3rd International Congress Design and Modelling of Mechanical Systems CMSM’2009 
 
March 16-18, 2009 Hammamet, Tunisia 
1 
 
 
 
Kinematic and Dynamic Analysis of the 2-DOF Spherical Wrist of 
Orthoglide 5-axis 
Raza UR-REHMAN, Stephane CARO, Damien CHABLAT and Philippe WENGER 
 
Institut de Recherche en Communication et Cybernétique de Nantes, UMR CNRS 6597, 1 rue de la 
Noë, 44321, Nantes Cedex 03, France 
Raza.Ur-Rehman@irccyn.ec-nantes.fr 
Stephane.Caro@irccyn.ec-nantes.fr 
Damien.Chablat@irccyn.ec-nantes.fr 
Philippe.Wenger@irccyn.ec-nantes.fr 
Abstract - This paper deals with the kinematics and dynamics of a two degree of freedom 
spherical manipulator, the wrist of Orthoglide 5-axis. The latter is a parallel kinematics machine 
composed of two manipulators: i) the Orthoglide 3-axis; a three-dof translational parallel 
manipulator that belongs to the family of Delta robots, and ii) the Agile eye; a two-dof parallel 
spherical wrist. The geometric and inertial parameters used in the model are determined by means 
of a CAD software. The performance of the spherical wrist is emphasized by means of several test 
trajectories. The effects of machining and/or cutting forces and the length of the cutting tool on the 
dynamic performance of the wrist are also analyzed. Finally, a preliminary selection of the motors 
is proposed from the velocities and torques required by the actuators to carry out the test 
trajectories. 
KEYWORDS: Spherical mechanism/Agile eye/Orthoglide 5-axis/kinematics/dynamics 
Résumé - L’objet de ce papier est de présenter une étude cinématique et dynamique d’un 
mécanisme sphérique à deux degrés de liberté (ddls), le poignet de l’Orthoglide 5 axes. Ce dernier 
est une machine à cinématique parallèle composée de deux manipulateurs: i) l’Orthoglide 3 axes, 
manipulateur parallèle à trois ddls de translation qui appartient à la famille des robots de type 
Delta, et ii) l'Oeil Agile; poignet sphérique à deux ddls. Les paramètres géométriques et inertiels 
sont déterminés à l’aide d’un logiciel de CAO. La performance du poignet sphérique est soulignée 
par plusieurs trajectoires tests. Les effets de l’effort d'usinage et de la longueur de l'outil sur la 
performance dynamique du poignet sont également analysés. Enfin, une sélection préliminaire des 
moteurs est proposée à partir des vitesses et des couples requis par les actionneurs afin d’effectuer 
les trajectoires désirées  
MOTS-CLÉS: Mécanisme sphérique/Oeil Agile /Orthoglide 5-axis/cinématique/dynamique 
 
 
 
