933  
Abstract— A classification of a family of 3-revolute (3R) 
positining manipulators is established. This classification is based 
on the topology of their workspace. The workspace is 
characterized in a half-cross section by the singular curves. The 
workspace topology is defined by the number of cusps and nodes 
that appear on these singular curves. The design parameters 
space is shown to be divided into nine domains of distinct 
workspace topologies, in which all manipulators have similar 
global kinematic properties. Each separating surface is given as 
an explicit expression in the DH-parameters. 
 
Keywords—Classification, 
Workspace, 
Singularity, 
Cusp, 
node, orthogonal manipulator. 
I. INTRODUCTION 
A positioning manipulator may be used as such for 
positioning tasks in the Cartesian space or as the regional 
structure of a 6R manipulator with spherical wrist. Most 
industrial regional structures have the same kinematic 
architecture, namely, a vertical revolute joint followed by two 
parallel joints, like the Puma. Such manipulators are always 
noncuspidal (i.e. must meet a singularity to change their 
posture) and they have four inverse kinematic solutions (IKS) 
for all points in their workspace (assuming unlimited joints). 
This paper focuses on alternative manipulator designs, namely, 
positioning 3R manipulators with orthogonal joint axes 
(orthogonal manipulators). Orthogonal manipulators may have 
different global kinematic properties according to their link 
lengths and joint offsets. They may be cuspidal, that is, they 
can change their posture without meeting a singularity [1, 2]. In 
1998, ABB-Robotics launched the IRB 6400C, a 6R 
manipulator to be used in the car industry and designed to 
minimize the swept volume. The only difference with the Puma 
was the permutation of the first two link axes, resulting in a 
manipulator with all its joint axes orthogonal, and cuspidal. 
Commercialization of the IRB 6400C was finally stopped one 
year later. Exact reason is beyond the knowledge of the authors 
but the cuspidal behavior is likeable to have disappointed the 
end users. Cuspidal robots were unknown before 1988 [3], 
when a list of conditions for a manipulator to be noncuspidal 
was provided [4, 5]. This list includes simplifying geometric 
conditions like parallel and intersecting joint axes [4] but also 
nonintuitive conditions [5]. A general necessary and sufficient 
condition for a 3-DOF manipulator to be cuspidal was 
 
This work was supported in part by C.N.R.S. MathStic program "Cuspidal 
robots and triple roots".  
established in [6], namely, the existence of at least one point in 
the workspace where the inverse kinematics admits three equal 
solutions. The word “cuspidal manipulator” was defined in 
accordance to this condition because a point with three equal 
IKS forms a cusp in a cross section of the workspace [4, 7]. 
The categorization of all generic 3R manipulators was 
established in [8] based on the homotopy class of the singular 
curves in the joint space. [9] proposed a procedure to take into 
account the cuspidality property in the design process of new 
manipulators. More recently, [10] applied efficient algebraic 
tools to the classification of 3R orthogonal manipulators with  
no offset on their last joint. Five surfaces were found to divide 
the parameters space into 105 cells where the manipulators 
have the same number of cusps in their workspace. The 
equations of these five surfaces were derived as polynomials in 
the DH-parameters using Groebner Bases. A kinematic 
interpretation of this theoretical work was conducted in [11] : 
the authors analyzed general kinematic properties of one 
representative manipulator in each cell. Only five different 
cases were found to exist. However, the classification in [11] 
did not provide the equations of the separating surfaces in the 
parameters space for the five cells associated with the five 
cases found. On the other hand, [11] did not take into account 
the occurrence of nodes, which play an important role for 
analyzing the number of IKS in the workspace. 
The purpose of this work is to classify a family of 3R 
positining manipulators according to the topology of their 
workspace, which is defined by the number of cusps and nodes 
that appear on the singular curves. The design parameters space 
is shown to be divided into nine domains of distinct workspace 
topologies, in which all manipulators have similar global 
kinematic properties. This study is of interest for the design of 
new manipulators. 
The rest of this article is organized as follows. Next section 
presents the manipulators under study and recalls some 
preliminary results. The classification is established in section 
III. Section IV synthesizes the results and section V concludes 
this paper.   
II. PRELIMINARIES 
A. Manipulators under study 
The manipulators studied in this paper are orthogonal with 
their last joint offset equal to zero. The remaining lengths 
parameters are referred to as d2, d3, d4, and r2 while the angle 
parameters α2 and α3 are set to –90° and 90°, respectively. The 
A Classification of 3R Orthogonal Manipulators 
by the Topology of their Workspace 
Maher Baili, Philippe Wenger and Damien Chablat 
Institut de Recherche en Communications et Cybernétique de Nantes, UMR C.N.R.S. 6597 
1, rue de la Noë, BP 92101, 44321 Nantes Cedex 03 France 
Maher.Baili@irccyn.ec-nantes.fr 
 
