Conference PaperPDF Available

# Robust adaptive sliding mode control of a redundant cable driven parallel robot

Authors:

## Abstract and Figures

In this paper we consider the application problem of a redundant cable-driven parallel robot, tracking a reference trajectory in presence of uncertainties and disturbances. A Super Twisting controller is implemented using a recently proposed gains adaptation law [1], thus not requiring the knowledge of the upper bound of the lumped uncertainties. The controller is extended by a feedforward dynamic inversion control that reduces the effort of the sliding mode controller. Compared to a recently developed Adaptive Terminal Sliding Mode Controller for cable-driven parallel robots [2], the proposed controller manages to achieve lower tracking errors and less chattering in the actuation forces even in presence of perturbations. The system is implemented and tested in simulation using a model of a large redundant cable-driven robot and assuming noisy measurements. Simulations show the effectiveness of the proposed method.
Content may be subject to copyright.
Robust Adaptive Sliding Mode Control of a
Redundant Cable Driven Parallel Robot
Christian Schenk
Max Planck Institute
for Biological Cybernetics
Spemannstraße 38
72076 Tübingen, Germany
christian.schenk@tuebingen.mpg.de
Prof. Dr. Heinrich H. Bülthoff
Max Planck Institute
for Biological Cybernetics
Spemannstraße 38
72076 Tübingen, Germany
heinrich.buelthoff@tuebingen.mpg.de
Carlo Masone
Max Planck Institute
for Biological Cybernetics
Spemannstraße 38
72076 Tübingen, Germany
carlo.masone@tuebingen.mpg.de
Abstract—In this paper we consider the application problem
of a redundant cable-driven parallel robot, tracking a reference
trajectory in presence of uncertainties and disturbances. A Su-
per Twisting controller is implemented using a recently proposed
gains adaptation law [1], thus not requiring the knowledge of
the upper bound of the lumped uncertainties. The controller
is extended by a feedforward dynamic inversion control that
reduces the effort of the sliding mode controller. Compared to
a recently developed Adaptive Terminal Sliding Mode Controller
for cable-driven parallel robots [2], the proposed controller
manages to achieve lower tracking errors and less chattering
in the actuation forces even in presence of perturbations.
The system is implemented and tested in simulation using a
model of a large redundant cable-driven robot and assuming
noisy measurements. Simulations show the effectiveness of the
proposed method.
Index Terms—Parallel Robots, Adaptive Control, Robust
Control, Gain, Sliding mode control
I. INTRODUCTION
Cable-Driven Parallel Robots (CDPRs) are parallel robotic
manipulators in which the motion of the end-effector is
controlled through cables that are pulled by actuators placed
offboard the robot (see the sketch in Fig. 1). The advantage
of this architecture, is that CDPRs not only retain the high
stiffness, accuracy and payload typical of rigid parallel ma-
nipulators (e.g. Stewart platform) but also have less moving
mass in comparison to manipulators with rigid links. There-
fore, by using CDPRs it is possible to attain faster dynamics
and far larger workspaces than the ones achievable by serial
manipulators or parallel rigid manipulators. These properties
make CDPRs excellent for performing a wide range of
applications such as pick-and-place of heavy loads [3], large
scale automated construction [4], sandblasting [5], sensing of
large outdoor environments [6], large radiotelescopes [7] and
suspended actuated cameras for sport events [8].
The adoption of CDPRs in real world applications requires
robust control algorithms that are able to cope with uncer-
tainties in the model of the robot (e.g. dynamic parameters,
unmodelled effects), changes in the system during operations
H. H. Bülthoff is also with the Department of Brain and Cognitive
Engineering, Korea University,Seoul, 136-713 Korea.
(e.g. load changes during pick-and-place) and external dis-
turbances (e.g. wind gusts in outdoor operations).
In the literature of CDPRs, few papers have tackled some
of the aforementioned problems by applying robust or adap-
tive controllers. In [3] the authors propose a dual closed-
loop inverse dynamic controller with online adaptation of
the dynamic and kinematic parameters of the model, but
without considering external disturbances. In [9] a robust PID
controller for a planar CDPR is presented in which the gains
of the controller are tuned according to the known bounds
on norm of the disturbances and of the uncertainties in the
Jacobian and all dynamic matrices. In [10], the authors design
an adaptive controller with dynamic inversion in which the
dynamic parameters are updated online and the uncertain
Jacobian matrix is supposed to have a known bound on the
norm. Finally, the authors in [2] discribed an Adaptive Termi-
nal Sliding Mode Controller (ATSMC) that uses a continuous
control plus a closed-loop dynamic inversion control with
adaptation of the model parameters. Bounded disturbances
are accounted for, by choosing sufﬁciently high control gains
to reduce the boundary layer of the sliding mode.
All the methods cited present some limitations, either
i) only parametric uncertainties are considered [3], [10], or
ii) it is required the knowledge of an upper bound on the
uncertainties and disturbances [9], [10], or iii) aggressive
adaptation laws can introduce chattering [2], [3], or iv) to
guarantee a certain performance in presence of disturbances
the gains must be tuned to a worst case scenario which
could lead to unnecessary high control actions and ampliﬁed
noise [2]. To overcome these limitations, in this paper we
implement an Adaptive Super Twisting Controller (ASTC)
with the following properties:
1) It considers all the uncertainties (parametric, model,
disturbances) lumped together.
2) It does not require the knowledge of the upper bound of
the uncertainties.
3) Rather than adapting the parameters of the model, it
adapts the gains according to the method recently pro-
posed in [1]. In this way, the gains are reduced whenever
2015 19th International Conference on System Theory, Control and Computing (ICSTCC), October 14-16, Cheile Gradistei, Romania
possible, thus reducing control actions, chattering and
noise ampliﬁcation;
4) It uses a feedforward dynamic inversion (FF) to reduce
the discontinuous control, thus improving performance
and further reducing chattering.
This paper is organized with the following structure. First,
in Sec. II we introduce the model of the system. Then,
in Sec. III we describe the controller. Finally, simulation
results of the proposed ASTC are presented in Sec. IV. The
simulations include both external disturbances and parametric
uncertainties and also show a comparison of the ASTC and
the ATSMC [2].
