singularities and mobility of a class of 4-dof mechanisms

SINGULARITIES AND MOBILITY OF
A CLASS OF 4-DOF MECHANISMS
Dimiter Zlatanov 1 , Matteo Zoppi 2 and Clément M. Gosselin 3
1 Universität Innsbruck, Technikerstr. 13, A-6020, Innsbruck, Austria
2 Università di Genova, via Opera Pia 15A, 16145, Genova, Italia
3 Laboratoire de robotique, Université Laval, Québec, Canada, G1K 7P4
[zlatanov, gosselin]@gmc.ulaval.ca, zoppi@dimec.unige.it
Abstract
The paper discusses mobility and singularities of a class of 4-dof parallel
wrists with four legs. The singular configurations are classified in detail
and their geometric interpretation is discussed. The velocity kinematics
and the Jacobian operator are formulated via a screw-system approach.
A fully parameterized package of Maple tools was developed and used
to visualize singularities and their consequences.
Keywords: singularities, 4dof mechanisms
1. Introduction
The paper studies the mobility and singularities of a class of parallel
mechanisms (PMs), Figure 1. When first introduced (Zlatanov and Gosselin,
2001) these were the only known fully-parallel 4-dof chains. Later,
other 4-dof PMs have been presented (Huang and Li, 2002).
Four 5R legs, labelled A, . . .,D, connect base and platform. Consider
leg L, L ∈ {A, B, C, D}. The joint screws (and their axes), ξ L 1 , . . .,ξ L 5 ,
as well as their direction vectors, k L 1 , . . .,kL 5 , are numbered from the
D
C
B
ξ L 5
O
A
π
45
L
O
π 0
ξ L 4
ξ L 3
ξ L 45
ξ L 2
ξ L 1
(a)
(b)
Figure 1.
The mechanism architecture with 4-dof.

ase. The first three axes intersect in O, the fixed rotation centre of the
PM, common to all legs. The fourth and fifth are parallel. The fifth
axes, ξ A 5 , . . .,ξ D 5 , are parallel to a common platform plane, π h (fixed in
the platform), but are not all parallel to each other. Then, the platform
has four dof: it rotates arbitrarily around O and translates orthogonally
to π h . The extrusion (heave), h, of the platform is the distance between
π h and its parallel 0-extrusion plane, π 0 , through O.
Singularities are classified using six types, as in Zlatanov et al., 1995:
redundant input/output (RI/RO), impossible output/input (IO/II), redundant
passive motion (RPM), increased instantaneous mobility (IIM).
A fully parametric mathematical model of the mechanism kinematics
has been developed in Maple to support the analysis and illustrate and
verify the obtained conditions for singularity. The created library of
graphical and computational tools simplifies the analysis and provides a
sophisticated and effective graphic output. All presented 3D figures have
been automatically generated by Maple and are exact. The images can
be modified immediately for different values of the geometric parameters.
2. Analysis of the platform constraint
We use the rotating reference frame Oijk with a fixed origin and
vectors constant in the platform, k ⊥ π 0 . For any frame, we associate
a standard basis, {ϱ x , ϱ y , ϱ z , τ x , τ y , τ z }, of the space of twists, S. The
basis elements are the three rotations and three translations about the
coordinate axes. The dual wrench basis is {ϕ x , ϕ y , ϕ z , µ x , µ y , µ z }.
For leg L, the twist systems T L ⊂ P L are spanned by all joint screws
and by the passive (non-actuated) joint screws, respectively. The reciprocal
system W L = TL
⊥ consists of the platform wrenches the leg can
resist (transmit to the base) when all its joints are free to move. The
wrenches resisted when the actuated joint is locked form V L = PL ⊥.
For any nonsingular leg posture (with linearly independent leg joint
screws) Span (ϕ L 0
) = W L V L , dim V L = 2, where ϕ L 0
is the force at O
with direction k L 5 (kL 4 ). In any leg singularity, Span (ϕL 0
) W L ⊂ V L .
The spaces of structural constraints, W = ∑ L W L, and actuated constraints,
V = ∑ L V L, represent the wrenches the PM can resist, when the
actuated joints are free and blocked, respectively. If no leg is singular,
W = Span (ϕ A 0
, . . .,ϕ D 0
) = Span (ϕ x , ϕ y ). (Since k A 5 , . . .,kD 5 , are coplanar
and not all parallel, the four ϕ L 0
span the planar pencil of forces,
W 0 = Span (ϕ x , ϕ y ), at O in π o . The PM cannot have a constraint
singularity (Zlatanov et al., 2002), where dim W > 2.)
With no leg singular, the platform freedoms, T = ∩ L T L = W ⊥ , are
given by M = W ⊥ 0
= Span (ϱ x , ϱ y , ϱ z , τ z ) (M ∗ = Span (ϕ z , µ x , µ y , µ z )

