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On the Effects of the Design of Cable-Driven Robots on
Kinematics and Dynamics Models Accuracy
A. Gonzalez-Rodrigueza, F. J. Castillo-Garciab,∗
, E. Ottavianoc, P. Reac, A.G.
Gonzalez-Rodriguezd
aSchool of Industrial Engineering, University of Castilla-La Mancha. Av. Camilo Jose Cela
SN, 13071, Ciudad Real, Spain
bSchool of Industrial Engineering, University of Castilla-La Mancha. Av. Carlos III SN,
45071, Ciudad Real, Spain
cDICeM, University of Cassino and Southern Lazio, Via G. Di Biasio 43, 03043 Cassino,
Italy
dDepartment of Electric Engineering and Automation, University of Jaen, Campus Las
Lagunillas SN,23071 Jaen, Spain
Abstract
This paper tackles the problem of improving the connection between the fixed
frame and the end-effector of planar and spatial cable-driven robots. A new
design concept is detailed, which consists in adding pulleys to the attachment
between the cables and the end-effector. These reflective pulleys must have the
same radius as the ones at the frame in order to compensate their geometry.
Without this modification in the end-effector, the usual simplification of the
point-to-point method used to model the connection between frame and end-
effector leads to significant errors in the Kinematics and Dynamics of the system
due to the fact that the geometry of the frame pulleys is disregarded. By adding
the reflective pulleys in the end-effector, the equations of the Kinematics and
Dynamics of the real system are equivalent to those of the point-to-point model,
and therefore, this simplified method can be used without inherent errors. This
solution may be of great importance for computational issues because it leads
to codes that may run at real time. An analytical proof of the equivalence of
both models is presented. Finally, experimental results have been conducted for
∗Corresponding author
Email addresses: antonio.gonzalez@uclm.es (A. Gonzalez-Rodriguez),
fernando.castillo@uclm.es (F. J. Castillo-Garcia), ottaviano@unicas.it (E. Ottaviano),
ottaviano@unicas.it (P. Rea), agaspar@ujaen.es (A.G. Gonzalez-Rodriguez)
Preprint submitted to Mechatronics January 3, 2017
both, planar and spatial cable-robots, in order to illustrate the advantages of
using this novel design concept for the connection between the end-effector and
the fixed frame.
Keywords: cable-driven robot, kinematics model, dynamics model,
positioning accuracy, end-effector attachment
1. Introduction
Cable-driven robots are a class of parallel manipulators [1] that consist of a
moving end-effector connected to a fixed frame by means of cables [2, 3]. The
main components are the fixed frame (or structure), the end-effector, cables,
and the actuation and transmission system. A cable-based manipulator can5
operate the end-effector by changing the lengths of cables while preventing any
cable of becoming slack.
During the last decade, several planar and spatial versions of cable-driven
manipulators have been proposed [4, 5, 6, 7, 8, 9, 10]. In all these prototypes
and design solutions, cables are commanded by electrical actuators, mainly DC-10
motors due to the necessary application of a torque control [11] to properly
regulate the position/orientation of the end-effector [12]. The usual solution to
connect DC-motors and end-effector by means of cables is to use winches that
roll each cable in and out, as in [9]. They are attached at the motor shaft and
they require the so called guiding pulleys to change the direction of the cables15
and to transmit force [13]. While the guiding pulleys are used in all cases for
guiding cables from the winch to the end-effector, in most of the works the cable
attachments on the mobile platform is found in the form of cable attachment
pegs, ideally represented by a single point [3, 4, 7, 8, 9, 10, 12, 13, 14]. Although
guiding pulleys are always used, few works describe their influence [15, 16], or20
reduce the positioning error by taking them into account in the kinematic model
(e.g. [17]).
A second way to connect a cable at the fixed frame is the rail-based system
in which cables are considered as links of constant length, driven by a skid-rail
2
system. Although in most of proposed designs of this type [18] each two skids25
share a common rail, every skid is separately operated by a DC motor through
a drive belt. This makes this design solution similar to a linear drive [19].
On the other hand, most of the published works present cable manipulator
designs assuming that no sensor is available for a direct measurement of the
end-effector position owing to the fact that this kind of sensors (which provide a30
precise real time measurement) are expensive [20]. Therefore, the most extended
solution consists in estimating the end-effector pose by means of the position
of the motor/gearbox or pulleys [21, 22] and/or the cables tensions [23, 16].
