Massachusetts Institute of Technology
Department of Electrical Engineering and Computer Science
Thesis in Partial Fulfillment of the Requirements for the Degree of Master of Science
Title: Computer Simulation of the neuromuscular reaction to electrical
stimulation of the spinal cord of a spinalized frog .
Submitted by: Ari Wilkenfeld
34 The Fenway
Boston, MA 02215
Date of Submission:
Work done in the Eric P and Evelyn E Newman Laboratory for Biomechanics and Human
Rehabilitation with partial support from a grant by the National Science Foundation
@ 1997 Massachusetts Institute of Technology. All rights reserved.
(Afied by Neville Hogan, lrof.Mechanical
Engineering and Brain and Cognitive Science
Thesis AdYisor
_
Accepted by Arthur C. Smt<Chairman, Committee on Graduate Students
Department of Electrical Engineering and Computer
Science
OCT 2 9199
For L.F. and the people who keep him out of trouble
Table of Contents
Page
Abstract
4
Chapter 1: Introduction
5
Chapter 2: Theoretical Background of the Equilibrium Point Hypothesis
9
a. Alpha and Lambda Hypothesis
9
b. The Equilibrium Point Hypothesis in Motor Control
15
Chapter 3: Experimental data available and data processing
18
a. Experimental Data Available
18
b. Processing of Force Data
22
Chapter 4: Testing Potential Models
37
a. Modeling Goal
37
b. Possible Models
37
c. Testing Model one
38
d. Testing Model two
42
Chapter 5: Future Work, Discussion
47
a. Future Work
47
b. Discussion
50
References
51
Appendix 1: Verification of the software for integration of forward dynamics
53
Appendix 2: Figures
56
Title: Computer Simulation of the neuromuscular reaction to electrical
stimulation of the spinal cord of a spinalized frog
Submitted by: Ari Wilkenfeld, September 1997
Thesis in Partial Fulfillment of the Requirements for the Degree of Master of Science for
the Department of Electrical Engineering and Computer Science,
Massachusetts Institute of Technology
@ 1997 Massachusetts Institute of Technology. All rights reserved.
Abstract
This thesis presents models of the mechanical behavior of a frog limb when the
lumbar region of the spinal cord is activated using electrical stimulation. Data was
obtained from the electrical stimulation of several spinalized preparations. Isometric data
was available from measuring the force at the ankle while the limb was held isometrically
with a force transducer and the spinal cord stimulated. Free limb trajectories were also
available using the same electrical stimulation patterns and attaching the frog's ankle to a
small two link mechanism with the ability to sense position.
Possible models were considered based on the equilibrium point hypothesis- the
idea that movements are planned as a series of stable postures with some impedance
between the motion of the limb and the trajectory of equilibrium postures- the so-called
"virtual trajectory". Two possible classes of models were eliminated. Any model
consisting solely of an elastic element and a moving mechanical equilibrium point was
eliminated by comparing isometric forces on the limb to inertial forces required to move it
along its actual trajectory. Any model relying solely on a viscous element between the
limb and the ground frame for energy dissipation was eliminated by noting that such an
element would have to be active for some portion of the motions . This was determined
by comparing the isometric forces measured with the forces required to actually move the
limb and noting the difference. For these purposes, the frog leg was modeled as a two link
manipulandum and dynamic equations were formulated. Future work is suggested.
Supervised by: Neville Hogan, Prof. of Mechanical
Engineering and Brain and Cognitive
Science
Chapter 1: Introduction
A classic problem in neural control of movement is to determine at what level
movements are controlled. It is generally assumed that there is some hierarchy in this
process. The brain is the highest level, planning out the movement based on primarily
visual data. It is possible that all calculations are in fact done at this level with the brain
explicitly calculating necessary joint torques from required hand accelerations (via inverse
dynamics) and then firing muscles accordingly via alpha motor neurons, taking all the
nonlinear properties of muscle into account. This is unlikely for a variety of reasons,
among them the fact that small errors in the calculations, in the brain's knowledge of the
system properties (limb inertias, etc.), or from external perturbation, could compromise
performance. For example, it can be shown that for a movement starting and ending with
zero velocity, infinite precision is needed if pure torque control is utilized. From a
computational standpoint, the inverse dynamic calculations required for a multi-joint
system are also extremely complex (Hogan, 1997).
The "equilibrium point hypothesis" (actually a family of related theories) states in
essence that some responsibility for motion is relegated to lower in the system. The
theory states that movement is generated by commanding a series of postures, which is
then translated into firing commands (Feldman, 1966). Each of these postures is stable
due to the apparent elastic and viscous nature of muscle and/or the effects of spinal
feedback circuitry.
Experimentally, it has been confirmed by Emilio Bizzi and colleagues
that microstimulation of the gray matter of the lumbar spinal cord of a decerebrated frog
preparation will in the majority of cases cause the hind limb to converge to a
stable posture within the frog's reach. This leads to the possibility that the equilibrium
point phenomenon is hardwired into the gray matter of the spinal cord.
A further result noted by researchers (Mussa-Ivaldi, 1990, Lemay, 1997) was that
the force measured at a point in the frog's workspace when two points in the spinal cord
were simultaneously stimulated was the vector sum of the force produced by each site
stimulated alone. It was also observed that the relative contribution of the two force
vectors could be scaled by scaling the relative stimulation of the two electrode sites.
The motivation behind this thesis's research was to develop a competent computer
model to allow prediction of the motion of a frog's leg due to spinal stimulation. The
ultimate goal of such a simulation, aside from purely scientific interest, is for application to
human neural prosthetics.
Current rehabilitation technology allows for Functional Electrical Stimulation
(FES) of muscles in spinal cord injured patients to restore functionality to areas no longer
receiving descending neural commands. There are, however, several problems limiting
clinical practicality of the technology, among them a) it requires 48 implanted electrodes
to achieve something approaching functional gait, b) the control algorithm is extremely
complex and c) because the muscles are not generally recruited in the most efficient
possible manner, fatigue results very quickly (Chizeck, 1992).
An FES system using electrodes implanted in the spinal cord could more simply
drive the limb to an equilibrium position. Fewer electrodes would probably be needed and
the control could be extremely simplified compared to current strategies since individual
muscles would not need to be innervated by separate electrodes.
Theoretically, any posture in the limb's workspace could be reached by stimulating
the combination of spinal sites whose force vectors sum up to produce a stable equilibrium
point at the desired location. By stimulating multiple sites in the spinal cord and
controlling the magnitude of stimulation, one could conceivably move the limb stably from
one posture to another. These moves could then be used to build up complex motions
such as locomotion.
The specific goal of this thesis was to develop a model that would competently
reproduce trajectory data of a frog hind leg under electrical stimulation of the lumbar
spinal cord given measurements of the isometric force when the ankle was held at different
spots under that same stimulation. Such a model would be useful for prediction of motion
given different combinations of multiple electrodes stimulating simultaneously. It would
eliminate the need to run an experiment to determine the motion obtained from each
combination of stimulus levels in a multiple electrode system. This would facilitate
investigation of possible limb control strategies using this technique.
In order to produce such a model, it is necessary to make assumptions about the
specific structure of the system made up of the limb and the equilibrium point. The major
contribution of this work is to evaluate some of the common assumptions in the literature
regarding such a system. Specifically, the competence of models based on dissipating
energy by modeling a damper between the limb and ground is investigated at some length.
The major conclusion of this work is that this class of models is not tenable. For it
to be a viable model would require violating the properties of damping elements. This
casts doubt on previous work in the field which assumed a model of this structure.
In chapter two, a brief summary of the relevant literature regarding motion control
based on attractor trajectories is presented. Chapter three describes the experimental data
available for this work as well as the processing algorithms used. Chapter four reviews
the results of testing the competence of the two simplest models- the ones which most
easily lend themselves to parametrization based on isometric data. Chapter five discusses
the results and suggests future work to test more complex models.
Chapter 2: Theoretical Background of the Equilibrium Point Hypothesis
Alpha and Lambda Hypothesis
The theory that that the spring-like properties of muscles led to stable postures
which could be used for motion control was first put forth by Feldman and colleagues
(Feldman, 1966, 1986), who formulated it in a version now known in the literature as the
'lambda hypothesis'.
The lambda hypothesis relies heavily on the reflex loop to generate the equilibrium
phenomenon. In essence, it states that a) the stretch reflex threshold is an independent
measure of central commands descending to alpha and gamma motor-neurons b) the
nervous system uses the stretch reflex to perform movements and c) muscle together with
reflex and central control mechanisms behave like a nonlinear spring, the zero length of
which is a controllable parameter. (Feldman, 1986). When the zero lengths of agonist and
antagonist muscles (and the associated reflexes) about the same joint are changed, the
force-length relation of each muscle is shifted and the position of the joint at which they
balance is changed (see figure 1).
Two major criticisms of this model have been cited. First, the inherent
propagation delays in the reflex loop would likely lead to instability if the gain of the reflex
loop was sufficient to produce the impedance commonly observed, and second,
experiments done with deafferented animals have shown that they are still capable of
stable motions.
a
c
..L
S
length
x
Figure 1: The 'lambda hypothesis': (Taken from Feldman, 1986). The Torque vs. length curves show
that torque produced by an agonist or antagonist at a given position can be modulated by shifting the rest
length of the muscle. The experimental subject was able to move one joint, which was connected to a
servo motor. The limb was placed at a given position and held against a constant torque. The torque was
then suddenly reduced. The subject was told not to intervene, and the final angle of the limb was
measured. Each curve represents a series of experiments with the limb starting at the black dot and
moving to the different open dots as the torque was lowered by different amounts. All torque length
curves were found to be repeatable and further to have the same shape (referred to as an "invariant
characteristic") even when shifted. The dotted line represents passive muscle forces. The point where one
curve meets the curve of passive forces allows the determination of Xon the scale of muscle lengths
An alternative hypothesis, referred to by Feldman as the 'alpha hypothesis' was
developed by Emilio Bizzi and colleagues (Polit, 1979). This theory states that
equilibrium can be shifted by altering the impedance of the muscle directly via the alphamotor-neurons. This is a result of impedance (apparent stiffness and viscosity) changing
with the level of activation. Muscles are arranged about joints in agonist/antagonist pairs.
By changing the slope of the force-length curves of agonists and antagonists, the system
can change the position of the joint at which the forces balance. Likewise, by cocontracting both agonist and antagonist, the same posture can be maintained while
increasing stiffness about the joint (see figure 2). Critics of the 'Alpha' model generally
point to the fact that it makes no direct prediction about EMG activity, therefore making it
less based in physiology than the 'Lambda' model.
CanaPaWoos
W tV. Angle U
0o
B
anragonzs
1o
Angle 0
Figure 2: The 'alpha hypothesis': (taken from Won, 1983) The torque vs. length curves show (in a
simplified version) that torque produced by an agonist or antagonist at a given length can be modulated by
increasing or decreasing the stiffness of the muscle. Agonist curves start on the left. Antagonist curves
start on the right and are made positive so their point of intersection with the agonist curve can be seen.
A: By changing the stiffnesses of the opposing muscles (from the solid lines to the dotted lines) one can
change the equilibrium position about the joint. B: By changing the stiffnesses of the opposing muscles
(from the solid lines to the dotted lines) one can keep the same position and change the stiffness about the
joint
The debate between 'alpha' and 'lambda' has not yet been settled. A distinct
possibility is that the truth lies somewhere in the middle, with impedance modulated
equilibrium being fine tuned or 'conditioned' by afferent feedback (Hogan, 1997).
It was originally assumed that motion could be performed by instantly shifting the
equilibrium point and having the limb follow as soon as its dynamics would allow. An
experiment performed by Bizzi, Hogan, and colleagues (Bizzi, 1984) showed that it was
more likely that there was in fact a continuous transition of the equilibrium point from
start to end position.
The experiment consisted of training a monkey to point at a series of illuminated
panels (see figure 3 for experimental setup). The animal's upper arm was immobilized and
its lower arm attached to a torque motor and position sensor. A cover prevented the
animal from seeing its own arm. The animal was trained to point at a light until it went
out and another lit, at which point it was to shift its arm to point to the new target.
