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On Control of Flexible Robots
Antonio Ficola, Mario Luca Fravolini, and Michele La Cava
Dipartimento di Ingegneria Elettronica e dell’Informazione
Universit`a degli Studi di Perugia
http://www.diei.unipg.it/RICERCA/rautom.htm
This chapter deals with control of robots with flexible links. The most recent
results about modelling and control of flexible robots are examined; in par-
ticular, the results obtained by the authors and concerning predictive control
are described in detail.
1 Introduction
A mechanical structure with holonomic constraints can be represented by the
following equations:
B(q)¨q+h(q, ˙q)+Kq +g(q)=τ
0q∈Rn,τ ∈Rm(1.1)
where B(q) is a symmetric positive definite matrix, h(q, ˙q) includes the Cori-
olis and centrifugal generalised forces, Kis a constant positive (semi)definite
matrix that model the elasticity, g(q) is the vector of gravitational generalised
forces, τis the control vector.
In case of robots with elastic joints and/or flexible links the number of
control inputs is smaller than the number of Lagrangian coordinates m<
n, making the system underactuated. In fact, the elements of the flexible
links are coupled by the elastic forces/torques Kq, which can be considered
generated by proportional regulators with zero set points. Therefore, a flexible
link without gravity has a unique equilibrium configuration of the neutral
axis, determined only by the term Kq; in this case the elasticity reduces the
degree of underactuation. The term g(q) can be regarded as a further input
or a disturbance.
In fact, the “underactuation” of the flexible robots is not of the most
general kind. For instance, in case of rigid walking machines, only the gravity
can be used as a further input; in case of rigid robots moving in a horizon-
tal plane with some underactuated joints, not even the gravity is present,
causing a “true underactuation”. Therefore, the wider class of underactuated
holonomic systems comprises systems without gravity and elastic elements,
and includes the class of flexible robots.
S. Nicosia et al. (Eds.): RAMSETE, LNCIS 270, pp. 103−120, 2001.
Springer-Verlag Berlin Heidelberg 2001
104 A. Ficola, M.L. Fravolini, M. La Cava
2 Modelling
Modelling structures with flexible elements requires a preliminary examina-
tion of a unique flexible element, since a distributed parameters body is more
complex to be described than a rigid body. For this reason, in the following
section some results in the analysis of a single flexible link are presented.
2.1 Single Link
The models are usually developed referring to the Saint Venant assumptions:
i) the material behaves according the Hooke’s law; ii) the neutral axis is a
straight line; iii) only small deflections are considered and the end point of
the beam moves along a straight line orthogonal to the undeformed axis of
the beam. In order to reduce the model complexity, it is usually assumed
that the shear does not give contribution to the elastic energy and only the
effect of the bending moment is considered; the rotational energy could not
be considered in case of slender structures with small deflections; in case of
beams moving in three dimensions, there is no coupling between the motion
in orthogonal directions. The model employed to design the control law must
be carefully chosen, since in some case these hypothesis are not verified. For
instance, iii) is not verified for very slender structures like fishing-rods.
2.2 Exact Model
When all the above reported hypotheses are verified, the Euler Bernoulli
equation of the single link can be derived:
∂2
∂x2EJ ∂2w(x, t)
∂x2=−ρ∂2w(x, t)
∂t2+p(x, t) (2.1)
where w(x, t) is the deflection at abscissa xand time t,p(x, t) is the external
load, %is the mass per unit length, Eis the Young’s modulus and Jis the
cross section momentum of inertia. This result can be obtained using the
Hamilton principle. The same results can be obviously achieved using the
Lagrange equation directly [44]. If the cross section inertia and the Young
modulus are constant along the beam, it follows:
EJ ∂4w(x, t)
∂x4+ρ∂2w(x, t)
∂t2=p(x, t).(2.2)
Damping is not taken into account but it can be introduced by means
of p(x, t) assuming a proper law. If no external loads act on the beam, the
equation admits at least two kinds of solutions.
