pt
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4
Workspace Augmentation of Spatial 3-DOF Cable Parallel Robots
Using Differential Actuation
4-
18
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Hamed Khakpour and Lionel Birglen, Member, IEEE,
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Cable robots are a special type of parallel robots in which
cables are used instead of rigid limbs to manipulate the
moving platform (MP) with respect to the base platform (BP)
[1]. Using cables in the architecture of these mechanisms
brings interesting characteristics. For instance, in addition to
the advantages of linkage-driven parallel robots with respect
to their serial counterparts (e.g. higher dexterity, payload,
and precision), cable-driven designs offer a simpler structure,
lighter moving components, low friction (generally), and
larger workplaces [2], [3], [4], [5], [6], [7].
Due to these properties and especially the achieved larger
workspaces, cable robots are preferred to other robotic manipulators in some applications such as construction sites,
shipyards and airplane hangars [8], [9]. Amongst the many
practical examples of these robots, one can cite the wellknown NIST Robocrane [10], Cablecam [11], Falcon [12],
and Skycam [13]. On the other hand, parallel cable robots
suffer from disadvantages such as limited cable tensions,
vibration and cable interference, etc [3], [4].
Because of unidirectional nature of the forces in cables,
they can only pull never push on the MP. Therefore, to fully
constrain this MP, redundancy in the number of cables is
a necessity. This means that with a n-degrees-of-freedom
(DOF) mechanism, at least m = n + 1 cables are required
[2], [14], [15]. As mentioned in the literature, if more cables
(i.e., m > n + 1) are used with a proper arrangement in the
architecture of the cable-driven mechanism, then, a larger
workspace and better performance can be expected [6], [8].
The unilateralism of cables also results in major differences
in the analysis of cable-driven and linkage-driven robots
d
Sy
I. INTRODUCTION
Pr
e
(e.g., how to compute the workspaces of these robots [4],
[6], [8], [16]).
In general, cable robots are classified as either incompletely or fully restrained [4], [17]. In the former, either the
MP is suspended from the ceiling (a.k.a. cable suspended
robots [4], [18]) or only m ≤ n cables are in tension. While
in the latter, regardless of the external wrench, the forceclosure condition is satisfied [4].
Typically, different aspects of cable-driven robots including their kinematics, wrench-closure and wrench-feasible
workspaces (WCW & WFW), cable arrangement and force
distribution have been analyzed in the literature. For example, Tadokoro et al. [19] optimized the cable distribution
of a cable mechanism to improve its workspace. Pusey
et al. [16] proposed an incompletely restrained 6-6 cable
suspended robot and optimized its workspace by considering
the global dexterity of the mechanism. Shiang et al. [9]
designed a 3-DOF cable suspended robot and optimized its
force distribution. Jiang et al. [20] analyzed the kinematics
of cable suspended mechanisms with two to six cables driven
by several aerial robots. Yang et al. [6] introduced a 7-DOF
cable-driven robotic arm and studied its workspace.
Moreover, Bouchard and Gosselin [1] provided a geometrical approach to study the WFW of two to six-DOF cabledriven mechanisms. Gouttefarde and Gosselin [7] proposed
several new theorems to determine the WCW of planar
cable parallel robots. Mao and Agrawal [21] designed a 5DOF cable robot with adjustable cable attachment points and
improved its workspace by optimizing the locations of the
cable connection points. Gouttefarde et al. [22] investigated
the WFW of a n-DOF cable-driven mechanism by using an
interval analysis based approach.
Due to the redundancy in the number of cables of these
robots, to independently manipulate them redundancy in
actuation is also necessary. Therefore, the cost of the mechanism and its control strategy can become quite challenging
especially when the number of cables is increased to obtain
better workspaces [6], [8]. Thus, it would be very interesting
to keep the number of actuators at minimum while the
number of the cables is increased. For this, the idea of using
cable differentials in the structure of a cable mechanism was
presented by the authors and the properties and requirements
of such systems to be used with planar cable architectures
have been described in [23]. A closely related design to
differentials has been proposed in [24], [25] for building
cable robots. Although, it was not recognized as a part of
a much larger family of architectures based on differentials
as demonstrated in [23]. In this paper, the authors focus
,1
Abstract— In this paper, it is proposed to use spatial differentials instead of independently actuated cables to drive cable
robots. Spatial cable differentials are constituted of several
cables attaching the moving platform to the base but all
of these cables are pulled by the same actuator through a
differential system. To this aim, cable differentials with both
planar and spatial architectures are first described in this work
and then, their resultant properties on the force distribution
is presented. Next, a special cable differential is selected and
used to design the architecture of two incompletely and fully
restrained robots. Finally, by comparing the workspaces of
these robots with their classically actuated counterparts, the
advantage of using differentials on their wrench-closure and
wrench-feasible workspaces is illustrated.