Figure 3: Experimental setup for monkey experiment (taken from Polit, 1979)
After it had been trained, the animal was deafferented, depriving it of
proprioceptive feedback from the upper limb. The experiment was then repeated. In a
variant of this experiment, however, the arm was servoed to the second target while the
first target was still lit and held there. The animal was unaware of this due to the lack of
afferent feedback. The second light was then lit, and the arm released.
Surprisingly, the arm started to move back towards the first target before turning
around and finishing at the second target. This data was interpreted to mean that the
equilibrium position was still moving from the first to the second target at the time the
limb was released from the second target. From this experiment came the concept of the
"virtual trajectory" or "equilibrium point trajectory", the time history of the attractor point
during a motion.
The Equilibrium Point Hypothesis in Motor Control
The Equilibrium Point hypothesis is a highly controversial theory. One of the
major criticisms is that there is no recordable physiological phenomenon related to the
proposed "Equilibrium Point" that attracts the limb, so it is impossible to either prove or
disprove as a hypothesis (Smith, 1991). However, work by Bizzi and colleagues (Giszter,
1993) shows that the limb at least demonstrates stability properties which must be
explained in some manner.
One test that has been applied to test the theory is to examine the "virtual" and
"actual" trajectories for the same movement and to see which is more complex. It is
generally assumed that if the theory is valid, the virtual trajectory should be simpler than
the actual trajectory, which will be modified by the dynamic effects created by moving the
limb. If the virtual trajectory were in fact more complex, then there would be no
computational advantage to using it rather than computing the inverse dynamics for the
move directly (Hodgson, 1994, Uno, 1993).
The typical approach to applying this test has been to model the impedance
between the attractor point and the limb and between the limb and ground. This has been
done in both the single joint (Hogan, 1984) and multi-joint (Flash 1987) case by assuming
a model with an elastic element between the limb and the attractor point and a viscous
element between the limb and the ground reference frame. Both concluded that there was
good evidence that actual motions could be reproduced using relatively simple virtual
trajectories.
These results, however have been widely challenged. Kawato and colleagues
(Katayama, 1993) cite the unnaturally high values for the elastic element in the model used
by Flash. They instead used the empirically measured values for dynamic stiffness
reported by Bennett and colleagues (Benett, 1991) and calculated a virtual trajectory far
more complex than the actual motion. Gribble and colleagues countered that if a more
complex and physiologically plausible model of muscle and its associated neural anatomy
was used, including afferent feedback, a simple virtual trajectory control signal could be
obtained from the same data (Gribble).
Hodgson attempted to estimate the path of the virtual trajectory without modelbased assumptions by providing the necessary inertial forces to move the limb with a small
robotic arm. This meant that no force was being applied by the impedance element
between the attractor point and the limb and the path of the limb tracked that of the
attractor point. He found that in general the path of the equilibrium point was no simpler
than that of the limb itself. However, his estimates of the profile of the virtual trajectory
were also very different from those calculated by Kawato et al (Hodgson, 1994).
Different sources have brought forward evidence both for and against the
assumptions of high stiffness in motion. It is not reviewed in detail in this work since it
will be shown in chapter four that the structure of the model is untenable regardless of
parameters.
In a later work, Won (Won, 1993) attempted a similar modeling based on a virtual
trajectory with both an elastic and damping element between the virtual and actual
trajectories. Chapter five will briefly the difficulties in obtaining the virtual trajectory
needed to predict motion from such a model from isometric data.
It should be noted once again that the use of attractor trajectories for motor
control is still highly debated. Kawato and colleagues (Uno, 1989) have proposed an
alternate hypothesis based on the minimization of torque change over the course of the
movement. This would make an equilibrium based model unnecessary as there would be
no computational advantage if the torques had to be directly computed.
Lackner and Dizio (Lackner, 1994) have brought a significant objection to the
entire notion of the equilibrium point hypothesis by noting that when placed in a spinning
room, subjects attempting to reach a point will end up with a steady state error which does
not appear when the room is not rotating. This should not be the case in an equilibrium
point model as the Coriolis force driving the limb away from its target should go to zero at
steady state when the limb is not moving'
In this work, it is generally assumed that there is some validity to the equilibrium
hypothesis and concentration is placed on what specific form of the model will give the
best convergence with observed data.
1 Coriolis forces are proportional to the product of the velocity of the moving object and the angular
velocity of the reference frame in which they are moving. At steady state, the velocity of the limb goes to
zero, as should the Coriolis force
Chapter 3: Experimental data available and data processing
Experimental Data Available
Experimentally, it has been shown by Emilio Bizzi and colleagues (Giszter, 1993),
that microstimulation of the lumbar spinal cord of a decerebrated frog preparation will in
the majority of cases cause the hind limb to converge to a stable posture within the frog's
reach. It has been shown that it is extremely unlikely that this tendency of the limb to
develop a stable point of equilibrium upon stimulation is random, but rather that it is likely
an integral part of the neural control system (Loeb, 1995, Giszter, 1993, Bizzi, 1995).
Several experiments were undertaken to determine the location of these
equilibrium points. The nature of the experiment was to clamp the ankle of the frog in a
fixed location with a force transducer. The spinal cord was then stimulated electrically
with a pulse train and the isometric force measured in both the x and y (orthogonal
horizontal axis) directions. This was repeated at nine locations within the frog's
workspace (see figure 4 (Giszter, 1993)).
INTRmPOLATD FIED
ANMEQILIEIUM
,KmSUI~
X Y
A
*
*
*
Tmo
KInIoEKCTRnO
Tn
di.
%
FIELD IN RELATION
TO FROG BODY
r"
STICK rFIRE
DIAGRAM OF LEG
AN FROG BODY
APPARATUS
m
Wrmulb
Figure 4- Experimental Setup (taken from Bizzi, 1995) A: Orientation of frog preparation B: Measured
force vectors C: Force vectors interpolated between measured locations and calculated location of
equilibrium point D: Field in relation to frog body and stick diagram of leg and frog body E: Experimental
setup- Note that our setup, unlike his, did not include an electromyogram recorder
Forces between measured points were linearly interpolated and the equilibrium
point was defined to be the point at which both the x and y forces were zero. Giszter
reported that as stimulation proceeded, the equilibrium point moved smoothly from
it's prestimulation value to a spot within the reach of the frog. This motion of the
equilibrium point during the course of (and after) stimulation was referred to as an
equilibrium trajectory. He further reported that when the limb was released (i.e., not
held isometrically) it's actual trajectory generally followed this 'virtual trajectory'.
Experiments similar to those reported in (Bizzi, 1995) are currently being repeated
by Dr. Michel Lemay in the Newman laboratory. Each frog is spinalized. Microelectrodes
are then inserted into the lumbar region of the spinal cord. The time varying isometric
field of forces is then measured using the method described above. Several free motion
trajectories are also recorded for each frog. A typical field for a frog is shown in figure 5
(Lemay, 1996) The field shown is the maximum force at each location over the course of
stimulation 2 (and linearly interpolated points in between in space). Three trajectories
when the limb is released from different spots are also shown. The vectors represent the
magnitudes and directions of the forces. The three curves represent measured free limb
trajectories.
2On
average, the maximum force at each point was reached 0.45 seconds after onset of stimulation. The
standard deviation of this time over the set of measured points was 0.17 seconds
;
\
.
I
40
\
Newton
60
\
\
\
\
\
I
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IN
I
'Ilk
E
E
'1.11-4
Il
0
CL
8 80
100
-.
192
0
20
~---
40
60
80
100
x position (mm)
Figure 5: A typical force field for a frog. The field shown is the maximum force at each location over the
course of stimulation (and linearly interpolated points in between in space). The vectors represent the
magnitudes and directions of the forces. The three curves represent measured free limb trajectories
(starting at the x's). The asterisk is the Equilibrium Point (where the x and y forces would go to zero)
Seventeen trajectories were available, measured from two frog preparations. The
relevant parameters for each are shown in Table 1. The force data was measured at 50 Hz
with an ATI Industrial Automation force transducer with a precision of 0.02N. The
trajectories were measured by connecting the limb to a small, 2 link mechanism with no
actuators but the ability to measure its position based on 1% accuracy linear Bourns
potentiometers at the joints . Position data was gathered at 997 Hz.
Table 1: Trajectory information
Trajectory
Number
Frog Number
Force Field
Number
Date
Stimulation
pulse width
Stimulation
pulse
amplitude
Stimulation
Frequency
Length of
Stimulation
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
ffl4
ff14
ffl6
ffl6
ff16
ff16
ff16
ffl6
ff17
ff17
ffl7
ffl7
ffl7
ff17
ff22
ff22
ff22
5/27/96
5/27/96
5/29/96
5/29/96
5/29/96
5/29/96
5/29/96
5/29/96
5/30/96
5/30/96
5/30/96
5/30/96
5/30/96
5/30/96
6/10/96
6/10/96
6/10/96
variable
variable
0.5 msec
0.5 msec
0.5 msec
0.5 msec
0.5 msec
0.5 msec
0.5 msec
0.5 msec
0.5 msec
0.5 msec
0.5 msec
0.5 msec
0.5 msec
0.5 msec
0.5 msec
5gA
5tA
5tA
5gA
5tpA
5gA
5pA
5pA
5gA
5gA
6pA
6gA
7pA
7.tA
5pA
5.tA
5.tA
40 Hz
40 Hz
10 Hz
10 Hz
10 Hz
10 Hz
10 Hz
10 Hz
40 Hz
40 Hz
40 Hz
40 Hz
40 Hz
40 Hz
40 Hz
40 Hz
40 Hz
300 msec
300 msec
500 msec
500 msec
500 msec
400 msec
400 msec
400 msec
300 msec
300 msec
300 msec
300 msec
300 msec
300 msec
400 msec
400 msec
400 msec
Processing of Force Data
Several different types of forces need to be distinguished in these experiments.
Those directly measured are isometric forces at a given location at sampled intervals of
time. The software used for this thesis interpolates linearly between these data points in
both space and time, giving an approximation of what forces the limb would be exerting if
held at a given position in space at any given time (i.e., Fisometric(space,time)).
From these forces, it is possible to calculate the equilibrium point at any given
time, as the location at which the force in both the x and y direction go to zero. This
defines an equilibrium trajectory as a function of time which can be compared to the actual
motion of the limb when released.
Both the trajectories and the measured forces were in general extremely repeatable
over multiple trials for a given frog for a given set of conditions. Figure 6 shows three
trajectories measured using nominally the same starting point on the same frog with the
same stimulation parameters. Part A shows the trajectories separately. Part B shows the
extreme points for each time sample. The dashed lines connect points from the same time
sample. Part C shows the distance over time of between the two extremes shown in part
B. The sum of this distance at each time point divided by the number of time points yields
an average distance of approximately 2.4 mm.
Figure seven shows multiple measurements of the time course of the force
magnitude using the same location for the same frog with the same stimulation
parameters3 . The sum of the difference between the minimum and maximum force at each
time point, divided by the number of time points, yields an average range of approximately
0.06 Newtons.
3 'Force Units' are in hundredths of a a Newton
80
S85
E0
0.
> 90
95
"II(
5
50
55
60
50
55
60
65
70
75
80
85
90
65
70
75
80
85
90
x position (mm)
B
-E85
0
S90
95
'II
45
x position (mm)
M
0
0.2
0.4
0.6
0.8
Time(sec)
1
1.2
1.4
1.6
Figure 6: Three trajectories measured using the same starting point on the same frog with the same
stimulation parameters (starting at the x's). Part A shows the trajectories separately. Part B shows the
extreme points for each point in time. The dashed lines connect points from the same time sample. Part C
shows the distance over time of between the two extremes shown in part B. On average, there is a
difference of approximately 2.4 mm..
40
(o
30
E 20
E
0
>
10
0
0.0
0.2
0.4
0.6
0.8
1.0
time (sec)
Figure 7: Multiple measurements of the time course of the force magnitude using the same location for
the same frog with the same stimulation parameters.
