the European projects 
NEXT (“Next Generation of Productions Systems”). 
I. INTRODUCTION 
ARALLEL kinematic machines (PKM’s) are well known 
for their high structural rigidity, better payload-to-
weight ratio, high dynamic performances and high 
accuracy [1-3]. Thus, they are prudently considered as 
attractive alternatives designs for demanding tasks such as 
high-speed machining [4]. 
A well-known feature of PKM is the existence of 
multiple solutions to the direct kinematic problem. That is, 
the moving platform can admit several positions and 
orientations (poses) in the workspace for one given set of 
input joint values [5]. Moreover, parallel manipulators exist 
with multiple inverse kinematic solutions. This means that 
the moving platform can admit several input joint values 
corresponding to one given pose of the end-effector [6]. 
For industrial PKMs, the resolution of the direct and 
inverse kinematic problems is usually made using iterative 
methods. With such methods, the calculation time can 
change as function of the configuration of the PKM. A 
major drawback of PKMs is a complex workspace. The 
workspace of a PKM has not a simple geometric shape, and 
its functional volume is reduced as compared to the space 
occupied by the machine [7]. 
 
D. Kanaan is with the Institut de Recherche en Communications et 
Cybernétique de Nantes (IRCCyN) (UMR CNRS 6597), France 
(corresponding author to provide phone: 33-240376958; fax: 33-
240376930, e-mail: Daniel.Kanaan@irccyn-nantes.fr); P. Wenger is with 
the 
IRCCyN 
(UMR 
CNRS 
6597), 
France 
(e-mail: 
Philippe.Wenger@irccyn-nantes.fr); D. Chablat is with the IRCCyN 
(UMR CNRS 6597), France (e-mail: Damien.Chablat@irccyn-nantes.fr). 
Because many industrial tasks require less than six 
degrees of freedom, several lower-DOF PKMs have been 
developed. For some of theme, the reduction of the number 
of DOFs can result in coupled motions of the moving 
platform [8-10]. This is the case, for example, in the RPS 
manipulator [9] and in the parallel module of the VERNE 
machine. The kinematic modeling of these PKMs must be 
done case by case according to their structure. 
Many researchers have contributed to the study of the 
kinematics of lower-DOF PKMs. Many of them have 
focused on the discussion of both analytical and numerical 
methods [11-12]. This paper investigates the workspace 
and the inverse and direct kinematics analysis of the 
VERNE machine (Fig. 1).  
 
Fig. 1. Overall view of the VERNE machine 
The following section describes the VERNE machine. In 
section III, we study the kinematics of the VERNE 
machine. In section IV, we present the workspace 
calculation for different tool length and for different 
orientation angle of the tool. Finally Section V concludes 
this paper. 
II. DESCRIPTION OF THE VERNE MACHINE 
The VERNE machine consists of a parallel module and a 
tilting table as shown in Fig. 2. The vertices of the moving 
platform of the parallel module are connected to a fixed-
base plate through three legs Ι, ΙΙ and ΙΙΙ. Each leg uses a 
pair of rods linking a prismatic joint to the moving platform 
through two pairs of spherical joints. Legs ΙΙ and ΙΙΙ are 
two identical parallelograms. Leg Ι differs from the other 
two legs in that 
11
12
11
12
A A
B B
≠
, where 
ij
A  (respectively 
Workspace and Kinematic Analysis of the VERNE machine 
Daniel KANAAN, Philippe WENGER and Damien CHABLAT 
P
 
 
 
ij
B ) is the center of spherical joint number j on the 
prismatic joint number i (respectively on the moving 
platform side), i = 1..3, j = 1..2. The movement of the 
moving platform is generated by three sliding actuators 
along three vertical guideways. 
ΙΙ
Ι
ΙΙΙ
x
y z
xp
yp
zp
xt
zt
yt
da
dt
Tilting axis
Δ
U
 
Fig. 2: Schematic representation of the VERNE machine 
Due to the arrangement of the links and joints, legs ΙΙ 
and ΙΙΙ prevent the platform from rotating about y and z 
axes. Leg Ι prevents the platform from rotating about z-axis 
(Fig. 2) because 
11
12
11
12
A A
B B
≠
, however, a slight coupled 
rotation α  about x-axis exists. The tilting table is used to 
rotate the workpiece about two orthogonal axes. The first 
one, the tilting axis, is horizontal and the second one, the 
rotary axis, is always perpendicular to the tilting table. This 
machine takes full advantage of these two additional axes 
to adjust the tool orientation with respect to the workpiece. 
III. KINEMATIC ANALYSIS OF THE VERNE MACHINE 
A. Kinematic equations of the VERNE machine 
In order to analyze the kinematics of the VERNE 
machine, three relative coordinates are assigned as shown 
in Fig. 2. A static Cartesian frame 
( , , , )
b
R
O x y z
=
 is fixed 
at the base of the machine tool, with the z-axis pointing 
downward along the vertical direction. Two mobile 
Cartesian frame, the first frame 
( ,
, 
, 
)
pl
P
P
P
R
P x
y
z
=
, is 
attached to the moving platform at point P and the second 
frame, 
( ,
, , )
t
t
t
t
R t x
y
z
 is attached to the tilting table at 
point t. Let us b
pl
T  define the transformation matrix that 
brings the fixed Cartesian frame 
b
R  on the frame 
pl
R  
linked to the moving platform. 
(
)
(
)
, 
, 
, 
b
pl
p
p
p
T
Trans x
y
z
Rot x α
=
 
(1) 
We use this transformation matrix to express 
ij
B  as 
function of 
, 
, 
 and 
p
p
p
x
y
z
α  by using the relation 
b
pl
ij
pl
ij
B
T
B
=
 where 
pl
ij
B  represents the point 
ij
B  
expressed in the frame 
pl
R . 
-600
-400
-200
0
200
400
600
-800
-600
-400
-200
0
200
400
600
800
P
A11
B11
A12
B12
A21
B21
A22
B22
A31
B31
A32
B32
r4
y
x
yP
xP
R2
 d2
D2
R1
D1
r1
d1
2r3
2r2
 
Fig. 3: Dimensions of the parallel kinematic structure 
Using the parameters defined in Figs. 2 and 3, the 
constraint equations of the parallel manipulator are 
expressed as: 
(
)
(
)
(
)
(
)
2
2
2
2
0 
1..3, 
1..2
Bij
Aij
Bij
Aij
Bij
Aij
i
x
x
y
y
z
z
L
i
j
−
+
−
+
−
−
=
=
=
 
(2) 
Leg Ι is represented by two different Eqs. (3-4). This is due 
to the fact that 
11
12
11
12
A A
B B
≠
 (figure 3). 
(
)
(
)
(
)
2
2
1
1
1
1
2
2
1
1
1
cos( )
sin( )
0
P
P
P
x
D
d
y
R
r
z
R
L
α
α
ρ
+
−
+
+
−
+
+
−
−
=
 
(3) 
(
)
(
)
(
)
2
2
1
1
1
1
2
2
1
1
1
cos( )
sin( )
0
P
P
P
x
D
d
y
R
r
z
R
L
α
α
ρ
+
−
+
−
+
+
−
−
−
=
 
(4) 
Leg ΙΙ is represented by a single Eq. (5).  
(
)
(
)
(
)
2
2
2
2
2
4
2
2
2
2
2
cos( )
sin( )
-
0
P
P
P
x
D
d
y
R
r
z
R
L
α
α
ρ
+
−
+
−
+
+
−
−
=
 
