Efficacy of Hammerstein Models in Capturing the
Dynamics of Isometric Muscle Stimulated at
Various Frequencies
by
Danielle S Chou
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree ofMASSACHUSETTS
OF ECHNOLOGY
Master of Science in Mechanical Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
. LIBRARIES
September 2006
@ Massachusetts Institute of Technology 2006. All rights reserved.
Author .....
..............................
. .......
Department of Mechanical Engineering
August 11, 2006
C e r ti fi e d by . . . . . . . . . . . . . . . . . . . . . . . . . .
H u g h -M-H e r
Hugh M Herr
Associate Professor
Thesis Supervisor
Certified by......................................................
David L Trumper
Associate Professor
~A
Thesis Reader
A ccepted by ............................
Lallit Anand
Chairman, Department Committee on Graduate Students
BARKER
Efficacy of Hammerstein Models in Capturing the Dynamics
of Isometric Muscle Stimulated at Various Frequencies
by
Danielle S Chou
Submitted to the Department of Mechanical Engineering
on August 11, 2006, in partial fulfillment of the
requirements for the degree of
Master of Science in Mechanical Engineering
Abstract
Functional electrical stimulation (FES) can be employed more effectively as a rehabilitation intervention if neuroprosthetic controllers contain muscle models appropriate
to the behavioral regimes of interest. The goal of this thesis is to investigate the
performance of one such model, the Hammerstein cascade, in describing isometric
muscle force. We examine the effectiveness of Hammerstein models in predicting the
isometric recruitment curve and dynamics of a muscle stimulated at fourteen frequencies between 1 to 100Hz. Explanted frog plantaris longus muscle is tested at
the nominal isometric length only; hence, the muscle's force-length and force-velocity
dependences are neither assumed nor ascertained. The pilot data are fitted using ten
different models consisting of various combinations of linear dynamics with polynomial nonlinearities. Models identified using data with input stimulation frequencies
of 20Hz and lower generate an average RMS error of 12% and are reliably stable (87
of 90 simulations). Between 25 to 40Hz, the average error generated by the estimated
models is 10%, but the estimated dynamics are less stable (16 of 30 simulations).
Above 40Hz, linear and Hammerstein nonlinear models fail to consistently generate
stable dynamic estimates (11 of 30 simulations), and errors are large (ERMS = 44%).
Simulations also suggest Hammerstein models found iteratively do not perform much
better than linear dynamic systems (in general, ERMS = 10 to 15%). In addition,
simulations using iterated nonlinearities generate RMS errors that are comparable to
those simulations using a fixed nonlinearity (both about 16%). These preliminary
results warrant further investigation into the limits of Hammerstein models found
iteratively in the identification of isometric muscle dynamics.
Thesis Supervisor: Hugh M Herr
Title: Associate Professor
Thesis Reader: David L Trumper
Title: Associate Professor
3
4
0.1
Acknowledgments
To the members of Biomechatronics, particularly Max and Sam, who helped provide structure and focus when I began to feel lost, and Waleed, who answered my
many questions and toughened me up; Biomech's veterinarians, Katie and Chris, for
their hours of dissection; Tony, Angela, Sungyon, Tina, and the rest of the Perfect
Strangers, the people on whom I've unburdened myself most in the past two years;
and last but certainly not least, Mom, Dad, and Jeff for always being there when I
felt I had no one-thank you all for helping me stay sane in your various capacities.
5
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6
Contents
0.1
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
1 Introduction
2
1.1
Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
1.2
Thesis organization . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
23
Background
2.1
2.2
2.3
3
5
Motor unit physiology
. . . . . . . . . . . . . . . . . . . . . . . . . .
23
2.1.1
Neuron physiology
. . . . . . . . . . . . . . . . . . . . . . . .
23
2.1.2
From neurons to muscle fibers . . . . . . . . . . . . . . . . . .
25
2.1.3
Muscle fibers and the actin-myosin complex
. . . . . . . . . .
26
2.1.4
Muscle feedback sensing . . . . . . . . . . . . . . . . . . . . .
28
Muscle behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
2.2.1
Summation of forces
. . . . . . . . . . . . . . . . . . . . . . .
29
2.2.2
Muscle recruitment . . . . . . . . . . . . . . . . . . . . . . . .
30
2.2.3
Non-isometric muscle contraction
. . . . . . . . . . . . . . . .
32
Muscle models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
2.3.1
Hill-type models
. . . . . . . . . . . . . . . . . . . . . . . . .
36
2.3.2
Biophysical models . . . . . . . . . . . . . . . . . . . . . . . .
39
2.3.3
Mathematical models . . . . . . . . . . . . . . . . . . . . . . .
40
2.3.4
Hammerstein model
. . . . . . . . . . . . . . . . . . . . . . .
41
47
Methods and Procedures
3.1
Hardware
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
47
3.1.1
Experimental set-up
. . . . . . . . . . . . . . . . . . . . . . .
47
3.1.2
Muscle set-up . . . . . . . . .
. . . . . . . . . . . . . . . . . .
49
. . . . . . . . . . . . . . . . . . . . . . . . .
50
. . . . . . . . . . . . . . . . . .
51
3.3.1
The static input nonlinearity . . . . . . . . . . . . . . . . . . .
52
3.3.2
The linear dynamic system . . . . . . . . . . . . . . . . . . . .
55
3.3.3
Bootstrapping . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
3.3.4
Simulation order
59
3.2
Experimental procedure
3.3
System identification and simulation
. . . . . . . . . . . . . . . . . . . . . . .. .
4 Results and Discussion
61
4.1
A prototypical system identification . . . . . . . . . . . . . . . . . . .
62
4.2
Is the input pulse-width range adequate to reach force saturation?
65
4.3
Can the algorithm identify the appropriate recruitment curve?....
67
4.4
What basis functions should be used to fit the input nonlinearity?
67
4.5
What is the appropriate input delay? . . . . . . . . . . . . . . . . . .
68
4.6
How does a dynamic system with no input nonlinearity perform?. . .
72
4.7
How do different combinations of input polynomial nonlinearities and
dynamic systems perform? . . . . . . . . . . . . . . . . . . . . . . . .
78
4.8
How does a system with fixed input nonlinearity perform?
81
4.9
How does a system with fixed input nonlinearity and fixed dynamics
. . . . . .
perform for high stimulation frequency input? . . . . . . . . . . . . .
83
4.10 Comparison of all simulation results . . . . . . . . . . . . . . . . . . .
86
4.10.1 Simulation errors . . . . . . . . . . . . . . . . . . . . . . . . .
86
4.10.2 Simulation stability . . . . . . . . . . . . . . . . . . . . . . . .
90
5 Conclusion
5.1
95
Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
A Experimental Protocol
97
B MATLAB code
99
8
107
C Simulation Log
9
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10
List of Figures
2-1
Neuron physiology
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2-2
Sarcom ere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
2-3
Actin-myosin cycle from http://www.essentialbiology.com . . . . . . .
27
2-4
Isometric twitch and tetanus . . . . . . . . . . . . . . . . . . . . . . .
29
2-5
Force-length relationships for pennate-fibered gastrocnemius and parallelfibered sartorius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
2-6
Force-velocity relationship and the underlying actin-myosin mechanism
35
2-7
Surface created by force-length and force-velocity relationships . . . .
35
2-8
Hill-type model used by Durfee and Palmer (1994) . . . . . . . . . . .
37
2-9
Hill-type model used by Ettema and Meijer (2000) . . . . . . . . . . .
38
2-10 Hill-type rheological model used by Forcinito, Epstein, and Herzog (1998) 38
. . . . . . . . . . . . . . . . . . . . . .
41
3-1
Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
4-1
Hammerstein model structure
. . . . . . . . . . . . . . . . . . . . . .
62
4-2
Example identification after 1 and 2 iterations (2Hz data) . . . . . . .
63
4-3
Example identification after three, four, and twelve iterations (2Hz data) 64
4-4
Isometric recruitment relationship - 1Hz
. . . . . . . . . . . . . . . .
66
4-5
Isometric recruitment relationship - 2Hz
. . . . . . . . . . . . . . . .
66
4-6
Delay selection: 3Hz data
. . . . . . . . . . . . . . . . . . . . . . . .
69
4-7
Delay selection: 7Hz data (continued on next page) . . . . . . . . . .
70
4-8
Delay selection: 7Hz data (continued from previous page) . . . . . . .
71
4-9
Simulations with no nonlinearity: muscle response at 3Hz stimulation
73
2-11 Hammerstein model structure
11
4-10 Simulations with no nonlinearity: muscle response at 40Hz stimulation
74
4-11 Simulations with no nonlinearity: muscle response at 100Hz stimulation (all have unstable dynamics) . . . . . . . . . . . . . . . . . . . .
4-12 Simulation errors with no nonlinearity
. . . . . . . . . . . . . . . . .
4-13 Difference equation denominator coefficient variation
. . . . . . . . .
75
76
77
4-14 Simulation errors of different dynamic systems with different nonlinearities (in scatter and bar graph format) . . . . . . . . . . . . . . . .
79
4-15 Comparison of second- and third- order dynamic fits (2Hz data identified iteratively with third-order nonlinearity and five delays)
. . . .
80
4-16 Simulation errors for fixed and iteratively-found recruitment nonlinearity (third-order dynamics, third-order polynomial . . . . . . . . . .
82
. . . . . . . . .
82
4-17 Difference equation denominator coefficient variation
4-18 Muscle response estimates for high stimulation frequency input using
fixed nonlinearity and fixed dynamics trained from 2Hz data . . . . .
84
4-19 Simulations with fixed third-order polynomial nonlinearity and thirdorder dynam ics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-20 Summary of simulation errors (scatter and bar graphs)
85
. . . . . . . .
87
4-21 Average simulation errors across stimulation frequencies . . . . . . . .
88
4-22 Average model errors across seven data sets below 25Hz . . . . . . . .
88
4-23 Pole-zero maps for third-order dynamic systems with different input
polynomial nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . .
91
4-24 Fluctuating passive muscle force: 1Hz data . . . . . . . . . . . . . . .
92
4-25 Average stability of ten estimated models across stimulation frequency
(1 = stable, 0 = unstable) . . . . . . . . . . . . . . . . . . . . . . .
92
C-1 No nonlinearity, second-order dynamics (Experiments 1 to 8: 1, 20, 3,
...........................
50, 10, 75, 15, 100Hz).......
108
C-2 No nonlinearity, second-order dynamics (Experiments 9 to 15: 7, 30,
1, 5, 40, 25, 2Hz)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
109
C-3 No nonlinearity, third-order dynamics (Experiments 1 to 8: 1, 20, 3,
110
50, 10, 75, 15, 100Hz).................................
C-4 No nonlinearity, third-order dynamics (Experiments 9 to 15: 7, 30, 1,
111
5, 40, 25, 2Hz).....................................
C-5 No nonlinearity, fourth-order dynamics (Experiments 1 to 8: 1, 20, 3,
112
50, 10, 75, 15, 100Hz).................................
C-6 No nonlinearity, fourth-order dynamics (Experiments 9 to 15: 7, 30, 1,
113
5, 40, 25, 2Hz).....................................
C-7 Difference equation denominator coefficient parameter variation
. . .
114
C-8 Nonlinear recruitment curves from iterative identification with thirdorder dynamics (Experiments 1 to 9: 1, 20, 3, 10, 15, 7Hz) . . . . . . 115
C-9 Nonlinear recruitment curves from iterative identification with thirdorder dynamics (Experiments 11 to 15: 1, 5, 40, 25, 2Hz) . . . . . . . 116
C-10 Simulations with fixed third-order polynomial nonlinearity and thirdorder dynamics (Experiments 1 to 6: 1, 20, 3, 50, 10, 75Hz)
. . . . . 117
C-11 Simulations with fixed third-order polynomial nonlinearity and thirdorder dynamics (Experiments 7 to 12: 15, 100, 7, 30, 1, 5Hz) . . . . . 118
C-12 Simulations with fixed third-order polynomial nonlinearity and thirdorder dynamics (Experiments 13 to 15: 40, 25, 2Hz) . . . . . . . . . . 119
C-13 Pole-zero maps for second-order dynamic systems with different input
polynomial nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . 120
C-14 Pole-zero maps for third-order dynamic systems with different input
polynomial nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . 121
C-15 Pole-zero maps for fourth-order dynamic systems with different input
polynomial nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . 122
13
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14
List of Tables
4.1
Average linear model RMS errors across seven data sets below 25Hz
.
89
C.1 RMS errors for all simulations . . . . . . . . . . . . . . . . . . . . . . 123
C.2 Stability for all simulations . . . . . . . . . . . . . . . . . . . . . . . . 124
15
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16
Chapter 1
Introduction
Today more than five million Americans suffer from stroke [1] with the associated
side-effects of movement impairment. An additional 250,000 Americans are living
with spinal cord injuries [40]. With such a population in potential need of movement
aids, research has intensified in the field of neuroprosthetics and functional electrical
stimulation (FES) control.
FES is used in cases when the skeletal muscle tissues are intact, but their neural
activation has been impaired due to nervous tissue damage resulting from traumatic
injury or disease. When such pathologies occur, an external electrical stimulation can
be applied to a motoneuron to generate movement. The electrical signal depolarizes
the nerve, and an action potential propagates down the nerve to the neuromuscular
junction. The muscle then becomes depolarized, resulting in calcium release from
the sarcoplasmic reticulum. The calcium release triggers the interaction of actin and
myosin proteins and causes the muscle to generate force.
FES controllers contain muscle models that allow them to determine what input
would be appropriate to elicit a particular muscle behavior. For demonstrations of
FES control, it is not imperative that a prosthesis controller use an accurate muscle
model. As shown by multiple researchers [46, 9, 14, 10], feedback controllers require
little knowledge of the muscle system in advance, yet are still able to modulate force.
However, this brute force method requires constant re-stimulation of muscle to achieve
the desired force and would be unsuitable for any realistic long-term prosthesis. In17
stead, a good muscle model should help minimize the amount of stimulation to which
we subject the muscle, thereby preventing fatigue and tissue damage. In better understanding the relationship between electrical activation and muscle force behavior,
it is likewise possible to better and more effectively employ neural prostheses for the
restoration of movement in a paralyzed limb.
The better the model linking input stimulation to force output, the better the
control exerted by a neural prosthesis. This search for a sufficiently descriptive model
has driven past and ongoing research into muscle modeling. One of the more popular
muscle models has been the nonlinear Hammerstein system. The Hammerstein model
consists of a static input nonlinearity followed by linear dynamics. It is popular
because it readily lends itself to a physiological analogy: the input nonlinearity may
be interpreted as muscle recruitment and the linear dynamics as the dynamics of
cross-bridge formation.
Beginning in 1967, Hammerstein models have been used in practical prostheses.
That year, Vodovnik, Crochetiere, and Reswick [46] developed a closed-loop controller
for an elbow prosthesis. Feeding back position, their controller linked elbow angle
to input voltage via joint torque and contractile force. The model accounted for
recruitment voltage threshold and saturation by using a piecewise linear recruitment
nonlinearity.
In 1986, Bernotas, Crago, and Chizeck [4, 27] used a Hammerstein system to model
isometric muscle. They explored force dependence on muscle length and stimulation
frequency and found the relationships were different for different muscles depending on
slow- versus fast-twitch fiber composition. Though at the time they did not use their
estimated model as part of a controller, the use of recursive least squares estimation
meant online identification was possible.
Two years later, Chizeck, Crago, and Kofman [9] created a closed-loop muscle force
controller. The pulse-width modulated controller included a muscle model comprised
of a memoryless nonlinearity in series with a continuous second-order dynamic system.
The group demonstrated that though their controller did not require a precise model
of muscle dynamics, it was still able to regulate force while maintaining stability and
18
repeatability.
In 1998, Bobet and Stein [6] modeled dynamic isometric contractions with a modified Hammerstein model. Dynamic in their case meant an input modulated by pulse
rate rather than pulse-width, and the Hammerstein modification consisted of a linear
filter preceding the traditional Hammerstein cascade. In addition, the output force
dynamics depended on a force-varying rate constant b. Inputting the same pulse train,
estimated force profiles were compared to experimental force measurements; results
showed that unfused and fused tetanus were well-simulated though isometric twitch
was not. No active controller was attempted though the authors suggested that for an
adaptive controller of isometric muscle, only one parameter-the force-varying rate
constant b-need be adjusted.
In 2000, Munih, Hunt, and Donaldson [38] used a Hammerstein model-based controller to help normal and impaired subjects maintain balance. After obtaining recruitment curves based on Durfee and MacLean's twitch-response method [15], Munih
et al. then had subjects stand rigidly on a wobbling platform. The patients' bodies
thus acted essentially as inverted pendulums, and balancing torques were achieved by
stimulation of ankle plantarflexors.
Because of its ready translation into a well-established physiological phenomenon
and its relative speed in identification compared to other mathematical models, the
Hammerstein-based isometric model has been employed time and time again throughout the decades of research into FES-motivated muscle models. However, the model
has its shortcomings. The foremost two flaws are its assumption of linear dynamics
and its disregard of any recruitment-dynamics coupling. Recruitment and dynamics
are coupled in a manner described by Henneman's principle: small neurons innervate slow-twitch fibers, and large neurons innervate fast-twitch fibers. Thus, natural
muscle recruitment, which begins with the smallest and progresses to the largest
neurons, affects dynamics by recruiting the slowest to fastest muscles [25]. Yet, the
Hammerstein model continues to be used. The question then arises, to what extent
is the model appropriate in describing the force behavior of pulse-width modulated
isometric muscle? And what are the limits for the identification process?
19
1.1
Objectives
When is it appropriate to use a Hammerstein model to describe pulse-width modulated isometric muscle force? What are the limits to the identification of such a
model? This thesis seeks to address these questions in one aspect: that of stimulation frequency. It aims to assess the efficacy of a Hammerstein model in iterative
identification of muscle undergoing a range of stimulation frequencies between 1 and
100Hz. Specifically, it attempts to determine the maximum stimulation frequency at
which a Hammerstein model identified iteratively may still accurately describe muscle
behavior. Only isometric muscle contraction is considered; while muscle clearly does
not operate only in this regime, this simplification allows the use of a traditional Hammerstein model with no consultation of muscle length or shortening velocity. Different
combinations of dynamics and polynomial nonlinearity are tested. The best combination is then simulated across all data sets with each data set allowed to choose and
validate its own dynamic system and regressor coefficients. Simulations are also run
for each data set with a fixed input nonlinearity and dynamics chosen by MATLAB's
oe command. Errors and dynamic system parameter variation are compared across
experiments run with different stimulation frequencies.
Stability of the estimated
systems and model iterative convergence are also examined.
1.2
Thesis organization
Chapter 2 reviews muscle physiology, then describes the different types of muscle models popular in the literature. The structure of the model of interest, the Hammerstein
cascade, is discussed in greater detail along with methods of identifying Hammerstein
systems and convergence of the iterative method. Chapter 3 begins with a description
of the experimental hardware, followed by a discussion of the procedures and simulation methods. Chapter 4 presents the simulation results and attempts to explain
them in the context of muscle mechanics. Chapter 5 summarizes the results and ends
with potential future work and future questions. Appendix A contains experiment
20
notes, Appendix B contains the MATLAB iterative code used to run simulations, and
Appendix C is comprised of simulation data.
21
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22
Chapter 2
Background
This chapter contains background on the relevant biology and modeling history essential to exploring the use of Hammerstein models in muscle identification. A brief
description of muscle physiology and motor units is given, followed by §2.2 on muscle
phenomenological behavior. The chapter then embarks on a review of different approaches to muscle modeling and concludes with a summary of the history and issues
encountered specifically by Hammerstein muscle models.
2.1
Motor unit physiology
Most muscle models are steeped in understanding of the underlying physiological
mechanics behind muscle innervation. Hence, this section discusses two components
comprising a single motor unit: a motoneuron and the multiple muscle fibers it can
innervate. Both are fundamental to understanding muscle contraction.
2.1.1
Neuron physiology
A motoneuron consists of a soma (cell body), a long axon, and dendrites branching
from the cell body, as shown in Figure 2-1. Each motor neuron has only one axon,
but the axon may branch into a maximum of 100 axon terminals. Synapses connect
the transmitting axon terminals of one neuron to either the receiving dendrites of
23
adjacent neurons or the motor end plates of multiple muscle fibers. A motor end
plate is an area of muscle cell membrane that contains receptors for the excitatory
neurotransmitter, acetylcholine.
Weak signals from the individual dendrites collect near the base of the axon at a
location called the axon hillock until a temporal and/or spatial integration of the signals results in a voltage exceeding the threshold voltage (-55mV). At that point, the
neuron depolarizes and an action potential propagates down the axon to the axon terminal where it will release neurotransmitter molecules. The neurotransmitters cross
the synapse and temporarily bind to the corresponding neurotransmitter receptors of
the next neuron. This signal continues from one neuron to the next until it reaches a
neuron whose axon is connected to a motor end plate. An axon may contact several
muscle fibers, but each muscle fiber is innervated by a single axon. That single axon
and the muscle fibers it contacts form a motor unit (Kandel et al. [32, pp 28-31]).
An action potential occurs when the sodium ion channels open. The behavior
of ion channels also explains why action potentials may only travel in one direction,
from dendrite to axon terminal: the ion channels have refractory periods of several
milliseconds during which they cannot open or close, no ions can travel in or out of
the axon, and therefore the voltage at that point along the axon cannot change. Refractory periods partly dictate conduction velocities. Signal conduction velocities also
vary with the diameter of the neuron; small neurons conduct more slowly than large
ones. In addition, conduction speed is effected by nerve myelination: nerves coated
with the fatty insulating substance myelin have much faster conduction velocities (4
to 120 m/s) than unmyelinated nerves (0.4 to 2.0 m/s) (Kandel et al. [32, p 678]).
When studying muscle mechanics, nerve conduction speeds and delays are relevant
as they have an effect on muscle system dynamics.