{e ∉ B and π L 45 ≢ π0} O ∉ πL 45 {e ∈ B or π L 45 ≡ π0} O ∈ πL 45
π L 45
ϕ L
π L 45
µ L
ϕ L ⊥
π L 45
O
ϕ L ⊥
π
45
L
π
23
L
µ L
(1)
π L 23
O ϕ L 0
(2)
π L 23
O ϕ L 0
(3)
π L 23
ϕ L 0
(4)
O ϕ L 0
k L 1 ∦πL 23 k L 1 ‖πL 23 k L 1 ∦πL 23 k L 1 ‖πL 23
Figure 2.
Wrench spaces and leg geometry for each posture class.
is the dual). In IO-type singularities T is smaller. A leg singularity leads
to a loss of a platform freedom (IO) only if W L − W 0 ≠ ∅ (W L ⊄ W 0 ).
Depending on the relative location of the passive joints in a leg, we distinguish
four basic classes of leg postures, i.e. subsets, C L i , i = 1, . . .,4,
of the feasible leg configuration space, C L . To describe geometrically a
leg posture, q L , we refer to two planes: π23 L through ξL 2 and ξ L 3 ; and π45
L
defined by ξ L 4 and ξ L 5 . The link angle αi L i+1 , i < 3, is between kL i and
k L i+1 , while αL 34 is between kL 3 and the plane defined by ξL 4 and O.
For each class, C L i , a subclass, C L i1, is defined by the additional condition
k L 1 ‖ πL 23 (i.e., when the first three joint axes are coplanar).
Class 1. C L 1 = {q L | k L 4 ∦ πL 23 and O ∉ πL 45 } (Figure 2–1).
V L = Span (ϕ L 0
, ϕ L ), where ϕ L is the force along π23 L ∩ πL 45 . The force
ϕ L is not through O and not on π 0 , therefore the intersection V L ∩W 0 =
Span (ϕ L 0
), and V L is a cylindroid. Since O ∉ π45 L , then πL 45 ≢ π 0. If
q L ∈ C L 11 (k L 1 ‖ πL 23 ), W L = V L and W L ⊄ W 0 , else W L = Span (ϕ L 0
).
Class 2. C L 2 = {q L | k L 4 ‖ πL 23 and O ∉ πL 45 } (Figure 2–2).
V L = Span (ϕ L 0
, µ L ) is the 2-system of the pure forces parallel to k L 4 and
lying on π23 L (µL is a pure moment orthogonal to π23 L ). We note that
V L ∩ W 0 = Span (ϕ L 0
). Since O ∉ π45 L , πL 45 ≢π 0. W L =V L and W L ⊄ W 0
iff q L ∈ C L 21 (k L 1 ‖ πL 23 ), else W L = Span (ϕ L 0
).
Class 3. C L 3 = {q L | k L 4 ∦ πL 23 and O ∈πL 45 } (Figure 2–3).
V L = Span (ϕ L 0
, ϕ L ⊥
) is the planar pencil of forces centred at O and lying
in π45 L . ϕL ⊥ is a pure force through O orthogonal to kL 4 and lying on
π45 L . If πL 45 ≡ π 0, then V L =W 0 ; else, V L ∩ W 0 = Span (ϕ L 0
). W L =V L
everywhere in C L 3 . Iff π45 L ≢π 0, W L ⊄ W 0 .
Class 4. C L 4 = {q L | k L 4 ‖ πL 23 and O ∈πL 45 } (Figure 2–4).
V L = Span (ϕ L 0
, µ L , ϕ L ⊥
) is a 3-system spanned by a planar pencil in
π45 L and a moment perpendicular to πL 23 . (If αL 34 ≠ 0, then πL 23 ≢ πL 45
and V L has screws of all pitches. Else, if α34 L = 0, then πL 23 ≡ πL 45
and V L is spanned by the pure forces in a plane.) If π45 L ≡ π 0, then
W 0 ⊂ V L ; else V L ∩ W 0 = Span (ϕ L 0
). W L =V L if q L ∈ C L 41 (k L 1 ‖ πL 23 ),