H. Khakpour and L. Birglen are with the Department of Mechanical
Engineering, Ecole Polytechnique de Montreal, Montreal, QC, Canada.
Fax number: (514) 340-5867, e-mails: hamed.khakpour@polymtl.ca, lionel.birglen@polymtl.ca.
A
f s=- f r
B'
B
Pulley attached to
a single-point MP
Spring
Cables
S
i
Cable
C
D
4
01
Elliptical
curve
pt
.2
fr
S
i+1
D'
Se
f
f
Elliptical
curve
BP
f
4-
Actuator
Fig. 2: Direction of the resultant force of the two cables
of a planar differential when its actuator is locked and the
elliptical curve along which the pulley can move.
,1
BP
85
f
18
Resultant force directions
pp
Pulley attached to
a single-point MP
MP
),
f s=- f r
on spatial architecture and investigate the effect of using
spatial differentials in 3-DOF spatial cable-driven robots.
For this, the properties of planar and spatial differentials
are first presented. Then, the designs of two robots with
a single-point MP, the first one with three and the second
one with four cable differentials are presented. Finally, two
types of workspaces of these novel designs are computed and
compared with these of fully actuated cable mechanisms.
.3
88
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Fig. 1: Single-point MP actuated by a cable and pulley
differential.
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3-elliptical
curve
f
ff
f
14
Spring
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BP
II. CABLE DIFFERENTIAL SYSTEMS
Actuator S1
f
st
S2
S3
f
(a)
S3
S2
S1
(b)
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Fig. 3: (a) Single-point MP connected to a planar q = 3 cable
differential when the actuator is locked; (b) the routing of the
cable.
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Differentials are used in many different machines to either
distribute the input power to two outputs or combine two
inputs into a single output. Since each individual differential
has 2-DOF, to obtain a larger number of outputs, these
differentials are attached together in serial or parallel patterns, or even combination of them [26], [27]. Examples
of commonly found differentials are seesaw mechanisms,
planetary gear differentials, bevel gearboxes, and tendonpulley systems, all of which have been noticeably successful
in driving underactuated robotic hands [28].
A cable differential consists in several cables connecting
the MP to the BP while the power of a single actuator is
distributed amongst them through a differential mechanism,
while in typical cable robots each cable is driven independently. Since, only one actuator is used in the architecture of
a typical differential, only a single force can be controlled
and applied to the MP of the robot. This property leads to
some differences between the analyses of differentially and
individually driven cable manipulators. Mainly, instead of
a single tension force produced by each individual cable,
the resultant force of all cables of each differential and its
limits should be considered. The simplest cable differential
might be the cable and pulley system presented in Fig. 1
(the spring is used to fully constrain the MP). As can be
seen in this figure, the two cables are pulling on both sides
of the pulley which is attached to a single-point MP. In an
ideal frictionless case and when the radius of the pulley is
negligible, the tension forces in the two sides of the cable are
equal. Consequently, the resultant force on the MP always
lays on the bisector line of both forces and the MP can be
constrained only in this particular direction but free to move
in the normal direction of the resultant force. This means
that if the actuator of this differential is locked the MP can
then move on an elliptical curve (compared to a circle in the
case of a non-differentially driven design).
The direction of this resultant force yields some interesting
properties. As illustrated in Fig. 2, when the pulley is at point
A (i.e., equidistant of the attachment points Si and Si+1 on
the BP), the resultant force intersects the line segment Si Si+1
at its midpoint C. In this configuration, the mechanism is
similar to a single cable attached between points A and C.
If the pulley moves towards point B, the intersection then
lies on point D. This means that the attachment point on the
BP of the “virtual” single cable equivalent to the differential
system is not stationary but moves along the line Si Si+1 .
This phenomenon leads to improvement of the size of the
workspaces of cable robots as will be shown. If the planar
differential illustrated in Fig. 1 is used in a spatial robot,
then the resultant unconstrained elliptical curve of the MP
becomes an ellipsoid (made by rotating this curve around the
axis defined by Si Si+1 ). In this case, although the MP can
move on the surface of the ellipsoid, the intersection of the
resultant force still moves along the line segment Si Si+1 .