Neuron input can be either excitatory (depolarizing in which axon voltage becomes
less negative) or inhibitory (hyperpolarizing in which axon voltage becomes more
negative) depending on the type of neurotransmitter. For muscle, acetylcholine (ACh)
is the neurotransmitter. Signals received through dendrites closer to the axon hillock
are weighted more heavily in the decision to send an action potential.
24
-Dendrile
Endopiasrrlic
Goig, apparatus
Transport
vesicle
Axon
T ransport
vesicle
synapt
vesacle
c
Synaptlc
terminal
Muscle
Figure 2- 1: Neuron physiology
2.1.2
From neurons to muscle fibers
At the terminus of a motoneuron, the neurotransmitter acetylcholine is released.
Muscle fibers on the other side of the synapse have acetylcholine receptors, and when
the receptors bind acetylcholine molecules, they cause sodium ion channels to open.
An action potential spreads via the transverse tubules (T-tubules) within the muscle
fiber, and the muscle fiber sarcoplasm (cytoplasm) depolarizes. As a result of the depolarization, calcium ion channels in the muscle fiber's sarcoplasmic reticulum open,
and calcium ions diffuse into the sarcoplasm. These ions are key to the interaction of
25
muscle fibers' actin and myosin filaments, which brings us to the next section.
2.1.3
Muscle fibers and the actin-myosin complex
Muscle fibers contain clusters of sarcomeres, the basic contractile unit. A sarcomere
is 1.5-3.5 ptm long and consists of three primary components as shown in Figure 2-2:
the thin actin filament (in green), the thick myosin filament (in red), and the Z-disks
defining the length of a sarcomere when they are arranged end on end to form a
muscle fiber. Myosin heads stick out from the myosin filament along the length of the
myosin filament except for a region in the middle. Several theories have been put forth
Z line
A band
II
I band
H zone
Cross bridges
Actin (thin) filament
Myosin (thick) filament
Figure 2-2: Sarcomere
regarding muscle contraction, but the widely accepted theory was proposed by A.F.
Huxley in 1957. He speculated that unbonded myosin heads are in a cocked position,
as shown in Figure 2-3(a), due to a molecule of ADP attached to the myosin head.
26
Meanwhile, an action potential causes calcium ions to be released by the sarcoplasmic
reticulum, and these ions bond to parts of the actin filament (see Figure 2-3(b)). In
so doing, the actin filament's conformation has been altered and bonding sites are
exposed. The myosin heads attach to these exposed sites to form cross-bridges, which
have a different preferred structure that requires the myosin head to rotate while
releasing the molecule of ADP (see Figure 2-3(c)). As the myosin head rotates, the
overlap between the actin and myosin filaments increases and results in the overall
length of the sarcomere shortening
(~ 0.06
pm). This is called the cross-bridge power
stroke. To release the bonds between the myosin and actin filaments at the end of
the power stroke, ATP is required. It binds to the myosin head, which then detaches
from the actin filament and the muscle fiber relaxes (see Figure 2-3(d)). Subsequently,
the ATP is dephosphorylated (releases a phosphate molecule) to become ADP, which
once again cocks the myosin head for future attachment to another actin binding site
(Kandel et al. [32, p 678], see also [31, 24]).
HYDROLYSIS
actin filament
plus
minu
end
en d
ADP
-D
myosin head
(b) Force-generating
(a) Cocked
m
myosin
filament
(d) Released
(c) Attached
Figure 2-3: Actin-myosin cycle from http://www.essentialbiology.com
If ATP or calcium ions are lacking, such as when an animal dies, rigor mortis
sets in because the released ADP can no longer be phosphorylated to become an
27
ATP molecule. Thus, the cross-bridges cannot be broken and the muscles remain
contracted.
Muscle fibers come in two types: fast-twitch white (fast-fatigue) and slow-twitch
red (fatigue-resistant) fibers. White and red fibers metabolize oxygen and glucose
differently. Depending on the species and the muscle in question, the composition of
red and white fibers will vary. Fiber composition affects muscle dynamics and thus
dictates the system time constants.
Muscle contraction dynamics are not only dependent on fiber composition and
voltage activation, they are also dependent on length, shortening velocity, and tension history. These dependences are suggested by physiological evidence of muscle
feedback sensing, which leads to the next subsection.
2.1.4
Muscle feedback sensing
Learning what properties muscles use for feedback allows biomechanists to determine
what states are relevant in their modeling and, ultimately, for control. Muscle feedback sensing is conducted via a number of sensory neurons that originate in skeletal
muscle. Of the six types, three are directly related to muscle contraction mechanics.
These sensory organs are the primary spindle endings (type Ia), which sense muscle
length and strain rate; golgi tendon organs (type Ib), which sense tension; and secondary spindle endings (type II), which sense muscle length. Dysfunctions in any of
these sensing organs can result in abnormal system dynamics such as spasticity and
oscillatory tremor [52, 43].
Physiological feedback sensing hints at what states affect muscle behavior, but
what are these behaviors? The next section discusses three dominant behaviors in
some detail.
2.2
Muscle behavior
As a result of its underlying microscopic mechanisms, muscle exhibits certain macroscopic behaviors. For example, sarcomere length dictates muscle's force-length rela28
tionship, and fiber composition can affect recruitment. This section details three specific behaviors: force summation, muscle recruitment, and force-length and -velocity
dependences. The latter, though not pertinent to this thesis' study of isometric contraction, is crucial to the understanding of muscle in different behavioral regimes and
would need to be accounted for by any practical neuroprosthetic controller. We begin
with the first behavior, force summation.
Summation of forces
2.2.1
For muscle, a single action potential causes an isolated twitch contraction analogous
to a system impulse response, as shown in Figure 2-4(a). A twitch results in relatively
little force being produced by the muscle because the action potential does not release
enough calcium ions to form a considerable number of cross-bridges, and the number
of cross-bridges is directly proportional to the force produced.
A
A
A
A
A
(b) Summation of successive isometric twitch
(a) Successive isometric twitches
contractions
(d) Fused tetanus
(c) Unfused tetanus
Figure 2-4: Isometric twitch and tetanus
When action potentials occur more often, cross-bridge formation will increase,
and individual twitches will overlap and begin to add (see Figures 2-4(b) and 2-4(c)).
This is called unfused tetanus. Eventually, when the calcium ion release rate is greater
29
than the rate at which the ions are reuptaken, a fused tetanus will occur in which a
constant force is sustained over a period of time, as illustrated in Figure 2-4(d).
Fiber length also affects the maximum force produced. With an increased number
of cross-bridges, more overlap between filaments will occur. If the fibers are very
short, then the amount of permissible overlap quickly gets saturated and no additional
force may be produced. Fiber length, too, varies across species, and a single muscle
may have non-uniform muscle fiber lengths in different sections of the muscle belly.
In muscle modeling, the non-uniformity is almost always neglected in an effort to
simplify the modeling problem.
Overlapping twitch responses is not the only way to generate forces greater than
that of a single twitch. It is also possible to generate higher forces by contracting
multiple muscles. This is done via muscle recruitment.
2.2.2
Muscle recruitment
A single muscle fiber is able to produce only a limited amount of force. For higher
force production to occur, multiple muscle fibers must be recruited. In vivo, muscle
fiber recruitment begins with the smallest motoneurons and gradually increase to
larger ones. According to Henneman's principle [25], this means slow-twitch fibers
are recruited before fast-twitch ones. This is metabolically more efficient and is
why animals can demonstrate very fine motor control. In contrast, experimental
stimulation generally is unable to follow this recruitment sequence. Current electrode
technology remains relatively crude and bulky when compared to the size scale of
nerves and muscle fibers, so it has been difficult electrically stimulating anything but
the largest motoneurons. Thus, when muscle recruitment is mentioned in the context
of functional electrical stimulation (FES), it generally refers to the number of active
muscle fibers rather than the increasing size of the motoneurons recruited.
Muscles can be recruited either via spatial or temporal summation. Physiologically, this hearkens back to the cross-bridge theory: either the area of stimulation is
increased or stimulation is sustained in order to increase the concentration of calcium
ions required to form cross-bridges. Spatial recruitment via pulse amplitude or pulse
30
period is more common, but some researchers have explored temporal recruitment
[6, 10].
The isometric recruitment curve (IRC) may be obtained a few different ways.
Durfee and MacLean [15] suggested four, the first three of which they discussed in
great detail: step response, peak impulse response, ramp deconvolution, and stochastic iteration [39, 29].
Each have their advantages and disadvantages, as discussed
below.
" Step response methods may fatigue the muscle but make no assumptions about
linear dynamics.
" Peak impulse response experiments, on the other hand, fatigue muscles less but
rely on assumed linear dynamics.
" Ramp deconvolution, the authors' proposed method, requires an estimate of
system dynamics based on an averaged impulse (twitch) response; the measured
force is then deconvolved with the impulse response to get the recruitment curve.
However, since deconvolution amplifies noise, a noise-free dynamic estimate was
enforced by restricting the impulse response to be a critically damped second
order system with a double real pole. This assumption allowed very rapid
identification of the IRC.
" The iterative process makes no assumptions as to the form of the nonlinearity or
the dynamics, but it is not guaranteed to converge and can be computationally
expensive.
Durfee and MacLean concluded that each method resulted in slightly different
curves, but no one method was more correct than the others: method selection should
vary with the factors most important to the experiment, such as muscle fatigue or
identification time.
31
2.2.3
Non-isometric muscle contraction
The previous two subsections described ways of generating forces greater than that of
a single muscle twitch. Yet, those are not the only two factors contributing to muscle's force output. The magnitude of force is also dependent on length and velocity; a
muscle with changing length behaves differently than a muscle whose length is fixed.
Though this thesis deals exclusively with isometric muscle stimulation, a brief review
of non-isometric behavior is included here to highlight the potential complexity demanded by an all-encompassing muscle model. However, limiting the scope of the
muscle identification problem to isometric contraction is not a wasted exercise. As
others have pointed out, muscles whose series elasticity approaches infinite stiffness
will have output behavior approaching that of isometric muscle [5], and the isometric
model may be a foundation upon which to build non-isometric ones [19].
We begin with a description of the force-length relationship and follow with a
discussion on the force-velocity relationship.
Force-length relationship
Muscle fiber force depends on sarcomere length and, correspondingly, muscle fiber
length. Force is also affected by the configuration of the muscle; that is, whether the
muscle fibers are exactly parallel to the axis of the muscle or at an angle (pennate).
Pennate muscles, due to their fiber configuration, allow more muscle fibers to work
in parallel, so the forces generated are usually larger [36].
Figure 2-5, from McMahon's book [36], illustrates force-length relationships for
two different types of muscles, gastrocnemius and sartorius. The gastrocnemius, a
large calf muscle, is composed of short, pennate fibers while the sartorius, which is
a thin and very long thigh muscle, is comprised of parallel fibers. Three curves are
shown for each muscle: passive force occurs when the muscle is simply stretched
but not contracting; active force, labeled 'Developed,' accounts only for contractile
force generation; and total force is a summation of both active and passive contributions. The force-length relationship describes the steady-state force produced by
32
Soriorius
Gostrocnemius
Totol
Total
Tenson
Tension
Possive
Pvssive
D*4loed
Oeveloped
to
to
Length
Length
Figure 2-5: Force-length relationships for pennate-fibered gastrocnemius and parallelfibered sartorius
cross-bridges when muscle length is constant, i.e., an isometric response. At low
strains, the active force is low because the number of cross-bridges is low. As the
strain increases, the number of cross-bridges increases and therefore isometric force
increases despite overlap between the thin actin filaments. At yet higher strain rates,
the peak active force is produced and plateaus slightly because the middle region of
the thick myosin filament lacks myosin heads, as discussed in §2.1.3. This strain is
often termed LO and corresponds to muscle rest length. At rest length, peak power is
produced. Straining further results in a steady decline in force as the overlap between
thick and thin filaments decreases. This decline is further accelerated when the thick
filaments collide with the bounding Z-discs, and a force against the contractile strain
is produced (Kandel et al. [32, p 682]).
Meanwhile, passive forces are also at work. Sarcomeres are active components,
but there are also several passive elements. Tendon is one such element, and it can
significantly affect muscle force behavior. In addition to tendon, connecting filaments
attach the thick myosin filaments to the Z-discs at each end of a sarcomere, while
connective tissue surrounds each muscle fiber to aid in even distribution of force and
strain along the entire muscle fiber length.
Passive components can contribute a
considerable amount of force depending on the muscle strain. At low strains, the
connecting filaments and tissue exert no force as they are not yet in tension. At
33
approximately LO, the passive elements begin to produce a force that increases exponentially with strain until at very high strains, the force produced is entirely due to
passive components. This can be seen in Figure 2-5.
Force-velocity relationship
Besides muscle's force-length dependence, it also exhibits force-velocity dependence.
For muscles, contraction velocity is proportional to the rate of energy consumption.
During muscle shortening when the contractile force exceeds external load and concentric work is done, the velocity is positive and high but there is little force produced because the myosin heads must rotate to be recocked for future cross-bridge
formation.
In contrast, during muscle lengthening when the external load exceeds
contractile forces and eccentric work is done, the velocity is negative and force output
is high because the myosin heads do not need to be recocked. Instead, as the actin
filament slides past, the head rotates past its neutral location.
When the bond is
released, the head merely rotates back to its neutral location where it is again ready
for cross-bridge formation with another actin binding site. Figure 2-6 illustrates this
action.
The force-velocity relationship acts independently of the force-length relationship.
Together they form a surface of muscle behavior shown in Figure 2-7.
2.3
Muscle models
Any sort of muscle control requires a working model of the muscle to be controlled,
and the better the model, the more efficiently functional electrical stimulation can
elicit desired behavior. There are many types of models, but they may be lumped
into three categories: Hill-type mass-spring-damper models, Huxley-based biophysical
models, and what some call the "model-free" purely mathematical model [12].
No one muscle model is perfect; all of them make simplifications, and each has its
advantages and disadvantages. While biophysical models seek to explain macroscopicscale behavior through microscopic-scale mechanics, they are generally not compu-
34
Force
Shortening
Lngth,flon
Figure 2-6: Force-velocity relationship and the underlying actin-myosin mechanism
0
3
05
-3
et
L
Figure 2-7: Surface created by force-length and force-velocity relationships
tationally tractable, especially when applied to practical controllers that may be implemented on mobile prosthetic and orthotic systems. Meanwhile, macroscopic-scale
models often generalize based on isometric observations or Hill-type relationships and
end up missing behaviors that are unique to muscle's nonlinearities. Amongst these
features of muscle which scientists and engineers try to explain are:
35
* the tendency to fatigue over time depending on a particular muscle's white
(fast-fatigue) versus red (fatigue-resistant) fiber composition;
* the ability to potentiate, i.e., produce more than double the force output of one
single pulse when the muscle is subjected to two rapid stimulations in succession;
* the catch-like effect, similar to potentiation but longer-lasting, in which an extra
pulse during a regular pulse train results in lasting force enhancement. This is
especially evident in fatigue-resistant muscle; and
" stretch-induced force enhancement and shortening-induced force depression.
A potential solution to modeling these behaviors is using a purely mathematical
model. However, purely mathematical models can be free of any physical mechanism,
it may be highly difficult to gain intuition that would enable better understanding
and prediction of muscle behavior.
Though this thesis addresses one specific example from the last category of models, a few examples from each category are reviewed so as to illustrate the breadth
of muscle modeling and the alternatives to the chosen Hammerstein model. For a
more comprehensive history and review of muscle models, please consult [24, 44]. A
literature review of Hammerstein muscle models concludes this chapter.
2.3.1
Hill-type models
Hill-type models are by far the most popular of the three categories due to their relative simplicity and their ability to be analyzed by classical mechanical methods. These
models consist of masses, springs, dampers, and black-box contractile elements and
generally concern themselves with describing force-length and -velocity dependences.
A.V. Hill's original model [26] was comprised of a spring in series with a contractile
element; the contractile element is a force generator whose behavior is described by
what has become the rudimentary hallmark of force-velocity relationships:
(F+a)-v=(F 0 -F)-b
36
Most Hill-type models are variations of the contractile element with springs and
dampers in series and/or parallel. For example, Durfee and Palmer used a contractile
element in series with a nonlinear spring (active elements) and both in parallel with
a second nonlinear spring and a nonlinear dashpot (passive elements).
The total
force output of the muscle is modeled as a sum of the forces from active and passive
elements.
[16], as shown in Figure 2-8. Physiologically, the model is a mechanical
Xmt
.a---
Xce
Xs&
K
s
Ave (AE)
CE
PE
Kp
ssve
..........
Figure 2-8: Hill-type model used by Durfee and Palmer (1994)
decomposition meant to imitate the cross-bridge force production mechanism in series
with tendon. Length and velocity-dependences are ascribed to the nonlinear spring
and damper. The mass-spring-damper system is what frequently gives rise to muscle's
description as a second order dynamic system.
Ettema and Meijer chose a different configuration of Hill-type models.
They
compared three models, one of them a Hill-type model consisting of a contractile
element with a series spring and both in parallel with another spring (see Figure 2-9);
the second, a modified Hill model whose contractile element has a contraction history
dependence; and a third called the exponential decay model.
This last is a third
order linear model with no explicit force-velocity relationship; instead, its contractile
element acts based on both contraction history and force-length history. Previous
changes in length exponentially decay with time [17]. Like many researchers, Ettema
and Meijer tailored their models to capture a specific behavior: in this case, stretchinduced force enhancement and shortening-induced force depression for subsequently
37
isometric muscle.
CE
SEE
PEE
Figure 2-9: Hill-type model used by Ettema and Meijer (2000)
Forcinito, Epstein, and Herzog [20] also sought to duplicate non-isometric force
enhancement and depression using a Hill-based model. They chose what they called
a rheological model, as shown in Figure 2-10(c); it is distinct mechanically in its employment of an elastic rack-and-pinion type of setup that allows the stiffness of the
model to vary as a function of muscle length. This parallel elastic element is engaged
only when the muscle is activated. Again, Hill-type contractile elements, springs, and
L
(a) Contractile element
(b) Elastic rack
relay
C
k
(c) Rheological model
Figure 2-10: Hill-type rheological model used by Forcinito, Epstein, and Herzog (1998)
dampers are present. In this case, Hill's force-velocity relationship is not embedded
38
in the contractile element but rather falls out of the combined behavior of the mechanical components. At the end of their discussion, Forcinito et al emphasize that
they make no attempt to provide an all-compassing muscle model covering different
behavioral regimes; rather, they point out that many simple models can represent
different aspects of muscle behavior.
2.3.2
Biophysical models
Hill-type models are useful for gaining intuition about muscle as mass, spring, and
damper-like systems, but the models are purely phenomenological; that is, they are
based only on output behavior and make no reference to the relatively well-understood
underlying cross-bridge mechanics causing the behavior. Some researchers have developed models based on Andrew Huxley's 1957 cross-bridge theory [31] and have
explored the chemical reactions required for myosin head attachment. Calcium ion
concentration tends to be the most common independent variable as it affects the
binding of troponin and the formation of cross-bridges.
Otazu, Futami, and Hoshimiya [41] chose to explain potentiation and the catchlike effect based on the existence of calcium-induced calcium channels and the existence of two stable calcium concentration equilibrium points. In addition to normal
voltage-gated calcium channels, these calcium-induced channels are encouraged by
the presence of increased calcium levels to release additional calcium ions.
Likewise, Lim, Nam, and Khang modeled muscle fatigue by altering the crossbridge cycling velocity. Thus, reduced ATPase activity, increased H+ concentration,
and decreased intracellular pH may slow calcium reabsorption into the sarcoplasmic
reticulum. The reduced calcium levels then cause a delay in activation accordingly
[34].
Wexler, Ding, and Binder-Macleod developed a model with three coupled nonlinear differential equations. The first described calcium release and uptake by the
sarcoplasmic reticulum. The second filter modeled calcium and troponin binding and
release, and the third concerned itself with the force mechanics of cross-bridges. The
first two differential equations with respect to calcium concentration. To model force
39
mechanics, the third relation was based on a Hill-type model with a spring, damper,
and motor in series [48]. Bobet, Gossen, and Stein did a model efficacy comparison in
2005 [5] and showed that the model by Ding et al. was one of the best at fitting isometric force. However, the parameter identification of such a procedure is nonlinear
and may thus get stuck in local minima.
The drawback of biophysical models lies in their complexity and computational
intractability. To address this, Zahalak and Ma attempted to describe macroscopic
behavior based on microscopic mechanics using their bond distribution moment (DM)
model. The bond distribution function developed by A.F. Huxley relates the fraction
of cross-bridges formed to filament displacement from equilibrium (bond length).
Zahalak and Ma showed that assuming cross-bridge stiffness is constant, the first three
moments of the bond distribution function are proportional to muscle stiffness, muscle
force, and cross-bridge elastic energy [51, 49, 50]. Thus, macroscopic outputs such
as force and stiffness are explained on a molecular basis. Still, the bond distribution
moment model has a large number of parameters that must be solved nonlinearly;
though tractable, it lessens the DM model's appeal.
Though physiochemical models can be good descriptors of muscle contractile dynamics, their numerical complexity makes them slow and difficult to realistically
implement. Instrumentation, which may involve chemical sensors measuring calcium
concentrations, is also more difficult and time-consuming.
2.3.3
Mathematical models
Muscle often exhibits non-linear behavior that is not easily described by simple physical models. This category lumps together non-physiologically based models. Some
of these models are linear, while others are nonlinear. Despite muscle's obvious nonlinear behaviors, linear models have long been used to characterize muscle under
certain behavioral regimes. The advantage of linear models lies in their simplicity;
with them, it is easier to develop a physical intuition for predicting muscle behavior
without knowing the exact model or requiring a computer to crunch through the calculations. Moreover, linear models result in speedy calculations that make the models
40
practical for implementation. On the other hand, nonlinear models can capture some
details in behavior that are easily overlooked by linear models. For instance, fatigue,
stretch and shortening effects, and potentiation are all nonlinear behaviors.