II. SYSTEM MODELING
The robot considered in this paper is a redundant CDPR with
n=6DoF, that consists of a rigid body (end-effector) whose
motion is controlled by the wrench exerted via m>ncables.
We assume that the i-th cable is ﬁxed at one end at a point Bi
on the end-effector (onboard connection) and that at the other
end it comes from a point Aiﬁxed in the workspace (offboard
connection), as shown in Fig. 1. In the rest of this section
we introduce a model of this system in which the cables are
considered massless and inextensible. The inclusion of an
accurate model of the cables is an open topic of research and
it is outside the scope of this paper.
FW
FE
Ai
Bi
Ebi
Wli
Wai
p
Figure 1: Sketch of the cable robot.
A. Kinematics
The pose of the robot is described by the vector xν=
[pTνT]TSE(3), which represents the position p
R3and orientation νSO(3) of a frame FE=
OE,
XE,
YE,
ZEﬁxed on the end-effector w.r.t. an inertial
world frame FW=OW,
XW,
YW,
ZW(see Fig. 1).
Hereinafter we describe the orientation using the roll-pitch-
yaw angles ν=(φθψ)T. Therefore, the rotation matrix that
expresses the end-effector orientation in FWis
WRE=
cψcθcψsθsφsψcφcψsθcφ+sψsφ
sψcθsψsθsφ+cψcφsψsθcφcψsφ
sθcθsφcθcφ
(1)
where c= cos()and s=sin().
The generic i-th cable satisﬁes a loop closure constraint
given by the triangle with vertices i) the offboard connection
point Ai, ii) the onboard connection point Bi, and iii) the
point OE(see Fig. 1). Namely, the loop closure constraint is
Wli=ρiWni=WaipWREEbi(2)
where
WaiR3is the position of Aiin FW;
EbiR3is the position of Biin FE;
WliR3is the vector BiAiin FW, which is factored
as the product of the unit vector WniR3in FW
(cable direction) and the scalar ρi>0(cable length).
The loop constraint (2) can also be formulated in terms of
cable lengths as
ρi=WaipWREEbi2.(3)
To model the differential kinematics, we describe the
rotation rate of FEw.r.t. FWby means of the angular
velocity Wωand deﬁne1˙
x=[˙
pTWωT]T. With this setting,
differentiating (2) for all the cables yields the well-known
differential relation ˙
ρ=[˙ρ1... ˙ρm]T=J˙
x[9], [11], [12],
where the Jacobian matrix Jhas the structure
J=
WnT
1(WREEbi×Wni)T
.
.
..
.
.
WnT
n(WREEbn×Wnn)T
.(4)
The transpose Jacobian JTprovides the mapping from cable
tensions to wrenches at the end effector, as shown in the next
section.
B. Dynamics
The dynamic equations of the system depend on the physical
properties of the end-effector, i.e., its total mass m, the
position Wc=[cxcycz]TR3of the center of mass
in FW2, and the 3×3inertia matrix EIEin FE.Given
these parameters, the dynamic equations obtained through the
Newton-Euler approach or the Euler-Lagrangian formulation
have the following well known form [3]
B(xν)¨
x+C(xν,˙
x)˙
xg(xν)=u=JTt,(5)
1Note that ˙
x=d
dt xνbecause Wω=˙
ν.
2The position of the center of mass of the end-effector is typically
expressed in FEwhere it is constant, i.e., Ec. Moving to FWis straight-
forward, i.e. Wc=WREEc
428
where ¨
x=[
¨
pTW˙
ωT]T,uRnis the controller wrench
resulting from the tensions tRmexerted by the cables,
and B,Cand gare deﬁned as
B(xν)=mI3mWc×T
mWc×H,(6)
C(xν,˙
x)˙
x=mWω×Wω×Wc
Wω×HWω,(7)
g(xν)=00mg mcygmc
xg0T
(8)
H=WREEIEERW+mWc×Wc×T.(9)
InSecs. II-B to II-B I3indicates the 3×3identity matrix
and ×is the cross product operator deﬁned for an arbitrary
vector vR3as
v×=
0v3v2
v30v1
v2v10
The dynamic model here discussed requires a few remarks.
Firstly, we recall that the inertia matrix BSec. II-B is symmet-
ric, positive deﬁnite and therefore invertible [13]. Secondly,
we observe that the control wrench uin (5) is achieved
by the mapping JTof the vector tof the tensions that
are applied by the cables. These tensions, must be positive3
and are bounded in a certain range to prevent slackness or
ruptures, i.e.,
0<tmin ttmax.(10)
To ensure that uis feasible with respect to the mapping (5)
and under the limit (10), we make one assumption.
Assumption 1. The control wrench is bounded, i.e., u∈U=
{u[umin,umax ]}.
Under Assumption 1, feasibility of uis guaranteed by
restricting the motion of the end-effector to the so-called
Wrench-Feasible-Workspace (WFW) [14], i.e. the set of all
poses in which, for any wrench u∈U, there exists a
tensions vector t[tmin,tmax ]such that JTt=u.
Furthermore, provided that Ucontains a neighborhood of
the origin, Jhas full rank [15]. Regarding Assumption 1,
we further note that in practice it is easily veriﬁed because
the WFW is typically over-dimensioned with respect to the
task. Nevertheless, a saturation of the control wrench can be
C. Regular Form
The kinematic and dynamic equations developed in Secs. II-A
and II-B can be combined together to obtain the full state
3The cables can only pull the end-effector, not push it.
space model of the system. Taking the 12 ×1state vector
¯
x=[xT
ν˙
xT]T=[pTνT˙
pTWωT]T,wehave
˙
¯x=
˙
p
νEωWω
B1(C˙
xg)
+
03×n
03×n
B1
u,(11)
where νEωis the transformation matrix from ωto ˙
ν, i.e.,
νEω=
cθcψsψ0
cθsψcψ0
sθ01
(12)
Taking the vector xν=[pTνT]as output, it is easy to
see that the relative degree for all the six components of the
output is r1=r2=... =r6=2, i.e., 6
i=1 ri=12,
therefore there exists a diffeomorphism Φ(¯
x)such that with
the change of coordinates z(
¯
x)the system is exactly
feedback linearizable [16]. Consider the change of coordinate
z=z1
z2(
¯
x)=
I606×6
06×6I303×3
03×3
νEω
 
A(xν)
xν
˙
x.
