This document summarizes a study that examined how electrical stimulation of the tibialis anterior and gastrocnemius muscles in rats affects ankle joint impedance. The study found that: 1) Ankle joint impedance and net torque at a given velocity were linearly related to the stimulation amplitude and frequency delivered to the two muscles. 2) Ankle net torque depended on the difference between tibialis anterior and gastrocnemius stimulation, while ankle impedance depended on the sum of their stimulation. 3) Ankle net torque and impedance increased linearly with increased velocity of ankle joint movement.

1. ACTIVE CONTRIBUTIONS TO ANKLE-JOINT IMPEDANCE IN RATS
by
David Morrison
has been approved
April 2010
APPROVED (printed name, signature)
___________________________________,__________________________________, Director
___________________________________,_____________________________, Second Reader
___________________________________,______________________________, Third Reader
Honors Thesis Committee
ACCEPTED:
_______________________________________
Dean, the Barrett Honors College

2. Abstract
During legged locomotion, the mechanical properties of joints and legs (i.e. force,
impedance) are modulated to achieve task-level movement dynamics. A ‘tuned’
musculoskeletal system can contribute to stable locomotion. Joint and leg mechanics
reflect both passive properties and the properties of active muscle. However, the
sensitivity of joint-level impedance to different patterns of muscle activity have not been
well characterized in rodents. Therefore, we examined the relationship between muscle
activity and ankle joint impedance in the laboratory rat. In deeply anesthetized animals,
we stimulated the tibialis anterior (TA) and gastrocnemius (GAS) muscles with acute
implanted electrodes, using 40 Hz trains of 0.2 ms square wave pulses while the ankle
joint was attached to a position-controlled force transducer. The ankle was subject to
sinusoidal angular displacements at 5 frequencies between 1-10 Hz. TA and GAS were
stimulated using currents ranging from 0.0-5.0 mA for 3 seconds, with 30 seconds rest
periods between trials. Static forces were estimated by averaging transducer output
during the last 0.5 seconds, and impedance properties calculated from force changes and
phase relative to the position changes. Net ankle torque depended on the difference
between TA and GAS stimulation while ankle impedance depended on the sum of TA
and GAS stimulation. This behavior is predicted by the pre-stressed two-spring joint
model.
Introduction
Biologically-inspired SLIP-based (spring-loaded inverted pendulum) walking
mechanics have proven effective for stable, high performance locomotion in legged
robots while also simplifying the controller (Raibert 1986). These robots, such as the

3. “Big Dog” from Boston Dynamics, are capable of stabilizing large disturbances and
handling rough terrain. Despite these successes, the principles that motivated the design
of these robots and their control have not been applied to neuroprosthesis to restore
locomotion to individuals with spinal cord injury (SCI).
Tuning leg compliance allows SLIP-based robots to locomote over a variety of
terrains with simplified control. Given that joint torque is required to generate
movement, a key control objective for an impedance-based controller should be
generating a range of leg compliances for a given joint torque. Previous studies
examining the relationship between joint-level impedance and electrical stimulation have
only stimulated one muscle (Flaherty 1994). Therefore, this study intends to characterize
how extensor and flexor muscle activity can act to tune leg compliance in biological
neuromechanical systems by measuring ankle-joint impedance in rats, whose locomotion,
like humans, can be described by the SLIP model.
Towards this end, we tested the hypotheses that (i) ankle-joint impedance and (ii)
net torque at a given velocity are a linear function of the stimulation amplitude and
frequency delivered to the gastrocnemius (GAS) and tibialis anterior (TA) muscles.
Furthermore, we hypothesize that ankle net torque and impedance will increase linearly
with velocity.
We moved the ankle joint through a small sinusoidal movement using a range of
speeds (5 even intervals between 1-10 Hz) and stimulation patterns (Current: 5 even
intervals between 0.0-5.0 mA) to the TA and GAS muscles and collected force data.