This section describes one popular math model, the bilinear muscle model. In the
next section, the models of interest to this thesis-the Hammerstein or HammersteinWiener nonlinear cascade with static nonlinearities acting upon dynamic linear systemswill be reviewed in detail.
Bilinear model
Perhaps the simplest mathematical muscle model is the bilinear model. It uses a
least-squares fit to determine the weighting of different terms and cross terms:
F = Ax + BJ + Cu+ Dx + Exu+ G.u+ ...,
(2.1)
where x represents length, J represents shortening velocity, and u is the input or
activation, depending on the modeler's choice. The cross terms enable an easy and
quick representation of activation-varying stiffness and impedance. The simplicity of
this model makes it a popular choice amongst muscle modelers.
2.3.4
Hammerstein model
The Hammerstein model is comprised of essentially two blocks, a static input nonlinearity followed by a linear dynamic system [23]:
U(t)
V(t)
G()
Input Nonlinearity
Linear Dynamics
f(u)
G(q)
z (t)
Figure 2-11: Hammerstein model structure
The benefit of breaking the system model into discrete, independent blocks is
that the individual blocks may correspond to different natural phenomena whose
41
interactions we may understand better. The alternative to such discretized blocks is
a purely computational black box approach that may work but whose inner mechanics
are incomprehensible.
History of Hammerstein cascade in muscle modeling and prostheses
As early as 1967, scientists and engineers building prostheses began to use the Hammerstein model structure to describe muscle behavior. Vodovnik, Crochetiere, and
Reswick [46] were amongst these pioneers. Their model consisted of a first-order
nonlinearity that exhibited saturation and had a non-zero threshold voltage. Using
this basic Hammerstein model, they developed a closed-loop controller for an elbow
prosthesis.
In 1986, Hunter tested both Hammerstein and Wiener cascades in a biological
setting. Whereas the Hammerstein model uses a static nonlinearity before passing
through linear dynamics, the Wiener model passes the input through linear dynamics followed by a static nonlinearity. Hunter tried both models in describing muscle.
However, unlike most muscle modelers, he attempted to emulate non-isometric muscle
fiber, rather than whole muscle, behavior. His results showed that with the addition
of the nonlinearity, both models could solve the problem of what seemed to be input
amplitude-dependent time constants. He also found the Wiener model outperformed
the Hammerstein one [30]. Yet, this result does not detract from the value of Hammerstein models versus Wiener since Hunter experimented with non-isometric muscle
fibers rather than whole muscle. Muscle fiber is binary with either complete or no activation and therefore does not undergo muscle recruitment, the critical phenomenon
that Hammerstein models seek to describe. In a separate publication the same year,
he and Korenberg mentioned that the least-squares computations were more straightforward for Hammerstein than for Wiener systems [29]-another motivation for using
Hammerstein models.
Also that year, Bernotas, Crago, and Chizeck developed a discrete-time Hammerstein model for isometric muscle. They tried three different discrete models and settled upon a second-order system with two poles and one zero. Varying muscle length
42
and stimulation frequency, the authors found that for two different muscles, soleus
and plantaris, the length and frequency dependences were different: soleus acted like
an overdamped second-order system at high stimulation frequencies and short muscle
lengths. As the length increased and the stimulation frequency decreased, the system
became less damped. Plantaris, on the other hand, was underdamped at high stimulation frequencies. This was attributed to fast versus slow muscle fiber composition.
Also, the authors used recursive least squares estimation, so online identification was
possible [4, 27].
Two years later, Chizeck, Crago, and Kofman used a Hammerstein model in their
closed-loop pulse width-modulated controller. Their control loop was tested on two
isometric cat muscles, soleus (slow-twitch) and plantaris (fast-twitch) and was robust
against time-variant effects such as potentiation and fatigue. The researchers pointed
out their controller design did not require a very good description of the system's
dynamics or frequency response, yet was able to perform adequately [9].
Bobet and Stein [6] investigated a variation of the Hammerstein model in 1998;
their Hammerstein model was preceded by a first-order linear low-pass filter representing calcium release. The input was modulated by pulse period, which is different
than most as this adds a temporal aspect to recruitment and may subsequently affect dynamics. The traditional static nonlinearity was interpreted to be saturation
of calcium-troponin binding while the first-order linear output dynamics represented
cross-bridge dynamics. Their model simulated both fused and unfused tetanus well
but predicted isolated twitches poorly.
In 2005, Bobet, Gossen, and Stein compared simulations from seven different
models to experimental data collected on isometric ankle contraction. Again, the
muscles were stimulated with pulse period modulation rather than by amplitude or
duration. The models, all limited to six or fewer parameters to be identified, were:
(1) a critically-damped linear second-order model; (2) a general linear second-order
model; (3) a third-order linear model with one zero used by Zhou et al. [52]; (4) a
general linear model using Hsia's least squares weighting function method to determine best possible impulse response assuming linearity [27]; (5) a Wiener model with
43
second-order dynamics and third-order polynomial output nonlinearity; (6) Bobet
and Stein's 1998 model; (7) the biophysical model of Ding et al.'based on two differential equations for calcium activity and one for force dynamics. Bobet et al. found
that the Bobet-Stein and the Ding et al. models performed best. They also stated
that a critically damped second-order model was the best possible linear model, with
linear models having errors of 9% or higher of maximum stimulated force. Bobet et al.
concluded that a Hammerstein model in conjunction with a series spring and appropriate force-velocity relationship could yield a Hill-type model effective at simulating
non-isometric as well as isometric contraction [5].
Hammerstein model shortcomings
The Hammerstein model has shortcomings, of course. For one, it assumes that the
dynamics are linear, which is not necessarily true. It also assumes calcium dynamics are independent of recruitment. Physiologically, this is not true: Henneman's
principle dictates that small neurons innervate slow-twitch motor units, while large
neurons innervate fast-twitch [25]. Thus, there is a coupling between the motor units
recruited and the system dynamics. This coupling presents another system identification challenge as well: physiological recruitment begins with the finest motor units
to the largest (slow to fast), while experimental recruitment, due to the crudeness of
electrode stimulation, begins at the level of the largest motor units to the smallest
(fast to slow). Therefore, experimental stimulation may lead to identification of a
neuromuscular system that is unlike the real system in its natural working environment.
Hunt, Munih, Donaldson, and Barr [28] proposed an alternative to the Hammerstein model that addressed this coupling issue. They proposed the use of local
linearized models about different activation levels and a final interpolation between
local models to create a single standalone model. Gollee, Murray-Smith, and Jarvis
followed up in 2001 [21] with a model which used a scheduler to weight each of the
local models according to its relevance at the current operating point.
It is important to note that while such criticism is valid, the proposed alterna44
tive of multiple local linearized models is less tractable as an algorithm for practical
implementation. In fact, the very same critics-Hunt et al.-chose to employ a Hammerstein model-based controller in their experimental orthotic device two years later
[38].
Another pitfall of the classical Hammerstein model is its inability to describe nonisometric muscle behavior.
However, the Hammerstein model may nonetheless be
useful by providing an isometric model upon which a non-isometric model may be
built. This approach was explored by Farahat and Herr [19]; their model combined
the Hammerstein structure in series with a static output nonlinearity. The output
nonlinearity has three inputs: position, velocity, and the output from the isometric
Hammerstein subsystem, z. This is similar to a Hammerstein-Wiener cascade except
that the output nonlinearity is linear in z; that is, h(x, ., z) = z. h(x, i), where z, x,
and - represent activation, muscle length, and shortening velocity, respectively.
Convergence of Hammerstein iterative identification procedures
In 1966, Narendra and Gallman were the first to explore an iterative procedure able
to identify Hammerstein models [39]. Their strategy lay in bootstrapping: estimating
one function at a time, whether it was the static nonlinearity or the linear dynamic
system, then using the new estimate to determine the other function. Both the nonlinearity and dynamic difference equation were estimated by choosing linear coefficients
that minimized the squared error. Narendra and Gallman did make clear that their
method did not always converge to the model minimizing the mean squared error,
but they showed that for some cases of polynomial nonlinearities followed by linear
dynamics, their procedure could be effective and result in rapid convergence.
Stoica provided a counter-example demonstrating that the procedure was not
generally convergent but commented that Narendra and Gallman's method could
still be useful if few counter-examples existed [45].
Bai and Li then proceeded to refute Stoica's specific counter-example by illustrating how Stoica's choice of parameter normalization resulted in unbounded errors, yet
with a subtle change in normalization, the errors reach a global minimum.
45
While
Bai and Li made no claims as to universal error convergence, they did illustrate how
minor changes in iterative procedure may lead to very different convergence results,
and they emphasized the utility of such an iterative identification procedure [2].
Rangan, Wolodkin, and Poolla reinforced the importance of an iterative versus
correlative identification method. They proved that, provided the input is white noise
and the data set is sufficiently long, the iterative result will be a global minimum [42].
However, since for this thesis, there is no guarantee that our input is white, likewise,
covergence is not guaranteed.
Also, though it is not applied in this thesis, using constrained rather than regular
least squares methods should improve the iterative convergence of the model. By
constraining the nonlinear recruitment curve to be a monotonically increasing and
slowly time-varying (due to fatigue) polynomial, Chia, Chow, and Chizeck presented
results which showed a recruitment curve more in line with the sigmoidal shape found
in muscle literature. Their simulations with the constrained recruitment curve also
showed better estimates of muscle force [11, 8].
Given the convergence issues of iterative Hammerstein identification, some have
attempted to find non-iterative methods. Chang and Luus [7] did so by converting
a single input-single output Hammerstein system into a multiple input-single output
model. The input nonlinearity regressors constitute the multiple inputs, each filtered
by the linear time-invariant dynamics. These filtered regressors are then least-squares
weighted to the output. The proposed method greatly reduces computation time and
can be as good an estimate as the iterative process, but it requires that disturbances be
relatively low [27]. Moreover, Chang and Luus demonstrated only offline identification
as their model assumed the previous outputs were all known, not estimated. Since
the outputs were not rapidly changing and the true outputs were continually fed back
through the system, the estimated output was not allowed to stray from the true
output. Thus, it is difficult to gauge the performance of the estimated system over
longer time periods, e.g., more than three time samples for a third-order difference
equation.
46
Chapter 3
Methods and Procedures
This chapter is comprised of two main sections: experimental procedure and simulation methods. Experimental procedure includes hardware setup and experimental
protocol; it begins this chapter.
3.1
Hardware
3.1.1
Experimental set-up
The non-biological hardware was the same as that from Farahat and Herr [18] and
consisted of eight fundamental components:
" two computers, one of them the user interface and the second a machine that
did all of the processing;
" a voice coil motor which enforced a muscle length boundary condition-in this
case, a constant length;
" a load cell which measured muscle force output;
" a linear encoder which measured the position of one end of the muscle and
whose reading was fed back to maintain the isometric boundary condition;
* a suction electrode comprised of a syringe entwined with fine silver electrode
wire that directly contacted the sciatic nerve of the muscle. The syringe tip's
47
pressure difference helped maintain electrode contact by sucking the nerve end
close to the electrode wire;
" an electrical power supply providing up to at least 18V;
" a circuit board specially built and programmed to deliver bipolar stimulator
pulses between 0 to 18V with l[s timing resolution.
The circuit board was connected to a data acquisition board in a specially dedicated computer; this machine was booted to exclusively run the MATLAB xPC Target
kernel and handled all the data acquisition as well as commands to the stimulator
board. The user issued high-level commands via a GUI on the xPC host machine.
These commands were then communicated to the target machine through a TCP/IP
connection. Lower-level functions were directly programmed into a microcontroller
on the circuit board.
Host
Computer
TCP/IP Connection (xPC Target)
0-1
Target
Computer
DAQ Cable
Suction
Electrode
Sciatic Nerve
oice Coil
Mo)tor
usc egl
Load
Ringer's Solution Bath
Figure 3-1: Experimental setup
Experiments were run by an identification experiment model built in the MATLAB Simulink xPC Target environment. The Simulink model dictated stimulation
48
parameters and onset of stimulation while recording muscle output force and length
via a load cell in series with the muscle and via a linear encoder, respectively. Time
and the varying stimulation parameter were also recorded.
3.1.2
Muscle set-up
The experiments were performed on leopard frog (Rana pipiens) plantaris longus
muscles. To extract the muscles, the frogs were anesthetized in icy water, then euthanized via a double pithing procedure (severance of the brain and spinal cord). Next,
the skin was removed from the thigh and calf. Then the thigh was positioned at a
ninety-degree angle from the body, and the calf was positioned perpendicular to the
thigh. Silk suture was tied around each end of the plantaris longus muscle as close
as possible to the hip and knee joints. The distance between sutures was measured
in vivo and recorded as the rest length of the muscle. Although the suture distance
was not equivalent to muscle belly length since tendon length was inevitably included
in the measurement, tendon has experimentally been shown to be much stiffer than
muscle [3]. Therefore, any changes in suture distance were practically equivalent to
changes in muscle belly length. Moreover, suture distance was readily identifiable
whereas the muscle belly length was not quite as apparent since the muscle and tendon fused into one another. Having a known rest length was important because of
muscle's force-length dependence and because maximal force was produced at rest
length when cross-bridge formation could be maximized.
After measuring the suture distance, the muscles were cut out of the body with
small bone chips attached to either end. These chips helped prevent suture slippage.
A length of the sciatic nerve attached to the muscle was also preserved. The sutures
were tied to small dove-tailed acrylic mounts, and the mounts were affixed, the knee
mount to the load cell and the ankle mount to the voice coil motor (VCM). The VCM
was then calibrated so its reference position held the muscle at rest length. Last, the
suction electrode was positioned in direct electrical contact with the sciatic nerve but
not the rest of the muscle; instead of applying charge directly to the muscle tissue,
this emulated natural recruitment as much as possible.
49
Throughout extraction and experimentation, the muscles were kept moist with
commercial amphibian Ringer's solution (Post Apple Scientific, Inc.).
Please see
Figure 3-1 for a diagram of the experimental hardware.
3.2
Experimental procedure
One frog provided two plantaris longus muscles. Experiments were conducted on one
muscle at a time: while the first muscle underwent trials, the second muscle was
refrigerated to slow its metabolism and thus minimize necrosis of the core tissue.
Fourteen to fifteen tests were performed on each muscle.
Each test was conducted in three stages. The first stage measured the current
position of the linear encoder. The second stage compared the current and reference
positions, then commanded the voice coil motor to make a smooth, linear transition
to the desired position. The third stage used the voice coil motor to maintain the
reference position during isometric stimulation of the muscle. Note that for higher
muscle force output, some muscle shortening occurred due to VCM compliance.
The stimulation input consisted of a bipolar pulse train with varying pulse width;
amplitude and pulse count remained constant. The choice of pulse width, rather than
amplitude, as the varying parameter originated from literature demonstrating that
pulse width modulated force just as well as pulse amplitude [9] and, furthermore, was
less damaging to tissue [10, 37]. Bipolar pulses were recommended as they lessened
tissue damage as well as prevented electrode corrosion [33, 13].
Since the Simulink model recorded actual stimulator triggers (which could include
random pulse period variation, or "jitter") rather than pre-programmed pulse periods
(no jitter), pulse period was not necessarily fixed. Pulse period jitter was added as
a possible method of enhancing the identification process as it could cover a slightly
wider range of dynamic response in a single experiment.
Moreover, pulse period
jitter could have helped prevent muscles from fatiguing due to very regular, cyclic
stimulation. Owing to the stimulator cards' design and programming, pulse period
was interpreted as the time between the end of one pulse and the onset of a second
50
rather than the traditional definition as the time from the onset of one pulse to the
onset of a second pulse.
Experiments lasted seven seconds: three to allow the VCM to servo to its reference isometric position and four for actual muscle stimulation (unless otherwise noted
in Appendix A). Tests were conducted with 1OV-pulses at frequencies between 1Hz
to 100Hz. Each stimulation consisted of only one pulse to avoid modulation simultaneously by pulse width and pulse period as demonstrated by [10, 6] and to avoid
temporal effects on dynamics. A few tests were conducted with frequency jitter in
which pulses could be delayed up to 90% of the stimulation period minus maximal
pulse width. Pulse width varied between 0 to ims with a resolution of Lys.
For further specifics on each experimental run, Appendix C includes notes from
each test.
3.3
System identification and simulation
The bulk of this thesis' work was on system identification of isometric muscle. Once
the experiments were completed, curve-fitting was conducted on data from the one
muscle that underwent different stimulation frequencies. Each data set was divided
into two sections: a training portion to estimate the static nonlinearity as well as the
linear dynamic system (recall Figure 4-1) and a validation portion which used the
same static nonlinear function and linear dynamic system but with a different input.
The input was the electrical pulse train delivered to the muscle via suction electrodes in contact with the sciatic nerve. In our model, this signal was then sent
through the input nonlinearity block, which may be interpreted as an isometric recruitment curve (IRC) [15]. The output signal was filtered by a dynamic system.
Employing the common simplification that the muscle was purely isometric, the output of the dynamic filter was equivalent to force produced by the muscle.
Each data set was identified in two parts. The first involved training the model
to determine what coefficients and systerh orders were appropriate. The second half
involved inputting the experimental stimulation into the model trained by the first
51
half. This would produce a predicted output force. For the figures in this thesis, blue
lines plot the measured force, while both red and green lines plot the estimated force.
The red lines illustrate the interval over which training of the model occurred.
Identifying each data set's Hammerstein system involved two main challenges:
determining the static input nonlinearity and the linear dynamics.
3.3.1
The static input nonlinearity
The static input nonlinearity was estimated using linear combinations of third- to
fifth-order Chebyshev polynomials. The coefficients of each of the Chebyshev polynomials were determined using a least-squares regression. There are several ways to do
least squares regression, but a linear (in the coefficients) least-squares is the simplest.
It is known as the Moore-Penrose pseudoinverse.
Use of the Moore-Penrose pseudoinverse
For systems Ax = b where the A-matrix is not square or full rank or for systems where
Ax only approximates b so there is no exact solution, the regular inverse A- 1 cannot
be found. The pseudoinverse is calculated instead. In curve fitting-an instance
of Ax ~ b--an effort is made to determine the best x such that the error Ax - b
is minimized. However, instead of using a direct measure of error, which may be
positive or negative and thus result in some cancellation, the square of the Euclidean
norm ("length") of the error is minimized:
d
-d Ax - b 12 =
dx
=
__
-((Ax - b) T - (Ax - b))
dx
d
-(( Ax )TAx
dx
=
dx
-
(Ax )Tb
-
bT(Ax )
((Ax)T Ax - 2(ATb)x - bTb)
-
b'b)
(3.1)
= 2AT Ax - 2AT b = 0
This leads to the solution x = (ATA)-'ATb, where (ATA)--'AT is the pseudoinverse.
In the curve-fitting procedures for this thesis, the pseudoinverse was used to de52
termine coefficients that weight components depending on their contribution to the
final output. In this case, the output was force. Note that this iterative procedure
remained a linear least squares fit: it was linear in the coefficients though the basis
functions themselves were nonlinear.
Basis functions for the input nonlinearity
Despite physiological evidence that supports a sigmoidal-shaped isometric recruitment relationship between input parameter value (one of a few different things: pulse
amplitude, pulse width, pulse count, or pulse frequency) and activation, the basis
functions used in this thesis were not sigmoids. Sigmoids may be represented by a
function of the form:
1
S =(3.2)
1 - exp(-a(x - xo))'
where a changes the slope of the sigmoid and xO changes the point at which the
sigmoid passes through 0.5. To change the shape of this sigmoid would require a
nonlinear regression in a and xO, and of course, a nonlinear regression involves more
difficult computations than a linear regression. Thus, instead of attempting to directly
fit a sigmoid, two other types of basis functions were tried: radial basis functions and
Chebyshev type I unshifted polynomials.
Radial basis functions
Radial basis functions (RBFs), which are essentially Gaussians with specified variances and centers, are described mathematically by:
A(u)
=
exp (-U2
,
(3.3)
where C and o are the centers and spreads of the Gaussians. They are often used
because they are simple to implement in multiple dimensions, and the variances and
centers can be tweaked to be nearly, but not completely, orthogonal and independent
of each other. Moreover, it is relatively simple to get an intuition for how all the
weighted RBFs can be summed to resolve a curve.
53
Chebyshev polynomials
Chebyshev polynomials are also popular for curve-fitting procedures and were also
tried during the curve-fitting procedures. Chebyshev polynomials obey the following
orthogonality condition on the [-1,1] interval:
ifm # ri,
1
/11
0
if m = nf0
Orthogonal polynomials are desirable in curve-fitting for a number of reasons; most
importantly, they run into fewer problems when calculating the matrix's inverse. The
factor
1
implies that Chebyshev polynomials are only orthogonal on the [-1,1]
interval for a given input probability distribution. Experimentally, this is impossible
to achieve: there is never infinite probability at the extreme ranges of the data set.
However, though Chebyshev polynomial regressors will not be completely orthogonal,
they have similar variances (their amplitudes on the [-1,1] interval are roughly the
same), so a regression matrix composed of Chebyshev polynomials should be relatively
well-conditioned when taking the pseudoinverse (Westwick and Kearney [47, pp 2933]). In addition, because they are roughly orthogonal, Chebyshev polynomials may
approximate a curve more concisely using fewer numbers of polynomials.
There are several types of Chebyshev polynomials. Those of the first kind, T(x),
are more frequently used than those of the second, U(x). The two kinds have different input probability distributions: T(x) uses
i
V/(1-X)2
while U(x) uses V(1 - x) 2 .
Both kinds may also be shifted to be orthogonal on the [0,1] interval. In this thesis,
unshifted Chebyshev polynomials of the first kind were used. They are generated by
54
the recurrence relation:
To(x) = 1,
T1(x)
T,1(x)
=,
=
2x -T,(x) - Ti-1(x).
(3.5)
When constructing the regression matrix, each column was composed of one Chebyshev polynomial whose input x was merely the input to the system at that time
sample. The length of each column was equivalent to the length of the data set.
