The 2nd Joint International Conference on Multibody System Dynamics
May 29-June 1, 2012, Stuttgart, Germany
Wave-based control of under-actuated flexible structures with strong external
disturbing forces
William J O’Connor, Hossein Habibi
School of Mechanical & Materials Engineering
UCD, Belfield, Dublin 4, Ireland
william.oconnor@ucd.ie; hossein.habibi@ucd.ie
ABSTRACT
Wave-based control (WBC) of underactuated, flexible systems considers actuator motion as launching a
mechanical wave into the flexible system which it then absorbs on its return to the actuator. The
launching and absorbing proceed simultaneously. This simple, intuitive idea leads to robust, generic,
highly efficient, precise, adaptable controllers, allowing rapid and almost vibrationless re-positioning of
the system, using only sensors colocated at the actuator-system interface. These wave-based ideas have
already been shown to work on simple systems such as mass-spring strings, systems of Euler-Bernoulli
beams, and flexible space structures undergoing slewing motion (rotation with lateral translation). The
current work extends this strategy to systems experiencing external disturbing forces, whether body
forces which endure over time, such as gravitational effects which change with system orientation, or
transient forces such as from impacts or external viscous damping. The revised strategy additionally
provides robustness to some sensor errors. The strategy has the controller learn about the disturbances
and compensate for them, yet without needing new sensors or measurements beyond those of standard
WBC.
1. INTRODUCTION
As described in the literature [1, 2, 3], wave-based control (WBC) of under-actuated systems consists of a
directly-controlled actuator which is indirectly controlling an attached flexible system. To move the
system through a target displacement from rest (see Fig.1), the requested motion input to the actuator,
c(t), is set to be the sum of a “launch” displacement a(t) of half the reference displacement, ½r(t), and a
measured “return” displacement, b(t). The returning motion component b(t) provides active vibration
damping while also causing a net displacement which, in the absence of external disturbances, equals the
second half of the target displacement, ½r(∞). b(t) is determined from two interface measurements, here
taken as the actuator position, x(t), and the force, f(t), which the actuator applies to the flexible system.
For brevity we assume ideal actuator behaviour, so x(t) = c(t). The actuator control law is then:
c(t )  x(t )  a (t )  b(t );
a (t ) 
1
2
r (t );
b(t ) 
1
2
x(t ) 
1
Z

f (t ) dt

(1)
Z is an interface impedance, the value of which is not critical. For a lumped system it can be set to the
square root of the product of the spring stiffness at the interface and the first mass element. For rotational
motion, x, a & b will be angular displacements and f a torque.
This control strategy has been thoroughly tested on flexible systems of many kinds and sizes, lumped and
distributed, undergoing different kinds of motion, including 1-D and 2-D translation, rotation, and
simultaneous translation and rotation. In the absence of external disturbances and changing external
forces, Eq.(1) gives very rapid, rest-to-rest motion to target, with no steady-state error. Note that all
measurements are made at the interface, and so are colocated with the actuator. Also no system model is
required, making the strategy inherently robust to system changes and unknown system dynamics.
r
1/2
+
a
c
+
k
m
x
b
Return
wave
f
Fig.1 A lumped 1-D system where the returning wave b(t) will manifest changing, gravity-effects.
2. MODIFIED WBC STRATEGY
In Eq.(1), the force (or torque) f(t) is the “dynamic” force experienced by the actuator, due to the
interaction of its own motion with the resulting mass-acceleration and internal vibration of the attached
flexible system. For rest to rest motion, this dynamic force starts and ends at zero, and its integral over
time must also be zero (since there’s no net momentum change). If there is a static force present at the
interface, e.g. due to weight under gravity, it should be subtracted from the measured force to yield f(t).
Frequently, external forces are negligible in comparison with the forces internal to the actuator and
attached flexible system. In other words the actuator-system interaction, f(t), is dominated by the internal
flexible dynamics. This is the case in many crane systems, for example, or in many robotic applications.