4. Materials and Methods
Ankle joint impedance data were collected in four female Wistar rats (200-240 g)
aged 3 to 12 months over a period of six months. The rats were housed individually in a
university animal care facility with 12-hour light and dark cycles and provided access to
food and water ad libitum. The animals were treated in accordance with US Public
Health Service Guide for the Care and Use of Laboratory Animals and the Institutional
Animal Care and Use committees at ASU approved all surgical and experimental
protocols.
2.1 Experimental Setup
Designing a device capable of accurately measuring the impedance and providing
consistent force mapping of the electrically stimulated ankle-joint in the laboratory rat
was central to the success of this experiment. However, there are three primary problems
facing effective data collection in an impedance experiment. First, the potential for high-
frequency noise in impedance data collection is greater than in static force data collection
because the limb is moved through a small range of motion at high accelerations.
Second, the moments generated by muscles are dependent in part on force-length
properties of the muscles involved. Third, a predictable angular displacement of the
ankle must be achieved to calculate impedance. Therefore, our experimental device was
designed to reduce noise in the data, control the muscle length of the tibialis anterior
(TA) and gastrocnemius (GAS), and provide a predictable angular displacement at the
ankle.
In order to reduce noise in the impedance data, efforts were made to constrain the
ankle and knee and to create a stable scaffolding for the force transducer. For ankle

5. constraint, we custom-designed an adjustable foot grabber to anchor the foot shank to the
force lever. This confined movement to the saggital plane and gave the foot a firm
connection to the force collection device. To reduce knee movement, the ankle’s axis of
rotation was placed at the axis of rotation of the force transducer. Assuming a rigid foot
shank and sufficient inertia coming from the hip joint, vertical displacement of the knee
and rotation about the frontal axis of the knee was avoided. Finally, the force transducer
was screwed into a rigid scaffolding to avoid noise that could result from an unstable data
collection device.
Fig. 1. (a) Picture of the rat in the experimental rig. Rat is suspended in a sling while its ankle is
connected to the force transducer by a custom-designed adjustable foot grabber. (b) Shows experimental
rig without the rat. The metal dowel extending to the superior edge of the right-leg hole was intended to
constrain the knee
To control the length of the TA and GAS, the experimental rig was designed to
control the angle of the ankle and knee. The orientation of the ankle was effectively
controlled by the orientation of the adjustable force transducer arm. The knee angle
could be modified by adjusting the vertical position of the force transducer or the
horizontal position of the rat.
In order to produce a predictable angular displacement of the ankle, we placed the
ankle’s axis of rotation in line with the axis of rotation of the force transducer. Provided
the knee and tibial leg shank did not move during the trial, the angular displacement of
a b

6. the rat’s ankle was assumed to be the same as the displacement of the force transducer
(about 2o
).
2.2 Experimental Procedures
Rats were anesthetized using 2% isoflurane and 2% oxygen, enough to suppress
the toe pinch reflex. Once anesthetized, the rat’s right leg was shaved and cleaned using
isopropyl alcohol and iodine. The lower one-third of the tibia, medial malleolus,
calcaneus, and first metatarsal were marked in ink and photographed so that ankle angle
could be later determined (adapted from Varejao 2002).
The rat was then placed in the experimental rig (Fig. 1a) while still under
anesthesia. The rig attached the rat’s foot to a force transducer with a rotational
component (Aurora Scientific 305 C-LR), placing the malleolus at the center of rotation
of the force transducer, and the foot along the central axis of the force-transducer lever.
Once in the rig, sterilized acute intramuscular electrodes were inserted in the proximal
and distal third of the tibialis anterior (TA) and lateral gastrocnemius (GAS) (four in
total), the primary muscles for ankle dorsiflexion and plantarflexion.
The TA and GAS were subjected 3-s trials of monophasic cathodic stimulation
while the force transducer moved the ankle through a small sinusoidal range of motion,
with 30 s between stimulation trials. The stimulation train used a pulse duration of 200
μs at 40 Hz at an amplitude ranging from 0.0-5.0 mA in five even intervals. This range
of stimulation amplitude extended to about ten times over twitch threshold measures for
the TA and GAS, as measured by Jung et al. (2009). Twitch threshold was not measured
in this experiment. However, minimal recruitment threshold, detailed later, was tested
for. The force transducer moved the ankle at a frequency ranging from 1-10 Hz in five