3.3.2
The linear dynamic system
The second challenge in Hammerstein system identification was determining the linear
dynamics best describing each data set. The linear dynamic system was estimated
using various functions from the MATLAB System Identification toolbox.
The most important modeling choice was the disturbance model selection. Several disturbance models are available from the System ID toolbox, such as ARX
(autoregressive with exogeneous input), ARMAX (autoregressive with moving average exogeneous input), and OE (output error) models. The output error model was
chosen on the basis of noise characteristics; it assumes noise is added directly to the
measurement of output force and does not enter into any of the system dynamics. A
mathematical description of the discrete-time model is:
B(q)
(3.6)
u(t - nk) + e(t),
y(t) = B(q)
F(q)
where nk is the delay and q is the shift operator (commonly also denoted as Z-1
in the signal processing field or Z by controls engineers). Thus,
)
represents a
discrete-time difference equation. The orders of B(q) and F(q) are specified by the
user; if the user dictates that both orders be equivalent, MATLAB fits a F(q) of the
specified order and generally a B(q) of one order less than specified. Using the output
force data and the guessed v = f(u), a dynamic model-linear in its coefficients-was
55
estimated by the oe command.
MATLAB's oe command
A crucial part of the identification process occurs within MATLAB's oe command,
so the command warrants some discussion. The oe code finds parameter coefficients
using the prediction error method in which the minimized criterion is the square of the
filtered error, normalized by the length of the data set. The linear filter is created by
multiplying the input frequency spectrum by the inverse noise model and is referred
to by MATLAB as prediction weighing. This filter essentially weights the transfer
function fits more heavily for frequencies containing more spectral power.
1
ON
In the above equation,
6N
N- 1
= argmin -
(L(q) - e(t, 0))2
(3.7)
refers to the parameter estimates (e.g., difference equation
coefficients), N to the length of the data set, DM to the set of models to which
the estimated model belongs, L(q) to the prediction weighing filter mentioned above,
and e to the errors between estimates and actual data. Note that this is a nonlinear
optimization as e is both a function of time t and parameter estimate 0.
There are several algorithm options available for oe. The algorithm itself is iterative, and by default, the maximum number of iterations is twenty. For this thesis,
the only additional specified property was that the focus be on stability. MATLAB's
estimation of a stable system uses the prediction weighing method mentioned above.
For more information on the prediction error method algorithm, please consult
Ljung's book, System Identification: Theory for the User [35] and MATLAB documentation for the System Identification toolbox.
Simulation initial conditions
Once the coefficients of the discrete transfer function M)
F(q) were found, the transfer
function system representation was converted to a state-space representation. Statespace was preferred over a transfer function representation as it allowed specification
56
of non-zero initial states for the validation half of the data. The initial conditions
for the training half of the data were zero. For a discrete system, the states could
be previous values of the output. However, in simulation, when the states and the
force estimate were plotted, the states were indeed previous force estimates, but they
were also multiplied by a gain. This gain may be explained by examining the generic
continuous-time state-space representation:
x(t)
= Ax(t) + Bu(t)
y(t) = Cx(t) + Du(t -
tdelay)
(3.8)
+ e(t)
The value of gain observed is the reciprocal of the coefficient C. Thus, the non-zero
initial states of the validation data were more easily calculated using a state space
representation.
3.3.3
Bootstrapping
Before the iterative process began, a number of critical modeling decisions must be
made. What order should the dynamic system be? What should be the largest order Chebyshev polynomial to be included in the nonlinear fit? Muscle is sometimes
described as a critically-damped second-order dynamic system, but higher order behavior would be neglected by such a simplistic model. It was unclear what order
the model and the nonlinear fits should be, so combinations of dynamic systems
between second- to fourth-order and Chebyshev polynomials between first- to thirdorder were tested on sixteen data sets. Purely dynamic systems between second- and
fourth-order were also simulated. Root mean-squared errors were recorded for each
simulation, and based on the errors and quality of fits, the orders of the dynamics
and the polynomial nonlinearity were chosen.
Hammerstein model structures require two functions: a static input nonlinearity
and a linear dynamic system. When iterating to estimate the model, either the static
nonlinearity or the linear dynamics may first be estimated. In this case, the initial
estimate was of the static nonlinearity; it merely guessed v(t)
57
=
f(u) =
u.
0
Having assumed a v(t), the linear dynamic system was then estimated between
v(t) and f(t) using MATLAB's oe command to create an output error model. The
algorithm was implemented with an emphasis on identifying stable systems, but stability was not guaranteed.
Finding the nonlinearity composed of weighted Chebyshev polynomials was the
next step. Calculating the inverse dynamic system, estimating the new v(t) based
on f(t), and then finding the polynomial coefficients would require three steps. Additionally, it would require fitting polynomial weighting coefficients to a v(t) that
would be heavily reliant on the estimated dynamic system. Instead, system linearity
was exploited. It presented a less computationally expensive alternative that fit the
regression matrix directly to measured output y(t) rather than an estimated v(t). In
other words, it was assumed that the model was still linear: a gain at the beginning
of the system was equivalent to a model with the same gain being multiplied once
anywhere else in the system. In this case, the regression matrix was passed through
the newly-estimated dynamic system, and a least-squares fit was then conducted directly between that filtered matrix and the measured output force. This produced an
estimate of the regressor coefficients, which were then weighted (a = 0.5) and added
to weighted (a = 0.5) previous regressor coefficients. a could be changed between 0
(use only new coefficients) and 1 (use only old coefficents), and was enacted to lessen
rapid fluctuation of the input nonlinearity. On the down side, it slowed down model
convergence.
The matrix product of the regression matrix and the weighted-sum
coefficients constituted the new estimate of v(t).
The process then repeated by re-estimating the dynamic system based on the new
v(t) and the measured force f(t). The iterative procedure continued until the error
metric reached a pre-specified threshold. Here, the stopping criterion required that
the standard deviation of the last five errors, normalized by the last error, be less
than five percent. Stability of the converged estimate was also recorded: if the H2
norm of each dynamic estimate was not infinite, the estimate was stable.
As a cautionary note, this iterative procedure is not guaranteed to find a global
minimum.
Narendra and Gallman [39] employed an iterative process to identify
58
known Hammerstein systems, and they showed it worked for their four test cases. A
counter-example was provided in 1981 [45], but the same was refuted in 1995 by Rangan et al. [42]. Additionally, Rangan et al. claimed that convergence was guaranteed
provided the data set was sufficiently long and the input was white. In summary, error convergence of iterative Hammerstein identification remains debatable, and since
we had no guarantee that our input ws white, neither could we ensure simulation
convergence.
As standard practice, the low frequency gain of either the nonlinearity or the
dynamic system is normalized. The gain is preserved by multiplying it with the
other function. Here, the recruitment curve was normalized to unity to represent full
activation.
3.3.4
Simulation order
With several available parameters with which to play, the first question became, what
orders of dynamic system and polynomial nonlinearity were appropriate? The first
batch of simulations were conducted on second-, third-, and fourth-order dynamic
systems without any input nonlinearity. From the results, it became easier to justify
the choice or elimination of a particular dynamic system order. The next step was to
add the nonlinearity to the choice(s) of dynamic systems. These simulations were run
with first-, second-, and third-order polynomial nonlinearities. Finally, simulations
were run with fixed nonlinearities and/or dynamics to determine how well one model,
trained by one data set, could estimate the output of another data set.
59
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60
Chapter 4
Results and Discussion
This chapter presents results and discusses them immediately. The first section uses
a prototypical example to review the identification process. The ensuing sections
examine questions presented by the identification process; in order to determine the
effectiveness of Hammerstein models across stimulation frequencies, several additional
questions must be asked. For instance, is the input range adequate to cover the force
behaviors in which we are interested? What order of linear dynamics is appropriate?
What should the highest-order polynomial regressor be for the input recruitment
nonlinearity? Can fixing the nonlinearity produce better curve fits when the data is
unable to inform the recruitment curve? Such are the questions asked and answered
in this chapter. By incrementing the modeling degrees of freedom, it is possible to
see which, if any, are drastically beneficial to the curve-fitting process. The best fits
that the algorithm can produce may then be compared across input frequencies to
determine up to what frequency the models generate reasonable estimates.
Note that all plots, unless otherwise noted, show only results from stable estimated
systems. Logs of experimental and simulation notes are contained in Appendices A
and C. Meanwhile, specific examples and figures are contained in this chapter to
illustrate particular points of discussion.
61
4.1
A prototypical system identification
To review, the Hammerstein model structure is shown below. The notation, as in-
Utt)
V (t)
-
)
Input Nonlinearity
Linear Dynamics
f(u)
G(q)
z (t)
Figure 4-1: Hammerstein model structure
troduced in §2.3.4 is as follows: u is the system input to the static nonlinearity, v is
the output of the nonlinearity and input to the dynamics, z is the measured system
output, and Z^is the estimated output from the dynamics. Physiologically, these variables may be interpreted as the input pulse train, the activation level, the measured
contractile force, and the estimated force, respectively.
The following sample identification process was conducted with third-order dynamics, third-order Chebyshev polynomial regressors, and a pure delay of 2.5ms. The
first step in the identification process was making an initial guess of the nonlinearity
or dynamics. For this thesis, the initial guess of the nonlinearity was v = f(u) = uin other words, a linear recruitment curve with no gain. The algorithm began by
making an estimate of the dynamics between v = u and z using the oe command.
The command outputted a discrete-time filter through which the Chebyshev polynomial regressors of v are passed. A least-squares fit was made between the regressors
and the measured data, and those coefficients dictated the new v. The new v was
then passed through the filter to create an estimated force i. This completed the
first iteration; the resulting recruitment nonlinearity and estimated force are shown
in Figure 4-2(a).
The second iteration then used the newly-estimated v and measured force z to
get a new estimate of the dynamics, and the process began anew. After the second
iteration, the recruitment nonlinearity and estimated force were as shown in Figure 42(b).
Figure 4-3 shows the recruitment nonlinearities and estimated forces after three,
62
IRC recruitment cure (2Hz): iteration #1
Force estimate (2Hz): iteration #1
0.3
1.
F
-measured
estimated
025
0.8
0.2
-3 0.6
0.15
0.4
0.1
stimulation inputs
continuous estimate
0.2
0.05
----
0
0
5
4.5
4
35
3
-0 21 )
5.5
0.1
0.2
0.3
0.4
0.5
0.6
0.7
pulse width (seconds)
0.8
0.9
u=
time (s)
X 10-
(a) Force and isometric recruitment curve after one iteration
IRC recruitment curve (2Hz): iteration #2
Force estimate (2Hz): iteration #2
12
140
120 -
_
mard
100
0.8)
80
0.6
60
0.4
40
0.2
20
=
--
0
0
3
3.5
4
4.5
5
0
55
time (s)
0.1
0.2
0.3
0.4 05
u = pulse width
stimulation
confinuous
0.6
0.7
(seconds)
0.8
Input s
estim ate
(b) Force and isometric recruitment curve after two iterations
Figure 4-2: Example identification after 1 and 2 iterations (2Hz data)
63
1
0.9
X 103
IRC recruitment curve (2Hz): iteration #3
Force estimate (2Hz): iteration #3
nq
0.25
1.2
-
-
measured
estimated
--
0.2
0.8
0.15
0.6
0.1
0.4
0.05
0.2
+
--
O
0
-U us
3
3.5
4
5
4.5
-0.2 '
0
55
III
01
0.2
0.4
03
u = pulse
time (s)
stimulation inputs
continuous estim ate
06 0.7
0.5
width (seconds)
0.8
0.9
1
X 10
(a) Force and isometric recruitment curve after three iterations
IRC recruitment curve (2Hz): iteration #4
Force estimate (2Hz): iteration #4
1.2 .
0.4
036
-
---
measured
0.8
0.25
0.6
0.2
0.16
0.4
0.2
006
0,05
-
0-0,06
3
5
4.5
4
3.5
stimulation inputs
estimate
continuous
--
0
--
-0 2'
55
0
0.1
0.2
0.3
time (s)
u
=
0.7
05
0.6
0.4
pulse width (seconds)
0.8
1
0,9
3
X 10
(b) Force and isometric recruitment curve after four iterations
Force estimate (2Hz) iteration #12 (converged)
E,,
0.029961
IRC recruitment curve (2Hz): teration #12 (converged)
0 45
0.4
-_----__
-
0 35
03
measured
training fit
validation fit
stimulation points
0.8
2 0.6
0.25
0.2
0.4
0.15
*
0.2
0.1
0.05
+
training set
validation set
continuous estimate
0
3
3.5
4
45
5
5.5
6
6.5
-0 2
7
0
01
0.2
0.3
0.4
u = pulse
time (s)
0.7
0.6
0.6
width (seconds)
0.8
0.9
1
X 10
(c) Force and isometric recruitment curve after twelve iterations (converged result)
Figure 4-3: Example identification after three, four, and twelve iterations (2Hz data)
64
four, and twelve iterations. By the twelfth iteration, the algorithm converged; that
is, the standard deviation of the last five RMS error values reached less than 5% of
the last RMS error value.
This identification procedure included a number of parameters which were specified by the user, such as input range, dynamic system order, the existence and
duration of pure delays, the choice of regressor basis functions, and so forth. In order to determine what each of these parameters should be, studies were conducted
beforehand to explore the options for each of these parameters. The ensuing sections
detail the investigations into each of these questions. We begin with the question of
whether or not force saturation was being reached with the input range used.
4.2
Is the input pulse-width range adequate to
reach force saturation?
One of the questions that arose was whether or not the range of pulse widths was
adequate to reach saturation of the contractile force and whether or not the polynomial estimates of the recruitment curves truly reflected the recruitment behavior of
the muscles. Assuming the dynamics were purely linear, each pulse should elicit a
force peak of proportional magnitude.
These questions were addressed by plotting the magnitude of the individual force
peaks versus the inputted pulse widths for the corresponding twitches. The force
peaks were obtained after low pass filtering the raw data using a second-order Butterworth low pass filter with a cutoff frequency of 100Hz, then differentiating and
looking for zero crossings. Since each twitch must be isolated in order that the dynamics are not affected by a previous input, the following exercise was conducted
primarily for low stimulation frequency data sets. At high frequencies, it was more
difficult to ascertain the discrete peaks. Figures 4-4(a) and 4-5(a) illustrate such a
relationship between force and pulse width for two stimulation frequencies, 1 and
2Hz. The relationships were obtained in a manner similar to Durfee's peak impulse
65
method [15] (see §2.2.2).
Isometric recruitment curve for
22jun06/lD 001 .sbr
IRC: ID 001 (3rd order nonlinearity)
experiments
1 2
4 .5
3 .5
0.8
3 0.6
2 .5
E
.2
-
04
1
/
1
0.2
0.5
0
0.1
0.2
0.3
0.4
u
=
05
06
0.7
08
0.9
pulse width (seconds)
0
1
01
0.2
0.3
0.8
0.7
0.4
0,5
pulse width (seconds)
0.8
0.9
u=
X 10
3
x 10
(b) IRC found by iterative algorithm
(a) Pulse width and corresponding force
Figure 4-4: Isometric recruitment relationship - 1Hz
Isometric recruitment curve for
experiments 22jun06/lD 015.sbr
IRC: ID 015 (3rd order nonlinearity)
1.2
0 25
0.2
-
0. 15
06
046
8
0.2
005
0
0
0.1
0.2
0.3
0.4
0.5
08
u = pulse width (seconds)
0.7
0.8
09
0
x
10"
01
0.2
0.3
0.6 0.7
05
04
u = pulse width (seconds)
0.8
0.9
x 103
(b) IRC found by iterative algorithm
(a) Pulse width and corresponding force
Figure 4-5: Isometric recruitment relationship - 2Hz
Figures 4-4(a) and 4-5(a) demonstrate that increasing pulse width did indeed
result in recruitment of motor units to produce higher force levels. All three plots
also confirm the pulse width range between 0.1 to ims was adequate to attain force
saturation.
66
4.3
Can the algorithm identify the appropriate recruitment curve?
Figures 4-4(b) and 4-5(b) are third-order polynomials obtained, in concert with thirdorder dynamic systems, by the iterative identification algorithm used by Hunter and
Korenberg [29] and mentioned by Durfee and MacLean [15]. The figures show the
algorithm produces recruitment nonlinearities resembling those found by the peak
impulse method (see Figures 4-4(a) and 4-5(a)). In addition, they are consistent with
the recruitment curves found in the literature.
4.4
What basis functions should be used to fit the
input nonlinearity?
One of the crucial decisions was what basis function to use as a regressor for the static
nonlinearity. Three types of basis functions could potentially be tried: sigmoids, radial basis functions, and Chebyshev polynomials. Sigmoids, though similar in shape
to experimental recruitment curves, were scrapped because they require nonlinear regressions rather than straightforward least-squares estimates. Likewise, radial basis
functions (RBFs) often produced very wavy recruitment curves with widely fluctuating coefficients. This was especially true if the variances were relatively small and
the centers not particularly numerous. Such fluctuations are indicative of a large
condition number-meaning, the coefficients have great relative sensitivity to noise
(Westwick and Kearney [47, p 29]). Intuitively, this may be explained by the lack of
data points near a particular RBF center. Thus, the uninformed RBF center may be
heavily weighted; instead of considering its weighting coefficient based on contribution of the peak to the overall curve, the pseudoinverse determines the weight based
on contribution of the RBF's low amplitude edges to the overall curve. These weights
often lead to drastically inaccurate curve fits. Due to the waviness and unreliability of
using RBFs for fitting, they were scrapped in favor of Chebyshev polynomials, which
67
were then used across all the simulations described below.
4.5
What is the appropriate input delay?
Delay selection was important as the algorithm often estimated non-minimum phase
responses when given inappropriate delays, as seen in Figures 4-6 and 4-8 (blue =
measured force, red = estimated force over training interval, green = estimated force
over validation interval, - = onset of stimulation).
To determine what number of
delays would suit best, the delay parameter of Matlab's oe command was increased
from between one to thirty-five delays for 3, 5, 7, 10, 20, 75, and 100Hz data. The
algorithm was then run with a second-order polynomial nonlinearity and third-order
dynamics. The delay that generated least simulation error was noted for each frequency. One and five delays seemed to produce lower errors. With the data sampled
at 2kHz, five delays (2.5ms) seemed like a more reasonable delay than 1 delay (0.5ms).
Physiologically, such a time scale is reasonable since the chemical processes involved
in muscle contraction-transmission across the motor end plate, muscle fiber conduction, and excitation-contraction coupling [52]-take a little time. As an example,
Wexler, Ding, and Binder-Macleod chose 4ms as the refractory period of calcium ion
channels [48]. Hill concluded that frog muscle at 0' C takes less than 20ms to shorten
(McMahon [36, pp 17-19]); however, the muscle used in this thesis' experiments were
stimulated at room temperature, so one would expect a shorter delay.
Frequency (Hz)
Delay for least RMS error
3
5
5
15
7
1
10
1
20
5
75
1
100
5
68
Isometric ID: experiments 22jun06/ID 003,sbr
0ns 0,32148
Isometric ID: experiments 22jun06/ID 003. sbr
E0m =0,29339
1 delays
5 delays
2.5
5
1.5
00
0.5
0
0-
-0.5
3
3.5
4
4.5
6
5.5
5
lime CS)
7
6.5
3
35
45
4
5
time (s)
5.5
6
6.5
7
(b) 5 delays
(a) 1 delay
Isometric ID: experiments 2jun06/ID 003.sbr
Isometric ID: experiments 22junO6/ID 003.sbr
Er =0.30011
15 delays
10 delays
3
3 -
5
2.5
2
2
72
t
1.5
0.5
iI;~j
3
3.5
4
4.5
5
rime
55
s)
6
65
3
7
3.5
4
45
(c) 10 delays
time (s)55
(d) 15 delays
Isometric ID: experiments 22jun06/ID
Emis
44645
35 delays
003.sbr
25
2
1.5
0.5
3
3.5
4
4.5
5
( ime (s)
5.5
6
6.5
7
(e) 35 delays
Figure 4-6: Delay selection: 3Hz data
69
6
65
7
Isometric ID: experiments 2
5m =006n
Isometric ID: expenments 2 2un(O6/ID 009 sbr
E,
= 0.076965
1 delays
un06/D
009.sbr
1 delays
0.14
0.7
measured
- training ft
validation fit
stimulation points
-___
-
0.6
-
0.12 -
0.5
01
0.4
0.08
0.3 10.2
0.05
01
3
I5
3.5
8
55
4.5
5
time (S)
4
(a)
0.021
4.8
7
6.5
4.85
4.9
495
time (s)
5
5.05
51
5.05
5.1
5.05
5.1
1 delay (force estimate and detail)
Isometric ID experiments 22jun08/D 009.sbr
Ermns 0.078249
Isometric ID: experiments 22junO6/AD 009 sbr
Erms =0.078249
5 delays
5 delays
0 14
0.6
-V
012
0.5
0 1
0.4
0.08
03
01
02
0.06
0.1
0.04
3
3,5
4
4.5
5
tme (s)
5.5
6
65
0.02 L
4.8
7
4.85
4.9
4 (95
time (S)
(b) 5 delays (force estimate and detail)
Isometric
Isometric ID: experiments 22jun06/D 009.sbr
ID. experiments 22jun06/D 009.sbr
=0.07978
10 delays
1
10 delays
014
0.6
0,1205
U.
0.4
0,08
0.3
0.2
006-
0.1
0.04
0
3
3.5
.
time (s)
5
6
6.5
0.024,
7
5
4.85
4.9
495
time (S)
5
(c) 10 delays (force estimate and detail)
Figure 4-7: Delay selection: 7Hz data (continued on next page)
70
Isometric ID: experiments 22junO6/ID
Er =0.082842
18 delays
009.sbr
Isometric ID: experiments 22junO/1D 009.sbr
1 = 0.082842
15 delays
0.1 4
0.6
0.1
.~0.4
00
03
0t
0.2
0.0
0.1
0.0
0
3
3.5
4
4.5
(5
5.