But not always. Where there are significant external forces, particularly when they are not constant, under
WBC they cause a change in the measured return wave, b(t), which in turn leads to errors in the final
position, if not properly handled.
This paper considers how to adjust the control law, Eq.(1), to take account of changing external forces of
different kinds, including external damping and elastic forces, contact or impact forces, and forces due to
net changes in gravity effects. Sometimes such forces are predictable or quantifiable beforehand,
sometimes entirely unpredictable. Likewise auxiliary sensor information may, or may not, be available. In
the sprit of WBC, it is here assumed that no such foreknowledge or extra measurements are available. As
in Eq.(1), the measured variables remain the interface motion and force only. This restriction increases the
challenge considerably. If however the control law can meet this challenge, it will be exceptionally
robust.
The effects of the changing external forces show up in the measured b(t), which in turn depends mainly
on the force integral. Depending on the type of external force, the effects will be different, and so the
action required to remedy the control law will be different.
2.1.
Enduring, non-impulsive external disturbing forces
First we consider a disturbing force which arises during the manoeuvre and endures after the system has
settled, becoming constant then. This might be due to an external elastic force which is active at the final
position, a continuing contact force, or a gravity force the net effect of which has changed as a result of
the manoeuvre. Figure 1 shows an example where this might happen in a 1-D lumped chain. A slewing
motion of a flexible arm as shown in Figure 2 in the vertical plane under gravity will also experience
changing gravity effects. Indeed if the system’s mass centre passes over the actuator, the gravity-induced
torque will change quite dramatically, on passing through the vertical position, as the system strain flips
over and the torque changes sign. The effect is like a change in the DC (non oscillatory) force level
present in the measured force f(t).
Reference flexible
system motion
[x, y, θ]Ref
l
A
k
m
½
+
B
Actuator
+
Actual
actuator motion
[x, y, θ]0
Measured interface
reactin loads
[Fx, Fy ,M]0
Returning Motions
g
[x, y, θ]b
Returning
wave calculator
Fig.2 Representation of WBC of a 2-D beam-like structure moving in a vertical plane under gravity
To cope with such changes, the control law of Eq.(1) is modified as follows:
a(t )  12 r(t );
b(t )  g1 (t ) * a(t )  12
 p(t) f (t)  1t  f (t) dt dt/ Z
.
(2)
The launch wave, a=½r, is as before. * indicates convolution. The reference, r, might be a ramp up to the
target displacement, r(∞), with a being half of this. The g1(t) term is chosen so that, when convolved with
a, it produces a delayed version of a. The delay is chosen to mimic, approximately, how b would behave
in the absence of external forces, as happened in Eq.(1). It corresponds roughly to the time for the launch
wave to enter the flexible system, reach the extremity, and return to the actuator. So it supplies the
missing half target displacement to the actuator motion, at approximately the right time, but without the
full, active vibration damping of Eq.(1).
The term 1
t
 f (t ) dt
is a steadily improving estimate of the final DC force in f, because the contribution
to the time integral of the oscillatory components in f will approach zero, leaving the constant component
to dominate. This revised DC value, which will change smoothly, is subtracted from f, to leave the rapidly
changing, dynamic component, which can still bring about active vibration damping, as before, in Eq.(1).
The multiplying function, p(t), is chosen to remain close to zero for the initial transient, corresponding to
the delay of g1, and then to becomes unity, to allow f(t) to then act as in Eq.(1). Meanwhile both the outer
integrand and the outer integral gradually return to zero, leaving the correct final actuator displacement,
x(∞)=r(∞).
The gravity-induced change in the strain of the flexible system will also change the relative position of
the system tip with respect to that of the actuator. So if the controller is being used to achieve a desired tip
position, the target displacement of the actuator should obviously be adjusted to allow for the effect of
this changed strain on the tip displacement. This question is separate from the repositioning and active
vibration control problems, which are the focus of this paper, and so will not be considered explicitly
here.
