THE ORTHOGLIDE: 
KINEMATICS AND WORKSPACE ANALYSIS 
A. Pashkevich 
Robotic Laboratory, Belarusian State University of Informatics and Radioelectronics  
6 P. Brovka St., Minsk, 220027, Belarus 
E-mail: pap@ bsuir.unibel.by  
D. Chablat, P. Wenger 
Institut de Recherche en Communications et Cybernétique de Nantes 
1 rue de la Noë, BP 92 101, 44321 Nantes, France 
E-mails:Damien.Chablat@irccyn.ec-nantes.fr, Philippe.Wenger@irccyn.ec-nantes.fr 
Abstract 
The paper addresses kinematic and geometrical aspects of the Orthoglide, a 
three-DOF parallel mechanism. This machine consists of three fixed linear 
joints, which are mounted orthogonally, three identical legs and a mobile 
platform, which moves in the Cartesian x-y-z space with fixed orientation. 
New solutions to solve inverse/direct kinematics are proposed and a detailed 
workspace analysis is performed taking into account specific joint limit 
constraints. 
Keywords: 
Parallel manipulators; Workspace; Inverse and direct kinematics. 
1. Introduction 
For two decades, parallel manipulators attract the attention of more 
and more researchers who consider them as valuable alternative design 
for robotic mechanisms (Asada et al, 1986, Fu et al., 1987, Craig, 1989). 
As stated by a number of authors (Tsai, 1999), conventional serial 
kinematic machines have already reached their dynamic performance 
limits, which are bounded by high stiffness of the machine components 
required to support sequential joints, links and actuators. Thus, while 
having good operating characteristics (large workspace and high 
flexibility), serial manipulators have disadvantages of low precision, low 
stiffness and low power. Also, they are generally operated at low speed to 
avoid excessive vibration and deflection. 
Conversely, parallel kinematic machines offer essential advantages 
over their serial counterparts (lower moving masses and higher rigidity) 
that obviously should lead to higher dynamic capabilities. However, most 
existing parallel manipulators have limited and complicated workspace 
with singularities, and highly non-isotropic input/output relations 
 
 
    
2 
(Angeles, 2002). Hence, the performances may significantly vary over the 
workspace and depend on the direction of the motion, which is a serious 
disadvantage for machining applications. Research in the field of parallel 
manipulators began with the Stewart-platform used in flight simulators 
(Stewart, 1965). Many such structures have been investigated since then, 
which are composed of six linearly actuated legs with different 
combinations of link-to-platform connections (Merlet, 2000). In recent 
years, several new kinematic structures have been proposed that possess 
higher isotropy. In particular, a 3-dof translational mechanism with 
gliding foot points was found in three separate works to be fully isotropic 
throughout the Cartesian workspace (Carricato et all, 2002 and Kong et 
all, 2002). It consists of a mobile platform, which is connected to three 
orthogonal linear drives through three identical planar 3-revolute jointed 
serial chains. Although this manipulator behaves like a conventional 
Cartesian machine, bulky legs are required to assure stiffness because 
these legs are subject to bending.  
In this paper, the Orthoglide manipulator proposed by Wenger et all, 
2000, is studied. As follows from previous research, this manipulator has 
good kinetostatic performances and some technological advantages, such 
as (i) symmetrical design consisting of similar 1-d.o.f. joints; (ii) regular 
workspace shape properties with bounded velocity amplification factor; 
and (iii) low inertia effects (Chablat et all, 2003). This article analyses 
the kinematics and the workspace of the Orthoglide. Section 2 describes 
the Orthoglide geometry. Section 3 proposes new solutions for its inverse 
and direct kinematics. Sections 4, 5 present a detailed analysis of the 
workspace and jointspace respectively. Finally, Section 6 summarises the 
main contributions of the paper. 
2. Manipulator geometry 
The kinematic architecture of the Orthoglide is shown in Fig. 1. It 
consists of three identical kinematic chains that are formally described 
as PRPaR, where P, R and Pa denote the prismatic, revolute, and 
parallelogram joints respectively. The mechanism input is made up by 
three actuated orthogonal prismatic joints. The output body is connected 
to the prismatic joints through a set of three kinematic chains. Each 
chain includes a parallelogram, so that the output body is restricted to 
translational movements. To get the Orthoglide kinematic equations, let 
us locate the reference frame at the intersection of the prismatic joint 
axes and align the coordinate axis with them (Fig. 2), following the 
“right-hand” rule. Let us also denote the input vector of the prismatic 
joints variables as 