Traveling waves — A possible solution is
w(x, t)=f(kx ±ωt) = sin (kx ±ωt) (2.3)
On Control of Flexible Robots 105
where kand ωcan be determined substituting in (2.2). For instance, is the
beam has length Land is hinged at the extremities, it follows:
v=nπ
LsEJ
ρ.(2.4)
The higher the frequency, the higher is the wave speed. This particular solu-
tion should be considered for slender structures, when the propagation delay
is significant.
The (differential) eigenvalues problem — The following stationary solu-
tion is considered
w(x, t)=φ(x)ψ(t).(2.5)
Substituting in (2.2) yields
EJψ (t)∂4φ(x)
∂x4+ρφ (x)∂2ψ(t)
∂t2= 0 (2.6)
The solution for the eigenvalues problem is
ψ(t)=Asin (ωt)+Bcos (ωt) (2.7)
with the following eigenfunction(s) (β4=ω2ρ/EJ):
φ(x)=c1sin (βx)+c2cos (βx)+c3sinh (βx)+c4cosh (βx).(2.8)
The constants can be determined assuming the proper boundary conditions.
The eigenfunctions are orthogonal with respect to the inner product and can
be normalised:
ZL
0
φi(x)φj(x)dx =0 i6=jZL
0
φi(x)φi(x)dx =1.(2.9)
In case of a clamped beam the boundary conditions are (deflection and
rotation at the clamped extremity, shear and moment at the other)
w(0,t)=0 dw (0,t)
dx =0 EJ d3w(L, t)
dx3=0 EJ d2w(L, t)
dx2=0 ∀t. (2.10)
These boundary conditions are verified if the following characteristic equation
holds:
cos (βL) cosh (βL)=−1 (2.11)
which is satisfied by (infinite and numerable) βi, determining the natural
frequencies:
ωi=β2
isEJ
ρ.(2.12)
In case of a hinged beam the boundary conditions are (deflection and moment
at the hinge, shear and moment at the free extremity)
106 A. Ficola, M.L. Fravolini, M. La Cava
w(0,t)=0 EJ d2w(0,t)
dx2=0 EJ d3w(L, t)
dx3=0 EJ d2w(L, t)
dx2=0 ∀t
(2.13)
with the following characteristic equation:
sin (βL) cosh (βL) = sinh (βL) cos (βL).(2.14)
which gives the natural frequencies, different from (2.11). The normalised
eigenfunctions can be determined using the second of (2.9) and (2.10)
or (2.13). Further details can be found in some classical books [44], [43],
[59] or specific articles, e.g. [9], [29], [55].
2.3 Rayleigh-Ritz Methods
When the solution to the eigenvalue problem is not feasible or the closed
form solution cannot be computed, approximate solutions can be obtained
by means of the Rayleigh-Ritz methods. These methods assume a certain
function to describe the shape of the neutral axis of the link. The following
ones could be employed:
Eigenfunctions — They must satisfy both the differential equations and
the boundary conditions;
Comparison functions — They are 4 times differentiable, satisfy all the
boundary conditions but not necessary the differential equations; this class
includes the eigenfunctions;
Admissible functions — They are 2 times differentiable and satisfy only
the geometrical boundary conditions; this class includes the comparison func-
tions.
The geometric boundary conditions reflect the geometric constraints; the
natural ones reflect the constraints on forces and moments [44].
In the Rayleigh-Ritz method, the comparison functions of a complete-in-
energy set should be employed. If the interest is focused in the approximate
solution of the eigenvalue problem, the solution can be found in the space of
the admissible solutions instead of the comparison functions.
A series of polynomial can be employed, which can be properly trun-
cated to a certain order. These solutions satisfy all the geometric boundary
conditions but not the differential equations.
w(x, t)=
∞
X
i=0
ψi(t)xi.(2.15)
A suitable truncation is performed to determine the functions ψi(t) assum-
ing the proper geometric boundary conditions. Further details can be found
in [46], [56]. It can be proved that, notwithstanding this approximation does
not verify the differential equations of motion, the solution converges to the
exact one as the order of the truncation increases [27].
On Control of Flexible Robots 107
2.4 Finite Element Method
The finite element method can be regarded as a Rayleigh-Ritz method; in-
stead of global admissible solutions extended over the entire domain (length)
of the beam, the admissible functions are defined in subdomains (elements of
the beam) and are usually low-degree polynomials [44], [43], [59].