k
✛ rL 4
✛l L 45✲
✛
r L 5
✲
✲
♣ O
ξ 3
(a)
O
h
(b)
(c)
Figure 3. (a) The heave section of leg L. (b) Fixed platform and moving base at
maximum heave. (c) A leg allowing π45 L ≡π 0.
else W L = Span (ϕ L 0
, ϕ L ⊥ ). In CL 4 , we have extra constraints, W L ⊄ W 0 ,
unless both π L 45 ≡π 0 and k L 1 ∦ πL 23 .
Note that a leg singularity always leads to extra structural constraints
(and an IO singularity), W L − W 0 ≠ ∅, except if π L 45 ≡π 0 and q L ∉ C L 41.
3. The workspace and its boundary
Let X ⊂ SO(3)×R be the 4-dimensional workspace of the mechanism.
The extrusion boundary, B ∂X, consists of all feasible platform
poses, e = (R, h), where the heave, h, achieves an extremal value for
some L. The PRRR planar loop in Figure 3(a) represents the heave
section of the PM along one of the leg chains. Extremal extension occurs
when O ∈ π 45 , so a platform pose is in B iff for at least one leg π45 L ≢π 0
and O ∈π45 L (qL ∈ C L 3 ∪C L 4 ). Figure 3(b) shows a mechanism in different
configurations, with extremal heave and different platform orientations.
(A PM may not reach extremal values of h for some platform rotation.
In fact, if α12 L and αL 23 are sufficiently small, B can be empty.)
The remainder of the boundary, ∂X − B, consists of mechanism configurations
where, at least in one leg k L 1 ‖πL 23 , i.e., qL ∈ C L 11 ∪ C L 21.
It can be proven rigorously that every configuration with e ∈ ∂X
is an IO-type singularity. Conversely, an IO-type singularity generally
implies e ∈ ∂X, the only exception being configurations where ξ L 1 ≡ ξ L 3 .
Thus, only if α12 L = αL 23 can there be IO singularities for e ∈ int (X).
4. The equivalent mechanism and π L 45 ≡π 0
If all π L 45 ≢ π 0, the PM is instantaneously equivalent to a simplified
mechanism (Figure 1b). The fourth and fifth joints of each leg are replaced
by a single revolute joint, ξ L 45, at the intersection of π L 45 and π 0.

η B
B
A
ζ A
D
η D
D
C
A
B
C
(a)
(b)
(c)
Figure 4. (a) An (IIM, RI, RO)-singularity with π45 D ≡ π 0; ζ D is through O on
π 0. (b) An (RPM, IIM, IO, II)-singularity with two C 4 legs. (c) An (RPM, RI, IIM,
IO)-singularity with e ∈ B, q A ∈ C A 4 , q C ∈ C C 1 , q L ∈ C L 11, L = B, D.
(If π45 L ‖ π 0, the new joint is prismatic.) The new joint velocity is the
sum of the velocities it replaces.
When π45 L ≡π 0, there is no equivalence. This occurs when O ∈π45 L and
the distance, r5 L, from ξL 5 to Oz equals r4 L ±lL 45 , where rL 4 is between Oz
and ξ L 4 , and l45 L is the length of the last leg link (Figure3c).
Suppose π45 L ≡π 0, while the other legs are nonsingular. An IO occurs
only if q L ∈ C L 41. Then, there is also RPM and RI. The 5 joints of
leg L span a 3-system and the platform has 3-dof. Therefore, the PM
has 5-dof (like a closed loop with 8 joints spanning a 3 system) and
IIM is present. The configuration also belongs to both or neither of
types II and RO depending on the other 3 leg chains. (Indeed, replace
leg L with a passive nonsingular leg with the same 3-dof. Then this
equivalent PM can have only a singularity of class (RO, II).) Therefore,
the configuration is either (RI, RPM, IO, IIM) or with all six types.
Otherwise, if q L ∉ C L 41, although leg L is singular, the platform is
constrained only by W 0 and dim T = 4 (no IO). Either q L ∈ C L 4 − C L 41
and we have an RPM singularity, or q L ∈ C L 3 and there is RI. In either
case, since there is no IO, there must be an IIM-type singularity. (This
follows from the interdependence rules of the singularity types, Zlatanov
et al., 1995.) Moreover, if q L ∈ C L 3 and there is no RPM, an RO-type
singularity must be present as well. Thus we have at least (RPM, IIM)
or (RI, RO, IIM) and each can be augmented with (RO, II) if the other
three legs cooperate. Figure 4(a) shows an example of a singularity of
class (IIM, RI, RO). Thus, π45 L ≡π 0 allows the six singularity classes with
IIM and without IO, impossible for “usual” PMs (Zlatanov et al., 1994).
Below, we generally assume that no π45 L ≡π 0.