3rd International Congress Design and Modelling of Mechanical Systems CMSM’2009 
 
March 16-18, 2009 Hammamet, Tunisia 
2 
 
 
1 Introduction 
A spherical mechanism can orient or move the end-
effector about the center of rotation of the mechanism. 
A typical 3-DOF spherical manipulator or wrist 
provides the three dimensional (3D) rotations like a 
human hand but in most robotics applications only 2D 
rotations are sufficient. Rotation about the axis of 
symmetry of the end-effector is not necessary and if 
required, can be provided independently.  
Several researchers have worked in the domain of 
spherical mechanisms mainly for the applications of 
end-effector orientations. One of the earlier spherical 
mechanisms is that presented in the work of Asada 
and Cro Granito [1], which is a three-degree-of-
freedom (dof) spherical wrist with coaxial motors and 
three kinematics chains. In 1989, W. M. Craver [2], 
analysed a spherical robotic shoulder module. Other 
major publications to the research and development of 
the spherical mechanism are the work of Gosselin et al 
[3, 4, 5] where optimum kinematic designs of 
different types of spherical parallel mechanisms are 
presented. The Agile Eye, one of the most famous 
spherical mechanisms designed by Gosselin and 
Hamel [6], is a 3-dof parallel mechanism developed to 
control the orientation of a camera. Several 
mechanisms 
have 
been 
designed 
for 
diverse 
applications 
from 
the 
Agile 
Eye. 
Spherical 
mechanisms can be implemented in radar applications 
[7], 
camera 
manipulations 
[8] 
and 
surgical 
applications 
[9]. 
Cavallo 
and 
Michelini 
[10] 
introduced a 3-dof spherical parallel mechanism 
composed of three identical kinematic chains to orient 
the propeller and duct of a small autonomous 
underwater vehicle (AUV). Main contributions for the 
design and analysis of spherical mechanisms are 
reported in [1, 3, 4, 5, 6, 11, 12, 13, 14, 15, 16]. 
This paper focuses on the kinematic and dynamic 
analysis of the two dof spherical wrist of Orthoglide 
5 - axis, a five dof parallel kinematics machine 
developed for high speed operations. Here, we focus 
on the evaluation of the velocities, accelerations and 
the torques required by the actuators of the spherical 
wrist. First, the kinematics of the spherical wrist is 
studied and then its dynamics is analyzed by means of 
the Newton-Euler approach. The geometric and 
inertial parameters are determined with a CAD 
software. The performance of the manipulator is 
emphasized by means of several test trajectories. 
Finally, the actuators are selected in the catalogue 
based on the velocities and torques required by the 
actuators to carry out the test trajectories. 
2 Orthoglide 5-axis 
Orthoglide 5-axis, illustrated in Figure 1, is derived 
from a 3-dof translating manipulator, the Orthoglide 
3-axis and a 2-dof spherical wrist [17]. 
Orthoglide 3-axis, is a Delta-type PKM [18] dedicated 
to 3-axis rapid machining applications developed at 
the Institut de Recherche en Communications et 
Cybernétique de Nantes (IRCCyN) [19]. This 
mechanism is composed of three identical legs. Each 
leg is made up of a prismatic joint, a revolute joint, a 
parallelogram joint and another revolute joint. Only 
the prismatic joints of each leg are actuated.  
It gathers the advantages of both serial and parallel 
kinematic architectures such as regular workspace, 
homogeneous 
performances, 
good 
dynamic 
performances and stiffness. The interesting features of 
Orthoglide 3-axis are a regular dexterous workspace, 
uniform kinetostatic performances in all directions, 
good compactness [20] and high stiffness [21]. 
 
Figure 1. Orthoglide 5-axis 
The two-dof spherical wrist that is implemented in 
Orthoglide 5-axis is derived from the Agile Eye, a 
three-dof spherical wrist [6]. Here, the two-dof 
spherical wrist is designed to obtain high stiffness 
[22]. A CAD model of the latter is shown in Figure 2 
 
Figure 2. Spherical wrist of Orthoglide 5-axis 
3 Spherical wrist kinematic model 
3rd International Congress Design and Modelling of Mechanical Systems CMSM’2009 
 
March 16-18, 2009 Hammamet, Tunisia 
3 
 
 
 Reference frames and DH parameters 
The spherical wrist of Orthoglide 5-axis is composed 
of a closed kinematic chain made up of five 
components: proximal-1, proximal-2, distal, terminal 
and the base. These five links are connected by means 
of revolute joints, of which axes intersect at the center 
of the mechanism. Besides, only the two revolute 
joints connected to the base of the wrist are actuated. 
The distal has an imaginary axis of rotation passing 
through the intersection point of other joint axis and 
perpendicular to the plane of proximal-2. A unit 
vector ei and a reference frame Ri are associated to 
each joint with Zi-axis and ei being coincident. The 
angle between e1 and e2 is denoted by α0 and the angle 
between ei and ei+2 is denoted by αi, for i=1… 4. 
Reference frame R1 is defined in such a way that Z1-
axis coincides with e1 and e2 lies in the X1Z1-plane. 
Similarly R2 has its Z2-axis in the direction of e2 and 
e1 lies in the X2Z2-plane. Reference frame Ri (i =3, 4, 
5, 6) with Zi = ei are defined by the rotation of frame 
Ri-2 
and 
following 
the 
Denavit-Hartenberg 
conventions. DH-conventions for Orthoglide wrist 
mechanism are summarized as follow: 
Zi: axis of the ith joint; 
Xi: common perpendicular to Zi-2 and Zi; 
Yi: respecting the right hand rule;  
ai: distance between Zi and Zi+2;  
bi: distance between Xi and Xi+2;  
αi: angle between Zi and Zi+2 about Xi+2; 
θi: angle between Xi and Xi+2 about Zi. 
 