,
,
x
y
z





 and the output position vector of the 
tool centre point as 


,
,
x
y
z
p
p
p

p
. Taking into account obvious 
 
 
    
3 
properties of the parallelograms, the Orthoglide geometrical model can 
be presented in a simplified form, which consists of three bar links 
connected by spherical joints to the tool centre point at one side and to 
the corresponding prismatic joints at another side.  
 
 
x 
y 
z 
(x, 0, 0) 
(0, y, 0) 
(0, 0, z) 
 (px, py, pz) 
O 
 z 
 x 
 y 
 
Fig. 1. Orthoglide architecture 
Fig. 2. Orthoglide geometrical model 
Using this notation, the kinematic equations of the Orthoglide can be 
written as follows 
 






2
2
2
2
2
2
2
2
2
2
2
2
,
,
x
x
y
z
x
y
y
z
x
y
z
z
p
p
p
L
p
p
p
L
p
p
p
L















 (1) 
where L is the length of the parallelogram principal links and the 
“zero” position 


0
0, 0, 0

p
 corresponds to the joints variables 


,
,
L L L


. It should be stressed that the Orthoglide geometry and 
relevant manufacturing technology impose the following constraints on 
the joint variables 
 
0
2
; 0
2
; 0
2
x
y
z
L
L
L









, 
(2) 
which essentially influence on the workspace shape. While the upper 
bound (
2L

) is implicit and obvious, the lower one (
0

) is caused by 
practical reasons, since safe mechanical design encourages avoiding risk 
of simultaneous location of prismatic joints in the same point of the 
Cartesian workspace (here and in the following sections, while referring 
to symmetrical constraints are subscript omitted, i.e. 


,
,
x
y
z





).  
3. Orthoglide Kinematics 
3.1 Inverse kinematics 
For the inverse kinematics, the position of the end-point (
,
,
x
y
z
p
p
p ) is 
treated as known and the goal is to find the joint variables (
,
,
x
y
z


) 
that yield the given location of the tool. Since in the general case the 
inverse kinematics can produce several solutions corresponding to the 
same tool location, the solutions must be distinguished with respect to 
the algorithm “branch”. For instance, if the aim is to generate a sequence 
of points to move the tool along an arc, care must be taken to avoid 
branch switching during motion, which may cause inefficient (or even 
 
 
    
4 
impossible) manipulator motions. Moreover, leg singularities may occur 
at which the manipulator loses degrees of freedom and the joint variables 
become linearly dependent. Hence, the complete investigation of the 
Orthoglide kinematics must cover all the above-mentioned topics.  
From the Orthoglide geometrical model (1), the inverse kinematic 
equations can be derived in a straightforward way as: 
 
2
2
2
x
x
x
y
z
p
s
L
p
p




 
2
2
2
y
y
y
x
z
p
s
L
p
p




2
2
2
z
z
z
x
y
p
s
L
p
p




 
(3) 
where sx, sy, sz are the branch (or configuration) indices that are equal 
to 1. It is obvious that (3) yields eight different branches of the inverse 
kinematic algorithm, which will be further referred to as PPP, 
MPP…MMM following the sign of the corresponding index (i.e. the 
notation MPP corresponds to the indices 
1;
1;
1
x
y
z
s
s
s


). The 
geometrical meaning of these indices is illustrated by Fig. 2, where x, y, 
z are the angles between the bar links and the corresponding prismatic 
joint axes. It can be proved that 
1
s  if 
o
o
(90 ,180 )

 and 
1
s  if 
o
o
(0 ,90 )