The i-th element of length Liis characterised by the rotations and dis-
placements at the extremities:
Ψi(t)=[ziLiθizi+1 Liθi+1 ]T(2.16)
In case of bending vibrations, the Hermite cubics can be employed as
interpolation functions; the functions in the i-th subdomain are
Φi(x)=[3ξ2−2ξ3ξ2−ξ31−3ξ2+3ξ3−ξ+2ξ2−ξ3]T(2.17)
where ξ=xi/L;xi∈[0,L
i]. The admissible function in the i-th subdomain
is
wi(x, t)=ΦT
i(x)Ψi.(t) (2.18)
This function allows to compute the kinetic and elastic energy of the element:
Ti(t)=1
2
L
Z
0
ρ(x)∂wi(x, t)
∂t 2
dx =˙
ΨT
i
Mi
2˙
Ψi.(2.19)
Ui(t)=1
2
L
Z
0
EJ (x)∂2wi(x, t)
∂x22
dx =ΨT
i
Ki
2Ψi(2.20)
Matrices Miand Kiare constant positive definite and depend on the
geometrical and mechanical parameters of the beam. The internal geometric
boundary conditions are satisfied by imposing the congruence of the displace-
ments and rotations at the element extremities. In practice the equations of
the whole structure are obtained by a proper assembly of submatrices Mi
and Ki.
The method can be extended also to three dimensional cases and to trusses
and frames. The hierarchical finite elements method combines the advantages
of both the Rayleigh-Ritz and the finite element methods [43].
2.5 Multiple Links
The exact model of a robot with a rigid link and a flexible forearm can
be derived referring to the Hamilton principle, including the kinetic energy
of the rigid link and the kinetic and potential energies of the flexible one.
This approach can be generalised also to more complex structures, but the
results are quite involved and are difficult to be employed for control design
purposes. The model is described by a system of non linear partial differential
108 A. Ficola, M.L. Fravolini, M. La Cava
equations. An example is reported in [1]. Approximate models can be derived
by means of the Rayleigh-Ritz method [58]. In the paper the authors develop
the model of the rigid robot; the last link is modelled using the clamped-free
eigenfunctions.
The extension to multiple flexible link is plain, in theory. An example is re-
ported in [11], where an automatic symbolic modelling method is introduced.
In [15] the dynamics of multi-link spatial manipulators with flexible links
and joints is modelled by a redundant Lagrangian/finite element approach.
The elastic deformations of the links are expressed in their tangential local
floating frames. Both the rigid and elastic degrees of freedom of the system
are treated as generalised coordinates, taking into account the coupling be-
tween rigid body and elastic motion. The constraint equations, representing
kinematical relations among different coordinates due to connectivity of the
links, are added to the equations of motion of the system by using Lagrange
multipliers. The model includes nonlinear ordinary differential equations and
nonlinear algebraic equations depending on the coordinates and Lagrange
multipliers. It can be converted into a set of differential equations, which is
solved numerically to predict the dynamic behaviour of the system.
Finally, a finite element approach can always determine a model almost
automatically, also for three dimensional structures. The drawback is the
huge dimension of the model. On the other hand a proper truncation could
achieve smaller dimensions; in particular, examining the single body of the
structure could give rise to suitable approximation also before the assembly
process.
2.6 Remarks
The exact solution is useful to compute the “true” vibration frequencies of
the beam. In control application it is usually approximated by some eigen-
functions, derived assuming suitable boundary conditions. It should be noted
that in case of multiple links, the natural boundary conditions depends on
the control. The Rayleigh-Ritz methods allows dealing only with geometrical
boundary conditions, which depend on the configuration of the structure. For
instance when two links are orthogonal, the joint can be considered a hinge;
when the links are parallel, since deflection is possible, the hinge constraint
is only an approximation, and the constrained extremities tend to be free as
the stiffness of the first link decreases. The finite element approach can deal
with all these cases; on the other hand, a large number of elements could
be necessary to approximate the natural modes, giving rise to large order
model. The model order can be reduced by means of low frequency approx-
imations, but the employed algorithm should be carefully chosen, since the
linearised model has the eigenvalues near the imaginary axis and could be
non minimum phase.
