Biomedical Engineering Reference
In-Depth Information
driven by control actuators
[45-47]
have also
been published. The problem of motion control
of flexible multilink manipulators has recently
been considered by Wang
et al
.
[48]
. In the early
1990s, papers on the influence of nonlinear elastic
effects first began to appear. Fenili
et al
.
[49]
developed the governing equations of motion for
a damped beamlike slewing flexible structure
driven by a DC motor and presented discussions
of nonlinear effects.
Optimal control techniques
[50]
based on opti-
mization of
H
2
and
H
∞
norms of a cost functional
have emerged as reliable and computationally
fast methods for the synthesis of feedback con-
trol laws for systems that are essentially described
by a set of linear differential equations. However,
most engineering systems, including robotic
manipulators, particularly those involving some
form of energy conversion or systems associated
with biological processes, are patently nonlinear
and must be dealt with in their natural form. The
primary advantage of a nonlinear regulator is
that closed-loop regulation is much more uni-
form and, consequently, the response is always
closer to the set point than a comparable linear
controller. Although the roots of nonlinear opti-
mal control theory were established a long time
ago when Bellman
[51]
formulated his dynamic
programming approach and characterized the
optimal control as a solution of the Hamilton-
Jacobi-Bellman equation
[52]
, computational
methods for rapid computation of the feedback
control laws are not easily available.
Several other methods, such as the use of con-
trol Lyapunov functions and model predictive
control, have also emerged
[53, 54]
. One
approach, first suggested by Cloutier
et al
.
[55]
,
is nonlinear regulation and nonlinear
H
∞
con-
trol. A related technique due to Vlassenbroeck
and van Dooren
[56]
is based on an orthogonal
series expansion of the dynamical equations fol-
lowed by a sequential numerical solution of the
conditions for optimality. Another approach is
based on replacing the nonlinear optimal con-
trol problem by sequence of linear quadratic
optimal control problems. Several researchers
have addressed the issue of the synthesis of full-
state feedback control laws when all states are
measurable
[57-60]
. However, there are rela-
tively fewer examples of the application of non-
linear
H
∞
control
[61, 62]
to systems where only
a limited number of states are available for
measurement.
4.3.1.2 Modeling of a Multilink Serial
Manipulator
Let us apply the
H
∞
controller synthesis algorithm
to the position control of a flexible, three-link serial
manipulator. The traditional
H
∞
controller for a
linear system consists of a full-state estimate linear
feedback control law and a
H
∞
state estimator. The
feedback control law is synthesized by solving
an algebraic Riccati equation
[63]
, whereas the
estimator is similar in form to a
Kalman filter
(KF)
[64]
. The nonlinear controller is obtained
by synthesizing a
frozen estimated state
optimal
control law and replacing the linear estimator
by a nonlinear filter known as the
unscented
Kalman filter
(UKF) that is obtained by applying a
nonlinear transformation known as the
unscented
transformation
[65-67]
while propagating the
estimates and covariance matrices. The resulting
controller is a nonlinear controller.
The unscented
H
∞
estimator, a nonlinear esti-
mator that bears the same relationship to the
linear
H
∞
estimator as the relationship of the
UKF to the linear KF, is constructed by the same
process as the UKF, by employing a weighted
combination of estimates and process covariance
matrices evaluated at a finite set of sigma points.
The modeling of a multilink manipulator can
be done by adopting the Lagrangian formulation
[1]
. With the correct choice of reference frames,
the dynamics can be reduced to a standard form.
A typical three-link serial manipulator used to
represent a human prosthetic limb is illustrated
in
Figure 4.7
. The Euler-Lagrange equations
may be expressed as a standard set of three cou-
pled second-order equations for the joint varia-
bles
θ
i
that are defined in
Figure 4.7
. The nonlinear