(13)
Applying (13) to (11) the system becomes
˙
z1=z2
˙
z2=f(¯
x)+h(¯
x)u(14)
with:
f(¯
x)=A(xν)B1(xν)[C(xν,˙
x)˙
xg(xν)](15)
+˙
A(xν,˙
x)˙
x
h(¯
x)=A(xν)B1(xν)(16)
For the state space representation given by (14)
and Sec. II-C to hold, we make the following assumption:
Assumption 2. The pitch angle θis limited to (π/2/2).
Assumption 2 ensures the matrix νEωin (12) is nonsin-
gular and thererefore that h(¯
x)has always rank m, i.e., is
invertible.
D. Uncertainties
The model presented in (14) depicts the system in absence
of uncertainties. To incorporate the effect of inexact knowl-
edge of the parameters and of disturbances, we consider that:
1) the robot is subject to disturbances ζthat act on the
end-effector as wrenches. The dynamic equation of
motion (5) becomes
B(xν)¨
x+C(xν,˙
x)˙
xg(xν)=u+ζ;(17)
2) only the dynamic parameters m,H,care uncertain,
whereas the kinematic parameters (and therefore the
Jacobian J) are known accurately.
429
Following these assumptions the model (14) becomes
˙
z1=z2
˙
z2=fnf+hn(u+ζ)+Δb(u+ζ)
=fn+hnu+ξ
(18)
where
fnand hndescribe the nominal model of the robot;
Δaand Δhcontain the parametric uncertainties;
ξ=hnζfh(u+ζ)is the vector of lumped
perturbations.
Note that hnis always full rank (Assumption 2), so
the lumped perturbations satisfy the matching condition.
Moreover, we make an additional assumption:
Assumption 3. ξis a bounded as ξ2ξmax, but the bound
ξmax 0is unknown.
Assumption 3 is motivated by the fact that in practice it
can be difﬁcult to estimate the upper bound on ξ, which
could lead to over-conservative gain tuning and consequently
to unnecessary high control actions, chattering and noise
ampliﬁcation.
Finally, we want to stress that we consider the case that
only the dynamic parameters are uncertain, because the
onboard/offboard connection points usually do not change
signiﬁcantly after they are calibrated. On the other hand,
the dynamic parameters may change with every operation
(as speciﬁc hardware is removed or installed onto the end-
effector) or even during the same task (i.e., in pick and place
III. CONTROL
In this section we tackle the problem of letting the end-
effector track a desired trajectory4z1,d =[pT
dνT
d]TSE(3)
that is given together with its derivatives z2,d =[
˙
pT
d˙
νT
d]T
and ˙
z2,d =[
¨
pT
d¨
νT
d]T, and in presence of the lumped
disturbance ξ. We assume that the full state z=[zT
1zT
2]T
is available.5
The tracking controller is designed as a robust law uof
the form
u=uSM +uFF,(19)
where
uSM is a discontinuous term based on a sliding mode
approach;
uFF is a feedforward term based on the dynamic
inversion of the nominal model.
The vector of cable tensions to be applied is then achieved
by inverting the relation u=JTtand then resolving the
redundancy with one of the tension distribution algorithms
known in literature, e.g., [18], [19]. In the remaining of this
section we detail the two terms that compose uin (19).
4The reference trajectory can be expressed as xν,d,˙
xdand ¨
xdand
transformed with the change of coordinates given by (13).
5If the velocity z2is not available then it can be estimated with a robust
sliding mode observer [17].
The sliding mode control term uSM is designed to steer to
zero the tracking errors e=z1z1,d and ˙
e=z2z2,d in
presence of the uncertainties ξ. For this purpose, the sliding
variable is chosen as
σ=˙
ee(20)
where ΛRn×nis a positive deﬁnite diagonal matrix. The
time derivative of σyelds
σ=˙
ee
˙
σ=f(z)˙
z2,d +Λ(z2z2,d)+h(z)u,(21)
showing that σhas relative degree one with respect to u.
To achieve the 2-sliding mode σ=˙
σ=0, we implement
uSM according to the well known Super Twisting controller
(STC) [17], [20]. The expression of the standard STC is
uSM =α|σ|
1
2sign (σ)+v(22)
˙
v=uSM if|uSM |>u(23a)
βsign (σ)if |uSM |≤u(23b)
.
Here, udenotes an upper bound for uSM and α,β
are deﬁnite positive diagonal matrices of gains. The control
law (22) has two remarkable properties, i) it does not require
the knowledge of ˙
σand therefore of the acceleration ˙
z2,
and ii) the discontinuous function sign(σ)is integrated, thus
signiﬁcantly attenuating chattering.
It is proven (see [17]) that the STC controller achieves
ﬁnite-time convergence to the 2-sliding manifold under few
mild assumptions. In particular, it is necessary to choose the
gains αand βhigh enough, according to the upper bound on
ξ. Since the upper bound on ξis not known (Assumption 3)
we adapt the gains online according to the law proposed
in [1], [21],
˙
α=
ωαγ
2sign (|σ|−μ),if α>αm(24a)
η,if ααm(24b)
β=2α(25)
where
ωα,γ,ηare arbitrary positive constants;
αmis an arbitrary small positive constant introduced to
keep the gains positive;
μis a positive parameter that deﬁnes the boundary layer
for the real sliding mode.
Under few mild assumptions [1], the STC with adaptive
gains (24) achieves ﬁnite-time convergence to a real 2-sliding
mode σ≤μ1and σ≤μ2, with μ1μand μ20.
Note also that in (24) the choice of the parameter μis critical,
gains increasing to inﬁnity, or bad accuracy [22]. Here, we
430
Table I: Kinematic and dynamic parameters
Kinematic Parameters
x y z x y z
a1[m]5 5 10 b1[m]-0.5 0.4 0
a2[m]5 5 1 b2[m]0 0.5 -0.4
a3[m]5 -5 10 b3[m]-0.5 -0.4 0
a4[m]5 -5 1 b4[m]0 -0.5 -0.4
a5[m]-5 5 10 b5[m]0 0.5 0.4
a6[m]-5 5 1 b6[m]0.5 0.4 -0
a7[m]-5 -5 10 b7[m]0 -0.5 0.4
a8[m]-5 -5 1 b8[m]0.5 -0.4 0
Dynamic Parameters
mn[kg]50
cn[m]0
EIE,nkgm2diag{[50,50,50]}
m[kg]40
c[m]0
EIEkgm2diag{[40,40,40]}
Table II: ASTC parameters and limits
ASTC Parameters and Limits
ωα1e5αmin [10,10,10,4,4,4]T
γ10.75
tmin [N]10 tmax [N]10.000
choose μaccording to [22] as a time-varying parameter
function.
