Kinematic analysis of a family of 3R manipulators, IFToMM, Problems of Mechanics, M. Baili, P. Wenger and D.
Chablat, Vol. 15, N°2, pp 27-32, juillet 2004.
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Kinematic Analysis of a Family of 3R Manipulators

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

Abstract— The workspace topologies of a family of 3-revolute (3R) positioning manipulators are
enumerated. The workspace is characterized in a half-cross section by the singular curves. The
workspace topology is defined by the number of cusps that appear on these singular curves. The
design parameters space is shown to be divided into five domains where all manipulators have
the same number of cusps. Each separating surface is given as an explicit expression in the DH-
parameters. As an application of this work, we provide a necessary and sufficient condition for a
3R orthogonal manipulator to be cuspidal, i.e. to change posture without meeting a singularity.
This condition is set as an explicit expression in the DH parameters.

Keywords—Workspace, Singularity, 3R manipulator, Cuspidal 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]. 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 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 quaternary 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
Kinematic analysis of a family of 3R manipulators, IFToMM, Problems of Mechanics, M. Baili, P. Wenger and D.
Chablat, Vol. 15, N°2, pp 27-32, juillet 2004.
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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.
The purpose of this work is to classify a family of 3R positioning manipulators according to the
topology of their workspace, which is defined by the number of cusps appear on the singular
curves. The design parameters space is shown to be divided into five domains where all
manipulators have the same number of cusps. As an application of this work, a necessary and
sufficient condition for a 3R orthogonal manipulator to be cuspidal is provided as an explicit
expression in the DH parameters. 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 states the necessary and sufficient condition and section V concludes this paper.
II. MANIPULATOR 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 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 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
θ1
θ2
θ3
d 2
r 2
d4
d3
.

Figure 1 : Orthogonal manipulators under study.
Kinematic analysis of a family of 3R manipulators, IFToMM, Problems of Mechanics, M. Baili, P. Wenger and D.
Chablat, Vol. 15, N°2, pp 27-32, juillet 2004.
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The singularities of general 3R manipulators can be determined by calculating the determinant of
the Jacobian matrix. For the orthogonal manipulators under study, the determinant of the
Jacobian matrix takes the following form [15]:

det(J) = (d3 + c3d4)(s3d2 + c2(s3d3 – c3r2))
(1)

where ci=cos(θι) and si=sin(θι). A singularity occurs when det(J)=0. Since the singularities are
independent of θ1, the contour plot of det(J)=0 can be displayed as curves in
2
3
,
.
π
θ
π
π
θ
π
−
≤
<
−
≤
<
The singularities can also be displayed in the Cartesian space by
plotting the points where the inverse kinematics has double roots [13]. Thanks to their symmetry
about the first joint axis, it is sufficient to draw a half cross-section of the workspace by plotting
the points (
2
2
ρ =
+
x
y , z).
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 [15]. When the manipulator is in such a singularity, there is line that
passes through the operation point and that cuts all joint axes [4]. 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). Figure 2(a) shows the
singularity curves when d3=2, d4=1.5 and r2=1. For this manipulator, the internal boundary WS1
has four cusp points, where three IKS coincide. It divides the workspace into one region with
two IKS (the outer region) and one region with four IKS (the inner region).

4 IKS
2 IKS

(a) d3=2, d4=1.5, r2=1

(b) d3=3, d4=4, r2=3
Figure 2 : Singularity curves when d3>d4 (a) and when d3<d4 (b)
If d3 ≤ d4, the operation point can meet the second joint axis whenever θ3 = ±arccos(-d3/d4) and
two horizontal lines appear in the joint space. No additional curve appears in the workspace
cross-section but only two points. This is because, since the operation point meets the second
Kinematic analysis of a family of 3R manipulators, IFToMM, Problems of Mechanics, M. Baili, P. Wenger and D.
Chablat, Vol. 15, N°2, pp 27-32, juillet 2004.
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joint axis when θ3 = ±arccos(-d3/d4), the location of the operation point does not change when θ2
is rotated. Figure 2 (b) shows the singularity curves of a manipulator such that d3=3, d4=4, r2=3.
III. WORKSPACE TOPOLOGIES
The workspace is defined by the topology of the singular curves, which we characterize by the
number of cusps. A cusp is associated with one point with three equal IKS. These singular points
are interesting features for characterizing the workspace shape and the accessibility in the
workspace.
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. 3). 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. Figure 4 shows a manipulator of
this case with a hole. Transition between case 1 and case 2 is a manipulator with two points with
four equal IKS, where one node and two cusps coincide [15].