6
6.5
2
4.8
7
41E
49
time (S)
4.9
time (s)
5
5.8
1
(a) 15 delays (force estim ate and detail)
Isometric ID, experments 22jun06/ID 009.sbr
rn = 0,10811
22junO6/ID 009.sbr
Isometric ID: experiments
5
rms =01011
38 delays
35 delays
01
086
01
085
0.
80.4
40.3
00 18
0.2
00
0.1
4 -
3
35
4
4.5
8
time (S)
55
6
65
00 2
4.8
7
4
4.85
49
4.95
8
808
time (s)
(b) 35 delays (force estim ate and detail)
Figure 4-8: Delay selection: 7Hz data ( continued from previous page)
71
8. 1
4.6
How does a dynamic system with no input
nonlinearity perform?
The next step was to test how well a simple, linear time-invariant dynamic system
with no input nonlinearity would perform.
The dynamic estimates were obtained
using Matlab's oe command for output error model identification. Second-, third-,
and fourth-order models were applied. Figures 4-9, 4-10, and 4-11 illustrate simple
dynamic fits on three data sets in which muscle was stimulated at 3, 40, and 100Hz.
Experimental force data is shown in blue, estimated force in red, and asterisks mark
the onset of stimulation pulses.
For isolated muscle twitch response at 3Hz stimulation, the second- and fourthorder estimates resulted in non-minimum phase zeros. The fourth-order estimate also
oscillated rather than returned to its zero-force rest state.
For unfused tetanic muscle response due to 40Hz stimulation, second-order dynamics tended to follow low frequency behavior less well (eRMS
=
10.9% of Fma,)
than third- or fourth-order dynamics (eRMs = 8.5% and 6.9%, respectively).
For fused tetanus due to 100Hz stimulation, all fits failed. Only the fourth-order
estimate slightly resembled the measured force, but all of the estimated systems were
unstable.
Behavior due to 100Hz stimulation was nonlinear; with nearly continual
stimulation, one would expect a step response from the muscle, but this is not what
happened.
Instead, the muscle response violated one of the Hammerstein model
tenets-that the dynamics are linear. If they were linear, the force should be sustained. Realistically, muscle fatigues during prolonged tetanus, thus introducing a
nonlinearity that the identification algorithm is, at present, incapable of estimating.
In order to model fatigue, a time-dependent nonlinearity would need to be added to
the Hammerstein cascade. That the dynamics were not even able to construct the
first-order step response may be attributed to the lack of information content when
identifying tetanic contraction:
the algorithm observed that regardless of the level
of input, the force response was nearly constant.
As a result, noise behavior may
easily have dominated the identification algorithm, and unsurprisingly, the resulting
72
Isometric D experiments 22jun06/ID 003.sbr
training error= 1147 .0054, erms =0.37863
2.5
21.5
-
0,5
0
-0.5
3
3.5
4
4.5
5
time (s)
5.5
6
6.5
7
(a) second-order dynamics
Isometric ID: experiments 22junO6i 003. sbr
training error = 994.3635, erms 0035253
3.5
3
2.5 2 0
C
1.5
0.5 0 -
3
3.5
4
45
5
time (s)
55
6
6.5
7
6.5
7
(b) third-order dynamics
Isometric ID: experiments 22jun126I 003. sbr
training error= 795.6698, erms = 0,31535
3
2.5
2
1.5-
0.5
0-
3
3.5
4
4.5
5
time (s)
55
6
(c) fourth-order dynamics
Figure 4-9: Simulations with no nonlinearity: muscle response at 3Hz stimulation
73
Isometric ID: experiments 22jun06/ID 013 sbr
training error = 1047.1039, erms = 0.36176
3.5
3
2,5
2
1.5
1
0.5
0
3
4
3.5
5.5
5
time (s)
4.5
6
6.5
7
(a) second-order dynamics
22 0 6
Isometric ID: experiments jun /ID 013. sbr
training error = 640.3417, erms = 0,2829
3,
3
2.51
2
1.51
1
+
051
measured
training fit
stimulation points
_
fli
4
3.5
3
4.5
5
time (s)
5.5
6.5
6
7
(b) third-order dynamics
experiments 22jun06/D 013 sbr
training error = 421.1195, erms = 0 22942
Isometric ID:
2
1
3.5
0
.5
3
3.5
4
5
4.5
5.5
6
6.5
7
time (s)
(c) fourth-order dynamics
Figure 4-10: Simulations with no nonlinearity: muscle response at 40Hz stimulation
74
Isometric ID: experiments 22jun08/0 006. sbr
training error = 499135.2351, erms = 7.8984
6
4
2
E8
-2
-4
measured
training fit
stimulation points
------
*
4.5
4
3.5
3
5
time (s)
5.5
6.5
6
7
(a) second-order dynamics
22
0
Isometric ID: experiments jun 6/[D 008 sbr
training error = 742322.7252, erms = 9.6322
3
3.5
4
4.5
6
5.5
5
time (s)
6.5
7
(b) third-order dynamics
Isometric ID: experiments 22jun06/ID 008. sbr
training error = 2266.2552, er = 0.53221
7,
6
4
5
S
2
1
0
. L
2.5
3
3,5
4
4.5
5
time (s)
5.5
6
6.5
(c) fourth-order dynamics
Figure 4-11: Simulations with no nonlinearity: muscle response at 100Hz stimulation
(all have unstable dynamics)
75
estimates were very poor.
---
- ---
30
+no nonlinearity, 2nd
order dynamics
25
n no nonlinearity, 3rd
order dynamics
E20
no nonlinearity, 4th
dynamics
%order
0
0
*
5.
0
0
20
60
40
80
100
Frequency (Hz)
Figure 4-12: Simulation errors with no nonlinearity
Errors based on linear dynamics-only curve fits (shown in Figure 4-12) helped
determine what dynamic system would be most appropriate for the Hammerstein
muscle model. The variation of difference equation coefficients (see Figure 4-13) also
informed the choice of dynamic system order.
Although fourth-order models generated the least error, particularly at high frequencies, it was not apparent that the physical system was predominantly fourthorder or that the identification algorithm was capable of identifying a consistent
fourth-order model for the data. The relatively large amount of difference equation
coefficient variation for fourth-order models (average standard deviation & = 1.3 over
four coefficients versus & = 0.0095 and 0.027 for second- and third-order models,
respectively) seemed to suggest the algorithm was capable of finding many different
solutions to fit the data but which did not necessarily uniquely describe the system.
Thus, a decision was made to proceed simulating with only second- and third-order
dynamic models.
76
120
Parameter vanation: 2nd order dynamics, no nonlinearity
1.5 1-
05
C
20
-0.5
40
80
60
120
100
0o
-2
-2.5
-2.5
-..-.
-
-
Frequency (Hz)
(a) second-order dynamics with no nonlinearity
Parameter variation: 3rd order dynamics, no nonlinearity
-
4 -
E
-U
-
-
-
2
' *fl ~
i
M*f2
f3
T0
g801
20
40
60
so
10
100
02 0
-2
-3
o
-4
Frequency (Hz)
(b) third-order dynamics with no nonlinearity
Parameter variation: 4th order dynamics, no nonlinearity
8 +f1
;5
S 6
maf2
f3
4
of4
C
2
-4
-6
Frequency (Hz)
(c) fourth-order dynamics with no nonlinearity
Figure 4-13: Difference equation denominator coefficient variation
77
4.7
How do different combinations of input polynomial nonlinearities and dynamic systems perform?
Since it remained unclear what combination, if any, of polynomial nonlinearities and
linear dynamics would consistently produce the best estimates of force, all combinations of linear second- and third-order dynamic systems with first- through thirdorder polynomial nonlinearities were simulated. Figure 4-14 plots, as scatter and bar
graphs, the simulation errors across all stimulated frequencies for one muscle.
Figure 4-14 shows that at frequencies below 20Hz, linear and nonlinear models
produced simulations errors that were relatively comparable
(eRMS
= 12.4% and
12.3%, respectively) and dynamically stable (26/27 = 96% of linear simulations and
52/54 = 96% of nonlinear simulations). For frequencies between 20 to 40Hz, the addition of a nonlinearity resulted in errors (ERMs = 10.5% of Fmax) comparable to those
of linear models
(eRMS =
10.2% of Fmax), but algorithm convergence and dynamic
stability was not as reliable (6/18 = 33% of nonlinear simulations versus 8/9 = 90%
of linear simulations). For frequencies above 40Hz, neither the linear nor nonlinear
models fared well; dynamic estimates were often unstable or the algorithm did not
converge (2/9 = 22% of linear simulations, 8/18 = 44% of nonlinear simulations).
It should be noted that dynamics order selection based purely on errors would
be naive. Based on strictly RMS errors, it would seem that second-order dynamics
sometimes describe the system better than third-order. However, third-order models
should not be discounted; as Figure 4-15 reveals, there can be discrepancies in which
the dynamic system producing lower errors has a poorer qualitative fit. In this case,
the second-order system generates lower errors, but has larger non-minimum phase
dips.
Both second- and third-order dynamics are reasonable.
Though many muscle
models assume critically-damped second-order dynamics, a third-order model is not
highly unusual. Ettema and Meijer used one in their exponential decay model [17].
78
Comparison of different polynomial nonlinearities with second- and third-order dynamics
------------------
90
no nonlinearity, 2nd
order dynamics
80
* no nonlinearity, 3rd
order dynamics
70
> 1st ord nonlinearity,
60
2nd order dynamics
50
0 1st ord nonlinearity,
3rd order dynamics
0£ 40
4*
+ 2nd ord nonlinearity,
2nd order dynamics
2 30
2nd ord nonlinearity,
3rd order dynamics
20
3rd ord nonlinearity,
2nd order dynamics
*o
.
10
0
3rd ord nonlinearity,
Frequency (Hz)
120
100
80
60
40
20
0
3rd order dynamics
Comparison of different polynomial nonlinearities with second- and third-order dynamics
--9 0 -r--
80
mno nonlinearity, 2nd
70
, no nonlinearity, 3rd
60
1st ord nonlinearity,
2nd order dynamics
5L
1st ord nonlinearity,
3rd order dynamics
40
* 2nd ord nonlinearity,
order dynamics
order dynamics
E
U-
L
0
0
2nd order dynamics
Cl)
30 -
* 2nd ord nonlinearity,
3rd order dynamics
20 -
" 3rd ord nonlinearity,
2nd order dynamics
10 -
* 3rd ord nonlinearity,
3rd order dynamics
-- --
U
1
1
2
3
5
7
10
15
20
25
30
40
50
75
100
Frequency (Hz)
Figure 4-14: Simulation errors of different dynamic systems with different nonlinearities (in scatter and bar graph format)
79
2
Isometric
1D:
2nd ord dynamics, 3rd ord nonlinearity,
0.4
---
Isometric ID: esperiments 2 junO6/ID 015. sbr
Eis = 0.029961
015.sbr
experiments 2OiunO6/D
4
Em0029 78
3rd ord dynamics, 3rd ord nonlinearity, 5 delays
5 delays
045
measured
training
-
0.3
fit
04
validation fit
stimulation points
035
0.3 -
0.22
2
~0.2
0.1
-0.1
05
-0.15
-0 2 -
0
3
35
4
45
5
time (s)
55
6
65
3
7
35
4
45
5
time (s)
55
6
&5
7
Figure 4-15: Comparison of second- and third- order dynamic fits (2Hz data identified
iteratively with third-order nonlinearity and five delays)
Zhou et al. chose to describe thier muscle as a second-order system, but the system
became third-order when modeling the muscle-joint in order to take into account the
viscoelastic properties of tendon and ligament [52]. Meanwhile, Donaldson et al. chose
a third-order linear transfer function to model isometric force due to a pulse train
with constant amplitude and pulse width but random stimulation periods between
1 to 70ms [12].
In general, when researchers have used third-order relationships,
the transfer functions have been between output force and position for their Hilltype mass-spring-damper models; however, in our case, the third-order dynamics are
between output force and a voltage-based pulse train. Physiologically, voltage drives
the release of ions and incites a diffusion-based reaction. Non-steady state diffusion
is a second-order phenomenon, as described by Fick's second law:
du
dt
k
&2 u
2(4.1)
Ox2 '
where u may be ion concentration, k is a diffusivity constant, and x represents the
direction(s) of diffusion [22].
A second-order diffusion-based system may be cas-
caded with a first-order relationship similar to the force generator model proposed
by Wexler, Ding, and Binder-Macleod, who modeled the force mechanics as a motor
with a linear force-velocity curve and who related shortening velocity with calcium
concentration [48].
80
4.8
How does a system with fixed input nonlinearity perform?
Given that muscle modelers already have an idea what the recruitment nonlinearity
should be, some of the modeling degrees of freedom could be removed by fixing
the nonlinearity. Figures 4-4 and 4-5 demonstrated that the iterative identification
algorithm was capable of finding the recruitment nonlinearity, so the first step was to
determine what recruitment curves were generated across all data sets, then to pick a
suitable one based on the widely-accepted view of recruitment as a sigmoidal curve.
The selected nonlinearity would be used as the recruitment curve for all data sets to
further obtain estimates of the dynamics.
Most muscle models in the literature create the recruitment as either a piece-wise
linear or a polynomial-composed sigmoidal curve. Of the third-order polynomial recruitment curves obtained in conjunction with third-order dynamic estimates (shown
in Figures C-8 and C-9), the one from experiment 15 at 2Hz was most sigmoid-like
and resembled recruitment curves discussed in the literature. The polynomial regressor coefficients obtained from that fit were then applied across all data sets to
fix the nonlinearity while allowing the algorithm to find dynamic estimates tailored
for each data set. The resulting curve fits may be seen in Figures C-10, C-11, and
C-12. A comparison of simulation errors for third-order dynamics with both fixed
and iteratively-found third-order polynomial input nonlinearities may be seen in Figure 4-16, and Figure 4-17 shows difference equation denominator coefficient variation.
Figure 4-16 suggests that at frequencies of 20Hz or lower, fixing the recruitment
nonlinearity produced comparable results
(ERMS =
16.3% over 12 simulations) as
iteratively finding both the nonlinearity and dynamics (aRMs
= 16.6% over 66 simu-
lations). Figure 4-17 shows how little variation in difference equation coefficients there
is across data sets identified with a fixed nonlinearity (& = 0.020); this demonstrates
that the estimated dynamics were consistent across simulations and were therefore relatively reliable. This is a useful conclusion, as the estimation process may be reduced
from a potentially time-consuming bootstrapping procedure to one step: identifica81
Benefit of iterative procedure versus fixing nonlinearity
60
*3rd ord nonlinearity, 3rd
order dynamics
U
50
* 3rd ord fixed nonlinearity,
3rd ord dynamics
E 40
0
30
0
20
EU
10
E
+
M*
0
0
20
60
Frequency (Hz)
40
80
100
12
0
Figure 4-16: Simulation errors for fixed and iteratively-found recruitment nonlinearity
(third-order dynamics, third-order polynomial
3rd order (fixed) nonlinearity, 3rd order dynamics
4 -
0
o
C
0)
D
*
.-
20
0
40
60
80
100
-1 -4
-4
-
Frquncy(Hz
Frequency (Hz)
Figure 4-17: Difference equation denominator coefficient variation
82
10
tion of the dynamics. At stimulation frequencies above 40Hz, fixing the recruitment
nonlinearity did not result in better dynamic stability or algorithm convergence (one
stable estimate with eRMS = 53.6% of the maximum force for that data set). One
would expect the simulations with the fixed nonlinearity to be somewhat better since
the algorithm only needs to concentrate on dynamics. However, as in the other identification attempts, the estimated force often does not follow the measured behavior
at all. This was probably because the information content of the data was low, so
the resulting lousy dynamic estimate was dominated by noise behavior. Potentially,
the fixed nonlinearity may have been inappropriate at higher stimulation frequencies,
but this remains unclear.
4.9
How does a system with fixed input nonlinearity and fixed dynamics perform for high stimulation frequency input?
Muscle's force response due to high frequency stimulation demonstrates two dominating nonlinear behaviors: force potentiation and fatigue.
Since simulation of high frequency stimulation input with the fixed nonlinearity
still produced poor results, the question was brought up that perhaps the data contained too little information and too much noise to estimate the dynamics adequately.
In that case, it was interesting to see if applying the recruitment nonlinearity and
dynamics found from a data set with well-defined twitches to a data set with high
frequency stimulation input would be effective in describing muscle behavior under
high frequency input. The nonlinearity regressor coefficients and third-order difference equation coefficients from 2Hz data (Experiment ID-015) was applied to 50, 75,
and 100Hz data, and the resulting simulated responses are shown in Figure 4-18.
While the estimates showed improvement
found with linear models
( RMS =
(ERMS
= 65.7% of Fma,) over those
100.4%), fixing the dynamics and nonlinearity
did not show marked improvement over nonlinearities and dynamic estimates found
83
Isometric ID: e x eriments 22un06AD 004.sbr
fixed 3rd order nonlinearity and 3rd order dynamics, 6 delays
7 6
measured
.l 4
- -estimate
stimulation points
3
2
2.5
3
3.5
4
4,5
5
time (s)
5.5
6
66
7
(a) 50Hz stimulation frequency
ID:
experiments 22junO6/D 004 sbr
E
= 5.343
fixed 3rd order nonlinearity and 3rd order dynamics, 5 delays
Isometric
7 6
5
4___measured
i *i4
estimate
stimulation
ai 3 -
points
2
0
2.5
3
3.5
4
4.6
5
time (s)
5-5
6
6.5
7
(b) 75Hz stimulation frequency
experiments 22junC6/lD 004 sbr
Erms = 5.3438
fixed 3rd order nonlinearity and 3rd order dynamics, 5 delays
Isometric
ID:
6
5 -4
e
4
measured
3-+
stimulation points
estimate
2 -
0-1
2.5
3
3.5
4
6
4.5
time (s)
5.5
6
6.5
7
(c) 100Hz stimulation frequency
Figure 4-18: Muscle response estimates for high stimulation frequency input using
fixed nonlinearity and fixed dynamics trained from 2Hz data
84
iteratively
( RMS
= 49.6%). Figure 4-19 shows simulation errors for responses due to
high frequency stimulation.
400
350
300
E
250
0I
#no nonlineanty, 2nd
Simulations at high frequencies
-
450
order dynamics
* no nonlinearity, 3rd
order dynamics
A no nonlinearity, 4th
order dynamics
o st
ord nonlinearity,
2nd order dynamics
0 1st ord nonlinearity,
3rd order dynamics
*2nd ord nonlinearity,
2nd order dynamics
200
2nd ord nonlinearity,
3rd order dynamics
WU
1 50
100
ord nonlinearity,
2nd order dynamics
50
3rd ord nonlinearity,
3rd order dynamics
K3rd
03rd ord fixed nonlin,
3rd ord dynamics
0
40
50
60
80
70
Frequency (Hz)
90
110
100
Simulations at high frequencies (DETAIL)
*3rd ord fixed nonlin,
3rd ord fixed dynamics
*no nonlinearity, 2nd
order dynamics
100
90
nonlinearity, 3rd
order dynamics
80
80
*no nonlinearity, 4th
order dynamics
70
ost ord nonlinearity,
60
ci 1st ord nonlinearity,
3rd order dynamics
no
2nd order dynamics
E
0
lip
02nd
i2
+2nd
a>
U
30
20
A
0
40
,
50
60
2nd ord nonlinearity,
3rd order dynamics
3rd ord nonlinearity,
2nd order dynamics
3rd ord nonlinearity,
3rd order dynamics
A
10
ord nonlinearity,
order dynamics
70
80
Frequency (Hz)
,
90
--
100
a 3rd ord fixed nonlin,
3rd ord dynamics
-,
110
*3rd ord fixed nonlin,
3rd ord fixed dynamics
Figure 4-19: Simulations with fixed third-order polynomial nonlinearity and thirdorder dynamics
There was no consistent combination of nonlinearity and dynamics that produced
the best fits. With fixed nonlinearities and/or dynamics, the quality of fits probably
depended on the data set from which the nonlinearity and dynamics were obtained. If
two data sets had output noises that correlated highly, fits would probably be better.
However, if they had very different noise characteristics, the fits were probably very
poor.
85
Iterative convergence and stability of the estimated dynamics were also issues for
data sets with high stimulation frequency input. Of 150 simulations run on data
sets with between 1 to 100Hz input stimulation frequency, 130 simulations (87%)
converged. 114 (76%) resulted in stable estimated dynamic systems. However, for
data sets with input stimulation frequency of 50Hz or more, 20 of 30 simulations
(67%) converged, while only 11 cases (37%) were stable.
4.10
Comparison of all simulation results
After running ten model types on the fifteen sets of pilot data, the model performances
were assessed. In this section, simulation errors and stability are compared across
input stimulation frequencies and between linear and nonlinear models.
4.10.1
Simulation errors
Figure 4-20 compares all simulation errors, including those without the static input nonlinearity, those with an iteratively-found nonlinearity, and those with a fixed
nonlinearity.
For stimulation frequencies of 20Hz and below, the RMS errors average 12%.
Between 20 to 40Hz, simulations produced errors of 10%, and above 40Hz, errors
average around 44%.
Table 4.1 shows, for each model, the RMS errors averaged across seven different
frequencies. The averages are also plotted in Figure 4-22. The frequencies (1, 3, 5,
10, 15, and 20Hz) were selected on the basis that they converged and were stable for
simulations of all ten model types. We see the average errors generated by all the
model types are relatively consistent and generally fall within 10 to 14%; no model
stands out in its ability to estimate contractile forces.