. The branch transition (
o
90

) corresponds to the serial 
singularity (where the leg is orthogonal to the relevant translational axis 
and the input joint motion does not produce the end-point displacement). 
It is obvious that if the inverse kinematic solution exists, then the target 
point (px, py, pz) belongs to a volume bounded by the intersection of three 
cylinders  
 


2
2
2
2
2
2
2
2
2
;
;
L
x
y
x
z
y
z
C
p
p
L
p
p
L
p
p
L







p
 
(4)  
that guarantees non-negative values under the square roots in (3). 
However, it is not sufficient, since the lower joint limits (2) impose the 
following additional constraints 
 
2
2
2
x
x
y
z
p
s
L
p
p



; 
2
2
2
y
y
x
z
p
s
L
p
p



; 
2
2
2
z
z
x
y
p
s
L
p
p



 (5)  
which reduce a potential solution set. For example, it can be easily 
computed that for the “zero” workspace point 


0
0, 0, 0

p
, the inverse 
kinematic equations (3) give eight solutions 


,
,
L
L
L




 but only one 
of them is feasible. To analyse in details the influence of the joint 
constraints impact, let us start from separate a study of the inequalities 
(5) and then summarise results for all possible combinations of the three 
configuration indices. If 
1
xs , then consideration of two cases, 
0
xp 
 
and 
0
xp 
, yields the following workspace set satisfying the constraint 
0
x

  
 




2
2
2
2
|
0
|
0;
x
L
L
x
L
x
x
y
z
W
C
p
C
p
p
p
p
L








p
p

 
(6)  
which consists of two fractions (½ of the cylinder intersection denoted 
L
C  and ½ of the sphere whose geometric center is (0,0,0) and radius is L). 
 
 
    
5 
If 
1
xs , then the second case 
0
xp 
 does not give any solution and the 
joint constraint 
0
x

 is expressed in the workspace as 
 


2
2
2
2
|
0;
x
L
L
x
x
y
z
W
C
p
p
p
p
L






p
. 
(7)  
The latter defines a solid bounded by three cylindrical surfaces and 
the sphere. The remaining constrains 
0
y

 and 
0
z

 can be derived 
similarly, which differ from (6), (7) by subscripts only.  
Then, there can be found intersection of the obtained sets for different 
combinations of the configuration indices. It can be easily proved that the 
case “PPP” yields 
 




2
2
2
2
|
,
,
0
|
PPP
L
L
x
y
z
L
x
y
z
W
C
p
p
p
C
p
p
p
L







p
p

 
(8)  
while the remaining cases give 
 




2
2
2
2
...
|
,
,
0
|
MPP
MMM
L
L
L
x
y
z
L
x
y
z
W
W
C
p
p
p
C
p
p
p
L









p
p

 (9)  
Expressions (8) and (9) can be put in the form  
 
 
  
;
...
PPP
MPP
MMM
L
L
L
L
L
L
W
S
G
W
W
G





 
(10)  
where  


2
2
2
2
L
L
x
y
z
S
C
p
p
p
L





p
;   


2
2
2
2
|
,
,
0;
L
L
x
y
z
x
y
z
G
C
p
p
p
p
p
p
L






p
;   
L
L
S
G 

. 
Therefore, for the considered positive joint limits (2), the existence of 
the inverse kinematic solutions may be summarised as follows (i) inside 
the sphere SL there exist exactly one inverse kinematic solution PPP 
with positive configuration indices sx, sy, sz, (ii) outside the sphere SL , but 
within the positive part of the cylinder intersection CL, there exist 8 
solutions of the inverse kinematics (PPP, MPP, … MMM) corresponding 
to all possible combinations of the configuration indices sx, sy, sz . These 
conclusions may be illustrated when 
1
L  by numerical examples. If the 
target point 
=(-0.5, 0.4, 0.3)
p
 is within the sphere SL, then the joint 
coordinates 
must 
be 
taken 
from 
the 
sets 