The apparent number of output cables of a differential, q,
can be increased to 3, 4 or more (in the previous example,
one had q = 2). The arrangement of cables of differentials
with q = 2, 3, and 4 cables were synthesized in [23] for
the planar case. It should be noted that a q = 2 differential
is necessarily planar since it is defined by three points. On
S4
f s =- f r
S2
z
3
o
S1
S7
S2
S5 S6
BP
y
o
Cables
x
S1
S7
Cables S2
S3
S9
S8
Single-point
MP
Cables
85
,1
S12
38
S10
88
Fig. 4: Single-point MP connected to a spatial q = 3 cable
differential in three different positions when the actuator is
locked.
BP
Weighted
single-point MP
g
BP
(a)
(b)
pp
.3
f S3
S11
0,
S9
4-
f
Midpoint S1 Actuator
f
S3
S8
z
(IR
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14
),
Fig. 5: Schematic of (a) a cable suspended mechanism with
three q = 3 spatial differentials; (b) a fully constrained cable
mechanism with four q = 3 spatial differentials.
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Thus, as mentioned earlier the three output cables of the
differential can also be arranged in a 3-D fashion. The
schematic of an example of a spatial q = 3 differential is
illustrated in Fig. 4. In this case, the routing of the cable
though the pulleys on the BP and MP are the same as the
schematic presented in Fig. 3b, while the location of point
S1 is offset from the previous planar arrangement to obtain
a spatial structure.
As shown in this figure, the single-point MP is connected
to the vertices of a triangle through the three parts of a cable.
When the actuator is locked, the MP is free to move on
all directions which are perpendicular to the resultant force
of the three cable tension forces. In this case, the resultant
surface on which the MP can freely move is a 3-ellipsoid.
When the MP moves on this surface, the intersection of the
resultant force and the BP accordingly moves inside this
triangle. As it can be easily seen in this figure, the main
difference between a planar and a spatial differential is that
in the first case, the aforementioned intersection moves along
a line while in the latter it moves on a surface.
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the other hand, if q > 2 then the differential can be either
planar or spatial. As an example, a planar q = 3 differential
is depicted in Fig. 3.
In this system, a cable is attached to the single-point MP
at one end, then after passing around the pulleys respectively
at points S2 , S3 and the MP, it is connected to an actuated
winch at its other end (for better understanding, the routing
of the cable is illustrated in Fig. 3b). Similarly to the previous
cases, only the resultant force of all three output cables needs
to be considered to compute the WCW and WFW. It should
be noted that if the actuator is locked, the direction of this
force changes when the MP moves along a 3-ellipse not
a simple two focal points ellipse. A 3-ellipse is a planar
closed curve consisting of all the points on which if the MP
is placed, the sum of its distances from the three attachment
points on the BP is constant [29]. Due to the use of a spool in
the winch (connected to the actuator) of the cable differential
system there is no theoretical limit in the total length of
its cable and accommodate any location of the MP. Thus,
the aforementioned elliptical curve can have an arbitrary
configuration. The location, p(x, y), of the points of this 3ellipse for a known total distance d from the three foci is
obtained as [29]:
p(x, y) = det A
(1)
III. PERFORMANCE OF DIFFERENTIALLY
DRIVEN SPATIAL CABLE ROBOTS
A. Architecture of the differentially driven cable robots
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where A is an 8 × 8 symmetric matrix in which the diagonal
elements are functions of d, x, x1 , x2 , and x3 while the
other components are function of y, y1 , y2 , and y3 . Also,
S1 (x1 , y1 ), S2 (x2 , y2 ), and S3 (x3 , y3 ) are the coordinates
of the attachment points of the cables on the BP in the local
coordinate system of the differential. Since this matrix is
quite large, it is not presented here. The obtained components
of p(x, y) are polynomials of degree eight [29].
The effect of this planar q = 3 differential on the workspace
of a planar cable robot (made of three similar differentials)
was presented by the authors in [23]. In this paper, spatial differentials are taken into account and their ability to improve
the performance of 3-D cable-driven robots is investigated.
4
x
01
S4
2
f ff
fr
y
BP
S6
18
1
S5
pt
.2
Spring
Se
Pulley attached to
a single-point MP
To evaluate the effect of using spatial differentials in the
performance of cable robots, two mechanisms are introduced.
The schematic of these two designs are presented in Fig. 5.
For the sake of simplicity a single-point MP is used in these
mechanisms. The first one is a 3-DOF cable suspended robot
in which the MP is suspended from the ceiling by three
q = 3 spatial differentials. In this robot, the weight of the
MP is used to fully constrain the system and all the three
differentials are attached to the top of the robot base. The
second one is a 3-DOF fully constrained cable robot with
a similar MP but four q = 3 spatial differentials. In this
case, the MP is assumed to be weightless and thus, in its
analysis, only the tensions of the cables and the external
wrench exerted to the MP are considered.
