UNIVERSITY OF CASTILLA-LA MANCHA Escuela T ´ ecnica Superior de Ingenieros Industriales Departamento de Ingenier ´ ıa El ´ ectrica, Electr ´ onica, Autom ´ atica y de Comunicaciones Vibration Control Strategies for a Very Lightweight One Degree-Of-Freedom Flexible Arm Built with Composite Material Ph. D. Thesis Supervisor Vicente Feliu Batlle Author Francisco Ramos de la Flor Ciudad Real, 2009
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UNIVERSITY OF CASTILLA-LA MANCHA
Escuela Tecnica Superior de Ingenieros Industriales
Departamento de Ingenierıa Electrica, Electronica,
Automatica y de Comunicaciones
Vibration Control Strategies for a
Very Lightweight One Degree-Of-Freedom
Flexible Arm Built with Composite Material
Ph. D. Thesis
SupervisorVicente Feliu Batlle
AuthorFrancisco Ramos de la Flor
Ciudad Real, 2009
”If you are out to describe the truth, leave elegance to the tailor.”
Albert Einstein.
Acknowledgements/Agradecimientos
Ha sido un largo camino, sembrado con dificultades pero tambien con numerosas
alegrıas y experiencias de las que completan a una persona y hacen que se sienta
viva. He conocido mucha gente buena (de la mala por suerte no me acuerdo) que me
ha ayudado, apoyado, animado, escuchado, y tantos otros -ados, y a la que quiero
agradecer su tiempo, su esfuerzo y, en la mayor parte de casos, su amistad.
En primer lugar debo agradecer la posibilidad de haber realizado esta Tesis a mi
director, el Profesor Vicente Feliu Batlle, quien me enseno el camino de baldosas amar-
illas de la investigacion cientıfica y me ha guiado por el con una paciencia que roza
lo ilimitado. Del mismo modo, agradezco a la Junta de Comunidades de Castilla-La
Mancha la financiacion recibida en forma de beca predoctoral puesto que sin dicho
apoyo no hubiese podido terminar estos estudios.
Tambien tengo que dar las gracias a mis antes profesores, a los que ahora llamo
companeros e incluso amigos: Luis Sanchez, Pedro Roncero, Jose Andres Somolinos y,
especialmente, a Daniel Cortazar, quien ha intentado hacer de mı un buen docente,
ademas de regalarme valiosısimas ensenanzas acerca de la vida.
Y ahora viene la parte mas extensa, porque he tenido muchos y muy buenos
companeros durante esta etapa y se lo quiero agradecer aunque solo sea con una lınea.
A Rafa, porque fue el primero que recorrio el camino conmigo y despues de tantos
anos se que aun puedo confiar en el, y a Ismael, el hombre que siempre suma, quien
incluso me ha abierto las puertas de su casa cuando me ha hecho falta, no me olvido
de ello. A Fernando, quien me ha hecho sonreır en momentos muy difıciles y siempre
tiene la solucion para el problema, da igual cual sea el problema. A Virginia, que
ha compartido mil y un cafes conmigo aportando ideas, animandome y dandome la
calma cuando me he encontrado mal. A Juanra, quien ha escuchado pacientemente
5
todas mis historias y mis histerias y se ha reıdo con todos mis chistes malos: eso es
amistad. Y especialmente a Emiliano, el de Badaho, mi hermano pacense, que se ha
convertido en una parte muy importante de mı. Nunca antes haba encontrado una
persona tan dispuesta a ayudar y tan sacrificada por los demas salvo mi madre. Y
hay mucha gente mas que lleva tiempo por aquı: Jose Antonio, Elisa, Pedro, Gabi,
Johnny, Ivan, Andres... O que hace menos que llego: Raul, Vıctor Hugo, Xavi, Salva,
Juanmi, Carlos... Gracias a todos. Siento que parezca la guıa de telefonos, pero aun
ası se me olvidara gente. Como se me olvidaba Shigueo, el brasileno mas exotico que
nunca conocere y el peor jugador de futbol, muito obrigado por sua amizade.
I will not miss my stay at University College Dublin, and all the fantastic people
I met there. First of them, my tutor, Dr. William J. O’Connor, who received me
with open hands, a big smile and a handful of ideas to solve the problem I came up
with. He always had time to help me to absorb his wave control ideas: “Engage brain,
Fran”. Also the lads at the office: Johhny, John, Barry and James for the laughs and
for repeating three times each joke so I could laugh with them. I will not forget to
Tang-Wen Yang, good colleague and better person. And finally, to David Joseph, not
nice at all, not even polite some times, but one of my best friends so far. I still keep
the Santa’s hat (without ball). It was great luck to find you, buachaill beag, and yes,
I know: You finished first!
En el aspecto personal, quiero agradecer a mis amigos que hayan aguantado, mejor
o peor, mi mal humor durante algunos periodos, ellos me han ayudado a “desfruncir”
el ceno. A Quiteria quiero agradecerle de corazon que haya sacrificado tanto (quiza
demasiado) por darme fuerzas en los momentos mas delicados para que llegase este
dıa, y pedirle perdon porque le ha tocado vivir lo peor de mı. Y como no, a mi familia,
que me ha apoyado todo el tiempo en las decisiones difıciles o incluso en las erroneas.
Gracias a mis sobrinas por sonreırme tanto y hacerme sonreır tanto a mı.
Esta Tesis ha necesitado muchos agradecimientos y falta el mas importante. A mi
madre, que me dio la vida hace ya muchos anos y no pasa un dıa sin que me la vuelva
a dar. Si el mundo estuviese lleno de personas como ella, serıa infinitamente mejor.
1.2.3 Flexible boom: have your own flexible robot!
The importance and convenience of studying the control of vibrations in flexible
manipulators maneuvers had been discussed and unquestionably demonstrated. The
pioneers of the 80’s had lead the way to construct lighter robots with high performance.
Some classic control techniques had been applied with success to this problem, theo-
retically first, and experimentally in one and two flexible link devices later. However,
the sensitivity to parameters variations was still too great for practical applications
and further work was required in improving the robustness of the regulation. Still, an
effective multi-link solution needed to be found (and, much more complex, to be built),
and many new, modern control schemes could still be implemented.
Since more than a thousand documents were published on these topics during the
90’s, attempting to document all these results is not practical. Hence, only the signi-
ficative advances, in the writer’s opinion, both in control theory and in flexible manip-
ulators will be simply mentioned here.
In (Book, 1993), a review on the elastic behavior of manipulators was meticulously
performed. In his conclusions, Prof. Book remarks that the exponential growth in the
number of publications and also the possibility of corroborating simulation results with
experiments is what turns a flexible arm into a ”...one test case for the evaluation of
control and dynamics algorithms.”. And so it was.
Control theory became one of the platinum clients of flexible robots. The con-
structive easiness and the relatively reduced price of the materials involved in the con-
struction of a real platform (at least a single-link one) caused that many researchers
developed his own manipulator or recreated any of the existent in literature, turn-
ing this equipment into a control theory test bench as foreseen by Prof. Book in his
survey. Thus, during the 90’s a huge number of control schemes were tested on a
flexible manipulator: PD-PID (DeLuca and Siciliano, 1993; Tokhi and Azad, 1996),
8 Introduction
feedforward (Tzes and Yurkovich, 1993; Singhose et al., 1994; Feliu and Rattan, 1999),
adaptive (Feliu et al., 1990; Damaren, 1996; Yang et al., 1997; Apkarian and Adams,
1998), intelligent (Moudgal et al., 1995; Gutierrez et al., 1998; Talebi et al., 1998),
robust (Banavar and Dominic, 1995), strain feedback (Luo, 1993; Ge et al., 1998),
energy-based (Ge et al., 1996), wave-based (O’Connor and Lang, 1998), etc.
But not only control theory advanced during this decade. A number of different
sensorial systems were also tried on experimental platforms: gauges (Luo, 1993), ac-
celerometers (Feliu et al., 1999), cameras (Feliu et al., 1990), piezoelectric (Choi et al.,
1994), optical fiber, etc. Also innovative materials in actuators and/or links were used:
shape memory alloys (Baz et al., 1990; Choi and Cheong, 1996), piezoelectric (Choi
et al., 1994), composites (Choi et al., 1995), electro-rheological fluids(Choi et al., 1996),
etc.
In addition, models for flexible robots were standardized and divided into two main
groups: lumped masses and distributed masses models. Obligatory reads are (Bellezza
et al., 1990), where the differential partial equation of a slewing link is solved in pseudo-
pinned and pseudo-clamped formulations to obtain the resonant frequencies from the
analytical expression, (Feliu et al., 1992) where a simple and efficient lumped masses
model which gives very good results in the case of small link mass (compared to pay-
load) is presented and (DeLuca and Siciliano, 1991) where the closed-form dynamics
equation of multi-link flexible robots are deducted using Lagrangian formulation.
Concerning robotics this period brought some very significative advances. While
most of the research had been performed on single flexible link manipulators, a few
two degrees of freedom flexible robots, double or simple leaned on an air table, had
been built in the past decade. Now the challenge was constructing a three-degrees-
of-freedom (3-dof) flexible robot. On this topic, (Wang and Vidyasagar, 1991a; Wang
and Vidyasagar, 1991b) discussed the dynamics and control of a 3-dof robot with the
last link flexible applying results to a 5-bar-linkage mechanism, while (Yoshikawa et al.,
1990) presented a 3-dof robot with two flexible links which later on would transform,
aiming for precise trajectory tracking control, into a new, interesting concept: the
macro(flexible)/micro(rigid) manipulator (Yoshikawa et al., 1993). Some time later,
a new development was reported in (Fattah et al., 1995) using a parallel manipulator
with flexible links, but results were only presented in simulation. Last but not least, at
1.2. A brief history on flexible robotics 9
the end of the 90’s, the first spanish design of a 3-dof robot with all links flexible was
built and controlled in the Ph.D. Thesis of Prof. Jose Andres Somolinos (Somolinos,
1999) under the supervision of Prof. Vicente Feliu. Modeling and control issues were
documented in subsequent publications (Somolinos et al., 2002; Feliu et al., 2003).
1.2.4 Next generation: the search for new applications
After the huge amount of literature published on this topic during the past century,
flexible robotics was at a stalemate. Some books had been already published on the
subject (Tokhi and Veres, 2002; Wang and Gao, 2003), what indicates that it was a
deeply studied field.
New control laws could still be studied (and they are, actually) due to simplicity
of the physical platform, but, as discussed in (Benosman and LeVey, 2004), most of
the topics on modeling or controllability had been satisfactorily addressed in previous
literature. It is remarkable, however, the appearance of some model-free controls based
in energy considerations (Sanz and Etxebarrıa, 2007) or neural networks (Su and Kho-
rasani, 2001), which lead to generic controls that need to know very little about the
system and still provide a good response in terms of vibration control. In this direction,
wave-based control (Hu, 2005; O’Connor, 2006; O’Connor, 2007; ?) has provided very
good performance since its recent development.
Benosman and LeVey considered some topics still open due to their complexity:
application of closed-loop control strategies to multi-link models; increasing robustness
of the feedforward control schemes; controllability problems on large 3D motions; and
large elastic displacements. Specifically, some effort has been devoted to the study
of geometrical nonlinearities (those due to large elastic displacements of the links) of
flexible robots. In (Belendez et al., 2002) the large displacement topic is investigated on
a very flexible beam, and this study was afterwards applied to the dynamic modeling
(Payo et al., 2005) which is part of the present Thesis. Another different model for
this phenomena is given in (Lee, 2005).
On the other hand, a search for new applications has also concerned researchers.
In (Feliu, 2006), flexibility is considered as a potential benefit instead of a disadvan-
tage, showing some examples of improvement in assembling (Whitney, 1982), collision
(Garcıa et al., 2003), sensors (Ueno et al., 1998) or mobile robots (Kitagawa et al.,
10 Introduction
2002).
1.3 Motivation
The Thesis attempts to solve one of the open topics proposed in (Benosman and
LeVey, 2004). The large elastic displacements that lead to geometrical nonlinearities
have been little studied in the field of flexible robotics. Before this Thesis, very few
works had dealt with this topic (Damaren and Sharf, 1995; Al-Bedoor and Hamdan,
2001; Rixen, 2002), and it has not been thoroughly researched since (Lee, 2005; Abe,
2009).
The obvious question is: are there any benefits in controlling robots undergoing
large displacements? Besides the philosophical, intrinsic joy of describing and solving
a physical phenomena, the problem of large displacements could give us a foundation for
advances in other topics such as human-robot safe (for the human at least) cooperation,
necessary in key engineering fields such as medical or service robotics. In (Zinn et al.,
2004) this topic is approached from a different point of view regarding the actuation
control, but the work also remarks the influence of the robot’s effective inertia (directly
related to its mass) and interface stiffness (material), in terms of an empirical index that
correlates head acceleration to injury severity known as the head injury criteria (HIC).
According to this criteria, the use of lighter materials diminish the harm caused by the
impact because of the smaller inertia, while the deflection of the link provides enough
time to detect this impact and control the manipulator before it causes more damage.
But not only this: the bigger the deflection, the more kinetic energy is transformed
into potential energy of deformation, making the hit less destructive. Apparently there
are only advantages! Obviously, the nonlinearity presents an enormous difficulty on
the control of the system, as usual control proves to be inefficient, and therefore an
innovative approach needs to be taken.
Other advantages of flexible manipulators are the reduction in the amount of energy
needed for driving them. This in addition to its low weight, might lead to the use of
these devices in autonomous mobile robots where power limitations are imposed by
battery autonomy.
1.4. Objectives of the Thesis 11
1.4 Objectives of the Thesis
Hence, the objective of the Thesis is the control of flexible beams constructed with
composite materials. Depending on the length and the section of the beam the behavior
will be linear or nonlinear. Control schemes for both cases are to be applied.
The main contribution of the Thesis will be the control of a long arm exhibiting
big deflections, but the following issues are also addressed:
• Creation of an appropriate non linear model for describing the phenomena.
• Design of smooth trajectories for a better performance.
• Robustness to changes in the parameters of the robot, focusing on changes in the
payload.
1.5 Organization of the manuscript
The manuscript will be divided in seven chapters:
• Chapter 1 presents the state-of-the-art from a chronological point of view and
explains the framework and objectives of the Thesis.
• Chapter 2 describes the lumped masses and distributed masses models used
throughout the Thesis. It also contains the description of an innovative non-
linear model for flexible arms undergoing large displacements.
• Chapter 3 details some concepts of smooth trajectory generation that will be
used throughout the Thesis and discusses the advantages of these references when
performing open-loop control based on dynamic inversion.
• Chapter 4 shows a robust control scheme with application to a conventional
metallic flexible robot.
• Chapter 5 introduces a new adaptive control based in the estimation of the ma-
nipulator payload with application to a composites flexible robot.
12 Introduction
• Chapter 6 presents an innovative control scheme based in wave motions through
lumped masses systems. The application of this method to the geometric nonlin-
ear problem brings to light some precision problems that can be solved applying a
modified scheme, hence achieving the control of the nonlinear single-link system.
• Chapter 7 discusses presented results and proposes some future works on this
topic.
Chapter 2
Dynamic Models for Single-Link
Flexible Arms
The robotic systems with flexible links are continuous dynamic systems character-
ized by an infinite number of vibration modes and are governed by nonlinear, cou-
pled, partial differential equations. The exact solution of such systems is not feasible
practically and infinite dimensional models impose severe constraints on the design of
controllers as well. Hence, they are discretized using assumed modes, finite elements
or lumped parameter methods.
This chapter presents a selection of different mathematical models which have been
applied to describing flexible arms in previous work. These have been chosen on the
basis of embracing all the models used throughout the Thesis. For the interested
reader, a comprehensive survey on the topic has been recently carried out (Dwivedy
and Eberhard, 2006), with an extensive bibliographic search including more than four
hundred papers discussing this subject.
2.1 Generic description
In general, there are three basic components of a single-link flexible arm:
• A payload/tool which is the target to be moved from one place to another.
• This is attached by a joint, articulated or not, to a light beam/bar/string/link
14 Dynamic Models for Single-Link Flexible Arms
Actuator
Flexible beam
Payload
Figure 2.1: Scheme of a single dof flexible robot arm.
that exhibits some flexibility due to its shape and reduced weight.
• Finally, the beam is driven by some kind of actuator, typically a DC motor, that
provides the motion of the arm.
A fourth item could be added to previous parts when considering controlled motion
(which will be in most cases): a sensorial system is also required in order to monitor the
maneuver and allow corrections of any deviations from desired behavior. An outline of
this scheme is displayed in Figure 2.1.
Even if there are mathematical models for robots with an arbitrary number of
degrees of freedom, single dof arms are most commonly used in the literature and most
flexible platforms existent in the world are of this kind. Hence, this will be the case
used to study the different control strategies proposed in the Thesis.
In addition for simplicity’s sake, gravity effects will be neglected throughout the
Thesis. This can be achieved by performing the movements in an horizontal plane,
either by using a beam which is very stiff in the direction gravity is acting while flexible
in perpendicular directions, or using the support of an horizontal air table where the
payload could slide with no friction, as will be shown in the real platforms description.
2.2. Actuator model 15
The physical drawback of using light beams for robotics is very simple: beam
flexibility produces a deflection in the link which causes a misalignment between motor
angle, θm, and tip angle, θt. However, the modeling and control of this kind of robotic
systems is not that easy, and therefore, a number of different mathematical templates
have been used to describe this phenomenon. Some of them have been chosen because
of their simplicity or completeness in idealizing the actual platforms. They will be
presented subsequently.
The model used for the actuator is presented in Section 2.2, as it is common to
all of the platforms described afterwards. Subsequently, some classical approaches for
modeling flexibility in beams are detailed. First, the distributed masses equivalent
model is presented in Section 2.3, jointly with the procedure to obtain the transfer
functions of the system truncated to a finite number of modes of vibration. Next,
Section 2.4 presents a concentrated masses model for describing the beam+payload
system, concentrating on two cases: negligible beam mass and beam mass concentrated
at its middle point. To finish, Section 2.5 introduces a novel model for flexible systems
subject to strong geometric non-linearities which is based on a duffing-like equation
(Thompson and Stewart, 2002) with the addition of variant coefficients.
These models lie beneath the real platforms that are to be detailed next. All used
platforms are listed and detailed in Section 2.6. Namely, an old-generation duralu-
minium arm, which is driven by a DC-motor and allows a payload of up to 5 kg,
and a much lighter and safer new-generation composites arm, also DC-motor driven
that allows movement up to 200g. In the same Section, the sensorial system used to
instrument the platforms is also described.