7. even intervals. Single-muscle control trials (2.5 mA TA/0 mA GAS, 0 mA TA/2.5 mA
GAS at 5.5 Hz) were run every 20 trials to assess the role of fatigue and electrode
stability during the course of the experiment. There were 137 trials run each completed
experiment.
Two adjustments were made from initial trials (pre-11/09) and later trials (post
11/09)—the foot attachment device was changed and attempts were made to constrain the
knee (Fig. 1b), which was unconstrained in initial trials. The hip was unconstrained for
all trials. One experiment was captured by high-speed camera (Miro Phantom) at 100
frames per second to verify to effectiveness of the experimental device.
2.4 Data Analysis
Average static force
Average static force was measured each trial by averaging the force during the
last .5 s of the trial, thereby eliminating the force transient at the beginning of each trial
(Jung et. al, 2009). Trials were zeroed according to the passive trials, and therefore
average static force represents only the active contribution to force, ignoring gravity,
inertia, and passive viscoelastic characteristics of the ankle.
Impedance
Impedance was measured by using principal components analysis. The slope of
the first principal component describes the best fit to the in-phase relationship between
force and length. The slope of this line was taken to represent mechanical stiffness.
Impedance properties were also calculated by fitting a Voigt model to the following
equation.
θθθ kbJ ++

8. Controls
In order to improve the predictability of the average static force and impedance
data, a series of controls were run to account for the shortcomings in the stimulation
protocol for this experiment. Those controls are the following
i) Experiment by experiment comparison: Since the electrode placement varied from
experiment to experiment, specific parameters (e.g. the relationship between current
and force) also differed among experiments. For this reason, parameters were
calculated separately for each experimental session.
ii) Errant trials: Since the TA and GAS current was set manually in the 30 s in between
each trial, errors sometimes occurred, and were recorded in a notebook. These trials
were removed from the data
iii) Electrode stability: Since the electrodes could potentially move during the
experiment, electrode stability was assessed by comparing control trials across the
experiment (Fig. 2) and by examining recruitment curves in MATLab (Appendix Id).
Stable electrodes showed a consistent recruitment curve and stable force production
throughout the experiment. Experiments that showed electrode instability were not
used.
Fig. 2. Control trials for the GAS in
our 1/30 experiment. Reveals
electrode stability throughout the
experiment and a steady-state
fatigue setting in at the time of the
second control trial.
Gastrocnemius
0
0.1
0.2
0.3
0.4
0.5
0 2 4 6 8 10
Control trial number
Force

9. iv) Steady-state fatigue (App. Ic): Since 30 s may not have been enough time for
stimulated muscles to fully recover, we based the figures in our experiment off of
muscles in a predictable state of metabolic fatigue. Fig. 2 shows that this steady-state
fatigue sets by the second set of control trials, or twenty trials in. Therefore, the first
20 trials of each experiment were eliminated for mapping purposes.
v) Minimal recruitment threshold testing (App. Ib): Due to the minimally-invasive
methods used to implant the intramuscular electrodes, we were not able to place the
electrodes directly on the motor point of the target muscles. So, although our
stimulation levels were up to ten times above twitch threshold for the pulse frequency
we used (Jung 2009), we did not always see recruitment at our lowest stimulation
current, 1.2 mA. To test for minimal recruitment threshold, averaged single-muscle
forces at 1.2 mA were tested for statistical difference from passive trials. For
experiments that did not pass threshold testing, single-muscle trials at 1.2 mA or dual-
muscle trials containing a muscle at 1.2 mA were relabeled as passive.
vi) Recruitment of antagonistic muscles (App. Ia): single-muscle stimulation data were
examined for “force reversal”, or a reversal in the force from a demonstrated linear
trend to detect recruitment of antagonistic muscles. The only stimulation setting that
revealed this tendency was the TA at 5.0 mA. These trials were removed from
single-muscle average force data for the TA, but kept in all other data sets.
Fig. 3. Graph plotting net force against TA current.
5.0 mA stimulation condition shows a reversal in a
demonstrated trend in linear, owed to and termed
"antagonistic muscle recruitment"
Tibialis Anterior
-0.20
-0.15
-0.10
-0.05
0.00
0.0 1.0 2.0 3.0 4.0 5.0 6.0
TA current (mA)
Netforce