S'2
z
o
x
In this paper, the effect of using differentials on cable
robots is investigated through measuring the sizes of their
WCW and WFW. The WCW is a workspace in which the
MP can be located while all cables are in tension and their
force vectors can fully constrain the motion of the MP. In
cable suspended robots, the weight of the MP is used as a
passive constraining force and thus, the term static workspace
(SW) is used instead of WCW to get rid of any confusion
with the usual definition found in the literature. The WFW
is a space where the tensions of all the cables are within safe
pre-specified limits. To calculate the SW of the robots, the
methodology presented in [4] is used. For this, unit vectors
along the resultant forces of cables of each differential are
first defined as:
01
4
B. Workspace analysis
S'1
y
BP
S'2
S'3
Cables
x
38
BP S'4
Weighted
single-point MP
.3
88
g
85
,1
Cables
0,
S'3
4-
18
Single-point
MP
Se
S'1
y
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pt
.2
z
BP
pp
(a)
(b)
ec(3i−2)+ec(3i−1)+ec(3i)
er(i) = ec(3i−2)+ec(3i−1)+ec(3i) for i = 1, 2,· · ·, k
(IR
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14
),
Fig. 6: Schematic of a fully driven (a) cable suspended robot
with three cables and (b) a fully constrained cable robot with
four cables.
particular pose. If it is larger than the minimal required force,
that pose is then inside the WFW of the mechanism. In the
presented cable suspended robot, the MP should be heavy
enough to bring the origin of the zonotope to its center and
consequently maximize the volume of the sphere inside this
geometry. In the proposed fully restrained robot, this weight
is not useful and could be minimized.
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In
Z = α1 ∆f1 ⊕ α2 ∆f2 ⊕ · · · ⊕ αk ∆fk +
s
X
fimin +fg
(3)
i=1
where αi ∈ [0, 1], s = k × 3, and fg = mg is the weight
vector of the MP. Finally, the radius of the largest sphere
which can be contained by this zonotope while its center is
attached to the origin is the magnitude of the maximum force
that can be resisted by the cable robot in all directions in that
em
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C. Architecture of similar fully actuated cable robots
To serve as a basis for comparison between the spatial
differentially driven designs and the independently actuated
ones, four fully driven cable mechanisms are considered.
Two of these mechanisms have exactly the same architectures
as the differentially actuated cases of Fig. 5 but all the cables
in these designs are assumed to be independently driven.
In the other two fully actuated cases, each differential is
replaced by a single actuated cable which is connected to
the center of the triangles as illustrated in Fig. 6.
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where er(i) is the unit vector along the resultant force of
the ith differential; ec(3i−2) , ec(3i−1) , and ec(3i) are the unit
vectors along the forces of the three output cables of the ith
differential; also, k is the total number of differentials. With
the first robot, k = 3 and er(4) is along the weight vector of
the MP while in the second mechanism, one has k = 4.
Next, as described in [4], the cross products cij = er(i) ×
er(j) for i, j = 1, 2,· · ·, 4 and i < j are calculated. Afterwards,
the dot products dijm = eTr(m) cij for m 6= i, j are obtained.
If for each cij there is at least one change in the sign of the
dijm and this is valid for all cij s the corresponding pose is
then included in the SW of the cable robot.
To obtain the WFW, a geometrical method similar to the
one described in [1] is used. In this method, the tension
limits of all cables are assumed to be between fmin and
fmax . Next, the tension vectors ∆fc(j) = (fmax −fmin )ec(j)
of all cables are obtained. Since the resultant force of
each differential should be considered, the vectors ∆fr(i) =
∆fc(3i−2) +∆fc(3i−1) +∆fc(3i) are calculated. These vectors
are then considered as line segments and by computing their
Minkowski sum [1] an initial zonotope (a type of convex
polytope) is generated. Next, taking into account only the
lower limit of the cable tensions (namely fmin ) the position
of this polytope is modified. The final situation of this
polytope can be used to evaluate the WFW of the robot.
The mathematical equation for obtaining this zonotope can
be formulated as [1]:
s
(2)
IV. IMPLEMENTATION AND RESULTS
An algorithm was developed to find the SW and WFW of
all these robots. To do this, all the parameters are first chosen
(cf. Table I). In table I, a is the length of the edges of the
(equilateral) base triangles and the hexagon on the BP (c.f.