(5) 
The leg ІІІ, which is similar to leg ІІ (figure 3), is also 
represented by a single Eq. (6). 
(
)
(
)
(
)
2
2
2
2
2
4
2
2
2
3
3
cos( )
sin( )
0
P
P
P
x
D
d
y
R
r
z
R
L
α
α
ρ
+
−
+
+
−
+
+
−
−
=
 
(6) 
θ1
φ2
θ2
φ1
Tilting axis
Tool
 
Fig. 4: Draw of the tilting table where the tool orientation is defined by 
(
1φ , 
2
φ ) relative to 
tR  and the orientation of the tilting table is defined by 
(
1θ ,
2
θ ) relative to 
b
R  
 
 
 
Let b
tT  define the transformation matrix that brings the 
fixed Cartesian frame 
b
R  on the frame 
tR  linked to the 
tilting table.  
1
2
( ,
)
( ,
)
( ,
)
( , )
( ,
)
b
t
a
t
T
trans z d
rot x
trans z d rot x
rot z
θ
π
θ
=
 (7)  
Let t
pl
T  define the transformation matrix that brings the 
frame 
tR  linked to the tilting table on the frame 
pl
R  linked 
to the moving platform; where 
,
u
X
 
uY  and 
u
Z  are the 
coordinates of the tool centre point (TCP), U, in 
tR . 
2
1
(
,
,
)
( ,
)
( ,
)
( ,
)
t
pl
u
u
u
T
trans X
Y Z
rot z
rot x
trans z
φ
π
φ
=
+
−Δ
 
(8) 
We use transformation matrices from Eqs. (7) and (8) in 
order to express 
ij
B  as function of 
,
u
X
 
,
uY
 
u,
Z
 
1,
φ  
2,
φ
 
1θ  and 
2
θ  by using the relation 
b
pl
ij
pl
ij
B
T
B
=
 where  
b
b
t
pl
t
pl
T
T T
=
 and pl
ij
B  represents the point 
ij
B  expressed 
in the frame 
pl
R . 
Using Eq. (2) and the parameters defined in Figs. 2, 3 
and 4, we can express all constraint equations of the 
VERNE machine. However knowing that 
1
1
i
i
A B  and 
2
2
i
i
A B  are parallel for i=1..2, we can prove that: 
2
2
θ
φ
= −
 
(9) 
Substituting the above value of 
2
θ  in all constraint 
equations resulting from Eq. (1), we obtain that leg Ι is 
represented by two different equations (10) and (11) while 
leg ΙΙ (respectively leg ΙΙΙ) is represented by only one 
equation (12) (respectively equation 13)).  
Equations (10-13) are not reported here because of space 
limitation. They are available in [14].) 
If we identify Eqs. (10), (11), (12) and (13) with Eqs. 
(3), (4), (5) and (6) respectively, we conclude that: 
1
1
α
θ
φ
=
+
 
(14) 
The constraint equations of the VERNE machine will be 
used in order to obtain the inverse kinematic models of the 
full VERNE machine. 
B. Coupling between the position and the orientation of 
the platform 
The parallel module of the VERNE machine possesses 
three actuators and three degrees of freedom. However, 
there is a coupling between the position and the orientation 
angle of the platform. The object of this subsection is to 
study the coupling constraint imposed by leg I.  
By eliminating 
1
ρ  from Eqs. (3) and (4), we obtain a 
relation (15) between 
, 
 and 
P
P
x
y
α  independently of 
Pz . 
(
)
(
)
(
)
(
)
2
2
2
2
2
2
1
1
1
1
1 1
1
2
2
2
2
2
1
1
1
1
1 1
sin ( )
2
cos( )
sin ( )
2
cos( )
0
P
P
R
x
D
d
r
R r
R
y
R
L
R
r
R r
α
α
α
α
+
−
+
−
+
−
−
+
−
=
 
(15) 
We notice that for a given α , Eq. (15) represents an 
ellipse (16). The size of this ellipse is determined by a  and 
b , where a  is the length of the semi major axis and b  is 
the length of the semi minor axis.  
(
)
2
2
1
1
2
2
1
P
P
x
D
d
y
a
b
+
−
+
=  
(16) 
where 
(
)
(
)
(
)
(
)
(
)
2
2
2
1
1
1
1 1
2
2
2
2
2
1
1
1
1
1 1
2
2
1
1 1
1
 
2
cos( )
                
sin ( )
2
cos( )
 
 
2
cos( )
a
L
R
r
R r
R
L
R
r
R r
b
r
R r
R
α
α
α
α
⎧
=
−
+
−
⎪
⎪⎨
−
+
−
⎪
=
⎪
−
+
⎩
 
These ellipses define the locus of points reachable with 
the same orientation α . 
C. The Inverse Kinematics 
For the inverse kinematic problem of the parallel 
module, the position coordinates (
, 
, 
P
P
P
x
y
z ) are given but 
the coordinates 
(
1..3)
i i
ρ
=
 of the actuated prismatic joints 
and the orientation angle α  of the moving platform are 
unknown. The problem consists in solving the system ( 1)
S
 
of 4 equations (3-6) for only 4 unknowns (
(
1..3)
i i
ρ
=
 and 
α ). Thus, the position of the TCP (
, 
, 
u
u
u
X
Y
Z ) and the 
orientation of the tool (
1
2
and 
φ
φ ) are given relative to the 
frame 
tR , but the joint coordinates, defined by the position 
(
1..3)
i i
ρ
=
 of the actuated prismatic and the orientation 
(
1
2
and 
θ
θ ) of the tilting table in the base frame 
b
R  are 
unknown. Knowing that 
2
2
θ
φ
= −
 from Eq. (9), the 
problem consists in solving the system ( 2)
S
 of 4 
equations ((10), (11), (12) and (13)) for only 4 unknowns 
(
(
1..3)
i i
ρ
=
 and 
1θ ). 
In both cases, we follow the same reasoning to solve the 
inverse kinematics. Due to space limitation, we present 
only the inverse kinematic model of the full VERNE 
machine. The inverse kinematic model of the parallel 
module can be obtained easily by considering system ( 1)
S
 
instead of system ( 2)
S
 (see [13]). 
First, we eliminate 
1
ρ  from Eqs. (10) and (11) in order 
to obtain a relation (17) between the TCP position and 
orientation (
,
u
X
 
,
uY
 
,
u
Z
 
1φ , 
2
φ ) and the tilting angle 
1θ . 
Then, we find all possible orientation angles 
1θ  for 
prescribed values of the position and the orientation of the 
tool. These orientations are determined by solving a six-
degree-characteristic polynomial in 
1
tan(
/ 2)
θ
 derived 
from Eq. (17). This polynomial can have up to four real 
solutions. This conclusion is verified by the fact that 
1
1
θ
φ
α
=
−
 from Eq. 30 where α  can have only four real 
solutions (which can be proved by drawing together all 
ellipses of iso-values of α , see [13]). After finding all the 
possible orientations, we use the system of equations ( 2)
S
 
to calculate the joint coordinates 
iρ  for each orientation 
angle 
1θ . 
For 
1,
ρ  we must verify that the values of 
1
ρ  obtained 
from Eqs. (10) and (11) are the same. As a result, we 
eliminate one of the two solutions. 
 