 
 
three joint variables are referred to as θ1, θ2 and θ3, 
respectively. They will be assumed unlimited in this study. 
Figure 1 shows the kinematic architecture of the manipulators 
under study in the zero configuration. The position of the end-
tip (or wrist center) is defined by the three Cartesian 
coordinates x, y and z of the operation point P with respect to a 
reference frame (O, x, y, z) attached to the manipulator base as 
shown in Fig. 1.  
 
z 
y
O 
x 
P 
θ 1 
θ 2 
θ 3 
d 2 
r 2
d 4 
d 3 
. 
 
 
Figure 1.  Orthogonal manipulators under study. 
B. Singularities and aspects 
The determinant of the Jacobian matrix of the orthogonal 
manipulators under study is 
 
det(J) = (d3 + c3d4)(s3d2 + c2(s3d3 – c3r2))  
(1) 
 
where ci=cos(θi) and  si=sin(θi). A singularity occurs when 
det(J)=0. Since the singularities are independent of θ1, the 
contour 
plot 
of 
det(J)=0 
can 
be 
displayed 
in 
2
3
,
π
θ
π
π
θ
π
−
≤
<
−
≤
<
 where they form a set of curves. If 
d3>d4, the first factor of det(J) cannot vanish and the 
singularities form two distinct curves S1 and S2 in the joint 
space [12]. S1 and S2 divide the joint space into two singularity-
free open sets A1 and A2 called aspects [1]. The singularities 
can be also displayed in the Cartesian space [13, 14]. Thanks to 
their symmetry about the first joint axis, a 2-dimensional 
representation in a half cross-section of the workspace is 
sufficient. The singularities form two disjoint sets of curves in 
the workspace. These two sets define the internal boundary 
WS1 and the external boundary WS2, respectively, with 
WS1=f(S1) and WS2=f(S2). Fig. 2 (left) shows the singularity 
curves when d2=1, d3=2,  d4=1.5 and r2=1. For this 
manipulator, the internal boundary WS1 has four cusp points. It 
divides the workspace into one region with two IKS (the outer 
region) and one region with four IKS (the inner region). 
 
4 
2 
 
Figure 2.  Singularity curves in joint space (left) and workspace (right, 
number of IKS in each region is indicated). 
If d3≤d4, the operation point can meet the second joint axis 
whenever θ3=±arccos(-d3/d4) and two horizontal lines appear, 
which may intersect S1 and S2 depending on d2, d3, d4 and r2 
[12]. The number of aspect depends on these intersections. 
Note that if d3<d4, no additional curve appears in the workspace 
cross-section but only two points where the operation point 
meets the second joint axis and the manipulator has an infinite 
number of IKS. Fig. 3 shows the singularity curves when d2=1, 
d3=3,  d4=4 and r2=2. The singular line defined by θ3=+arccos(-
d3/d4) maps onto one singular point in the workspace cross-
section, which is located at the self-intersection of the internal 
singular boundary. The remaining singular line θ3=–arccos(-
d3/d4) maps onto an isolated singular point in the workspace. 
The workspace topology of this manipulator features two cusps 
and three nodes, two regions with two IKS and two regions 
with four IKS. In the following section, the complete 
classification is established. 
 