In order to get velocity and acceleration, the trajectory data was digitally
differentiated. The finite difference method was used followed by low pass filtering with a
100th order filter with a cutoff frequency of 10 Hz4 The effectiveness of this filter is
shown in figure 8. Part A shows the step response of the filter. Part B shows the
magnitude of the velocity obtained by differentiating the three trajectories from figure 6
(which were obtained using supposedly identical experimental conditions). Part C shows
the magnitude of the acceleration obtained by differentiating again and reapplying the
filter. The average range of the calculated velocity for any given time sample is 13.4
mm/sec. The average range of the calculated acceleration for any given time sample is
237 mm/sec 2.
4The
filter used was generated with the 'firl' command in MatlabTM . It followed the windowing method
with a hamming window. The filter itself was an implementation of Program 5.2 in the IEEE programs
for digital signal processing tape.
I
r
I
r
I
r
r
I
II
0
100
200
·
300
·
400
500
·
600
·
700
·
800
·
900
1000
C
B
tI
I.
I Ci
S':
-I,I
I I:
II
IlI.
Time(sec)
Time(sec)
Figure 8: The effectiveness of the low pass filter used after differentiation: Part A shows the step
response of the filter. Part B shows the magnitude of the velocity obtained by differentiating the three
trajectories from figure 6 (which were obtained using supposedly identical experimental conditions). Part
C shows the magnitude of the acceleration obtained by differentiating again and reapplying the filter. As
can be seen, the velocities and accelerations are very similar
One difference between these experiments and those previously reported is the
lack, in general, of a smooth path of the equilibrium point during the course of (and after)
stimulation from the starting point to a stable equilibrium, and then back to the 'passive'
position (Giszter, 1993). In fact, for 75% of reported trials, actual and virtual paths were
reported to differ by less than 8mm (in a work space of around 500 square mm) in
previous work (Giszter, 1993). Even when smoothed5 , the path of the equilibrium point in
the current data shows significant deviation from that of the free limb trajectory. Figure 9
shows a typical example of the distance, from the actual position of the limb to where the
equilibrium point is calculated at that time. Part A shows the actual (solid) and virtual
(dashed) trajectories, each marked off in increments of 0.1 seconds. Part B shows the
magnitude of the distance between the two over time. Similar plots for all 17 trajectories
are included in appendix 2. Figure 10 shows the minimum, maximum, and average
distance over time between the actual and virtual trajectories for all 17 trajectories
(average is solid, minimum and maximum are dashed).
5 The path of the EP is linearly interpolated between times when its value can be calculated. If there is no
initial EP calculated, it is assumed that the EP starts at the first place one appears. If there is a time
segment where no EP is calculated, the previous and subsequent time samples are spatially averaged. If
there is no EP calculated at the end of the data, it is assumed the EP stays at its last calculated position
until the end of the data set.
A: Trajectory 7: Actual and Equilibrium Trajectories
rr\
140
-
~
Ad
50
60
-
'-I
E
70
E
80
90
-
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50
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35
-
30
-
25
20
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E
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0.5
Time(sec)
Figure 9 shows a typical example of the distance over time for a trajectory, from the actual position of the
limb to where the equilibrium point is calculated at that time. Part A shows the actual (solid) and virtual
(dashed) trajectories, each marked off in increments of 0.1 seconds. Part B shows the magnitude of the
distance between the two over time.
60
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Figure 10 shows the minimum, maximum, and average distance over time between the actual and virtual
trajectories for all 17 trajectories (average is solid, minimum and maximum are dashed).
For each available trajectory, the average (over time) and standard deviation of this
error was calculated. This data is summarized in Table 2. If the limb in fact followed
closely to the virtual trajectory, it would be possible to predict movement by simply
predicting the motion of the EP. The large errors, relative to the length of the frogs leg
(around 12 cm total on average- roughly the same as those used in (Giszter, 1993)),
underscore the need to create a model that takes the dynamics of the limb into account.6
6It
is unknown at this point why such a large discrepancy was obtained with past results. It could be
related to a change in experimental protocol. In (Giszter, 1993), the limb was suspended in a sling from
the ceiling, whereas in Lemay's work, it was not.
Table 2: Distance between actual limb position and calculated equilibrium point
Trajectory
Average of distance
between actual limb position
and calculated equilibrium
point (over time) in mm
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
Average
27.5
32.7
28.1
27.6
21.9
26.5
26.0
20.0
27.2
32.4
32.2
36.3
30.9
36.1
19.8
31.0
29.1
28.5
Standard deviation of
distance between actual limb
position and calculated
equilibrium point in mm
11.6
12.4
12.5
10.5
9.1
14.7
9.0
8.5
11.9
15.4
13.7
12.3
12.9
12.2
8.4
15.9
12.0
11.9
From measured trajectories and knowledge of the inertial properties of the limb, it
is also possible to perform inverse dynamic calculations to determine the forces required
to drive the limb along that trajectory (i.e. Finertial(trajectory,time)).
Inertial data for the frog leg (masses, lengths, etc.) were taken from actual post
operative measurements. It was assumed for the purposes of this project that the frog leg
consisted of a two bar mechanism, with each bar consisting of a homogenous cylinder at
the density of water. The lengths and masses of the cylinders were taken from the upper
and lower shanks of the leg. Table 3 shows the lengths and masses for each frog.
Table 3: Inertial Parameters of Frog legs
Length of upper
shank (cm)
6.5
6.0
Frog 1
Frog 2
Mass of upper
shank (grams)
43.0
39.0
Length of lower
shank (cm)
6.2
4.5
Mass of lower
shank (grams)
16.0
14.0
The software written for this project (using the MatlabTM programming language)
allowed for both calculation of accelerations given torque functions (forward dynamics)
and torques given accelerations (inverse dynamics). Both were calculated using the
general form:
t
= I(0)
+ C(,o)
(1)
where t is the vector of joint torques, I is the position dependent inertia tensor, 0 is the
vector of joint positions, co is the vector of joint velocities, x is the vector of joint
accelerations, and C is the vector of nonlinear centripetal and Coriolis forces.
Several factors could effect the accuracy of inverse dynamic calculations. One
source of potential error is inaccuracy in the measurement of the trajectory. Figure 11 a
shows the magnitudes of the inertial forces required to move the limb through the three
trajectories in figure 6a. The average range of calculated forces for any given time is 0.05
Newtons.
Two other possible sources of error in the calculation of inertial forces are
mismeasurements of the geometry or the inertial properties of the frog leg. To get an idea
of the range of errors likely as a result of a plausible mismeasurement, inverse dynamic
calculations were performed using a trajectory obtained from one frog but using either a)
the limb lengths or b) the limb masses from a different frog in the calculations. Figure 1lb
shows the calculated inertial forces under three different assumptions, with the solid line
representing the calculation using all the parameters from the correct frog, the dashed line
using the limb lengths from a different frog and the dash-dotted line using the limb masses
from a different frog. The average range of forces for any given time is 0.03 Newtons.
To be conservative, the greater of the two possible error sources is assumed for
the rest of this work, and a range of 0.05 Newtons is taken into account in the calculations
in chapter 4 of this thesis.
__1C
0
z
0
U.
Time(sec)
B
U.U
0.0
o
C
0
z
0)
UR
LL.
0.0
Time(sec)
5
Figure 11 A: Shows the magnitudes of the inertial forces required to move the limb through the three
trajectories in figure 6a. As can be seen, they are very similar, as they should be considering the similarity
of the original trajectories. B: Shows the calculated inertial forces under three different assumptions, with
the solid line representing the calculation using all the parameters from the correct from, the dashed line
using the limb lengths from a different frog and the dash-dotted line using the limb masses from a
different frog.
Another source of potential error in the calculation of inverse dynamics is the
introduction of bias. The nonlinear matrix in equation (1) represents centripetal and
Coriolis forces. The terms accounting for centripetal forces include functions of angular
velocity squared for a given joint (i.e. t 2 , where o, is the angular velocity of the nth joint
and n is either 1 or 2). The terms accounting for Coriolis forces include functions of the
angular velocity of one joint multiplied by the angular velocity of the other (i.e. 0t *0,
where o~ is the angular velocity of the nth joint). Because of this nonlinearity, an evenly
distributed error about a nominally measured angular velocity will result in calculated
Coriolis and centripetal forces which are biased towards larger magnitudes.
The problem is demonstrated in figure 12, where angular velocity (omega) is
plotted against angular velocity squared. On the horizontal axis, a nominal value for
omega is marked as well as points evenly spaced by some delta to each side. As can be
seen, the regular distribution in angular velocity one the horizontal axis maps to a biased
distribution in angular velocity squared on the vertical axis.
Angular velocity versus Angular velocity squared
omegeO
AnWAW ve"cx
Figure 12: Angular velocity (omega) is plotted against angular velocity squared. On the horizontal axis,
a nominal value for omega is marked as well as points evenly spaced by some delta to each side. As can be
seen, the regular distribution in angular velocity on the horizontal axis maps to a biased distribution in
angular velocity squared on the vertical axis.
To estimate the bias introduced, a typical value of angular velocity was obtained by
averaging the magnitudes of the angular velocities of the trajectories in figure 6a (which
came to 0.6 rad/sec). The bias was then calculated as the ratio of the average of this
nominal velocity plus some delta 7 squared and minus the same delta squared to the value
of the nominal velocity itself squared. That is:
bias=
(omega(nominal) + delta)2 + (omega(nominal) - delta)2
2*omega(nominal) 2
7The average range of velocity error (as shown in figure 8) is 13.4mm/sec and frog shank lengths (as
shown in table 3) are on the order of 62mm. A reasonable estimation of error range in angular velocity is
linear velocity error / shank length. This yields a range of 0.22 rad/sec or +/- 0.11 rad/sec. 0.11 rad/sec
was used for delta in this case.
The estimated value for the bias yielded by this method was 1.03. Due to the
complex and nonlinear nature of the dynamic equations, it is not immediately clear how
the introduction of a bias factor will affect the calculation of inertial forces. To be
conservative, calculations in chapter 4 will be done both with this nominal value of bias
and assuming no bias at all. It will become obvious that bias has no relevant effect for this
data.
The software used a fixed step numerical integrator for the forward dynamics and
utilized routines to convert from Cartesian to joint space (and vice versa) for the 2 link
mechanism representation of the frog's leg. The software was tested using simple physical
examples of two-bar mechanism motion, conservation of energy, and other tests (see
appendix 1).
Chapter 4: Testing Potential Models
Modeling Goal
The purpose of the proposed model is to represent the dynamic behavior of the
frog leg with a relatively simple mechanical analog. The requirements for such a model
include the ability to predict free-limb motion based on a knowledge of the inertial
parameters and measured isometric forces.
Such a model is necessary to both verify the basic concept of controlling limb
movement based on an attractor trajectory and to simulate possible control algorithms
without having to run an experiment for each potentially required movement.
Possible Models
There are several configurations one could envision to represent the impedance
relating the limb, the attractor point, and ground. The limb itself is an inertia. In the spirit
of the equilibrium hypothesis, it is assumed that there is some 'desired' or 'rest' position
for the endpoint of the limb. There is also a ground reference frame as there is in any
mechanical system. Possible mechanical elements to connect these three include
elasticities (elements that produce a force proportional to their displacement) and
viscosities (elements that produce a force proportional to their rate of change of
displacement). Both elasticities and viscosities can be either linear or nonlinear. They can
also be arranged in multiple combinations.
Several of these combinations are shown in Figure 13 where M represents the
inertia of the limb, springs represent elastic components (possibly nonlinear), dashpots
represent viscous components (again, possibly nonlinear), and Xd is the attractor point.
Note that the models shown are one dimensional representations of what is actually a two
dimensional system.
I
Xd
Xd
Xd
Xd
5.
6.
Xd
Figure 13: Diagram representing the structure of some possible models
Testing Model one
To test model one, I compared magnitudes of Fisometric(space,time) and
Finertial(space,time) for a series of trajectories. For each trajectory, the average of
Fisometric and Finertial (as well as their ratio) was calculated. A typical example of the
time course of Fisometric(space,time) and Finertial(trajectory,time) is shown in figure 14.
The dashed line in figures 14c and 14d represents the magnitude of the isometric force.
Xd
The confidence band shown around it represents the expected error range as calculated in
chapter 3. Likewise, the solid line represents the magnitude of the inertial force. The
small size of the confidence band surrounding the inertial force plot makes it hard to see
on the scale of the graph. Figure 14d is the equivalent to figure 14c with the exception
that the bias factor calculated in chapter 3 is used in calculating the inertial forces in 14d
but not in 14c. Similar plots for all 17 trajectories are included in Appendix 2.