 
 
Observing the above remark and knowing from [16] that 
Eqs (10-11), (12), (13) are defined as two-degree-
polynomials in 
, 
1..3
i i
ρ
=
 respectively, we conclude that 
there are four solutions for leg Ι and two solutions for leg ΙΙ 
and ΙΙΙ. Thus there are sixteen inverse kinematic solutions 
for the VERNE machine.  
From the sixteen theoretical inverse kinematics solutions 
shown in Fig. 5, only one is used by the VERNE machine: 
the one referred to as (p) in Fig. 5, which is characterized 
by the fact that each leg must have its slider attachment 
points above the moving platform attachment points. 
 
 
(a) 
(b) 
(c) 
 
 
(d) 
(e) 
(f) 
 
 
(g) 
(h) 
(i) 
 
 
 
(j) 
(k) 
(l) 
 
 
 
(m) 
(n) 
(o) 
 
 
(p) 
Fig 5: The sixteen solutions to the inverse kinematics problem when 
-240 mm,
P
x =
 
-86 mm
P
y =
 and 
1000 mm
Pz =
 
For the remaining 15 solutions one of the sliders leaves 
its joint limits or the two rods of leg I cross. Most of these 
solutions are characterized by the fact that at least one of 
the legs has its slider attachment points lower than the 
moving platform attachment points. To prevent rod 
crossing, we also add a condition on the orientation of the 
moving platform. This condition is 
1
1
1
1
cos(
)
R
r
θ
φ
+
>
. 
Finally, we check the joint limits of the sliders and the 
serial singularities [10]. 
Applying the above conditions will always yield a 
unique solution for practical applications. The proposed 
method for calculating the various solutions of the inverse 
kinematic problem has been implemented in C++. We are 
working with AMTRI, a UK industry, on implementing 
this code in a simulation package of PKMs, Visual 
Components. 
D. The Forward Kinematics  
For the forward kinematics of our spatial parallel 
manipulator, 
the 
values 
of 
the 
joint 
coordinates 
(
1..3)
i i
ρ
=
 are known and the goal is to find the 
coordinates 
P
x , 
P
y  and 
Pz  of the centre of the moving 
platform P. 
To solve the forward kinematics, we successively 
eliminate variables 
P
x , 
P
y  and 
Pz  from the system ( 1)
S
 
of four equations (3-6) to lead to an equation function of 
the joint coordinates 
 (
1..3)
i i
ρ
=
 and function of the 
orientation angle α  of the platform. To do so, we first 
compute 
P
y  as function of 
Pz  by subtracting equation (3) 
from equation (4) and we replace this variable in system 
( 1)
S
 to obtain a new system ( 3)
S
 of three equations (18), 
(19) and (20) derived from equations (3), (5) and (6) 
respectively. We then compute 
Pz  as function of 
(
1..3)
i i
ρ
=
 and α  by subtracting equation (19) from 
equation (20). We replace this variable in system ( 3)
S
 to 
obtain a new system ( 4)
S
 of two equations (21) and (22) 
derived from equations (18) and (19) respectively. Finally, 
we compute 
P
x  as function of 
 (
1..3)
i i
ρ
=
 and α  by 
subtracting equation (21) from equation (22) and we 
replace this variable in the system ( 4)
S
 in order to 
eliminate 
P
x . Equations of system (
) (i=3..4)
Si
 are not 
reported here because of space limitation. They are 
available in [14]. 
For each step, we determine solutions existence 
conditions by studying the denominators that appear in the 
expressions of 
P
x , 
P
y  and 
Pz . These conditions are: 
1
1
cos( )
0
R
r
α −
≠
 
(23) 
(
)(
)
(
)
2
3
1
1
4
1
1
2
cos( )
2sin( )
0
R
r
r R
r R
ρ
ρ
α
α
−
−
+
−
≠
 
(24) 
Equation (23) implies that 
1
1
A B  is perpendicular to the 
slider plane of leg І. In this case equation (16) represents a 
circle because a
b
=
. 
When 
2
3
= 
ρ
ρ  in equation (24), we have 
{0, }
α
π
=
. 
This means that 
0
P
y =
 (obtained from Equations. 
(5) −(6)). 
To finish the resolution of the system, we perform the 
 
 
 
tangent-half-angle 
substitution 
tan(
/ 2)
s
α
=
. 
As 
a 
consequence, the forward kinematics of our parallel 
manipulator 
results 
in 
a 
eight-degree-characteristic 
polynomial in s , whose coefficients are relatively large 
expressions in 
1
ρ , 
2
ρ  and 
3
ρ . 
 
(a) 
(b) 
 
 
(c) 
(d) 
Fig. 6: The four assembly-modes of the VERNE parallel module for 
1
674 mm,
ρ =
 
2
685 mm
ρ =
 and 
3
250 mm.
ρ =
 only (a) is reachable by 
the actual machine 
For the forward kinematics of the VERNE machine, the 
values of the joint coordinates, defined by the position 
 (
1..3)
i i
ρ
=
 of the actuated prismatic and the orientation 
(
1
2
 and 
θ
θ ) of the tilting table in the base frame 
b
R  are 
known and the goal is to find the position of the TCP 
(
, 
, 
u
u
u
X
Y
Z ) and the orientation of the tool (
1
2
and 
φ
φ ) in 
the frame 
tR . 
Knowing that 
2
2
φ
θ
= −
 and 
1
1
φ
α
θ
=
−
 from Eqs (9) and 
(14), we solve this problem by first solving the forward 
kinematics of the parallel module of the VERNE machine 
in order to find the coordinates 
P
x , 
P
y  and 
Pz  of the 
centre of the moving platform P and the orientation α  of 
the moving platform in term of the joint coordinates 
 (
1..3)
i i
ρ
=
. We then use transformation matrices from 
Eqs. (1) and (7) in order to express the tool position and 
orientation 
(
1
2
, 
, 
, 
 and 
u
u
u
X
Y
Z
φ
φ ) 
as 
function 
of 
(
)
1
2
,
,
,
,
P
P
P
x
y
z θ θ
. 
1
t
t
b
t
b
pl
b
b
pl
b
b
pl
t
pl
U
T U
T T
U
T
T
U
−
=
=
=
 
(25) 
where 
[
]
0
0
1
T
plU =
Δ
 and 
[
]1
T
t
u
u
u
U
X
Y
Z
=
 
represent the TCP, 
,
U  expressed in frames 
pl
R  (linked to 
the moving platform) and the base frame 
b
R  respectively. 
Finally we obtain: 
1
1
2
2
2
2
2
2
1
 and 
sin(
)
1cos(
)
  
cos(
)
1sin(
)
sin(
)
2
u
p
u
p
u
p
Y
x
V
X
x
V
Z
y
V
φ
α
θ
φ
θ
θ
θ
θ
θ
θ
=
−
= −
= −
+
⎧
⎧
⎨
⎨
=
+
=
+
⎩
⎩
 
(26) 
where 
(
)
1
1
1
V1= sin(
)
cos(
)
sin( )
p
p
a
y
z
d
α
θ
θ
θ
Δ
−
−
−
−
 and 
1
1
1
2
cos(
)
cos(
)
cos(
)
t
p
a
V
d
z
d
θ
θ
α
θ
=
−
+
−Δ
−
 