Figure 3 : Manipulator of case 1

Figure 4 : 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
Kinematic analysis of a family of 3R manipulators, IFToMM, Problems of Mechanics, M. Baili, P. Wenger and D.
Chablat, Vol. 15, N°2, pp 27-32, juillet 2004.
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2
2
2
2
3
2
3
2
(
1)
and
(
1)
A
d
r
B
d
r
=
+
+
=
−
+

(3)
Note that there exist two other instances of case 2: the manipulator shown in Fig. 2a with no
hole, and a manipulator where the upper and lower segments of the internal boundary cross,
forming a ‘2-tail fish’ [15].
The third case is a manipulator with only two cusps on the internal boundary, which looks like a
fish with one tail (Fig. 2b). 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, where A is given by (3) :

3
4
3
1
d
d
A
d
=
⋅
+

(4)
The fourth case is a manipulator with four cusps. Unlike case 2, the cusps are not located on the
same boundary (Fig. 5). 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, where B is given by
(3):

3
4
3
3
and
1
1
d
d
B
d
d
=
⋅
>
−

(5)
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. 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)
where B is given by (3).
Kinematic analysis of a family of 3R manipulators, IFToMM, Problems of Mechanics, M. Baili, P. Wenger and D.
Chablat, Vol. 15, N°2, pp 27-32, juillet 2004.
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Figure 5: Manipulator of case 4

Figure 6 : Manipulator of case 5
We have provided the equations of four surfaces that divide the parameters space into five
domains where the number of cusps is constant. Figure 7 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.

Fig. 7 : Plots of the four separating surfaces in a section (d3, d4) of the parameter space for r2=1.
IV. NECESSARY AND SUFFICIENT CONDITION FOR A MANIPULATOR TO BE CUSPIDAL
The above classification provides a means to derive an explicit DH parameter condition for an
orthogonal manipulator to be cuspidal, i.e., to change posture without meeting a singularity. In
effect, as shown in Fig. 7, any cuspidal manipulator belongs to domains 2, 3 or 4. Thus, the DH-
parameters must satisfy d4 > C1 and (d4< C4 or d3 > 1). Thus, a necessary and sufficient condition
for an orthogonal manipulator to be cuspidal is (by dividing the parameters by d2, one gets the
general formula for manipulators such that d2≠1)
Kinematic analysis of a family of 3R manipulators, IFToMM, Problems of Mechanics, M. Baili, P. Wenger and D.
Chablat, Vol. 15, N°2, pp 27-32, juillet 2004.
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2
2
2
2
2
2
2
2
3
2
3
2
2
4
3
2
2
2
2
2
3
2
2
3
2
2
1
(
)
(
)
2
(
)
(
)
d
r
d
r
d
d
d
r
d
d
r
d
d
r
⎛
⎞
+
−
+
⎜
⎟
>
+
−
⎜
⎟
+
+
−
+
⎝
⎠

and
2
2
3
3
2
3
2
4
3
2
2
2
3
or
and
(
)
)
d
d
d
d
d
d
d
d
r
d
d
⎛
⎞
≥
<
<
−
+
⎜
⎟
−
⎝
⎠

(7)
This condition is explicit and can be checked very easily.
V. CONCLUSION
A family of 3R manipulators was classified according to the topology of the workspace, which
was defined as the number of cusps. The design parameters space was shown to be divided into
five domains where all manipulators have the same number of cusps. Each separating surface
was given as an explicit expression in the DH-parameters. An interesting application result of
this work is the establishment of a necessary and sufficient condition for a manipulator to be
cuspidal, i.e., to change posture without meeting a singularity. This condition was set as an
explicit expression in the DH parameters.
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, pp. 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
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[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,"
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Kinematic analysis of a family of 3R manipulators, IFToMM, Problems of Mechanics, M. Baili, P. Wenger and D.
Chablat, Vol. 15, N°2, pp 27-32, juillet 2004.
p8
[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.

Kinematic analysis of a family of 3R manipulators, IFToMM, Problems of Mechanics, M. Baili, P. Wenger and D.
Chablat, Vol. 15, N°2, pp 27-32, juillet 2004.
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SUMMARY

Kinematic Analysis of a Family of 3R Manipulators

Maher Baili, Philippe Wenger and Damien Chablat
Institut de Recherche en Communications et Cybernétique de Nantes, u.m.r. C.N.R.S. 6597
1, rue de la Noë, BP 92101, 44321 Nantes Cedex 03 France

Keywords—Workspace, Singularity, 3R manipulator, Cuspidal manipulator.