The strategy based on Eq.(2) produced very good results, when tested in 1-D and 2-D systems under
gravity, especially for translation and rotation (slewing) in the vertical plane under gravity, or with
external viscous damping. The flexible system quickly arrives at the target and settles.
The same scheme simultaneously provides a practical solution to a problem that arises in implementing
WBC. A force sensor will inevitably have a static offset, or zero error, however small. Strain guage or
potentiometer measurements, for example, will have some DC bias. Because the control law involves
integrating the force over time, even when the force is nominally zero again, any small bias will cause the
force integral to grow, contributing to b(t), and so causing the system to drift from target (if not switched
off, or otherwise dealt with). The modified control law automatically compensates for any such static bias
in the measurement of f(t). The second force integral treats this bias like an enduring gravity effect,
integrates it over time, and subtracts it from f(t) before it is integrated in the first integral, thereby
canceling its effects.
2.2.
Impulsive disturbing external forces
When it was tested with impact forces however, the control law of Eq.(2) behaved less well. The system
moved to a new position and apparently settled, but frequently with large position errors, which then
slowly crept back to zero over a long time. This initial error and subsequent slow drift back to target was
attributed to the impulsive jump picked up by the outer and the inner force integrals in Eq.(2) associated
with the change in momentum with impact. Even with moderate impact, the outer integral did not
approach zero again until t became very large, implying an unacceptably long wait.
Knowing this, the solution was simply to add, after a suitable delay, a canceling offset for the slowly
decaying jumps in the integrals. As before, the delay can be achieved by convolution with a delay
function, g2(t), as follows (now shown without the explicit (t) time dependencies):
b  g1 * a  12
 p f  1t  f dt dt/ Z

 12 g2 *  p f  1
t
 f dt

dt / Z
(3)
This control law gave very good results for all cases. Note that it is still based on measuring only the
interface force f(t). It does require various functions and parameters to be estimated, including Z, g1, g2
and p, but none of these is critical and the properties they should have are well defined. The desired
robustness is achieved.
When more information is available about disturbing forces, it can be used to improve things further, but
Eq.(3) provides a bottom-line controller which always delivers position and vibration control.
3. Sample results
The new approach was tested on a slewing 2-D beam-like mass-spring array shown in Fig. 2. It is moving
in a vertical plane under gravity. The actuator has its own subcontroller which can set x and y (horizontal
and vertical) translation and  (rotation) in the plane. Each of these three variables is treated as if it were
independently controlled, using a control law such as Eq.(1), (2) or (3). In other words, the three inputs to
the actuator subcontroller are each made up of half the corresponding reference variable plus the
corresponding measured return variable, “b”.
The array mass and spring values can be chosen to simulate real systems with different values for mass,
moments of inertia and elastic moduli. Table 1 shows a sample set used in the simulation results below in
which the chosen mesh is uniform, with kx, ky and kd the spring stiffness in horizontal, vertical and
diagonal directions respectively, m the mass values and l the spacing between masses.
Table 1. Parameter values for the model of Fig.2
kx (kN/m)
8
ky (kN/m)
7
kd (kN/m)
10
m (kg)
0.1
l (m)
0.25
As the first test, the reference input is a ramped motion, in the vertical direction, up to one meter
displacement, under gravity. Although apparently simple, this motion involves a strong slewing action of
a very flexible system with many degrees of freedom, with internal and external forces acting. Figure 3
shows how the actuator responds to this input under different control strategies.
In the absence of gravity, standard WBC, described in Eq.(1), works well. The system quickly reaches the
target displacement while absorbing the flexible system vibrations. When gravity effects are incorporated,
however, the same controller fails badly. Following an initial, poor, attempt to follow the reference, there
is then a dramatic falling away, as the control system interprets the weight of the system as a returning
wave, which it tries to absorb, leading to an ever-increasing error. The third curve shows the response
under gravity when the WBC system is modified using Eq.(2). The system travels quickly to target, with
no overshoot, and rapid settling. This implies that the controller is managing to detect and separate the
additional force at the actuator due to the system weight under gravity, and to compensate for this,
automatically, without impairing the active vibration damping action associated with the vibratory
dynamics of the system. Thus, at the cost of a modest increase in the settling time, the revised strategy
achieves a combination of position control and active vibration damping, in a robust way, using only
forces measured at the interface.