2.2 Actuator model
As mentioned in previous Section, a mathematical definition for the actuator must
be provided for any complete model of a flexible arm. It is and is common and necessary
for all models.
The actuator chosen to drive the links in all platforms is a DC motor which has a
reduction gear with a reduction relation nr. That increases the strength of the system
while diminishing its velocity. Taking into account that the maximum angular velocity
16 Dynamic Models for Single-Link Flexible Arms
of a DC motor is usually much higher than needed, this gear-box reduces the size of
the selected motor for driving a specific payload while keeping a sufficient maximum
output velocity. It also reduces the effect of the beam-payload coupled inertia in the
motor. Note that the magnitudes seen from the motor side of the gear will be written
with an upper hat, while the magnitudes seen from the link side will be denoted by
standard letters.
The dynamics of the motor with a closed loop current control system (where the
voltage Vm is assumed to be proportional to the current) is given by
Γm = KmVm = Jm¨θm + ν
˙θm + Γcoup + ΓCoul (2.1)
where · denotes differentiation with respect to time, Γm is the torque produced by the
motor, Km is a motor constant, Vm is the voltage signal that controls the motor, Jm
is the motor inertia, ν is the viscous friction coefficient, Γcoup is the coupling torque
between the motor and the link, and ΓCoul is the Coulomb friction. The conversion
equations between magnitudes, valid for any angle or torque involved in the actuator
model equations, are given by
θ = nrθ Γ =Γ
nr
(2.2)
The last term of equation (2.1) is assumed to be negligible or compensated, as
shown in (Feliu et al., 1993), with a compensation term of the form
VCoul =ΓCoul
Km
sign(˙θm
)(2.3)
where ΓCoul is an estimation of the Coulomb friction value. On the other hand, as-
suming that the coupling torque can be measured or estimated from measurements,
another compensation term of the form
Vcoup =Γcoup
Km
=Γcoup
Kmnr
(2.4)
that counteracts the effect of the arm inertia and decouples motor and link dynamics,
can be added to the control signal, as shown in Figure 2.2, yielding the model to
2.2. Actuator model 17
V
coup
Coul !
mV
" #$%sJsK
m
m
m&!
m&%
%%rmnK
'
rn
'
mK
' CoulV
coupV
Figure 2.2: Model for DC motor actuator.
KmV = Jm¨θm + ν
˙θm (2.5)
where V = Vm − VCoul − Vcoup. Expression (2.5) can be transformed into the Laplace
domain to obtain the transfer function between control signal, V , and the motor angle,
θm, resulting inθm(s)
V (s)=
Km
s (Jms+ ν)(2.6)
Finally, adding the conversion equations due to the reduction gear (2.2), the sim-
plified, decoupled dynamic model for the actuator is
Gm(s) =θm(s)
V (s)=
Km/nr
s (Jms+ ν)(2.7)
which is a typical second order system with a pole in the origin. This kind of systems
can be easily regulated with a simple PD controller, as will be shown in Section 3.2.1.
In the case of motors with reduction gears of large value nr, the compensation
term can be removed without producing significant changes in the closed loop motor
dynamics as its effects are divided by nr, the amplitude of the control action generated
by it thus being very small, as will be demonstrated in Section 4.6.
18 Dynamic Models for Single-Link Flexible Arms
2.3 Distributed masses model
These models consider flexible links as a continuum. They are calculated as the
solution of the partial differential equation that characterizes the system which can be
obtained e.g. applying variable separation and modal expansion.
Modal expansion method has been widely used in previous work (Nicosia et al.,
1986; Bellezza et al., 1990; Boyer and Coiffet, 1996) for modeling flexible manipula-
tors. Starting from the Euler-Bernouilli equation, and assuming the link possesses an
infinite number of natural frequencies, we obtain a truncated model with n modes of
vibration. These are usually the lowest frequency modes as they are the most signifi-
cant, (biggest amplitudes), to the system dynamics. Once the modes are known, the
link displacements are presented as the product of two terms, a spacial term (modal
functions, ϕi(x)), and a temporal term (generalized coordinates, qi(t)) as expressed in
following equation:
w(x, t) =∑i
ϕi(x)qi(t) (2.8)
The modal functions must fulfill three conditions:
• They must constitute a complete coordinate base, that is, this set of functions
must be able to express the displacement of any point of the link.
• The functions must satisfy the geometric boundary conditions.
• They must be differentiable over the defined domain, at least up to the degree of
the differential equation that rules the model.
In addition, modal functions fulfill some ortogonality conditions which lead to some
simplifications in the model (Clough and Penzien, 1993). On its part, generalized
coordinates compose a set of time-dependent parameters which are independent among
them.
Once modal functions are calculated, there exist two ways to address the dynamic
modeling of the system: by means of Lagrange Equations (Book, 1984; DeLuca and
Siciliano, 1991) or through Newton-Euler Equations (Rakhsha and Goldenberg, 1986;
Boyer and Coiffet, 1996). In this section, the first method is presented using the
2.3. Distributed masses model 19
example to obtain a model with three modes of vibration. We will assume that the
manipulator consists of a distributed mass link with a point mass attached to its end
and whose movement is restricted to an horizontal plane. In addition, the pseudo-
pinned formulation will be adopted for solving the Euler-Bernouilli equation, that is,
the x-axis of the rotary frame, X−Y, intersects the center of mass of the arm as shown
in Figure 2.3 (Bellezza et al., 1990). Following hypotheses are adopted:
• The material of the link is continuous, uniform, homogeneous and isotropic.
• Small transversal displacements.
• Navier hypothesis are assumed to be valid, that is, flat sections remain flat after
deformation.
• Torsional effects are negligible.
Next the dynamic model will be obtained.
2.3.1 Solution of the Euler-Bernouilli Equation
Taking into account the previously adopted hypothesis and the Hamilton’s princi-
ple (Meirovitch, 1997), the behavior of an Euler-Bernouilli beam is governed by the
following fourth order partial differential equation
EIz∂4w(x, t)
∂x4+ ρL
∂2w(x, t)
∂t2= fd(x, t) (2.9)
where w(x, t) is the deflection at point x of the beam, and fd(x, t) is the external
distributed force.
Defining the position of a point x of the link with respect to the fixed frame as
y(t, x) = θ(t)x+ w(x, t) (2.10)
and applying expression (2.9) to the case of a flexible arm under the effect of an
input torque, Γm, due to motor action, the following boundary value problem can be
formulated
EIzwiv(x, t) + ρLy(x, t) = 0 (2.11)
20 Dynamic Models for Single-Link Flexible Arms
Y0
X0
!"
E, I, l, !L
#$%
Jm, nr, Km
mp, Jp
Y
X
w &'!"
y &'!"
Figure 2.3: Scheme of a single dof flexible arm with distributed link mass
Γm(t) = Jmy′(0, t) + ρL
∫ l
0
xy(x, t)dx+mpy′(l, t) + Jpy
′(l, t) (2.12)
with the following boundary conditions
w(0, t) = 0 (2.13a)
EIzw′′(0, t) = Jmy
′(0, t)− Γm (2.13b)
EIzw′′(l, t) = −Jpy
′(l, t) (2.13c)
EIzw′′′(l, t) = mpy(l, t) (2.13d)
These sort of partial differential equations with boundary conditions (2.11)-(2.13)
can be solved using variable separation with an expression of the form given in (2.8),
usually truncated to the first n vibrational modes. The general solution is of the form
(Bellezza et al., 1990)
ϕi(x) = A sin(λx) + B cos(λx) + C sinh(λx) + D cosh(λx) + F x (2.14)
qi(t) = K1 sin(ωt) +K2 cos(ωt) (2.15)
2.3. Distributed masses model 21
being λ4 = ω2mEIz
. The values of K1 and K2 depend on the initial conditions of position
and velocity: K1 = q(0)/ω and K2 = q(0). On the other hand, A, B, C, D and F are
obtained from the boundary conditions of Eq. (2.14). They are
y(t, 0) = 0 (2.16a)
EIzy′′(t, 0) + Γm − Jmθ = 0 (2.16b)
EIzy′′(t, l) + Jey
′(t, l) = 0 (2.16c)
EIzy′′′(t, l)−my(t, l) = 0. (2.16d)
2.3.2 System model in space-state form
Rearranging previous expressions (Feliu, 1997), the solution to the Euler-Bernouilli
equation in a space-state form is
Xs = AXs +BΓm
Ys = CXs +DΓm
(2.17)
where the state variables are the generalized coordinates qi and the matrix dimensions
depend on the number of vibration modes considered. Taking n modes into account,
the state vector is Xs = [q0 . . . qn q0 . . . qn]T , T denoting matrix transpose, and the
space-state matrixes are
A =
[0(n+1)×(n+1) In+1
Ω(n+1)×(n+1) 0(n+1)×(n+1)
]B =
1
J
0(n+1)×1
1
ϕ′1(0)...
ϕ′n(0)
(2.18)
where 0p×n is a matrix of p rows and n columns of zeros, In is the identity matrix of
dimension n, and Ω(n+1)×(n+1) is a matrix defined as follows
22 Dynamic Models for Single-Link Flexible Arms
Ω(n+1)×(n+1) =
0 0 · · · 0
0 ω21 · · · 0
......
. . ....
0 0 · · · ω2n
(2.19)
Matrixes C and D depend on the desired output. Specifically, we will consider the
motor angular position, the payload angular position and the coupling torque of the
system, that is Ys = [θm θt Γcoup]T . To calculate this outputs, the values for C and D
are
C =
1 ϕ′1(0) · · · ϕ′
n(0) 0 · · · 0
1 ϕ′1(l) · · · ϕ′
n(l) 0 · · · 0
0 −EIϕ′′1(l) · · · −EIϕ′′
n(l) 0 · · · 0
D =
000
(2.20)
The dimensions of the matrixes are A ∈ ℜ(2n+2)×(2n+2), B ∈ ℜ(2n+2)×1, C ∈ℜp×(2n+2) and D ∈ ℜp×1, the state vector is Xs ∈ ℜ(2n+2)×1 and outputs vector is
Ys ∈ ℜp×1.
2.4 Concentrated masses model
In this case, the system we want to model consists of a flexible link whose mass
is assumed to be concentrated in a certain number of points along itself (Feliu et al.,
1991). The number of vibration modes in the structure match the number of point
masses along the beam. Advantages of this method are
• Dynamics are simpler to model in comparison with distributed mass arms.
• Payload changes are straightaway applied to the model.
• Since the distributed mass model of a flexible link is usually truncated to a finite
number of vibration modes, we can find a concentrated masses model with similar
characteristics.
These models are useful when the beam mass is not very significant but cannot be
neglected.
2.4. Concentrated masses model 23
m1
,-
./
.
Y
X
Y0
X0
0$
, 1
$
21/
31
31)!31)4
$
,15!/
,1
,1)!
m2
Figure 2.4: Outline of a link represented by a finite number of point masses.
2.4.1 General model for an arbitrary number of masses
Let us consider the system in Figure 2.4, which represents a flexible beam, of length
l, whose total mass, mb, is assumed to be divided into n point masses distributed along
the link, the last of them being the payload. Coefficients mi represent the value of the
i− th mass, li is the distance between masses i and i− 1 being l1 the distance between
motor shaft and first mass and Li is the distance between mass mi and motor shaft.
Let us assume that arm displacements for i − th element of the link, yi(x), are very
small, so that distances between consecutive masses (measured along the beam length)
are equal to their projection on the X axis. The angle rotated by the motor shaft is
θm, while the angle rotated by any mass mi is θi.
The picture displays two coordinate frames with the origin at the motor: frame
X0-Y0 is fixed while frame X-Y rotates jointly with the motor angle. Hence, y(x) is
the distance between any point of the link, x, and the X axis.
Let us also define F (x) and Γ(x) as the force and the torque, respectively, at point
x. For generality, another force Ft and another torque Γt are exerted on the link tip as
the resultant of the external forces and torques. These effects can be produced either
for many reasons: the joint to another link, the reaction forces of pushing a surface,
24 Dynamic Models for Single-Link Flexible Arms
or, typically, a load placed at the tip, among others.
From the general equation of Euler-Bernouilli given in (2.11), and neglecting the
mass of the structure, the equation describing dynamics of each element is
EIzd4y
dx4= 0 (2.21)
where E denotes the Young’s modulus and Iz the cross section inertia. If we assume
that both values are constant along the beam and then we integrate the equation,
completing the model’s equations. Equations (2.40) and (2.37) of the massless link
model shown in Section 2.4.2 can be easily deduced from (2.48) and (2.52) by calcu-
lating the limit of these latter expressions when Jp and m1 tend to zero.
2.5 Nonlinear model for a very flexible manipulator
with geometric non linearities
The previously derived models are adequate to describe links exhibiting a moderate
degree of flexibility, assuming the small displacements hypothesis is correct. But when
32 Dynamic Models for Single-Link Flexible Arms
the link presents large tip displacements, some of assumptions adopted in the resolution
of the Euler-Bernouilli Equation are no longer valid.
In previous literature, some work has also been devoted to model large elastic dis-
placements. Based on the Euler-Bernouilli equation, numerical algorithms have been
proposed to estimate the curvature and tip deflection of a static elastic beam. These
methods rely on the solution of complicated integral equations by numerical meth-
ods (Wang, 1981; Belendez et al., 2002), numerical solution of nonlinear differential
equations ((Lee, 2002) for fixed cross sections and nonlinear elastic materials and (Lee
et al., 1993) for variable cross section), or the use of sensors (strain gauges) that mea-
sure the curvatures at certain points of the beam followed by a polynomial interpolation
(Gu and Piedboeuf, 2003).
Some of the modeling techniques mentioned above give precise descriptions of the
geometrical nonlinear dynamics of the flexible beam. These models are based on nu-
merical approximations of differential or integral equations that have to be solved at
every considered time, or they are represented by means of complicated analytical dif-
ferential equations. They are well suited for numerical simulations or for calculating
command profiles (usually motor torques) to be applied in an open loop fashion to the
arm in order to follow a desired tip trajectory. But these models can hardly be applied
to analyze and design nonlinear closed loop control systems for these arms. At most,
linearized models of local validity around the desired trajectory can be derived from
the previous models, which may lead to local linear controllers that need to be updated
in real time in response to the state of the arm.
However, we are interested in simple nonlinear dynamic models, that capture the
most important dynamics of the arm while remaining useful for the design of arm tip
position nonlinear controllers. That is, the desired nonlinear model should still remain
computationally light, allowing it to be applied to real platforms.
For these reasons, a new model with these features is presented (Payo et al., 2005).
A general model is attained from the Euler-Bernouilli beam equation and then the
equations are particularized for a concentrated masses model with a single point mass
at the tip which is subject to a force. These equations are applied to the linear sys-
tem, obtaining an equivalent expression to that of Section 2.4.2, and to the non-linear
system, which yields a non-linear expression computationally light and much more rep-
2.5. Nonlinear model for a very flexible manipulator with geometric nonlinearities 33
resentative of the actual behavior of the real platform, as is demonstrated latter by
means of latter experimentation.
2.5.1 On the Euler-Bernouilli beam
A flexible beam exhibits a nonlinear behavior when it is under the effects of large
forces (Landau and Lifshitz, 1969). Assuming that our composite material remains
linearly elastic and there is not a change of status, this nonlinear behavior is due to
a change in the geometric layout of the bar (geometric nonlinearities) due to bending
and/or torsional moments. Under the assumption that torsional effects are negligible,
the equation of Euler-Bernoulli for large deflections is
dθpds
=d2y′
dx′2
(1 + ( dy′
dx′ )2)32
=Mz
EIz(2.53)
where θp is the orientation of the beam at point p ′, and s is the arc length over the
beam, x′ and y′ are the coordinates of a point of the beam expressed in a cartesian
frame, X−Y, according to Figure 2.7, Mz is the bending moment on any section of
the beam and EIz is the stiffness of the bar. For small deflections s is equal to x′,
and the rotation angle θp can be approximated by dy′/dx′ yielding that dθp/ds can be
approximated by d2y′/dx′2. Then, equation of Euler-Bernoulli (2.9) simplifies to the
well-known small displacements equation used for previous models (2.21).
However, for large deflections these simplifications are not valid and it is necessary
to use the above equation (2.53) in its complete form. The solution of this equation
can be obtained by calculating some elliptic integrals, which can be evaluated using
numerical methods (Wang, 1981; Belendez et al., 2002). Equation (2.53) does not have
an analytical solution.
Let us consider Figure 2.7, where a force F is applied at the tip of the beam. This
force produces a torque along the beam which is equal to the bending moment Mz. Let
us denote the magnitude of this force as F and its orientation as φ. Then, the static
(kinematic) equation of the beam is obtained from (Landau and Lifshitz, 1969)
d2θpds2
=F
EIzsin(θp − φ) 0 ≤ s ≤ l (2.54)
34 Dynamic Models for Single-Link Flexible Arms
Y
X
!"#
$%& ' !(&) *&#
F
+
,"
l
$%&-' !(&- ) *
&-#
Figure 2.7: Large deflection of a one side clamped beam
where l is the beam length.
The solution of this differential equation is based on an elliptical integral and its
numerical solution leads to a generic expression, (particularized at the tip position pt),
of the form
F = Φ (pt′, l, EIz) = Φ (ρ′t, θ
′t, l, EIz) (2.55)
where ρ′t and θ′t are the polar coordinates of the tip position in the frame X−Y.
Defining a normalized force Fn =Fl2
EIzand a normalized arc length sn =
s
land
substituting into (2.54) yields
d2θpds2n
= Fn sin (θp − φ) 0 ≤ sn ≤ l (2.56)
which is a normalized equation independent of the geometric dimensions of the bar and
its elasticity coefficient E. This equation has a general solution of the form
Fn = Φn (ρ′tn, θ
′t) (2.57)
being ρ′tn =ρ′tl.
2.5. Nonlinear model for a very flexible manipulator with geometric nonlinearities 35
2.5.2 General model
Let us consider the model of Fig. 2.8, where the frame X0 −Y0 is fixed and frame
X−Y is aligned with the beam base (it rotates with motor angle). Assuming we have
a massless link, that is, all the system mass is concentrated at the tip, and that payload
can be considered as a point mass (with zero rotational inertia and, consequently, no
torque at the tip), the dynamic model for a very flexible nonlinear arm can be expressed
by the following general equation:
−Φ(ρt(t), θt(t)− θm(t), l, EIz) = md2pt(t)
dt2(2.58)
obtained from (2.55) by simply taking into account that ρ′t = ρt, θ′t = θt − θm and
F = −md2ptdt2
, where ρt is the tip radius, θt and θm are the tip and the motor angles
respectively, m denotes the tip mass and pt the tip position, all of them expressed in
the cartesian frame X0 −Y0 of Figure 2.8. If we expand the acceleration in polar
coordinates, the expressions obtained are
ptx = ρt cos θt − 2ρt sin θtθt − ρt cos θtθ2t − ρt sin θtθt (2.59a)
pty = ρt sin θt + 2ρt cos θtθt − ρt sin θtθ2t + ρt cos θtθt (2.59b)
By substituting these equations into (2.58) we obtain the final expression of this model.