Fig. 5), h is the distance between the upper and lower parts of
the BP (with the fully constrained designs), α is the rotation
angle of the equilateral triangles with respect to the hexagon
around their shared edges on the top side of the BP; wmp
is the weight of the MP with the cable suspended robots,
and fW F W is the minimal magnitude of the external force
which should be resisted in all directions by the MP. Also,
the algorithm searches for the workspaces of the robots in a
cylindrical space with a diameter of dc and an height of hc
(c.f. Table I). The base surface of this imaginary cylinder is
attached to the bottom side of the BP while its axis coincides
with the origin of the inertial frame of the robot.
With these numerical values, the workspaces of all robots
are obtained and illustrated in Figs. 7-12. Also, the ratios
TABLE I: Parameters used in the calculations
Values
100 N
0.05
1.2
0.3
−20
WFW
−40
−60
SW
,1
4-
0
−80
−40
85
40
X
−40
0
0
Y
−20
20
40
−40
.3
WFW
0,
SW
38
20
−20
88
−20
Z
4
0
01
Par.
fmax
fmin /fmax
wmp /fmax
fW F W /fmax
pt
.2
Values
40 cm
60 cm
Se
Par.
dc
hc
18
Values
40 cm
80 cm
π/4 rad
Z
Par.
a
h
α
pp
Fig. 9: SW and WFW of the cable suspended robot driven
by nine independent cables.
−80
−40
20
14
),
−60
40
−20
O
Y
0
Fig. 7: SW and WFW of the cable suspended robots driven
by three independent cables.
em
s
−40
−40
WFW
s
an
Z
−20
st
40
Sy
20
(IR
0
0
d
X
S
20
−20
SW
ob
ot
−60
tR
0
en
40
X
on
ce
−60
re
n
SW
on
fe
−80
−40
40
Y
−40
Fig. 10: SW and WFW of the fully restrained cable robot
driven by four independent cables.
40
20
0
0
−20
20
−40
0
Y
rn
at
40
io
na
X
lC
−20
−20
20
In
−40
0
0
te
WFW
20
−20
llig
−20
Z
−80
−40
−20
Z
WFW
−40
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Fig. 8: SW and WFW of the cable suspended robot driven
by three q = 3 spatial differentials.
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of the volumes of the workspaces to the volume of the
cylindrical configuration space are presented in Table II. As
can be seen in this table, with both suspended and fully
constrained cable robots, the SW and WFW of the differentially driven designs are larger than these with the same
number of actuators but each only driving a single cable.
On the other hand, the workspaces of differentially driven
designs are smaller than these of fully driven mechanisms
with the same number of cables when they are all actuated
individually. This means that using differentials, the size
of the workspaces of cable robots can be expected to be
improved. Nevertheless, one should note that using differential may also result in some drawbacks in practice. For
instance, the friction between the cables and pulleys may lead
SW
−60
−80
−40
40
20
−20
0
0
−20
20
X
40
−40
Y
Fig. 11: SW and WFW of the fully restrained cable robot
driven by four q = 3 spatial differentials.
to unbalance distribution of tension forces and using more
cables can increase the possibility of interference of cables.
But, with a proper design, differentials can indeed increase
the performance of these robots. The concept of differentially
−40
WFW
SW
−80
−40
,1
4-
18
−60
85
40
20
−20
\\\\\\\\
\\\\\\\\
\\\\\\\\
Y
\\\\\\\\
\\\\\\\\
\\\\\\\\
X
\\\\\\\\
\\\\\\\\
88
40
−40
0,
Y
−20
20
38
0
0
X
20
14
),
pp
.3
Fig. 12: SW and WFW of the fully restrained cable robot
driven by twelve independent cables.
O
(IR
s
an
d
Sy
st
em
s
W F W/Clr
0.0107
0.0855
0.3487
0.0426
0.1163
0.3472
In
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driven cable manipulator can be used in different fields such
as manipulation, rehabilitation, cargo, and space robots.
tR
ob
ot
SW/Clr
0.4469
0.5263
0.9054
0.1583
0.1987
0.5443
S
TABLE II: Comparing the SW and WFW of the cable-driven
robots.
Type of the cable robot
Three cables
Suspended
Three differentials
with:
Nine cables
four cables
Fully restrained
four differentials
with:
twelve cables
on
V. CONCLUSIONS
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In this paper, spatial cable differentials were presented
and their properties compared with common independently
actuated cables. Their atypical force distribution was first
described. Then, two cable robots either suspended or fully
restrained with a similar single-point MP were introduced
and driven by respectively three and four q = 3 spatial
differentials. The SW and WFW of these robots were then
compared with four more classically driven robots. The
results revealed that by replacing a single cable with a spatial
differentials both types of workspaces can be improved.
R EFERENCES
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