B: Trajectory 16
:Tajcoy1
A:Trajectory 16
A:Taetry1
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84
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7
8
8
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70
75
80
85
90
45
50
55
60
65
70
75
80
85
90
x-pos (mm)
16
D:Trajectory
x-pos (mm)
16
C:Trajectory
1
III
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III \\\
r,\ \l
z
U-
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S0.5
0
L.
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II
II II
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III
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II
Ilt
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_
n
0
0.1
0.2
0.3
0.4
0.5 0.6
time(sec)
0.7
0.8
0.9
1
Figure 14: Example of the time course of Fisometric(space,time) and Finertial(trajectory,time). Part A:
Trajectory marked off in 100 msec increments with vectors representing directions and magnitudes of
Fisometric(space,time) Part B: Trajectory marked off in 100 msec increments with vectors representing
directions and magnitudes (scaled larger than part A so as to be visible) of Finertial(trajectory,time) Part
C: Magnitude of Fisometric(space,time) (dashed) and Finertial(trajectory,time) (solid) vs. time.
Confidence bands are included for both based on the estimated errors from chapter 3. Note that
Fisometric at all times and places is MUCH larger than what would be needed to actually accelerate the
limb along the trajectory (Finertial) Part D: Same as part C but with inertial forces calculated with bias
factor from chapter 3
The ratio of the average isometric force to the average inertial force in Figure 14c
is 42.6. Taking the extreme case of using the minimum edge of the confidence band for
the isometric forces and the upper edge of the confidence band for the inertial forces, the
ratio of the average isometric force to the average inertial force is still 26.1. For the data
using the bias factor in figure 14d, the ratio of the average isometric force to the average
inertial force is 41.1 (26.1 for the extreme case) 8.
For each trajectory, the average isometric force, average inertial force, and their
ratio were calculated. These results are shown in table four.
Table 4: Difference in Fisometric(space,time) and Finertial for several trajectories
Trajectory
Average of
magnitude of
Fisometric
Average of
magnitude of
Finertial
(space,time)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0.1730
0.0555
0.6685
0.6471
0.4348
0.5432
0.4994
0.3448
0.2379
0.2485
0.3068
0.3042
0.3513
0.3821
0.2395
16
0.3497
17
Average
0.2979
0.3579
0.0226
0.0227
0.0130
0.0112
0.0067
0.0073
0.0088
0.0077
0.0081
0.0066
0.0089
0.0087
0.0103
0.0104
0.0162
0.0082
0.0116
0.0111
Ratio of
average of
Fisometric to
average of
Finertial
7.65
2.44
51.42
57.78
64.90
74.41
56.75
44.78
29.34
37.65
34.47
34.97
34.11
36.74
14.78
42.65
25.68
38.27
8 Note that the bias has almost no effects on the results
One observations is immediately obvious. Fisometric(space,time) is much larger
than Finertial(trajectory,time) for almost all trajectories and all times. On average, the
force being measured is between one and two orders of magnitude higher than that
actually required to move the limb.
Model one (inertia connected to attractor point without any viscous elements) is
clearly not a usable model. If it were, the distance from the actual position to the desired
would be the sole determination of force and Fisometric would always equal Finertial in
both magnitude and direction. This is not at all surprising- it would have been very odd
indeed if a model incorporating no dissapative element at all were able to capture the
physical behavior of the system.
Testing Model two
To test Model two (inertia connected to attractor point with elastic element,
viscous element to ground), the power that would go into the hypothetical viscous
element in the model was examined. If this were a competent model, the force on the mass
would be the elastic force between the desired and actual position minus viscous forces to
ground (e.g. friction). That is:
Finertial(trajectory,time) = Fisometric(space,time)-Fviscosity (3)
For a given trajectory, one can calculate the power over time that would have to
go into the proposed damper by taking the dot product of the velocity with Fviscosity as
calculated by equation (3)9. An example of this data is plotted as figure 15. Part A shows
the trajectory marked off in 100 msec increments. Part B shows the power that would
have to go into the damper vs. time. By convention, negative power means that the
damper was GENERATING energy. A line is drawn at zero on the power axis for all
time. The confidence band shown around the power plot represents the combined error
ranges for inertial forces and isometric forces as calculated in chapter 3. Figure 15c is the
equivalent to figure 15b with the exception that the bias factor calculated in chapter 3 is
used in calculating the inertial forces in 15c but not in 15b. Similar plots for all 17
trajectories are included in appendix 2.
9 Calculations were done in both joint and Cartesian coordinates for verification purposes, yielding
identical results, as they must.
A: Trajectory 16
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80
82
84
86
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88
90
92
94
96
98
100
* 45
45
50
55
60
65
70
75
80
85
90
50
55
60
65
70
75
80
85
90
x-pos (mm)
B:Trajectory
16
0.14
C:Trajectory
16
-
0.14
0.12
I\
0.12
S,\
I
\
r
0.1
0.1
0.08
0.08
Z 0.06
P 0.06
0.04
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-
0.02
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,
--
- - -----
-
-
-
0
-
-0.02
-0.02
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\
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0
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I
0.1 0.2
0.3
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0.4 0.5 0.6
time(sec)
0.7 0.8 0.9 1
-
40 0.1 0.2 0.3 0.4 0.5 0.6
time(sec)
0.7 0.8 0.9 1
Figure 15: An example of power into the proposed damper in model 2 for a typical trajectory. Part A
shows the trajectory marked off in 100 msec increments, and part B shows the power that would have to
go into the damper vs. time. By convention, negative power means that the damper was GENERATING
energy. A line is drawn at zero on the power axis for all time. Note that the damper would in fact be
generating power for part of the trajectory. The confidence band shown around the power plot represents
the combined error ranges for inertial forces and isometric forces as calculated in chapter 3. Figure 15c is
the equivalent to figure 15b with the exception that the bias factor calculated in chapter 3 is used in
calculating the inertial forces in 15c but not in 15b.
The power was negative 12.4% of the time for the example shown in figure 15b.
Values were only counted as negative if they were lower than negative one times 2% of
the absolute value of the peak power (this is to eliminate the 'noise' when the power is
very close to zero). Taking the extreme case of using the upper edge of the confidence
band for power, it is still negative for 9.2% of the time. For the data using the bias factor
in figure 15c, the percentage of time power was negative was identical to the non-biased
case up to 4 significant figures for both the normal and extreme case. The percentage of
time that the damper would be generating power for each trajectory is shown in table 4.
Table 4: Model-2- Power flow through hypothetical damper for several trajectories
Trajectory
Percentage of time hypothetical damper
1
2
3
4
24.4%
37.8%
6.8%
9.2%
5
6
7
8
9
10
5.6%
14.8%
5.1%
1.3%
15.6%
0%
would be generating power
11
12.5%
12
13
14
15
16
17
Average:
Standard Deviation:
0%
6.7%
10.4%
15.9%
12.4%
16.3%
11.5%
9.4%
The data shows that for a significant portion of the trials tested, the proposed
damper is GENERATING power for some portion of the trajectory. For Model 2 to be
valid, it is essential that the damper to ground be passive- it cannot generate power. It is
therefore clear that any model based solely on damping to ground (e.g. friction) to
accommodate the difference between Fisometric(space,time) and Finertial(trajectory,time)
cannot work.
Several other models are shown in figure 13, but all have an element in common- a
viscous element between the attractor point and the inertia. As shown in the next section,
this makes it difficult to determine the location of the attractor point and the values of the
impedances based on present data.
To summarize, the power argument has shown that any model relying solely on a
viscous element to ground (e.g. friction) for energy dissipation cannot work. The data
does not show whether damping to ground will get within allowable tolerances of
matching the simulated velocity profile to the actual for a working model.
Chapter 5: Future Work, Discussion
Future Work
With the exception of the two models from Figure 12 examined in chapter 4, all
other models shown involve a viscous element between the attractor point and the limb.
This raises a subtle but critical issue. If the impedance between the ankle and the attractor
point is dependent on their relative velocities, it is impossible to determine the path of
the attractor point from isometric force trials without assuming knowledge of the stiffness
and damping parameters. Notice that I distinguish between "equilibrium point" which is
where the statically measured forces go to zero and "attractor point" which is where the
forces would actually be zero during motion. The "attractor point" cannot be directly
measured in isometric experiments.
This can be seen in a linear case by looking at four data points in an isometric
measurement of the force assuming that the system looks like model 3 in Figure 13.
Assume that the measurements are taken at the same time in each case (time T) and hence
the attractor point position (Xd) and attractor point velocity (vd) are the same in each
trial. Further assume that the mass is moved to a different location (xl->x4) for each trial
and Fisometric(space,T) measured (see figure 16).
Fiso(x 1,T)=k(Xd(T)-x 1)+b(vd(T)-v 1)
Fiso(x2,T)=k(Xd(T)-x2)+b(vd(T)-v2)
Fiso(x3,T)=k(Xd(T)-x3)+b(vd(T)-v3)
Fiso(x4,T)=k(Xd(T)-x4)+b(vd(T)-v4)
(4a)
(4b)
(4c)
(4d)
d,vd
Xd,v A
XX.vO-•V kW.
xi
AU,VU
-- " "-
Yý
S. .
AGVG
xci,vd
Figure 16: Proposed Experiment: Move the mass (the limb) to four different locations. Stimulate the
spinal cord and record the isometric force at the limb at some time T for each location. Xd and vd are the
position and velocity of the equilibrium point which should be the same for each trial but are unknown. b
and k are the damping and spring constant which should be the same for each trial but are unknown. xl>x4 are the position of the mass (the limb) which is different for each trial but is known. vl->v4 are the
velocities of the mass (the limb) which is zero for each trial since the trials are isometric. Fiso(xl->x4,T)
are the isometric forces at the limb which should be different for each trial but can be measured
experimentally. That leaves four unknowns at time T: Xd, vd, k and b
It can be seen fairly easily that it is impossible to determine both the damping
coefficient (b) and the velocity of the attractor point (vd) from an isometric trial. In other
words, if b is non-zero and vl->v4 are zero (i.e. the trial is isometric) the equations 4a>4d are non-linear and b and vd will always be multiplied together. Taking more
measurements would not solve this, as it would still be impossible to separate out b and
vd.
It will therefore be necessary to use a different method to determine the attractor
trajectory. An appropriate starting point was proposed by Anthony Hodgson (Hodgson,
1994). The theory is to use an actuated robot to apply the force necessary to move the
inertia. This means that no force is being applied by the impedance element between the
attractor point and the limb and the path of the limb tracks that of the attractor point.
Using an actuated small animal robot and appropriate software that is being
designed in our lab, it should be possible to track the attractor trajectory. A simple
experimental protocol would be to
a) Stimulate the animal and track its motion
b) Use the robot to move the limb through the same motion without spinal
stimulation and measure the inertial, Coriolis, centripetal, and frictional forces
c) Stimulate the animal again while applying the forces measured in (b) with the
robot. Measure the force at the robot/animal interface and the new trajectory.
d) Repeat part (b) with the new trajectory
e) If the force measured in (d) is the same as the force measured in (c), the
trajectory measured in (c) is the attractor trajectory i". Otherwise, iterate until
convergence is reached.
Discussion
An attempt was made to model the movement of a frog limb as a result of spinal
stimulation. Although no suitable model emerged, several possibilities have been
eliminated, most notably any model relying solely on damping relative to the ground frame
for energy dissipation.
The major relevance of this is that it makes it extremely unlikely that an adequate
dynamic model can be determined based on isometric data alone. It also throws doubt on
previously published arguments both pro and con the equilibrium point hypothesis as a
motor control method. Work in the past to both prove and disprove the equilibrium point
hypothesis of motor control has often relied on models assuming a structure with
dissipation only to ground and examined the required complexity of a virtual trajectory
control for a movement. If the model itself is incorrect, of course, then these predictions
as to the virtual trajectories will be wrong as well.
Future work should most likely include experiments to directly measure the
attractor trajectory by active perturbance of the limb.