μ(t)=4α(t)Te,(26)
where Teis the sampling period for the controller.
Finally, we remark that the gain adaptation law (24) not
only does not need the knowledge of the upper bound of
ξ, but it also helps to further reduce chattering of the STC
because the gains αand βare not chosen according to a
worst case uncertainty, but rather they are increased only
when necessary.
B. Feedforward Control
The second term of (19), uFF is the wrench that should be
applied to the nominal model of the robot to follow the refer-
ence trajectory in absence of initial errors. This feedforward
term is included in the control wrench to decrease the action
of the sliding mode control uSM , thus helping reducing the
gains of the ASTC and attenuating chattering. The expression
of uFF is obtained by dynamic inversion of (14), i.e.,
uFF =b1
n(˙
z2,d an).(27)
IV. SIMULATIONS
The proposed strategy has been implemented and tested in
simulation on a redundant CDPR with m=8cables (see
Fig. 1) and modelled according to the formulation in Sec. II.
The ﬁxed positions of the onboard and offboard connection
points, respectively biand aiwith i=1...8, are displayed
Table III: ATSMC parameters as deﬁned in [2].
ATSMC Parameters
K1[3000,2500,3000,150,300,500]T
K2[1200,700,1000,150,300,500]T
β1
γ1.5
ρ0.33
Λ1000 ·16×1,500 ·14×1
0 2 4 6 8 10
2
0
2
4
6
8
time [s]
xyzφθψ
Figure 2: Desired trajectory.
in Tab. I. The nominal values of the parameters mn,cnand
EIE,n of dynamic model (5) and Secs. II-B to II-B are shown
in Tab. I. The actual values of these parameters, i.e., m,c
and EIEare also presented in Tab. I.
The parameters of the controller and of the observer are
shown in Tab. II. The minimal and maximal cables tensions,
tmin and tmax respectively, are also shown in Tab. II.
The results presented in Sec. IV-A show the behaviour
trajectory z1,d (see Fig. 2) which is given together with its
derivatives z2,d and ˙
z2,d.
During the task, unknown bounded external wrenches ζ
(cfr. (17)) are applied to the system at times, as shown in
Fig. 3. Furthermore, a zero mean Gaussian noise is added
to the measurements. The results of the proposed ASTC are
compared to those of the ATSMC proposed in [2], whose
parameters are chosen according to Tab. III.
A. Simulation Results and Discussion
The evolution of the sliding variable σand of its derivative
˙
σduring the execution of the task is shown in Fig. 4. Note
that initially σand ˙
σare almost zero, because the robot is
starting from the correct initial conﬁguration z(0) = zd(0).
Nevertheless the ASTC controller manages to keep the 2-
sliding mode despite the perturbation, as it can be seen from
Figs. 4a and 4b that the spikes that arise when external
wrenches ζare applied (cfr. Fig. 3) rapidly vanish.
In Fig. 5 we compare the performance of i) the proposed
ASTC controller with the feedforward term, and ii) the
ATSMC controller.
431
0 2 4 6 8 10
0.01
0
0.01
0.02
time [s]
e
[
]
x
y
z
φ
θ
ψ
(a) ASTC + FF
0246810
0.01
0
0.01
0.02
time [s]
e
[
]
x
y
z
φ
θ
ψ
(b) ATSMC
0 2 4 6 8 10
0.1
0
0.1
0.2
time[s]
x
y
z
φ
θ
ψ
(c) ASTC + FF
0246810
0.1
0
0.1
0.2
time [s]
x
y
z
φ
θ
ψ
(d) ATSMC
0 2 4 6 8 10
1000
2000
3000
4000
5000
6000
7000
8000
time [s]
t
[
N
]
(e) ASTC + FF
0 2 4 6 8 10
1000
2000
3000
4000
5000
6000
7000
8000
time [s]
t
[
N
]
(f) ATSMC
Figure 5: Comparison: error e( Subﬁgs. a and b ); error ˙
e( Subﬁgs. c and d); cables tension t( Subﬁgs. e and f ).
0246810
100
0
100
200
300
time [s]
ζ
[
N,Nm
]
x
y
z
φ
θ
ψ
Figure 3: External Disturbance ζ.
Before commenting the results we recall that in the
ATSMC [2] the dimension of the boundary layer of the
sliding manifold depends on a ﬁxed gain and on the upper
bound on the disturbance ζ. This creates a tradeoff because
to achieve certain performance the gains must be tuned ac-
cording to the worst case disturbance, yet high gains amplify
noise and therefore cause chattering which could damage the
mechanical parts of the system. In this simulation, the gains
of the ATSMC have been tuned such that the errors as well
as chattering effect can be kept small. Now, let us look at the
results of the comparison in Fig. 5. In particular, we look at
the tracking error e(Figs. 5a and 5b), the error ˙
e(Figs. 5c
and 5d) and the tension tthat the controller requires to the
cables (Figs. 5e and 5f). We clearly see that the ASTC has
smaller tracking errors eand ˙
e, but also less chattering in
432
0 2 4 6 8 10
0.1
0
0.1
time [s]
σ
x
y
z
φ
θ
ψ
(a)
0 2 4 6 8 10
40
20
0
20
40
time [s]
˙σ
x
y
z
φ
θ
ψ
(b)
Figure 4: Sliding variable during the task with the ASTC. a)
σ.b) ˙
σ.
the required cable tensions in comparison to the ATSMC.
Indeed, the ATSMC only adapts the dynamic parameters
that are used in the dynamic inversion (feedforward), but
to counteract external wrenches requires high gains of the
between small tracking errors (in presence to disturbances)
and low chattering. On the other hand, in the ASTC it is
not the dynamic parameters of the model that are adapted,
but rather the gains of the Super Twisting controller. This
allows to have better rejection of the disturbances (not only
of parametric uncertainties) and less chattering by avoiding
to always keep high gains. In particular if we look in Fig. 6 at
the evolution of the gains αin the ASTC (22). Note that the
gains increase only during the transients of the disturbances
(cfr. Fig. 3), but decrease afterwards.