Hence, a very accurate kinematic relation between end-effector position and
actuator position is needed to estimate the end-effector pose. Depending on the35
demanded accuracy, two kinds of approaches can be found:
a) Works in which the kinematic and dynamic effects of the pulleys are ne-
glected and they are only taken into account to calculate the relation
between the motor torque and the cable tension, and to estimate cables
length variation [24, 25]. In this case, these modeling assumptions may40
cause errors in the estimation of the end-effector pose (which is used as a
reference in the control loop) and inherent errors will appear.
b) Works in which the kinematic effects of the pulley inclusion is assumed
and included into the model [26, 17]. In this case, the reference error
described in a) is highly reduced, although the model complexity consid-45
erably increases.
It is less frequent to find works focused on improving mechanical designs
in order to reach more accurate models (e.g. [27, 28]). In this line, this paper
introduces a new design concept that overcomes the problems described in a) and
b), without increasing the complexity of such models. It consists in including50
pulleys in the end-effector that compensates the geometry of the pulleys in the
frame. This way, the point-to-point model can be used to obtain an accurate
relationship between end-effector pose and joints angles without introducing
complex equations that take into account the non-punctual geometry of the
3
pulleys/drums/winches in the frame. Preliminary results were presented in [29].55
The remaining of the paper is organized as follows: Section 2 details the con-
ventional modeling for cable robots and the Kinematics and Dynamics errors
that originate. Section 3 presents the new design concept for the connection be-
tween frame and end-effector, which allows using the simplified models without
affecting to Kinematics and Dynamics estimation. Section 4 supports the novel60
proposal by means of experimental results for both, a planar and a spatial case.
Finally, Section 5 summarizes the main conclusions obtained in this work.
2. Problem Statement
2.1. Conventional point-to-point model: spatial case
This subsection is devoted to detail the conventional point-to-point model65
for spatial cable robots. This model assumes that cables connect the frame
(from fixed points placed at pulleys centroid) to fixed points at the end-effector,
neglecting the effects of the pulley radius.
In a first stage let’s assume the planar ncable robot of Fig.1, in which
every cable go from a winch/drum attached to the actuator (Fig.1 detail 1),70
then to a pulley (detail 2), and, finally, to a fixed point at the end-effector
(detail 3). The lengths of the cables are denoted by Li, and the elevation angles
by θi. The rotation angle of each actuator is designed as αi. Positive values
originate negative changes in the cables length, ∆Li, that can be expressed as
∆Li=−αir, being rthe radius of the winches. Therefore, actuators angles can75
be obtained as
α=−1
r∆L(1)
In the case of a spatial robot (see Fig.2), the end-effector pose is defined as
Q= [x, y, z, ϕx, ϕy, ϕz]T, being ϕx,ϕyand ϕzthe Tait-Bryan angles used to ex-
press the end-effector orientation [30]. Cables connect the frame at [xfi , yf i, zfi]
to the end-effector at [xei, yei , zei]. In the same way, the orientation of cables80
is expressed in spherical coordinates by means of angles θiand φi, where θiis
4
attachment
point
Figure 1: Conventional planar point-to-point model for ncable-driven robot
the angle between the cable iand its horizontal projection, and φiis the angle
between this horizontal projection and the xaxis. In a similar way, and for the
segment ilinking the end-effector centroid to the attachment point i, the angle
ζiis the angle between this segment and the horizontal projection, and δiis the85
angle between this horizontal projection and the xaxis.
The length of each cable is
Li=h(xfi −x−dicos(ζi) cos(δi))2+
(yfi −y−dicos(ζi) sin(δi))2+
(zfi −z−disin(ζi))2i
1
2
(2)
which are controlled by modifying the set of actuator angles αthrough (1).
5
Figure 2: Spatial cable robot: frame/end-effector connection for point-to-point modeling.
In the planar case δi= 0. We can also use the ycoordinate as the vertical one
Li=h(xfi −x−dicos(ζi))2+ (zfi −z−disin(ζi))2i
1
2→
Li=h(xfi −x−dicos(ζi))2+ (yfi −y−disin(ζi))2i
1
2(3)
It is worth noting that the angles ζi,δi(only ζiin the planar case) can be90
obtained from the end-effector pose Qand the relative positions between the
attachment points [xfi, yf i , zf i ] and the centroid [x, y, z]. These positions only
depends on the end-effector geometry, and hence they are constant. Once known
these angles, the cable lengths can be obtained from (2), and finally, the actuator
angles from (1). This procedure corresponds to the inverse Kinematics, which95
starts from the end-effector pose to obtain the actuator angles. It is denoted
as α=ΛI(Q). The forward Kinematics starts from the actuator angles and
obtain the end-effector pose. It is denoted as Q=ΛF(α).