For the VERNE machine, only 4 assembly-modes have 
been found (figure 6). It was possible to find up to 6 
assembly-modes but only for input joint values out of the 
reachable joint space of the machine. Only one assembly-
mode is actually reachable by the machine (solution (a) 
shown in Fig. 6) because the other ones lead to either rod 
crossing, collisions, or joint limit violation. The right 
assembly mode can be recognized, like for the right 
working mode, by the fact that each leg must have its slider 
attachment points above the moving platform attachment 
points 
The proposed method for calculating the various 
solutions of the forward kinematic problem has been 
implemented in Maple. 
IV. WORKSPACE CALCULATION OF THE VERNE MACHINE 
A. Workspace calculation of the parallel module [10] 
The parallel module of the VERNE machine possesses 3 
degrees of freedom. A complete representation of the 
workspace is a volume. Consequently this workspace can 
be defined by the positions in space reachable by the point 
P, centre of the mobile Cartesian frame 
pl
R . However the 
parallel module undergoes a complex motion, thus to 
overcome the complexity caused by the presence of the 
coupled rotation, we propose the following method. 
Step 1. We virtually cut the leg І, which is constituted of 
rods 11 and 12 (rod ij denotes AijBij) by supposing that 
11
ρ  
is independent of 
12
ρ , where 
11
ρ  and 
12
ρ  are respectively 
the joint coordinates of the prismatic joints linking rods 11 
and 12 to their guideways. The parallel module of the 
VERNE machine possesses now 4 degrees of freedom 
instead of 3. These degrees of freedom are defined by 
coordinates 
,
P
x
 
P
y  and 
Pz  of the point P and by the 
orientation α  of the moving platform. Thus we consider 
that the orientation α  of the platform is given, and we 
geometrically model constraints limiting the workspace of 
the new parallel architecture. These constraints are: (i) 
Interference between links, (ii) Leg Length limits, (iii) 
Serial Singularity, (iv) Mechanical limits on passive joints 
(we take into consideration the type of joints as well as the 
location of these joints in the machine) and (v) Actuator 
stroke. The intersection between these models is a volume. 
Step 2. We consider the interdependence between rods 
11 and 12 of leg І characterized by the fact that 
11
12
1
ρ
ρ
ρ
=
=
. This will allows us to determine the 
geometric shape for which the coupling between the 
position and the orientation of the platform exists. Thus for 
a given orientation 
,
α  the point P describes a surface 
defined by a hollow cylinder whose base is an ellipse as 
shown in subsection III.B. 
 
 
 
The intersection between geometric models defined at 
step 1 and the surface defined at step 2 represents the 
constant orientation workspace of the new 4 degree-of-
freedom parallel module. We calculate a horizontal cut of 
this workspace when the point P, center of the mobile 
Cartesian frame, moves in a known horizontal plane. Then 
we proceed by discretization to determine the complete 
workspace of the parallel module of the VERNE machine 
(see Figure 7). To calculate the workspace for a given tool 
length, we only need to express the TCP, U, in the base 
frame as function of (
)
,
,
,
P
P
P
x
y
z
α  by using the 
transformation matrix from Eq. (1), 
b
pl
pl
U
T
U
=
: 
, 
sin( ) and 
cos( )
U
P
U
P
U
P
x
x
y
y
z
z
α
α
=
=
−Δ
=
+ Δ
 
(27) 
 
 
 
 
Fig. 7: Workspace defined as the 
set of points P in the fixed Cartesian 
frame 
b
R  
Fig 8: The manufacturing 3D 
workspace for given tool length of 
50 mm 
B. The manufacturing 3D workspace [14] 
In this subsection, we calculate the manufacturing 3D 
workspace of the VERNE (the tool axis remains 
perpendicular to the part base) by considering that the 
tilting table rotates about its axis and follows the 
orientation α  of the moving platform, which means that 
1θ
α
=
 and 
1
0
φ =
, we suppose also that 
2
2
0
θ
φ
=
=
. 
To calculate this workspace we use Eq. (26) after 
replacing 
1θ , 
2
θ , 
1φ , and 
2
φ  by their values. This will give 
us the position and the orientation of the TCP in the frame 
Rt (
,
u
X
 
,
uY
 
,
u
Z
1,
φ  
2
φ ) expressed as function of (
,
px
 
 
p
y , 
pz , α ). A graphical representation of the workspace 
of PKMs with more than three degrees of freedom is only 
possible if we fix parameters representing the exceeded 
degrees of freedom. Thus to calculate the workspace for 
various orientations of the tool, we first fix 
1φ  and 
2
φ , then 
we calculate the position of the TCP (
,
u
X
 
,
uY
 
u
Z ) in the 
frame Rt by using Eq. (26). 
The proposed method for calculating the workspace for 
various tool lengths and for different tool orientation angles 
has been implemented in Maple and displayed in the CAD 
software CATIA. The obtained results show a workspace 
larger than the one currently used by the VERNE machine 
(for some given tool orientation angles). So this will able us 
to improve the productivity of the VERNE machine and to 
reach the limit of its capacity without risk of collision or 
damage of the VERNE machine. 
V. CONCLUSIONS 
This paper was devoted to the kinematic and workspace 
analysis of a 5-DOF hybrid machine tool, the VERNE 
machine. This machine possesses a complex motion caused 
by the unsymmetrical architecture of the parallel module 
where one of the legs is different from the other two legs. It 
was shown that the inverse kinematics has sixteen solutions 
and the forward kinematics may have six real solutions. 
The workspace was calculated by using a combination of 
discretization and geometrical method. This work is of 
interest as it may improve the efficiency of the machine 
because the controller of the actual VERNE machine 
resorts to an iterative Newton-Raphson resolution of the 
kinematic models.  
ACKNOWLEDGMENT 
This work has been partially funded by the European 
projects NEXT, acronyms for “Next Generation of 
Productions Systems”, Project no° IP 011815. The authors 
would like to thank the Fatronik society, which permitted 
us to use the CAD drawing of the Machine VERNE what 
allowed us to present well the machine. 
REFERENCES 
[1] 
J.-P Merlet, Parallel Robots. Springer-Verlag, New York, 2005. 
[2] 
J. Tlusty, J.C. Ziegert and S. Ridgeway, Fundamental comparison of 
the use of serial and parallel kinematics for machine tools, Annals of 
the CIRP, 48 (1) 351–356, 1999. 
[3] 
Ph. Wenger, C. Gosselin and B. Maille, A comparative study of 
serial and parallel mechanism topologies for machine tools. In 
Proceedings of PKM’99, pp.  23–32, Milan, Italy, 1999. 
[4] 
M. Weck, and M. Staimer, Parallel Kinematic Machine Tools – 
Current State and Future Potentials. Annals of the CIRP, 51(2), 
pp. 671-683, 2002. 
[5] 
J-P. Merlet, “Parallel robots”, Kluwer, June 2000. 
[6] 
Chablat D., Wenger Ph., Working Modes and Aspects in Fully-
Parallel Manipulator, Proceeding IEEE International Conference on 
Robotics and Automation, pp. 1964-1969, May 1998 
[7] 
Majou F., Wenger Ph. and Chablat D., The Design of Parallel 
Kinematic Machine Tools Using Kinetostatic Performance Criteria, 
3d Int. Conference on Metal Cutting, Metz, France, June, 2001. 
[8] 
H. S. Kim, L-W. Tsai, Kinematic Synthesis of a Spatial 3-RPS 
Parallel Manipulator. In Journal of mechanical design, volume 125, 
pp. 92-97, March 2003. 
[9] 
O. Ibrahim and W. Khalil, Kinematic and dynamic modelling of the 
3-RPS parallel manipulator. In 12 IFToMM World Congress, 
Besancon, June 2007. 
[10] D. Kanaan, Ph. Wenger and D. Chablat, Workspace Analysis of the 
Parallel Module of the VERNE Machine. Problems of Mechanics, № 
4(25), pp. 26-42, Tbilisi, 2006. 
[11] X.J. Liu, J.S. Wang, F. Gao and L.P. Wang, On the analysis of a new 
spatial three-degree-of freedom parallel manipulator. IEEE Trans. 
Robotics Automation 17 (6), pp. 959–968, December 2001. 
[12] Nair R. and Maddocks J.H. On the forward kinematics of parallel 
manipulators. Int. J. Robotics Res. 13 (2), pp. 171–188, 1994. 
 