Equation (2.58) allows a subsequent normalization making tn = t/√m. Then
−Φn(ρtn, θt − θm) =d2ptndt2n
(2.60)
where ptn = pt/l.
2.5.3 Linear model
To obtain the linear model, which was previously derived from the dynamic simpli-
fied equations in Section 2.4.2, from equation (2.58), the assumption that ρt = l ∀t isadopted. Using polar coordinates to represent the dynamics, only the angle is variable
and Φn results in an scalar function of the form
36 Dynamic Models for Single-Link Flexible Arms
Y
X
! " #$
%&'!(%)&
*+
'!( , -.( / 0(1Y0
X0
232+
435
Figure 2.8: Dynamic deflection model
Φl =3EIzl
(θt − θm) (2.61)
Using the expression for natural frequency in (2.41) and taking into account that
ρt = l, combining expressions (2.58) and (2.59) yields to
θt + ω20 (θt − θm) = 0 (2.62)
which is a analogous expression to (2.39). If we wish to take into account the friction
in the payload and the internal energy dissipation of the beam, a friction term, ξlθt,
that depends on velocity can be added for completion. Therefore, the final expression
for the dynamics is
θt + ξlθt + ω20 (θt − θm) = 0 (2.63)
2.5.4 Non linear model
As long as our arm exhibits large deformations, the tip radius can not be considered
constant, and the linear deflection model (2.61) no longer holds true. Next we propose
a new nonlinear model that approximates these effects.
2.5. Nonlinear model for a very flexible manipulator with geometric nonlinearities 37
Nonlinear model for the tip angle
We assume here the equation typically used to model the stiffness of nonlinear
springs
Φnl = αθ′t + βθ′3t (2.64)
In this case, the resulting dynamic equation of the beam is
Experimentally identified parameters for the nonlinear model
α β ξ γ8.7907 0.5139 0.0458 0.4462
Table 2.6: Data of the glass fiber composites link and its payload
The payload can be exchanged using the same mechanism described previously in
Section 2.6.2. The variation of the tip mass attached to this slender glass fiber link
ranges from 43.76 g to 101 g. Table 2.6 details the values of its physical parameters and
the values obtained through identification for the parameters of the nonlinear model
described in Section 2.5.4.
2.7 Note on the software used in the Thesis
Throughout the Thesis, there are a number of simulations for the mathematical
models and/or control schemes designed/used within chapters 3 to 6. All of them have
been carried out using the numerical methods package MATLAB r and the toolboxes:
Control System ToolboxTM and Simulinkr. Some different versions of this software have
been used in the preparation of different simulations included in the Thesis, starting
from MATLAB 6.1 to MATLAB 7.6, but, for homogeneity, every simulation included
in this Thesis has been run under the latest version of them, that is MATLAB 7.6.
46 Dynamic Models for Single-Link Flexible Arms
Chapter 3
Open loop control based on system
inversion
This Chapter presents a simple, low computational cost scheme for open-loop tra-
jectory tracking control of single link flexible manipulators. The control is performed
by direct dynamic inversion of the mathematical model of the system. This inversion is
non-causal, therefore it is necessary to calculate the trajectory beforehand in an off-line
stage. In addition, the trajectory needs to be differentiable the same number of times
as the order of the differential equation utilized for modeling.
Due to the particular characteristics of the trajectory needed for inversion, a family
of polynomial trajectories is defined in this Chapter. They depend on a small number of
parameters which are particular to the specifications (maneuver time, angle described)
or dependent on the physical limitations of the platform (acceleration and snap). They
also provide objective criteria for determining the parameters of the motor controller.
3.1 Problem description
The vibrations of flexible systems when they are performing movements are tightly
related with the reference signals that we provide to these structures. That is, if we
set a reference that is extremely demanding (e.g. a linear reference requiring a high
speed in a very short time or even a step input) for a mechanical system it results in
a motion exhibiting, almost surely, high overshoot and, most probably, some vibration
48 Open loop control based on system inversion
until we reach the target position of the system. Why does it happen?
Any text book in basic mechanics teaches that any real mechanical structure has,
as was exposed in the modeling Chapter, infinite vibration modes and, equally, infinite
vibration frequencies. Whenever a reference maneuver excites any of these frequencies,
a vibration appears in our system compromising its steadiness. Fortunately, we do
not need to take care about an infinite number of frequencies, as most of the high fre-
quencies vibrations are either negligible in amplitude or quickly damped by structural
properties. We just need to cancel/control/eliminate those frequencies too stubborn to
disappear naturally, that is, the lowest natural frequencies of the structure. Typically,
and depending on its nature, a flexible link (of the kind used in flexible robotics) can
exhibit from one up to three significant modes of vibration. Actually, the platforms
presented in this Thesis can be rigourously modeled with a single natural frequency,
and the target model for which the inputs are to be designed is the single mass lumped
model presented in Section 2.4.2.
3.1.1 Open loop control approach
Between the number of different control schemes that have been previously applied
to cancel the vibrations in flexible robotic arms, considerable attention has been de-
voted to the open loop control of this kind of systems. There exist two main approaches:
based on the inversion of the flexible arm dynamics (Bayo, 1988; Feliu and Rattan,
1999; Piazzi and Visioli, 2000) or on the adjustment of the input reference (Singer and
Seering, 1990; Pao, 1999; Zanasi and Morselli, 2002; Mohamed et al., 2006).
The first consists on passing the desired reference through a inverted system model
and obtaining and equivalent reference that, when is applied to the system, reverts (in
the output) to the initial desired reference. While conceptually simple, this scheme
needs a complete model of the system to be inverted and usually has any or various
of the following undesirable effects: it takes a considerable amount of resources if the
model is complex; or must be performed previously if the inversion is non causal (Piazzi
and Visioli, 2000); or needs of an undetermined time of computation if it follows an
iterative process (Bayo, 1988). The latter control strategy, often called input shaping,
is a technique based on generating adequate input references that inherently reduce
the oscillation of a flexible structure without needing a feedback control or needing it
3.1. Problem description 49
only for refinement. The main drawback of this method is that the command signal is
lengthened by an amount of time equal to the duration of the input shaper. Actually,
both apparently different methods are two sides of the same theory as shown in (Feliu
and Rattan, 1999).
Lately, a new, different approach, which follows an intuitive, simple, but unconven-
tional idea, called wave-echo control, has also been developed (O’Connor, 2006). This
strategy sets the first half of the maneuver to a predefined trajectory and measures
the actual motion performed by the manipulator. Then, the controller adjusts the
input reference for the second half to describe an ”echo” of that movement, which, to
some extent, includes the dynamics of the manipulator. While this method is model
free, inherently robust and achieves vibrationless point-to-point motion, the reference
tracking problem has not been addressed yet.
In present Chapter, a non-causal, dynamic inversion approach has been adopted.
The stability of a system inversion has been investigated since the 60’s, mainly in linear
multi-input multi-output (MIMO) systems (Silverman, 1969), or, in recent years, in
nonlinear systems also (Devasia and Paden, 1994; Marro and Piazzi, 1996). A relevant
concept that appears when dealing with invertibility of a system is the functional
reproducibility, meaning the capacity of reproducing an output from a suitable input
function, introduced in (Brockett and Mesarovic, 1965). This property has been the
subject to many studies when the construction of noncausal inverse is carried out in
presence of unstable zero dynamics (Marro and Piazzi, 1996). For the model adopted in
this Chapter, functional reproducibility is guaranteed because the model of the system
dynamics is scalar and minimum phase.
3.1.2 Influence of trajectories
The selection of an appropriate reference is crucial and has been studied in literature
a long time since (Bayo and Paden, 1987). The more demanding a trajectory is, the
wider is the range of frequencies that are excited when it is introduced to the system.
As an example, the step reference excites all vibration frequencies from zero to infinite.
Therefore, if a smooth (high continuity and differentiability) reference is provided to
the manipulator, the necessary control effort to canceling vibrations diminishes, as the
residual vibration remaining in the system is smaller.
50 Open loop control based on system inversion
This chapter presents a kind of smooth reference trajectories that allows the use
of an open-loop control based on the inversion of the system dynamics. They also
take into account some physical limitations of our electro-mechanical system, as the
saturation of the motor control signal, which could lead to undesired vibrations even
when we are using appropriate references.
3.2 System inversion based control scheme
The general control scheme is shown in Figure 3.1, and consists simply of three
consecutive open-chain blocks. Namely, F (s) represents the dynamic inversion of the
system, M(s) is an equivalent transfer function of an actuator control loop, and G(s)
denotes the beam dynamics/model. The selected model for this experience corresponds
to the lumped masses model considering a single point mass at the tip and negligible
link mass that was presented in Section 2.4.2. Therefore, link dynamics is given by
G(s) = Gt(s) =ω20
s2 + ω20
(3.1)
M(s)t m m t
G(s)F(s)
Dynamic
Inversion
Controlled
Motor
Beam
dynamicsr rθ θ θ θ
Figure 3.1: Block diagram of the dynamic inversion
3.2.1 Actuator control scheme
As described in Section 2.2, a DC motor is used for the actuation of the various
platforms used in the Thesis. Due to the pole at the origin, a PD controller (Ogata,
2001) is an adequate, simple controller for closing the inner loop at the motor. This
controller possesses two adjustable constants: Kpm for the proportional branch, and
Kvm for the derivative. However, the derivative term might cause some problems when
applied directly to the error signal, e, as it adds a zero to the closed-loop equivalent
transfer function. A zero-cost improvement to this issue is placing the derivative branch
3.2. System inversion based control scheme 51V m m sJs Km m rnsK v mK p m rmn rrm eFigure 3.2: Inner control loop
in the motor inner feedback as shown in Figure 3.2. This control scheme also amends
the performance of the motor in presence of friction (Coulomb and/or viscous), and
allows to decouple the dynamics of the robot in the rigid (or inner) and the flexible (or
outer) parts, as was demonstrated in (Feliu et al., 1993), dividing the whole control
problem into two easier sub-problems, as will be seen in Chapters devoted to control.
Operating expression (2.7) jointly with this motor controller, we obtain following
relation between input, θrm, and output, θm,
θmθrm
=θm
θrm=
1
Jm
KpmKm
s2 +νm +KvmKm
KpmKm
s+ 1(3.2)
We can now tune the PD controller parameters for achieving a critically damped,
second order system dynamics for the motor of the form
M(s) =θm(s)
θrm(s)=
1
(as+ 1)2(3.3)
where a is a parameter that represents the velocity of response of the motor. This
is the most desirable behavior for a second order system, as avoids overshoot while
minimizes settling time. Equaling terms between equations (3.2) and (3.3), following
expressions are obtained for controller parameters depending on parameter a,
Kpm =Jm
a2Km
(3.4a)
Kvm = 2aKpm − νmKm
(3.4b)
it is theoretically possible making motor as faster as desired. However, due to the
52 Open loop control based on system inversion
physical limitations of the actuator that is a very unrealistic hypothesis.
Basing upon previous equations, it is theoretically possible to obtain a motor dy-
namics as fast as desired by simply making a → 0. However, that is a very unrealistic
hypothesis as it leads to very large controller gain Kpm, hence the controller demanding
high values of the control signal of the motor with the consequences of saturation and
malfunction of the motor. In Section 3.3.1 these physical constraints are incorporated
to the motor control design to obtain a realistic and reachable value a. Therefore, al-
though the motor dynamics can be made quite fast compared to that of the mechanical
part of the arm, in general, we cannot consider it negligible.
3.2.2 Noncausal dynamic inversion
The system inversion is performed easily by inverting the model transfer function
of the manipulator dynamics. Considering only the link dynamics, the inversion block
yields
Fb(s) = G−1t (s) =
s2 + ω20
ω20
(3.5)
Applying the inverse Laplace transform to Fb, the modified reference that must be
provided to the motor to cancel the vibration of the link can be calculated as
θrm(t) =1
ω20
θrt (t) + θrt (t) (3.6)
Equation (3.6) implies that θrt must be, at least, two times differentiable to obtain
a finite reference that achieves the desired vibrationless tracking of the reference.
However, even if this inversion cancels the beam vibration, still a perfect tracking is
not achieved, as the equivalent motor control loop dynamics introduces a (small) delay
in the response. This can be corrected if we include the equivalent transfer function
for the actuator control loop (3.3) in equation (3.5), resulting
Fs(s) = M−1(s)G−1t (s) = (1 + as)2
s2 + ω20
ω20
(3.7)
and, applying again the inverse Laplace
3.3. Constrained trajectory design 53
θrm(t) =a2
ω20
....θ
rt (t) +
2a
ω20
...θrt (t) +
(
a2 +1
ω20
)
θrt (t) + 2aθrt (t) + θrt (3.8)
Hence, achieving a perfect tracking of a desired trajectory is restricted to those
references that are four times differentiable.
3.3 Constrained trajectory design
We have finally defined the scenario which the trajectories must be defined for: a
flexible arm modeled as a second order system with two poles in the imaginary axis, as
given by equation (2.40), attached to a DC motor with an internal control loop con-
sisting of a PD controller which has two degrees of freedom for design (constants Kpm
and Kvm). The coupling between both attached systems has proven to be removable
by means of the compensation term shown in Figure 2.2.
These features determine the definitive form and coefficients of the trajectory equa-
tions as stated in following subsections.
3.3.1 Control signal saturation
A previous step to the definition of the trajectories, the possible saturation of the
control signal must be studied from a high-level point-of-view. Depending on the ref-
erence shape, the controller might demand to the motor very high control signals (that
is, unrealistic, high motor torques). However, due to the electromechanical limitations
of DC motors, the maximum torque that can be delivered, and, hence the input control
signal that can be provided to the motor has an upper limit. This constraint provides
us with a criterion for choosing appropriately the PD controller constants in order to
avoid saturations that cause the apparition of non-linearities. Thus, we divide the
capacity of the signal between the two main tasks we must consider to perform the
manoeuvre: firstly, the manoeuvre itself, as if the tracking were perfect; and secondly,
the correction of position errors due to perturbations. Therefore, we split our prob-
lem in other two which are simpler. From now on we will denote Vt as the maximum
amount of control signal dedicated to the trajectories, and Vp to the maximum amount
dedicated to correct errors such that Vmax = Vt + Vp.
54 Open loop control based on system inversion
tθ
≡
θ∆
1 ≡2 ≡3
tθ
tθ
mθ
θ∆θ∆
tθ
tθ
mθ
mθ
tθ
rr
r
Figure 3.3: Perturbation types
Error correction
Once we have decided how much of the control signal should be used for this task,
we proceed to describe the different types of perturbations we may find in the system
during the manoeuvre, which are outlined in Figure 3.3. They can contribute to the
control signal in two ways: with the coupling torque due to the link flexibility and with
the error feedback in the tip position. Their effects are studied below.
Case 1 In this case the coupling torque is zero while the tip is not at the desired
position. Therefore, the control signal to correct this perturbation is given by
Vp = Kpm∆θ −Kvmθms (3.9)
where ∆θ is the maximum admissible/expected displacement of tip mass. Applying the
initial value theorem when a step input is introduced to the motor velocity expression
given by
˙θmV
=Km/Jm
s+ νm/Jm
(3.10)
we can observe that the velocity tends to be zero at the initial moment, so it will be
neglected. Hence, the required control signal will be
Vp = Kpm∆θ (3.11)
3.3. Constrained trajectory design 55
Case 2 The second kind of perturbation considered occurs when the bar is deflected
but the tip position is at the right place. In this case, the error e will be zero, and the
only contribution for correcting the perturbation will be given by the coupling torque
according to
Vp =c
Kmnr
∆θ (3.12)
Case 3 The last considered perturbation includes both effects: tip position error and
link deflection. Combining them, the expression for the control signal will be
Vp =
(
Kpm +c
Kmnr
)
∆θ (3.13)
where the velocity term has been again canceled for the aforementioned reason. It
seems quite obvious that it represents the worst case and, therefore, it is the only one
we need to consider.
3.3.2 Motor controller tuning
In equation (3.13), we have found objective criteria for calculating Kpm from the
control signal limitations. In addition, from the restriction proposed in Section 3.2.1,
equation (3.3), of adapting the motor to a critically damped system, a new restriction
for determining Kvm is introduced. Combining this requirement with equations (3.13)
and (3.4), we obtain the following equations for determining the controller parameters
Kpm =Vp
∆θ− c
Kmnr
(3.14a)
Kvm = 2
√JmKpm
Km
− ν
Km
(3.14b)
With these equations we have completed the design of our motor control system
guaranteeing that there will be no saturation in our control signal, and, therefore,
linearity in motor operation.
56 Open loop control based on system inversion
θ m (
rad)
ωm
(ra
d/s)
α m (
rad/
s2 )
α m (
rad/
s3 )
t (s)
δ m(r
ad/s
4 )t4 t
6t10
t8t
1t3 t
5t7
t9 t
11t2
⋅
Figure 3.4: Generic trajectory outline
3.3.3 Trajectory definition
The trajectory has been defined for the dynamic inversion of the complete arm
model. Therefore, as commented in Section 3.2.2, it is evident that, in order to obtain
a finite control input, the selected trajectory should be at least four times differentiable
(same degree as our model).
A simple and low order trajectory that adjusts to this premise is a fourth-order
polynomial (Feliu et al., 1993). In Fig.3.4, it is qualitatively outlined the shape of
the manoeuvre and its derivatives. This input has been picked out with three main
purposes: 1) that the trajectory be invertible; 2) that the manoeuvre be symmetrical;
and 3) that zero acceleration interval be maximized.
This is a piecewise function whose coefficients change for each segment, θt,i(t),
which is comprised within the characteristic times ti−1 and ti, to a total of 12 parts.
The general form of any of these segments of the reference is
θt,i(t) = ci,4t4 + ci,3t
3 + ci,2t2 + ci,1t+ ci,0 ∀ i ∈ [1, 12] (3.15)
where ci,j is the j-th order coefficient of the i-th piece of the trajectory.