2.1.1. Dynamics
The end-effector dynamics can be expressed by100
6
M¨
Q=F(4)
where Mis a diagonal matrix being m11 =m22 =m33 =m(i.e. the end-effector
mass), m44,m55 ,m66 are the momenta of inertia along the respective axes, and
F= [Fx, Fy, Fz, τx, τy, τz]T, being Fx,Fyand Fzthe Cartesian components of
the force applied to the end-effector, and τx,τy,τzthe respective torques.
On the other hand, Fcan be computed by means of the equilibrium of the105
end-effector forces and torque as
F=JsT+Fg(5)
where Fg= [0,0,−mg, 0,0,0]T,T= [T1, ..., Tn]T, i.e. the array of cables ten-
sion, and Jsis the static Jacobian matrix, that yields
Finally, the static Jacobian matrix, Jscan be written as
Js=
J11 J12 ... J1n
J21 J22 ... J2n
J31 J32 ... J3n
J41 J42 ... J4n
J51 J52 ... J5n
J61 J62 ... J6n
(6)
where110
J1i= cos(θi) cos(φi)
J2i= cos(θi) sin(φi)
J3i= sin(θi)
J4i= (yei −y) sin(θi)−(zei −z) cos(θi) sin(φi)
J5i= (zei −z) cos(θi) cos(φi)−(xei −x) sin(θi)
J6i= (xei −x) cos(θi) sin(φi)−(yei −y) cos(θi) cos(φi)
(7)
In the particular case of a planar robot, φi= 0 and the Jacobian matrix is
7
reduced to a 3x3 one. It yields:
J1i= cos(θi)
J2i= sin(θi)
J3i= (zei −z) cos(θi)−(xei −x) sin(θi)
(8)
2.2. Modeling errors
It is well-known that providing a direct measurement of the end-effector
pose, Q, to be fedback in the control scheme, is expensive [20]. Therefore,115
most of the works assume that no sensors are available to provide Qby direct
measurement. On the contrary, it is estimated from the forward Kinematics
Q=ΛF(α) and the actuator positions, α, which can be directly measured
by means of conventional encoders placed at the motors or at the winches.
As a consequence, accurate kinematic transformations are required in order to120
properly control the end-effector pose [31, 32].
As it was mentioned in Section 2.1, the conventional point-to-point model
assumes that cables goes from the pulleys centroid, [xfi, yf i , zf i], to the attach-
ment points at the end-effector, [xei, yei , zei], although in a practical realiza-
tion with pulleys in the frame, the cable ends at the exit point of this pulley125
hxr
fi, yr
fi, zr
fii. Figure 3 illustrates these modeling errors for a planar cable robot,
in which superindex rstands for the real variables. This assumption simplifies
the model complexity but originates errors on cable lengths, Li6=Lr
i, and on
cable angles, θi6=θr
i. These deviations are not constant values since the po-
sition of the exit point of the cable from the frame pulley changes when the130
end-effector pose does. As a consequence, these deviations give rise to errors
when formulating the Kinematics and Dynamics of the system.
As previously mentioned, some works modify the conventional modeling by
including the effects of the frame pulleys of Fig. 3, but obviously the model
complexity grows in comparison to the conventional models [26, 15]. The fol-135
lowing Section presents a novel mechanical design to attach cables to end-effector
which allows using the conventional models without getting into the mentioned
Kinematics and Dynamics errors.
8
Ti
r
end-effector
frame
[x , y ]
f i f i
[x , y ]
ei ei
q
i
q
i
r
Li
r
Li
X
Y
Ti
Figure 3: Deviation of the cable length and angle of the real configuration compared to the
point-to-point model.
3. The novel mechanical design solution
A pulley (or also a drum or winch) located in the frame is required in almost140
all the cable-driven robot configurations. This non-punctual geometry originates
a deviation with respect to the point-to-point model, which is usually preferred
to model the drive. This section analyses the effect of adding a pulley in the
end-effector, to compensate the pulley geometry in the frame. This modification
must satisfy the following:145
•The cables that connect the frame to the end-effector go from a pulley
placed at a fixed point in the frame, to a second pulley placed at a point
of the end-effector. Both pulleys have the same radius.