 
Figure 3.   Singularity curves when d3<d4. The two horizontal singular lines 
maps onto isolated singular points in the workspace.   
III. WORKSPACES CLASSIFICATION 
A. Classification criteria 
The classification is conducted on the basis of the topology 
of the singular curves in the workspace, which we characterize 
by (i) the number of cusps and (ii) the number of nodes or 
intersecting points. A cusp (resp. a node) is associated with one 
point with three equal IKS (resp. with two pairs of equal IKS). 
These singular points are interesting features for characterizing 
the workspace shape and the accessibility in the workspace. 
B. Number of cusps 
For now on and without loss of generality, d2 is set to 1. 
Thus, we need handle only three parameters d3, d4 and r2. 
Efficient computational algebraic tools were used in [10] to 
provide the equations of five separating surfaces, which were 
shown to divide the parameter space into 105 cells. But [11] 
showed that only 5 cells should exist, which means that one or 
more surfaces among the five ones found in [10] are not 
relevant. However, [11] did not try to find which surfaces are 
really separating. To derive the equations of the true separating 
surfaces, we need to investigate the transitions between the five 
cases. First, let us recall the five different cases found in [11]. 
The first case is a binary manipulator (i.e. it has only two IKS) 
with no cusp and a hole (Fig. 4). The remaining four cases are 
quaternary manipulators (i.e. with four IKS). The second case 
is a manipulator with four cusps on the internal boundary.  Fig. 
5 shows a manipulator of this case with a hole and two nodes. 
Note that the manipulator shown in Fig. 2 is another instance of 
 
 
case 2, although it has no node and no hole (see section C). 
Transition between case 1 and case 2 is a manipulator having a 
pair of points with four equal IKS, where two nodes and one 
cusp coincide [15]. 
 
Figure 4.  Manipulator of case 1. 
 
Figure 5.  Manipulator of case 2. 
Deriving the condition for the inverse kinematic polynomial 
to have four equal roots yields the equation of the separating 
surface [15] 
 
2
2
2
2
2
2
2
3
2
3
2
4
3
2
(
)
1
2
d
r
d
r
d
d
r
AB
⎞
⎛
+
−
+
=
+
−
⎟
⎜
⎝
⎠
 
(2) 
where 
 
2
2
2
2
3
2
3
2
(
1)
and
(
1)
A
d
r
B
d
r
=
+
+
=
−
+
. 
(3) 
The third case is a manipulator with only two cusps on the 
internal boundary, which looks like a fish with one tail (Fig. 6). 
As shown in next section, an intermediate state exists between 
the manipulator shown in Fig. 5 and the one depicted in Fig. 6. 
This intermediate state is a variant of case 2 with two nodes and 
no hole (the upper and lower segments of the internal boundary 
cross, forming a ‘2-tail fish’, see Fig. 11). 
 
 
Figure 6.  Manipulator of case 3. 
As shown in [15], transition between case 2 and case 3 is 
characterized by a manipulator for which the singular line 
given by θ3=–arccos(-d3/d4) is tangent to the singularity curve 
S1. Expressing this condition yields the equation of the 
separating surface  
 
3
4
3
1
d
d
A
d
=
⋅
+
 
(4) 
where A is given by (3).  
The fourth case is a manipulator with four cusps. Unlike case 2, 
the cusps are not located on the same boundary (Fig. 7).  
 
Figure 7.  Manipulator of case 4. 
Transition between case 3 and case 4 is characterized by a 
manipulator for which the singular line given by θ3=–arccos(-
d3/d4) is tangent to the singularity curve S2 [15]. Expressing this 
condition yields the equation of the separating surface 
 
3
4
3
3
and 
1
1
d
d
B
d
d
=
⋅
>
−
 
(5) 
where B is given by (3). As shown in next section, an 
intermediate state exists between the manipulator shown in Fig. 
6 and the one depicted in Fig. 7. This intermediate state is a 
variant of case 3, which features two additional nodes that 
result from the intersection of the two workspace boundaries 
(like in Fig. 3). 
 Last case is a manipulator with no cusp. Unlike case 1, the 
internal boundary does not bound a hole but a region with 4 
IKS. The two isolated singular points inside the inner region 
are associated with the two singularity lines. 
 
Figure 8.  Manipulator of case 5. 
 Transition between case 4 and case 5 is characterized by a 
manipulator for which the singular line given by θ3=+arccos(-
d3/d4) is tangent to the singularity curve S1 [15]. Expressing this 
condition yields the equation of the separating surface  
 
3
4
3
3
and 
1
1
d
d
B
d
d
=
⋅
<
−
 
(6) 
We have provided the equations of four surfaces that divide 
the parameters space into five domains where the number of 
cusps is constant. Fig. 9 shows the plots of these surfaces in a 
section (d3, d4) of the parameter space for r2=1. Domains 1, 2, 
3, 4 and 5 are associated with manipulators of case 1, 2, 3, 4 
and 5, respectively. C1, C2, C3 and C4 are the right hand side of 
 
 
(2), (4), (5) and (6), respectively. 
 