The family of polynomials (segments) that describes the trajectory is completely
3.3. Constrained trajectory design 57
defined using four parameters: maximum allowed snap (second acceleration derivative),
δM ; maximum allowed acceleration, αM ; target tip angle θtf ; and duration of the
maneuver, tf . The last two are defined by the desired trajectory, while αM depends on
the mechanical constraints of the link, and, lastly, δM is chosen to avoid saturations of
the control signal to the motor.
Generic equations depending on these four parameters are detailed subsequently.
• First segment: 0 ≤ t ≤ t1
....θ t(t) = δM ⇒
....θ t(t1) = δM
...θ t(t) = δM t ⇒
...θ t(t1) =
√
δMαM
θt(t) =1
2δM t2 ⇒ θt(t1) =
1
2αM
θt(t) =1
6δM t3 ⇒ θt(t1) =
1
6αMTm
θt(t) =1
24δM t4 ⇒ θt(t1) =
1
24αMT 2
m
• Second segment: t1 < t ≤ t2
....θ t(t) = −δM ⇒
....θ t(t2) = −δM
...θ t(t) = −δM (t− 2Tm) ⇒
...θ t(t2) = 0
θt(t) = −1
2δM (t− 2Tm)2 + αM ⇒ θt(t2) = αM
θt(t) = −1
6δM (t− 2Tm)3 + αM t− αMTm ⇒ θt(t2) = αMTm
θt(t) = − 1
24δM (t− 2Tm)4 +
1
2αM t2 − αMTmt+
7
12αMT 2
m ⇒ θt(t2) =7
12αMT 2
m
• Third segment: t2 < t ≤ t3
....θ t(t) =
...θ t(t) = 0 ⇒
....θ t(t3) =
...θ t(t3) = 0
θt(t) = αM ⇒ θt(t3) = αM
θt(t) = αM t− αMTm ⇒ θt(t3) = αM (Tma + Tm)
θt(t) =1
2αM t2 − αMTmt+
7
12αMT 2
m ⇒ θt(t3) = αMTma
(1
2Tma + Tm
)
+7
12αMT 2
m
58 Open loop control based on system inversion
• Fourth segment: t3 < t ≤ t4
....θ t(t) = −δM ⇒
....θ t(t4) = −δM
...θ t(t) = −δM (t− 2Tm − Tma) ⇒
...θ t(t4) = −
√
δMαM
θt(t) = −1
2δM (t− 2Tm − Tma)
2 + αM ⇒ θt(t4) =1
2αM
θt(t) = −1
6δM (t− 2Tm − Tma)
3 + αM t− αMTm ⇒ θt(t4) = αM
(
Tma +11
6Tm
)
θt(t) = − 1
24δM (t− 2Tm − Tma)
4 +1
2αM t2 − αMTmt+
7
12αMT 2
m ⇒
θt(t4) = αMTma
(1
2Tma + 2Tm
)
+49
24αMT 2
m
• Fifth segment: t4 < t ≤ t5
....θ t(t) = δM ⇒
....θ t(t5) = δM
...θ t(t) = δM (t− 4Tm − Tma) ⇒
...θ t(t5) = 0
θt(t) =1
2δM (t− 4Tm − Tma)
2 ⇒ θt(t5) = 0
θt(t) =1
6δM (t− 4Tm − Tma)
3 + αM (Tma + 2Tm) ⇒ θt(t5) = αM (Tma + 2Tm)
θt(t) =1
24δM (t− 4Tm − Tma)
4 + αM (Tma + 2Tm) t− αMTma
(1
2Tma + 3Tm
)
− 4αMT 2
m
⇒ θt(t5) = αMTma
(1
2Tma + 3Tm
)
+ 4αMT 2
m
• Sixth segment: t5 < t ≤ 12tf
....θ t(t) =
...θ t(t) = θt(t) = 0 ⇒
....θ t(t6) =
...θ t(t6) = θt(t6) = 0
θt(t) = αM (Tma + 2Tm) ⇒ θt(t6) = αM (Tma + 2Tm)
θt(t) = αM (Tma + 2Tm) t− αMTma
(1
2Tma + 3Tm
)
− 4αMT 2
m ⇒ θt(t6) =1
2θf
This defines the first half of the trajectory. The remaining half can be calculated
using symmetry: velocity and jerk possess even symmetry, while position, acceleration
and snap exhibit odd symmetry.
In these equations two new constants, Tm and Tma, have been defined, whose values
3.3. Constrained trajectory design 59
are
Tm =
√αM
δM(3.16)
Tma =
(1
2tf − 3Tm
)
−
√(1
2tf − Tm
)2
− θfαM
(3.17)
Tmb =1
2tf − 4Tm + Tma (3.18)
where Tm represents the duration of each of the segments in which there is a change
in the acceleration (δM 6= 0 in first, second, fourth, fifth, eighth, ninth, eleventh and
twelfth), Tma is the duration of the segments of constant acceleration (third and tenth),
and Tmb has been defined for explicitly delimiting the duration of each of the two
segments of constant velocity (sixth and seventh). From these constants the trajectory
characteristic times can be derived
t1 = Tm
t2 = 2Tm
t3 = 2Tm + Tma
t4 = 3Tm + Tma
t5 = 4Tm + Tma
t6 =1
2tf
The rest of the times (t7 − t11) can be calculated by symmetry, while t12 = tf .
3.3.4 Kinematic limits
Once the trajectories have been defined, next step is the calculation of the valid
values of acceleration, αM , and snap, δM . This involves some massaging of the kine-
matic equations of the trajectory obtained in Section 3.3.3 to look for conditions that
must be fulfilled by these two constants. In subsequent paragraphs, these restrictions
are obtained.
60 Open loop control based on system inversion
Condition 8Tm ≤ tf
Assuming Tma = Tmb = 0, that is, that the motor is continuously changing acceler-
ation, the total time of the trajectory would be divided in eight equal-time segments,
8Tm = tf . This is the upper limit for the value of Tm and gives the following relation
between the values of δM and αM
δM ≥ 64αM
t2f(3.19)
The lower limit of Tm is obviously 0, giving the trivial condition that both δM and
αM must be positive (so that Tm ∈ ℜ+).
Condition Tma ≥ 0
Mathematically, a solution with a negative Tma can be obtained for the trajectory
equations under certain circumstances, but that is not physically feasible. Substituting
equation (3.17) in condition and rearranging terms yields
T 2m − 1
4Tmtf +
1
8
θfαM
≥ 0
which is a second order polynomial in Tm. Solving the equation and substituting Tm
with its value (3.16), δM can be isolated for obtaining another relation
δM ≥ αM(1
8tf −
√1
64t2f −
1
8
θfαM
) (3.20)
This equation has the particularity that can be applied only if αM ≥ 8θft2f
. Other-
wise, δM would have imaginary part different from 0. Moreover, this condition is more
restrictive than the previous given in (3.19).
Condition Tma ∈ R
Complex values are not valid for Tma. Hence, the square root in (3.17) must be real
for the existence of the trajectory, that is, the radicand must be positive
3.3. Constrained trajectory design 61
(1
2tf − 3Tm
)2
− θfαM
≥ 0
Rearranging terms and clearing δM we obtain the following new restriction between
the values of acceleration and snap
δM ≥ αM(1
2tf −
√θfαM
)2 (3.21)
Under this limit, Tma is a complex number and the trajectory cannot be described.
Condition Tma ≤ 1
2tf − 4Tm
Tma is obviously limited by the total maneuver time and the time devoted to change
the acceleration. Again, substituting (3.16) and (3.17) and isolating the snap, the last
inequality is obtained, being
δM ≥ αM
t2f
(1
4t2f −
θfαM
) (3.22)
It can be proven that this equation is a stronger restriction than previous condition
Tma ∈ ℜ. It also has a vertical asymptote in
αM =4θft2f
(3.23)
that corresponds to the acceleration of a bang-bang profile that covers θf radians in tf
seconds. Below that acceleration limit, the trajectory cannot be performed.
3.3.5 Physical limits
These restrictions apply because of the physical limitations of the elements that
the manipulator consists of. Namely, the motor has an internal dynamics that, even
if can be noticeably quickened, cannot be neglected, and its signal control is bounded
depending on the speed and/or output torque that can supply; and the material of
62 Open loop control based on system inversion
the link, when subjected to bending moments surpassing its elastic limit, might suffer
permanent deformation.
Link material limit
This factor throws a limit for the maximum acceleration a material can experiment
without showing plastic deformation or excessive displacements at the tip. Depending
on if the material is metallic or composite, the calculations are different.
Metallic material In this case, the maximum acceleration allowed to the payload
is determined by the bending moment that we may reach in the bar before it exceeds
the elastic limit, which yields from the following expression (Berrocal, 2002)
σe =Mz
Izy (3.24)
being σe the elastic limit, Mz the bending moment, Iz the cross sectional inertia of the
beam, and y the distance between a point of the beam and its neutral axis.
On the other hand, the bending moment will be equal to the coupling torque in
equation (2.36). Then, if we substitute the maximum value admissible for Mz in the
equation and reorder terms, the following relation for the acceleration limit is obtained
αM =σeIz
ml2yM(3.25)
where yM represents the maximum distance to the neutral axis. In the case of a
cylindric beam, it corresponds to its radius.
Composite material When dealing with composites such as carbon or glass fibers,
the elastic limit has no meaning, as these materials remain elastic until they break.
Instead, these materials might enter in the large deflections zone, meaning that their ge-
ometry would be nonlinear (small deflections hypothesis being no longer valid). There-
fore, our acceleration limit will depend on this boundary. Directly from equation (2.36),
it yields
αM =c∆θMml2
(3.26)
3.3. Constrained trajectory design 63
where ∆θM is typically assumed to be 0.2 radians, as explained in (Belendez et al.,
2002).
Control signal limit
This value is slightly more difficult to obtain. First of all, we need to find where is
the maximum value reached by the control signal during the trajectory, which should
be smaller than the upper limit of the voltage supplied to the motor for avoiding
saturations in the control (which complicate the transfer function inversion). Because
of the fragmented nature of our expression for the reference we have, a priori, more than
one candidate. As a first approach, we calculate voltage reference at all the changes of
segment in the first half of the trajectory t1, t2, t3, t4 and t5. In the second half the
absolute value of the control signal demanded is smaller, as the viscous friction “helps”
the robot to decelerate.
The equation for the voltage, in terms of the desired trajectory, is obtained by apply-
ing the inverse Laplace transform to whole open loop model given by equation (2.42).
This operation yields
V (t) =nr
Kω20
(
Jm
....θ t(t) + ν
...θ t(t) + Jmω
20θt(t) + νω2
0 θt(t))
(3.27)
From here, and applying the correspondent values of θt and its derivatives as stated
in trajectory equations defined in Section 3.3.3, we obtain the equations for each value
of the voltage reference
V (t1) =nr
Km
(Jm
ω20
δM +ν
ω20
√
αMδM + Jm
1
2αM + ν
1
6αMTm
)
(3.28a)
V (t2) =nr
Km
(JmαM + ναMTm) (3.28b)
V (t3) =nr
Km
(JmαM + ναM (Tm + Tma)) (3.28c)
V (t4) =nr
Km
(Jm
ω20
δM − ν
ω20
√
αMδM + Jm
1
2αM + ν
1
6αM
(11
6Tm + Tma
))
(3.28d)
V (t5) =nr
Km
(Jm
ω20
δM + ναM (2Tm + Tma)
)
(3.28e)
64 Open loop control based on system inversion
The procedure is the following:
1. The smallest value of αM such that max V = Vt and the trajectory be feasible
is searched between the boundary couples (αM ,δM) obtained with the kinematic
restrictions (3.19) to (3.23).
2. A new value, greater than the previous by a desired step, is assigned to αM while
the reference voltage is set to Vt for each equation.
3. The resultant functions f(δM) are solved numerically (δM cannot be isolated in
some of the above expressions).
4. The minimum value among the five δM ’s is chosen as the maximum value of δM
that does not saturate the motor for the specified αM .
5. The process is repeated from step 2 until we calculate the desired interval of
accelerations, which can range from the smallest value obtained in step 1 to the
maximum admissible acceleration calculated in (3.25) or (3.26).
This procedure yields a numerical function δM (αM) that bounds the upper limit
of the feasibility region of pairs of values (δM , αM) that can be used for defining a
trajectory of the family previously described. This manner, any of the pairs within the
defined feasibility region avoids the undesirable effects listed in this Section: torques
and control signal are bounded within the limits and the trajectory equations fulfill
the required specifications.
3.4 Simulation results
The dynamic inversion scheme is here designed and studied in simulation for the
experimental platform defined in Section 2.6.2, (Flexible arm with composites link),
with the choice of a short carbon fiber link, in order to remain in the small displacements
(linear) zone, whose parameters can be found in Tables 2.4 and 2.5. Specifically, the
nominal payload used for simulations and experiments is mp = 60.82 · 10−3 kg.
In Section 3.3 we have described an objective method for determining both the
trajectories and the inner control loop of a flexible manipulator. To achieve this we
3.4. Simulation results 65
have assumed a couple of values Vt and Vp for the amounts of control signal dedicated to
each task: performing the trajectory and correcting perturbations (due, for example, to
the link flexibility). This share-out of the control signal depends on experience and on
the specific characteristics of the considered platform. In our case we have decided to
give the trajectory a 60% of the total capacity of the control signal while the remaining
40% will be used for correcting the perturbations. Thus, it can be concluded that
Vt = 1.2 V and Vp = 0.8 V .
Following the procedure detailed in Section 3.3.2, the PD controller parameters
of the actuator control loop can be calculated using equations (3.14), obtaining the
following values: Kpm = 3.86 and Kvm = 0.0639. With these values, the parameter in
equation (3.3) is a = 0.00888, making the motor very fast in comparison to the beam
dynamics.
3.4.1 Reference trajectory
Once the mathematical model for our platform have been totally defined, we can
build up the trajectory following the steps detailed in Sections 3.3. First we state the
target position and the target time to the most common values θtf = 1 radian and
tf = 1 second. Then, the kinematic restrictions, described in Section 3.3.4, are applied
giving the following inequations that partially conform the feasibility region
δM ≥ 64αM (3.29a)
δM ≥ αM(1
8−√
1
64− 1
8αM
) (3.29b)
δM ≥ 4α2M
αM − 4(3.29c)
αM > 4 (3.29d)
which are graphed in Figure 3.5.
Regarding physical restrictions of Section 3.3.5, and as long as we are using a
composite material for our beam, equation (3.26) throws a maximum value of αM =
21.65 rad/s2. Along this interval, the DC motor limitation regarding maximum torque
66 Open loop control based on system inversion
4 5 6 7 8 9 10 11 12 13 140
1000
2000
3000
4000
5000
6000
7000
αM
(rad/s 2)
δ M (
rad/
s4 )
Tm
≤ 1/8 tf
Tma
≥ 0
4Tm
+ Tma
≤ 1/2 tf
Figure 3.5: Feasibility region of acceleration and snap after kinematic limitations
delivered is studied in terms of the control signal, whose maximum was determined
to Vt = 1.2 V. Then, following the procedure proposed in Section 3.3.5, the max-
imum value of snap associated to a particular acceleration is studied by means of
equations (3.28), where the lower value of acceleration that guarantees the existence of
Tma ∈ ℜ as stated in (3.21) while keeps max(V ) = 1.2 is αM = 4.5 rad/s2. Hence, the
maximum snap for the interval of valid values of acceleration αM ∈ (4, 21.65) rad/s2 is
depicted in Figure 3.6. Most of the maximum values correspond to the instant t5, but
with the lowest allowed values of acceleration the maximum voltage demand happens
at time t1.
Joining both requirements, the feasibility region is bounded and completely delim-
ited, as can be observed in Figure 3.7
This region gives us still considerable freedom in the choosing of the parameters
for the trajectory and a question might arise: which of them are the best? Among the
numerous valid choices of pairs (αM , δM), the ”best” depends on what we are looking
for. Some examples are:
• Attending to minimizing the strain at the beam, the lowest bending moment
3.4. Simulation results 67
4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 100
500
1000
1500
2000
2500
δ M (
rad/
s4 )
4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 101.195
1.2
1.205
αM
(rad/s 2)
VM
AX
(V
)
Figure 3.6: Maximum values of snap due to the control signal limitation
(coupling torque) and the less inertia at the payload is attained at the lowest
admissible acceleration (4.88, 2401.2). Trajectory is depicted in Figure 3.8.
• Attending to the lowest maximum velocity along the trajectory that corresponds
to the highest acceleration possible (9.69, 1829). Trajectory is shown in Fig-
ure 3.9.
• If the rule is getting a smoother control signal with the smallest bumps due to the
changes in the trajectory segment and the lowest demand of voltage reference,
we can choose the smallest snap value: (8, 512). Trajectory can be seen in
Figure 3.10.