•The disposition between cable and pulleys responds to one of the first two
situations shown in Fig. 4.150
The kinematic equivalence for two robot configurations will be investigated:
planar robots with two translational DoFs in a vertical plane; and spatial robots,
with DoFs in the three Cartesian axes and a rotational DoF about the vertical
9
Figure 4: Situations in which a pulley in the end-effector compensates the errors originated
by a pulley in the frame.
axis. In the third subsection the dynamic equivalence will be demonstrated for
the more general spatial case.155
3.1. Planar robots
In the following we demonstrate that, under certain conditions, the proposed
modification, i.e. adding a fixed pulley at the end-effector, makes the physical
system behaves as the point-to-point model detailed in the previous Section.
These conditions are:160
•Only translations along the horizontal (x) and/or vertical (y) directions
are allowed. Rotation is forbidden.
•At the end-effector, the relative position between the attachment point
[xai, yai ] and the pulley is fixed.
This last condition implies that the cable exit points remain constant re-165
gardless of the end-effector position, and therefore (see Fig. 5) εA
i,xei −xai,
and yei −yai are constant values. In the frame, the relative position between
winch and pulley is fixed, and therefore Lfr and εD
iare also constant.
In addition, this figure shows that Li=Lr
iand θi=θr
i, and also
εA
i+εB
i=π−εC
i+εD
i⇒εB
i+εC
i=constant (9)
10
eAi
eBi
eCi
eDi
[xai, yai]
DoF_y
DoF_x
Lfr
Lir
Li
qir
qi
xei - xai
yei - yai
Figure 5: Equivalence of angles and length in a planar configuration.
On the other hand, the total cable length Lrt
iwhich can be measured through170
(1) consists of:
Lrt
i=Lee+εB
ir+Lr
i+εC
ir+Lfr with Lee =p(xei −xai)2+ (yei −yai )2(10)
Since Lee,εB
i+εC
i, and Lf r are constant, and also Li=Lr
i, then from
the geometrical point of view, the value of Lirequired to operate in the point-
to-point model can be obtained in a straightforward way by measuring the
total cable length from (1) and applying (10). Therefore, including a fixed175
pulley in the end-effector allows using the equations for the inverse and forward
kinematic transformations obtained for the point-to-point model without the
errors illustrated in Fig.3. In addition, θi=θr
i.
3.2. Spatial robots
In an analogous way that in the case of planar robots, and under similar180
conditions, the real system behavior and the point-to-point model detailed in
11
the previous Section are equivalent. These conditions are
•Translations are allowed along the three Cartesian axes. Orientation of
the end-effector is allowed about the vertical axis (see Fig. 6a).
•The pulley at the end-effector can rotate (φ) about its vertical axis to185
accommodate the connection between pulleys when the end-effector moves
in the horizontal plane, or it rotates about the vertical axis.
•At the end-effector, the attachment point must belong to the axis of ori-
entation of the pulley (as in Fig. 6a).
Figure 6b) compares the point-to-point model scheme to the proposed one.190
(a) (b)
Figure 6: Spatial point-to-point scheme (solid red) and the proposed scheme (dashed blue)
In the spatial case, the reflective pulley is allowed to orientate such that the
following items are all in the same vertical plane:
•in the end-effector: the attachment point, the pulley center, and the exit
point [xr
ei, yr
ei, z r
ei]
•the cable195
•in the frame: the exit point [xr
fi, yr
fi, zr
fi], and the pulley center
12
Since all of the involved elements are in the same plane, the previous rea-
soning for planar robots is valid for spatial ones. This way, adding reflective
pulleys in the end-effector guarantees that the lengths Lr
iare the same as in
the point-to-point model, Li. This means that the inverse and forward kine-200
matics transformations ΛIand ΛFare equivalent to the ones obtained in the
previous Section for the point-to-point model. The cables lengths Lr
ican be
obtained from the motor angle through (1) and (10). In addition, the angles
satisfy θr
i=θiand φr
i=φi.