 
 
[13] Kanaan D., Wenger P. and Chablat D., Kinematics analysis of the 
parallel module of the VERNE machine, 12th World Congress in 
Mechanism and Machine Science, Besançon, Juin, 2007, IFToMM 
[14] Kanaan D., Wenger Ph. and Chablat D., Workspace and Kinematic 
analysis of the VERNE machine, Project NEXT internal report, 
February 2007. 
 
 
 
  
Abstract— This paper describes the workspace and the 
inverse and direct kinematic analysis of the VERNE machine, 
a serial/parallel 5-axis machine tool designed by Fatronik for 
IRCCyN. This machine is composed of a three-degree-of-
freedom (DOF) parallel module and a two-DOF serial tilting 
table. The parallel module consists of a moving platform that 
is connected to a fixed base by three non-identical legs. This 
feature involves (i) a simultaneous combination of rotation 
and translation for the moving platform, which is balanced by 
the tilting table and (ii) workspace whose shape and volume 
vary as a function of the tool length. This paper summarizes 
results obtained in the context of the European projects 
NEXT (“Next Generation of Productions Systems”). 
I. INTRODUCTION 
ARALLEL kinematic machines (PKM’s) are well known 
for their high structural rigidity, better payload-to-
weight ratio, high dynamic performances and high 
accuracy [1-3]. Thus, they are prudently considered as 
attractive alternatives designs for demanding tasks such as 
high-speed machining [4]. 
A well-known feature of PKM is the existence of 
multiple solutions to the direct kinematic problem. That is, 
the moving platform can admit several positions and 
orientations (poses) in the workspace for one given set of 
input joint values [5]. Moreover, parallel manipulators exist 
with multiple inverse kinematic solutions. This means that 
the moving platform can admit several input joint values 
corresponding to one given pose of the end-effector [6]. 
For industrial PKMs, the resolution of the direct and 
inverse kinematic problems is usually made using iterative 
methods. With such methods, the calculation time can 
change as function of the configuration of the PKM. A 
major drawback of PKMs is a complex workspace. The 
workspace of a PKM has not a simple geometric shape, and 
its functional volume is reduced as compared to the space 
occupied by the machine [7]. 
 
D. Kanaan is with the Institut de Recherche en Communications et 
Cybernétique de Nantes (IRCCyN) (UMR CNRS 6597), France 
(corresponding author to provide phone: 33-240376958; fax: 33-
240376930, e-mail: Daniel.Kanaan@irccyn-nantes.fr); P. Wenger is with 
the 
IRCCyN 
(UMR 
CNRS 
6597), 
France 
(e-mail: 
Philippe.Wenger@irccyn-nantes.fr); D. Chablat is with the IRCCyN 
(UMR CNRS 6597), France (e-mail: Damien.Chablat@irccyn-nantes.fr). 
Because many industrial tasks require less than six 
degrees of freedom, several lower-DOF PKMs have been 
developed. For some of theme, the reduction of the number 
of DOFs can result in coupled motions of the moving 
platform [8-10]. This is the case, for example, in the RPS 
manipulator [9] and in the parallel module of the VERNE 
machine. The kinematic modeling of these PKMs must be 
done case by case according to their structure. 
Many researchers have contributed to the study of the 
kinematics of lower-DOF PKMs. Many of them have 
focused on the discussion of both analytical and numerical 
methods [11-12]. This paper investigates the workspace 
and the inverse and direct kinematics analysis of the 
VERNE machine (Fig. 1).  
 
Fig. 1. Overall view of the VERNE machine 
The following section describes the VERNE machine. In 
section III, we study the kinematics of the VERNE 
machine. In section IV, we present the workspace 
calculation for different tool length and for different 
orientation angle of the tool. Finally Section V concludes 
this paper. 
II. DESCRIPTION OF THE VERNE MACHINE 
The VERNE machine consists of a parallel module and a 
tilting table as shown in Fig. 2. The vertices of the moving 
platform of the parallel module are connected to a fixed-
base plate through three legs Ι, ΙΙ and ΙΙΙ. Each leg uses a 
pair of rods linking a prismatic joint to the moving platform 
through two pairs of spherical joints. Legs ΙΙ and ΙΙΙ are 
two identical parallelograms. Leg Ι differs from the other 
two legs in that 
11
12
11
12
A A
B B
≠
, where 
ij
A  (respectively 
Workspace and Kinematic Analysis of the VERNE machine 
Daniel KANAAN, Philippe WENGER and Damien CHABLAT 
P
 
 
 
ij
B ) is the center of spherical joint number j on the 
prismatic joint number i (respectively on the moving 
platform side), i = 1..3, j = 1..2. The movement of the 
moving platform is generated by three sliding actuators 
along three vertical guideways. 
ΙΙ
Ι
ΙΙΙ
x
y z
xp
yp
zp
xt
zt
yt
da
dt
Tilting axis
Δ
U
 
Fig. 2: Schematic representation of the VERNE machine 
Due to the arrangement of the links and joints, legs ΙΙ 
and ΙΙΙ prevent the platform from rotating about y and z 
axes. Leg Ι prevents the platform from rotating about z-axis 
(Fig. 2) because 
11
12
11
12
A A
B B
≠
, however, a slight coupled 
rotation α  about x-axis exists. The tilting table is used to 
rotate the workpiece about two orthogonal axes. The first 
one, the tilting axis, is horizontal and the second one, the 
rotary axis, is always perpendicular to the tilting table. This 
machine takes full advantage of these two additional axes 
to adjust the tool orientation with respect to the workpiece. 
III. KINEMATIC ANALYSIS OF THE VERNE MACHINE 
A. Kinematic equations of the VERNE machine 
In order to analyze the kinematics of the VERNE 
machine, three relative coordinates are assigned as shown 
in Fig. 2. A static Cartesian frame 
( , , , )
b
R
O x y z
=
 is fixed 
at the base of the machine tool, with the z-axis pointing 
downward along the vertical direction. Two mobile 
Cartesian frame, the first frame 
( ,
, 
, 
)
pl
P
P
P
R
P x
y
z
=
, is 
attached to the moving platform at point P and the second 
frame, 
( ,
, , )
t
t
t
t
R t x
y
z
 is attached to the tilting table at 
point t. Let us b
pl
T  define the transformation matrix that 
brings the fixed Cartesian frame 
b
R  on the frame 
pl
R  
linked to the moving platform. 
(
)
(
)
, 
, 
, 
b
pl
p
p
p
T
Trans x
y
z
Rot x α
=
 