• If we are looking for a trajectory that fulfil the restrictions even if there are errors
in the calculus of boundaries, we need to select a point as far as possible of any of
the limits of the feasibility region, for example: (7, 1400). Trajectory is displayed
in Figure 3.11
68 Open loop control based on system inversion
5 6 7 8 9 10 110
500
1000
1500
2000
2500
3000
αM
(rad/s 2)
δ M (
rad/
s4 )
T
m ≤ 1/8 t
f
Tma
≥ 0
Tma
∈ ℜ
4Tm
+ Tma
≤ 1/2 tf
VMAX
= Vt
Feasibility region
Figure 3.7: Feasibility region for pairs acceleration-snap
0 0.2 0.4 0.6 0.8 10
0.5
1
Tip
ang
le θ
t (ra
d)
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
Tip
vel
ocity
(ra
d/s)
t (s)
0 0.2 0.4 0.6 0.8 1−5
0
5
Tip
acc
eler
atio
n (r
ad/s
2 )
0 0.2 0.4 0.6 0.8 1−200
−100
0
100
200
t(s)
Tip
jerk
θ3 (
rad/
s3 )
0 0.2 0.4 0.6 0.8 1−4000
−2000
0
2000
4000
t (s)
Tip
sna
p θ4 (
rad/
s4 )
0 0.2 0.4 0.6 0.8 1−1
0
1
2
Con
trol
sig
nal V
(V
)
Figure 3.8: Trajectory with minimum acceleration in the feasibility region
3.4. Simulation results 69
0 0.2 0.4 0.6 0.8 10
0.5
1
Tip
ang
le θ
t (ra
d)
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
t(s)
Tip
vel
ocity
(ra
d/s)
0 0.2 0.4 0.6 0.8 1−10
−5
0
5
10
Tip
acc
eler
atio
n (r
ad/s
2 )0 0.2 0.4 0.6 0.8 1
−200
−100
0
100
200
Tip
jerk
θ3 (
rad/
s3 )
0 0.2 0.4 0.6 0.8 1−2000
−1000
0
1000
2000
t(s)
Tip
sna
p θ4 (
rad/
s4 )
0 0.2 0.4 0.6 0.8 1−1
0
1
2
Con
trol
sig
nal V
(V
)
Figure 3.9: Trajectory with maximum acceleration in the feasibility region
0 0.2 0.4 0.6 0.8 10
0.5
1
Tip
ang
le θ
t (ra
d)
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
t(s)
Tip
vel
ocity
(ra
d/s)
0 0.2 0.4 0.6 0.8 1−10
−5
0
5
10
Tip
acc
eler
atio
n (r
ad/s
2 )
0 0.2 0.4 0.6 0.8 1−100
−50
0
50
100
Tip
jerk
θ3 (
rad/
s3 )
0 0.2 0.4 0.6 0.8 1−1000
−500
0
500
1000
t(s)
Tip
sna
p θ4 (
rad/
s4 )
0 0.2 0.4 0.6 0.8 1−0.5
0
0.5
1
Con
trol
sig
nal V
(V
)
Figure 3.10: Trajectory with lowest maximum in the control signal demand
70 Open loop control based on system inversion
0 0.2 0.4 0.6 0.8 10
0.5
1
Tip
ang
le θ
t (ra
d)
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
t(s)
Tip
vel
ocity
(ra
d/s)
0 0.2 0.4 0.6 0.8 1−10
−5
0
5
10
Tip
acc
eler
atio
n (r
ad/s
2 )
0 0.2 0.4 0.6 0.8 1−100
−50
0
50
100
Tip
jerk
θ3 (
rad/
s3 )
0 0.2 0.4 0.6 0.8 1−2000
−1000
0
1000
2000
t(s)
Tip
sna
p θ4 (
rad/
s4 )
0 0.2 0.4 0.6 0.8 1−1
−0.5
0
0.5
1
Con
trol
sig
nal V
(V
)
Figure 3.11: Trajectory with (αM ,δM) in the center of the feasibility region
Another possible rule, which will be the adopted one, consists on selecting a trajec-
tory that minimizes the energy provided by the motor to perform the maneuver. The
mechanical power delivered by the motor for fulfilling the trajectory is given by
Pm(t) = Γmθm = KmV (t)θm(t) (3.30)
Then, the energy provided can be calculated by integration of that expression,
yielding
Pm(t) =dWm
dt⇒ Wm =
∫ tf
0
KmV (t)θm(t)dt (3.31)
Taking this numerical study within the feasibility region, the best pick turns out
to be the pair (9.69, 1829), that is, the maximum acceleration choice. The trajectory
has already been drawn in Figure 3.9.
However, for leaving a security margin, αM = 9 rad/s2 is used instead. For the
snap, any value in the interval (1296, 1899) can be used. Again, for leaving a margin,
an intermediate value, δM = 1600 rad/s4, has been selected. For these parameters,
3.4. Simulation results 71
the desired trajectory and the control signal scheduled for the motor are depicted in
Figure 3.12.
0 0.2 0.4 0.6 0.8 10
0.5
1
Tip
ang
le θ
t (ra
d)
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
t(s)
Tip
vel
ocity
(ra
d/s)
0 0.2 0.4 0.6 0.8 1−10
−5
0
5
10
Tip
acc
eler
atio
n (r
ad/s
2 )
0 0.2 0.4 0.6 0.8 1−200
−100
0
100
200
Tip
jerk
θ3 (
rad/
s3 )
0 0.2 0.4 0.6 0.8 1−2000
−1000
0
1000
2000
t(s)
Tip
sna
p θ4 (
rad/
s4 )
0 0.2 0.4 0.6 0.8 1−1
0
1
2
Con
trol
sig
nal V
(V
)
Figure 3.12: Selected reference trajectory with αM = 9 rad/s2 and δM = 1600 rad/s4
3.4.2 Trajectory inversion
Once the reference has been completely defined, the two proposed dynamic inver-
sions (Fs and Fb) are applied as described in 3.2.2, giving the modified reference inputs
shown in Figure 3.13.
These inputs have been introduced in a MATLAB Simulink model following the
open-loop control scheme shown in Figure 3.1, and the system responses are depicted
in Figure 3.14 jointly with the errors of each output compared to the target trajectory,
which has been displayed in Figure 3.13. Both trajectories, linear and fourth order
polynomial produce a similar response, but the advantage of fourth order is noticeably
appreciated in the responses of the system-inverted references. The response to the
beam inversion achieve a vibrationless steady-state, while a small delay appears with
respect to the desired output causing a constant error along the motion. This delay is
72 Open loop control based on system inversion
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
1.2
t(s)
θ mr (
rad)
Linear
4th order
Beam inversion of the 4th order
System inversion of the 4th order
Figure 3.13: Input references for the flexible arm
due to the controlled motor equivalent dynamics and is directly related with constant
a. Once the trajectory is subjected to the complete system inversion, the trajectory
tracking is perfect (error is lesser than 2 · 10−5 rad).
3.5 Experimental validation
In our experimental setup, the Coulomb friction has been experimentally measured
and counterbalanced with a compensating term VCoul = 0.5V , while the coupling torque
has been neglected, since it is divided by the reduction relation of the gear being its
influence over the motor dynamics very small (Feliu and Ramos, 2005).
3.5.1 Nominal case
Experiments carried out in this section take as reference the trajectory designed in
Section 3.4.1 for the nominal payload mp = 60.82 · 10−3.
3.5. Experimental validation 73
0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1
1.2
1.4
θ t (ra
d)
0 0.5 1 1.5 2 2.5 30
0.05
0.1
0.15
0.2
0.25
t (s)
|θtr −
θt| (
rad)
Linear
4th orderBeam inversionSystem inversion
Figure 3.14: Top: Simulated response to different inputs. Bottom: Error of the re-sponses with respect to the reference
Ramp input
The maximum angle we can rotate the arm is constrained by the size of the table,
while the time it takes the robot to perform the manoeuvre is limited by the control
signal saturation (as this final time lowers, the demanded control signal increases).
Taking into account these two restrictions, the trajectory proposed in simulation, which
rotates the beam 1 radian in a total time of 1 second, is feasible. For testing the
oscillatory response of the system, a ramp input with constant slope has been tried.
In this case, both tip and motor references are equal, as there is no system inversion
in process. The results of this movement are illustrated in Figure 3.15, where we
can observe that, as predicted in simulation, our arm is subjected to big deflections,
near the linear-nonlinear behavior limit. Moreover, it is also noticeable the lag of the
motor angle with respect to the input reference. This delay is due to the second order
dynamics of the actuator control loop determined by equation (3.3), which, despite it
is very fast in comparison to the beam, it is not negligible.
74 Open loop control based on system inversion
0 0.5 1 1.5 2 2.5 3−0.2
0
0.2
0.4
0.6
0.8
1
1.2
angl
e (r
ad)
t (s)
Tip referenceMotor referenceMotor angleTip angle
Figure 3.15: Experimental response to a linear trajectory input
Fourth order trajectories
The polynomial trajectory used in simulation has been applied to our experimental
setup with the results displayed in Figure 3.16. Again, both tip and motor references are
identical. Apparently no benefits have been obtained from using a smoother trajectory,
but the advantage of this reference will be evident when applying dynamic inversion.
Dynamic inversion of the beam model
Now the desired 4th order polynomial reference is precalculated using the noncausal
expression given by (3.6). The modified trajectory is the new reference for our DC
motor, θrm.
The results for this experiment are shown in Figure 3.17, where we can see the
outstanding trajectory tracking achieved. There is still some lag between angles and
references. Note that the ramp input cannot be inverted in this way because the
acceleration is undefined at the beginning and end of the input trajectory. The selected
trajectories avoid this problem by defining the accelerations first.
3.5. Experimental validation 75
0 0.5 1 1.5 2 2.5 3−0.2
0
0.2
0.4
0.6
0.8
1
1.2
angl
e (r
ad)
t (s)
Tip referenceMotor referenceMotor angleTip angle
Figure 3.16: Experimental response to a 4th order trajectory
0 0.5 1 1.5 2 2.5 3−0.2
0
0.2
0.4
0.6
0.8
1
1.2
angl
e (r
ad)
t (s)
Tip referenceMotor referenceMotor angleTip angle
Figure 3.17: Experimental response to a reference that inverts the beam dynamics
76 Open loop control based on system inversion
Dynamic inversion of the system model
We can also correct this delay if we apply the whole system (beam plus actuator
equivalent dynamics) inversion to the desired trajectory as stated in equation (3.8).
In this case, the system response to this input is displayed in Figure 3.18. Now we
have not only canceled, to a great extent, the oscillation of the tip mass, but also the
lag has been corrected. This feature can only be achieved by references, at least, four
times differentiable with the proposed model. The election of the polynomial grade is
now fully substantiated.
0 0.5 1 1.5 2 2.5 3−0.2
0
0.2
0.4
0.6
0.8
1
1.2
angl
e (r
ad)
t (s)
Tip referenceMotor referenceMotor angleTip angle
Figure 3.18: Experimental response to a reference inverts the system dynamics
Figure 3.19 compares the different references used and the payload angles recorded
with the position measurement system, respectively. The effectiveness of the dynamic
inversion joined to the proposed trajectories is unquestionable.
3.5.2 Changes of the payload
Being proposed scheme an open-loop controller without any trace of feedback, it
cannot detect any change in the parameters/constants of the system. Hence, when a
3.5. Experimental validation 77
0 0.5 1 1.5 2 2.5 3−0.5
0
0.5
1
1.5
θ t (ra
d)
0 0.5 1 1.5 2 2.5 30
0.05
0.1
0.15
0.2
0.25
t (s)
|θtr −
θt| (
rad)
Ramp4th order polyBeam inversionSystem inversion
Figure 3.19: Experimental response to different inputs. Top: System tip angle per-formed. Bottom: Error of the responses with respect to the reference
significant parameter, such as the payload, differs from expected value, the performance
of the system worsens. In Figure 3.20, a range of values for this parameter has been
tested. If the value is considerably different from the nominal as for 20 or 100 grams,
the reference obtained by means of the dynamic inversion gives the same results as
the linear trajectory, with oscillations of about 0.2 radians of amplitude. But also
when the change is small, just a 10% over or under the nominal value, the oscillations
become inadmissible, in the range of 0.02-0.03 radians (what turns into a displacement
of around 2 cm of amplitude at the payload).
In an application where the payload is known, this issue is not crucial, as we can
have a number of different trajectories and select the appropriate for the specific opera-
tion. For example, Figure 3.21 demonstrates that, choosing an adequate reference, the
system behavior continue being remarkably good. In this case, the chosen references
for the other masses have the same maneuver time and described angle parameters
(tf = 1 s and θf = 1 rad/s), while αM = 13 rad/s2 and δM = 7000 rad/s4 for the 20 g
tip mass, and αM = 8 rad/s2 and δM = 1000 rad/s4 for the 60 g payload.
78 Open loop control based on system inversion
0 0.5 1 1.5 2 2.5 3−0.5
0
0.5
1
1.5
thet
at (
rad)
Tip referencem
t = 60g
mt = 20g
mt = 54g
mt = 66g
mt = 100g
0 0.5 1 1.5 2 2.5 30.8
0.9
1
1.1
1.2
θ t (ra
d)
0 0.5 1 1.5 2 2.5 30
0.05
0.1
0.15
0.2
0.25
|θtr −
θt| (
rad)
t (s)
Figure 3.20: System performance when payload changes or is not accurately measured.Top: System response. Middle: Zoom of response. Bottom: Error with reference.
0 0.5 1 1.5 2 2.5 3
0
0.2
0.4
0.6
0.8
1
θ t (ra
d)
mt = 60g
mt = 20g
mt = 100g
0 0.5 1 1.5 2 2.5 30
0.005
0.01
0.015
0.02
|θtr −
θt| (
rad)
t (s)
Figure 3.21: System response to trajectories defined for actual payloads. Top: Systemresponse. Bottom: Error with respect to reference.
3.6. Summary and conclusions 79
3.6 Summary and conclusions
This chapter presented an open-loop, noncausal dynamic inversion control scheme
based on the precalculation of a modified input that, when applied to the system,
provides the desired trajectory in the output. This direct, easy inversion has been
feasible because of the simplicity of the model used for describing the robot, which is
scalar and minimum phase.
In addition, a family of piecewise, 4th order polynomial trajectories that are suit-
able for dynamic inversion in simple, lumped masses models of flexible arms has been
developed. The order of the polynomial is in concordance with the relative order of the
manipulator model. Only four parameters need to be specified for defining the maneu-
ver during the design, namely, time, angle, maximum acceleration and maximum snap.
An objective process has been provided for calculating those maximums depending on
the physical constraints of the system, and also the actuator controller constants.
The open-loop dynamic inversion has been tested on a flexible arm with a carbon
fiber link. Despite its simplicity, the control scheme has provided superb experimental
results with very little computation effort. Simply by using a previously calculated off-
line stage to produce a modified input which provides perfect tracking of the desired
trajectory.
However, these open-loop schemes prove to be very sensitive to uncertainties in the
system parameters, degrading the performance abruptly when a change is introduced
in the system. An experimental example has been performed with different payloads
where the system has behaved as expected. To maintain the optimal performance, we
would need to know in advance the mass so we could switch to the adequate reference
for tracking the original reference.
Evidently, in the pursuit of controllers whose performance is not affected by changes
in the parameters of the system, we need to know how good is the system output dur-
ing maneuvers. Therefore, the necessity of closed-loop schemes that intelligently take
advantage of the sensorial system in a feedback loop is unquestionable, and motivates
the development of the robust/adaptive controllers described in following Chapters.
80 Open loop control based on system inversion
Chapter 4
Robust control
4.1 Introduction
In the previous Chapter, the vibrationless motion of a flexible manipulator was
achieved by a dynamic inversion based open-loop controller. While simple, this scheme
is very sensitive to any changes in the plant, such as the variation of the tip mass,
and depends strongly on the exact modeling of the mechanical system. Any change
in the plant parameters reduces the controller performance, making the system lose
the desired specifications. Actually, the payload, which is one of the most determinant
parameters in robot dynamics, may change very often in a robotic system under normal
operation, either when changing the tip tool being used, or when grasping an object
with non negligible weight.
A number of different strategies, which can be grouped into three big categories of
control theory, have been applied to overcome this drawback. Namely, intelligent con-
trol, based on techniques with the ability to learn, such as neural networks or genetic
algorithms, which are usually very demanding computationally due to its nonlinear
intrinsic nature; robust control, e.g. H∞ control, which provides schemes that perform
correctly under a range of parameter uncertainties; and adaptive control, which pro-
vides controllers that are capable of adjusting their values depending on some criteria,
e.g. the estimation of the uncertain parameters or a cost function representing the
system performance.
This chapter presents a closed-loop robust control scheme for governing single-link
82 Robust control
flexible very lightweight robots maneuvers. First, a discussion on the inappropriateness
of the general purpose PID controllers is carried out in Section 4.2. Because of the
lack of robustness of these classical controllers, which also loose performance rapidly as
the system drifts away from the design point, the scheme has been evolved to a more
robust version.
Then, subsequent sections present a scheme for controlling rest-to-rest manoeuvres
of single-link flexible arms with varying payload. It uses the measurement of the link
deflection provided by two strain gauges placed at the base of the link. It is shown
that this control scheme is more robust to payload changes than control schemes based
on accelerometer measurements, as was demonstrated in (Feliu et al., 2002), whereas
strain gauges are simpler to instrument. The robustness of this control scheme will
also be studied when other robot parameters such as motor friction or bar stiffness are
modified.
Along the chapter we illustrate theoretical conclusions with sets of simulations in
which we compare the results of our schemes with those of classical controllers. Then,
the control is put into practice and some experiments validate the numerical results.
Finally, some conclusions are drawn in light of the experiences.
4.2 PID controllers and their drawbacks
Despite the advent of many sophisticated control theories and techniques, proportional-
integral-derivative (PID) control is still one of the widely used control structures in
industrial applications. The popularity of PID control is mainly due to its structural
simplicity, demonstrated reliability, and broad applicability. The most usual control
law for these controllers is the following:
y(t) = Kpx(t) +Kv
dx(t)
dt+Ki
∫ t
0
x(t)dt, (4.1)
which, transformed into the Laplace domain, results
Y (s) =
(
Kp +Kvs +Ki
s
)
X(s). (4.2)
They consist of three different blocks comprising three different effects on the system
4.2. PID controllers and their drawbacks 83
plant. TheP stands for the proportional effect which is the product of the error between
input (or reference) and actual output and a constant, Kp. Hence, it corrects the error
in the output by setting the input signal of the plant system to a value proportional
to the error. However, it does not take into account the dynamics of the error, i.e.
its variation ratio, and it can only control very stable and well-behaved systems by
itself. To improve the regulation properties, a derivative effect, denoted by the D, is
added to a proportional controller, with derivative constant Kv, adding some sort of
sensibility to the controller. On the one hand, this action responses to the velocity
of change in the error signal, and corrects it quicker, the bigger is the error variation
ratio (error velocity), providing a more reliable manner to control a wide range of
physical systems. On the other hand, this effect is very sensitive to noises in the error
signal (coming from sensing, for example) and may lead to an steady-state error in the
target output. Finally, the integrative effect, denoted by I, and with constant Ki, adds
steady-state accuracy for those systems that exhibit difficulties for exactly matching the
input reference. In exchange, this integrative term slows down the transitory response
of the system. Hence, no one of these three terms or even the joint action of them
guarantees that the physical plant performs a perfect behavior, as we always have to
find a not-so-optimal trade-off between stability, rapidness and steady-state error.
Differently from dynamic inversion technique, which is open-loop, PIDs close a
control loop what confers them some capacity of reacting to perturbations and/or
deviations from desired trajectory. Nevertheless, one of the most serious drawbacks
of the PID controllers is the small ability to cope with changes in the plant. Any
change that occurs in the system usually causes the lost of the design specifications,
e.g. settling time, for which the controller was tuned. This is serious if we take into
account that many robotic applications are based on manipulating objects and, hence,
the payload of the arm might often change even during normal operation. This change
in the payload, as was seen in Chapter 2, changes dramatically the arm dynamics,
producing a bad behavior or even instability in the PID controlled system.