3.3. Equivalence of the Dynamics205
In the more general case of spatial configurations, the static Jacobian matrix,
Jr
s, can be written as (6, 7), in which the point-to-point variables must be
changed into real variables (with superindex r), resulting in
Jr
1i= cos(θr
i) sin(φr
i)
Jr
i2= cos(θr
i) cos(φr
i)
Jr
i3= sin(θr
i)
Jr
4i= (yr
ei −y) sin(θr
i)−(zr
ei −z) cos(θr
i) sin(φr
i)
Jr
5i= (zr
ei −z) cos(θr
i) cos(φr
i)−(xr
ei −x) sin(θr
i)
Jr
6i= (xr
ei −x) cos(θr
i) sin(φr
i)−(yr
ei −y) cos(θr
i) cos(φr
i)
(11)
The exit points in the end-effector pulley result (see Fig. 6)
xr
ei
yr
ei
zr
ei
=
xei +rcos(θr
i) cos(φr
i)
yei +rcos(θr
i) sin(φr
i)
zei +rcos(θr
i)
(12)
13
Replacing (12) in Jr
4i,Jr
5iand Jr
6iterms of (11) Jr
sresults210
Jr
1i= cos(θr
i) sin(φr
i)
Jr
i2= cos(θr
i) cos(φr
i)
Jr
i3= sin(θr
i)
Jr
4i= (yei −y) sin(θr
i)−(zei −z) cos(θr
i) sin(φr
i)
Jr
5i= (zei −z) cos(θr
i) cos(φr
i)−(xei −x) sin(θr
i)
Jr
6i= (xei −x) cos(θr
i) sin(φr
i)−(yei −y) cos(θr
i) cos(φr
i)
(13)
that are equals to the point-to-point model ones due to θr
i=θiand φr
i=φi.
As it has been demonstrated, inverse and forward kinematics transforma-
tions, ΛIand ΛFare equivalent to the point-to-point model ones, and static
Jacobian matrix, Js, is also equivalent. Therefore, the point-to-point Dynam-
ics model can be also accurately used for the new proposed spatial cable robot215
using pulleys at the end-effector.
3.4. Effects of the reflective pulley on the Dynamics
Pulleys in the frame of cable-driven parallel robot are necessary in their
practical realization to guide cables from winches to the end-effector, but they
introduce modifications in the applied force due to the presence of friction and220
inertia. Modeling and compensation of friction effects in cable-driven parallel
robot design are proposed in [16], in which Coulomb and Dahl models are derived
and used to compensate errors in cable forces measurements. The inertia effect
is discussed in [33]. The case of the reflective pulley in the end-effector as
proposed in this paper is a different issue. In the planar case, it is fixed since it225
does not roll or rotate. In fact, it can be substituted by a circular shape where
the cable lay on, as will be seen in Section 4. Therefore, inertia and friction are
not of consideration.
With regard to the spatial configuration, variations in the fourth DoF (the
end-effector orientation δof Fig. 5) leads to variations (with opposite sign) in230
φi. This angle also varies when the end-effector translates in the horizontal
plane. It is worth noting that this pulley does not roll, and only the rotation
14
about the vertical axis is allowed. Due to the small weight of the rotating part
and the fact that this rotation ˙
φiis slow, the effect of its inertia or friction is
very reduced. Nevertheless, a complete study of these effects should be deeply235
performed in the Dynamics analysis in order to estimate the loss of accuracy
that they originate, although it is out of the scope of this analysis.
4. Experimental results
This section presents the experiments to obtain the positioning error made
when using the new proposal to connect the frame to the end-effector. The240
first subsection presents the experimental platform in which planar and spatial
experiments have been conducted. The second and the third subsections present
the planar and the spatial results, respectively.
4.1. Experimental test-bed
The prototype has been designed with Solidworks R
and built with alu-245
minium profiles and 3D printed plastic parts (PLA). The final design and pro-
totype are shown in Fig. 7.
(a) (b)
Figure 7: Experimental platform: a) Solidworks design; b) real prototype
15
The overall dimensions of the prototypes are 760 ×760 ×1000 mm. In the
frame, four stepper motors (NEMA 17 ), move four mechanisms which allow the
cables to be rolled in and out on the drums. This roller mechanism makes that250
the exit point of cables remains invariant when the drum rotates (see Fig. 8a).
Figures 8b and 8c show the actuator design and the final aspect of the roller
mechanism. It is worth noting that each mechanism rolls a pair of cables.