86
RMS errors from all simulations
90
+no nonlinearity, 2nd
order dynamics
80
* no nonlinearity, 3rd
order dynamics
70
* no nonlinearity, 4th
order dynamics
60
c 1st ord nonlinearity,
50
o 1st ord nonlinearity,
2nd order dynamics
E
0
0IQ
CO
3rd order dynamics
40
ord nonlinearity,
2nd order dynamics
+2nd
YE'
30
2nd ord nonlinearity,
3rd order dynamics
U
0 3rd ord nonlin earity,
2nd order dynamics
20
10
3rd ord nonlinearity,
3rd order dynamics
0
60
40
20
0
100
80
12
0
* 3rd ord fixed nonlin,
3rd order dynamics
Frequency (Hz)
RMS errors from all simulations
90
-
, no nonlinearity, 2nd
order dynamics
Sno nonlinearity, 3rd
order dynamics
80
70
- -----
-
----
no nonlinearity, 4th
order dynamics 3rd
- -n-- -i-ty,
1st ord nonlinearity,
2nd order dynamics
60
E
* 1st ord nonlinearity, 3rd
order dynamics
0 50
Cn
40
* 2nd ord nonlinearity,
30
* 2nd ord nonlinearity,
2nd order dynamics
Ul)
3rd order dynamics
m 3rd ord nonlinearity,
2nd order dynamics
m 3rd ord nonlinearity,
10
Ithmil
0
1
1
2
3
3rd order dynamics
i
5
7
10
15
20
25
30
40
I
n 3rd ord fixed nonlin,
50
75
100
3rd order dynamics
Frequency (Hz)
Figure 4-20: Summary of simulation errors (scatter and bar graphs)
87
Average RMS errors for ten models across fourteen frequencies
- - - ----
-
80
m 70E
050
0
40-
~30 -
0
80
60
40
20
0
120
100
Frequency (Hz)
Figure 4-21: Average simulation errors across stimulation frequencies
25 -
U)
I
0(U
u--
0
o1
E
U0-
5
0
Ne
(0
6
10PI
I:.
x
0
M
Model type
Figure 4-22: Average model errors across seven data sets below 25Hz
88
Freq (Hz)
Experiment
1
ID-001
1
ID-011
3
ID-003
5
ID-012
10
ID-005
15
ID-007
20
ID-002
model average error
Linear models (no nonlinearity)
4th ord dyn
3rd ord dyn
2nd ord dyn
5.86
6.21
7.94
3.84
3.88
4.30
14.96
16.73
17.97
11.43
11.67
11.92
10.33
10.85
12.63
12.87
14.34
13.49
12.56
27.45
26.87
10.26
13.02
13.59
00
Experiment
ID-001
ID-011
ID-003
ID-012
ID-005
ID-007
ID..002
model avg error
Freq (Hz)
1
1
3
5
10
15
20
1st ord nonlin,
2nd ord dyn
7.97
4.94
15.96
12.66
12.26
13.04
9.49
10.90
1st ord nonlin,
3rd ord dyn
6.36
4.04
15.82
12.87
12.39
10.80
8.49
10.11
2nd ord nonlin,
2nd ord dyn
6.06
4.94
14.27
17.34
11.46
58.10
36.15
21.19
Nonlinear models
3rd ord nonlin,
2nd ord nonlin,
2nd ord dyn
3rd ord dyn
6.00
3.43
4.91
3.98
13.79
13.92
22.25
17.87
11.47
11.81
12.98
11.21
10.86
8.18
11.75
10.06
3rd ord nonlin,
3rd ord dyn
3.26
3.98
13.08
22.14
11.79
12.40
7.68
10.62
Table 4.1: Average linear model RMS errors across seven data sets below 25Hz
3rd ord fixed nonlin,
3rd ord dyn
7.57
3.01
18.69
12.87
14.01
13.10
26.69
13.70
The continuous-time system poles and zeros were also examined across all simulations. Since the discrete difference equation denominator coefficients had been
relatively consistent across frequencies for second- and third-order dynamics, the continuous poles were expected to be relatively consistent as well. Figure 4-23 (also found
in Appendix C) plots the poles and zeros for third-order dynamic systems. The real
parts of the poles are relatively consistent across all data sets. The imaginary parts
are non-zero. Since the estimates were not confined to the critically damped case,
the poles were frequently estimated to be complex for both second- and third-order
dynamics. As a result, oscillation could arise in the estimated force profile, yet this
was not seen in the experimental evidence. When oscillatory behavior does arise in
muscle, such as in patients with spasticity or tremor, it may be attributed to defects in the feedback sensing organs [52, 43]. However, this is clearly not the case
for our healthy muscle. It is possible the complex poles may be attributed to noise
and/or changes in the passive muscle force between stimulations: often, the baseline
for inactive muscle fluctuates, as shown in Figure 4-24.
4.10.2
Simulation stability
Stability of the estimates was also compared across frequencies as well as between
linear systems and nonlinear Hammerstein models. A model was ranked unstable
(0) if it either had an infinite H2 norm, or the algorithm did not converge after over
one hundred iterations. Typically, the algorithm took eleven to twenty iterations to
converge. If the H2 norm was finite and the model converged, the model was ranked
stable (1).
The stability rankings across all ten (linear and nonlinear) simulations
were then averaged for data at each stimulation frequency. As shown in Figure 4-25,
stability of the estimated dynamics in general declined for frequencies above 40Hz (11
of 30 simulations), while data with stimulation frequencies of 20Hz and less generated
more stable dynamic estimates (87 of 90 simulations). Between 20 to 40Hz, dynamic
stability occurred in 53% (16 of 30) of the simulations.
To address differences in stability between linear dynamic estimates and Hammerstein dynamic estimates, a Student's t-test was performed to compare the stability
90
S-plane: 1et order nonlinearity, 3rd order dynamics
S-plane: no nonlinearity, 3rd order dynamics
,4in
-.--...------.---
200
----------
100
_ . .- . .- -
200
--- -- -- -- --
.
-.
--
-0--
-30-0
-
-5000
100
.
--
-00
----
x
K
0
0
*F
---------- --
-----------
E
-100
-100
-------
- --
-200
-200
X
a
I --------------- -------- ----------_------------------------------- --- -----
....-------.---...------- ..--- ------
----- ------t
3
-
----- ----- ---800
600
400
Real
2A
Jo
10C
500
S-plane 2nd order nonlinearity, 3rd order dynamics
-3000
-4M
-2000
Real
0
-1000
1000
S-plane 3rd order nonlinearity, 3rd order dynamics
3W0
300
1<
100
200
-- -- .. ....----...
- . .......
----
-- --
----- - ----
200
100
-
------
7 --
-
-
4 -,
-----
0
-100
+---- ------------------+------------
L"
0
X
6
-100
--------------------------
-200
-- ------
-200
-300
2.5
2
1.5
Real
1
0.5
0
-b.5
3.5
3
4
1=
0
1000
3000
Real
2000
4000
5000 6000
S-plane: 3rd order fixed nonlinearity, 3rd order dynamics
:0
200
200 -------- ------- ------ ------
2100 --------- --------- )-------
0 -----------
------
.... . .---. .. .. .
-------. -..
X
----- -
--
-- ---
- ---
------------- -------
-------------
-100 -------
*
X
*
X
---------- ----
X
-200 .------
----- +
...
--- + -----
- --------- ------
-------
0
0
-300C
400
1
-300
-200
-100
100
0
200
X
X
X
1 Hz
1Hz
2Hz
2Hz
3Hz
5Hz
7Hz
(ID001)
(ID011)
(10015)
(1D016)
(ID003)
(ID012)
(ID009)
10Hz (ID005)
15Hz (ID007)
20Hz (ID002)
25Hz (ID014)
30Hz (10010)
40Hz (10013)
50Hz (10004)
75Hz (10006)
100Hz (ID008)
1
300
400
500
Real
Figure 4-23: Pole-zero maps for third-order dynamic systems with different input
polynomial nonlinearities
91
7000
experiments 22jun06/ID 001. sbr
5
4.5
4
3.5
3
(a
E
2.5
0
I',
(a
0
2
1.5
1
0.5
0
2.5
3
3.5
4
5
4.5
5.5
6
6.5
7
7.5
time (s)
Figure 4-24: Fluctuating passive muscle force: 1Hz data
Average stability of all dynamic estimates (1 = stable, 0 = unstable)
1.2
1.0
1 .0
-
0
0.84
0
0
-J
S0.4
0.2
0.0
I
0
20
40
60
Frequency (Hz)
80
100
Figure 4-25: Average stability of ten estimated models across stimulation frequency
(1 = stable, 0 = unstable)
92
120
measures of the three linear versus seven nonlinear models. The standard deviation
between the means for linear and nonlinear models was 12.2%, and the t-value was
0.47. To be statistically significant, the critical t for p = 0.05 is 2.05; therefore, the
stability of the linear versus nonlinear models was not significantly different.
93
THIS PAGE INTENTIONALLY LEFT BLANK
94
Chapter 5
Conclusion
For the pilot data of this thesis, the Hammerstein iterative identification algorithm
seemed to be effective at identifying dynamically stable models (87 of 90 simulations)
for muscle subjected to 20Hz or less pulse train stimulation frequency. The estimates
generated an average RMS error of 12%. Between 20 to 40Hz, the simulation errors
were of similar magnitude (10%), but the dynamic stability of the estimated models
declined (16 of 30 simulations). Beyond 40Hz, the curve fits were very poor
(IRMS =
44%, 11 of 30 simulations dynamically stable); the information content of the data
was too low to estimate the dynamics and/or recruitment curve adequately, and noise
likely dominated the dynamic characterization.
Iterative convergence and stability were issues for data with high input stimulation
frequencies. Of 150 simulations run on data with input frequencies ranging from 1
to 100Hz, 87% converged, often within the first eleven to twenty iterations, while
the remaining 23% hadn't converged by at least the
10 0
th iteration.
Of the 150
simulations, 76% were dynamically stable with finite H2 norms. When considering
only the 30 simulations for data with input frequencies above 40Hz, the numbers
drop: 67% converged, while 37% of the 30 simulations were dynamically stable.
Since data with stimulation frequencies of 20Hz and less generated reliable and
stable estimates, the simulations run on these data sets allowed comparison between
the different model types. Using a fixed nonlinearity produced comparable simulation errors as the bootstrapping procedure that simultaneously finds the recruitment
95
nonlinearity and the dynamics. This suggests that a straightforward estimate of the
dynamics in series with a known recruitment nonlinearity may be applied in lieu of
the potentially time-consuming bootstrapping procedure.
Likewise, linear dynamic and nonlinear Hammerstein models performed comparably. This is possibly because the recruitment curves of the two types of models are
not necessarily so very different. Hammerstein models add a nonlinear recruitment
curve normalized to unity, while linear models assume a monotonically increasing
recruitment curve with no gain. However, if the zeros of the estimated dynamics are
appropriately manipulated, linear models can make up for changes in gain that would
otherwise be provided by Hammerstein recruitment nonlinearities.
However, by no means do these results confirm that the Hammerstein model
is inappropriate.
Rather, this thesis argues that the Hammerstein muscle model
should be further explored to determine and understand what its limitations are.
Once it is determined at what frequency the Hammerstein model ceases to be an
effective description of isometric muscle behavior, the use and range of the model in
neuroprosthetic controllers may be better gauged.
5.1
Future work
Since this thesis conducted system identification on one group of pilot data sets, to
reach any firm conclusion would require additional testing and simulation. Moreover,
a few modifications to the model might be useful. For instance, constraining the nonlinearity to be monotonically increasing may help the algorithm find better solutions
[11, 8]. Adding another linear filter to precede the recruitment curve may also be a
useful way of better modeling calcium release
of freedom in the modeling process.
96
[6]
though it presents additional degrees
Appendix A
Experimental Protocol
97
Purpose of Experiments
Hammerstein system ID; see effect of stimulation frequency on recruitment curve
Frog
Rana pipiens (female)
Mass
85.96 grams
Solution
Ringers
No glucose
Oxygenation
Circulation
Off
Off
Muscle
Plantaris longus (muscle belly only)
36.05 (33.75 sut) mm
Left Lo
0.972 grams
Left mass
Right Lo
Right mass
35.7 (sut) mm
0.986 grams
Electrode
Suction
Experiment duration
Varying PW limits
4sec isometric, 0 sec dynamic (unless otherwise noted)
100 to 1000us (0.0001 to 0.001sec)
Isometric Stimulation
Test ID
Bioreactor
Condition
Freq (Hz)
Forcemax (N)
Comments
LEFT MUSCLE
IDA
Secondary
ID_2
Secondary
ID_3
Secondary
IDA
Secondary
ID-5
Secondary
ID_6
Secondary
ID-7
Secondary
ID_8
Secondary
ID-9
Secondary
ID-10
Secondary
ID-11
Secondary
ID-12
Secondary
10V,
10V,
1OV,
1OV,
1OV,
10V,
1OV,
1OV,
1OV,
1OV,
1OV,
1OV,
4.5N
8N
2N
6N
1.5N
6N
1.25N
5N
0.5N
2.5N
0.2N
0.5N
exper duration too short (few data pts)
ID-13
Secondary
1OV, 1 pulse, random PW, w/ jitter
ID-14
Secondary
10V, 1 pulse, random PW, w/ jitter
ID-15
Secondary
1OV, 1 pulse, random PW, w/ jitter
1
20
3
50
10
75
15
100
7
30
1
5
max jitter
40
max jitter
25
max jitter
2
max jitter
1
1
1
1
1
1
1
1
1
1
1
1
pulse,
pulse,
pulse,
pulse,
pulse,
pulse,
pulse,
pulse,
pulse,
pulse,
pulse,
pulse,
random
random
random
random
random
random
random
random
random
random
random
random
PW,
PW,
PW,
PW,
PW,
PW,
PW,
PW,
PW,
PW,
PW,
PW,
no jitter
no jitter
no jitter
no jitter
no jitter
no jitter
no jitter
no jitter
no jitter
no jitter
no jitter
w/ jitter
RIGHT MUSCLE
ID-16
Secondary
ID-17
Secondary
ID-18
Secondary
ID.19
Secondary
ID-20
Secondary
ID-21
Secondary
ID_22
Secondary
ID-23
Secondary
ID-24
Secondary
ID.25
Secondary
ID-26
Secondary
1OV,
1OV,
1OV,
1OV,
1OV,
1OV,
1OV,
1OV,
1OV,
1OV,
1OV,
ID-27
Secondary
10V, 1 pulse, random PW, w/ jitter
ID-28
Secondary
10V, 1 pulse, random PW, w/ jitter
ID-29
Secondary
1OV, 1 pulse, random PW, w/ jitter
1
1
1
1
1
1
1
1
1
pulse,
pulse,
pulse,
pulse,
pulse,
pulse,
pulse,
pulse,
pulse,
1 pulse,
1 pulse,
random
random
random
random
random
random
random
random
random
random
random
PW,
PW,
PW,
PW,
PW,
PW,
PW,
PW,
PW,
PW,
PW,
no jitter
no jitter
no jitter
no jitter
no jitter
no jitter
no jitter
no jitter
no jitter
no jitter
w/ jitter
2
20
10
50
3
75
15
100
7
30
5
max
40
max
25
max
85
max
5sec
of stimulation
12sec of stimulation
0.1791s
3.5N
0.0216s
1N
0.0351s
0.25N
0.4491s
3N
3N
1.25N
3N
0.75N
3.25N
0.75N
3N
6sec of stimulation
5sec of stimulation
0.4N
1.5N
0.1N
jitter
jitter
jitter
jitter
0.1791s
1.5N
0.0216s
0.5N
0.0351s
2.5N
0.0097s
NOTES:
* max jitter = 1/f - PWmax so that pulse widths don't overlap; for safety, make that 0.9*(l/f - PW-max)
* let stim frequency (1/pulse period) = 200Hz, train duration = 0.005sec so that pulse count = 1
* primary stimulator - very low voltages, so used secondary card for all experiments
* secondary stimulator - fuzz seen on oscilloscope as VCM servos to Lo, then diminishes as stimulation begins (cross-talk?)
* -15N voltage as baseline in data (though load cell reads 2.17N passive right)
98
Appendix B
MATLAB code
99
% dataFit .m
clear all
close all
file = 'experiments_22junO6/ID_015.sbr';
load(file ,'-mat')
'_);
file (underscInd)
underscInd = find(file
';
time = FileData.TestData.Time;
forceUnfilt = FileData.TestData.Data(: ,4)
mean(FileData . TestData. Data(end-4000:end - 1000,4));
posn = FileData . TestData. Data (:
,1);
figure (1) ,plot (time , forceUnfilt ) ,hold on
xlabel('time'),ylabel('force')
title
({'measuredudatauanducorrespondingustimulation:u',file})
%ccutoff freq
wn = 100/(0.5*2000);
[B,A] = butter (2 ,wn, 'low','z');
force = filt filt (B,A, forceUnfilt )
plot (time , force , 'k')
normalized
to Nvquist
%ciose 2nd
order
freq
=
/2*Fs
BuItterworthP
%i&§$§Sicc %iCREATING INPUT'c% S(< (( (
amplitude = str2num(FileData. StimulatorParameters . Amplitude);
pperiod = 1/str2num(FileData. StimulatorParameters . Frequency);
pwidth = str2num( FileData. StimulatorParameters . StimulationTime )*le -6;
pcount
str2num( FileData. StimulatorParameters . TrainDuration ) * ...
str2num( FileData. StimulatorParameters . Frequency);
whichStimPar = FileData. TestData.Data(1000 ,10); %det varying param
%voltage
if whichStimPar =1
amplitude = FileData. TestData. Data(: ,9);
end
if whichStimPar = 2 %pulse period
pperiod = FileData.TestData.Data(: ,9); %excl intentional freq jitter
end
if whichStimPar = 3 %pulse widtli
pwidth = FileData. TestData. Data(: ,9); %Aisgen smaller than 1 sample
end
if whichStimPar = 4 %puIse count
FileData. TestData. Data(: ,9);
pcount
end
vti=pwidth/1e6;
stimIndices = find(diff(FileData.TestData.Data(: ,3)))+1;
100
filter
zeros(size(vti));
u(stimIndices) = vti (stimlndices
u =
);
plot (time , u*5000 , 'r' ,' LineWidth' ,2)
legend('measureduforce','LPufiltereduforce','pulseuwidthu(ms)')
%to save mernory/ spare confusion , clear variables that won' t be called
clear FileData vti whichStimPar amplitude pperiod pwidth pcount
STRUMRMi'
ysheTN ;pt1.
'
basis
'Chebyshev'
;
%/coptions
: 'RBF'
basis =
e
C/he
'Chebyshev
(RBF fitting var i ab 1 es
chosen spacing -4; %/(rbitrarily
numZeta
c h o i c e - can twea
%arbitrary
0.01;
stdDev
%Polvnomial fit t ing variables
polynOrder = 3; %highest polyn
%Dynatmic
denOrder
numOrder
delays =
again
Legendre
can tweak
P-polynOrder , #
of polyn terms
polynOrder+1
system variables
nniumber of system poles
= 3;
%nmnber of system zeros
= denOrder ;
%number of delays in discrete system
5;
errorTrain(1:6)
= [10000
1000 100 10 1
0.1]; %(arbitrarily chosen errors
runNum = 1;
numIter = 50;
isomTrBounds = [6000 11000]; %6000:14000 for ID-011 . 7000:1.1000
[11001 14000]; %14001:30000 for ID-01.1
isomValBounds
for
ID-012
u(isomTrBounds (1): isomTrBounds (2));
uTrain
force (isomTrBounds (1): isomTrBounds (2));
zTrain
uValidate = u(isomValBounds (1): isomValBounds (2));
zValidate = force (isomValBounds (1): isomValBounds (2));
K
IDENTIFICATION -- TRAINING DATA SET1M'>MA
W4A
(FSO\IETIC
min(min(uTrain),min(uValidate));
uIsomMin
max(max(uTrain) ,max(uValidate));
uIsomMax
if strcmp(basis , 'RBF') = 1
[aInpTrain , isomTrCenters] = rbfSetup (uTrain ,stdDev ,numZeta);
else i f strcmp( basis , ' Chebyshev') =
uTrainScaled = (uTrain-uIsomMin) / (uIsomMax-uIsomMin)*2-1;
aInpTrain = chebyshevFit(uTrainScaled ,polynOrder ,1); (1 b/c sc ale d
else uTrainScaled = (uTrain-uIsomMin)/(uIsomMax-uIsomMin)*2-1;
aInpTrain = legendreFit (uTrainScaled , polynOrder ,1);
end
101
end
v100 = linspace (ulsomMin , uIsomMax) ';
=1
if strcmp(basis , 'RBF')
aContCurve = rbfSetup (v1O0 , stdDev , numZeta, isomTrCenters);
1
else if strcmp(basis , 'Chebyshev') =
aContCurve = chebyshevFit (v100 , polynOrder , 0);
else aContCurve = legendreFit (v100 , polynOrder , 0);
end
end
% ESTIMATE V FIRST
vEstimTrain{1} = uTrain; %cstart w/ assumption v =:
h2norm(1) = 1000; %arbitrary choice
vError(1) = 1000; %arbitrary choice
alpha = 0.5; %A0 if
all new,
11
1 if all previous coeffs
cInpTrain{1}= pinv(aInpTrain)*vEstimTrain{1};
%coeffs
of
normalized
liuea r IBi
vContCurve{1}= aContCurve*cInpTrain{1};
recrNorm = max(vContCurve{1});
cInpTrain{1} = cInpTrain{1}/recrNorm;
cHybrid{1} = cInpTrain{1};
vContCurve{1}= aContCurve*cHybrid{1};
vEstimTrain{1}= aInpTrain*cHybrid{1};
while std ( errorTrain (end-5:end))/ errorTrain (end) >0.05
lti = iddata(zTrain,vEstimTrain{runNum},1/2000);
sysOE =oe ( lti , [ numOrder denOrder delays ], 'Focus', 'Stability' );
sysD
tf(sysOE.b,sysOE.f,1/2000,'variable','z');
[a,b,c,d] = tf2ss(sysOE.b,sysOE.f);
sys = recrNorm*ss(a,b,c,d,1/2000);
impulse (sys);
[impulseSave {runNum}, tlmpSave{runNum}]
h2norm(runNum+1) = norm(sys );
for ind = 1: size (aInpTrain ,2)
vEstimMx(: ,ind) = lsim (sys , aInpTrain(: ,ind));
end
cInpTrain{runNum+1} = pinv(vEstimMx)*zTrain;
cHybrid{runNum}* alpha + cInpTrain {runNum+1}*(1- alpha);
cHybrid{runNum+-1}
vCont Curve {runNum+1} = aContCurve* cHybrid {runNum+ 1};
recrNorm = max(vContCurve{runNum +1});
cHybrid{runNum+1} = cHybrid{runNum+1}/recrNorm; %iiorniahize i1on linearitv
vEstimTrain{runNum+1}
aInpTrain*cHybrid{runNum+1};
vCont Curve {runNum+1} = aContCurve* cHybrid {runNum+ 1};
coid later
%/init
= sim(sys ,vEstimTrain{runNum+1});
[zEstimTrain ,t ,vStates]
sum ((zTrain-zEstimTrain ). ^2);
errorTrain (runNum)
Inf)
(norm(tf (sysOE.b,sysOE. f ,1/2000,'variable' ,'z'))
stability (runNum)
102
-1
runNum = runNum+1
end
runNum = runNum- 1;
figure (3) ,plot (uTrain , vEstimTrain{runNum+1},'. '),hold on
figure (4) , subplot (311) , bode (sys ),subplot (312) , zplane (sysOE. b, sysOE. f)
subplot (313) ,impulse (sys)
figure , plot ([ 1: 1: runNum] , errorTrain (1: end) , ' -')
xlabel ( 'rununumber') , ylabel ( 'error')
title('forceuestimateusquareduerroru-utrainingudata')
figure , plot ([1:1:runNum] , s t abilit y , ' . -')
xlabel ( 'rununumber') , ylabel ( 'stability')
title ('stability:uyesu=ul,unou=uO')
[gainM,phaseM,wg,wp] = margin(sys);
disp ( 'Gainumargin: ') ,disp (gainM)
disp ( ' Phase amargin: '), disp (phaseM)
disp ('Gainucrossoverufrequency: '),disp (wg)
disp ('Phaseucrossoverufrequency: ') , disp (wp)
clear gainM phaseM wg wp
clear t tImpSave impulseSave
S
IDENTIFICATION - VALIDATION DATA SET% (A ( j4%/M%/5
%K %X
M%/[SOMETRIC
if strcmp(basis ,'RBF' ) -- 1
rbfSetup (uValidate , stdDev , numZeta, isomTrCenters);
aInpValidate
else if strcmp(basis , 'Chebyshev')
-= 1
uVaIScaled = (uValidate-uIsomMin) / (ulsomMax-ulsomMin)*2 -1;
chebyshevFit (uVaIScaled , polynOrder ,1);
aInpValidate
(uValidate-uIsomMin) /(uIsomMax-uIsomMin)*2 -1;
else uValScaled
legendreFit (uValScaled , polynOrder ,1);
aInpValidate
end
end
aInpValidate*cHybrid{runNum+1};
Isim (sys , vEstimValidate , ...
time (isomValBounds (1): isomValBounds (2)) , vStates (end,:))
errorValidate = surn(( zValidate (2:end-1)-zEstimValidate (2:end-1)).