0.37,-1.37
x

, 


 1.21,  -0.41  
x

, 


1.07,-0.47
x

, which allow only one positive 
combination. In contrast, for the target point 
=(0.7, 0.7, 0.7)
p
, which is 
outside the sphere, the inverse kinematics yields solutions with two 
positive values 


,
,
0.84, 0.56
x
y
z



 that allow 8 positive combinations 
of the joint variables. An interesting feature is that intermediate cases 
(with 2 or 4 feasible solutions) are not possible. 
3.2 Direct kinematics 
For the direct kinematics, the values of the joint variables (x, y, z) 
are known and the goal is to find the tool centre point location (px, py, pz) 
that corresponds to the given joint positions. While, in general, the 
inverse kinematics of parallel mechanisms is straightforward, the direct 
 
 
    
6 
kinematics is usually very complex. The Orthoglide has the advantage 
leave an analytical direct kinematics. Like for the previous section, the 
solutions must be distinguished with respect to the algorithm “branch” 
that should be also defined both geometrically and algebraically, via a 
configuration index. 
To solve the system (1) for px, py, pz, first, let us derive linear relations 
between the unknowns. By subtracting three possible pairs of the 
equations (1), we leave 
 
2
2
2
2
2
2
2
2
2
2
2
2
 , 
 , 
x
x
y
y
x
y
x
x
z
z
x
z
y
y
z
z
y
z
p
p
p
p
p
p





















 
(11) 
As follows from these expressions, the relation between px, py, pz may 
be presented as  
 
/ 2
/
;
/ 2
/
;
/ 2
/
x
x
x
y
y
y
z
z
z
p
t
p
t
p
t












, 
(12) 
where t is an auxiliary scalar parameter. From a geometrical point of 
view, the expression (12) defines the set of equidistant points for the 
prismatic joint centres (Fig. 6). Also, it can be easily proved that the full 
set of equidistant points is the line perpendicular to  and passing 
through (
,
,
x
y
z


)/2, where.  
 


|
/
/
/
1
x
x
y
y
z
z
p
p
p







p
 
(13) 
After substituting (12) into any of the equations (1), the direct 
kinematic problem is reduced to the solution of a quadratic equation in 
the auxiliary variable t, 
 
2
0
At
Bt
C



, 
(14) 
where
2
2
2
(
)
(
)
(
)
x
y
x
z
y
z
A






,
2
(
)
x
y
z
B


,
2
2
2
2
(
4
)
/ 4
x
y
z
C
L B







. 
The quadratic formula yields two solutions 
 
2
(
4
)/(2 );
1
t
B
m B
AC
A
m



 
(15)  
that geometrically correspond to different locations of the target point 
P (see Fig. 6) with respect to the plane passing through the prismatic 
joint centres (it should be noted that the intersection point of the plane 
and the set of equidistant point corresponds to 


0
2
t
B
A

). Hence, the 
Orthoglide direct kinematics is solved analytically, via the quadratic 
formula (14) for the auxiliary variable t and its substitution into 
expressions (12). The direct kinematic solution exists if and only if the 
joint variables satisfy the inequality 
2
4
B
AC

, which defines a closed 
region in the joint variable space  
 





2
2
2
2
2
2
2
|
4
1
L
x
y
z
x
y
z
L

















 
(16)  
Taking into account the joint limits (2), the feasible joint space may be 
presented as 


|
,
,
0
L
L
x
y
z








. Therefore, for the considered 
 
 
    
7 
positive joint limits (2), the existence of the direct kinematic solutions 
may be summarised as follows: (i) inside the region 
L

, there exist 
exactly two direct kinematic solutions, which differ by the target point 
location relative to the plane  (Fig. 7a). (ii) On the border of the region 
L

 located inside the first octant, there exist a single direct kinematic 
solution, which corresponds to the “flat” manipulator configuration, 
where both the target point and prismatic joint centres belong to the 
plane  (Fig. 7b). 
 