10. Results
(Note—Any data reported but not displayed here is displayed in the Appendix)
I) EXPERIMENTAL APPARATUS GOALS—ANALYSIS BY HIGH-SPEED
CAMERA
A high-speed camera was employed on one of our trials to assess whether or not
we successfully constrained the ankle and the knee. The camera revealed that at high
GAS stimulations, the ankle lifted very slightly out of line with the rotational axis of the
force transducer, revealing a degree of foot shank compliance that had not been
anticipated. It is possible that this could have resulted in up and down oscillation of the
knee, but further examination by software is needed to confirm this. The knee, however,
remained steady on most trials and oscillated back and forth on a few trials. The
relationship between this behavior and stimulation and perturbation parameters could not
be discerned. To determine actual ankle angular displacement, frames at the top and
bottom of the force transducer’s oscillation were selected and measured. Unfortunately,
the standard of error for the methods used (± 5o
) was considerably greater than the
displacement we were attempting to measure (about 2o
).
II) AVERAGE STATIC FORCE
Single-muscle
Current
Average force of the GAS and TA calculated from single-muscle stimulation
trials (Fig. 4) showed a strong linear correlation with stimulation amplitude (TA, r2
= .95,
.87, GAS, r2
= .92 for both). One set of GAS stimulation trials was better described by a
quadratic model (r2
= .96). In the TA trials, the 5.0 mA stimulation condition was
removed because of non-linearity due to antagonistic muscle recruitment (the complete

11. Gastrocnemius
y = 0.0709x + 0.0095
R
2
= 0.9232
0.00
0.10
0.20
0.30
0.40
0.50
0 2 4 6
GAS current (mA)
Netforce
Tibialis Anterior
y = -0.0312x + 0.0107
R
2
= 0.8686
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0 1 2 3 4
TA current (mA)
Netforce
Fig. 4. Single-muscle stimulation trials plotting current amplitude against net ankle force. (a) is the GAS
while (b) is the TA, both well described by a linear relationship. (b) has the 5.0 mA stimulation condition
removed due to antagonistic recruitment.
data can be viewed in Appendix, Ia). For the GAS trials, the 1.2 mA stimulation
condition was removed because of non-linearity due to minimal threshold recruitment.
Trials here included all force transducer velocities.
Velocity
Force transducer velocity had no discernible relationship with average static
force. If differences existed, the data set was too small to detect the differences.
Dual-muscle
Current
Dual-muscle force (Fig. 5) was well predicted by a plane (r2
= .89, .84) based on
the difference between flexor and extensor stimulation.
F = -k1(TAstim) + k2(GASstim)
Where k1 and k2 are constants and TAstim and GASstim are current amplitudes.
III) IMPEDANCE
Single-muscle
Current
Current demonstrated a strong linear relationship with impedance (TA, r2
= .75, .
79, GAS, r2
= .86, .92) in single-muscle trials (Fig. 6). It is important to note that the TA
a b

12. Fig. 5. A 3-D plot of the TA and GAS
current against net ankle force. The
data is well-characterized by a plane,
reflecting to the linear nature of the
single-muscle stimulation trials. The
relationship between net force and TA
and GAS is described by the function of
the difference between the two control
variables. Figure does contain the 5.0
mA stimulation condition for the tibialis
anterior.
5.0 mA stimulation condition, excluded from average static force analysis due to
antagonistic muscle recruitment, was kept for this analysis. This is because of correlation
between impedance and net recruitment, and not net force. Additionally, it is important
to note that the GAS variance appears inflated due to the grouping of the 1.2 mA
stimulation condition as “passive.”
Tibialis Anterior
y = 0.0018x + 0.0079
R
2
= 0.7909
0
0.005
0.01
0.015
0.02
0.025
0 2 4 6
TA stim (mA)
Instantaneous
impedance
Gastrocnemius
y = 0.0112x + 0.0109
R
2
= 0.9197
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0 1 2 3 4 5 6
GAS current (mA)
Impedance
Fig. 6. Graphs plotting TA (a) and GAS (b) current amplitude against instantaneous impedance. Both are
well characterized by linear relationships, including the tibialis anterior (a), which displayed antagonistic
recruitment at the 5.0 mA stimulation condition .
a b