(a) (b) (c)
Figure 8: Roller mechanism: a) fixed location of the exit point; b) actuation design; c) real
actuator
(a) (b)
Figure 9: a) Planar (two motors) and b) spatial (four motors) configurations
The designed manipulator can be configured to be used as a planar one (two
motors which two pairs of cables), or as an spatial one (four motors with four255
pairs of cables). In the planar case, the end-effector has two degrees-of-freedom,
which are the translations in a vertical plane. The spatial case has four degrees-
16
of-freedom, allowing the end-effector positioning, and the rotation about the
vertical axis; the rotations in the xand yaxes are blocked. Figure 9 details
both configurations. The workspace of each robot (see Fig. 10) can be obtained260
assuming an end-effector pose and checking if all cable tensions remains positive
(see [10] for more details).
(a) (b)
Figure 10: Workspaces for a) planar and b) spatial configuration. Values are given in m
(a) (b)
Figure 11: Compared end-effectors: a) without pulleys and b) with fixed pulleys
In the planar configuration, a more complete experiment has been conducted
in order to verify the reduction of the positioning error if fixed pulleys are
included at the end-effector connection. To this end, two different end-effectors265
17
have been compared: without pulleys as in the traditional case, and with fixed
pulleys. Figure 11 shows both end-effectors.
With regard to the spatial configuration, only the proposed modification has
been tested. The end-effector is shown in Fig. 12. with the pulleys rotating
about vertical axes.
Figure 12: Spatial manipulator: end-effector
270
The main parameters of the robot are summarised in Table 1, specifying the
differences between both experiments.
4.2. Case 1: Two DoF planar cable robot
4.2.1. Trajectories
This subsection compares the accuracy of the point-to-point Kinematics275
model using, on the one hand, a conventional punctual attachment in the end-
effector (Fig. 11a) and, on the other, including a pulley in the end-effector side
(Fig. 11b). To this aim, two planar trajectories have been tested for end-effector
positioning: a circle (100 mm of diameter) and a rhombus (50 mm of side).
The time profile of both trajectories are a 4th order Bezier one to ensure the280
differentiability of its velocity and acceleration. The centre of both trajectories
has been placed at the centre of the workspace and are 10 s duration.
Figures 13 and 14 respectively shows the sequence of movements of the circle
trajectory and the rhombus one, with and without pulleys.
18
Table 1: Parameters of the manipulator with planar configuration
Symbol Description Value
Hheight of the frame 600 mm
Wwidth of the frame 566 mm
Llength of the frame 566 mm
hheight of the end-effector 60 mm
wwidth of the end-effector (planar) 80 mm
wwidth of the end-effector (spatial) 90 mm
llength of the end-effector (spatial) 45 mm
mmass of the end-effector (planar) 112 g
mmass of the end-effector (spatial) 248 g
rradius of pulleys (end-effector) 30 mm
rradius of drums/winches (frame) 30 mm
4.2.2. Results285
The accuracy of our robot is estimated by means of vision techniques. Com-
puter vision cannot be used to feedback the real end-effector position (as oc-
curs in conventional serial robot), but it is a suitable technique to measure the
tracking error. MATLAB Image Toolbox R
has been used to obtain an image
binarization, compensate the tangential error and compare the real trajectory290
to the reference for both cases. By overlapping the reference and the real tra-
jectory, it is possible to measure the perpendicular distance between them and
proceed to directly obtain the error.
Due to the resolution of the camera, each image pixel corresponds to 0.085
mm approximately.295
By defining the error in the normal direction to the reference trajectory, it
is possible to obtain the number of pixels from the desired reference to the real
trajectory and therefore obtain the tracking error. The compared results are
shown in Fig. 15.
19
(a)
(b)
Figure 13: Front view of the circle trajectory on a vertical plane: a) without pulleys and b)
with fixed pulleys
The results show that with the conventional point attachment, the maximum300
tracking error is 6.89mm (and the average one is 2.24 mm), whereas with the
proposed pulley attachment, the maximum tracking error is 1.26mm (and the
average one is 0.28 mm).
Finally, the same procedure has been followed for the rhombus trajectory.