10 Since the force being applied by the robot equals the force moving the limb, there is no work being done
by the impedance between the attractor point and the limb and hence no change in "length" of the
impedance elements
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Hodgson, Anthony J. Inferring Central Motor Plans from Attractor Trajectory
Measurements. Doctoral Thesis. Massachusetts Institute of Technology, July 1994.
Hogan, Neville. Lecture Notes for Course 'Biomechanics and Neural Control of
Movement'. 1997
Hogan, Neville. An Organizing Principle for a Class of Voluntary Movements. Journal of
Neuroscience, November 1984, Volume 4, number 11: 2745-2754
Katayama, Masazumi, Kawato, Mitsuo. Virtual Trajectory and Stiffness Ellipse During
Multijoint Arm Movement Predicted by Neural Inverse Models. Biological Cybernetics,
1993, Volume 69: 353-362
Kearney, Robert E., Hunter, Ian W. System Identification of Human Joint Dynamics.
Biomedical Engineering, 1990, Volume 18, issue 1: 55-87
Koebetic, Rudi, Triolo, Ronald, Marsolais, Byron. Muscle Selection and Walking
Performance of Multichannel FES systems for Ambulation in Paraplegia. IEEE
Transactions on Rehabilitation Engineering, March 1997, Volume 5, number 1: 23-29
Lemay, Michel. Personal communication and unpublished work. 1996, 1997
Loeb, Eric. Uses of Statistical Muscle Models, including a Test of an Equilibrium Point
Control Theory of Spinal Cord Function in Rana Catesbiana. Doctoral Thesis.
Massachusetts Institute of Technology, May 1995.
Mussa-Ivaldi, F.A., Hogan, Neville, Bizzi, E. Neural, Mechanical, and Geometric Factors
Subserving Arm Posture in Humans. Journal of Neuroscience, October 1985, Volume 5,
number 10: 2732-2743
Mussa-Ivaldi, F.A., Giszter, S.F. and Bizzi, E. Proc. Natl Acad. Sci. USA 91: 7534-7538
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Smith, A.M., Humphrey, D.R., What Do Studies of Specific Motor Acts Such as
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John Wiley and Sons, 1991:357-81.
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Multijoint Arm Movement. Biological Cybernetics, 1989, Volume 61: 89-101
Won, Justin. The Control of Constrained and Partially Constrained Arm Movements.
Masters Thesis, Massachusetts Institute of Technology, February 1993.
Appendix 1: Verification of the software for integration of forward dynamics
Two fairly simple tests were conducted to verify that the software worked based
on simple physical properties, one based on simplifying the dynamics to the point where
the behavior could be predicted and the other on conservation of energy arguments.
To start, the dynamic properties were entered as though the two bar linkage were
massless and there was a single point-mass at the end of the second link. The mechanism
was given an initial velocity in both joints but no applied torque. Since the mechanism
must obey the simple laws of particle physics under these conditions, the angle of
incidence when it reaches a kinematic extreme must always equal the angle of reflection.
As can be seen in figure 17, this is in fact the case.
As a second test, the dynamics were set as typical for a frog leg and a constant
torque of 1 mili-newton-meter was applied to each joint from 0.5 seconds to 1.0 seconds,
with zero torque being applied at all other times and with the system starting at rest. The
results can be seen in figure 18, where part A is the trajectory and part B is the energy of
the manipulandum at each point in time as calculated by the equation:
Energy= 1/2 o T I(0)co
(5)
where 0o is the vector of angular velocities, I is the position dependent inertia matrix, and T
is the transpose operator. As can be seen, the energy stays at zero until the beginning of
the applied torque, rises while torque is applied, and then stays constant afterwards. As
there is no dissipation in the system, this is as it should be.
2
1
0
0
-1
-2
-?
-3
-2
-1
0
1
2
3
x position
Figure 17: Trajectory of a two bar linkage with dynamic properties were entered as though the two bar
linkage were massless and there was a single point-mass at the end of the second link. The mechanism
was given an initial velocity in both joints but no applied torque. Since the mechanism must obey the
simple laws of particle physics under these conditions, the angle of incidence when it reaches a kinematic
extreme must always equal the angle of reflection. As can be seen, this is in fact the case.
160
140
120
100
20
-
-
SI
0
20
40
60
80
I
I
I
100
120
140
160
8
7
6
5
cO
34
3
2
1
1
Time(sec)
2
Figure 18 Conservation of Energy test. The dynamics were set as typical for a frog leg and a constant
torque of 1 mili-newton-meter was applied to each joint from 0.5 seconds to 1.0 seconds, with zero torque
being applied at all other times and with the system starting at rest. Part A is the trajectory of the endpoint
and part B is the energy of the manipulandum at each point in time. Note that the energy is zero until the
torque is applied, rises while power is put into the system, and stays constant thereafter
Appendix 2: Figures
Distances from actual to virtual position over time for each trajectory
A: Trajectory 1: Actual and Equilibrium Trajectories
II
//
-/
/I
.-
-
--
--
-- /
6
·-
-/1
LI
" . ,',
.40
45
50
55
60
65
70
B: Trajectory 1
75
80
85
,1/
90
50
40
E 30
20
10
I
0.5
Time(sec)
Figure 19: Distance over time for Trajectory 1, from the actual position of the limb to where the
equilibrium point is calculated at that time. Part A shows the actual (solid) and virtual (dashed)
trajectories, each marked off in increments of 0.1 seconds. Part B shows the magnitude of the distance
between the two over time.
A: Trajectory 2: Actual and Equilibrium Trajectories
bu
65
70
1
75
80
-
/
-
--
//a"
G 1//
-1-
-
S//
6
85
---
//
I
,•i
/
95
/
,
/
100
105
II
I
i-
~//
-
%-
/v\
/6
4
-
90
•
"'"
/
,. ./
£-..>
'1 /
-/ ";I
I
I
I
1ln
30
40
50
60
B: Trajectory 2
70
80
90
E
E
0.5
0.5
Time(sec)
1
Figure 20: Distance over time for Trajectory 2, from the actual position of the limb to where the
equilibrium point is calculated at that time. Part A shows the actual (solid) and virtual (dashed)
trajectories, each marked off in increments of 0.1 seconds. Part B shows the magnitude of the distance
between the two over time.
A: Trajectory 3: Actual and Equilibrium Trajectories
,,
OV
60
70
.,1
-
-
-
,/
ý3
90
100
110
r
-
120
-
I
'
30
40
I
I
50
60
I
70
80
B: Trajectory 3
I
I
I
90
100
110
II I
50
E
E 40
E
20
E
0.5
1
Time(sec)
Figure 21: Distance over time for Trajectory 3, from the actual position of the limb to where the
equilibrium point is calculated at that time. Part A shows the actual (solid) and virtual (dashed)
trajectories, each marked off in increments of 0.1 seconds. Part B shows the magnitude of the distance
between the two over time.
A: Trajectory 4: Actual and Equilibrium Trajectories
5U
-1
r
I 'A
40
i40
I
I
50
60
80
70
B: Trajectory 4
90
100
110
45
40
E
30
E
E 25
E
20
E
10
E
0
0.5
Time(sec)
1
Figure 22: Distance over time for Trajectory 4, from the actual position of the limb to where the
equilibrium point is calculated at that time. Part A shows the actual (solid) and virtual (dashed)
trajectories, each marked off in increments of 0.1 seconds. Part B shows the magnitude of the distance
between the two over time.
A: Trajectory 5: Actual and Equilibrium Trajectories
~-1
60
E
90
E
100
E
J-
P
80
B: Trajectory 5
50
100
110
35
E
30
25
-
-
E 20
0
0.5
1
Time(sec)
Figure 23: Distance over time for Trajectory 5, from the actual position of the limb to where the
equilibrium point is calculated at that time. Part A shows the actual (solid) and virtual (dashed)
trajectories, each marked off in increments of 0.1 seconds. Part B shows the magnitude of the distance
between the two over time.
A: Trajectory 6: Actual and Equilibrium Trajectories
1·-I
80
-
o
S
•}
100
14 "
IIv
5O
-
..
;7-
--
/3
-
--
o
6/
80
B: Trajectory 6
100
110
It
0.5
Time(sec)
1
Figure 24: Distance over time for Trajectory 6, from the actual position of the limb to where the
equilibrium point is calculated at that time. Part A shows the actual (solid) and virtual (dashed)
trajectories, each marked off in increments of 0.1 seconds. Part B shows the magnitude of the distance
between the two over time.
A: Trajectory 7: Actual and Equilibrium Trajectories
,01
50
I-'
1-
,'
60
70
-
,of
80
90
-
-
----
---,-•-,
•'- -7
.....--
AI
50O
11 0
I
I
I
80
B: Trajectory 7
I
I
I
100
110
E
E
0
0.5
1
Time(sec)
Figure 25: Distance over time for Trajectory 7, from the actual position of the limb to where the
equilibrium point is calculated at that time. Part A shows the actual (solid) and virtual (dashed)
trajectories, each marked off in increments of 0.1 seconds. Part B shows the magnitude of the distance
between the two over time.
A: Trajectory 8: Actual and Equilibrium Trajectories
-
'F~
90
E
8\/
n
60
I
I
I
I
65
70
75
80
I
I
85
90
B: Trajectory 8
I
I
I
I
95
100
105
110
40
35
30
25
20
15
10
5
0
0
01
0.5
Time(sec)
1
1
Figure 26: Distance over time for Trajectory 8, from the actual position of the limb to where the
equilibrium point is calculated at that time. Part A shows the actual (solid) and virtual (dashed)
trajectories, each marked off in increments of 0.1 seconds. Part B shows the magnitude of the distance
between the two over time.
A: Trajectory 9: Actual and Equilibrium Trajectories
,
-
I
,-
.. ,-
80
-
-
90
-
100
-
-I
-
L-
I
6
S4 J5
110
-
1 p3
40
I
I
I
50
60
70
I
I
80
90
B: Trajectory 9
I
I
I
100
110
120
hi I
40
L
E
E 30
0
0.5
Time(sec)
1
Figure 27: Distance over time for Trajectory 9, from the actual position of the limb to where the
equilibrium point is calculated at that time. Part A shows the actual (solid) and virtual (dashed)
trajectories, each marked off in increments of 0.1 seconds. Part B shows the magnitude of the distance
between the two over time.
A: Trajectory 10: Actual and Equilibrium Trajectories
,-0
9
-,6
-
I
-'
-
-c -
100
/
,.
110
E
19
40
50
60
70
80
90
B: Trajectory 10
100
110
120
60
50
E
40
E 30
E
20
0
0.5
Time(sec)
1
Figure 28: Distance over time for Trajectory 10, from the actual position of the limb to where the
equilibrium point is calculated at that time. Part A shows the actual (solid) and virtual (dashed)
trajectories, each marked off in increments of 0.1 seconds. Part B shows the magnitude of the distance
between the two over time.
A: Trajectory 11: Actual and Equilibrium Trajectories
,,
5U
r
-, -
\I
70
t\
\
9
90
I
110
\I '1
i93
50
60
50
80
B: Trajectory 11
100
110
-
-
E
0.5
Time(sec)
1
Figure 29: Distance over time for Trajectory 11, from the actual position of the limb to where the
equilibrium point is calculated at that time. Part A shows the actual (solid) and virtual (dashed)
trajectories, each marked off in increments of 0.1 seconds. Part B shows the magnitude of the distance
between the two over time.
A: Trajectory 12: Actual and Equilibrium Trajectories
"I
80
E
-
-
-
/2
I-
I nn
6C
0
6'
I
I
I
I
65
70
75
80
I
I
85
90
B: Trajectory 12
I
95
I
100
I
105
I
110
E
E3
2
2
0
0.5
Time(sec)
1
Figure 30: Distance over time for Trajectory 12, from the actual position of the limb to where the
equilibrium point is calculated at that time. Part A shows the actual (solid) and virtual (dashed)
trajectories, each marked off in increments of 0.1 seconds. Part B shows the magnitude of the distance
between the two over time.
A: Trajectory 13: Actual and Equilibrium Trajectories
4+v
50
K
60
//
70
80
90
100
110
139
40
701
50
60
70
80
90
B: Trajectory 13
100
110
120
60
50
40
E
E
30
20
10
0
0
0.5
Time(sec)
1
Figure 31: Distance over time for Trajectory 13, from the actual position of the limb to where the
equilibrium point is calculated at that time. Part A shows the actual (solid) and virtual (dashed)
trajectories, each marked off in increments of 0.1 seconds. Part B shows the magnitude of the distance
between the two over time.