V. CONCLUSIONS
In this paper we have considered the problem of a redundant
cable-driven parallel robot that is tasked to track a trajectory
in presence of uncertainties, external wrenches acting on the
robot and noise on the measurements. The overall dynamic
model of the robot has been reformulated in regular form
to design the controller. It has been implemented a robust
controller based on a Super Twisting architecture with adap-
tive gains. The controller is also extended with a feedforward
0 2 4 6 8 10
0
500
1000
1500
2000
2500
time [s]
α
x
y
z
φ
θ
ψ
dynamic inversion of the nominal model. The main features
of the proposed method are 1) the knowledge of the upper
bound of the perturbations is not needed, 2) chattering is
limited. Simulations show that the controller is effective, even
in comparison to a recently proposed terminal sliding mode
controller for cable-driven parallel robots. In the future, we
plan to continue this work by implementing the controller on
a real robot and running extensive experimental studies.
REFERENCES
[1] Y. B. Shtessel, M. Taleb, and F. Plestan, “A novel adaptive-gain
supertwisting sliding mode controller: Methodology and application,”
vol. 48, no. 5, pp. 759–769, 2012.
[2] G. El-Ghazaly, M. Gouttefarde, and V. Creuze, “Adaptive terminal
sliding mode control of a redundantly-actuated cable-driven parallel
manipulator: CoGiRo,” in Cable-Driven Parallel Robots, ser. Mecha-
nisms and Machine Science, A. Pott and T. Bruckmann, Eds. Springer
International Publishing, 2015, vol. 32, pp. 179–200.
[3] J. Lamaury, M. Gouttefarde, A. Chemori, and P.-E. Herve, “Dual-space
adaptive control of redundantly actuated cable-driven parallel robots,
in 2013, Nov 2013, pp. 4879–4886.
[4] P. Bosscher, R. L. Williams II, L. S. Bryson, and D. Castro-Lacouture,
“Cable-suspended robotic contour crafting system,” Automation in
Construction, vol. 17, no. 1, pp. 45–55, 2007.
[5] L.Gagliardini, S. Caro, , M. Gouttefarde, P. Wenger, and A. Girin, “A
reconﬁgurable cable-driven parallel robot for sandblasting and painting
of large structures,” in Cable-Driven Parallel Robots, ser. Mechanisms
and Machine Science, A. Pott and T. Bruckmann, Eds. Springer
International Publishing, 2015, vol. 32, pp. 275–291.
[6] P. H. Borgstrom, N. P. Borgstrom, M. J. Stealey, B. Jordan,
G. Sukhatme, M. A. Batalin, and W. J. Kaiser, “Design and im-
plementation of NIMS3D, a 3-D cabled robot for actuated sensing
applications,” vol. 25, no. 2, pp. 325–339, April 2009.
[7] B. Duan, Y. Qiu, F. Zhang, and B. Zi, “Analysis and experiment of the
feed cable-suspended structure for super antenna,” in 2008, July 2008,
pp. 329–334.
[8] Spydercam. [Online]. Available: http://dev.spidercam.org/
[9] M. A. Khosravi and H. D. Taghirad, “Experimental performance of
robust pid controller on a planar cable robot,” in Cable-Driven Parallel
Robots, ser. Mechanisms and Machine Science, T. Bruckmann and
A. Pott, Eds. Springer Berlin Heidelberg, 2013, vol. 12, pp. 337–
352.
[10] R. Babaghasabha, M. A. Khosravi, and H. D. Taghirad, “Adaptive con-
trol of KNTU planar cable-driven parallel robot with uncertainties in
dynamic and kinematic parameters,” in Cable-Driven Parallel Robots,
ser. Mechanisms and Machine Science, A. Pott and T. Bruckmann,
Eds. Springer International Publishing, 2015, vol. 32, pp. 145–159.
433
[11] R. Chellal, E. Laroche, L. Cuvillon, and J. Gangloff, “An identiﬁcation
methodology for 6-DoF cable-driven parallel robots parameters appli-
cation to the INCA 6D robot,” in Cable-Driven Parallel Robots, ser.
Mechanisms and Machine Science, T. Bruckmann and A. Pott, Eds.
Springer Berlin Heidelberg, 2013, vol. 12, pp. 301–317.
[12] P. Miermeister, W. Kraus, and A. Pott, “Differential kinematics for
calibration, system investigation, and force based forward kinematics
of cable-driven parallel robots,” in Cable-Driven Parallel Robots, ser.
Mechanisms and Machine Science, T. Bruckmann and A. Pott, Eds.
Springer Berlin Heidelberg, 2013, vol. 12, pp. 319–333.
[13] L. Sciavicco and B. Siciliano, Modelling and Control of Robot Manip-
ulators, 2005.
[14] P. Bosscher and I. Ebert-Uphoff, “Wrench-based analysis of cable-
driven robots,” in 2004, vol. 5, April 2004, pp. 4950–4955.
[15] M. Gouttefarde, D. Daney, and J. Merlet, “Interval-analysis-based
determination of the wrench-feasible workspace of parallel cable-driven
robots,” vol. 27, no. 1, pp. 1–13, Feb 2011.
[16] A. Isidori, Nonlinear Control Systems, 3rd edition, 1995.
[17] Y. Shtessel, C. Edwards, L. Fridman, and A. Levant, Sliding Mode
Control and Observation, 2014.
[18] L. Mikelsons, T. Bruckmann, M. Hiller, and D. Schramm, “A real-time
capable force calculation algorithm for redundant tendon-based parallel
manipulators,” in 2008, May 2008, pp. 3869–3874.
[19] J. Lamaury and M. Gouttefarde, “A tension distribution method with
improved computational efﬁciency,” in Cable-Driven Parallel Robots,
ser. Mechanisms and Machine Science, T. Bruckmann and A. Pott,
Eds. Springer Berlin Heidelberg, 2013, vol. 12, pp. 71–85.
[20] A. Levant, “Sliding order and sliding accuracy in sliding mode control,
vol. 58, no. 6, pp. 1247–1263, 1993.