13. Average static force
Correlation of average static force with impedance (Fig. 7) in single-muscle trials
ranged from weak in TA trials (r2
= .55, .21) to strong in GAS trials (r2
= .82, .94). This
reflects antagonist recruitment in the TA at high stimulation, and mostly agonist
recruitment for all GAS stimulation amplitudes. Like the current vs. impedance data, all
stimulation conditions were kept in the data set.
Velocity
A relationship between velocity and impedance could not be confirmed, although
the TA velocity vs. impedance graph seems to suggest a trend towards increasing
impedance with speed.
Dual muscle
Current vs. impedance
In dual-muscle trials, current vs. impedance data (Fig. 8) could be described by a
plane (r2
= .91, .85) based on the interaction of summed extensor and flexor stimulation.
F = k1(TAstim) + k2(GASstim)
Fig. 7. Graph plotting net force production
against impedance in the tibialis anterior. Weak
correlation coefficient is explained by
antagonistic recruitment, which decreases net
force production and increases instantaneous
impedance. Points deviating most from the line
to the top are exclusively from the 5.0 mA
stimulation condition.
Tibialis Anterior
y = -0.0668x + 0.0127
R
2
= 0.2122
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
-0.2 -0.15 -0.1 -0.05 0 0.05
Net force
Impedance

14. Fig. 8. A 3-D plot of the TA and GAS current
against instantaneous impedance. The data is
well described by a plane, reflecting the linear
nature of the single muscle trials. The
relationship between TA and GAS current and
instantaneous impedance is described by a sum
of the variables, whereas the relationship of the
said variables to net force was based upon the
difference between the variables.
Discussion
Our hypotheses that (i) ankle impedance and (ii) net ankle torque could be
described by linear functions of current to the ankle flexors (TA) and extensors (GAS) in
both single and dual-stimulation conditions were supported by the results of this study,
while the role of velocity in ankle impedance and net torque could not be verified. The
relationship between joint impedance and torque fit predictions described by a simple
pre-stressed two-spring model. The role of velocity matched expectations for average
static torque at the ankle, but impedance was not shown to respond as would be expected
by the force-velocity properties of the Hill model of muscle.
Ankle impedance, torque, and a pre-stressed two-spring model
A simple pre-stressed two-spring model can be used to predict joint torque and
impedance behavior. In work by Thales Souza (2009), a pre-stressed two-spring model
was successfully used to describe the behavior of passive properties in the human ankle.
Here, we adapted the same model to predict behavior in an active stimulation setting (Fig.
9).

15. Fig. 9. A drawing of the pre-stressed two-spring model
of the joint, used to predict torque and impedance
behavior at a two-muscle joint. F1 and F2 represent the
force generated by their respective springs (muscles). y
is the change in muscle length resulting from a position
change in the shank (foot shank) extending below the
fulcrum from T0 to T1 or T2, which reflect positions of
maximum and minimum torque.
In this figure, there are two springs (muscles) on either side of a "see-saw"
(muscle moment arms about a joint axis), while a third projection (in our case, the foot
shank) extends out below. If we ignore changes in l1 and l2,net torque T will be
proportional to the difference between the force generated by the right (F1)and left (F2)
springs.
T F1 – F2
This is intuitive and well recognized, and, indeed, our net torque model for dual-
stimulation trials follow this format, with F1 representing the GAS and F2 representing
the TA.
Impedance is less intuitive, and requires a longer derivation. The basic formula for
impedance, as described by Dudek (2006), is given by the equation,
Z = (Fmax-Fmin)/(xmax-xmin)
So, with respect to our model, let us consider a force maximum (T1) and force minimum
(T2), each occurring a distance ± x from a point T, directly resulting in a length change ± y
in the springs.. So, in our model, impedance will be given by