Figure 3. Orientations of reference frames according 
to DH-Conventions for Orthoglide Wrist 
As all joints axes intersect at a common point, the 
origin of all the frames are the same i.e., ai= bi= 0. 
Figure 3 shows the orientations of reference frames 
attached to the Orthoglide wrist according to the DH 
convention, αi being equal to π/2. In the home 
configuration, θ1=θ4= -π/2 and θ2=θ3=+π/2. 
 Kinematics of Orthoglide wrist 
The kinematics of the wrist of Orthoglide 5-axis is 
similar to the one of the Agile Eye introduced in [8]. 
However, the architecture is not the same as the distal 
and the proximal – 2 are different. Nevertheless, we 
can easily derive the kinematic and dynamic models 
of the wrist of Orthoglide 5-axis from [8]. Moreover, 
the influence of the external and/or machining forces 
on the dynamic performance is also considered. 
Kinematic equations of the Orthoglide wrist are 
developed with the help of the reference frames 
defined above and the DH-parameters. It is 
noteworthy that the camera manipulator [8] and the 
Orthoglide wrist are analogous. However, subsequent 
modifications are made. Vector v depicts the 
orientation of the wrist end-effector (cutting tool etc), 
which is defined in reference frame R5 by three angles 
β1, β2 & γ illustrated in Figure 4: 
 
β1 being the angle between e5 and the projection 
of v to e3-e5 plane; 
 
β2 being the angle between v and e3-e5 plane; 
 
γ being the angle between v and e3. 
The expression of v in R5 is then defined as, 



2
1
2
1
2
5
sin
sin
cos
cos
cos
T






v
 
Since vectors v and e5 coincide (Figure 3), i.e. v = e5 
hence β1 = β2 = 0 and γ = π/2. 
The orientation of vector v is defined in reference 
frame R1 by the pan (φ1) and tilt (φ2) angles as shown 
in Figure 4. With the definitions of φ1 and φ2, the 
components of v in R1 are given by:  



1
2
1
2
2
1
cos
cos
sin
cos
sin
T






v
 
 
Figure 4. Definition of v in R1 and R5 
Finally, the inverse kinematic problem of the wrist can 
be derived from v, α, β1, β2, γ, φ1 and φ2. The relations 
for the joints variables (θ1, θ2, θ3, θ4) are summarized 
below [8] (in these relations, c and s stands for cosine 
and sine functions respectively). 


1
2
1
tan
2
1
T
T






 
where 
1
1
2
1
1
1
4
2
2
z
y
x
z
y
A
v c
v s
c
B
B
AC
T
B
v s
A
C
v c
v c
c























 
1
3
tan
a
e
b
d
a
d
b e












 
where 
2
3
1
2
3
1
2
1
1
1
1
1
1
1
s
x
y
x
y
z
a
b
s
c
c
c
s
c
d
v c
v s
e
v s c
v c c
v c


















3rd International Congress Design and Modelling of Mechanical Systems CMSM’2009 
 
March 16-18, 2009 Hammamet, Tunisia 
4 
 
 




1
2
2
2
tan
2
1
T
T




 
where 




1
1
3
1
3
1
3
1
3
3
1
1
3
1
3
1
3
1
3
3
1
3
3
1
3
2
0
2
2
0
2
4
2
0
2
0
2
2
0
2
2
0
2
4
2
2
2
2
2
2
2
2
4
2
x
y
z
x
y
z
x
z
x
y
z
u
s s
c
s c
c
s
c s
s
u
c s
c
c c
c
s
s s
s
u
c
c
c
c
c
A
u s
c
u s
u c
c
c
B
u c
s
u s
s
C
u s
c
u s
u c
c
c
T
B
B
A C
A















































1
4
4
4
tan
s
c



 
where 




4
0
2
2
0
2
4
2
2
4
0
2
2
0
2
2
0
2
2
0
4
4
4
4
4
4
4
,
,
x
y
z
x
y
z
a
u
s
s
s
c
c
b
u c
c
d
u
c
c
s
s
c
u c
c
u s
u c
s
a
b
d
s
c
s
s






























Similarly, analytical relations of joints rates, i.e. joints 
velocities and acceleration can be obtained with the 
help of the previous relations of joint displacements 
and unit vectors ei. Otherwise, numerical techniques, 
like finite difference method can be used to obtain 
joint velocities and accelerations. 
4 Wrist dynamics  
Dynamic analysis is of primary importance to 
investigate the forces and moments applied to the 
actuators to carry out a desired task or motion by the 
manipulator. Dynamic analysis of 2-DOF camera 
manipulator is presented in [8] where Newton-Euler 
approach is used. A similar methodology is also used 
here for Orthoglide wrist. 
As a first step, wrist joints displacements are 
calculated from the kinematic model as discussed in 
the previous section. Velocities and accelerations of 
each component are obtained with the help of 
kinematic modeling, which has not been presented 
here because of space limitations. Also first and 
second derivatives of the unit vectors e1, e2, e3, e4 and 
e5 are calculated. For the detailed relations of these 
calculations reader is referred to [8]. Finally a system 
of equilibrium equations, obtained from the free body 
diagrams of each wrist component, is used to get the 
relations of actuators torques. A flow chart of the 
torques calculations is given in Figure 5 
 