[21] Y. B. Shtessel, J. A. Moreno, F. Plestan, L. Fridman, and A. S. Poznyak,
“Super-twisting adaptive sliding mode control: A Lyapunov design,” in
2010, Dec 2010, pp. 5109–5113.
[22] F. Plestan, Y. Shtessel, V. Brégeault, and A. Poznyak, “New methodolo-
gies for adaptive sliding mode control,” vol. 83, no. 9, pp. 1907–1919,
2010.
434
... In the present paper we consider the design of a closedloop control for the End Effector (EE) pose for general CDPRs, in the task space. Several works have dealt with this problem but, generally, with the need for a measure of the pose itself [3], [4], [5], [6], [7]. In order to obtain such a measure, in [3] the authors propose a vision-based algorithm which, however, increases overall complexity. ...
... However, this approach is infeasible when the pose is unknown and integration does not ensure convergence, due to initial errors. In [5], [6], [7] the pose of the EE is assumed to be known, without providing any solution to its measure. ...
... 5 The cables are considered as massless and without damping [15]. 6 Reference on Δq i is avoided to simplify the notation By considering the system coordinates ξ = (χ q) T the constraints are expressed as ...
... Additionally, sliding-mode controllers suffer from chattering effects [19] which can reduce the performance of the system and even damage its components. To overcome these limitations we proposed a robust sliding-mode controller with adaptive gains (ASTC) [16], based on [20], that does not require the knowledge of an upper bound of the perturbations. In [16] the controller was successfully validated in a numerical simulation, demonstrating the capability of tracking desired trajectories in operational space (i.e. ...
... To overcome these limitations we proposed a robust sliding-mode controller with adaptive gains (ASTC) [16], based on [20], that does not require the knowledge of an upper bound of the perturbations. In [16] the controller was successfully validated in a numerical simulation, demonstrating the capability of tracking desired trajectories in operational space (i.e. pose of the end-effector) in presence of parameter uncertainties and external disturbances. ...
... Indeed, the results of this paper show that the resilience of the controller to perturbations and the property of finite time convergence are preserved but with worse performance in terms of tracking error. In comparison to [16], another novelty of this paper is the introduction of an adaptive sliding mode differentiator [19] to indirectly derive the velocity of the end-effector using only encoders at the winches and the Forward Kinematics (FK) model [13,18], that is the barest minimum sensor information usually available in CDPR. The results demonstrate also the robustness and finite time convergence of the differentiator. ...
Chapter
In this paper we present preliminary, experimental results of an Adaptive Super-Twisting Sliding-Mode Controller with time-varying gains for redundant Cable-Driven Parallel Robots. The sliding-mode controller is paired with a feed-forward action based on dynamics inversion. An exact sliding-mode differentiator is implemented to retrieve the velocity of the end-effector using only encoder measurements with the properties of finite-time convergence, robustness against perturbations and noise filtering. The platform used to validate the controller is a robot with eight cables and six degrees of freedom powered by $$940\,{\text {W}}$$ compact servo drives. The proposed experiment demonstrates the performance of the controller, finite-time convergence and robustness in tracking a trajectory while subject to external disturbances up to approximately 400% the mass of the end-effector.
... It has been proven that there exists an RBF approximator of Eq. (35) such that it can uniformly approximate a nonlinear and even time-varying function Θ . Using the universal approximation theorem [28], there exists an optimal RBF approximator y * such that [29]: ...
... in which the computation controller is chosen as [35]: ...
... if and only if 2 κ − 1 ρ 2 ≥ 0 or 2ρ 2 ≥ κ [35][36][37]. Therefore, for a prescribed ρ in H ∞ tracking control, in order to guarantee the solvability of H ∞ tracking performance, the weight κ on control law u R of Eq. (59) should satisfy the above inequality. ...
Article
Full-text available
In this study, a robust neuro-adaptive controller for cable-driven parallel robots is proposed. The robust neuroadaptive control systemis comprised of a computation controller and a robust controller. The computation controller containing a neural-network-estimator with radial basis function activator is the principal controller and the robust controller is designed to achieve tracking performance. An on-line tuning method is derived to tune the parameters of the neural network for estimating the controlled system dynamic function. To investigate the effectiveness of the robust adaptive control, the design methodology is applied to control a cable-driven parallel robot. Simulation results demonstrate that the proposed robust adaptive control system can achieve favorable tracking performances for the robot.
... It has been proven that there exists an RBF approximator of Eq. (35) such that it can uniformly approximate a nonlinear and even time-varying function Θ . Using the universal approximation theorem [28], there exists an optimal RBF approximator y * such that [29]: ...
... in which the computation controller is chosen as [35]: ...
... if and only if 2 κ − 1 ρ 2 ≥ 0 or 2ρ 2 ≥ κ [35][36][37]. Therefore, for a prescribed ρ in H ∞ tracking control, in order to guarantee the solvability of H ∞ tracking performance, the weight κ on control law u R of Eq. (59) should satisfy the above inequality. ...
Conference Paper
Full-text available
This paper presents different classical control approaches for planar cable-driven parallel robots. For the proposed robot, PD and PID controllers are designed based on the concept of pole placement method. In order to optimize and tune the controller parameters of planar cable-driven parallel robot, Differential Evaluation, Particle Swarm Optimization and Genetic algorithms are applied as optimization techniques. The simulation results of Genetic algorithm, Particle Swarm Optimization and Differential Evaluation algorithms reveal that the output results of tunes controllers with Particle Swarm Optimization and Differential Evaluation algorithms are more similar than Genetic algorithm and the processing time of Differential Evaluation is less than Genetic algorithm and Particle Swarm Optimization. Moreover, performance of the Particle Swarm Optimization and Differential Evaluation algorithms are better than Genetic algorithm for tuning the controllers parameters.
... proposed an adaptive terminal SMC method to improve the robustness despite the endeffector inertial uncertainties [17]. In [18], trajectory tracking of cable robots in presence of model uncertainties and external disturbances is presented employing an adaptive super twisting controller which adapts the gains. Compared to the controller proposed in [17], this controller produces lower tracking error and less chattering in the control input. ...