Fig.3 WBC response to slewing reference motion, with and without gravity,
In the next sample result, this flexible arm is hit by an impulsive force, of unspecified magnitude,
representing an impact happening during a manoeuvre. In Fig. 4 the impact happens between about 2.6
seconds into the transit, causing a sudden, significant system disturbance. The reference input is again a
simple vertical translation of one meter in the y direction, during which the tip experiences a sharp shock.
Standard WBC (Eq. 1) absorbs the shock very well, but then settles in the wrong place. The large steadystate error is due to the integral of the effects of the external impulsive force arriving at the actuator and
changing b(t) inappropriately. Applying the first modification to WBC, Eq.(2) above (not shown), would
eventually cause this error to decay to zero, but only very slowly. To speed up the process, the second
modification, of Eq.(3) above, is seen to bring the system to target quite quickly, absorbing the shockinduced vibrations while eliminating the steady-state position error.
The system takes a little time to learn to distinguish between the different kinds of forces appearing at the
actuator, and then to deal with each of them appropriately. So the overall scheme takes longer to settle,
both before and after the impact. But this is the only cost, and it is modest. The benefits of standard WBC
are retained, including robustness, collocation of sensing, and no requirement for a system model. Note
that the controller does not need to know the size, location, nature or timing of the impact.
In the results in Fig.(4) the impulsive force was the only external force. The strategy was however fully
tested with gravity, impact and other forces present simultaneously, and it is found to cope very well.
Fig.4 System undergoing impact in the y-direction after a y-direction manoeuvre.
As a final example of external disturbing forces acting on the flexible system, external viscous forces
were added to the model. Each mass was given a velocity-dependent damping force, which therefore
changes magnitude and direction with the system motion. Unlike impact forces, viscous forces continue
for as long as the system is in motion, and they change direction due to the oscillatory motion. With
standard WBC the viscous dampers will absorb some of the motion which would otherwise return to the
actuator and be absorbed. This results in a final settling of the system short of the target displacement, as
seen in Fig.5. If the viscous damping coefficients are known, it is possible to predict the net shortfall, and
so compensate for it. But the application of the control law of Eq.(3) also works, as can be seen in Fig.(5).
An advantage of this strategy is that it does not need any information about the viscosity coefficients.
Fig.5 System experiencing external viscous forces, under standard and modified WBC.
4.
Concluding remarks
It is notable that single-actuator, motion control of a complex flexible system has been achieved using the
actuator’s own motion and only a single further measured parameter, namely the interface force, f(t). Also
it has been achieved without a system model and in a robust way. Admittedly two tuning functions g(t)
and p(t) are needed, but these are easily implemented. In any case, how exactly they are implemented is
not critical, provided they have steady-state gains of unity.
The effectiveness of these modifications to WBC has been investigated. They are found to maintain the
many advantages of standard WBC, including robustness, inherent adaptability, precision and speed of
response.
REFERENCES
[1] O'Connor, W.J., “Wave-Based Analysis and Control of Lump-Modelled Flexible Robots”, Robotics,
IEEE Transactions on, vol. 23, pp. 342-352, 2007.
[2] O’Connor, W.J., Ramos F, McKeown, D. J., Feliu V. “Wave-based control of non-linear flexible
mechanical systems”, Nonlinear Dynamics, Springer, Vol 57, No 1-2, pp 113-123, July, 2009
[3] O’Connor, W.J., Fumagalli A., “Wave-Based Control Applied to Nonlinear, Bending, and Slewing
Flexible Systems”, ASME Journal of Applied Mechanics, Vol 76, Issue 4, July 2009