On Control of Flexible Robots 109
3 Control
3.1 PD Regulation
The basis in control of robots with flexible links is the regulation of an equi-
librium point. It can be shown that the decentralised PD regulation of the
motors can achieve the asymptotic stability of the equilibrium point. The ef-
ficiency of the algorithm (i.e. the damping that can be introduced), depends
on the coupling between the motors and the link: only if the vibration of the
flexible link causes the motors to move, damping can be achieved; therefore
direct drive motors with colocated sensors are taken into account. Some pa-
pers are devoted to show that a PD decentralised control law can impose
and asymptotically stabilise a unique equilibrium point. In a first approach,
reference is made to a two link robots with the last one flexible. The model
includes the equations of the flexible beam in the Euler Bernoulli form and
neither discretisation nor truncation are required. Assuming that no gravity
is present, the global asymptotic stability of the unique equilibrium point
can be proved by means of the Lyapunov method. The candidate Lyapunov
function includes also the elastic and kinetic energy of the flexible beam and
other terms related to the rigid motion. The LaSalle-Krasowskii theorem can
be employed to prove asymptotic stability. The proof does not depend on the
particular discretisation/truncation of the model equations and therefore is
not sensitive to spillover. The result is also robust with respect to parameter
uncertainties [1]. Only motor speed and position are required, but the proof
can be easily extended to the case in which the PD regulator is replaced by
a lead compensator; in this case only the motor position is necessary. In case
of gravity a compensation term is introduced. The proof is reported in [14]
where the damped and undamped case are examined. The compensation term
can be computed either in the desired final position or along the trajectory.
Since no feedback exists on the tip position error and only motor position and
speed are employed, the payload and the masses must be known to achieve
the desired point.
3.2 Strain Feedback Control
This technique is also employed in the regulation and includes in the control
law some terms that depend on the strain of some points of the flexible
links [41], [49], [28].
3.3 Inverse Dynamics
In case of trajectory tracking, the inverse kinematics and/or dynamics of flex-
ible robots is required. Some algorithms have been investigated. References
can be found in [5], [8], [10].
110 A. Ficola, M.L. Fravolini, M. La Cava
3.4 Fuzzy Regulation
A colocated decentralised PD regulation can achieve global asymptotic sta-
bility of the equilibrium point. On the other hand it is not possible to impose
an arbitrary damping to all the modes. To improve the damping ratio and
reduce the overshoot, it is possible to modulate the gains of the PD regulator
in function of some outputs. A careful investigation of the behaviour of the
plant can carry to the definition of rules to adapt the PD gains [17], [18], but
the results strictly depend on the depth of the physical insight. The main
property of this approach is that asymptotic stability can be demonstrated.
The application to multi link flexible robots requires a huge set of rules
and membership functions [45]. The problem can be overcome introducing an
adaptation algorithm to synthesise and tune the rules on-line. Acceleration
feedback of the end effector is required. The adaptation scheme is of MRAS
kind. Wide explanations can be found in [45], [34], where the method is ap-
plied to a two flexible link robot. Another application can be found in [25],
where acceleration feedback is also required. Other references to fuzzy learn-
ing control are [37], [35], [36].
3.5 Robust Control
H∞techniques are employed to design robust controllers for flexible links
with uncertain flexible dynamics. If it is assumed that there is not damping,
the link has all the eigenvalues on the imaginary axis, and therefore the
necessary conditions to design an H∞controller are not satisfied. Safonov [48]
introduces a right shifting transformation in the s-plan to make the poles
with real positive part. Since the H∞design does not cancel unstable poles,
the controller is stable and remains stable after the inverse transformation.
In practice the right-shifting allows only poor performance. Left-shifting is
more feasible; on the other hand, since the regulator cancels the stable poles
of the shifted plant, it will have imaginary zeros. Another approach consists
in adopting a High Authority/Low Authority Control architecture: an inner
loop makes the controlled system asymptotically stable, even if with poor
performance (Low Authority Control); the external loop is designed referring
to a more suitable system, in order to achieve the control system specifications
(High Authority Control). Safonov [48] uses in the inner loop a PD regulator;
Banavar [7] uses LQG controllers. The first approach employs a colocated PD
regulator that is robust with respect to model uncertainties; the other method
gives rise to better performance, because it implements a state feedback, but
it is less robust in presence of relevant uncertainties; moreover it is sensitive
to spillover; anyway, experiments show that the LQR regulator is effective.