... In the papers [182][183][184][185][186][187][188][189][190][191][192][193][194][195][196], adaptive neuro-fuzzy, optimal assembly sequence planning using hybridized immune simulated annealing, restarted simulated annealing, neural networks, WNN approach with RBFNN and adaptive immune based motion planner have been implemented for the motion control of mobile robots. Papers [197][198][199][200][201][202][203][204][205][206][207][208][209][210][211][212][213][214] propose ant colony algorithm, improved ant colony algorithm, firefly algorithm, IWO based adaptive neuro-fuzzy controller, FIS and MANFIS, and robust adaptive sliding mode controller for obstacle avoidance and independent navigation of cellular robots. ...
Preprint
Full-text available
Experimental and simulation analysis of path architecture of mobile robotic platform with the help of Cuckoo-Neuro search algorithm have been carried out in this paper. Inputs in the form of obstacle distances from the sensors and target angle help the robot to decide the final steering angle for movement of the robot from goal position to target position while avoiding obstacles. Results are compared both in simulation and experimental modes and are found to be within average seven percentages.
... This approach was also employed for a cable based transmission system in a surgical robot [13]. Tracking control of the constrained cable robots in presence of noise and system uncertainties was also addressed by Schenk using a robust adaptive sliding mode control [14]. Control of linear systems in presence of measurement and process noise can be accomplished using LQG method. ...
Article
Full-text available
Cable robots are a type of parallel robots where the rigid links are replaced by flexible cables. This flexibility produces internal dynamic which challenges the rigid model based controller. In this paper, the dynamic equations of cable robots with viscoelastic cables are obtained. The Feedback Linearization (FL) method is used to provide a linearized dynamic error for the closed loop model of the system with rigid cables. Using the Lyapunov criterion, the stability analysis of the flexible system with the rigid FL control input is performed. It is shown that considering a minimum damping coefficient and employing the rigid FL controller, the system stability can be guaranteed. In order to achieve a trade-off between the control input and the tracking error, the FL gains are obtained using LQR method. In practice, measurement noise usually exists. On the other hand, the end-effector vibration caused by the cables elasticity can be considered as a process noise. Therefore, the LQG approach is used to estimate the states in presence of the process and measurement noise. Using simulation, it is shown that in presence of measurement noise, the LQG method effectively controls the system while the LQR and also the SMC approach, employed in Korayem et al. (Robotica 33(3), 578–598, 2015), lead to the system instability. Another simulation demonstrates that the system with damping less than the specified minimum value can be stable with the LQG approach, in contrary to the LQR controller. Moreover, in order to investigate the vibrational effect of the cable stiffness and damping coefficient, a frequency analysis is performed. Finally, experimental result obtained by implementation on a manufactured cable robot is presented and verified the approach.
... For these reasons, a large portion of research about CDPR is currently devoted to cables modelling. So far most papers on cable robots have (partially) disregarded cable dynamics, by considering them as massless strings [4], [9]- [11]. However, recently a few authors have proposed better approximations of the cable models based either on the linearization of the nonlinear PDE of motion [12]- [15] or on the discretization of the cable with finite element masses (FEM) [9], [10], [16], [17]. ...
Conference Paper
Full-text available
In this paper we study if approximated linear models are accurate enough to predict the vibrations of a cable of a Cable-Driven Parallel Robot (CDPR) for different pretension levels. In two experiments we investigated the damping of a thick steel cable from the Cablerobot simulator [1] and measured the motion of the cable when a sinusoidal force is applied at one end of the cable. Using this setup and power spectral density analysis we measured the natural frequencies of the cable and compared these results to the frequencies predicted by two linear models: i) the linearization of partial differential equations of motion for a distributed cable, and ii) the discretization of the cable using a finite elements model. This comparison provides remarkable insights into the limits of approximated linear models as well as important properties of vibrating cables used in CDPR.
Article
This paper has proposed a parallel-wire driven robot (PWDR) with an active balancer, which is notably useful for such applications as ceiling maintenance and object conveyance near a ceiling in a factory. Because this robot is an under-actuated system, the uncertainty of the inertial parameters of the load strongly affects the resultant motion and reduces the control accuracy because of the dynamics interference. However, to date, the dynamics of this robot has not been thoroughly elucidated. Thus, this study analyzes the dynamics of a PWDR that controls three degree-of-freedom using two wires and an active balancer. Moreover, based on the dynamic analysis, a model-based adaptive controller for the parameter uncertainty of a load is proposed, and its effectiveness is demonstrated through simulation.
Chapter
Full-text available
This paper addresses the design and implementation of adaptive control on a planar cable-driven parallel robot with uncertainties in dynamic and kinematic parameters. To develop the idea, firstly, adaptation is performed on dynamic parameters and it is shown that the controller is stable despite the kinematic uncertainties. Then, internal force term is linearly separated into a regressor matrix in addition to a kinematic parameter vector that contains estimation error. In the next step to improve the controller performance, adaptation is performed on both the dynamic and kinematic parameters. It is shown that the performance of the proposed controller is improved by correction in the internal forces. The proposed controller not only keeps all cables in tension for the whole workspace of the robot, it is computationally simple and it does not require measurement of the end-effector acceleration as well. Finally, the effectiveness of the proposed control algorithm is examined through some experiments on KNTU planar cable-driven parallel robot and it is shown that the proposed control algorithm is able to provide suitable performance in practice.
Chapter
Full-text available
In this paper dynamic analysis and experimental performance of robust PID control for fully-constrained cable driven robots are studied in detail. Since in this class of manipulators cables should remain in tension for all maneuvers through their whole workspace, feedback control of such robots becomes more challenging than conventional parallel robots. To ensure that all the cables remain in tension, a corrective term is used in the proposed PID control scheme. In design of PID control it is assumed that there exist bounded norm uncertainties in Jacobian matrix and in all dynamics matrices. Then a robust PID controller is proposed to overcome partial knowledge of robot, and to guarantee boundedness of tracking errors. Finally, the effectiveness of the proposed PID algorithm is examined through experiments and it is shown that the proposed control structure is able to provide suitable performance in practice.
Chapter
Full-text available
The research work presented in this paper introduces a Reconfigurable Cable Driven Parallel Robot (RCDPR) to be employed in industrial operations on large structures. Compared to classic Cable-Driven Parallel Robots (CDPR), which have a fixed architecture, RCDPR can modify their geometric parameters to adapt their own characteristics. In this paper, a RCDPR is intended to paint and sandblast a large tubular structure. To reconfigure the CDPR from one side of the structure to another one, one or several cables are disconnected from their current anchor points and moved to new ones. This procedure is repeated until all the sides of the structure are sandblasted and painted. The analysed design procedure aims at defining the positions of the minimum number of anchor points required to complete the task at hand. The robot size is minimized as well.