16. Imp = (T1 – T2)/2x
T1 and T2 will be given by the difference of the opposing springs at each displacement
T1 = F1' – F2' T2 = F1" – F2"
And the force of each spring at T1 and T2 will be given by
F1' = k1(x1 – y) F1" = k1(x1 + y)
F2' = k2(x2 + y) F2" = k2(x2 – y)
Substitute these into our impedance equation
Imp = [(F1' – F2') – (F1" – F2")]/2x
rearrange,
Imp = [(F1' – F1") + (F2" – F2')]/2x
substitute again and simplify
Imp = {[k1(x1 – y) – k1(x1 + y)] +[k2(x2 – y) – k2(x2 + y)]}/2y
Imp = –2y(k1 – k2)/2x
Assuming both lever arm displacement x and spring length change y are directly related
and constant from trial to trial, then
Imp = – (k1 – k2)
Imp = k1 + k2
and you have an equation which predicts that impedance in a two-muscle system will be
determined by the sum of the spring constants, which, in our case, is the stimulation
amplitude. This behavior describes the relationship seen in our 3D dual-muscle current
vs. impedance graphs (Fig. 8), and also describes the poor relationship observed between
static force and impedance in the single-muscle TA trials (Fig. 7), which displayed
antagonistic recruitment in our controls. However, with the same data in the impedance

17. vs. current graph, the TA yields linear behavior, further emphasizing the point that
impedance can be predicted by the summed stimulation, or recruitment about the joint.
Velocity, torque, and impedance
A relationship between velocity, torque, and impedance could not be discerned.
Through the lens of the Hill model of muscle, we would expect velocity and net torque in
our experiment to be unrelated considering our methods. In our experiment, we collected
average force data by oscillating the foot shank back and forth at a variety of speeds, and
Fig. 10. A force-velocity curve, used to describe the expected relationship between velocity, impedance,
and torque. Velocity was not expected to affect the average force values, since differences in eccentric and
concentric force production would be averaged out. The curve also predicts that impedance will increase
with respect to velocity, although this could not be verified in our data.
then averaged the last 0.5 s of data. On a force-velocity curve, what we did may have
looked like Fig. 10. Given the low velocities at work (maximal angular velocities were
100-400 times less than in vivo ankle joint velocities, calculated from position data in
Varejao, 2002), it is probable that the displacements here were traveling along the small,
linear region on either side of the isometric axis. Since the forces were averaged, any
effect of velocity on net torque would be averaged out.
However, force-velocity relationships do predict an effect on impedance. As
speeds increase, the difference between the maximum eccentric force exerted and lowest
concentric force exerted would grow. Additionally, this would create more velocity

18. dependent character, and increase our damping constant. The force-velocity data could
not be discerned from our impedance graphs, but it may be detectable in our
damping/stiffness data (not analyzed).
Implications of linear relationship between stimulation current and avg. force,
impedance
Since a plane could be fitted to dual-muscle stimulation trials to predict force
output (Fig. 5), there were a range of different flexor/extensor stimulation patterns that
yield the same net torque, as given by the intersection of an xy plane level with the torque
on the z axis with this plane. Take that line and impose it upon the dual-muscle
impedance graph, and you have the range of impedances possible for that torque. This
will be important so that individuals with this controller can adjust leg stiffness
depending on terrain, while still executing the necessary torque for locomotion.
Controller with adjustable leg stiffness still has much work to go before it is realized
(limb-level stiffness vs. joint level stiffness), but this experiment seems to suggest that a
key component of this puzzle may have a simple solution.
Conclusion
The results of this experiment indicate that while net ankle torque in the
laboratory rat is predicted by the difference between extensor and flexor stimulation,
ankle impedance is predicted by the summed stimulation (recruitment) about the joint.
These findings are supported by the pre-stressed two-spring joint model, which suggests
that, in spite of protocol limitations, the experiment succeeded in establishing general
relationships between impedance, torque, and flexor and extensor stimulation amplitude.