A: Trajectory 14: Actual and Equilibrium Trajectories
-
60
---
/
~d
~
"
~
90
100
-
110
6
L4
I
193
3
I
I
I
90
100
B: Trajectory 14
6(
I
I
110
120
40
35
E
E 30
25
20
10
+
c,
0
0.5
Time(sec)
1
Figure 32: Distance over time for Trajectory 14, from the actual position of the limb to where the
equilibrium point is calculated at that time. Part A shows the actual (solid) and virtual (dashed)
trajectories, each marked off in increments of 0.1 seconds. Part B shows the magnitude of the distance
between the two over time.
A: Trajectory 15: Actual and Equilibrium Trajectories
_-8
65
70
A
I
, >3
90
~6
95
Y\
100
E
I
1An
70
I
85
B: Trajectory 15
I
I
95
100
30
E
E 25
15
-
0
0.5
Time(sec)
1
Figure 33: Distance over time for Trajectory 15, from the actual position of the limb to where the
equilibrium point is calculated at that time. Part A shows the actual (solid) and virtual (dashed)
trajectories, each marked off in increments of 0.1 seconds. Part B shows the magnitude of the distance
between the two over time.
A: Trajectory 16: Actual and Equilibrium Trajectories
85
-
90
-
95
1 00
-
'
403
0
70
B: Trajectory 16
0.5
Time(sec)
90
100
1
Figure 34: Distance over time for Trajectory 16, from the actual position of the limb to where the
equilibrium point is calculated at that time. Part A shows the actual (solid) and virtual (dashed)
trajectories, each marked off in increments of 0.1 seconds. Part B shows the magnitude of the distance
between the two over time.
A: Trajectory 17: Actual and Equilibrium Trajectories
^^
t0u
r
7A
/3
90
1
"il
55
I
I
I
1
60
65
70
75
I
I
I
I
I
80
85
90
95
100
B: Trajectory 17
40
E
E
E 25
L;
0
0.5
1
Time(sec)
Figure 35: Distance over time for Trajectory 17, from the actual position of the limb to where the
equilibrium point is calculated at that time. Part A shows the actual (solid) and virtual (dashed)
trajectories, each marked off in increments of 0.1 seconds. Part B shows the magnitude of the distance
between the two over time.
Fisometric(space.time) vs. Finertial(trajectory,time) for all trajectories
Note that in some of the trials the measured free-limb trajectory temporarily left
the area where isometric force was measured. The software, in this case, arbitrarily sets
Fisometric to zero. Data in these regions is marked by asterisks and should be
disregarded.
A:Trajectory 1
B:Trajectory 1
1Nwton
i
30
I
40
50
I
i
·
80
70
60
90
30
40
x-pos (mm)
50
60
70
x-pos (mm)
80
90
C: Trajectory 1
,,
U.9
r
r
r
r
r
r
r
r
0.8
0.7
0.6
r
I
I
I
0•.5
" 0.4
0.3
r
i
I
r
r
r
r
r
r
r
1-
I
I
-
I
r
-~
€]
0
...........
0.1
I
0.2
0.3
0.4
0.5
time(sec)
0.6
0.7
0.8
0.9
1
Figure 36: Time course of Fisometric(space,time) and Finertial(trajectory,time) for Trajectory 1. Part A:
Trajectory marked off in 100 msec increments with vectors representing directions and magnitudes of
Fisometric(space,time) Part B: Trajectory marked off in 100 msec increments with vectors representing
directions and magnitudes (scaled larger than part A so as to be visible) of Finertial(trajectory,time) Part
C: Magnitude of Fisometric(space,time) (dashed) and Finertial(trajectory,time) (solid) vs. time. Note that
Fisometric at most times and places is MUCH larger than what would be needed to actually accelerate the
limb along the trajectory (Finertial). Areas marked by asterisks are when the free-limb trajectory goes
outside of the measured isometric force range and should be ignored.
A:Trajectory 2
B:Trajectory 2
60
60
70
70
80
E
o90
90
0.
100
100
110
110
120
120
30
40
50
60
80
70
90
30
40
x-pos (mm)
50
60
70
80
90
x-pos (mm)
C: Trajectory 2
U.4
II
0.3
II
III I
-
II
II
II
4 0.2
0
I I'
/
I
--
/
•
--
I
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
time(sec)
Figure 37: Time course of Fisometric(space,time) and Finertial(trajectory,time) for Trajectory 2. Part A:
Trajectory marked off in 100 msec increments with vectors representing directions and magnitudes of
Fisometric(space,time) Part B: Trajectory marked off in 100 msec increments with vectors representing
directions and magnitudes (scaled larger than part A so as to be visible) of Finertial(trajectory,time) Part
C: Magnitude of Fisometric(space,time) (dashed) and Finertial(trajectory,time) (solid) vs. time. Note that
Fisometric at most times and places is MUCH larger than what would be needed to actually accelerate the
limb along the trajectory (Finertial). Areas marked by asterisks are when the free-limb trajectory goes
outside of the measured isometric force range and should be ignored.
A:Trajectory
3
B:Trajectory 3
TI :Nwton
0.1Newton
0E 9C
90
100oo
30
40
50
60
70
80
90
30
40
x-pos (mm)
50
70
60
x-pos (mm)
80
90
C: Trajectory 3
I
n
0
I
0.1
I
0.2
I
0.3
I
0.4
0.5
time(sec)
0.6
0.7
0.8
I
0.9
---
I
1
Figure 38: Time course of Fisometric(space,time) and Finertial(trajectory,time) for Trajectory 3. Part A:
Trajectory marked off in 100 msec increments with vectors representing directions and magnitudes of
Fisometric(space,time) Part B: Trajectory marked off in 100 msec increments with vectors representing
directions and magnitudes (scaled larger than part A so as to be visible) of Finertial(trajectory,time) Part
C: Magnitude of Fisometric(space,time) (dashed) and Finertial(trajectory,time) (solid) vs. time. Note that
Fisometric at most times and places is MUCH larger than what would be needed to actually accelerate the
limb along the trajectory (Finertial). Areas marked by asterisks are when the free-limb trajectory goes
outside of the measured isometric force range and should be ignored.
B:Trajectory
4
A:Trajectory
4
0.1ton
ewon
V/
30
40
50
80
60
70
x-pos (rnm)
90
30
40
50
60
70
x-pos (mrm)
80
90
C: Trajectory 4
I
A
0
I
I
I
I
I
0.1
0.2
0.3
0.4
'
'
0.5
0.6
time(sec)
.
0.7
--
,•.-•'-
0.8
,
i
0.9
1
Figure 39: Time course of Fisometric(space,time) and Finertial(trajectory,time) for Trajectory 4. Part A:
Trajectory marked off in 100 msec increments with vectors representing directions and magnitudes of
Fisometric(space,time) Part B: Trajectory marked off in 100 msec increments with vectors representing
directions and magnitudes (scaled larger than part A so as to be visible) of Finertial(trajectory,time) Part
C: Magnitude of Fisometric(space,time) (dashed) and Finertial(trajectory,time) (solid) vs. time. Note that
Fisometric at most times and places is MUCH larger than what would be needed to actually accelerate the
limb along the trajectory (Finertial). Areas marked by asterisks are when the free-limb trajectory goes
outside of the measured isometric force range and should be ignored.
A:Trajectory
5
B: Trajectory 5
ewton
01Newton
Oe/
I
I
30
50
40
I
I
I
70
80
90
]
60
i
30
40
x-pos (mm)
I
50
I
I
60
70
x-pos (mm)
80
90
C: Trajectory 5
0.6
0 .s
U
0o
LL
0.4
0.2
-
I
0.1
I
0.2
~- 0.3
0.10.
.
0.4
.
0.5
.c-~0.2-
0.6
.
.
.
0.7
0.8
0.9
1
time(sec)
Figure 40: Time course of Fisometric(space,time) and Finertial(trajectory,time) for Trajectory 5. Part A:
Trajectory marked off in 100 msec increments with vectors representing directions and magnitudes of
Fisometric(space,time) Part B: Trajectory marked off in 100 msec increments with vectors representing
directions and magnitudes (scaled larger than part A so as to be visible) of Finertial(trajectory,time) Part
C: Magnitude of Fisometric(space,time) (dashed) and Finertial(trajectory,time) (solid) vs. time. Note that
Fisometric at most times and places is MUCH larger than what would be needed to actually accelerate the
limb along the trajectory (Finertial). Areas marked by asterisks are when the free-limb trajectory goes
outside of the measured isometric force range and should be ignored.
A:Trajectory
6
60
rl
B:Trajectory 6
ewton
0.1Nwtn
70
80
E
j
90
100
110
120
40
50
60
70
x-pos (mm)
90
80
30
40
50
60
70
x-pos (mm)
80
90
C: Trajectory 6
N
n~
0
t
0.1
I
0.2
I
0.3
i
0.4
I
0.5
• |
0.6
.
0.7
.
.
0.8
.
0.9
.
._
1
time(sec)
Figure 41: Time course of Fisometric(space,time) and Finertial(trajectory,time) for Trajectory 6. Part A:
Trajectory marked off in 100 msec increments with vectors representing directions and magnitudes of
Fisometric(space,time) Part B: Trajectory marked off in 100 msec increments with vectors representing
directions and magnitudes (scaled larger than part A so as to be visible) of Finertial(trajectory,time) Part
C: Magnitude of Fisometric(space,time) (dashed) and Finertial(trajectory,time) (solid) vs. time. Note that
Fisometric at most times and places is MUCH larger than what would be needed to actually accelerate the
limb along the trajectory (Finertial). Areas marked by asterisks are when the free-limb trajectory goes
outside of the measured isometric force range and should be ignored.
A:Trajectory
7
B:Trajectory 7
F1 3Nwton
30
40
50
60
70
80
x-pos (mm)
90
30
40
0.9
1
50
60
70
x-pos (mm)
80
90
C: Trajectory 7
I
0
0.1
0.2
0.3
0.4
\
0.5
time(sec)
0.6
0.7
0.8
Figure 42: Time course of Fisometric(space,time) and Finertial(trajectory,time) for Trajectory 7. Part A:
Trajectory marked off in 100 msec increments with vectors representing directions and magnitudes of
Fisometric(space,time) Part B: Trajectory marked off in 100 msec increments with vectors representing
directions and magnitudes (scaled larger than part A so as to be visible) of Finertial(trajectory,time) Part
C: Magnitude of Fisometric(space,time) (dashed) and Finertial(trajectory,time) (solid) vs. time. Note that
Fisometric at most times and places is MUCH larger than what would be needed to actually accelerate the
limb along the trajectory (Finertial). Areas marked by asterisks are when the free-limb trajectory goes
outside of the measured isometric force range and should be ignored.
A:Trajectory
8
60
B:Trajectory 8
eTwtc
n
Newton
70
80
,ýrý
E
090
0
I
30
I
40
I
l
70
60
x-pos (mm)
50
~'
i
I
80
90
30
40
50
60
70
x-pos (mm)
80
90
C: Trajectory 8
I
I
I
I
r
I
I
I
f
I
0
0.1
·
0.2
0.3
0.4
~c~----Tz-I~---T·--~
0.5
time(sec)
0.6
~
0.7
0.8
0.9
1
Figure 43: Time course of Fisometric(space,time) and Finertial(trajectory,time) for Trajectory 8. Part A:
Trajectory marked off in 100 msec increments with vectors representing directions and magnitudes of
Fisometric(space,time) Part B: Trajectory marked off in 100 msec increments with vectors representing
directions and magnitudes (scaled larger than part A so as to be visible) of Finertial(trajectory,time) Part
C: Magnitude of Fisometric(space,time) (dashed) and Finertial(trajectory,time) (solid) vs. time. Note that
Fisometric at most times and places is MUCH larger than what would be needed to actually accelerate the
limb along the trajectory (Finertial). Areas marked by asterisks are when the free-limb trajectory goes
outside of the measured isometric force range and should be ignored.