Figure 9.  Plots of the four separating surfaces in a section (d3, d4) of the 
parameter space for r2=1. 
It is interesting to see the correspondence between the 
equations found with pure algebraic reasoning in [10] and those 
provided in this paper. The five equations found in [10] are 
 
2
3
4 2
4
0
d
d r
d
−
+
+
=
 
(7) 
 
2
2
2
3
4
2
0
d
d
r
−
+
=
 
(8) 
2
6
4
4
2
4
2
2
4
4
2
4
2
2
4
3
4
3
4
3
2
4
3
4
3
4
3
2
2
2
2
2
4
2
2
4
2
4
4
4
2
6
4
3
4
3
2
3
2
4
2
4
2
4
4
2
2
2
2
4
4
2
4
2
3
2
2
2
3
2
2
0
d d
d d
d d r
d d
d d
d d r
d d
d d r
d r
d r
d r
d
d r
d r
d r
−
+
−
+
−
+
+
−
−
−
−
+
+
+
=
  (9) 
 
2
2
2
3
4
2
2
2
2
3
2
3
3
3
4
3
4
3
4
2
2
0
d r
d
d
d
d
d d
d d
+
−
+
−
+
−
=
 (10) 
 
2
2
2
3
4
2
2
2
2
3
2
3
3
3
4
3
4
3
4
2
2
0
d r
d
d
d
d
d d
d d
+
+
+
−
−
−
=
 (11) 
 
 
Equation (9) is a second-degree polynomial in d4
2. Solving 
this quadratics for d4
 shows that (9) can be rewritten as 
2
2
2
2
2
2
2
3
2
3
2
4
3
2
(
)
1
2
d
r
d
r
d
d
r
AB
⎞
⎛
+
−
+
=
+
−
⎟
⎜
⎝
⎠
 or 
2
2
2
2
2
2
2
3
2
3
2
4
3
2
(
)
1
2
d
r
d
r
d
d
r
AB
⎞
⎛
+
−
+
=
+
+
⎟
⎜
⎝
⎠
 
where A and B are defined in (3). The first branch is the 
separating surface d4=C1 between domains 1 and 2.  
Equation (10) is a second-degree polynomial  in d4. By 
solving this quadratics for d4
 and assuming strictly positive 
values for d4
 and r2, (10) can be rewritten as 
 
3
4
3
3
(
and
1)
1
d
d
B
d
d
=
⋅
>
−
or 
3
4
3
3
(
and
1)
1
d
d
B
d
d
=
⋅
<
−
 
where B is defined in (3). These two branches are the 
separating surfaces d4=C3 and d4=C4, respectively. 
In the same way, (11) can be rewritten as, 
 
3
4
3
1
d
d
A
d
=
⋅
+
 
which is the separating surface d4=C2.  
Thus, (7) and (8) found in [10] do not define separating 
surfaces, and only one branch of (9) defines a separating 
surface. 
C. Number of nodes 
In this section, we investigate each domain according to the 
number of nodes in the workspace.  
1) Domain 1 
Since all manipulators in this domain are binary, they 
cannot have any node in their workspace. Thus, all 
manipulators in domain 1 have the same workspace topology, 
namely, 0 node, 0 cusp and a hole inside their workspace. This 
workspace topology is referred to as WT1. 
2) Domain 2 
Figures 5 and 2 show two distinct workspace topologies of 
manipulators in domain 2, which feature 2 nodes and 0 node 
and which we call WT2 and WT3, respectively. Transition 
between these two workspace topologies is one such that the 
two lateral segments of the internal boundary meet tangentially 
(Fig. 10).  
 