x 
y 
z 
L 
L 
L 
x 
y 
z 
(x, y, z)/2 
p 
Equidistant 
 
 
(a)
x
y
z
P
m = -1
P
m = +1
(b)
x
y
z
P
m =  1
“flat”
 configuration
 
Fig. 6. Geometrical solution of 
the direct kinematics 
Fig. 7. Double (a) and single (b) solutions of 
the direct kinematics 
These conclusions may be illustrated by the following numerical 
examples (for the “unit” manipulator, L=1). Since the joint variables 
0.3
x
y
z






 are within 
L

, then the end-point coordinates are either 
0.46
x
y
z
p
p
p



 or 
0.66
x
y
z
p
p
p



. In contrast, for the joint variables 
1.5
x
y
z






, which are exactly on the surface 
L

, the direct 
kinematics yields a single solution 
1 6
x
y
z
p
p
p



 corresponding to the 
“flat” configuration (see Fig. 7b). 
3.3 Configuration indices 
As follows from the previous sub-sections, both the inverse and direct 
kinematics of the Orthoglide may produce several solutions. The problem 
is how to define numerically the configuration indices, which allow 
choosing among the corresponding algorithm branches. 
For the inverse kinematics, when the configuration is defined by the 
angle between the leg and the corresponding prismatic joint axis, the 
decision equations for the configuration indices are trivial: 
 






sgn
;
sgn
;
sgn
x
x
x
y
y
y
z
z
z
s
p
s
p
s
p









  
Geometrically, 
0
s 
 means that (see Fig. 2), 


,
,
/ 2  3 / 2
x
y
z




. 
For the direct kinematics, the configuration is defined by the end-point 
location relative to the plane that passes through the prismatic joint 
centres (see Figs. 6-7). Hence, the decision equation may be derived by 
analysing the dot-product of the plane normal vector 

1
1
1
,
,
x
y
z






 and 
the vector directed along any of the bar links (for instance, 
 
 
    
8 


,
,
x
x
y
z
p
p
p


 for the first link:  


sgn
1
y
x
z
x
y
z
p
p
p
m







 
which 
is 
equivalent to 


sgn
x
y
z
x
y
z
x
y
z
x
y
z
m
p
p
p









 for the positive 
joint limits. 
It should be stressed that the feasible solutions for the inverse/direct 
kinematics, located in the neighbourhood of the “zero” point, have the 
configuration indices 
1
x
y
z
s
s
s

 and 
1
m . 
4. Workspace analysis 
As follows from Eq. (10), Orthoglide workspace WL is composed of two 
fractions: (i) the sphere SL of radius L and centre point (0, 0, 0), and (ii) 
the thin non-convex solid GL, which is located in the first octant and 
bounded by the surfaces of the sphere SL and the cylinder intersection 
CL. It can be proved that the volume of CL, SL and WL is defined by the 
expressions 
 


3
(
) 8 2
2
L
Vol C
L


,


3
(
)
2
2
/ 6
L
Vol G
L




,


3
(
)
2 7
/ 6
2
L
Vol W
L




 (17) 
As follows from (17), the Orthoglide with the joint limits (2) uses about 
53% of the workspace 
PPP
V
 of its serial counterpart (a Cartesian PPP 
machine with 2
2
2
L
L
L


 workspace). Also, the volume of GL (
3
0.062 L ) is 
insignificant in comparison to the volume of sphere SL (
3
4.189 L ), which 
is equal to 52% of 
PPP
V
. On the other hand, releasing the lower joint limit 
(
0