Conference Paper
Full-text available
This paper presents an extended adaptive control scheme via terminal sliding mode (TSM) for cable-driven parallel manipulators (CDPM). Compared with linear hyperplane-based sliding mode control, TSM is able to guarantee high-precision and robust tracking performances which arise from its main feature of finite-time convergence. This motivates applying TSM to robotic manipulators in general and, as presented in this paper, to CDPM in particular. The scheme presented in this paper extends early developed TSM control schemes which are based on partial knowledge of system dynamics. Instead, making use of the property that the dynamic models of mechanical manipulators are linear in inertial parameters, an adaptive control law is synthesised based on an appropriate choice of Lyapunov function which guarantees finite-time convergence to neighborhood of sliding mode. A key challenge of the control of CDPM is that cable tensions must be admissible, i.e. lying in a non-negative range of admissible values. As long as cable tensions are admissible, the overall dynamics of CDPM can be easily written in either actuator space or operational space which in turn facilitates control system design. The extended adaptive control scheme has been applied to a large redundantly actuated CDPR prototype, CoGiRo. Simulation results show the effectiveness of the proposed control method.
Conference Paper
Full-text available
Cable-driven parallel robots (CDPR) are efficient manipulators able to carry heavy payloads across large workspaces. Therefore, the dynamic parameters such as the mobile platform mass and center of mass location may considerably vary. Without any adaption, the erroneous parametric estimate results in mismatch terms added to the closed-loop system, which may decrease the robot performances. In this paper, we introduce an adaptive dual-space motion control scheme for CDPR. The proposed method aims at increasing the robot tracking performances, while keeping all the cable tensed despite uncertainties and changes in the robot dynamic parameters. Reel-time experimental tests, performed on a large redundantly actuated CDPR prototype, validate the efficiency of the proposed control scheme. These results are compared to those obtained with a non-adaptive dual-space feedforward control scheme.
Article
Full-text available
A novel super-twisting adaptive sliding mode control law is proposed for the control of an electropneumatic actuator. The key-point of the paper is to consider that the bounds of uncertainties and perturbations are not known. Then, the proposed control approach consists in using dynamically adapted control gains that ensure the establishment, in a finite time, of a real second order sliding mode. The important feature of the adaptation algorithm is in non-overestimating the values of the control gains. A formal proof of the finite time convergence of the closed-loop system is derived using the Lyapunov function technique. The efficiency of the controller is evaluated on an experimental set-up.
Book
The sliding mode control methodology has proven effective in dealing with complex dynamical systems affected by disturbances, uncertainties and unmodeled dynamics. Robust control technology based on this methodology has been applied to many real-world problems, especially in the areas of aerospace control, electric power systems, electromechanical systems, and robotics. Sliding Mode Control and Observation represents the first textbook that starts with classical sliding mode control techniques and progresses toward newly developed higher-order sliding mode control and observation algorithms and their applications. The present volume addresses a range of sliding mode control issues, including: *Conventional sliding mode controller and observer design *Second-order sliding mode controllers and differentiators *Frequency domain analysis of conventional and second-order sliding mode controllers *Higher-order sliding mode controllers and differentiators *Higher-order sliding mode observers *Sliding mode disturbance observer based control *Numerous applications, including reusable launch vehicle and satellite formation control, blood glucose regulation, and car steering control are used as case studies Sliding Mode Control and Observation is aimed at graduate students with a basic knowledge of classical control theory and some knowledge of state-space methods and nonlinear systems, while being of interest to a wider audience of graduate students in electrical/mechanical/aerospace engineering and applied mathematics, as well as researchers in electrical, computer, chemical, civil, mechanical, aeronautical, and industrial engineering, applied mathematicians, control engineers, and physicists. Sliding Mode Control and Observation provides the necessary tools for graduate students, researchers and engineers to robustly control complex and uncertain nonlinear dynamical systems. Exercises provided at the end of each chapter make this an ideal text for an advanced course taught in control theory.
Chapter
The practical implementation of sliding mode controllers usually assumes knowledge of all system states. It also typically requires information (at least in terms of the boundaries) about the combined effect of drift terms, i.e., the internal and external disturbances of the system. In this chapter a feedback linearization-like technique is used for obtaining the input–output dynamics and reducing all disturbances to the matched ones. Then the sliding variables are introduced and their dynamics are derived. The higher-order sliding mode differentiator-based observer, which was discussed in Chap. 7, is used to the estimate system states, the derivatives of the sliding variables, as well as the drift terms. Therefore, in finite time, all information about the sliding variable dynamics becomes available. The estimated drift term is then used in the feedback loop to compensate the disturbances. The observed states are then used to design any (continuous) robust state-space controller while eliminating the chattering effect. Two case studies, launch vehicle and satellite formation control, illustrate the discussed robust control technique.
Chapter
This paper proposes a methodology for the identification of the combined kinematic and dynamic parameters of a 6-Degrees of Freedom (6-DoF) Cable-Driven Parallel Robots (CDPRs) model. This methodology aims to ensure that the errors on the kinematic parameters do not affect the performances of the dynamic parameters estimation step. The proposed methodology has been implemented on a 6-DoF INCA robot. The identified model fits the system behaviour with good accuracy, and should then be used for the synthesis and analysis of kinematic and dynamic position / vision control strategies.
Chapter
In this paper the differential kinematics for cable-driven robots is derived and the use for calibration, system investigation and a force based forward kinematics is shown. The Jacobians for each part of the kinematic chain are derived with respect to the platform pose and the most important system parameters. Beside the consideration of geometrical quantities, the differential relations between non-geometrical quantities such as cable stiffness and cable forces are determined. The decomposition in the most fundamental Jacobians allows to analyse and compute more complex relations by reassembling the Jacobians as needed. This approach allows more insight in the system behavior and enables the reuse of the individual modules. The purpose of this paper is to provide the framework and the key equations and to show the use for calibration, force based forward kinematics and system analysis as well as for control purposes.