(1) 
We use this transformation matrix to express 
ij
B  as 
function of 
, 
, 
 and 
p
p
p
x
y
z
α  by using the relation 
b
pl
ij
pl
ij
B
T
B
=
 where 
pl
ij
B  represents the point 
ij
B  
expressed in the frame 
pl
R . 
-600
-400
-200
0
200
400
600
-800
-600
-400
-200
0
200
400
600
800
P
A11
B11
A12
B12
A21
B21
A22
B22
A31
B31
A32
B32
r4
y
x
yP
xP
R2
 d2
D2
R1
D1
r1
d1
2r3
2r2
 
Fig. 3: Dimensions of the parallel kinematic structure 
Using the parameters defined in Figs. 2 and 3, the 
constraint equations of the parallel manipulator are 
expressed as: 
(
)
(
)
(
)
(
)
2
2
2
2
0 
1..3, 
1..2
Bij
Aij
Bij
Aij
Bij
Aij
i
x
x
y
y
z
z
L
i
j
−
+
−
+
−
−
=
=
=
 
(2) 
Leg Ι is represented by two different Eqs. (3-4). This is due 
to the fact that 
11
12
11
12
A A
B B
≠
 (figure 3). 
(
)
(
)
(
)
2
2
1
1
1
1
2
2
1
1
1
cos( )
sin( )
0
P
P
P
x
D
d
y
R
r
z
R
L
α
α
ρ
+
−
+
+
−
+
+
−
−
=
 
(3) 
(
)
(
)
(
)
2
2
1
1
1
1
2
2
1
1
1
cos( )
sin( )
0
P
P
P
x
D
d
y
R
r
z
R
L
α
α
ρ
+
−
+
−
+
+
−
−
−
=
 
(4) 
Leg ΙΙ is represented by a single Eq. (5).  
(
)
(
)
(
)
2
2
2
2
2
4
2
2
2
2
2
cos( )
sin( )
-
0
P
P
P
x
D
d
y
R
r
z
R
L
α
α
ρ
+
−
+
−
+
+
−
−
=
 
(5) 
The leg ІІІ, which is similar to leg ІІ (figure 3), is also 
represented by a single Eq. (6). 
(
)
(
)
(
)
2
2
2
2
2
4
2
2
2
3
3
cos( )
sin( )
0
P
P
P
x
D
d
y
R
r
z
R
L
α
α
ρ
+
−
+
+
−
+
+
−
−
=
 
(6) 
θ1
φ2
θ2
φ1
Tilting axis
Tool
 
Fig. 4: Draw of the tilting table where the tool orientation is defined by 
(
1φ , 
2
φ ) relative to 
tR  and the orientation of the tilting table is defined by 
(
1θ ,
2
θ ) relative to 
b
R  
 
 
 
Let b
tT  define the transformation matrix that brings the 
fixed Cartesian frame 
b
R  on the frame 
tR  linked to the 
tilting table.  
1
2
( ,
)
( ,
)
( ,
)
( , )
( ,
)
b
t
a
t
T
trans z d
rot x
trans z d rot x
rot z
θ
π
θ
=
 (7)  
Let t
pl
T  define the transformation matrix that brings the 
frame 
tR  linked to the tilting table on the frame 
pl
R  linked 
to the moving platform; where 
,
u
X
 
uY  and 
u
Z  are the 
coordinates of the tool centre point (TCP), U, in 
tR . 
2
1
(
,
,
)
( ,
)
( ,
)
( ,
)
t
pl
u
u
u
T
trans X
Y Z
rot z
rot x
trans z
φ
π
φ
=
+
−Δ
 
(8) 
We use transformation matrices from Eqs. (7) and (8) in 
order to express 
ij
B  as function of 
,
u
X
 
,
uY
 
u,
Z
 
1,
φ  
2,
φ
 
1θ  and 
2
θ  by using the relation 
b
pl
ij
pl
ij
B
T
B
=
 where  
b
b
t
pl
t
pl
T
T T
=
 and pl
ij
B  represents the point 
ij
B  expressed 
in the frame 
pl
R . 
Using Eq. (2) and the parameters defined in Figs. 2, 3 
and 4, we can express all constraint equations of the 
VERNE machine. However knowing that 
1
1
i
i
A B  and 
2
2
i
i
A B  are parallel for i=1..2, we can prove that: 
2
2
θ
φ
= −
 
(9) 
Substituting the above value of 
2
θ  in all constraint 
equations resulting from Eq. (1), we obtain that leg Ι is 
represented by two different equations (10) and (11) while 
leg ΙΙ (respectively leg ΙΙΙ) is represented by only one 
equation (12) (respectively equation 13)).  
Equations (10-13) are not reported here because of space 
limitation. They are available in [14].) 
If we identify Eqs. (10), (11), (12) and (13) with Eqs. 
(3), (4), (5) and (6) respectively, we conclude that: 
1
1
α
θ
φ
=
+
 
(14) 
The constraint equations of the VERNE machine will be 
used in order to obtain the inverse kinematic models of the 
full VERNE machine. 
B. Coupling between the position and the orientation of 
the platform 
The parallel module of the VERNE machine possesses 
three actuators and three degrees of freedom. However, 
there is a coupling between the position and the orientation 
angle of the platform. The object of this subsection is to 
study the coupling constraint imposed by leg I.  
By eliminating 
1
ρ  from Eqs. (3) and (4), we obtain a 
relation (15) between 
, 
 and 
P
P
x
y
α  independently of 
Pz . 
(
)
(
)
(
)
(
)
2
2
2
2
2
2
1
1
1
1
1 1
1
2
2
2
2
2
1
1
1
1
1 1
sin ( )
2
cos( )
sin ( )
2
cos( )
0
P
P
R
x
D
d
r
R r
R
y
R
L
R
r
R r
α
α
α
α
+
−
+
−
+
−
−
+
−
=
 
(15) 
We notice that for a given α , Eq. (15) represents an 
ellipse (16). The size of this ellipse is determined by a  and 
b , where a  is the length of the semi major axis and b  is 
the length of the semi minor axis.  
(
)
2
2
1
1
2
2
1
P
P
x
D
d
y
a
b
+
−
+
=  
(16) 
where 
(
)
(
)
(
)
(
)
(
)
2
2
2
1
1
1
1 1
2
2
2
2
2
1
1
1
1
1 1
2
2
1
1 1
1
 
2
cos( )
                
sin ( )
2
cos( )
 
 
2
cos( )
a
L
R
r
R r
R
L
R
r
R r
b
r
R r
R
α
α
α
α
⎧
=
−
+
−
⎪
⎪⎨
−
+
−
⎪
=
⎪
−
+
⎩
 
These ellipses define the locus of points reachable with 
the same orientation α . 
C. The Inverse Kinematics 
For the inverse kinematic problem of the parallel 
module, the position coordinates (
, 
, 
P
P
P
x
y
z ) are given but 
the coordinates 
(
1..3)
i i
ρ
=
 of the actuated prismatic joints 
and the orientation angle α  of the moving platform are 
unknown. The problem consists in solving the system ( 1)
S
 
of 4 equations (3-6) for only 4 unknowns (
(
1..3)
i i
ρ
=
 and 
α ). Thus, the position of the TCP (
, 
, 
u
u
u
X
Y
Z ) and the 
orientation of the tool (
1
2
and 
φ
φ ) are given relative to the 
frame 
tR , but the joint coordinates, defined by the position 
(
1..3)
i i
ρ
=
 of the actuated prismatic and the orientation 
(
1
2
and 
θ
θ ) of the tilting table in the base frame 
b
R  are 
unknown. Knowing that 
2
2
θ
φ
= −
 from Eq. (9), the 
problem consists in solving the system ( 2)
S
 of 4 
equations ((10), (11), (12) and (13)) for only 4 unknowns 
(
(
1..3)
i i
ρ
=
 and 
1θ ). 
In both cases, we follow the same reasoning to solve the 
inverse kinematics. Due to space limitation, we present 
only the inverse kinematic model of the full VERNE 
machine. The inverse kinematic model of the parallel 
module can be obtained easily by considering system ( 1)
S
 
instead of system ( 2)
S
 (see [13]). 
First, we eliminate 
1
ρ  from Eqs. (10) and (11) in order 
to obtain a relation (17) between the TCP position and 
orientation (
,
u
X
 