Figure 5. Flow chart of the dynamic model 
 Equilibrium equations 
In this section, the methodology to obtain equilibrium 
equations of the Orthoglide wrist is presented. The 
following assumptions are made: 
 
friction forces are neglected; 
 
a spherical joint is assumed between the distal 
and the terminal link to have isostatic mechanism; 
 
a planar joint between the distal and the 
proximal -2. 
With these assumptions, the free-body diagrams 
(FBD) of terminal, distal, proximal-1 and proximal-2 
are drawn, as shown in Figure 6 to 9. Forces and 
moments acting on four moving wrist components are 
summarized below:  
 
two forces A and B and one moment M are 
exerted to the terminal; 
 
two force C and D are exerted to the distal; 
 
two forces G and H and two moments N and P 
are exerted to the proximal-1; 
 
two forces E and F and a moment R are exerted 
to the proximal-2. 
3rd International Congress Design and Modelling of Mechanical Systems CMSM’2009 
 
March 16-18, 2009 Hammamet, Tunisia 
5 
 
 
 
 
Figure 6. CAD model and FBD of terminal 
 
Figure 7. CAD model and FBD of distal 
 
Figure 8.CAD model and FBD of proximl-1 
These forces and moments are resolved into 
orthogonal components with respect to the attached 
unit vectors Equilibrium equations are written for 
each moving component (resulting 24 equations). For 
instance, set of equilibrium equations for the terminal 
(Figure 6) is given by: 
3
3
3
3
3
5
5
5
5
5
3
5
3
5
3
5
3
5
3
5
3
5
3
5
3
5
5
5
3
5
3
5
5
3
3
5
3
3
5
3
5
3
5
3
5
3
5
5
e
e
e
gte
te
e
e
e
gte
te
e
e
e
e
e
e
gte
e
te
e
e
te
e
e
te
e
gte
te
gte
e
te
e
e
te
e
e
te
gte
e
te
e
gte
te
e
e
e
e
te
F
A
B
F
J
F
A
B
F
J
F
A
B
F
J
M
L
B
l
F
l
F
K
M
M
L
A
l
F
l
F
K
M
M
L










































3
3
5
3
5
5
3
3
5
e
te
e
te
gte
te
gte
te
e
B
L
A
l
F
l
F
K





Where Fgte3, Fgte5, Fgte3xe5 are the gravity terms, Kgte3, 
Kgte5, Kgte3xe5 are the inertial terms (function of angular 
velocity, acceleration and inertia matrix) and Jgte3, 
Jgte5, Jgte3xe5 are the inertial forces (function of linear 
acceleration and mass) of the terminal along e3, e5 and 
e3xe5 directions respectively. Other variables used in 
these equations are defined in the next section. 
Along with equilibrium equations, compatibility 
equations i.e., action-reaction equilibrium equations at 
terminal-distal, 
terminal-proximal-1 
and 
distal-
proximal-2 interaction points are also written 
(resulting 10 equations). The system of equations so 
obtained can be solved to obtain the torques 
experienced by the wrist actuators i.e. Pe1 and Re2. 
Figure 9. CAD model and FBD of proximl-2 
 Wrist dynamic parameters 
The input parameters of the dynamic model of the 
Orthoglide wrist are: 
 
mass of each component (mt, md, mp1, mp2); 
 
distance between wrist geometric centre and 
centre of mass of each component (l); 
 
distance between wrist geometric centre and the 
point of application of force (L); 
 
inertia matrices of wrist component (It, Id, Ip1, Ip2); 
 
inertia of actuators (Km1, Km2); 
 
reduction ratios of actuators (ηm1,ηm2). 
The mass and inertial parameters are determined by 
means of a CAD software. The geometric parameters 
(l or L) are determined from the drawings or CAD 
models of wrist components along with corresponding 
unit vectors. 
 