Figure 10.  Transition between WT2 and WT3. 
Equation of this transition can be derived geometrically and 
the following equation is found [15]  
 
4
1 (
)
2
d
A
B
=
−
 
(12) 
where A and B are defined in (3). 
As noted in section B, a third topology exists in this 
domain, where the internal boundary exhibits a ‘2-tail fish’. 
This workspace topology, which we call WT4, features two 
nodes like in Fig. 5, but these nodes do not play the same role. 
They coincide with two isolated singular points, which are 
associated with the two singularity lines defined by  
θ3=±arccos(-d3/d4) (the operation point lies on the second joint 
axis and the inverse kinematics admits infinitely many 
solutions). Also, the nodes do not bound a hole like in Fig. 5 
but a region with four IKS (Fig. 10). 
 
Figure 11.  Workspace topology WT4. 
Transition between WT3 and WT4 is a workspace topology 
 
 
such that the upper and lower segments of the internal 
boundary meet tangentially (Fig. 12).  
 
Figure 12.  Transition between WT3 and WT4. 
As shown in [15], this transition is the occurrence of the 
additional singularity d3 + c3d4 = 0, that is 
 
d4=d3  
(13) 
3) Domains 3 and 5 
The internal boundary has either 2 cusp (domain 3) or 0 
cusp (domain 5). This boundary is either fully inside the 
external boundary (like in Figs 6 and 8), or it can cross the 
external boundary, yielding two nodes as in Fig. 3 and 13. 
Thus, domain 3 (resp. domain 5) contains two distinct 
workspace topologies, which we call WT5 (1 node) and WT6 
(resp. WT8 and WT9).  
 
Figure 13.  Workspace topology WT9. 
  Transition between WT5 and WT6 and transition between 
WT8 and WT9 are such that the internal boundary meets the 
external boundary tangentially (Fig. 14). 
 
Figure 14.  Transition between WT5 and WT6 (left) and between WT8 and WT9 
right). 
This transition can be derived geometrically and the 
following equation is found [15] 
 
4
1 (
)
2
d
A
B
=
+
 
(14) 
where A and B are defined in (3). 
4) Domains 4 
Manipulators in domain 4 have four cusps and four nodes. 
No subcase exist in this domain [15]. Such topologies are 
referred to as WT7. 
IV. RESULTS SYNTHESIS 
A. Parameter space partition 
Taking into account the nodes in the classification results in 
a new partition of the parameter space, as shown in Fig. 15, 
where E1, E2 and E3 are the right hand side of (12), (13) and 
(14), respectively. Figure 15 depicts a section (d3, d4) of the 
parameter space for r2=1. 
 
7 
 
Figure 15.  Parameter space partition according to the number of cusps and 
nodes (in a section r2=1). 
Plots of the separating surfaces in sections for different 
values of r2 are shown in Fig. 16. 
 
d3
d4
WT3 
WT9 
WT1 
WT8 
WT2 
WT5 
WT6 
WT7 
 
d3
d4 
WT3 
WT9 
WT1 
WT8 
WT2 
WT5 
WT6 
WT7 
 
r2=0.3  
 
 
 
 
 
 
r2=0.7 
 
d3
d4
WT4 
WT3 
WT9 
WT1 
WT8 
WT2 
WT5 
WT6 
WT7 
 
d3
d4 
WT4 
WT3 
WT9 
WT1 
WT8 
WT2 
WT5 
WT6 
WT7 
 
r2=1.5  
 
 
 
 
 
 
r2=2 
Figure 16.  Separating surfaces for different values of  r2. 
The areas associated with WT1, WT2, WT7 and WT9 decrease 
when r2 increases. The area associated with WT4 is very tiny, 
especially for small values of  r2. This means that few 
manipulators have a topology of the WT4 type.  
 