,
uY
 
,
u
Z
 
1φ , 
2
φ ) and the tilting angle 
1θ . 
Then, we find all possible orientation angles 
1θ  for 
prescribed values of the position and the orientation of the 
tool. These orientations are determined by solving a six-
degree-characteristic polynomial in 
1
tan(
/ 2)
θ
 derived 
from Eq. (17). This polynomial can have up to four real 
solutions. This conclusion is verified by the fact that 
1
1
θ
φ
α
=
−
 from Eq. 30 where α  can have only four real 
solutions (which can be proved by drawing together all 
ellipses of iso-values of α , see [13]). After finding all the 
possible orientations, we use the system of equations ( 2)
S
 
to calculate the joint coordinates 
iρ  for each orientation 
angle 
1θ . 
For 
1,
ρ  we must verify that the values of 
1
ρ  obtained 
from Eqs. (10) and (11) are the same. As a result, we 
eliminate one of the two solutions. 
 
 
 
Observing the above remark and knowing from [16] that 
Eqs (10-11), (12), (13) are defined as two-degree-
polynomials in 
, 
1..3
i i
ρ
=
 respectively, we conclude that 
there are four solutions for leg Ι and two solutions for leg ΙΙ 
and ΙΙΙ. Thus there are sixteen inverse kinematic solutions 
for the VERNE machine.  
From the sixteen theoretical inverse kinematics solutions 
shown in Fig. 5, only one is used by the VERNE machine: 
the one referred to as (p) in Fig. 5, which is characterized 
by the fact that each leg must have its slider attachment 
points above the moving platform attachment points. 
 
 
(a) 
(b) 
(c) 
 
 
(d) 
(e) 
(f) 
 
 
(g) 
(h) 
(i) 
 
 
 
(j) 
(k) 
(l) 
 
 
 
(m) 
(n) 
(o) 
 
 
(p) 
Fig 5: The sixteen solutions to the inverse kinematics problem when 
-240 mm,
P
x =
 
-86 mm
P
y =
 and 
1000 mm
Pz =
 
For the remaining 15 solutions one of the sliders leaves 
its joint limits or the two rods of leg I cross. Most of these 
solutions are characterized by the fact that at least one of 
the legs has its slider attachment points lower than the 
moving platform attachment points. To prevent rod 
crossing, we also add a condition on the orientation of the 
moving platform. This condition is 
1
1
1
1
cos(
)
R
r
θ
φ
+
>
. 
Finally, we check the joint limits of the sliders and the 
serial singularities [10]. 
Applying the above conditions will always yield a 
unique solution for practical applications. The proposed 
method for calculating the various solutions of the inverse 
kinematic problem has been implemented in C++. We are 
working with AMTRI, a UK industry, on implementing 
this code in a simulation package of PKMs, Visual 
Components. 
D. The Forward Kinematics  
For the forward kinematics of our spatial parallel 
manipulator, 
the 
values 
of 
the 
joint 
coordinates 
(
1..3)
i i
ρ
=
 are known and the goal is to find the 
coordinates 
P
x , 
P
y  and 
Pz  of the centre of the moving 
platform P. 
To solve the forward kinematics, we successively 
eliminate variables 
P
x , 
P
y  and 
Pz  from the system ( 1)
S
 
of four equations (3-6) to lead to an equation function of 
the joint coordinates 
 (
1..3)
i i
ρ
=
 and function of the 
orientation angle α  of the platform. To do so, we first 
compute 
P
y  as function of 
Pz  by subtracting equation (3) 
from equation (4) and we replace this variable in system 
( 1)
S
 to obtain a new system ( 3)
S
 of three equations (18), 
(19) and (20) derived from equations (3), (5) and (6) 
respectively. We then compute 
Pz  as function of 
(
1..3)
i i
ρ
=
 and α  by subtracting equation (19) from 
equation (20). We replace this variable in system ( 3)
S
 to 
obtain a new system ( 4)
S
 of two equations (21) and (22) 
derived from equations (18) and (19) respectively. Finally, 
we compute 
P
x  as function of 
 (
1..3)
i i
ρ
=
 and α  by 
subtracting equation (21) from equation (22) and we 
replace this variable in the system ( 4)
S
 in order to 
eliminate 
P
x . Equations of system (
) (i=3..4)
Si
 are not 
reported here because of space limitation. They are 
available in [14]. 
For each step, we determine solutions existence 
conditions by studying the denominators that appear in the 
expressions of 
P
x , 
P
y  and 
Pz . These conditions are: 
1
1
cos( )
0
R
r
α −
≠
 
(23) 
(
)(
)
(
)
2
3
1
1
4
1
1
2
cos( )
2sin( )
0
R
r
r R
r R
ρ
ρ
α
α
−
−
+
−
≠
 
(24) 
Equation (23) implies that 
1
1
A B  is perpendicular to the 
slider plane of leg І. In this case equation (16) represents a 
circle because a
b
=
. 
When 
2
3
= 
ρ
ρ  in equation (24), we have 
{0, }
α
π
=
. 
This means that 
0
P
y =
 (obtained from Equations. 
(5) −(6)). 
To finish the resolution of the system, we perform the 
 
 
 
tangent-half-angle 
substitution 
tan(
/ 2)
s
α
=
. 
As 
a 
consequence, the forward kinematics of our parallel 
manipulator 
results 
in 
a 
eight-degree-characteristic 
polynomial in s , whose coefficients are relatively large 
expressions in 
1
ρ , 
2
ρ  and 
3
ρ . 
 
(a) 
(b) 
 
 
(c) 
(d) 
Fig. 6: The four assembly-modes of the VERNE parallel module for 
1
674 mm,
ρ =
 
2
685 mm
ρ =
 and 
3
250 mm.
ρ =
 only (a) is reachable by 
the actual machine 
For the forward kinematics of the VERNE machine, the 
values of the joint coordinates, defined by the position 
 (
1..3)
i i
ρ
=
 of the actuated prismatic and the orientation 
(
1
2
 and 
θ
θ ) of the tilting table in the base frame 
b
R  are 
known and the goal is to find the position of the TCP 
(
, 
, 
u
u
u
X
Y
Z ) and the orientation of the tool (
1
2
and 
φ
φ ) in 
the frame 
tR . 
Knowing that 
2
2
φ
θ
= −
 and 
1
1
φ
α
θ
=
−
 from Eqs (9) and 
(14), we solve this problem by first solving the forward 
kinematics of the parallel module of the VERNE machine 
in order to find the coordinates 
P
x , 
P
y  and 
Pz  of the 
centre of the moving platform P and the orientation α  of 
the moving platform in term of the joint coordinates 
 (
1..3)
i i
ρ
=
. We then use transformation matrices from 
Eqs. (1) and (7) in order to express the tool position and 
orientation 
(
1
2
, 
, 
, 
 and 
u
u
u
X
Y
Z
φ
φ ) 
as 
function 
of 
(
)
1
2
,
,
,
,
P
P
P
x
y
z θ θ
. 
1
t
t
b
t
b
pl
b
b
pl
b
b
pl
t
pl
U
T U
T T
U
T
T
U
−
=
=
=
 