19. Limitations
In our experiment, our stimulation protocol and impedance device were sufficient
to characterize relationships between stimulation, torque, and impedance. However,
there are several issues with both our stimulation protocol and impedance device that
need to be addressed in order to get repeatable data for use in functional electrical
stimulation, which is the ultimate goal.
First, a couple of changes to the experimental rig need to be made in order to
control knee angle and introduce knee constraint. The revised rig ought to be as drawn
below.
Fig. 11. A rough schematic of a future rat-lever orientation, which will facilitate knee-angle control and
constraint.
In this configuration, a pin runs from the force transducer up to another lever, to which
the rat’s leg is fixed. The stimulated leg is now facing the user, making the angle of the
knee measurable, and moving the rat vertically from the lever gives a chance to introduce
knee constraint, perhaps by velcroing the thigh to a pummel horse. However, since the
force transducer will be less mobile (now attached to another fixed lever, the rat position
will now have to be adjustable up/down as well.
Next, the displacement of the ankle must be known. Although this could be
accomplished in the current rig, it requires writing computer software that tracks the

20. markings about the rat’s ankle. These displacements will need to be averaged and
considered for the impedance calculations.
Lastly, the stimulation protocol has to yield consistent results from trial to trial.
To do this, the rodent-model stimulation protocol outlined by Jung would be appropriate.
Following this protocol, it will also be possible to better characterize results at low-level
stimulations, which were not well-characterized here.
Future directions
The basic pre-stressed two-spring joint model outlined above has shown to be
useful for basic behavioral predictions at the ankle joint. It can be integrated with active
muscle force-length characteristics (variable x) and a pre-stressed two-spring model for
passive behavior to yield more realistic joint level behavior. This model should be
developed and tested experimentally.

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22. APPENDIX
I.) CONTROLS
a. Antagonistic recruitment
b. GAS and TA threshold stimulation test
c. Electrode stability and steady state fatigue
d. Erratic recruitment
II.) AVERAGE STATIC FORCE
a. Single-muscle
i. Vs. current
ii. Vs. velocity
b. Dual-muscle vs. current
III.) IMPEDANCE
a. Single-muscle
i. Vs. current
ii. Vs. static force
iii. Vs. Velocity
b. Dual-muscle vs. current
I.) CONTROLS
a. Antagonistic recruitment
Antagonistic recuitment: TA, 1/30
-0.16
-0.14
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0
0 1 2 3 4 5 6
TA current (mA)
Netforce
Series1
Antagonistic recruitment: TA, 10/23
y = -0.0156x - 0.0095
R2
= 0.4314
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0 1 2 3 4 5 6
TA current (mA)
Netforce
Net force
Linear (Net force)
b. GAS and TA threshold stimulation test
1/30 Threshold stimulation test: GAS
-0.004
-0.002
0
0.002
0.004
0.006
1
Passive vs. 1.2 mA
Forceaverage
Passive
1.2 mA
1/30 Threshold stimulation test: TA
-0.09
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
1
Passive vs. 1.2 mA
Forceaverage
Passive
1.2 mA

23. 10/23 Threshold stimulation test: GAS
-0.008
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
0.008
1
Passive vs 1.2 mA
Forceaverage
Passive
1.2 mA
10/23 Threshold stimulation test: TA
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
1
Passive vs 1.2 mA
Forceaverage
Passive
1.2 mA
c. Electrode stability and steady state fatigue
Gastrocnemius
0
0.1
0.2
0.3
0.4
0.5
0 2 4 6 8 10
Control trial number
Force
TA control trials
-0.1
-0.08
-0.06
-0.04
-0.02
0
0 2 4 6 8
Trial number
Force
TA 2.5 GAS 0.0
d. Erratic recruitment (Bad/Good)
II.) AVERAGE STATIC FORCE
a. Single-muscle
i. Vs. current
SM trial: TA (5.0 rmvd)
y = -0.0279x
R2
= 0.9537
-0.14
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0 1 2 3 4
TA current (mA)
Netforce
Net force
Linear (Net force)
SM Trials: GAS
y = 0.0909x - 0.0152
R2
= 0.9226
y = 0.0134x2
+ 0.0317x + 0.0045
R2
= 0.9642
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 2 4 6
GAS current (mA)
Netforce
Net Force
Linear (Net Force)
Poly. (Net Force)