 
B. Classification tree 
A multi-level classification of the 3R orthogonal 
manipulators under study can be established by the 
classification tree shown in Fig. 17. For more legibility, only 
the generic cases are reported on this tree (i.e. manipulators on 
the separating surfaces of the parameter space are not reported).  
The root of the tree is the set of all manipulators under study 
and each leave is the set of manipulators with a completely 
specified workspace topology. The first level of the 
classification tree shows that a 3R orthogonal manipulator has 
either 2 aspects (if d3>d4), or it is quaternary and has no hole in 
its workspace (if d3<d4). The second level shows that (i) a 3R 
orthogonal manipulator with 2 aspects is either quaternary with 
4 cusps (if d4>C1), or binary with no cusp, no node and a hole 
(if d4<C1) and (ii) a 3R orthogonal quaternary manipulator may 
have 4 cusps and 6 aspects (if d4>C3 or d4<C2), or 2 cusps and 
5 aspects (if C2<d4<C3 and d4<C4), or 0 cusp and 4 aspects (if 
d4>C4). 
3R orthogonal 
manipulator 
with r3=0
d4< d3
d4> d3
• 2 aspects
• 0 cusp
• 0 node
• (1, 1, 0)
• binary
• 4 cusps
• quaternary
• 2 nodes
• (1, 1, 2)
• 0 node
• (0, 1, 1)
d4 < C1
d4 < E1
d4 > E1
d4 > C1
• 2 cusps
• 5 aspects
• 0 cusp
• 4 aspects
• 4 cusps
• 6 aspects
• 2 nodes
• (0, 1, 3)
• 3 nodes
• (0, 2, 2)
• 1 node
• (0, 1, 2)
(d4 > C3)
or (d4 < C2)
(C2 < d4 < C3)
and (d4 < C4)
d4 > C3
d4 < C2
d4 < E3
d4 > E3
d4 > C4
d4 < E3
d4 > E3
• 2 nodes
• (0, 2, 1)
• 0 node
• (0, 1, 1)
• 0 hole
• quaternary
WT1
WT2
WT3
WT4
WT5
WT7
WT9
WT6
WT8
• 4 nodes
• (0, 2, 3)
2
2
2
2
2
2
2
3
2
3
2
1
3
2
(
)
1
2
d
r
d
r
C
d
r
AB
⎛
⎞
+
−
+
=
+
−
⎜
⎟
⎝
⎠
2
2
2
2
3
2
3
2
(
1)
and
(
1)
A
d
r
B
d
r
=
+
+
=
−
+
1
1 (
)
2
E
A
B
=
−
2
3
E
d
=
3
2
3
1
d
C
A
d
=
⋅
+
3
1 (
)
2
E
A
B
=
+
3
3
3
1
d
C
B
d
=
⋅
−
3
4
3
1
d
C
B
d
=
⋅
−
(1, 1, 0) means 
• 1 hole
• 1 region with 2 IKS 
• 0 region with 4 IKS
d4
C1
E1
E2
C2
E3
C3
Fig. 4
Fig. 5
Fig. 2
Fig. 7
Fig. 3
Fig. 6
Fig. 11
Fig. 8
Fig. 13
3R orthogonal 
manipulator 
with r3=0
d4< d3
d4> d3
• 2 aspects
• 0 cusp
• 0 node
• (1, 1, 0)
• binary
• 4 cusps
• quaternary
• 2 nodes
• (1, 1, 2)
• 0 node
• (0, 1, 1)
d4 < C1
d4 < E1
d4 > E1
d4 > C1
• 2 cusps
• 5 aspects
• 0 cusp
• 4 aspects
• 4 cusps
• 6 aspects
• 2 nodes
• (0, 1, 3)
• 3 nodes
• (0, 2, 2)
• 1 node
• (0, 1, 2)
(d4 > C3)
or (d4 < C2)
(C2 < d4 < C3)
and (d4 < C4)
d4 > C3
d4 < C2
d4 < E3
d4 > E3
d4 > C4
d4 < E3
d4 > E3
• 2 nodes
• (0, 2, 1)
• 0 node
• (0, 1, 1)
• 0 hole
• quaternary
WT1
WT2
WT3
WT4
WT5
WT7
WT9
WT6
WT8
• 4 nodes
• (0, 2, 3)
2
2
2
2
2
2
2
3
2
3
2
1
3
2
(
)
1
2
d
r
d
r
C
d
r
AB
⎛
⎞
+
−
+
=
+
−
⎜
⎟
⎝
⎠
2
2
2
2
3
2
3
2
(
1)
and
(
1)
A
d
r
B
d
r
=
+
+
=
−
+
1
1 (
)
2
E
A
B
=
−
2
3
E
d
=
3
2
3
1
d
C
A
d
=
⋅
+
3
1 (
)
2
E
A
B
=
+
3
3
3
1
d
C
B
d
=
⋅
−
3
4
3
1
d
C
B
d
=
⋅
−
(1, 1, 0) means 
• 1 hole
• 1 region with 2 IKS 
• 0 region with 4 IKS
2
2
2
2
2
2
2
3
2
3
2
1
3
2
(
)
1
2
d
r
d
r
C
d
r
AB
⎛
⎞
+
−
+
=
+
−
⎜
⎟
⎝
⎠
2
2
2
2
3
2
3
2
(
1)
and
(
1)
A
d
r
B
d
r
=
+
+
=
−
+
1
1 (
)
2
E
A
B
=
−
2
3
E
d
=
3
2
3
1
d
C
A
d
=
⋅
+
3
1 (
)
2
E
A
B
=
+
3
3
3
1
d
C
B
d
=
⋅
−
3
4
3
1
d
C
B
d
=
⋅
−
(1, 1, 0) means 
• 1 hole
• 1 region with 2 IKS 
• 0 region with 4 IKS
d4
C1
E1
E2
C2
E3
C3
d4
C1
E1
E2
C2
E3
C3
Fig. 4
Fig. 5
Fig. 2
Fig. 7
Fig. 3
Fig. 6
Fig. 11
Fig. 8
Fig. 13
 