(25) 
where 
[
]
0
0
1
T
plU =
Δ
 and 
[
]1
T
t
u
u
u
U
X
Y
Z
=
 
represent the TCP, 
,
U  expressed in frames 
pl
R  (linked to 
the moving platform) and the base frame 
b
R  respectively. 
Finally we obtain: 
1
1
2
2
2
2
2
2
1
 and 
sin(
)
1cos(
)
  
cos(
)
1sin(
)
sin(
)
2
u
p
u
p
u
p
Y
x
V
X
x
V
Z
y
V
φ
α
θ
φ
θ
θ
θ
θ
θ
θ
=
−
= −
= −
+
⎧
⎧
⎨
⎨
=
+
=
+
⎩
⎩
 
(26) 
where 
(
)
1
1
1
V1= sin(
)
cos(
)
sin( )
p
p
a
y
z
d
α
θ
θ
θ
Δ
−
−
−
−
 and 
1
1
1
2
cos(
)
cos(
)
cos(
)
t
p
a
V
d
z
d
θ
θ
α
θ
=
−
+
−Δ
−
 
For the VERNE machine, only 4 assembly-modes have 
been found (figure 6). It was possible to find up to 6 
assembly-modes but only for input joint values out of the 
reachable joint space of the machine. Only one assembly-
mode is actually reachable by the machine (solution (a) 
shown in Fig. 6) because the other ones lead to either rod 
crossing, collisions, or joint limit violation. The right 
assembly mode can be recognized, like for the right 
working mode, by the fact that each leg must have its slider 
attachment points above the moving platform attachment 
points 
The proposed method for calculating the various 
solutions of the forward kinematic problem has been 
implemented in Maple. 
IV. WORKSPACE CALCULATION OF THE VERNE MACHINE 
A. Workspace calculation of the parallel module [10] 
The parallel module of the VERNE machine possesses 3 
degrees of freedom. A complete representation of the 
workspace is a volume. Consequently this workspace can 
be defined by the positions in space reachable by the point 
P, centre of the mobile Cartesian frame 
pl
R . However the 
parallel module undergoes a complex motion, thus to 
overcome the complexity caused by the presence of the 
coupled rotation, we propose the following method. 
Step 1. We virtually cut the leg І, which is constituted of 
rods 11 and 12 (rod ij denotes AijBij) by supposing that 
11
ρ  
is independent of 
12
ρ , where 
11
ρ  and 
12
ρ  are respectively 
the joint coordinates of the prismatic joints linking rods 11 
and 12 to their guideways. The parallel module of the 
VERNE machine possesses now 4 degrees of freedom 
instead of 3. These degrees of freedom are defined by 
coordinates 
,
P
x
 
P
y  and 
Pz  of the point P and by the 
orientation α  of the moving platform. Thus we consider 
that the orientation α  of the platform is given, and we 
geometrically model constraints limiting the workspace of 
the new parallel architecture. These constraints are: (i) 
Interference between links, (ii) Leg Length limits, (iii) 
Serial Singularity, (iv) Mechanical limits on passive joints 
(we take into consideration the type of joints as well as the 
location of these joints in the machine) and (v) Actuator 
stroke. The intersection between these models is a volume. 
Step 2. We consider the interdependence between rods 
11 and 12 of leg І characterized by the fact that 
11
12
1
ρ
ρ
ρ
=
=
. This will allows us to determine the 
geometric shape for which the coupling between the 
position and the orientation of the platform exists. Thus for 
a given orientation 
,
α  the point P describes a surface 
defined by a hollow cylinder whose base is an ellipse as 
shown in subsection III.B. 
 
 
 
The intersection between geometric models defined at 
step 1 and the surface defined at step 2 represents the 
constant orientation workspace of the new 4 degree-of-
freedom parallel module. We calculate a horizontal cut of 
this workspace when the point P, center of the mobile 
Cartesian frame, moves in a known horizontal plane. Then 
we proceed by discretization to determine the complete 
workspace of the parallel module of the VERNE machine 
(see Figure 7). To calculate the workspace for a given tool 
length, we only need to express the TCP, U, in the base 
frame as function of (
)
,
,
,
P
P
P
x
y
z
α  by using the 
transformation matrix from Eq. (1), 
b
pl
pl
U
T
U
=
: 
, 
sin( ) and 
cos( )
U
P
U
P
U
P
x
x
y
y
z
z
α
α
=
=
−Δ
=
+ Δ
 
(27) 
 
 
 
 
Fig. 7: Workspace defined as the 
set of points P in the fixed Cartesian 
frame 
b
R  
Fig 8: The manufacturing 3D 
workspace for given tool length of 
50 mm 
B. The manufacturing 3D workspace [14] 
In this subsection, we calculate the manufacturing 3D 
workspace of the VERNE (the tool axis remains 
perpendicular to the part base) by considering that the 
tilting table rotates about its axis and follows the 
orientation α  of the moving platform, which means that 
1θ
α
=
 and 
1
0
φ =
, we suppose also that 
2
2
0
θ
φ
=
=
. 
To calculate this workspace we use Eq. (26) after 
replacing 
1θ , 
2
θ , 
1φ , and 
2
φ  by their values. This will give 
us the position and the orientation of the TCP in the frame 
Rt (
,
u
X
 
,
uY
 
,
u
Z
1,
φ  
2
φ ) expressed as function of (
,
px
 
 
p
y , 
pz , α ). A graphical representation of the workspace 
of PKMs with more than three degrees of freedom is only 
possible if we fix parameters representing the exceeded 
degrees of freedom. Thus to calculate the workspace for 
various orientations of the tool, we first fix 
1φ  and 
2
φ , then 
we calculate the position of the TCP (
,
u
X
 
,
uY
 
u
Z ) in the 
frame Rt by using Eq. (26). 
The proposed method for calculating the workspace for 
various tool lengths and for different tool orientation angles 
has been implemented in Maple and displayed in the CAD 
software CATIA. The obtained results show a workspace 
larger than the one currently used by the VERNE machine 
(for some given tool orientation angles). So this will able us 
to improve the productivity of the VERNE machine and to 
reach the limit of its capacity without risk of collision or 
damage of the VERNE machine. 
V. CONCLUSIONS 
This paper was devoted to the kinematic and workspace 
analysis of a 5-DOF hybrid machine tool, the VERNE 
machine. This machine possesses a complex motion caused 
by the unsymmetrical architecture of the parallel module 
where one of the legs is different from the other two legs. It 
was shown that the inverse kinematics has sixteen solutions 
and the forward kinematics may have six real solutions. 
The workspace was calculated by using a combination of 
discretization and geometrical method. This work is of 
interest as it may improve the efficiency of the machine 
because the controller of the actual VERNE machine 
resorts to an iterative Newton-Raphson resolution of the 
kinematic models.  
ACKNOWLEDGMENT 
This work has been partially funded by the European 
projects NEXT, acronyms for “Next Generation of 
Productions Systems”, Project no° IP 011815. The authors 
would like to thank the Fatronik society, which permitted 
us to use the CAD drawing of the Machine VERNE what 
allowed us to present well the machine. 
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