Biomimetic Robotics - Engineered Biomimicry - page 94

less than 0.05 nm in magnitude. The measurement
disturbances were assumed to be less than 0.5°.
The masses, joint stiffnesses, and lengths of the
three links are assumed to be equal in magnitude.
The three links were assumed to be uniform, with
the centers of mass located at the middle of the
links. Only the first two modes were included to
represent the elastic effects in each of the three
links. As a result, in the example considered, a
total of six modal amplitudes are included in the
model, whereas the total number of states is 18.
Only a typical result of the simulated, esti-
mated, and measured positions of the three links
obtained by using the frozen estimated state
approximation to compute the control gains is
shown in Figure 4.8 . The time step for the calcu-
lation is dt = 0.0002 s and the timeframe for the
calculation is 12,000 time steps. The manipulator
was commanded to move from a configuration
with all three angles equal to zero, to a configu-
ration where the bottom, middle, and top links
were at the angles 24°, 18°, and 12° to the hori-
zontal. On examining the results, it is observed
that the estimator convergence in the closed
loop is the most important requirement for long-
term stable performance. Thus, the estimator
was tuned by tuning the unscented transforma-
tion. All other parameters in the unscented filter
algorithm are maintained the same. The control
gains are again computed using the frozen esti-
mated state approximation and the simulations
repeated. Although the very small amplitude
oscillations (< 0.2°) and small steady-state point-
ing errors were unavoidable, the closed-loop
system was stable in the long term. Increasing
the number of time steps provided no difficul-
ties, and the unscented H ∞ estimator continued
to converge to the measurement in spite of small
variations in angular positions about the desired
set point because of the presence of the noise.
The results presented here indicate that in the
case of the closed-loop nonlinear H ∞ control, the
estimator tracks the angular positions of a flexi-
ble multilink serial manipulator to within ± 3%
of the set point (the initial error was 2%), and this

value increases considerably for other cases that
are not optimal. Further reductions in the track-
ing error are possible if one relaxes the bounds
on the maximum permissible magnitude of the
control input, which have been implicitly set by
scaling the inputs. Moreover, the influence of
nonlinear flexibility was relatively small, indicat-
ing that the estimator's performance did not
degrade due to the presence of small structural
nonlinearities in the dynamic model. Like its lin-
ear counterpart, the nonlinear H ∞ controller
seeks to maintain a balance between disturbance
rejection and stability. When the H ∞ estimator is
combined with methods involving the control
law synthesis based on replacing the nonlinear
optimal control problem by sequence of linear
optimal control problems, a powerful computa-
tional tool may be established for synthesizing
control laws for a variety of nonlinear controller
synthesis applications in biomimetic robotics.

4.3.2 Muscle Activation Modeling for
Human Limb Prosthesis and
Orthotics
4.3.2.1 Muscle Activation Dynamics Modeling

Let us consider a representation of a human limb
as an articulated linkage system of intercalated
bony segments. The bones, articulations, and
some of the ligaments (bone-to-bone joining tis-
sues) and tendons (muscle-to-bone joining tissues)
involved in flexion and extension of a limb serve
as the template for generic modeling of limbs.
Flexion is defined as the bending of, or decrease in
angle between, articulated body parts. Extension
is the opposite of flexion, resulting in a straighten-
ing of anatomic links. In regard to limbs, adduction
and abduction are defined, respectively, as motion
away from or toward the center line of the limb—
that is, the vertical axis of the limb. Limb motion
generated solely by muscle activation inputs is
referred to as active movement, whereas motion
generated without reference to muscle activation
inputs is referred to as passive . The terms proximal