Effect of machining force on actuators 
torques 
So far we have not taken into account the effect of the 
machining or cutting forces on the wrist dynamics. To 
cater for these forces we redraw the free body diagram 
of the terminal, with three components of cutting 
force 


c
3
5
3
5
ce
ce
ce
e
F
F
F


f
 
 
Figure 10. Terminal FBD with machining forces 
The distance between the tool tip and geometric center 
of wrist is taken as Lc (Figure 10). For this free body 
diagram, equilibrium equations are rewritten and new 
set of actuators equations are obtained. 
3rd International Congress Design and Modelling of Mechanical Systems CMSM’2009 
 
March 16-18, 2009 Hammamet, Tunisia 
6 
 
 
 Motors parameters 
In the preliminary design stage, FFA 20-80 harmonic 
drive motors are selected for both actuators. Motors 
specifications taken from the motor catalogue are 
given in Table 1 and theirs inertia is incorporated in 
the dynamic equations. 
Table 1. Motors parameters 
FFA 20-80 Harmonic Drive 
Nominal speed 
2500 rpm 
Maximum speed 
6500 rpm 
Maximum torque at output shaft 
74 Nm 
Continuous torque 
23 Nm 
Motor moment of inertia 
0.00262 kg.m4 
Motor power 
800 W 
 Trajectories generations 
Two test trajectories are introduced for the Orthoglide 
5-axis. Corresponding to these trajectories, inverse 
kinematic problem for the wrist is solved. The 
trajectories are defined as follows: 
 
Traj. I: semi-circular trajectory in vertical or YZ-
plane defined by radius R and trajectory angles ψ 
and δ (Figure 11) 
x
y
z
v
0
v
sin
v
cos

























v
 where δ varies from π/6 to 
5π/6 while ψ varies from 0 to π. 
 
Traj. II: circular trajectory in horizontal or XY-
plane defined by radius R, constant orientation 
angle γ of vector v with Z-axis and angle δ 
(Figure 12) 
x
y
z
v
sin
cos
v
sin
sin
v
-cos



























v
 
where δ varies from 0 to 2π 
 
Figure 11.Orientation of vector v (Traj I) 
5 Kinematic and dynamic analyses 
The kinematics of the Orthoglide wrist is analysed for 
the two test trajectories. The velocity of the end-
effecter, located on point P (on the tip of the wrist 
tool), throughout the trajectory, is taken as constant 
i.e. Vp= 1 m/s and accordingly trajectory time is 
calculated with constant velocity. 
 
Figure 12.  Orientation of vector v (Traj II) 
Figure 13 shows the plots of position, velocity and 
acceleration of the joints for Traj. II with R= 0.25 m 
and γ= 45°. For the same trajectory, dynamic model is 
used to calculate the actuators torques, as shown in 
Figure 14. 
 
Figure 13. Position, velocity and acceleration of 
articulations (Traj II) 
 
Figure 14. Actuators torques vs time (Traj. II) 
Table 2 shows the maximum values of the kinematic 
and dynamic parameters for Traj. II for R= 0.05, 0.1, 
0.15 and 0.25 m and for γ= 30˚, 45˚ and 60˚ where T1 
and P1 (resp. T2 and P2) is the torque and the power of 
actuators 1 (resp. 2). Results show that the trajectory 
with R= 0.5 m and γ= 60˚ is more critical as compared 
to the other trajectories; it requires higher maximum 
velocities, accelerations and torques. Maximum values 
of actuators torques are also shown in Figure 15. 
3rd International Congress Design and Modelling of Mechanical Systems CMSM’2009 
 
March 16-18, 2009 Hammamet, Tunisia 
7 
 
 
Table 2. Kinematics and dynamics peak values with no external forces (Traj II) 
 
0
2
4
6
8
10
12
0 .2 50
0 .150
0 .10 0
0 .0 50
Trajectory Radius [m]
Torque [Nm]
Actuator-1
Actuator-2
 
Figure 15. Actuators maximum torques vs trajectory 
radii (γ=45˚, Traj. II) 
In order to analyse the effects of machining or cutting 
forces (see Figure 10), three equal components of fc 
are assumed, i.e.,  
ce3
ce3×e5
ce5
F
= F
= F
= F
constant
c 
 
Actuators torques are calculated while considering the 
machining forces of different magnitudes and for 
trajectory radius of 0.15 m with γ= 45˚. Three values 
of machining forces moment arm Lc are taken, i.e. 
Lc= 0.06, 0.11 and 0.15 m. Results are shown in 
Figure 16 to 18. 
 