η B
D
C
B
v
A
B
D
η C
C
B
η A
A
A
(a)
(b)
η D
(c)
Figure 5. IIM-type singularity with e ∈ int(X) and ξ L 1 ≡ ξ L 3 , ∀L. In (b) the
wrenches η L of (a) are drawn with the lengths of their pitches, to show the linear
dependence. In (c) the moment components of all η L are coplanar.
5. Leg Singularities
5.1 RPM- and IIM-type singularities
An RPM-type singularity occurs iff a leg chain has a linear dependence
of the passive joints screws, dim P L < 4, i.e., dim V L > 2. According to
Section 2, this is equivalent to q L ∈ C L 4 for some L.
Each C 4 leg has one passive dof, so the redundant passive freedoms
of the mechanism are as many as the C 4 legs. Figure 4b represents a
configuration with two RPM freedoms.
Provided that all π45 L ≢π 0, a necessary condition for O ∈π45 L is e ∈ B.
Therefore, under this hypothesis, all RPM-type singular configurations
have an end-effector in B and the workspace interior int (X) is free of
them. (As we saw, if the geometry of some leg is such that π45 L ≡π 0, it
is possible to have an RPM-type singularity and e ∈ int (X).)
A method for the study of IIM-type singularities in certain parallel
mechanisms is given in Zlatanov et al., 1994. The approach can be
applied to the instantaneously equivalent PM. A necessary and sufficient
condition for IIM is the existence of four linearly dependant wrenches,
η A , . . .,η D , not all zero, such that η L ∈ W L ∩ M ∗ , ∀L.
If π45 L ≢ π 0 and α12 L ≠ αL 23 for all L, the mechanism configuration
has an end-effector in ∂X and int (X) is free from IIM-type singularity.
Examples of IIM-type configurations are presented in Figures 4–5.

C
D
A
C
D
A
C
B
D
A
B
D
A
(a)
B
B
(b)
(c)
C
(d)
Figure 6. Singularities with IO and RI. (a) e ∈ B, q L ∈ C L 3 , L = A, C. (b)q C ∈ C C 4
also RPM-type singular.(c)e ∈ ∂X − B, q A ∈ C A 11, q L ∈ C L 1 , L = B, C, D. (d)e ∈
int (X), ξ C 1 ≡ ξ C 3 , q L ∈ C L 1 , L = A, B, D.
5.2 Singularities on and off B
For configurations with e ∈ B, either q L ∈ C L 3 or q L ∈ C L 4 (π45 L ≢π 0)
for some L. All these are IO-type singularities with respect to extrusion.
If q L ∈ C L 4 an RPM-type singularity occurs. Else, if q L ∈ C L 3 , there is an
RI-type singularity. If q L ∈ C L 41, both RI- and RPM-type are present.
When e ∈ ∂X − B, q L ∈ C L 11 ∪ C L 21 for all legs. Assuming π45 L ≢ π 0,
such a configuration is an (RI, IO) singularity. Iff α12 L = αL 23 , there are
leg singularities with e ∈ int (X). They are (at least) in the IO and RI
types, but never in RPM. Figure 6 shows singularities of class (RI, IO).
6. Singular configurations with nonsingular legs
The analysis in the previous sections shows that if no leg is singular,
the only possible singularity class is (RO, II), Figure 7. When π45 L ≢π 0
and α12 L ≠ αL 23 , “no singular legs” is equivalent to e ∈ int (X).
By using the equivalent PM with 4R legs we obtain the 4×4 Jacobian:
⎡
− 1
r 45n A A 23 · ⎤
kA 4 | n A 23
− 1
r
Z =
45n B B 23 · kB 4 | n B 23
⎢ − 1
r
⎣ 45n C C 23 · kC 4 | n C (1)
23 ⎥
− 1
r45n D D 23 · ⎦
kD 4 | n D 23
The rows of Z represent wrenches in M ∗ . These can be transformed to
an equivalent 4-element spanning system with three (or four) pure moments.
The condition for singularity is obtained as a 3 × 3 determinant,
and in case of singularity, the RO-motion screw is derived symbolically.

(a)
ξ RO
ξ RO
(b)
(c)
Figure 7. (RO, II) singularities. (a) The RO motion , ξ RO
(pitch −0.00104) and the
forces ϕ L are shown. (b-c) ξ RO
and the velocities at the platform points are shown.
7. Conclusions
The paper studies the possible singularity types of a family of 4-dof
parallel wrists. Under certain hypotheses, the RPM-, RI-, IO- and IIMtype
singularities are restricted to the workspace boundary. The typical
interior singularities of type (RO, II) are identified by checking a system
of 4 wrenches. Under the most general assumptions, these PMs allow
all 21 singularity classes, including those 6 not possible for classic PMs.
The method of analysis is general. A fully parametric 3D virtual
mock-up of the mechanism, based on a strong mathematic model, has
been implemented in Maple via a library of computational and graphical
routines created to analyze the architecture, visualize the mechanism
behaviour and verify the results of the singularity analysis. Since most
such routines are general, they can be reused for different mechanisms.
The software is available from the authors.
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and interpretation of mechanism singularities”. Trans. of the ASME Journal of
Mechanical Design, Vol. 117, pp. 566-572, December 1995.
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of freedom”. In F.C. Park C.C. and Iurascu, editor, Computational Kinematics,
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