%
vEstimValidate
zEstimValidate
2);
figure (3) , plot (uValidate , vEstimValidate , 'r. ')
plot (uValidate , vEstimValidate , 'r. ')
plot (v100 , vContCurve{runNum+1}, 'k ) , xlabel ( 'u') ,ylabel ('estimateduvu=uf (u)')
legend( 'traininguset' , 'validationuset' , 'continuousuestimate')
figure , plot (time (isomTrBounds (1): isomValBounds (2)) , [ zTrain; zValidate]) ,hold
plot (time (isomTrBounds (1): isomTrBounds (2)) , zEstimTrain , 'r ')
103
on
plot (time (isomValBounds (1): isomValBounds (2)), zEstimValidate ' , 'g')
error = sqrt ((errorTrain (end)+errorValidate ) /(length (uTrain)+length (uValidate
['E_{rms}=' ,num2str(error)];
errStr
];
['IsometriculID: ',file
titleStr
[num2str(delays) ,'delays'];
delayStr
title ({titleStr ; errStr ;delayStr });
xlabel('time'),ylabel('force_{isometric}')
indices = [];
for counter = 1: length (stimIndices)
if (stimlndices (counter)>isomTrBounds (1))&( stimlndices (counter)<isomValBounJi5
indices = cat (1, indices , stimIndices (counter ));
end
end
plot (time (indices ) ,force (indices ) , 'k. ')
legend( 'measured', 'trainingjfit' , 'validationufit' , 'stimulationupoints')
disp ('rmsuerror: ')
disp (sqrt (( errorTrain (end)+errorValid ate )/(length( uTrain)+length (uValidate ))))
disp('maxuforce: ')
disp (max(force (isomTrBounds (1): isomValBounds (2))))
disp ('StableuestimateuforuLID00x?')
disp( stability (end))
disp
disp
'dynamicusystemjnumerator: '),disp (sysOE.b)
'dynamicLsystemudenominator: ') , disp (sysOE. f)
sysCont = d2c (t f (sysOE. b, sysOE. f ,1/2000 ,variable
=
','z'));
tfdata(sysCont);
[numer,denom]
[zeros , poles I = tf2zp(numer{1},denom{1});
,disp(zeros ');
disp ('zeros:
disp ('poles:
),disp (poles ');
charEqn = denom{1};
2
if denOrder =
charEqn(2 ) % t his is 1)/m
bnorm
is k/mn (use measured mass to get k and )
charEqn(3
knorm
disp( 'bnorm: '),d isp (bnorm )
disp ( 'knorm: L') , d isp (knorm )
end
if denOrder = 3
kbnorm = -poles (find (imag(poles ))); %Athis is k/b (for 3rd ord sys)
complexPoles = poles (find (imag( poles )));
bnorm = -2*real(complexPoles (1)); %t his is b /m
knorm = (abs(complexPoles(1)))^2; %this is k/m
disp ( 'kbnorm: ') ,disp (kbnorm)
disp ( 'bnorm:') ,disp (bnorm)
disp ( 'knorm:') ,disp (knorm)
104
end
%check for particularly big coeffs (bad)
figure , plot (cInpTrain) , title ( 'inputunonlinearityucoeff
s ')
for frequency jit ter (irregular) vs none (constant (lifference)
figure , plot((stimIndices (2:end)-stimIndices (1:end--1),
t it le (' jitterujcheck')
%to check
%
%
%
%
%
%
clear
clear
file inidersclnd counter in( numnilter indices %stimInlndices
vEstimIMx
aInpTrain aInpValid ate dc iti
clear v100 aContiCurve
clear
clear
clear
uIson-i\in nIsomNMax nTrainScaled
isomnirBoun(s isomValBouiinds
tlInpSave imnpulseSave
errTrStr errValStr titleStr
105
uValScaled
THIS PAGE INTENTIONALLY LEFT BLANK
106
Appendix C
Simulation Log
107
I
4
A2
.9
W
2
.2
0-
2
5
L~--L
6
1.5
4
4
2
3
~
2
0.5
2
0
0
-2
-2
-0.5
-4,
00
Isometric I: expeniments 22junO6/ID 004.sbr
training error = 462827.3252, er = 7.6057
Isometric ID: experiments 22junO6ID 103. sbr
training error = 1147.0054, em = 0.37863
2.5
Isometric ID: experiments 22jurdO/D 002.sbr
training error = 19966.1762, e = 1.5797
6
Isometric ID: experiments 22jun06/lD 001. sbr
training error = 1160.0935, e... = 0.38078
3
4
5
6
n
7
2
3
4
5
6
.1
2
7
4
8
6
time
time
time
Isometric ID: experiments 22junO6/ID 005.sbr
training error 371.1872, erm= 0.21539
2
Isometric ID: experiments 22un6/ID 006.sbr
training error = 377312.
, erms = 6.8672
8
Isometric ID: experiments 22junO6ID 007.sbr
training error = 406.0351, eM = 0.22527
I
2I
2
3
4
3
4
1.5 [
2
j
2
0
1
.9
0.5
GA
-2
C-,
0.-1
0
0
-4
-6
0.
2
I1
5
6
7
time
Isometric ID: experiments 22jun06/ID 008.sbr
training error = 499135.2351, ef = 7.8984
6
4
-0.5t
7
4
6
1.5-
4L
0
-2
3
4
time
5
6
7
-4'
2
-8
3
'
4
5
time
'
6
1
7
-0.5
3
4
5
6
7
-10'2
time
Figure C-1: No nonlinearity, second-order dynamics (Experiments 1 to 8: 1, 20, 3, 50, 10, 75, 15, 100Hz)
5
time
6
7
0.4
2.5
0.8
0.2.
0.3
2
0.6
0.1
1.5
0.4
JA
2
Isometric ID: experiments 22junO6ID 012.sbr
training error = 18.5107. erms = 0.05142
0.5
Isometric ID: experiments 22junO6D 011 .sbr
training error = 1.2389, e,, = 0.012444
0.3
Isometric ID: experiments 22junO6AD 010.sbr
training error = 455.0937, erms = 0.23849
Isometric ID: experiments 22junD6/ID 009.sbr
training error = 58.4187, em = 0.085448
1,
0.2
0
0.2
0
0.1
0
4
0
1- w
0.5
0
-0.1
0
-0.2.
-0.41
2
3
4
5
6
-0. 51
2
7
time
Isometric ID: experiments 22junOS/ID 013.sbr
training error = 1047.1039, erm = 0.36176
-0.1
.
3
4
5
6
-0.2'
7
2
3
4
5
'
6
7
'-0.2
L
3
4
time
Isometric ID: experiments 22jun06/ID 015.sbr
training error = 4.0374, en
0.022464
time
Isometric ID: experiments 22junrE/D 014. sbr
training error = 477.8353, er, = 0.24438
0.3-
3.5
0.25
3.
1.5
2.6.
0.2
0.15
2
1
0.1
ft-
1.5
2
2
U
0
0.06
0
0.6
-0.05
0
0
-0.1
2
4
5
timA
6
7
-n 5
2
3
4
6
time
6
-0.16
-
3
4
5
6
time
Figure C-2: No nonlinearity, second-order dynamics (Experiments 9 to 15: 7, 30, 1, 5, 40, 25, 2Hz)
5
time
6
7
Isometric ID: experiments 22jun6/ID 002.sbr
training error = 20841.7933. erms = 1.614
Isometric ID: experiments 22jun06/ID 001.sbr
training error = 709.5599, em = 0.2978
6
6
5
5
4
4
-2 3
3
Isometric ID: experiments 22jung6ID 004.sbr
training error = 818593.838, erm = 10.1149
Isometric ID: experiments 22jun6/ID 003.sbr
training error 994.3635, enm = 0.35253
3
5
2.6
2
2
03
.2
2
1
2
1.5
-5
0.5
0
0
0
3
4
5
time
6
7
3
1.5-
4
0
4
5
time
6
7
.9
-15
0
E
V
0.5
-10
3
4
5
.2
-5
0
5
6
7
time
Isometric ID: experiments 22junOA6ID 008.sbr
training error = 742322.7252, en = 9.6322
10
4
1.5
E
0.5 [-
3
3
5
6
7
time
Isometric ID: experiments 22junO6/ID 007.sbr
training error = 458.6852 e = 0.23943
2
5
1
-10
-0.5
5
6
7
time
Isometric ID: experiments 22junO6/ID 006.sbr
training error = 865212.3227, erm = 10.3989
Isometric ID: experiments 22jun61ID 005.sbr
training error = 273.5743, em = 0.18491
21
.I
2
0
-10
0
3
4
5
time
6
7
3
4
5
6
7
2
3
time
Figure C-3: No nonlinearity, third-order dynamics (Experiments 1 to 8: 1, 20, 3, 50, 10, 75, 15, 100Hz)
4
5
time
6
7
Isometric ID: experiments 22junO6AD 010.sbr
training error = 365.0695, e. = 0.21361
Isometric ID: experiments 22junD6/ID 009.sbr
training error= 47.8688, en = 0.077349
0.8
Isometric ID: experiments 22jun06/ID 011.sbr
training error= 3.0224, em = 0.011222
0.3
Isometric ID: experiments 22junO6ID 012.sbr
training error = 17.7498, em, = 0.050352
0.6
2.5-
0.25
0.5
2-
0.2
0.4
0.6
0.4
1.5
2
42 0.2
0.3
0.1
W_
0.2
0
2
3
4
time
5
6
-0.6
7
-
3.6
0.1
2
3
4
5
6
-0.05
7
0
0.26
2
.2
ti
0.5
0.5
10
-0.11
3
15
0. 15
0.3.1
05
-
-0. 0 -
-
0
0
3
4
5
time
6
7
3
4
5
time
6
7
3
4
5
time
4
5
time
36
2.65
1
5
Isometric ID: experiments 22junO6AD 015.sbr
training error = 3.7698, rm = 0.021706
)3.
00
0
MAIiL
time
31
1.5 1
t
0
time
Isometric ID: experiments 22jun6AD 014.sbr
training error = 467.0151, em = 0.2416
2
Isometric ID: experiments 22junO6AD 013.sbr
training error = 640.3417, er = 0.2829
2
0.05
0
AIJM
L)
0
0.5
-0.2
0.15
0
Oi~
6
7
Figure C-4: No nonlinearity, third-order dynamics (Experiments 9 to 15: 7, 30, 1, 5, 40, 25, 2Hz)
6
2
Isometric ID: experiments 22jur6/ID 002. sbr
training error = 4360.6015, er = 0.73825
Isometric ID: experiments 22jurn/ID 001.sbr
training error = 631.6445, erms = 0.28097
7
7
6
6
5
5-
4
.2
a-
3
2
1
2.5
2
.2
.2
4
15
3
1
2
0.5
U
A
0
2
3
4
5
6
7
n
2
3
4
time
5
6
-0.5 L
2
7
time
2
tN~
3
4. F
5
4
-
,
_ *
6
7.
1.5
2
93
1~
0.
0
'
3
'
'
4
5
time
'
6
Isometric ID: experiments 22junO/ID 008.sbr
training error = 2266.2552, en = 0.53221
7
2
0
0
2
6
time
93
0.5
2
-n5'
5
4
4
4
3
0.5
3
5
5
-2
0,
6
1.5
6-
1
7
time
Isometric 10: experiments 22jun06IAD 007.sbr
training error = 369.6517, er = 0.21494
Isometric ID: experiments 22jun06/ID 006.sbr
training error = 8560.7246, er,. = 1.0344
Isometric ID: experiments 22jun6/ID 005.sbr
training error = 248.014., em = 0.17606
Isometric ID: experiments 22junO6iD 004.sbr
training error 6131.0626 er = 0.87538
Isometric ID: experiments 22jurdJ6/lD 003.sbr
training error = 795.6698, e, = 0.3153
2
3
5
4
time
6
7
2
3
4
5
6
-1
2
3
4
time
Figure C-5: No nonlinearity, fourth-order dynamics (Experiments 1 to 8: 1, 20, 3, 50, 10, 75, 15, 100Hz)
5
time
6
Isometric ID: experiments 22junO6ID 010 sbr
training error= 219.934, en = 0.1658
Isometric ID: experiments 22junO6/lD 009. sbr
training error = 47.2162, en. = 0.07682
,
1.2
E
,
,
Isometric ID: experiments 22junf6/Ni 011.sbr
training error= 2.9656, em = 0.011116
,
1
2.5
0.8
2
0.6
E
I|
0.25 r
|1
0.5
0.2
1.5
0.4
0.15
Os-
0.4
Isometric ID: experiments 22jun06/D 012.sbr
training error = 17.0168, erms = 0.049301
0.6
0.3I
2
t
0.1
0.3
02
0
0.2
0.5
0.05
01'
0
0
0
0~
-. 2;
2
3
5
4
6
-0. 5.
2
7
3
4
5
6
-0.05'
0
7
time
time
Isometric ID: experiments 22jun6AID 013.sbr
training error = 421.1195, e, = 0.22942
10
16
-0.13
time
Isometric ID: experiments 22junD6/ID 014.sbr
training error = 275.181, e = 0.18545
2
4
Isometric ID: experiments 22junD6/ID 015.sbr
training error = 3.7807, er
0.021738
0.35
3.5-
0.31.5
3-
0.25
2.6
0.2
1
2
2
5
0.15
1.5
0
0.5
0.51
.9
0.1
0.05
0
0
0
3
4
5
time
6
7
3
4
5
time
6
7
-0.05
3
4
5
time
6
7
Figure C-6: No nonlinearity, fourth-order dynamics (Experiments 9 to 15: 7, 30, 1, 5, 40, 25, 2Hz)
4
5
time
6
7
1.50 1.5 lt order non linearity, 2nd order dynamics
4.00
3.00
3.00a
1.00
0.50
m t0.00
m~S-0.50 1
*2
-
3
n
Al
100jir
.f2
1.200
0.
in
n
R
-1.00
-1.00
o
-1.50
-2.00
-
-2.00
T
s order nonlinearty, 3rd order dynamics
-2.50
- --
.2tL
0-fl
-- 3.00
-4.00
-
Frequency (H z)
Frequency (Hz)
1.50
4.00 2nd ode nonlieany3rd
-2.,00
2nd order nonlinearity,2nd order dynamics
1.00
.
0.50
0.00
;n
An
"X)
-0
inn
Rn
10
-1.00
-1.50
-2.00
$
2.00
=
1.00
-
.
-
.
'f2
0.00
-,-"4'--
----
- -
-- An
4.
e
100
.
#20
Frequency (Hz)
Frequency (Hz)
501
.
ef1
-3.00
-4.00
*:
-
a
-
C: -1.00
-2.50
orde r dynarnics
3.00
3rd order nonlinearity,2nd order dynamics
3rd order nonlinearity, 3rd order dynamics
00
C 2300
fl
1.00
S0.50
2
S3 0.00 -
P
n
''' -0.50 1
1.700
R
n
I
1n
2
200
f2
100
f3
0 00
-1.00
t
Freue1c50
(
-20
'0
m
9n
in
-300
S-2.00 -w
400
-2.50
Frequency (Hz)
Frequency (Hz)
Figure C-7: Difference equation denominator coefficient parameter variation
114
i2n
IRC: ID001
1.4
(3rd
order nonlinearity)
IRC: ID 002 (3rd order nonlinearity)
1.2
IRC: ID003 (3rd order nonlinearity)
I
1.2
1
0.8 0.8
if 0.6 -
0.8
0.6
E 0.6
E 0.4
0.4
0.2
0.2
0
E 0.4-
+
+
training set
validation set
continuous estimate
-
0.2
1I
0.2
0.4
0.6
0.8
U
0
0.2
0.4
0.6
0.8
IRC: ID 005 (3rd order nonlinearity)
u
1
x10
U
x 10
0.2
0.6
0.4
0.8
1
x 10-
IRC: ID 007 (3rd order nonlinearity)
12
. ,
1
IRC: ID 009 (3rd order nonlinearity)
1
I
C1
1
0.8
0.80.6
0.8
:3
0.6
E
0.4
0.4
F
-
II
E 0.4
E 0.2-
a)
0
0.2
-0.2
0
0.21
0
0.6
a)
0.2
0.4
U
0.6
0.8
1
x10-1
-0.4
0
0.2
0.4
u
0.6
0.8
1
X10-
-0.21
0
0.2
0.4
U
0.6
0.8
Figure C-8: Nonlinear recruitment curves from iterative identification with third-order dynamics (Experiments 1 to 9: 1, 20, 3,
10, 15, 7Hz)
X
3
IRC: ID 011 (3rd order nonlinearity)
IRC: 10 012 (3rd order nonlineanty)
IRC: ID 013 (3rd order nonlinearity)
1.2
+
+
0.8
I
training set
validation set
continuous estimate
1-
0.8
0.6
0.5
II
0.6
-o
E 0.4
E 0.4
D
E
0'
0.2
0.2 [
0.2
0.4
0.6
0.8
1
-0.2 '
0
0.4
IRC: ID014 (3rd order nonlinearity)
1.21
0.6
0.8
1
X10-
U
x 10
U
I,
0.2
3
-0.5
0
0.2
0.4
0.6
u
0.8
X 10 3
IRC: ID015 (3rd order nonlinearity)
1.2
1
1
0.8-
0.80.6-
E 0.4
E 0.4
0.2
0.2
0
0
-0.2
0
0.2
0.4
0.6
-0.2 L
0
0.8
x 10-3
0.2
0.4
U
0.6
0.8
1
x 10e
Figure C-9: Nonlinear recruitment curves from iterative identification with third-order dynamics (Experiments 11 to 15: 1, 5,
40, 25, 2Hz)
Isometric ID: experiments 2un(6/D 002.sbr
Isometric ID: experiments 22junO6/D 001.sbr
Ern =0.36331
Isometric ID: experiments 22jun06ID 003. sbr
Em =0397
4
6
5
3.5
5
4
3
4
3
2.5
2
3
2
1.5
S2
0
0.5
0
0
3
time
Isometric ID: experments 22rnOD 004. sbr
=3.7619
4
Isometric ID: ex
3
5
6
7
time
ments 22jur6AD 005.sbr
7
time
Isometric ID: ex
nts
u22
6nO/D 006 sbr
0.238W
-KJ
6
1.5 -
4
1
10
0.6 -
2
0
0
0
-2
-0.5
3
4
t
time
6
7m
3
4
5
time
6
7
3
5
4
6
time
Figure C-10: Simulations with fixed third-order polynomial nonlinearity and third-order dynamics (Experiments 1 to 6: 1, 20,
3, 50, 10, 75Hz)
7
Isometric ID: experiments 22jun6/ID 007.sbr
Em = 0.21866
Isometric ID: experiments 22unO6AD 009.sbr
E =0.074622
Isometric ID: experiments 22'unO6/D 008. sbr
Em = 20.77A
D
10
2
0
1.5 I
0.6
-10
-
.9 0.4
ig
1
-20-
0.5 [
0.2
-30-
0
0
-40
6
5
time
Isometric ID: experiments 22jun6AID 010.sbr
Em = 0.23232
4
3
7
3
5
4
6
-0.2
7
time
Isometric ID: experiments 22jur36/ID 01 1.sbr
Erms = 0.008716
0.3
25 2.5-
0.25-
2-
0.2-
2e
0.4
.8
is 0.3
9
oil
0.1
3
4
5
time
6
7
_n m
2
4
6
8
0.1
0
10
time
0.2
4-
I
0
0
7
0.5
0.05
0.5
4
I IV%.