The H∞controller can be designed using the Mixed Sensitivity Approach. A
drawback consists in the fact that the whole system does not ensure zero error
for step input. This is implicit in the Glover Doyle theorem, which requires
that the input must be L2 integrable. A feedforward term could be employed,
On Control of Flexible Robots 111
but since it introduces implicitly a pole in the origin, the stabilising effect of
the inner loop is lost. In practice the feedforward action can only reduce the
steady state error.
Another technique is named combined pole placement/sensitivity function
shaping method [33], [32]. Reference is made to a discrete time representa-
tion of the flexible link, the model of which is suitably truncated to represent
the significant dynamics. After choosing the sensitivity and the complemen-
tary sensitivity functions, an iterative procedure is employed to design the
controller, achieving the desired specifications.
The µ-techniques have also been applied. In this case, since structured
uncertainties are taken into account, the controller is less conservative than
an H∞one. An example of application can be found in [30]. Improvements
of the µ-techniques, developed for structural control applications, can be
found in [60], where a mixed H2/H∞technique is employed. Here, a set of
mixed H2/H∞compensators are designed, which are optimised for a fixed
compensator dimension. The mixed norm recovers the H2design performance
levels, while providing the same level of robust stability as in the µdesign.
3.6 Singular Perturbation
When the robot dynamics can be divided in fast and slow, the singular pertur-
bation approach can be employed [20]. Firstly the robot is partially feedback
linearised [52], improving also the separation between the slow and fast dy-
namics. The fast dynamics depends on the flexibility and is considered as a
perturbation of the slow dynamics, which coincides with the rigid motion.
The controller is composed of a slow term that ensures tracking, and a fast
term that damps the vibrations. Exponential stability can be demonstrated.
A preliminary requisite is that the bandwidth for tracking must be well dis-
tinguished from the vibration modes; from another point of view, either the
robots is rather rigid or the required bandwidth for trajectory tracking must
be small. Some references can be found in [4], where the technique is exper-
imentally applied to a single flexible link. Applications to multilink robots
can be found in [3], [64].
3.7 Sliding
The standard sliding mode technique requires to define a sliding surface for
each Lagrangian coordinate; to drive the trajectories on the sliding surface
the same number of control inputs are required. In case of flexible links,
where there are more Lagrangian coordinates than inputs, it is required to
adopt combined sliding surfaces. It is necessary to show that the equivalent
motion converges to zero. This is easy in case of single arms [61], [62] or
multilink robots, if the coupling between the links is negligible [12]. In case
of two link robots with a rigid and a flexible link, it can be shown the dy-
namics on the sliding surfaces is linear and can be stabilised by a suitable
112 A. Ficola, M.L. Fravolini, M. La Cava
choice of the parameters of the sliding surfaces [16]. The control law requires
some deflection measurements on the flexible link. The control chattering is
avoided approximating the ideal relay with a high gain saturated amplifier.
Experimental tests can be found in [4]. Another approach is the continu-
ous sliding-mode [63], which can be applied to minimum phase system. An
example can be found in [26].
3.8 Input Shaping
Shaped commands profiles are generated by convoluting a sequence of im-
pulses with the desired command signal. One of the problem to be solved is
the insensitivity, that is the ability of the input shaper to reduce the residual
vibrations in presence of modelling error. Details and comparisons of some
techniques are reported in [54]. Experimental results of the shaping technique
are reported in [31], [13].
3.9 Cyclic Control
This approach is useful when the motion occurs between some equilibrium
points and the whole path is not specified [39], [40]. Moreover, the problem of
trajectory tracking for a flexible arm between two equilibrium points is made
more difficult by the non-minimum-phase behaviour of this system. The prob-
lem can be classified as a state steering one. On the basis of these motivations,
the problem of steering the state of a control system by learning has been
investigated theoretically. The algorithms are based upon the selection of a
finite-dimensional subspace of the linear space or all control functions defined
over a finite time interval. As the search for the steering control is restricted
to this subspace, the resulting algorithm is finite dimensional [38].