Therefore, smarter techniques must be applied if we pursue the robustness of the
system to these changes in their parameters (specially changes in the mass). The use
of robust controllers for active suppression of vibrations has been deeply studied and
developed for vibration control of flexible manipulators, attaining noticeable robustness
84 Robust control
properties.
Spatially robust H∞ (Halim and Moheimani, 2002a; Karimi et al., 2006) vibration
controllers have been tested experimentally. In these studies, vibration is eliminated by
selecting an appropriate cost functional indicating the vibration energy of the structure
and designing an optimal controller considering the limits on the actuator signals.
However, the use of H∞ norm in the design of robust controllers leads to conservative
control systems, whose behavior for the nominal case is usually far from optimal, and
the performance of such controllers is not often satisfactory. A more suitable criterion
for minimization is the H2 norm of performance index (Halim and Moheimani, 2002b).
However, design of robust H2 controllers is computationally intractable, and there is
no analytic solution for them (Paganini and Feron, 1999). To alleviate this problem
and obtain a controller with robust performance and stability properties, µ-synthesis
technique was investigated in (Karkoub and Tamma, 2001).
The intelligent control have also paid contributions to the robust control of flexible
robots. Specifically, neural networks have been studied in order to obtain model-free
controllers (Su and Khorasani, 2001), which, attending to their no-needed-model con-
dition, give out nice robustness properties but with high computational requirements.
Fuzzy logic has also given some contributions to this field (Lin and Lee, 1993; Lin and
Lewis, 2002), but tuning schemes involved are tedious, time consuming and usually
parameters are determined off-line and, hence, they might not provide optimal con-
troller performance during actual operation of the robot. Lately, fuzzy controllers have
been used jointly with another techniques such as sliding control or neural networks
again, introducing advanced concepts of intelligent control as the fuzzy-sliding mode
controller presented in (Chalhoub et al., 2006) or the hybrid fuzzy neural control dis-
cussed in (Subudhi and Morris, 2003; Wai and Lee, 2004), improving their performance
at the cost of complicated schemes.
All these robust controllers guarantee the stability of the system under payload
changes for a bounded, determined range of variation of the payload. The present
Chapter proposes a much easier to implement, new control idea which proves to be
robust independently of the tip mass transported by the link. The behavior of the
system is optimal for the nominal case and degrades very little when there are changes
even if these are significant. The robust controller has been developed ‘ad hoc’ for a
4.3. Robust controller 85
one degree-of-freedom flexible robot which can be modeled by a single vibration mode
(high payload/link mass ratio). It is based on a very simple, astute feedback of both the
strain measurements obtained from a couple of gauges and the motor angle provided
by the motor encoder, and presents inherent robustness to variations in the value of
the payload. In addition, the controller also exhibits robustness to uncertainties in the
estimated values of the parameters of the link, the motor or the motor controller.
4.3 Robust controller
This control scheme attempts to decouple motor and link dynamics in order to sim-
plify the whole problem. This is achieved making use of the motor controller described
in Section 3.2.1. Applying said scheme, the actuator (inner) loop can be assumed to
have a second order critically damped dynamics, M(s), given by (3.3), which is com-
pletely isolated from the vibration and displacements of the flexible link and, hence,
of the vibration (outer) controller. This manner, the inner loop rules the solid rigid
motion, while the outer controller deals with the deviations due to flexibility. The
general block diagram is displayed in Figure 4.1, where Ri(s) and Re(s) represent the
inner and outer loops controllers respectively, and its features were described in (Feliu
et al., 1990; Feliu et al., 1997). To mention the most important: a) the controller de-
sign is simplified to a great extent since it allows us to design the inner loop separately
from the outer one, thus dividing the control design process in two other much simpler
design processes, b) this scheme minimizes the effects on the motor angle of Coulomb
friction and unexpected changes in the dynamic friction (as demonstrated in (Feliu
et al., 1993)).
4.3.1 Outer loop
As was mentioned in the Section preamble, we will use measurements of the stress
provided by a couple of strain gauges placed at the base of the beam. These measure-
ments are used to estimate the coupling torque, Γcoup, between the arm and the motor,
which is used to implement:
a. the compensation term Vcoup in the motor control system, which decouples motor
86 Robust control
Re(s) Ri(s) Motor Gb(s)θtr θm
rθm θt+
-
+
-
Inner loop
Figure 4.1: General robust controller scheme.
M(s)θtr
_
+ θmr
θm θt
Γcoup
Re(s)+
+
P(s)
Gb(s)
H(s) C(s)
Figure 4.2: Basic scheme of the robust controller outer loop.
and link dynamics; and
b. the external control loop used to cancel the vibrations at the tip of the arm
(controller Re(s) of Figure 4.1)
Combining equations (2.37), (2.37) and (2.41), the relation between Γcoup and the
output θt is obtained
C(s) =Γcoup(s)
θt(s)= c
s2
ω20
(4.3)
This expression is used to close the outer loop as shown in Figure 4.2. Perturbations
are also considered, which are modeled in this figure as a signal P (s) which is added
to the motor position θm. We assume that this perturbation is a first order polynomial
in s, thus representing initial deviations in the angular position and velocity of the tip.
Blocks Re(s) and H(s) in Figure 4.2 are to be designed in order to achieve the required
dynamics.
4.4. Robustness to payload changes 87
Operating the blocks of Figure 4.2 we obtain the closed loop dynamics
θt(s) =Re(s)M(s)Gt(s)
1 +Re(s)M(s)Gt(s)H(s)C(s)θrt (s) +
Gt(s)
1 +Re(s)M(s)Gt(s)H(s)C(s)P (s)
(4.4)
that relates the output θt to the desired reference θrt , and the perturbation P (s). Sub-
sequently, design procedures will be proposed to calculate Re(s) and H(s).
4.4 Robustness to payload changes
This Section proposes an extremely simple control system that is completely in-
sensitive to payload changes. Let us substitute blocks Gt(s), M(s) and C(s) by their
transfer functions (2.40), (3.3), and (4.3) in expression (4.4) and rearrange terms. Then
it is obtained that
θt(s) =
Re(s)1
(1 + as)2ω20
s2 + ω20 +Re(s)
1
(1 + as)2cs2H(s)
θrt (s)+ω20
s2 + ω20 +Re(s)
1
(1 + as)2cs2H(s)
P (s)
(4.5)
Defining the control terms as
Re(s) =1
M(s)= (1 + as)2 (4.6)
H(s) = −1
c(4.7)
and operating, expression (4.5) becomes
θt(s) = θrt (s) + P (s) (4.8)
Then we have achieved an outer control loop that makes the output exactly fol-
low the reference in absence of perturbations. When there are perturbations of the
kind defined in the previous section (polynomial Laplace Transforms) the Final Value
88 Robust control
Theorem states that the error will become zero in the steady state. Moreover, notice
that the closed loop behavior defined by (4.8) is completely independent of the tip
payload. The parameters of the robot needed to design this control system are: the
stiffness of the bar, c, and the dynamics of the controlled motor, characterized by a,
both parameters being independent of the payload. It has to be mentioned as well that
the parameters of the inner loop control system are also independent of the payload
value as the compensation term, Vcoup, makes the motor dynamics independent of what
happens in the link.
Otherwise the resulting closed loop system is very critical as it has been obtained
by several cancelations of zeros and poles that cannot be exactly characterized in an
experimental setup. For example if we assume that the estimation of the bar stiffness,
ce, does not agree with its real value, c, the closed loop transfer function would be
(assuming no perturbations):
θt(s)
θrt (s)=
ω20
(
1− c
ce
)
s2 + ω20
(4.9)
This result shows that a) if ce > c the closed loop system exhibits two complex
conjugate poles located over the imaginary axis, the system then being marginally
stable; b) if ce < c the closed loop system exhibits two real poles of opposite sign,
the system then being unstable. The critical nature of this control scheme makes it
necessary to carry out a study of the stability of the system, and look for a control
solution robust to small changes in robot parameters (other than the payload).
4.5 Robustness to small changes in system param-
eters
In order to perform the stability study, we define the following control law, which
is a generalization of the one developed in the previous section:
Re(s) = (1 + βs)2 (4.10)
4.5. Robustness to small changes in system parameters 89
ω02
s2 + ω02
ω02
s2c ·
θtr
+
+ θmr
θm θt
Γcoup
++
P(s)
(1+βs)21
(1+αs)2
c
1 α2
β2µ
Figure 4.3: Outer loop scheme that keeps stability when parameters vary.
H(s) = −1
c
a2
β2µ (4.11)
where β and µ are the parameters to be designed in order to achieve closed loop stability
when changes are produced in robot parameters. Figure 4.3 shows the detailed block
diagram.
First, robustness is studied for the case of imperfect tuning of the two parameters
used in the design of the controller of the previous section: a and c. Then robustness
is studied in the case that motor parameters (Jm and ν) change. This causes the inner
control loop to become untuned making M(s) change to a system with two different
real poles, or two complex conjugate poles. Robustness will be studied in all cases in
the sense of assuring closed loop stability.
4.5.1 Robustness to errors in tuning the controller parameters
First we study the stability margins of the control system given by (4.10) and (4.7).
This control is a particular case of control (4.10) and (4.11) just making
µ =β2
a2(4.12)
Operating in (4.5) now we obtain
θt(s)
θrt (s)=
(1 + βs)2 ω20
(1 + as)2 (s2 + ω20)
2 − (1 + βs)2 s2(4.13)
The stability of this system is studied by applying the Routh criterion (Ogata,
90 Robust control
2001) to its characteristic polynomial:
Pc(s) =(a2 − β2
)s4 + 2 (a− β) s3 + a2ω2
0s2 + 2ω2
0as+ ω20 (4.14)
The obtained stability conditions are
a > 0 (4.15a)
−a < β <−a
ω20a
2 − 1(4.15b)
ω0a >√2 (4.15c)
which raise a serious problem. Section 4.4 proved that the proposed control works
correctly when the phase advance network is perfectly tuned with the motor dynamics
(β = a). Nevertheless condition (4.15b) states that β has to be of the opposite sign of
a in order to guarantee stability. In order to obtain good dynamic behavior robust to
payload changes, we can afford small deviations in the tuning of Re(s): β ≃ a, but not
a strong deviation as condition (4.15b) demands.
This fact justifies the inclusion of parameter µ as a new degree of freedom of the
control system. Controller H(s) in (4.11) also includes a normalization factor a2/β2.
If H(s) is of the form (4.11) the closed loop transfer function is
θt(s)
θrt (s)=
(1 + βs)2 ω20
(1 + as)2 (s2 + ω20)− (1 + βs)2 s2
a2
β2µ
(4.16)
and the characteristic polynomial is
P (s) = a2 (1− µ) s4 + 2a
(
1− µa
β
)
s3 +
(
1 + a2ω20 − µ
a2
β2
)
s2 + 2aω20s+ ω2
0 (4.17)
Routh criterion is applied again giving the following stability conditions:
1 > µ (4.18a)
4.5. Robustness to small changes in system parameters 91
β
a> µ (4.18b)
β2
a2+ β2ω2
0 > µ (4.18c)
a2ω20µ
(
1− a
β
)
+
(
1− µa
β
)(
1− µa2
β2
)
> 0 (4.18d)
(
β2ω20 +
β
a− µ
)(a
β
)2(
1− a
β
)
> 0 (4.18e)
which yield to
Case a < β. In this case conditions (4.18b), (4.18c) and (4.18d), are included
in (4.18a). Condition (4.18e) is also verified if (4.18a) is fulfilled, since it can be
simplified to
β2ω20 +
β
a> µ
In summary, in this case there is a single stability condition given by
µ ∈ (0, 1) (4.19)
Case a > β. In this other case, condition (4.18d) becomes
β2ω20 +
β
a< µ
which contradicts (4.18b). Then the closed loop system is always unstable in this case.
From this analysis we can conclude that choosing β ≥ a and µ as stated in (4.19)
makes the closed loop system stable. These two parameters can be tuned in such a way
that the system remains stable in the cases of: a) lack of precision in the knowledge of
parameter a, which describes the controlled motor dynamics; and b) lack of precision
in the estimation of parameter c. The cost to be paid is that closed loop dynamics
will become a bit worse than in (4.8), and we no longer get the perfect tracking of
92 Robust control
Section 4. But this ideal behavior can be approximated by choosing values of β close
to a and µ close to 1.
4.5.2 Robustness to changes of the motor parameters
Next the case in which motor parameters Jm and ν change will be considered.
Variation of these parameters produce changes in the inner loop transfer function in
the sense that now
M1(s) =1
(1 + a1s) (1 + a2s)(4.20)
where a1 and a2 are two different parameters that can be real or complex conjugate.
We mention that - in particular - friction coefficient ν often experiences large variations
through time.
The closed loop transfer function is now
θt(s)
θrt (s)=
(1 + βs)2 ω20
(1 + a1s) (1 + a2s) (s2 + ω20)
2 − (1 + βs)2 s2a2
β2µ
(4.21)
Application of the Routh criterion now leads to complicate stability conditions
that cannot be easily used for control design. Then we will use the Nyquist criterion in
order to derive sufficient conditions for stability. The characteristic equation of (4.21)
expressed in the frequency domain (s = jω) is
1 +ω2
ω20 − ω2
(1 + jβω)2
(1 + ja1ω) (1 + ja2ω)
a2
β2µ = 0 (4.22)
The resulting Nyquist plot is shown in Figure 4.4. This plot shows that the closed
loop system is stable if the following sufficient conditions are verified:
a. Denoting
L1(ω) =(1 + jβω)2
(1 + ja1ω) (1 + ja2ω), (4.23)
and ∠ the phase of a complex number:
∠L1(ω) ≥ 0, ∀ω. (4.24)
4.5. Robustness to small changes in system parameters 93
I
II
III
IV
V
VI
α2
α1α2µ
Figure 4.4: Nyquist plot when having two different poles in the motor dynamics.
b.
limω→∞
(
ω2
ω20 − ω2
(1 + jβω)2
(1 + ja1ω) (1 + ja2ω)
a2
β2µ
)
< 1 (4.25)
To fulfill condition (a) it is sufficient that: i) ∠L1(0+) ≥ 0; and ii) ∠L1(ω) does not
change its sign in 0+ ≤ ω < ∞. The phase ∠L1(0+) is positive if
β >a1 + a2
2(4.26)
Equation (4.23) can be expressed as
L1(ω) =2β (a1 + a2)ω
2 − (1− β2ω2) (1− a1a2ω2)
1 + (a21 + a22)ω2 + a21a
22ω
4+
+ jω [2β (1− a1a2ω
2)− (1− β2ω2) (a1 + a2)]
1 + (a21 + a22)ω2 + a21a
22ω
4(4.27)
whose numerator has a phase always greater than zero if
94 Robust control
β >2a1a2a1 + a2
(4.28a)
β >a1 + a2
2(4.28b)
It is easy to show that condition (4.28a) is included in condition (4.28b), which is
the same as (4.26). Then fulfillment of condition (4.26) guarantees the verification of
condition (a). Condition (b) is equivalent to
µ <a1a2a2
(4.29)
Sufficient conditions (4.26) and (4.29) are valid both in the case that the two poles
are real, and in the case that the two poles are complex conjugate. In this last case
a1,2 = ar ± jai and the previous conditions become
β > ar (4.30)
µ <(a2r + a2i )
a2(4.31)
Conditions obtained in this section must be verified for all the range of variation
of a coefficients. The extreme values of a1 and a2 are obtained from the range of
variation of the motor parameters and we use them to design the values of β and µ
that guarantee stability in that interval.
4.6 Simulation
Methods developed in the previous sections will be illustrated next with several
simulations. Simulations of the ideal case are omitted since we have proven that we
will get perfect tracking completely insensitive to payload changes.
Let us consider for simulation the flexible robot described in Section 2.6.1, whose
nominal characteristics are presented in Tables 2.2 (duraluminium beam and nominal
payload) and 2.3 (DC motor). The dynamic model values for the nominal case are
detailed in Table 4.1. Let us also assume that the tip mass vary between 1 and 5 kg,
Figure 4.7: System response of our robust control scheme designed to guarantee sta-bility in a margin of ±10% error in c estimation when payload changes.
negligible, and the tracking of the reference is excellent.
These results are compared with the ones obtained from two standard controllers.
First an ideal PD controller has been designed for our system. Its control law is given
by
θrm(t) = Kpe
(
θrt −Γcoup
ce
)
−Kve
1
ce
dΓcoup
dt(4.36)
The PD parameters have been tuned to place the two dominant poles as close as
possible to the dominant poles of our control scheme (s = −13.88 ± j38.09), in order
to compare the degradation of the response from the same ”starting point”. This is
achieved with PD parameters Kpe = 0.85 and Kve = −0.025. Figure 4.8 shows the
responses obtained under the same conditions as before. Now the closed loop system
exhibits a transient behavior that is remarkably worse than in our control scheme, and
it even becomes unstable for a payload of 1 kg.
Next we designed a PID controller with the same criterion used in the previous PD
controller: to place two poles of the system in the same place as the dominant poles of
our control scheme. The control law is now
4.6. Simulation 99
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
3
3.5
4
1.8 2 2.2 2.4 2.6 2.83
3.05
3.1
3.15
3.2
3.25
3.3
3.35
3.4
t (s )
θt
(ra
d)
θrt
m = 1 kg
m = 2 kg
m = 3 kg
m = 4 kg
m = 5 kg
Figure 4.8: System response with a PD controller when payload changes.
θrm(t) = Kp
(
θrt −Γcoup
ce
)
−Kv
1
ce
dΓcoup
dt−Ki
1
ce
∫
Γcoupdt (4.37)
and the PID parameters are tuned to Kp = 1.033, Kv = −0.004, and Ki = 31.022.
Simulation results are shown in Figure 4.9, where it is observed that the PID scheme
has a good behavior in terms of stability and damping, but the reference is tracked
with a delay that increases as the tip mass grows.
From previous simulations we can conclude that the control scheme proposed in
this paper exhibits a better dynamic behavior when payload changes than standard
controllers do, being simpler than adaptive control solutions.
Figure 4.10 shows the response of the control system to a perturbation given by an
initial deviation of the tip position. In order to avoid permanent deformations in the
bar, tip deflections have to be less than 5 degrees, (0.0872 rad) Then simulations are
carried out in the worst case of θt0 = 0.0872 rad, and for different values of the tip
mass. This figure shows that as the tip mass increases the response becomes slower,
but the overshooting experiences small variations.
All previous simulations have been carried out under the assumption that ce = c.
