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

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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.
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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
978-1-4799-8481-7/15/$31.00 ©2015 IEEE 427
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
adopted, if necessary.
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
tasks).
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].
A. Adaptive Super Twisting Control
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,
because a bad choice can lead to either instability and control
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]
z1,d[N,rad]
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
of our system when the end-effector is tasked to follow a
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
[
m, rad
]
x
y
z
φ
θ
ψ
(a) ASTC + FF
0246810
0.01
0
0.01
0.02
time [s]
e
[
m, rad
]
x
y
z
φ
θ
ψ
(b) ATSMC
0 2 4 6 8 10
0.1
0
0.1
0.2
time[s]
˙e[m/s, rad/s]
x
y
z
φ
θ
ψ
(c) ASTC + FF
0246810
0.1
0
0.1
0.2
time [s]
˙e[m/s, rad/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
sliding control, which leads to the aforementioned tradeoff
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
Figure 6: Adaptive gain α
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.
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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.