3.10 Predictive Control
Since in many applications the most of the vibration energy is not generated
by disturbances but by the actuators [31], great advantages can be obtained
by a feedforward action which, by suitable shaping of the input command,
can improve the performance of an existing feedback controller. Feedforward
controllers can be designed to implement an inverse model control scheme,
aiming at the cancellation of the known dynamics of the open or closed loop
system [51]; in this way, if a reliable model is available, perfect tracking of any
trajectory can be obtained. The drawback of this approach is the difficulty
in the design of general model inversion algorithms for non linear systems
and in the lack of robustness. A problem which arises in the design of a feed-
forward inverse filter for linear flexible links is related to the fact that the
transfer function between the motor and the tip is non minimum phase, im-
plying an unstable inverse controller. This fact can be partially overcome by
On Control of Flexible Robots 113
designing a stable approximation of the inverse model [57]. Another method
for the approximated model inversion is the Model Predictive Control ap-
proach [24]. Model Predictive Controllers (MPCs) plan on-line a suitable
sequence of future input commands on the basis of a prediction model, in
order to track a desired output trajectory and minimising a defined index of
performance. Although MPCs have been mainly used in process control, they
have been recently applied also to the control of mechanical systems char-
acterised by non minimum-phase transfer functions and with badly damped
poles [50], [2]. Another attractive feature of MPCs consists of their ability to
handle constraints on control signals and state variables; these aspects are
quite important because it is possible to generate sophisticated optimal con-
trol laws satisfying multi-objective constrained performance criteria. Closed
form MPCs are available only for linear systems minimising a quadratic cost
function. Today analytical solutions are not available for general nonlinear
MPC and efficient numerical procedures exist only for convex optimisation
problems. In several papers Lagrange operators and quadratic programming
have been used to manage nonlinear constrained optimisations.
In MPC the optimal command sequence is determined as the result of an
optimisation problem at each sampling instant. All the performance speci-
fications are quantified by means of a cost index. An index, which is often
employed, is:
J=P
N2
X
j=N1
kˆy(k+j|k)−yd(k+j)k2
P+Q
Nc
X
j=1
k∆u(k+j−1)k2
Q+J1.(3.1)
The first term of Jevaluates the square error between the desired future
output signal yd(k+j) and the predicted future output ˆy(k+j|k), estimated
on the basis of the prediction model and the feedback information available
at instant k. This error is evaluated over a defined prediction horizon, which
is a window of N2−N1samples. The second term of Jweights the con-
trol effort of the sequence of the input increments ∆(k)=u(k)−u(k−1)
over the control horizon window of Nusamples. The diagonal matrices P
and Qcomprise weight coefficients, which are used to give a different em-
phasis to the terms of J. The term J1is a further cost function, which
can be used to take into account other specifications or constraints. MPC
is mainly used in conjunction with constraints on the amplitude and the
rate of variation of the input and output variables. The minimisation of in-
dex Jis performed with regard to the sequence of the control increments
[∆u(k),∆u(k+1),···,∆u(k+Nu−1)],j =1, ..., NU.
A key aspect in predictive control is the application of the receding horizon
strategy. In this approach only the first sample of the optimal sequence is
applied to the system; subsequently the horizon is moved one step in the
future and the optimisation is repeated on the basis of the measured feedback
information.