A:Trajectory
9
8: Trajectory 9
0.1
Newton
Newon
16-0/
30
40
60
70
80
x-pos (mm)
C: Trajectory 9
50
90
40
30
·
50
·
60
·
70
x-pos (mm)
·
80
I
90
0.8
E
20.5
LL
I
i
-•
n'
0
.
..
.
.
0.1
0.2
0.3
0.4
•,,-•.-•
•.,.,•,•
0.5
0.6
time(sec)
I
0.7
b -•
•
r•Jm•le
0.8
0.9
i
1
Figure 44: Time course of Fisometric(space,time) and Finertial(trajectory,time) for Trajectory 9. Part A:
Trajectory marked off in 100 msec increments with vectors representing directions and magnitudes of
Fisometric(space,time) Part B: Trajectory marked off in 100 msec increments with vectors representing
directions and magnitudes (scaled larger than part A so as to be visible) of Finertial(trajectory,time) Part
C: Magnitude of Fisometric(space,time) (dashed) and Finertial(trajectory,time) (solid) vs. time. Note that
Fisometric at most times and places is MUCH larger than what would be needed to actually accelerate the
limb along the trajectory (Finertial). Areas marked by asterisks are when the free-limb trajectory goes
outside of the measured isometric force range and should be ignored.
A:Trajectory
10
B:Trajectory
10
A
Se1wton
Newton
E
E
, 90
0
100
30
40
50
90
60
70
80
x-pos (mm)
C: Trajectory 10
30
I
·
40
50
I
60
I
70
x-pos (mm)
I
80
I
90
I
I
I
r
r
I
I
r
r
r
I
i
0
0.1
0.2
0.3
0.4
0.5
0.6
time(sec)
0.7
0.8
0.9
1
Figure 45: Time course of Fisometric(space,time) and Finertial(trajectory,time) for Trajectory 10. Part A:
Trajectory marked off in 100 msec increments with vectors representing directions and magnitudes of
Fisometric(space,time) Part B: Trajectory marked off in 100 msec increments with vectors representing
directions and magnitudes (scaled larger than part A so as to be visible) of Finertial(trajectory,time) Part
C: Magnitude of Fisometric(space,time) (dashed) and Finertial(trajectory,time) (solid) vs. time. Note that
Fisometric at most times and places is MUCH larger than what would be needed to actually accelerate the
limb along the trajectory (Finertial). Areas marked by asterisks are when the free-limb trajectory goes
outside of the measured isometric force range and should be ignored.
A:Trajectory
11
B:Trajectory 11
01Newton
lT ewton
30
I
I
40
50
·
I
·
I
80
60
70
x-pos (mm)
C: Trajectory 11
90
30
40
50
60
70
x-pos (mm)
80
90
F0.5
o
L.
S0.4
0.4
A
~~~~~/
-
0
0.1
0.2
0 0.
0.
0.4
0.
0.3
0.4
0.5
0.6
0.7
.8
.9
0.5
0.6
0.7
0.8
0.9
1
time(sec)
Figure 46: Time course of Fisometric(space,time) and Finertial(trajectory,time) for Trajectory 11. Part A:
Trajectory marked off in 100 msec increments with vectors representing directions and magnitudes of
Fisometric(space,time) Part B: Trajectory marked off in 100 msec increments with vectors representing
directions and magnitudes (scaled larger than part A so as to be visible) of Finertial(trajectory,time) Part
C: Magnitude of Fisometric(space,time) (dashed) and Finertial(trajectory,time) (solid) vs. time. Note that
Fisometric at most times and places is MUCH larger than what would be needed to actually accelerate the
limb along the trajectory (Finertial). Areas marked by asterisks are when the free-limb trajectory goes
outside of the measured isometric force range and should be ignored.
A:Trajectory
12
B: Trajectory12
ewton
Newton
I
30
40
50
90
60
70
80
x-pos (mm)
C: Trajectory 12
30
40
i
50
i
70
60
x-pos (mm)
I
80
I
90
20.5
L.0.4
I
0
rl
0
0.1
.
0.2
0.2
='
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
time(sec)
Figure 47: Time course of Fisometric(space,time) and Finertial(trajectory,time) for Trajectory 12. Part A:
Trajectory marked off in 100 msec increments with vectors representing directions and magnitudes of
Fisometric(space,time) Part B: Trajectory marked off in 100 msec increments with vectors representing
directions and magnitudes (scaled larger than part A so as to be visible) of Finertial(trajectory,time) Part
C: Magnitude of Fisometric(space,time) (dashed) and Finertial(trajectory,time) (solid) vs. time. Note that
Fisometric at most times and places is MUCH larger than what would be needed to actually accelerate the
limb along the trajectory (Finertial). Areas marked by asterisks are when the free-limb trajectory goes
outside of the measured isometric force range and should be ignored.
A:Trajectory
13
I
ewton
0.1Newto
SI
30
B:Trajectory 13
I
40
50
I
60
70
x-pos (mm)
I
I
80
90
30
40
50
70
60
x-pos (mm)
80
90
C: Trajectory 13
-
-,
0.8 I-
0.6
L0
L.
I
I
I
I
I
I
I
I
I
I
I
0
n
0
0.1
~~~0.2
0.2
0.10.
0.3
.
0.4
.
.
.
.
.
0.5
0.6
0.7
0.8
0.9
1
time(sec)
Figure 48: Time course of Fisometric(space,time) and Finertial(trajectory,time) for Trajectory 13. Part A:
Trajectory marked off in 100 msec increments with vectors representing directions and magnitudes of
Fisometric(space,time) Part B: Trajectory marked off in 100 msec increments with vectors representing
directions and magnitudes (scaled larger than part A so as to be visible) of Finertial(trajectory,time) Part
C: Magnitude of Fisometric(space,time) (dashed) and Finertial(trajectory,time) (solid) vs. time. Note that
Fisometric at most times and places is MUCH larger than what would be needed to actually accelerate the
limb along the trajectory (Finertial). Areas marked by asterisks are when the free-limb trajectory goes
outside of the measured isometric force range and should be ignored.
A:Trajectory
14
B:Trajectory14
1Newton
SNewton
120
30
40
70
60
x-pos (mm)
50
80
90
30
40
50
60
70
x-pos (mm)
80
90
C: Trajectory 14
/
/ /
//
1
\
\
t
0
0.6
LL
0
0.1
0.2
0.3
0.4
0.5
0.6
time(sec)
0.7
0.8
0.9
1
Figure 49: Time course of Fisometric(space,time) and Finertial(trajectory,time) for Trajectory 14. Part A:
Trajectory marked off in 100 msec increments with vectors representing directions and magnitudes of
Fisometric(space,time) Part B: Trajectory marked off in 100 msec increments with vectors representing
directions and magnitudes (scaled larger than part A so as to be visible) of Finertial(trajectory,time) Part
C: Magnitude of Fisometric(space,time) (dashed) and Finertial(trajectory,time) (solid) vs. time. Note that
Fisometric at most times and places is MUCH larger than what would be needed to actually accelerate the
limb along the trajectory (Finertial). Areas marked by asterisks are when the free-limb trajectory goes
outside of the measured isometric force range and should be ignored.
A:Trajectory 15
160
,"
lNewton
70
70
80
80
-
B:Trajectory 15
Newton
0 90
100
100
110
110
120
120
30
40
50
60
70
80
x-pos (mm)
C: Trajectory 15
90
I
30
40
50
60
70
x-pos (mm)
80
90
0.5
Ir
/i
0.4
Ii
I/
I
I
I
\
I
/I I
II
0.3
/
/
z
I
I
I
I
I
-
I
I
I
0.2
/
-
I
--. I
0.1
I
,
I
/;
-
t
I
0l
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
time(sec)
Figure 50: Time course of Fisometric(space,time) and Finertial(trajectory,time) for Trajectory 15. Part A:
Trajectory marked off in 100 msec increments with vectors representing directions and magnitudes of
Fisometric(space,time) Part B: Trajectory marked off in 100 msec increments with vectors representing
directions and magnitudes (scaled larger than part A so as to be visible) of Finertial(trajectory,time) Part
C: Magnitude of Fisometric(space,time) (dashed) and Finertial(trajectory,time) (solid) vs. time. Note that
Fisometric at most times and places is MUCH larger than what would be needed to actually accelerate the
limb along the trajectory (Finertial). Areas marked by asterisks are when the free-limb trajectory goes
outside of the measured isometric force range and should be ignored.
B:Trajectory 16
A:Trajectory
16
0.1Newton
F 3ewton
8C
E
090
0
0.
I
30
40
I
I
60
50
80
70
I
I
90
30
40
x-pos (mm)
50
60
70
x-pos (mm)
80
90
C: Trajectory 16
0.8
0.7
1-
-
0
LL
0.4
r
I
0-
n
0
0.1
'I
'I
0.
0.2
0.1-.·0.3
.
.
.
-.
.
.
0.4
0.5
0.6
0.7
0.8
0.9
1
time(sec)
Figure 51: Time course of Fisometric(space,time) and Finertial(trajectory,time) for Trajectory 16. Part A:
Trajectory marked off in 100 msec increments with vectors representing directions and magnitudes of
Fisometric(space,time) Part B: Trajectory marked off in 100 msec increments with vectors representing
directions and magnitudes (scaled larger than part A so as to be visible) of Finertial(trajectory,time) Part
C: Magnitude of Fisometric(space,time) (dashed) and Finertial(trajectory,time) (solid) vs. time. Note that
Fisometric at most times and places is MUCH larger than what would be needed to actually accelerate the
limb along the trajectory (Finertial). Areas marked by asterisks are when the free-limb trajectory goes
outside of the measured isometric force range and should be ignored.
A:Trajectory
17
B: Trajectory 17
0.1
2Newto4,
Newton
30
40
50
70
60
x-pos (mm)
80
90
30
40
50
70
60
x-pos (mm)
80
90
C:Trajectory 17
/ \
0.6
0.5
z-0.4
/-
•
x
a,
LL
r
r
I
r
I
I
n0
0.1
· _·I_·
0.3
0.2
0.4
0.5
0.6
time(sec)
·
0.7,
·
0.8
0.9
1
Figure 52: Time course of Fisometric(space,time) and Finertial(trajectory,time) for Trajectory 17. Part A:
Trajectory marked off in 100 msec increments with vectors representing directions and magnitudes of
Fisometric(space,time) Part B: Trajectory marked off in 100 msec increments with vectors representing
directions and magnitudes (scaled larger than part A so as to be visible) of Finertial(trajectory,time) Part
C: Magnitude of Fisometric(space,time) (dashed) and Finertial(trajectory,time) (solid) vs. time. Note that
Fisometric at most times and places is MUCH larger than what would be needed to actually accelerate the
limb along the trajectory (Finertial). Areas marked by asterisks are when the free-limb trajectory goes
outside of the measured isometric force range and should be ignored.
Power into hypothetical damper in model 2 (figure 13) for all trajectories
Note that in some of the trials the measured free-limb trajectory temporarily left
the area where isometric force was measured. The software, in this case, arbitrarily sets
Fisometric to zero (although not necessarily Finertial). Data in these regions is marked by
asterisks and should be disregarded (in some cases this results in discontinuities in the
power vs. time plots- this is just an artifact of not having isometric force data for those
time points).
A: Trajectory 1
80
85
E 95
o100
I
I
45
50
1io
40
I
I
55
60
x-pos (mm)
I
I
I
65
70
75
B: Trajectory 1
U.U,
-
0.04
-
0.03
-
0.02
I
E
0.01
E
-0.01
H
' ^^~r
0
I
0.1
I
I
I
I
I
I
I
I
I
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
time(sec)
Figure 53: Trajectory 1- Power into the proposed damper in model 2 for a typical trajectory. Part A
shows the trajectory marked off in 100 msec increments, and part B shows the power that would have to
go into the damper vs. time. By convention, negative power means that the damper was GENERATING
energy. A line is drawn at zero power for all time. Areas marked by asterisks are when the free-limb
trajectory goes outside of the measured isometric force range and should be ignored.