Figure 17.  Classification tree. 
V.  CONCLUSIONS 
A family of 3R manipulators was classified according to the 
topology of the workspace, which was defined as the number 
of cusps and nodes. The design parameters space was shown to 
be divided into nine domains of distinct workspace topologies. 
Each separating surface was given as an explicit expression in 
the DH-parameters. Further work will investigate each domain 
according to various interesting design criteria.  
REFERENCES 
[1] C.V. Parenti and C. Innocenti, "Position Analysis of Robot Manipulators: 
Regions and Sub-regions," in Proc. Int. Conf. on Advances in Robot 
Kinematics, pp 150-158, 1988. 
[2] J. W. Burdick, "Kinematic analysis and design of redundant manipulators," 
PhD Dissertation, Stanford, 1988. 
[3] P. Borrel and A. Liegeois, "A study of manipulator inverse kinematic 
solutions with application to trajectory planning and workspace 
determination," in Proc. IEEE Int. Conf. Rob. and Aut., pp 1180-1185, 
1986. 
[4] J. W. Burdick, "A classification of 3R regional manipulator singularities 
and geometries," Mechanisms and Machine Theory, Vol 30(1), pp 71-89, 
1995. 
[5] P. Wenger, "Design of cuspidal and noncuspidal manipulators," in Proc. 
IEEE Int. Conf. on Rob. and Aut., pp 2172-2177., 1997 
[6] J. El Omri and P. Wenger, "How to recognize simply a non-singular 
posture changing 3-DOF manipulator," Proc. 7th Int. Conf. on Advanced 
Robotics , p. 215-222, 1995. 
[7] V.I. Arnold, Singularity Theory, Cambridge University Press, Cambridge, 
1981. 
[8] P. Wenger, "Classification of 3R positioning manipulators," ASME 
Journal of Mechanical Design, Vol. 120(2), pp 327-332, 1998. 
[9] P. Wenger, "Some guidelines for the kinematic design of new 
Manipulators," Mechanisms and Machine Theory, Vol 35(3), pp 437-449, 
1999. 
[10] S. Corvez and F. Rouiller,"Using computer algebra tools to classify serial 
manipulators,"in Proc. Fourth International Workshop on Automated 
Deduction in Geometry, Linz, 2002. 
[11] M. Baili, P. Wenger and D. Chablat, "Classification of one family of 3R 
positioning manipulators, "in Proc. 11th Int. Conf. on Adv. Rob., 2003. 
[12] J. El Omri, 1996, “Kinematic analysis of robotic manipulators,” PhD 
Thesis, University of Nantes (in french). 
[13] D. Kohli and M. S. Hsu, "The Jacobian analysis of workspaces of 
mechanical manipulators," Mechanisms an Machine Theory, Vol. 22(3), p. 
265-275, 1987. 
[14] M. Ceccarelli, "A formulation for the workspace boundary of general n-
revolute manipulators," Mechanisms and Machine Theory, Vol 31, pp 
637-646, 1996. 
[15] M. Baili, "Classification of 3R Orthogonal positioning manipulators, " 
technical report, University of Nantes, September 2003.