114 A. Ficola, M.L. Fravolini, M. La Cava
The computational load required by classical optimisation algorithms with
a great number of decision variables is the main difficulty encountered in non-
linear constrained MPC design. A way to overcome these problems is to fuse a
MPC scheme with intelligent control optimisation techniques. In this context
a very promising approach is to exploit Evolutionary Algorithms (EAs). EAs
are pseudo random parallel search algorithms, the inference engine of which
is based on the principle of biological evolution. One of the main advantages
of evolutionary optimisation is the only linear increase of computational time
with the number of decision variables; the main disadvantage, when employed
in feedback control, is the lack of results proving the stability of the control
law; on the other hand EAs showed to be quite appropriate optimisers for
MPC [47], [42]. In [21] an Evolutionary MPC (EMPC) is used as feedforward
controller in order to improve the trajectory tracking accuracy of a feedback
controlled mechanical system. In [22] a PD+H∞controller has been employed
(Figure 3.1), while in [23] only a PD regulator. The aim of the feedback loop
is to achieve the asymptotic stability even if with poor performance. To guar-
antee a certain robustness, some information from the plant can be employed
(realigment): the actual output error can be fed back and included in the op-
timisation index; the nominal model used for the prediction can be simulated
starting from the measured or estimated state of the plant; in this way the
nominal model behaves like a one step predictor (Figure 3.2).
Figure 3.1. Pure Feedforward MPC
The complete freedom in the choice of the multivariable objective function
and the possibility to handle constraints allow the fulfillment of additional
performance of the whole control system.
As far as the real time implementation is concerned, in [23] some tests
were carried out on an experimental set-up. In order to achieve the desired
performance, the Evolutionary algorithm must be carefully tuned, with suit-
able genetic operators and parameters. An increase in the computational
speed can be achieved by a proper coding of the decision variables, but a
deeper insight in the optimisation problem and in the genetic algorithm can
On Control of Flexible Robots 115
Figure 3.2. MPC with feedback information
give rise to a better performance. In particular, two improvements can be
introduced.
Steady state reproduction mechanism during the optimisation for the k-th
sample — Since a limited computation time is available, it is essential that
the good solutions founded during previous evolutions are not lost, but are
used as “hot starters” for the next generation. To implement this strategy,
the best chromosomes pass unchanged from the current generation to the
next one, ensuring a not decreasing fitness function for the best individual
of the population. The remaining part of the population is generated on the
basis of a rank selection mechanism: a certain number of the best individ-
uals constitute a mating pool; within the mating pool two random parents
are selected and two children are created applying crossover and mutation
operators; this operation lasts until the new population is entirely generated.
Heredity from time interval k to k+1 — The results of optimisations at the
kth instant are not wasted, but they constitute part of the starting popula-
tion for the next time step optimisation. This approach can be easily applied
in a receding horizon strategy; in fact the best chromosomes of the current
generation computed at the end of the k-th time interval can be exploited
as a privileged starting solution for the next time step optimisation. At the
beginning of the (k+ 1)-th time interval, since the horizon is shifted one
step ahead, the Ngenes of the best solutions are shifted forward to the next
location; in this way the first genes are lost and replaced by the second ones
and so on; the values of the last positions are filled by keeping unaltered the
corresponding values of the previous optimisation. The remaining chromo-
somes are randomly generated. The application of hereditary information is
of great relevance, because a dramatic improvement in the convergence speed
of the algorithm can be achieved and a good updated solution can be found
in few generations [22], [22], [23].
Some results are reported in the subsequent figures. Reference was made
to a single flexible link. The controller must damp the oscillations when the
link starts from a deformed condition. In Figure 3.3 the free response is re-
ported and compared with a well tuned PD controller. In Figure 3.4 a pure
116 A. Ficola, M.L. Fravolini, M. La Cava
Figure 3.3. Free and PD controlled flexible link
feedforward MPC is employed. An increase in damping is achieved, but the
mismatches between the nominal model employed for the prediction and the
real plant cause some residual oscillations. In fact the prediction model is
controlled to zero, contrary to the real plant. Finally, in 3.5 an intermittent
realignment is introduced; the residual oscillation are very small, but they are
not disappeared; this effect depends on the evolutionary algorithm that can-
not generate a zero command signal, because of the quantisation. A further
improvement consists in introducing an adaptation mechanism, which reduce
the quantisation effects as the error decrease, causing the residual oscillations
to vanish.
Figure 3.4. Flexible link with pure feedforward MPC
On Control of Flexible Robots 117
Figure 3.5. Flexible link with intermittent realignment
4 Conclusion
A brief survey on control of flexible robots has been reported. The literature
about these arguments is so huge that only some citations have been included.
For a deeper insight, reference could be made to the bibliography hereafter.
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