0
2
4
6
8
10
12
14
0
10
20
30
40
50
100
125
150
Machining Force [N]
Torques [Nm]
Actuator-1
Actuator-2
 
Figure 16.Actuators maximum torques with Fc 
(Lc= 0.06m, Traj. II) 
0
5
10
15
20
25
0
10
20
30
40
50
100
125
150
Machining Force [N]
Torque [Nm]
Actuator-1
Actuator-2
 
Figure 17. Actuators maximum torques with Fc 
(Lc= 0.11m, Traj. II) 
Figure 18 shows that for the machining forces of 
125 N and 150 N, actuators torques exceed the motors 
continuous torque (23 Nm) but still these are well 
below to the maximum motors torque (74 Nm). Hence 
for the given test trajectories considered motors can 
work for a range of machining forces. These results 
also represent the considerable influence of the length 
of the moment arm Lc of the machining forces (or the 
tool length) on the actuators torques. 
 
q
g
0
5
10
15
20
25
30
0
10
20
30
40
50
100
125
150
Machining Force [N]
Torque [Nm]
Actuator-1
Actuator-2
 
Figure 18. Actuators maximum torques with Fc 
(Lc= 0.15m, Traj. II) 
 
 
γ 
[deg] 
Radius 
[m] 
Max Velocity  
[rad/s] 
Max Acceleration  
[rad/s2] 
Max Torque  
[Nm] 
Max Power  
[W] 
1 
2
 
3
 
4

1
2

3

4

T1 
T2 
P1 
P2 
30 
0.250 
2.31 
2.31 
2.00 
2.00 
7.29 
7.29 
9.21 
9.21 
0.68 
1.22 
0.64 
1.20 
0.150 
3.84 
3.84 
3.33 
3.33 
20.24 
20.24
25.59 
25.59 
0.90 
1.67 
1.58 
2.77 
0.100 
5.77 
5.77 
5.00 
5.00 
45.54 
45.54
57.58 
57.59 
1.35 
2.55 
3.89 
6.43 
0.050 
11.53 
11.53 
10.00 
10.00 
182.15 182.15
230.30 
230.35 
3.75 
7.29 
24.33 
37.46 
45 
0.250 
3.99 
3.99 
2.83 
2.83 
13.96 
13.96
15.92 
15.92 
0.94 
1.73 
1.26 
2.43 
0.150 
6.65 
6.65 
4.71 
4.71 
38.78 
38.78
44.22 
44.22 
1.23 
2.37 
3.86 
5.97 
0.100 
9.98 
9.98 
7.07 
7.07 
87.26 
87.26
99.48 
99.48 
1.79 
3.62 
11.14 
14.69 
0.050 
19.96 
19.96 
14.14 
14.14 
349.03 349.03
397.94 
397.94 
5.58 
10.38 
80.14 
92.25 
60 
0.250 
6.90 
6.90 
3.46 
3.46 
34.05 
34.05
27.36 
27.36 
1.11 
2.16 
3.17 
5.22 
0.150 
11.49 
11.49 
5.77 
5.77 
94.59 
94.59
75.99 
75.99 
1.54 
3.03 
13.47 
15.78 
0.100 
17.24 
17.24 
8.66 
8.66 
212.82 212.82
170.98 
170.98 
3.31 
4.72 
44.17 
45.04 
0.050 
34.48 
34.48 
17.32 
17.32 
851.30 851.30
683.92 
683.92 
12.94 
13.84 
347.23 
320.89 
Motors continuous torque=23 Nm 
3rd International Congress Design and Modelling of Mechanical Systems CMSM’2009 
 
March 16-18, 2009 Hammamet, Tunisia 
8 
 
 
6 Conclusions 
This paper deals with the kinematics and dynamics of 
the spherical wrist of Orthoglide 5-axis. The 
kinematic and dynamic performances of the wrist 
were analyzed and its actuators primarily selection 
was proposed by means of several test trajectories. A 
methodology 
was 
introduced 
to 
evaluate 
the 
velocities, accelerations and torques required by the 
actuators. The influence of the machining forces as 
well as the tool length on the wrist actuators torques 
and powers was also studied. It turns out that the 
primarily selected motors with a continuous torque of 
23 Nm and of power equal to 0.8 kW are suitable for 
the prototype of Orthoglide 5-axis. Finally, the 
following points will be taken into account in future 
works: (i) the friction between links has to be 
considered, (ii) planar joint between distal and 
proximal-2 should be analysed more precisely, (iii) 
weight of the machining tool should also be taken into 
consideration, and (iv) a larger range of trajectories 
should be analysed. 
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