) leads to an increases the workspace volume of up to 59% of 
PPP
V
 
only, since the volume of the workspace is, then, equal to 
L
C . The mutual 
location of GL and SL (and their size ratio) may be also evaluated by the 
intersection points of the first octant bisector. In particular, for the 
sphere SL the bisector intersection point is located at distance 1
3
0.58

 
from the origin, while for the solid GL the corresponding distance is 
1
2
0.71

 (assuming that L=1). Moreover, GL touches the sphere by its 
circular edges, which are located on the borders of the first octant. 
5. Joint Space Analysis 
The properties of the feasible jointspace are essential for the 
Orthoglide control, in order to avoid impossible combinations of the 
prismatic joint variables x, y, z, which are generated by the control 
system and are followed by the actuators. For serial manipulators, this 
problem does not usually exist because the jointspace is bounded by a 
parallelepiped and mechanical limitations of the joint values may be 
verified easily and independently. For parallel manipulators, however, 
we needs to check both (i) separate input coordinates (to satisfy the joint 
limits), and (ii) their combinations that must be feasible to produce a 
direct kinematic solution. 
 
 
    
9 
As follows from Sub-Section 3.3, the Orthoglide jointspace 
L

 is 
located within the first octant and is bounded by a surface, which 
corresponds to a single solution of the direct kinematics. Therefore, the 
jointspace boundary is defined by the relation 
2
4
B
AC

 (see equation 
(14)), which may be rewritten as 
 



2
2
2
2
2
2
2
4
1
x
y
z
x
y
z
L














 
(18) 
and solved for x assuming that y, z are known, 
4
2
0
x
x
D
DE
E





, 
where 
2
2
y
z
D






; 
2
2
2
4
y
z
E
L





. However, this equation is non-
symmetrical with respect to 
,
,
x
y
z


 and, therefore, is not convenient 
the real-time control. An alternative way to obtain the jointspace 
boundary, which is more computationally efficient, is based on the 
conversion 
from 
Cartesian 
to 
spherical 
coordinates, 
,
,
x
x
y
y
z
z
e t
e t
e t






, where t  0 is the length of the vector , and 
(ex, ey, ez) are the components of the unit direction vector, which are 
expressed via two angles 
,
 with 
cos
cos
xe



, 
cos
sin
ye



, 
sin
ze


, where 


,
0,
/ 2



. For such a notation, the original equation 
(18) is transformed into a linear equation for 
2t , 

2
2
1
4
F
t
L F


, 
2
2
2
x
y
z
F
e
e
e






, with an obvious solution 


2
1
t
L F
F


. As follows 
from its analyses, the bounding surface is close to the 1/8th of the sphere 
S2L. At the edges, which are exactly quarters of the circles of the radius 
2L, the surface touches the sphere. However, in the middle, the surface is 
located out of the sphere. In particular, the intersection point of the first 
octant bisector is located at the distance 
3 2
1.22

 from the coordinate 
system origin for the jointspace border and at the distance 2
3
1.15

 for 
the sphere S2L (assuming L=1). 
6. Conclusions  
This article focuses on the kinematics and workspace analysis of the 
Orthoglide, a 3-DOF parallel mechanism with a kinematic behaviour 
close to the conventional Cartesian machine taking into account the 
specific manufacturing constraints in the joint variables. We proposed a 
formal definition of the configuration indices and developed new simple 
analytical expressions for the Orthoglide inverse/direct kinematics. It 
was proved that, for the considered joint limits, the Orthoglide 
workspace is composed of two fractions, the sphere and a thin non-convex 
solid in which there are 1 and 8 inverse kinematic solutions, respectively. 
The total workspace volume comprises about 53% of the corresponding 
serial machine workspace, where over 52% belongs to the sphere (for 
comparison, releasing of the joint limits yields to an increase of up to 59% 
in the workspace volume). It was also shown, that the Orthoglide 
jointspace is bounded by surface with circular edges, which is more 
 
 
    
10 
convex than the sphere but is rather close to it. These results can be 
further used for the optimisation of the Orthoglide parameters, which is 
the subject of our future research work. 
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L.W. Tsai, (1999), “Robot Analysis, The Mechanics of Serial and Parallel 
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J.-P. Merlet, (2000), “Parallel Robots,” Kluwer Academic Publishers.  
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