In recent years, there has been a great deal of interest in the modeling and control of flexible arms -. This
interest has been motivated by the prospect of fast, light, robot whose links, due to material characteristics, will
bend under heavy loads. As a first step towards designing controllers for such robots, researchers have begun
studying controllers for simple flexible links. These links, in most cases modeled as Euler-Bernoulli beams be-
cause of the small deflections involved, are often analyzed through an eigen-function series expansion of the
solution to beam equation. However, under author’s knowledge, there has not yet been a study of force control
of a flexible Timoshenko arm based on the infinite dimensional model. The effect of shear deformation and the
effect of rotary inertia are considered in Timoshenko beam theory and thus the Timoshenko beam theory is
modified for a non-slender beam and high-frequency response. This means that the Timoshenko beam theory
has a wider application area than the Euler-Bernoulli beam theory. So we discuss the force control problem for
the flexible Timoshenko arm.
Figure 1 shows a constrained one-link flexible Timoshenko arm. One-end of the arm is clamped to control ac-
tuators consisting of the rotational motor and the translational slider, and the other end has a concentrated tip
mass m. The tip mass makes contact with the surface of an object. The flexible arm translates and rotates in the
horizontal plane (the XY plane in Figure 1) by control actuators; it is not affected by the acceleration of grav-
ity. The flexible arm, with length l, mass per unit length
, mass moment of inertia I
, cross sectional area
A, area moment of inertia
, Young’s modulus E, shear modulus G, and shear coefficient
, satisfies the
Timoshenko beam hypothesis.
In Figure 1, XY is an absolute coordinate system and
y is a local coordinate system, whose origin is
fixed at the rotor of the rotational motor. In addition,
y translates with the slider and rotates with the rotor of
the motor. Let
t, and ()
t be the inertia moment of the motor, the torque gener-
ated by the motor at time t, the rotational angle of the motor, the mass of the slider, the force generated by the
slider, and the translational position of the slider, respectively. Further, let (,)wxt and (,)
be the trans-
verse displacement of the flexible arm at time t and spatial point
, and the rotation of the cross section due
to bending deformation, respectively.
Since the tip mass makes contact with the surface of the object, we obtain the following geometric constraint:
()(,)() 0lt wlt st
This constraint means that the Y-axis position of the tip mass is constrained on the surface of the object. The
E and the potential energy
E of the overall system are given by the following:
, a dot denotes the time derivative, and a prime denotes the partial derivative with respect to
. Here the virtual work is given by δ() δ()() δ()WttFtst
Under the above preparation, we can obtain the following equations of motion by applying Hamilton’s princi-
ple and Lagrange’s multiplier, and using the procedure described in :
[( ,)()()][( ,)( ,)]0,wxtxtstKxtw xt
[ (,)()][(,)(,)](,)0,IxttKxtw xtEIxt
(0,)(0, )(, )()(, )()0,wttltlt wlt st
Figure 1. Flexible Timoshenko arm making contact with an object.
st FtKwt Ft
with the algebraic relation
is Lagrange’s multiplier and is equivalent to the contact force, i.e., the shear force at the tip of the
flexible arm, which arises in the direction along the normal vector of the constraint surface.
The aim of this paper is to control the contact force at the tip of the flexible arm. In other words, the control
objective is to construct a boundary controller satisfying
(), (,)0, (,)0,
is the constant desired contact force. At the desired equilibrium point
), (,)wxt and (,)
become the function of the variable
t become constant. Thus, we describe them as ()
, and d
, respectively. By sub-
stituting these into (1)-(6), we obtain:
In these relations, ()
, and d
mean a static transverse displacement, a static rotation of the
cross section of the flexible arm, a static angle of the motor, and a static position of the slider in the case where
the contact force is converged to the desired value, respectively. Furthermore, d
are coupled through
, and thus we cannot set the desired angle d
and the desired position d
Based on these results, we set the control objective as follows: to construct a controller accomplishing:
(,)(), (,) 0,
(), ()0, (), ()0.
For this purpose, we propose the following controller:
( )[(0,)(0)](0,)[( )](),
where feedback gain i
, 1,,8i, is a positive constant. In these controllers, (10) is the controller for the
slider and (11) is the controller for the motor. In (10), the first and second terms are for the control:
wxtwx and (,) 0wxt
, and the third and the forth terms are for the position control: () d
. On the other hand, in (11), the first and second terms are for the control of the rotation of the cross
section: (,) ()
and (,) 0xt
, and the third and the forth terms are for () d
Here, note that if we use the strain gauge, rotary encoder, and speed reference type servo amplifier of the motor
and the slider, we can easily implement the controller.
Taking the Laplace transform of (1)-(11), we can get
,, ,0swxsK xsKwxssxsSs
,, ,0sIKxsKwxsEI xssIs
,0ls wls Ss
MS sFsKwsF s
Finally we get
sk s kk sks
2(32 1 4*
IKkkkk sEkkkk KkkkkI
81 3125771 3
kk kSskkklkKkk kSK
Kkl kF sFsF
where the constants
can be determined using the boundary conditions.
Numerical inversion of Laplace transform is used to obtain the results in the time domain. The computation of
the inverse Laplace transform is based on the paper of T. Hosono . In the computer simulation study, we
consider a typical arm whose parameters are given in the Table 1. To investigate the validity of the proposed
controller, we considered the step responses of the desired contact force, 100
N, and the desired posi-
tion of the slider, 0.1
s m. Here, note that 100
means that the flexible arm pushed the surface of the
environment by the force of 100 N. Figures 2-6 show the time response of the transverse displacement, the an-
gle of deflection, the slider position, the rotational angle and the contact force with adequate feedback gains
0.2kk , 37
using try and error method. The desired value of d
00.25 0.5 0.7511.25 1.5 1.752
Figure 2. Time response of transverse displacement of the arm.
00.25 0.50.7511.25 1.5 1.752
Figure 3. Time response of angle of deflection of the arm.
00.25 0.5 0.75 11.25 1.5 1.752
Figure 4. Time response of the slider position.
00.25 0.5 0.75 11.251.5 1.75 2
Figure 5. Time response of rotational angle.
00.25 0.5 0.7511.25 1.5 1.752
λ[ N ]
Figure 6. Time response of the contact force.
Table 1. System parameters.
Length l 1.0 [m]
Mass per unit length
Mass moment of inertia
2.79e (−4) [kgm]
Cross sectional area A 3.5e (−4) [m4]
Area moment of inertia
3.57e (−8) [m4]
Young’s modulus E 2.00e11 [Pa]
Shear modulus G 7.69e10 [Pa]
Desired contact force d
Desired translational position of the slider d
S 0.1 [m]
4.67 10 
It can be seen that the transverse displacement, the angle of deflection, the slider position, the rotational angle
and the contact force toward the desired their values. With the adequate feedback gains there are no residual vi-
brations and no over shoot.
A contact-force control problem with regards to a constrained one-link flexible Timoshenko arm was described.
The equations of motion and the boundary conditions of the overall system were derived. To solve the contact
force control problem of such a system, we have proposed a simple controller, which is easy to implement. Sev-
eral numerical simulations using a numerical inversion of Laplace transform were carried out. The simulation
results showed the validity of the proposed controller for the contact-force control problem with no residual vi-
brations and no overshoot.
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