A: Trajectory 2
•t't
r
ou
70
E
90
-
95
-
100
-
105
-
110
30
-
40
50
60
70
80
x-pos (mm)
B: Trajectory 2
V.VU
0.05
0.04
0.03
0.02
0.01
-
-
-
.V>
-
i
_A
l
0
I
0.1
I II
0.2
_X
I
0.3
•
I
0.4
I
0.5
I
0.6
I
0.7
I
0.8
I
0.9
P
1
time(sec)
Figure 54: Trajectory 2- Power into the proposed damper in model 2 for a typical trajectory. Part A
shows the trajectory marked off in 100 msec increments, and part B shows the power that would have to
go into the damper vs. time. By convention, negative power means that the damper was GENERATING
energy. A line is drawn at zero power for all time. Areas marked by asterisks are when the free-limb
trajectory goes outside of the measured isometric force range and should be ignored.
__
A: Trajectory 3
d5U
K
85
00
E
110
-
115
-
120
-
I
I
I
I
1•cr
35
I
I
50
x-pos (mm)
B: Trajectory 3
0.
0.
0.
0.
0.
-0.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
time(sec)
Figure 55: Trajectory 3- Power into the proposed damper in model 2 for a typical trajectory. Part A
shows the trajectory marked off in 100 msec increments, and part B shows the power that would have to
go into the damper vs. time. By convention, negative power means that the damper was GENERATING
energy. A line is drawn at zero power for all time. Areas marked by asterisks are when the free-limb
trajectory goes outside of the measured isometric force range and should be ignored.
A: Trajectory 4
70
E
95
100
105
110
E
i
45
50
I
0.07
0.06
II
65
70
75
I
I
80
85
x-pos (mm)
B: Trajectory 4
^ ^^
u.u0
I
60
55
-
-
-
0.05
0.04
0.03
0.02
0.01
0
_n nl
0
I
I
I
·
·
·
·
·
·
·
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
time(sec)
Figure 56: Trajectory 4- Power into the proposed damper in model 2 for a typical trajectory. Part A
shows the trajectory marked off in 100 msec increments, and part B shows the power that would have to
go into the damper vs. time. By convention, negative power means that the damper was GENERATING
energy. A line is drawn at zero power for all time. Areas marked by asterisks are when the free-limb
trajectory goes outside of the measured isometric force range and should be ignored.
A: Trajectory 5
100
1A0
I
I
I
80
85
90
6
60
55
65
70
75
x-pos (mm)
B: Trajectory 5
V.035
0.03
/4"
0.025
0.02
F
0.015
0.01
E
0.005 1-
-n
M00
0
i
I
i
I
0.1
0.2
0.3
0.4
I
I
0.5
0.6
time(sec)
I
I
I
I
0.7
0.8
0.9
1
Figure 57: Trajectory 5- Power into the proposed damper in model 2 for a typical trajectory. Part A
shows the trajectory marked off in 100 msec increments, and part B shows the power that would have to
go into the damper vs. time. By convention, negative power means that the damper was GENERATING
energy. A line is drawn at zero power for all time. Areas marked by asterisks are when the free-limb
trajectory goes outside of the measured isometric force range and should be ignored.
A: Trajectory 6
_A
8U
r
95
F
100
E
Inci
1-q
56··
56
x-pos (mm)
60
62
B: Trajectory 6
I
o
_1
mU.U
I
0
0.1
0.2
0.3
0.4
0.5
0.6
time(sec)
0.7
0.8
0.9
1
Figure 58: Trajectory 6- Power into the proposed damper in model 2 for a typical trajectory. Part A
shows the trajectory marked off in 100 msec increments, and part B shows the power that would have to
go into the damper vs. time. By convention, negative power means that the damper was GENERATING
energy. A line is drawn at zero power for all time. Areas marked by asterisks are when the free-limb
trajectory goes outside of the measured isometric force range and should be ignored.
A: Trajectory 7
E
E
o
0
0.
>r
5
x-pos (mm)
B: Trajectory 7
rr nrr
U.VO
0.05
0.04
E 0.03
o 0.02
0.01
0
_n ni
0
I
q
I
I
I
I
I
0.1
0.2
0.3
0.4
0.5
0.6
0.7
I
0.8
I
0.9
1
time(sec)
Figure 59: Trajectory 7- Power into the proposed damper in model 2 for a typical trajectory. Part A
shows the trajectory marked off in 100 msec increments, and part B shows the power that would have to
go into the damper vs. time. By convention, negative power means that the damper was GENERATING
energy. A line is drawn at zero power for all time. Areas marked by asterisks are when the free-limb
trajectory goes outside of the measured isometric force range and should be ignored.
A: Trajectory 8
E
E
CL
o
0
x-pos (mm)
B: Trajectory 8
U.U4
-
F0.035
K
0.03
0.025
c
0.02
-
o 0.015
0.01
0.005
0
0n ft0
0
I
I
I
I
I
I
I
I
I
i
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
time(sec)
Figure 60: Trajectory 8- Power into the proposed damper in model 2 for a typical trajectory. Part A
shows the trajectory marked off in 100 msec increments, and part B shows the power that would have to
go into the damper vs. time. By convention, negative power means that the damper was GENERATING
energy. A line is drawn at zero power for all time. Areas marked by asterisks are when the free-limb
trajectory goes outside of the measured isometric force range and should be ignored.
100
A: Trajectory 9
,,
r
80
85
-
-
90
E
100
incl
-
II
50
%50
60
65
70
x-pos (mm)
B: Trajectory 9
,,,
u.u0
III
I
m
55
III
II
75
80
m
85
-
0.025
-
0.02
0.015
-
0.01
-
0.005
0
_n nnr,
0
I
I
I
I
0.1
0.2
0.3
0.4
I
I
0.5
0.6
time(sec)
I
I
I
i
0.7
0.8
0.9
1
Figure 61: Trajectory 9- Power into the proposed damper in model 2 for a typical trajectory. Part A
shows the trajectory marked off in 100 msec increments, and part B shows the power that would have to
go into the damper vs. time. By convention, negative power means that the damper was GENERATING
energy. A line is drawn at zero power for all time. Areas marked by asterisks are when the free-limb
trajectory goes outside of the measured isometric force range and should be ignored.
A: Trajectory 10
,,
OU
r
65
F
70
E
E
I>75
80
-
80
x-pos (mm)
B: Trajectory 10
65
x 10-3
20U
r
15
E
t
0
0.1
I
0.2
0.3
0.4
0.5
time(sec)
0.6
0.7
0.8
0.9
1
Figure 62: Trajectory 10- Power into the proposed damper in model 2 for a typical trajectory. Part A
shows the trajectory marked off in 100 msec increments, and part B shows the power that would have to
go into the damper vs. time. By convention, negative power means that the damper was GENERATING
energy. A line is drawn at zero power for all time. Areas marked by asterisks are when the free-limb
trajectory goes outside of the measured isometric force range and should be ignored.
102
A: Trajectory 11
- v
v
x-pos (mm)
B: Trajectory 11
=,• fir-
U.UO
0.04
0.03
E
0.02
-
0.01
-
A_n
1
0
0.1
1
1
1
0.2
0.3
0.4
1
I
0.5
0.6
time(sec)
I
I
I
I
0.7
0.8
0.9
1
Figure 63: Trajectory 11- Power into the proposed damper in model 2 for a typical trajectory. Part A
shows the trajectory marked off in 100 msec increments, and part B shows the power that would have to
go into the damper vs. time. By convention, negative power means that the damper was GENERATING
energy. A line is drawn at zero power for all time. Areas marked by asterisks are when the free-limb
trajectory goes outside of the measured isometric force range and should be ignored.
103
A: Trajectory 12
65
I-
E
E
cn75
0..
80
E
65
70
75
80
85
90
95
x-pos (mm)
B: Trajectory 12
^ ^^
U.U03
r
0.025
-
0.02
-
0.015
-
0.01
-
0.005
-
0r000;
0
I
I
I
I
T
I
I
I
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
E
0.9
1
time(sec)
Figure 64: Trajectory 12- Power into the proposed damper in model 2 for a typical trajectory. Part A
shows the trajectory marked off in 100 msec increments, and part B shows the power that would have to
go into the damper vs. time. By convention, negative power means that the damper was GENERATING
energy. A line is drawn at zero power for all time. Areas marked by asterisks are when the free-limb
trajectory goes outside of the measured isometric force range and should be ignored.
104
A: Trajectory 13
80
-
85
-
90
-
100
105
-
-
II
I I
"11
45
I
I
50
55
I
I
60
65
x-pos (mm)
B: Trajectory 13
^ ^^
I
70
I
I
75
80
/A
(
i'
(
0
_1'
-- U.U
I
0
0.1
0.2
0.3
0.4
0.5
0.6
time(sec)
0.7
0.8
0.9
1
Figure 65: Trajectory 13- Power into the proposed damper in model 2 for a typical trajectory. Part A
shows the trajectory marked off in 100 msec increments, and part B shows the power that would have to
go into the damper vs. time. By convention, negative power means that the damper was GENERATING
energy. A line is drawn at zero power for all time. Areas marked by asterisks are when the free-limb
trajectory goes outside of the measured isometric force range and should be ignored.
105
A: Trajectory 14
65
)-
E
E
o 75
o
.
80
85
90
-
-
-
0r
60
I
I
I
I
I
I
65
70
75
80
85
90
95
x-pos (mm)
B: Trajectory 14
^
^^
0.04
0.031
0.02
0.01
-0.01
0
0.1
0.2
0.3
0.4
0.5
0.6
time(sec)
0.7
0.8
0.9
1
Figure 66: Trajectory 14- Power into the proposed damper in model 2 for a typical trajectory. Part A
shows the trajectory marked off in 100 msec increments, and part B shows the power that would have to
go into the damper vs. time. By convention, negative power means that the damper was GENERATING
energy. A line is drawn at zero power for all time. Areas marked by asterisks are when the free-limb
trajectory goes outside of the measured isometric force range and should be ignored.
106
A: Trajectory 15
100
E
I
r0
6
85
84
86
87
88
89
90
91
x-pos (mm)
B: Trajectory 15
U.UJ
-
N
0.02
-
0.01
-0.01
F
-0.02
_n
n13
0
f
I
0.1
0.2
0.3
I I
0.4
i
0.5
I
0.6
0.7
I
0.8
II
III
0.9
1
time(sec)
Figure 67: Trajectory 15- Power into the proposed damper in model 2 for a typical trajectory. Part A
shows the trajectory marked off in 100 msec increments, and part B shows the power that would have to
go into the damper vs. time. By convention, negative power means that the damper was GENERATING
energy. A line is drawn at zero power for all time. Areas marked by asterisks are when the free-limb
trajectory goes outside of the measured isometric force range and should be ignored.
107
A: Trajectory 16
84
86
90
92
I
94
96
Inn
45
I
I
I
I
I
I
50
55
60
65
70
75
x-pos (mm)
80
85
90
B: Trajectory 16
f% .
n
0U.12
0.1
E
S
0.08
I
0.06
0.04
0.02
-0.02
_0
nA
v0
I
0.1
0.2
0.3
0.4
0.5
0.6
time(sec)
0.7
0.8
0.9
1
Figure 68: Trajectory 16- Power into the proposed damper in model 2 for a typical trajectory. Part A
shows the trajectory marked off in 100 msec increments, and part B shows the power that would have to
go into the damper vs. time. By convention, negative power means that the damper was GENERATING
energy. A line is drawn at zero power for all time. Areas marked by asterisks are when the free-limb
trajectory goes outside of the measured isometric force range and should be ignored.
108
A: Trajectory 17
A--
65
r
75
-
80
90
-
95
-
IIWMr
55
60
65
70
75
80
x-pos (mm)
85
90
SS
n
B: Trajectory 17
U.UI
0.06
0.05
0.04
-
* 0.03
~o
a-0.02
0.01
0
-0.01
no
0A
0
I
0.1
0.2
0.3
0.4
0.5
time(sec)
0.6
0.7
0.8
0.9
1
Figure 69: Trajectory 17- Power into the proposed damper in model 2 for a typical trajectory. Part A
shows the trajectory marked off in 100 msec increments, and part B shows the power that would have to
go into the damper vs. time. By convention, negative power means that the damper was GENERATING
energy. A line is drawn at zero power for all time. Areas marked by asterisks are when the free-limb
trajectory goes outside of the measured isometric force range and should be ignored.
109