In Figure 4.11 the closed loop responses for different ce values and the nominal mass
100 Robust control
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
3
3.5
1.9 2 2.1 2.2 2.3 2.4
3.05
3.06
3.07
3.08
3.09
3.1
3.11
3.12
3.13
3.14
t (s )
θt
(ra
d)
θrt
m = 1 kg
m = 2 kg
m = 3 kg
m = 4 kg
m = 5 kg
Figure 4.9: System response with a PID controller scheme when payload changes.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.06
0.04
0.02
0
0.02
0.04
0.06
0.08
0.1
t (s )
θt
(ra
d)
m = 1 kg
m = 2 kg
m = 3 kg
m = 4 kg
m = 5 kg
0.0873 rad = 5 º
Figure 4.10: System response of our robust control scheme with initial deflection dif-ferent from zero.
4.6. Simulation 101
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
3
3.5
θrt
ce
= 0.9c
ce
= 0.95c
ce
= c
ce
= 1.05c
ce
= 1.1c
1.85 1.9 1.95 2 2.05 2.1 2.15
3.136
3.138
3.14
3.142
3.144
3.146
3.148
t (s )
θt
(ra
d)
Figure 4.11: System response of our robust control scheme designed to guarantee sta-bility in a margin of ±10% error in c estimation when c estimation is not correct.
are drawn. This figure shows that the transient response has practically imperceptible
variations due to these errors. Responses with the standard controllers will not be
displayed because, as happens with our robust control scheme, deviations from nominal
behavior are negligible.
4.6.2 Errors in viscous friction estimation
Here we study the behavior of our control system when there are variations in the
viscous friction parameter ν of up to 100% (from 0 to 2ν0 being ν0 the nominal value).
In this case the extreme values of the controlled motor parameters are
α1,2 = −0.02627± j0.00916 (4.38a)
α1 = −0.01996 α2 = −0.03880 (4.38b)
These values applied to equations (4.26), (4.29), (4.30), and (4.31) give that β > 0.0294
and µ < 1. We choose β = 0.03 and µ = 0.99. Simulation results are drawn in
102 Robust control
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
3
3.5
t (s )
θt
(ra
d)
θt
r
νe
= 0
νe
= 0.5ν
νe
= ν
νe
= 1.5ν
νe
= 2 ν
1.86 1.88 1.9 1.92 1.94 1.96 1.98 2 2.02 2.04
3.135
3.136
3.137
3.138
3.139
3.14
3.141
3.142
3.143
3.144
3.145
Figure 4.12: System response of our robust control scheme designed to guarantee sta-bility in a margin of ±100% error in ν value when ν estimation is not correct.
Figure 4.12 for different friction values and nominal payload. They show very small
variations in the transient response as ν varies.
We have to mention that effects of friction changes on the dynamics of the closed
loop robot are very reduced as a consequence of the two nested loop control scheme
described in Section 3 where the inner loop is closed with a high gain controller, as was
demonstrated in (Feliu et al., 1993). The parameters of the outer loop have been tuned
in this subsection in order to avoid instabilities produced by small changes in M(s)
which are a consequence of the large changes that friction coefficient ν experiences.
Simulations of the robot controlled with the previous two standard controllers (PD
and PID) when ν changes give responses that vary very slightly from the nominal
responses, so these plots will not be displayed.
4.6.3 Effects of noise in the feedback signal
In practice, gauge signals are usually very noisy. Therefore, the double differen-
tiation of this signal that Re(s) implementation involves can be the source of serious
problems for our control system. This is studied in the next simulation by adding a
4.6. Simulation 103
0 0.5 1 1.5 2 2.5 30. 5
0
0.5
1
1.5
2
2.5
3
3.5
t (s )
θt
(ra
d)
θt
θt
r
2 2.2 2.4 2.6 2.8 3
3.126
3.128
3.13
3.132
3.134
3.136
3.138
3.14
3.142
3.144
Figure 4.13: Noise effect of strain gauge signal in the system response.
white noise of zero mean and standard deviation σ = 0.4476 to the coupling torque
signal. These noise characteristics correspond to a real strain gauge measurement
equipment used in (Garcıa, 1999). Simulated results are drawn in Figure 4.13, which
shows that noise has a small effect, and behavior remains good. Then additional fil-
tering of this signal is not necessary (apart from the standard filtering of the signal
conditioning equipment).
4.6.4 Effects of using a more complex dynamic model
Next, the effects of supposing that link mass is not negligible are shown. To achieve
this, a two concentrated mass model is used, assuming the bar mass to be placed in the
middle of the bar, whose mathematical description can be obtained from the generic
lumped masses model presented in Section 2.4. The new model is described by the
following equations
θt(s)
θm(s)=
γ(−1
4m1s
2 + 74γ)
m1ms4 + (m1 + 8m) γs2 + 74γ2
(4.39a)
θtV (s)
=Kγ
(−1
4m1s
2 + 74γ)
∆(4.39b)
104 Robust control
θm(s)
V (s)=
K(m1ms4 + (m1 + 8m) γs2 + 7
4γ2)
∆(4.39c)
where
∆ = Jmm1ms6 −m1mνs5 + γ
(
Jmm1 + 8Jmm+1
2l2m1m
)
s4−
− (m1 + 8m) γνs3 +7
4γ2
[
Jm + l2(1
4m1 +m
)]
s2 − 7
4γ2νs
(4.40)
In this equation m1 refers to the link mass, and γ is a constant value given by
the expression γ = 32c7l2
. In our particular case, the mass of the aluminium bar used
is m1 = 0.3 kg. The dynamics of this model is given by the Table 4.2, where it is
shown that the additional poles and zeros are far away from the dominant mode and
its amplitudes in the response signal are much lower. These facts allow us to assume
the major order dynamics to be negligible. On the other hand, if we compare the
first mode values for this model with data in Table 4.1, we observe that they are very
similar, and the errors between them are very slight (less than 0.5% for the natural
frequency). In spite of this, we will carry out the simulations with the purpose of
checking the behavior of the proposed control system.
When this model is introduced in our control scheme, the system proves to be
unstable unless we select µ < 7/16, but this fact causes an important deterioration of
the input tracking. In order to solve this deterioration, the control term Re(s) (4.32)
is modified as follows
Re(s) =(1 + β ′s)2
(1 + δ′s)3(4.41)
With this controller, the new Nyquist plot has the pattern displayed in Figure 4.14,
where the influence of the zeros of Re(s) appears after the first resonant frequency
ω1. Subsequently, the poles of the controller become dominant, approaching zero by
−π2radians. From then on, the second natural frequency dominates the frequency
response. As can be seen in the figure, appropriately choosing parameters β ′ and δ′,
we can assure our system stability, even taking into account more natural frequencies.
However, the system response is modified by this controller, because β ′ parameter
Table 5.1: Identification results for the single mass model with the single mass estima-tor.
in Table 5.1 while the evolution along time of standard deviation of estimation and
payload are displayed in Figure 5.4 and Figure 5.5 respectively.
We observe in Figure 5.4 that mass estimation steadies very quickly, typically be-
fore 0.4 seconds, and its value is very near to the actual payload, always below the 4%
error. This percentage of error is not significant and is due to the use of statistical
measurements, which introduce some inaccuracies in addition to the criteria for stop-
ping the identification. The specific values can be checked in Table 5.1. In the graphs
it is also noticeable that the estimation seems not to work properly at some periodical
instants. These moments correspond to the zero crossings of the tip acceleration, which
can be observed in Figure 5.6 (which is used in the denominator of the mass estimator
in the general expression (5.4)), as our arm is simulated to maneuver in open loop
mode (only the motor position loop is closed). In a tip controlled motion, as it will be
shown afterwards with the addition of a PD adaptive controller, these zero crossings
rarely take place.
Concentrated masses model with single mass estimator
Once the correctness of the algorithm has been verified for the single mass model, a
more detailed model for the flexible robot is now analyzed. Our purpose here is to study
the performance of low order estimators when dealing with higher order models. To
carry out the simulation tests, the concentrated masses model outlined in Section 5.3.2
has been adopted, and a two masses state-space model has been calculated for the
flexible arm. The state-space equations are given by
128 Adaptive control
0 0.5 1 1.5 20
0.01
0.02(a)
0 0.5 1 1.5 20
0.01
0.02(b)
0 0.5 1 1.5 20
0.01
0.02(c)
Mass (
kg)
0 0.5 1 1.5 20
0.01
0.02(d)
0 0.5 1 1.5 20
0.01
0.02(e)
Time (s)
Standard deviation of last n samples of massAcceptance limit
test
= 0.200 s test
= 0.250 s
test
= 0.284 s test
= 0.316 s
test
= 0.334 s
Figure 5.4: Standard deviations of the identification process for a single mass modelfor the following masses: (a) 0.021 kg; (b) .061 kg; (c) 0.100 kg; (d) 0.150 kg; and(e) 0.200 kg.
θ1
θ2
θ1
θ2
=
0 0 1 0
0 0 0 1
−15367
EIλMpL3
4807
EIλMpL3 0 0
1207
EIMpL3 −96
7EI
MpL3 0 0
θ1
θ2
θ1
θ2
+
0
02887
EIλMpL3
−247
EIλMpL3
θm (5.38)
θ1
θ2
Γcoup
=
1 0 0 0
0 1 0 0
− 727L
247L
0 0
θ1
θ2
θ1
θ2
+
0
0487L
θm (5.39)
Then, the results of the tip mass identification are summarized in Table 5.2. Obvi-
ously, as the ratio between the tip mass and the beam mass decreases, the tip estimation
worsens and the relative error augments because the beam mass interferes and changes
significantly the main natural frequency of the system. However, the errors are still
acceptable and quite good in most of the studied cases. In these estimated values it
has been taken into account that the inertia seen from the basis of the beam is slightly
5.5. Simulation results 129
0 0.5 1 1.5 20
0.02
0.04
(a)
0 0.5 1 1.5 20
0.05
0.1(b)
0 0.5 1 1.5 20
0.05
0.1
0.15(c)
Mass (
kg)
0 0.5 1 1.5 20
0.1
0.2(d)
0 0.5 1 1.5 20
0.1
0.2
(e)
Time (s)
Estimated massLast n samples mean massActual tip mass
Figure 5.5: Mass estimation in open loop for a single mass model. Tip masses: (a) 0.021kg; (b) 0.061 kg; (c) 0.100 kg; (d) 0.150 kg; and (e) 0.200 kg.
greater than the actual tip inertia, due to the addition of the beam mass into the
model. Therefore, a correction factor has been applied to the estimations. Specifically,
the estimation of the real payload is given by m = me − 14mb, where me is the estima-
tion given by (5.30) for the payload. Figures containing the standard deviation and the
mass estimation of the identification process are not displayed as they are very similar
Table 5.5: Identification results for the distributed masses model with the two massesestimator.
Distributed masses model
The same analysis performed in previous section is now carried out for a distributed
masses model for the flexible link, which is truncated at the second vibrational mode.
This model is based on the pseudo-pinned formulation described in Section 2.3. The
evolution of the tip mass estimation is shown in the following graphs. Specifically,
Figure 5.7 and Figure 5.8 show the tip mass identification by means of a single mass
model estimator for a robot model truncated at its second vibrational mode and Table
5.4 presents the numerical results.
Finally, the estimator obtained from the two masses model has also been applied
to the distributed masses model truncated on the second mode of vibration, with the
results shown in 5.5. These results are very close to those achieved with a simpler
estimator. The graphs of the identification process are omitted again as they are again
very similar to those of Figures 5.7 and 5.8.
132 Adaptive control
0 0.5 1 1.5 20
0.01
0.02(a)
0 0.5 1 1.5 20
0.01
0.02(b)
0 0.5 1 1.5 20
0.01
0.02(c)
Mass (
kg)
0 0.5 1 1.5 20
0.01
0.02(d)
0 0.5 1 1.5 20
0.01
0.02(e)
Time (s)
Standard deviation of last n samples of massAcceptance limit
test
= 0.330 s
test
= 0.452 s
test
= 0.524 s test
= 0.592 s
test
= 0.638 s
Figure 5.7: Standard deviations of the identification process for a distributed massesmodel truncated in two vibration modes with a single mass estimator. Tip masses:(a) 0.021 kg; (b) .061 kg; (c) 0.100 kg; (d) 0.150 kg; and (e) 0.200 kg.
0 1 2 3 40
0.05
0.1(a)
0 1 2 3 40
0.1
0.2(b)
0 1 2 3 40
0.1
0.2
(c)
Mass (
kg)
0 1 2 3 40
0.1
0.2
0.3(d)
0 1 2 3 40
0.2
0.4(e)
Time (s)
Estimated massLast n samples mean massActual tip mass
Figure 5.8: Mass estimation for a distributed masses model truncated in two vibrationmodes with single mass estimator. Tip masses: (a) 0.021 kg; (b) .061 kg; (c) 0.100 kg;(d) 0.150 kg; and (e) 0.200 kg.
5.5. Simulation results 133
M(s) G(s)
m
t
Kvs
Kp
c1
c
1+Kp
Kp
t
ref
+ +
+
+
+
+
m
ref
Noise
_
_
_
_
System plant
c
Estimation
block
Figure 5.9: Adaptive outer control loop based on payload identification.
As a consequence of the simulation results presented in this section, it is derived
that both, the simple and the two masses estimators, produce a similar identification
of the payload. Theoretically, the complete estimator should approximate better the
solution, as it is based on the model used for simulating the link, but, in the end, the
numerical errors accumulate because the estimator becomes more complicated and,
hence, the final errors are similar. The obvious choice is then to use the simplest
estimator, and this solution will be adopted in the application example.
5.5.2 Application to adaptive control
To illustrate the usefulness of this identification algorithm, an adaptive PD con-
troller has been designed for the vibration control loop of the flexible robot, as shown
in Figure 5.9. The experimental rig described in Section 2.6.2 will be used in exper-
imentation. This platform can be accurately simulated with the single point mass
model of Section 2.4.2, so this will be the adopted model in this Section.
Firstly, we calculate the PD constants, Kpe and Kve, for the nominal mass, m =
60 g, with the following specifications: no overshoot, Mp = 0%, and small settling time,
ts = 1 s. The results are Kpe = −0.7963 and Kve = 0.07447. The tip position is not
directly measured, but estimated with the aid of the coupling torque and the motor
angle measurements by means of the following expression
134 Adaptive control
θte = θm − Γcoup
c(5.40)
which is obtained from (2.37).
A noise term is added to the coupling torque to simulate the experimental measures
of the strain gauges, which are very noisy. From the experimental measurement of our
sensing system, the variance of this noise have been set to 10−5 Nm. Therefore, we now
need to filter the mass estimation as explained in Section 5.4.3. The selected cut-off
frequency for the estimator filter is ωf = 10 rad/s, what, applied to (5.36) and (5.37),
and particularized for the a sample time T = 0.002 s, yields
F (s) =
(10
s+ 10
)2
⇒ Z (F (s))T=0.002∼=
0.01982z2
z2 − 1.961z + 0.961(5.41)
Z(s2F (s)
)
T=0.002∼=
98.0232z2 − 196.046z + 98.023
z2 − 1.961z + 0.961(5.42)
Now we will perform the maneuvers for each of the masses considered for the system,
that is, from three times lighter than nominal, mm = 20 g, to more than three times
heavier, mM = 202 g. In Figure 5.10 the tip mass position during the maneuver when
using the PD controller for the nominal case is displayed. It is noticeable the high
overshoot that appears for payloads bigger than nominal, reaching the 14% for the
200 g mass. On the other side, the lightest mass performs slightly slower than the
nominal one. These results advise the use of a control scheme insensitive to payload
variations, such as adaptive control, to fulfill the requirements imposed to the system
for the whole range of masses. Figure 5.11 shows the control signal applied for each
simulation. Due to the noisy nature of these signals, it is very difficult to discriminate
any differences between the different masses. This voltage control is actually very
similar for all of them, but slightly delayed in time for the bigger masses.
Taking advantage of the identification algorithm (5.30), we will determine the ap-
propriate controller for each experiment. The simulation begins with the nominal PD
parameters and, whenever the estimation process finishes, these parameters are tuned
according to the actual payload, as shown in Table 5.6. In case the estimated mass
were not one of the those values, it would be obtained by linear interpolation. The
results are displayed from Figure 5.12 to Figure 5.14. From comparing the system re-
5.5. Simulation results 135
0 1 2 3 4 5 6 7 8 9 10-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
Tip
angle
(ra
d)
0.021 kg0.061 kg0.100 kg0.150 kg0.200 kg
Figure 5.10: Simulation of the tip mass position with a nominal PD controller fordifferent tip masses.
sponses with and without adaptive control, the improvements are obvious. Overshoot
has been completely removed while settling time has been reduced and kept nearly the
same for every mass. The estimation time in closed loop maneuvers is very similar to
its counterpart in open loop maneuvers, (around 0.5 sec), while the estimation error is
slightly bigger as can be observed in Table 5.7.
136 Adaptive control
0 1 2 3 4 5 6 7 8 9 10-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
Time (s)
Contr
ol sig
nal (V
)
20 g60 g100 g150 g200 g
Figure 5.11: Simulation of the control signal with nominal PD controller for differenttip masses.
0 1 2 3 4 5 6 7 8 9 10-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
Tip
angle
(ra
d)
0.021 kg0.061 kg0.100 kg0.150 kg0.200 kg
Figure 5.12: Simulation of the tip mass position with adaptive PD controller for dif-ferent tip masses.
5.5. Simulation results 137
0 1 2 3 4 5 6 7 8 9 10-0.5
0
0.5
1
Time (s)
Contr
ol sig
nal (V
)
0.021 kg0.061 kg0.100 kg0.150 kg0.200 kg
Figure 5.13: Simulation of the control signal with adaptive PD controller for differenttip masses.
0 0.5 1 1.5 20
0.02
0.04
(a)
0 0.5 1 1.5 20
0.05
0.1(b)
0 0.5 1 1.5 20
0.05
0.1
0.15
Mass (
kg)
(c)
0 0.5 1 1.5 20
0.1
0.2(d)
0 0.5 1 1.5 20
0.1
0.2
Time (s)
(e)
Estimated massLast n samples mean mass
Figure 5.14: Simulation of the mass identification process when the system is governedby a PD controller with different payloads. Tip masses: (a) 0.021 kg; (b) .061 kg;(c) 0.100 kg; (d) 0.150 kg; and (e) 0.200 kg.