Figure 16 represents the error for both compared attachment methods.305
The results show again a significant reduction in the positioning error in
comparison to the conventional point attachment. The maximum tracking error
for the conventional point attachment is 9.00 mm (and the average one is 2.01
mm). With the proposed pulley attachment, the maximum tracking error is
0.76 mm (and the average one is 0.17 mm).310
Therefore, adding fixed pulleys to the end-effector (with the same radius as
the frame pulleys) significantly decreases the positioning error with regards to
the conventional point connection to the end-effector. It is worth noting that
20
(a)
(b)
Figure 14: Front view of the rhombus trajectory on a vertical plane: a) without pulleys and
b) with fixed pulleys
Figure 15: Circle trajectory comparison: tracking error
21
x(mm)
0
50
100
150
point connection
150
100
z(mm)
50
0
10
5
0
error(mm)
x(mm)
0
50
100
150
pulley connection
150
100
z(mm)
50
0
10
5
0
error(mm)
t(s)
012345678910
error (mm)
0
5
10
Conventional
Proposed
9.00mm
9.00mm
0.76mm
0.76mm
Figure 16: Rhombus trajectory comparison: tracking error
the error made using the conventional point attachment at the end-effector can
be also reduced by software compensation (e.g. [32]) or taking into account the315
radius of the pulleys in the kinematic model (e.g. [34, 17]), but the proposed
solution provides an easier way to use the conventional point-to-point kinematic
model without the error obtained when the frame pulleys radius is not taken
into account [13].
4.3. Case 2: Four Dof spatial cable robot320
4.3.1. Trajectories
In this case, only the new pulley connection will be checked. The trajectories
are a circumference (150 mm of diameter) and a rhombus (75 mm of side), both
executed into the XY plane.
The time profile of both trajectories are again a 4th order Bezier. The center325
of both trajectories has been placed at the center of the workspace and are 10 s
duration.
22
4.3.2. Results
Using an analogous procedure of the previous subsection (MATLAB Image
Toolbox R
), the error of both trajectories has been obtained. They are shown330
in Fig. 17 and 18.
x(mm)
0
100
200
pulley connection
200
100
y(mm)
0
0
5
10
error(mm)
t(s)
012345678910
error (mm)
0
5
10
1.64 mm
1.64 mm
Figure 17: Circle trajectory error
As in the planar case, the results show that the proposed frame/end-effector
connection gives rise to low values of positioning errors. For the circle trajectory,
the maximum tracking error is 1.64mm and its average one is 0.44 mm. For a
rhombus trajectory, the maximum tracking error for the conventional point335
connection is 1.01 mm and its average one is 0.46 mm.
5. Conclusions and future works
In this paper, a new design solution is proposed to connect the fixed frame
to the end-effector in cable-driven robots.
The conventional point-to-point model to describe the kinematic and dy-340
namic behavior of cable robots has been recall for both planar and spatial cables
23
x(mm)
0
100
200
200
100
y(mm)
0
0
5
10
error(mm)
t(s)
012345678910
error (mm)
0
5
10
1.01 mm
1.01 mm
Figure 18: Rhombus trajectory error
robots. This usual way of modeling assumes that no pulleys are placed at the
fixed frame, but the radius is only taken into account to compute cables lengths
variations when drums roll in or out them. Indeed, it is shown in this paper
that important inherent errors appears when this simplification is made due to345
the difference between the lengths and orientations of the cables in the model
and the real system, which in turn originate deviations in the robot Kinematics
and Dynamics.
In order to overcome the mentioned problems, we propose a novel design
solution that includes pulleys at the end-effector, with the same radius as the350
fixed pulleys or winches at the frame. This solution makes the cable lengths
and orientations equivalent to those of the point-to-point model. In addition,
the static Jacobian matrix is also analytically equivalent to the point-to-point
model which can then be used to describe the system dynamic behavior without
the mentioned errors.355
Finally, two application examples that support the proposal have been pre-
24
sented: a two DoF cable manipulator and a four DoF spatial one. In both cases,
using the point-to-point conventional model with the proposed mechanical mod-
ification, the maximum positioning error is about 1 mm.
As the theoretical analysis and experiments show, adding fixed pulleys to the360
end-effector allows using the conventional and simple point-to-point kinematic
model while avoiding the inaccuracy derived from disregarding the pulleys radius
(as detailed in [13]). This proposal can be considered an alternative way to the
actual techniques consisting in software compensation (e.g. [32]) or the inclusion
of the radius of the pulleys in the kinematic model (e.g. [34, 17])365
In its present form, the proposed modification of the attachment between
cable and end-effector is not of application in all of the cable-driven robot con-
figurations, but it is valid in the very typical configuration of 3-positioning DoFs
with vertical orientation DoF, which is the preferred configuration in many ap-
plications (e.g. storing or palletization).370
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