Biomedical Engineering Reference
In-Depth Information
described by nonlinear strain-displacement
equations and a linear stress-strain relation,
which results in an approximate nonlinear equa-
tion
[68-70]
for beam vibrations. The general-
ized
external
nonlinear forces due to stretching
of the midplane may then be determined. The
complete equations of motion may then be
expressed in standard state-space form.
Expressions for the additional coefficient
matrices in the dynamics due to the presence of
flexibility and midplane stretching may be
obtained from the expressions for the additional
kinetic and potential energies due to flexibility
and the additional generalized work done by the
nonlinear force distribution and structural damp-
ing forces. For brevity, the details are not pre-
sented here. The state space is now augmented
to include the amplitudes of the assumed flexible
displacement modes, and the corresponding
equilibrium and perturbation equations may be
found. The nonlinear
H
∞
controller synthesis
problem is solved with the plant assumed to be
nonlinear in terms of state-space description, but
with the measurements and the performance
assessment outputs assumed to be linear in terms
of the states and inputs. The adopted nonlinear
H
∞
controller synthesis method is based on the
linear
H
∞
controller synthesis method
[71-75]
.
It should be mentioned at this stage that both
nonlinear
H
2
control and the UKF are special
cases of the corresponding
H
∞
controller and the
unscented
H
∞
estimator. To get the benefits of
both
H
2
and
H
∞
control, one design approach is
to choose the free parameters in the
H
∞
control-
ler design within appropriate limits. Such an
approach will let the designer allow for the fact
that the nonlinear closed-loop system, with the
H
∞
controller in place, is generally a lot less
robust than the linear closed system with the
linear
H
∞
controller in place.
FIGURE 4.7
Typical three-link manipulator showing the
definitions of the degrees of freedom.
perturbation dynamics about the equilibrium
may be obtained from these equations.
In the perturbation equations, the external
moments acting on the links are the sum of the
control torques and the disturbance torques. The
nonlinear equations can be written in state-
space form by adopting standard techniques for
defining an augmented state vector.
The additional kinetic and potential energies
due to elastic transverse (normal) displacement
normal to the link are used to determine the
equations of motion of a flexible manipulator. A
beam that is simply supported at its edges expe-
riences a midplane stretching when deflected.
The influence of this geometric nonlinearity or
stretching on the response increases with increas-
ing amplitude of motion. The dynamics may be
4.3.1.3 Application to the Position Control of a
Three-Link Flexible Serial Manipulator
The three-link flexible serial manipulator con-
sidered in this section serves as a benchmarking