Figure 6.7: Effect of the actuator in the wave absorb control scheme. Dashed linesrepresent system behavior using an ideal actuator (M(s) = 1) while solid lines representreal actuator
6.4. WBC applied to a non-linear system 151
0 1 2 3 4 5 6 7 8 9 100
0.2
0.4
0.6
0.8
1
angl
e (r
ad)
0 1 2 3 4 5 6 7 8 9 100.95
1
1.05
t (s)
angl
e (r
ad)
mt = 0.020
mt = 0.060
mt = 0.100
mt = 0.200
Figure 6.8: Wave absorb robustness to changes in the payload. Top: System response.Bottom: Zoom of the target position
and Lang, 1998) or gantry cranes (O’Connor, 2003). However, we are interested in the
performance of the scheme when dealing with mechanical systems (manipulators) sub-
jected to large displacements and, hence, exhibiting geometrical non-linearities during
their motion.
In order to analyze the system response, a set of simulations have been carried
out using the numerical analysis package MATLABr. The model parameters have
been extracted from the experimental platform presented in Section 2.6.2. Data of
the arm were previously detailed in Table 2.6. The non-linear model used in simula-
tion was described in Section 2.5 and its parameters, obtained from numerical fitting
from experiments as detailed in (Payo, 2008) were summarized in Table 2.6, while
the WBC controller parameters are shown in Table 6.1. A non-linear model has been
used for simulating the physical model, and controller parameters have been calculated
neglecting the nonlinearities and, hence, assuming that ω20 = α.
For the input launching wave, simple ramps with different slopes, p, were used.
Specifically, in the simulations we will use ramps from p = 1/10 to p = ∞ (step input).
In Figure 6.9, both launching waves, desired and calculated, returning wave and
actuator position for each input reference are displayed. We can see that differences
152 Wave-based control
m k = α ·m c =√k ·m
m = 43.8 g k = 0.3850 N/m c = 0.0169 N·s/m
Table 6.1: Wave-absorb controllers parameters
0 2 4 6 8 10
0
0.2
0.4
0.6
0.8
1
t (s)
angl
e (r
ad)
s = 0.10
0 2 4 6 8 10
0
0.2
0.4
0.6
0.8
1
t (s)
angl
e (r
ad)
s = 0.50
0 2 4 6 8 10
0
0.2
0.4
0.6
0.8
1
t (s)
angl
e (r
ad)
s = 2.50
0 2 4 6 8 10
0
0.2
0.4
0.6
0.8
1
t (s)
angl
e (r
ad)
s = ∞
Input launching waveCalculated launching waveReturning waveActuator position
Figure 6.9: Waves for the different input ramps
between input and calculated launch waves are almost imperceptible. It is also notice-
able that in all cases expect for the least demanding reference (p = 0.1), the returning
wave has not fully developed by the time the launching wave has arrived at its desired
value, and, as a result, there is some residual oscillation of the tip on arrival at the
target position (which in any case quickly diminishes). With the less severe ramp, even
better vibration control is achieved.
Another relevant phenomenon appears in these graphs. At first sight we could think
that the performance of the control is exactly equal to the one achieved for the linear
systems, but the steady-state value of the returning wave has a small error when the
input is demanding. To examine this, the system responses for the different slopes are
compared in Figure 6.10, where we can see that the control strategy performs very well
for vibration reduction in all trajectories. However an accuracy problem arises when the
nonlinearity comes into play and becomes bigger as the reference is more demanding.
6.5. Correcting the steady-state error 153
0 5 10 150
0.2
0.4
0.6
0.8
1
angl
e (r
ad)
s = 0.1s = 0.25s = 0.5s = 2.5s = ∞
0 5 10 150.98
0.99
1
1.01
1.02
angl
e (r
ad)
t (s)
Figure 6.10: Wave absorb control of a nonlinear system with different references. Top:System response. Bottom: Zoom at the target position
The returning wave is not correctly calculated, and does not fulfils the basic condition
of WBC given by equation (6.6). For not very demanding demanding trajectories, the
error can be neglected in most applications, (smaller than 0.001 radians), but, as we
make the trajectory steeper it increases and reaches a maximum around 0.012 radians
for the step input, and this error could be unacceptable.
6.5 Correcting the steady-state error
As commented, the steady-state problem makes its appearance only when the tra-
jectory is extremely demanding, i.e. close to a step input. The use of such kind of
input is questionable, since there are no clear benefits in using steps as the system
then becomes noticeably more oscillatory while the improvement in the rise time is not
very significant, even if the actuator can actually achieve such a demanding maneuver.
It is preferable to use a ramp, or even a higher order trajectory to obtain better results.
However, apart from the error size, the occurrence of the error raises an issue about
the effectiveness of the wave-absorb method when applied to nonlinear systems. We
will therefore propose here three possible solutions for removing this error. All of them
154 Wave-based control
are tested with step inputs to analyze the worst case.
6.5.1 Addition of a linear element
It has been observed throughout simulation that, as long as the only measurements
used in our control are X0 and X1, if we have a linear element at the interface between
actuator and non-linear system, the control algorithm will perform correctly indepen-
dently of the behavior of the rest of the system. That is, the rest of the links may
have arbitrarily nonlinear behavior and the control will still work, provided only that
the system returns to its original state of strain, and so the initial and final spring
extensions are equal.
So the first solution to overcome final error is “simply” to add a linear mass-spring
element between the actuator and the beginning of the flexible system, whose parame-
ters approximately match those of the real system to keep the dynamic mismatch small.
For the system under test, this amounts to increasing it from one mass-spring model to
a chain of two subsystems, being the first linear and the second non-linear. The results
obtained in simulation are shown in Figure 6.11, where it can be observed that the
steady-state error has been eliminated. Besides the additional hardware required, the
extra degree of freedom causes the system to become slightly more oscillatory before
settling perfectly at the target.
This solution may present high hardware difficulties when applied to the experi-
mental setup, but it proves to be effective in theory and in simulation.
6.5.2 Performing a second movement
We may accept the error as a first approach to our target and then perform a second
manoeuvre to position the tip more accurately. The second motion does not have the
inaccuracy problem, as it is much less demanding than the first and the nonlinear
effects are now really negligible: that is, the system behaves, to all effects, as linear.
This solution takes a longer time and is not very elegant, but it still works. In most
cases it would be preferable to use a ramp reference, which would arrive at the desired
position quicker than the two-stage approach and would be less vibratory. However,
this two-stage solution could be of interest, for example, when dealing with actuators
6.5. Correcting the steady-state error 155
0 2 4 6 8 10 12 14 16 18 200
0.2
0.4
0.6
0.8
1
Ang
le (
rad)
0 2 4 6 8 10 12 14 16 18 200.98
0.99
1
1.01
1.02
Time (s)
Ang
le (
rad)
Figure 6.11: Top: System response when a linear element is added between actuatorand nonlinear element. Bottom: Detail of the response in the steady-state
whose velocities cannot be regulated. In this case we would need, in our experimental
platform, a sensor that reads the actual position of the tip. This measurement allows
us to quantify the required new movement. Supposing we can do that, the simulation
results are presented in Figure 6.12, where the adjusting motion has been carried out at
instant t = 14 s, when the system was nearly stabilized from the first manoeuvre. Note
that the input for this second step should, once again, be half the value of the error,
as we will still have the wave-absorb doubling effect, one due to the input launching
wave and the other to the returning wave.
Obviously, as this solution attempts not to enter into the nonlinear zone during
second maneuver, it is valid for any control designed for linear systems, not being a
property of WBC.
6.5.3 Force based redefinition of waves
The superposition principle and transfer functions have been utilized in the defini-
tion of the waves in the proposed controllers of WBC. As linearity is inherent to them,
the controller does not have the ability to cope with nonlinear systems. However, the
definition of waves in WBC is notional and not unique. Actually, it is possible to define
156 Wave-based control
0 2 4 6 8 10 12 14 16 18 200
0.2
0.4
0.6
0.8
1
Ang
le (
rad)
0 2 4 6 8 10 12 14 16 18 200.98
0.99
1
1.01
1.02
Time (s)
Ang
le (
rad)
Figure 6.12: Top: System response when performing a second manoeuver for accurateapproaching to the target. Bottom: Zoom of the response
launch and absorb waves, A and B, in a way that eliminates the need for WTFs.
If instead of a lumped system, we assume that the system is distributed in the first
mass-spring subsystem (at the interface with the actuator), it can be studied using the
classical partial differential equation of a wave in a continuum. The general solution
consists of two superposed waves that propagate in opposite directions. Adopting force
as the wave variable, they can be expressed as f+(x − vct) and f−(x + vct), where vc
is the wave propagation speed. Associated with each wave there is a medium velocity
waveform given by v = f/Z, Z representing the wave impedance. The total force of a
specific point is the sum of both force waves, while the velocity is equal to the difference
of velocities
f(t) = f+(t) + f−(t) (6.7a)
v(t) = v+(t)− v−(t) (6.7b)
Assuming that the boundary conditions are determined by the spring force, f , on
the right side, and the actuator motion on the left, v(t) = v0(t), rearranging terms
in (6.7) and integrating with respect to time, a new definition for the displacement
Figure 6.18: Top: Experimental responses of the linear system with the wave absorbscheme using different damping values in the G1(s) controller. Bottom: Detail of thesteady-state
inition of the WBC can be seen as an improved scheme of the wave absorb that presents
more robustness to uncertainties in the system/model and a natural management of
geometrical nonlinearities that appear in extremely flexible mechanical systems.
6.6 Experimental verification
After the discussion on the different methods to correct the steady-state error that
appears when controlling a nonlinear system with a wave-absorb strategy, some experi-
ments were performed on the real platform. These tests make use of the third proposed
solution, the wave redefinition, which turns out to be the most elegant and to give the
best performance.
6.6.1 Linear system
As a first step, the wave absorb scheme is revisited. A number of experiments with
different values of the damping value of G1(s) have been performed, and the results are
6.6. Experimental verification 161
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.2
0.4
0.6
0.8
1
θ t (ra
d)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.95
1
1.05
θ t (ra
d)
t (s)
mt = 44g
mt = 60g (Nominal)
mt = 100g
mt = 150g
mt = 200g
Figure 6.19: Top: Experimental responses of the linear system with the wave absorbcontrol and payloads different from nominal. Bottom: Detail of the steady-state
presented in Figure 6.18, all of them using the nominal mass of the carbon fiber setup,
mt = 60 g. The output of the system is essentially the same as is obtained through
simulation in Figure 6.8. Again, the best results are achieved with c =√k ·m. With
this value of damping for the controller, the robustness to changes in the payload has
been tested, and the results are shown in Figure 6.19. The experiments prove the
control system to perform as expected from simulations.
Then, similar experiments were carried out using the force definition of waves.
Figure 6.20 looks for the best wave impedance for the nominal mass, which turns out
to be the same value as in simulation: Z =√
k·m2. Once Z has been determined, the
robustness to changes in the payload is also proven in Figure 6.21. It is remarkable the
high similarity between the simulations and the experiments, which corroborates how
Figure 6.20: Top: Experimental responses of the linear system with the wave forcedefinition scheme using different impedance values. Bottom: Detail of the steady-state
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.2
0.4
0.6
0.8
1
θ t (ra
d)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.95
1
1.05
θ t (ra
d)
t (s)
mt = 44g
mt = 60g (Nominal)
mt = 100g
mt = 150g
mt = 200g
Figure 6.21: Top: Experimental responses of the linear system with the wave absorbcontrol and payloads different from nominal. Bottom: Detail of the steady-state
6.6. Experimental verification 163
0 1 2 3 4 5 6 7 8 9−0.2
0
0.2
0.4
0.6
0.8
1
θ t (ra
d)
0 1 2 3 4 5 6 7 8 90.9
0.95
1
1.05
θ t (ra
d)
t (s)
s = 1/10s = 1/4s = 1/2s = 2s = ∞
Figure 6.22: Top: Experimental responses of the nonlinear system with the wave absorbscheme using different references. Bottom: Detail of the steady-state
6.6.2 Non linear system
Subsequently, the same studies performed in Section 6.6.1 were also accomplished
using the composites nonlinear platform using a very long, very slender glass fiber
beam.
First, the wave absorb control was applied, using different references, with the re-
sults displayed in Figure 6.22. The phenomenon predicted by Figure 6.10 in Section 6.4
is now fully substantiated by experimentation: WBC in its wave absorb form is not
appropriate for controlling systems exhibiting geometrical nonlinearities. In addition,
a non desirable, non evaluated effect comes into play: the spillover. Due to the slen-
derness of the beam, the second mode is excited by demanding references. This is very
clear for the step and p = 2 inputs, but it is still noticeable for the rest references at a
smaller scale.
The same set of experiments was tested for the wave force control scheme with the
results shown in Figure 6.23, where the response to the step input presents a similar
behavior to that of the wave absorb. This is due not to the control scheme, but to
the second vibration mode which is enormously excited by the trajectory and drives
164 Wave-based control
0 1 2 3 4 5 6 7 8 9−0.2
0
0.2
0.4
0.6
0.8
1
θ t (ra
d)
0 1 2 3 4 5 6 7 8 90.95
1
1.05
θ t (ra
d)
t (s)
s = 1/10s = 1/4s = 1/2s = 2s = ∞
Figure 6.23: Top: Experimental responses of the nonlinear system with the wave forcedefinition controller using different references. Bottom: Detail of the steady-state
the system to an abrupt transient. Still, the good news is that the system is still
controlled to a vibration less state, but, in the middle, the wildly nonlinear behavior of
the system devaluates the performance of the control scheme. However, the difference
with respect to wave absorb is appreciable, specially, in the p = 2 maneuver, where
wave force control reaches the target position accurately while wave absorb scheme
does not.
To study the performance of the controller, again some experiments varying Z
value have been carried out. Figure 6.24 depicts the system responses, from where
it can be deduced that, as proved by simulation and experiments on a linear system,
the best value for wave impedance is Z =√
k·m2. Finally, in order to verify the
robustness to changes in the payload, a last set of tests changing the value of the tip
mass has been performed. The results can be consulted in Figure 6.25, where this
property is guaranteed for an interval between the nominal mass mt = 44 g and a
maximum of mt = 101 g. Unfortunately, the experimental platform designed for the
large displacements beam has a smaller range of variation of the payload than the small
Figure 6.24: Top: Experimental responses of the nonlinear system with the wave forcedefinition controller using different wave impedances. Bottom: Detail of the steady-state
0 1 2 3 4 5 6 7 8 90
0.2
0.4
0.6
0.8
1
θ t (ra
d)
0 1 2 3 4 5 6 7 8 90.95
1
1.05
θ t (ra
d)
t (s)
mt = 44g
mt = 60g
mt = 101g
Figure 6.25: Top: Experimental responses of the nonlinear system with the wave forcedefinition controller when facing errors in the tip mass value of the design Z. Bottom:Detail of the steady-state
166 Wave-based control
6.7 Conclusions
In this chapter, the wave-absorb algorithm for canceling vibrations has been used in
the rest-to-rest manoeuvering of a single-link flexible arm exhibiting large deflections
that cannot be accurately modeled by a linear system.
The wave-absorb control had been previously applied to a number of generic linear
systems exhibiting some sort of vibration/flexibility with great success. Now, it has
been demonstrated that very good performance is also obtained when it is applied to
nonlinear one degree of freedom systems. However, the simulations have exposed a rel-
evant accuracy problem in the method when the trajectories are very demanding. This
is due to the use of superposition’s principle when combining the wave components,
which is clearly linear. Nonetheless, it has been shown that there are various practical
ways to correct the small remaining error.
A variation of the WBC consisting of redefining the waves using some concepts of
waves propagating through distributed beams has demonstrated very good properties
when applied to the nonlinear problem, correcting the accuracy problem and even
improving the robustness to changes in the tip mass.
Considerable effort has been devoted to the experimental verification of the results
obtained in theory and simulation. The results obtained with both control schemes for
the linear flexible robot were quite similar although the wave force scheme presented
less residual vibration than the wave absorb. For the nonlinear system, however, the
outputs differ: while the wave force scheme still performs accurately, a steady-state
error appears in the wave absorb control. Therefore, the final experiments have been
performed with the force based controller. Experimentally it has been observed that
the use of very steep references produces the undesirable effect of the apparition of the
second mode of vibration, which distorts the results in the case of step inputs. Still,
the controller proves to be stable even when this phenomena appears.
Chapter 7
Conclusions, contributions and
suggested future research
7.1 Summary and conclusions
Flexible robotics still has not reached its limit. While the modeling and position
control of these devices has been widely studied over the past three decades, the con-
tributions are in crescendo every year and new fields of application are continuously
reported and studied in literature. Recent advances in materials science, control theory,
sensors and actuators promote this sustained interest.
As commented in the Introduction, flexible robots provide a wonderful and motivat-
ing platform for corroborating the performance of innovative control techniques in a
simple, yet efficient, way. The present document deals with some of these schemes
(namely, dynamic inversion, robust and adaptive controls) and demonstrates their
properties with experimental results obtained from these arms.
Besides the study of novel control schemes for conventional flexible manipulators,
this Thesis raises the topic of controlling flexible links which are so slender that large
displacements can appear during maneuvers. The vibrationless point-to-point motion
of a flexible arm with these characteristics made of fiber glass has been properly ad-
dressed by means of a modification to the original wave-based control reported by Prof.
O’Connor. The models used for simulations purposes proved to be computationally
light while significantly more accurate than linear ones. The experimental results are
168 Conclusions, contributions and suggested future research
promising and there is scope for future developments. Some suggestions are enumerated
in Section 7.4were remarkably good.
7.2 Original contributions
Main contributions of the Thesis are listed below. Regarding control of robot links
subject to small displacements:
• Family of fourth-order, smooth polynomial trajectories that take into account
the physical limitations of the platform at the design stage.
• Direct dynamic inversion of the lumped masses model of a flexible link considered
in the Thesis.
• Control scheme robust to changes in the parameters of the plant (with stress on
payload changes) and to external perturbation for flexible robots.
• Adaptive control based in a novel payload estimation technique with application
to lumped and distributed masses models.
On the other hand, when dealing with extremely flexible robot arms (presenting
large displacements and geometrical non-linearities), the following topics have been
addressed:
• Application of wave-based control theory to control of manipulators whose links
are subject to large displacements, hence exhibiting geometrical nonlinear behav-
ior.
• Redefinition of the mechanical waves in a force-based manner to solve the preci-
sion issues of the wave-absorb schemes.
• Robustness to changes in the carried payload is inherent to the wave-based control
strategy, and it is a feature of both wave-absorb and force-based schemes.
7.3. List of publications 169
7.3 List of publications
The contents of this Thesis has given rise to the following articles published in
international journals with peer-reviewed selection processes:
• V. Feliu and F. Ramos, “Strain gauge based control of single-link very lightweight