24. Gastrocnemius
y = 0.0709x + 0.0095
R
2
= 0.9232
0.00
0.10
0.20
0.30
0.40
0.50
0 2 4 6
GAS current (mA)
Netforce
Tibialis Anterior
y = -0.0312x + 0.0107
R
2
= 0.8686
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0 1 2 3 4
TA current (mA)
Netforce
ii. Vs. velocity
Effect of velocity on GAS force
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 1 2 3 4 5 6
GAS current (mA)
Netforce
1.0 Hz
3.2 Hz
5.5 Hz
7.8 Hz
10.0 Hz
Effect of velocity on TA force production
-0.16
-0.14
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0
0 1 2 3 4 5 6
TA current (mA)
Netforce
1.0 Hz
3.2 Hz
5.5 Hz
7.8 Hz
10.0 Hz
Effect of Velocity of TA Force
-0.16
-0.14
-0.12
-0.10
-0.08
-0.06
-0.04
-0.02
0.00
0.02
0 1 2 3 4 5 6
TA current (mA)
Netforce
1 Hz
3.2 Hz
5.5 Hz
7.8 Hz
10.0 Hz
Effect of Velocity of GAS force production
-0.10
0.00
0.10
0.20
0.30
0.40
0.50
0 1 2 3 4 5 6
GAS current (mA)
Netforce
1 Hz
3.2 Hz
5.5 Hz
7.8 Hz
10 Hz
b. Dual-muscle vs. current

25. III.) IMPEDANCE
a. Single-muscle
iii. Vs. current
Tibialis Anterior
y = 0.0018x + 0.0079
R
2
= 0.7909
0
0.005
0.01
0.015
0.02
0.025
0 2 4 6
TA stim (mA)
Instantaneous
impedance Gastrocnemius
y = 0.0112x + 0.0109
R
2
= 0.9197
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0 1 2 3 4 5 6
GAS current (mA)
Impedance
TA SM trials: Current vs. Impedance
y = 0.003x + 0.0086
R2
= 0.7536
0.00
0.01
0.01
0.02
0.02
0.03
0.03
0.04
0 2 4 6
TA current (mA)
Impedance
Net force
Linear (Net force)
GAS SM Trials: Current vs. Impedance
y = 0.0074x + 0.0135
R2
= 0.8592
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0 2 4 6
GAS current (mA)
Impedance
TA 0.0
Linear (TA 0.0)
iv. Vs. static force
TA SM Trials: Net Force vs. Impedance
y = -0.0619x + 0.0085
R2
= 0.5522
0
0.005
0.01
0.015
0.02
0.025
-0.2 -0.15 -0.1 -0.05 0 0.05
Net Force
Instantaneousimpedance
GAS 0.0
Linear (GAS 0.0)
GAS SM Trials: Net Force vs. Impedance
y = 0.1116x + 0.0148
R
2
= 0.8219
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
-0.2 0 0.2 0.4 0.6 0.8
Net force
Impedance
TA 0.0
Linear (TA 0.0)
Tibialis Anterior
y = -0.0668x + 0.0127
R
2
= 0.2122
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
-0.2 -0.15 -0.1 -0.05 0 0.05
Net force
Impedance
GAS SM Trials: Net Force vs. Impedance
y = 0.1049x + 0.0124
R2
= 0.9417
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
-0.2 0 0.2 0.4 0.6
Net force
Impedance
TA 0.0
Linear (TA 0.0)

26. v. Vs. velocity
Effect of velocity on TA impedance
y = 0.0016x + 0.0083
5.5Hz R2
= 0.513
y = 0.0006x + 0.0089
1.0Hz R2
= 0.7571
y = 0.0023x + 0.0071
3.2Hz R2
= 0.8941
y = 0.0018x + 0.0082
7.8Hz R2
= 0.8695
y = 0.0025x + 0.0053
10 Hz R2
= 0.8559
0
0.005
0.01
0.015
0.02
0.025
0 2 4 6
TA current (mA)
Impedance
1.0 Hz
3.2 Hz
5.5 Hz
10.0 Hz
7.8 Hz
Linear (5.5 Hz)
Linear (1.0 Hz)
Linear (3.2 Hz)
Linear (7.8 Hz)
Linear (10.0 Hz)
b. Dual-muscle vs. current