0.15
1.5
5
6
time
Isometric ID: experiments 22 unO6/ID 012.sbr
E = 0.05555
3
12
14
16
-0.1
3.5
4
4.5
5
5.5
time
6
6.5
Figure C-11: Simulations with fixed third-order polynomial nonlinearity and third-order dynamics (Experiments 7 to 12: 15,
100, 7, 30, 1, 5Hz)
7
Isometric ID: experiments 22*unAD 014.sbr
s 4
Isometric ID: experiments 22junr6/ID 013.sbr
En = 0.45574
4
-2
3
-4
2
4.)
0
-6
~it2
C.)
1
-8
-10
0
-12
3
5
4
6
6
4
3
7
6
7
time
time
Isometric ID: experiments 22jun06AD 015.sbr
Em=0.024405
0.4
0.3
0.25
0.2
0.16
10
0.1
0.056
0 -. 05
3
4
6
time
6
7
Figure C-12: Simulations with fixed third-order polynomial nonlinearity and third-order dynamics (Experiments 13 to 15: 40,
25, 2Hz)
14
S-plane:
no nonlinearity, 2nd
S-plane: 1st order nonlinearity, 2nd order dynamics
order dynamics
30
20
----- ---
20
10
_-+ ----- + ----- +--------- -- ----- +-----
-- --
101
'0
0
- ---------------
E
I -
T
------
-10
_10
I
__
I
-------_I
_ ~ _1t_ --_-T_------
x:
--
-20
-20
-2
---
-
- -
-------
-200
.150
-100
50
0
-50
100
150
-40
-4
200
20
0
20
40
120
1,
-
-
x-
--- ---
-
---
20
10 ------------------ ------ ------
-----
+------
------
+-------. .
10
-0
0
--------- ----- --- ----------------------------- --------------- -------- - ------ - -- - --------------
-30
3o
----
-
30
...
- ...
------------------_-----
20
_Mf
100
S-plane: 3rd order nonlinearity, 2nd order dynamics
40
_
30
80
s0
Real
S-plane: 2nd order nonlinearity, 2nd order dynamics
In
-20
------- -------
--. ..
---------
-30
Real
-10
-
-10
------ ---
-30 ---------- --- -----
---------0
50
100
150
Real
200
250
300
50
3
x
x
x
x
x
x
x
--------------------------
---
-20
0
-50
-- -
-...
I ------- -
50
0
100
Real
1Hz (ID001)
1Hz (D011)
2Hz (D015)
2Hz (ID016)
3Hz (ID003)
5Hz (ID012)
7Hz (1D009)
10Hz (ID005)
15Hz (ID007)
20Hz (ID002)
25Hz (ID014)
30Hz (ID010)
40Hz (ID013)
50Hz (ID004)
75Hz (10006)
100Hz (D008)
Figure C-13: Pole-zero maps for second-order dynamic systems with different input
polynomial nonlinearities
120
150
--------
S-plane: Ist order nonlinearity, 3rd order dynamics
S-plane: no nonlinearity, 3rd order dynamics
Mn
MOC
200
t0
100
100
--- ------ ------ ----- ----- -----
0
E
X0
4
-
af)
X
-100
600
00
.
~ ~ ~
--
-200
a Z
0
0
00
400
Real
--- --------- --
------------
O0
10
S-plane: 2nd order nonlinearity, 3rd order dynamics
400
- - --
----------- - -- - -- - - - - - - - - - - - -- - -- - - - -
-200
IM
10
-500
0 -20
-300
-1000
0
1000
Real
S-plane: 3rd order nonlinearty, 3rd order dynamics
300
.. ----------...
------
-4000
-- -
200----------
........-------- -------- -------- --------
200
------+
------
--------- ------- + - ----- +------ +----- -
1 ---------
100
---t----
10
-100
-300 --
0
0.5
-
----- --
----
-------
- -
25
2
15
Real
1
-
3
-
35
-200 -------
10M
ILI
0
-----------
--
1000
2000
3D)
4000
5000
S000
Real
S-plane: 3rd order fixed nonlineariy, 3rd order dynamics
var
0
2W0
--- --------------
-- -
------ . 0-.
---------------------------
X
*
X
X
100
X
-100
-----
.---------. -- ---------
-------..
..------- ------- ------X
-
-400
-300
-200
00 0
.
-100
0
100
20
X
X
30
401)
(ID003)
25Hz (1D014)
*
-200
(ID001)
(0011)
(10015)
(ID016)
(ID012)
(10009)
10Hz (ID005)
15Hz (1D007)
20Hz (ID002)
X
E
1 Hz
1Hz
2Hz
2Hz
3Hz
5Hz
7Hz
30Hz (1D010)
4OHz (D013)
5OHz (10004)
75Hz (10000)
100Hz (10008)
W
Real
Figure C-14: Pole-zero maps for third-order dynamic systems with different input
polynomial nonlinearities
121
703M
S-plane: no nonlinearity, 4th order dynamics
000)
6WO0
------- ------ ------- ----- ---- ----4000 ----2000
-
E
-----------------------------
-------------------
---
-
---- 6
-
----- ------ ----------- - ---------------
X
1Hz (ID001)
X
2Hz (ID015)
2Hz (ID016)
3Hz (ID003)
5Hz (ID012)
7Hz (ID009)
10Hz (ID005)
15Hz (ID007)
20Hz (ID002)
25Hz (ID014)
30Hz (ID010)
40Hz (ID013)
50Hz (ID004)
X
- ------ -
-
-20
------- ------- ------- ------
-4000 --------------------
------ ------ ------ ------
-----------
------
------ ------
------ ------------------
X
X
-0 ------ ------
00o
8%1'C
-&uoX
i-------------- ------- -------- -------- -----.----..-------
100
160W0 -14000 12000 10000 -8000 -6000 -4000 -2000
Real
0
X
H ID 11
75Hz
5 z (10006)
I0 6
100Hz (ID008)
2000
Figure C-15: Pole-zero maps for fourth-order dynamic systems with different input
polynomial nonlinearities
122
Experiment
ID-001
ID..011
ID-015
ID-003
ID-012
ID-009
ID-005
ID-007
ID-002
ID-014
ID.010
ID.013
ID-004
ID-006
ID-008
Experiment
ID-001
ID-011
ID-015
ID-003
ID-012
ID_009
ID-005
ID-007
ID-002
ID..014
ID-010
ID-013
ID-004
ID.006
ID-008
Freq (Hz)
1
1
2
3
5
7
10
15
20
25
30
40
50
75
100
1st ord nonlin,
2nd ord dyn
7.97
4.94
10.78
15.96
12.66
12.26
13.04
9.49
1st ord nonlin,
3rd ord dyn
6.06
4.94
11.09
14.27
17.34
15.48
11.46
58.10
36.15
Linear models (no nonlinearity)
3rd ord dyn
2nd ord dyn
Freq (Hz)
6.2078
7.938
1
3.879
4.301
1
9.4006
9.729
2
16.7266
17.97
3
11.6718
11.92
5
14.815
16.37
7
10.8458
12.63
10
14.3363
13.49
15
27.4536
26.87
20
14.1435
14.31
25
7.335
8.189
30
8.5375
10.92
40
50
75
100
Nonlinear models
2nd ord nonlin,
2nd ord nonlin,
3rd ord dyn
2nd ord dyn
6.36
6.00
4.04
4.91
9.79
12.78
15.82
13.79
12.87
22.25
13.85
12.39
11.47
10.80
12.98
8.49
10.86
4th ord dyn
5.857
3.8424
14.9625
11.4281
14.7137
10.3267
12.8699
12.5574
10.8565
6.9236
12.4702
16.8146
-
3rd ord nonlin,
2nd ord dyn
3.43
3.98
10.31
13.92
17.87
14.98
11.81
11.21
8.18
--
43.95
49.30
-
15.13
76.41
12.61
-
5.88
10.95
35.78
49.88
64.71
10.95
9.72
66.62
3rd ord nonlin,
3rd ord dyn
3.26
3.98
12.99
13.08
22.14
3.26
11.79
12.40
7.68
-7.28
-
3rd ord fixed nonlin,
3rd ord dyn
7.57
3.01
10.57
18.69
12.87
14.29
14.01
13.10
26.69
7.98
13.75
53.59
-
* average of RMS errors from both linear and nonlinear simulations (dynamically unstable and non-converging simulation errors denoted '-')
Average eRMS
12.43 between 1 and 20Hz, inclusive
10.36 between 25 and 40Hz, inclusive
43.57 between 50 and 100Hz, inclusive
16.34 fixed nonlinearity
16.62 iterated nonlinearity
Table C.1: RMS errors for all simulations
Avg eRMS
across freq*
6.07
4.18
10.83
15.52
15.30
13.47
11.90
17.23
17.44
13.10
7.34
10.78
36.45
31.43
69.24
16 om-60 M-001IMMIJIMOMi W,
Linear models (no nonlinearity)*
Experiment
ID_001
ID_011
ID-015
ID-003
ID-012
ID-009
ID-005
ID-007
ID-002
Freq (Hz)
2nd ord dyn
1
1
1
1
2
1
3
1
5
1
7
1
10
1
15
1
20
1
ID-014
25
1
ID_010
30
1
ID-013
40
1
ID..004
50
0
ID-006
75
0
ID.008
100
0
average linear model stability: 0.8
Experiment
1st ord nonlin,
2nd ord dyn
1
1
1
1
1
0
1
1
1
-
Freq (Hz)
ID-001
1
ID-011
1
ID-015
2
ID_003
3
ID-012
5
ID..009
7
ID-005
10
ID-007
15
ID-002
20
ID-014
25
ID-010
30
0
ID.013
40
ID.004
50
1
ID-006
75
1
ID-008
100
average nonlinear model stability: 0.74
*
1st ord nonlin,
3rd ord dyn
1
1
1
1
1
1
1
1
1
-
1
-
1
3rd ord dyn
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
Nonlinear models
2nd ord nonlin, 2nd ord nonlin,
2nd ord dyn
3rd ord dyn
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
4th ord dyn
1
1
0
1
1
1
1
1
1
1
0
1
1
1
0
3rd ord nonlin,
2nd ord dyn
1
1
1
1
1
1
1
1
1
-
Average
stability
1.00
1.00
0.67
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0.67
1.00
0.33
0.33
0.00
3rd ord nonlin,
3rd ord dyn
1
1
1
1
1
1
1
1
1
0
-
1
-
1
1
stable models denoted 1, unstable models denoted 0, non-converging models denoted -
** average stability of all ten linear and nonlinear models
Table C.2: Stability for all simulations
1
-
3rd ord fixed nonlin,
3rd ord dyn
1
1
1
1
1
1
1
1
1
0
1
1
1
0
0
Avg nonlin
stability
1.00
1.00
1.00
1.00
1.00
0.71
1.00
1.00
1.00
0.00
Avg overall
stability**
1.00
1.00
0.90
1.00
1.00
0.80
1.00
1.00
1.00
0.30
0.29
0.40
0.86
0.90
0.43
0.43
0.43
0.40
0.40
0.30
Bibliography
[1] American Heart Association Statistics Committee and Stroke Statistics Subcommittee. Heart Disease and Stroke Statistics: 2006 Update, February 2006.
[2] E.W. Bai and D. Li. Convergence of the iterative hammerstein system identification algorithm. Automatic Control, IEEE Transactions on, 49(11):1929-1940,
2004.
[3] R. Baratta and M. Solomonow. The effect of tendon viscoelastic stiffness on the
dynamic performance of isometric muscle. Journal of Biomechanics, 24(2):109116, 1991.
[4] L.A. Bernotas, P.E. Crago, and H.J. Chizeck. A discrete-time model of electrically stimulated muscle. IEEE Trans. Biomed. Eng., 33:829-838, 1986.
[5] J. Bobet, E.R. Gossen, and R.B. Stein. A comparison of models of force production during stimulated isometric ankle dorsiflexion in humans. Neural Systems
and Rehabilitation Engineering, IEEE Transactions on [see also IEEE Trans. on
Rehabilitation Engineering], 13(4):444-451, 2005.
[6] J. Bobet and R.B. Stein.
A simple model of force generation by skeletal
muscle during dynamic isometric contractions. Biomedical Engineering, IEEE
Transactions on, 45(8):1010-1016, 1998.
[7] F. Chang and R. Luus. A noniterative method for identification using hammerstein model. Automatic Control, IEEE Transactions on, 16(5):464-468, 1971.
125
[8] T.L. Chia, P.-C. Chow, and H.J. Chizeck. Recursive parameter identification of
constrained systems: an application to electrically stimulated muscle. Biomedical
Engineering, IEEE Transactions on, 38(5):429-442, 1991.
[9] H.J. Chizeck, P.E. Crago, and L.S. Kofman. Robust closed-loop control of isometric muscle force using pulsewidth modulation. Biomedical Engineering, IEEE
Transactions on, 35(7):510-517, 1988.
[10] H.J. Chizeck, N. Lan, L.S. Palmieri, and P.E. Crago. Feedback control of electrically stimulated muscle using simultaneous pulse width and stimulus period
modulation. Biomedical Engineering, IEEE Transactions on, 38(12):1224-1234,
1991.
[11] P.-C. Chow and H.J. Chizeck. Nonlinear recursive identification of electrically
stimulated muscle. In Engineering in Medicine and Biology Society, 1989. Images
of the Twenty-First Century. Proceedings of the Annual International Conference
of the IEEE Engineering in, pages 965-966 vol.3, 1989.
[12] N. de N. Donaldson, H. Gollee, K. J. Hunt, J. C. Jarvis, and M. K. N. Kwende.
A radial basis function model of muscle stimulated with irregular inter-pulse
intervals. Medical Engineering & Physics, 17(6):431-441, September 1995.
[13] N.de N. Donaldson and P.E.K. Donaldson. When are actively balanced biphasic
('lilly') stimulating pulses necessary in a neurological prosthesis? Medical and
Biological Engineering and Computing, 24:41-56, 1986.
[14] W.K. Durfee.
Task-based methods for evaluating electrically stimulated an-
tagonist muscle controllers.
Biomedical Engineering, IEEE Transactions on,
36(3):309-321, 1989.
[15] W.K. Durfee and K.E. MacLean.
Methods for estimating isometric recruit-
ment curves of electrically stimulated muscle. Biomedical Engineering, IEEE
Transactions on, 36(7):654-667, 1989.
126
[16] W.K. Durfee and K.I. Palmer. Estimation of force-activation, force-length, and
force-velocity properties in isolated, electrically stimulated muscle. Biomedical
Engineering, IEEE Transactions on, 41(3):205-216, 1994.
[17] Gertjan J. C. Ettema and Kenneth Meijer. Muscle contraction history: modified
hill versus an exponential decay model. Biological Cybernetics, 83(6):491-500,
November 2000.
[18] W. Farahat and H. Herr.
An apparatus for characterization and control
Neural Systems and Rehabilitation Engineering, IEEE
of isolated muscle.
Transactions on [see also IEEE Trans. on Rehabilitation Engineering], 13(4):473481, 2005.
[19] W. Farahat and H. Herr. A method for identification of electrically stimulated
muscle. In Engineering in Medicine and Biology Society, 2005. IEEE-EMBS
2005. 27th Annual International Conference of the, pages 6225-6228, 2005.
[20] M. Forcinito, M. Epstein, and W. Herzog.
predict force depression/enhancement?
Can a rheological muscle model
Journal of Biomechanics, 31(12):1093-
1099, December 1998.
[21] H. Gollee, D.J. Murray-Smith, and J.C. Jarvis. A nonlinear approach to modeling of electrically stimulated skeletal muscle. Biomedical Engineering, IEEE
Transactions on, 48(4):406-415, 2001.
[22] R. Haberman. Elementary Applied Partial Differential Equations with Fourier
Series and Boundary Value Problems. Prentice-Hall, Inc., 1983.
[23] A. Hammerstein.
Nichtlineare integralgleichung nebst anwendungen.
Acta
Mathematica, 54:117-176, 1930.
[24] H. Hatze. Myocybernetic control models of skeletal muscle: Characteristics and
applications. University of South Africa, 1981.
127
[25] E. Henneman and C.B. Olson. Relations between structure and function in the
design of skeletal muscles. J Neurophysiol, 28(3):581-598, 1965.
[26] A.V. Hill. The heat of shortening and the dynamic constants of muscle. volume
126 of Proceedings of the Royal Society of London, Series B. Biological Sciences,
pages 136-195. Royal Society of London, 1938.
[27] T.C. Hsia.
System Identification: Least-Squares Methods.
D.C. Heath and
Company, 1977.
[28] K.J. Hunt, M. Munih, N.de.N. Donaldson, and F.M.D. Barr. Investigation of
the hammerstein hypothesis in the modeling of electrically stimulated muscle.
Biomedical Engineering, IEEE Transactions on, 45(8):998-1009, 1998.
[29] I. W. Hunter and M. J. Korenberg. The identification of nonlinear biological
systems: Wiener and hammerstein cascade models. Biological Cybernetics, 55(2
- 3):135-144, November 1986.
[30] I.W. Hunter. Experimental comparison of wiener and hammerstein cascade models of frog muscle fiber mechanics. Biophys J, 49:81a, 1986.
[31] A.F. Huxley. Muscle structure and theories of contraction. Prog. Biophys. and
Biophys. Chem., 7:257-318, 1957.
[32] Eric R. Kandel, James H. Schwartz, and Thomas M. Jessell. Principles of Neural
Science. McGraw-Hill, 4th edition, 2000.
[33] J.C. Lilly. Electrical stimulation of the brain, chapter Injury and Excitation by
Electric Currents: The Balanced Pulse-Pair Waveform, pages 60-66. University
of Texas Press, 1961.
[34] J.K. Lim, M.H. Nam, and G. Khang. Model of activation dynamics for an fesinduced muscle fatigue. In Engineering in Medicine and Biology Society, 2000.
Proceedings of the 22nd Annual International Conference of the IEEE, volume 3,
pages 2251-2253 vol.3, 2000.
128
[35] L. Ljung. System Identification: Theory for the User. Prentice Hall, 2nd edition,
1998.
[36] T.A. McMahon. Muscles, Reflexes, and Locomotion. Princeton University Press,
1984.
[37] J.T. Mortimer, D. Kaufman, and U. Roessmann. Intramuscular electrical stimulation: Tissue damage. Ann. Biomed. Eng., 8:235-244, 1980.
[38] M. Munih, K. Hunt, and N. Donaldson. Variation of recruitment nonlinearity
and dynamic response of ankle plantarflexors. Medical Engineering & Physics,
22(2):97-107, March 2000.
[39] K. Narendra and P. Gallman. An iterative method for the identification of nonlinear systems using a hammerstein model. Automatic Control, IEEE Transactions
on, 11(3):546-550, 1966.
[40] National Spinal Cord Injury Statistical Center. Spinal cord injury: Facts and
Figures at a Glance., June 2005.
[41] Gonzalo H. Otazu, Ryoko Futami, and Nozomu Hoshimiya. A muscle activation
model of variable stimulation frequency response and stimulation history, based
on positive feedback in calcium dynamics. Biological Cybernetics, 84(3):193-206,
February 2001.
[42] S. Rangan, G. Wolodkin, and K. Poolla. New results for hammerstein system
identification. In Decision and Control, 1995., Proceedings of the 34th IEEE
Conference on, volume 1, pages 697-702 vol.1, 1995.
[43] R. Riener. Model-based development of neuroprostheses for paraplegic patients.
Phil. Trans. R. Soc. Lond. B, 354(1385):877-894, May 1999.
[44] R.B. Stein, P.H. Peckham, and D.B. Popovic, editors.
Neural Prostheses:
Replacing Motor Function After Disease or Disability. Oxford University Press,
1992.
129
[45] P. Stoica. On the convergence of an iterative algorithm used for hammerstein
system identification. Automatic Control, IEEE Transactions on, 26(4):967-969,
1981.
[46] L. Vodovnik, W.J. Crochetiere, and J.B. Reswick. Control of a skeletal joint
by electrical stimulation of antagonists. Medical and Biological Engineering.,
5:97-109, 1967.
[47] David T. Westwick and Robert E. Kearney.
Identification of Nonlinear
Physiological Systems. IEEE Press Series in Biomedical Engineering, 2003.
[48] A.S. Wexler, Jun Ding, and S.A. Binder-Macleod. A mathematical model that
predicts skeletal muscle force. Biomedical Engineering, IEEE Transactions on,
44(5):337-348, 1997.
[49] G.I. Zahalak.
Multiple Muscle Systems:
Biomechanics and Movement
Organization, chapter Modeling Muscle Mechanics (and Energetics), pages 123. Springer-Verlag, 1990.
[50] G.I. Zahalak. Neural Prostheses: Replacing Motor Function After Disease or
Disability, chapter An Overview of Muscle Modeling, pages 17-57. Oxford University Press, 1992.
[51] G.I. Zahalak and S.P. Ma. Muscle activation and contraction: constitutive relations based directly on cross-bridge kinetics. J. Biomech Eng., 112:52-62, 1990.
[52] B. H. Zhou, M. Solomonow, R. Baratta, and R. D'Ambrosia. Dynamic performance model of an isometric muscle-joint unit. Medical Engineering & Physics,
17(2):145-150, March 1995.
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