IN PRESS: JAAX & HANNAFORD, ANNALS OF BIOMEDICAL ENGINEERING, 2002
A BIOROBOTIC STRUCTURAL MODEL OF THE MAMMALIAN
MUSCLE SPINDLE PRIMARY AFFERENT RESPONSE
Kristen N. Jaax1 and Blake Hannaford2
BioRobotics Laboratory, Departments of Bioengineering1 and Electrical Engineering2
University of Washington, Seattle, WA 98195-2500
ABSTRACT
A biorobotic model of the mammalian muscle spindle Ia response was implemented in
precision hardware. We derived engineering specifications from displacement, receptor
potential and Ia data in the muscle spindle literature, allowing reproduction of muscle
spindle behavior directly in the robot’s hardware; a linear actuator replicated intrafusal
contractile behavior, a cantilever-based transducer reproduced sensory membrane
depolarization, and a voltage-controlled oscillator encoded strain into a frequency signal.
Aspects of muscle spindle behavior not intrinsic to the physical design were added in
control software using an adaptation of Schaafsma’s mathematical model. We tuned the
response to biological ramp and hold metrics including peak, mean, dynamic index, time
domain response and sensory region displacement. The model was validated against
biological Ia response to ramp and hold, sinusoidal and fusimotor inputs. The response
with dynamic or static gamma motorneuron input was excellent across all studies. The
passive spindle response matched well in 5 of the 9 measures. Potential applications
include basic science muscle spindle research and applied research in prosthetics and
robotics.
Key Terms: Biomimetics, Mechanoreceptor, Ia, Gamma Motorneuron, Dynamic Index,
Fusimotor
Address correspondence to: Kristen Jaax, c/o Blake Hannaford, Department of Electrical
Engineering, University of Washington, Seattle, WA 98195-2500. Phone (206) 543-2197
fax (206) 221-5264 email: [email protected], cc: [email protected]
IN PRESS: JAAX & HANNAFORD, ANNALS OF BIOMEDICAL ENGINEERING, 2002
1 INTRODUCTION
Investigators have been studying the muscle spindle for many years, developing and
testing theories about the physiological origins of its unique transducer properties. One
means of testing these theories has been synthesizing them into a structural model, a set
of mathematical expressions that have direct analogs in the physiological system, to see if
the model exhibits muscle spindle-like behavior. A number of researchers have
recognized the potential of building these structural models in robotic hardware. The
primary goal for this breed of biorobotics researcher is to test the viability of biological
hypotheses by testing their ability to drive real systems replete with physical obstacles
such as friction and inertia. Spin-off applications are inherent to the nature of such a
project. Biorobotic devices are attractive candidates for prosthetics as they are designed
to use the language of the body to replicate its behavior. These devices also offer novel
mechanisms for engineering applications.
The robotic muscle spindle project was thus conceived with the following objectives: (a)
implementing a state-of-the-art structural model in precision robotic hardware for
prosthetics, robotics and basic science applications and (b) testing the physiological
faithfulness of this model as well as the viability of the biological theories which drive
the model by rigorously testing the model’s behavior against biological data from a wide
range of experimental protocols.
1.1 Prior Literature
1.1.1
Biological Muscle Spindles
The mammalian muscle spindle, Fig. 1, is a mechanoreceptor that resides in the body of
extrafusal muscle and transduces muscle length. Intrafusal muscle fibers span the length
of the spindle and are divided anatomically and functionally into a sensory region and a
contractile region, which lie in series. The contractile region is a muscle fiber aligned to
generate tension along the long axis of the spindle. The sensory region is a linearly
elastic spring devoid of contractile tissue. Group Ia afferent neurons wrap around the
sensory region, linearly transducing sensory region strain into receptor potential. This
analog potential is then encoded into an action potential train, the Ia response, whose
frequency is thought to be a function of the receptor potential and its first derivative12,13.
This frequency modulated spike train then travels down the Ia axon to the spinal cord.
There are three types of intrafusal fibers: static nuclear bag and nuclear chain fibers
transduce primarily position information while dynamic nuclear bag fibers transduce
primarily velocity information. Commands from the gamma motorneuron (γmn) descend
from the spinal cord and control contraction of the intrafusal muscle. Two types of γ
motorneurons exist, static γmns and dynamic γmns, which innervate position sensitive
fibers and velocity sensitive fibers, respectively.
IN PRESS: JAAX & HANNAFORD, ANNALS OF BIOMEDICAL ENGINEERING, 2002
Static Nuclear
Bag Fiber
Dynamic
Nuclear Bag
Fiber
Nuclear Chain
Fibers
Capsule
B. H. C. Matthews first proposed in the
1930’s that the position and velocity
sensitivity of the muscle spindle could
arise from differing mechanical properties
between the intrafusal muscle and the
sensory region19. Studies using
stroboscopic photomicroscopy4,25 and force
transducers14 to study intrafusal fibers
support this hypothesis, which forms the
foundation of the structural model
presented here.
1.1.2 Modeling
Researchers have been developing models
of the muscle spindle for decades. Many
linear models have been developed, but
exhibit limited ranges due to the spindle’s
Ia
nonlinear behavior26. Empirical nonlinear
Neuron
models are in common use in large
Output
neuromuscular models31 due to their
computational simplicity and broader
range. Structural nonlinear models, though
computationally intensive, offer a unique
opportunity in that specific model
Gamma Motor
behaviors can be correlated to analogous
Neuron Input
physiological mechanisms. A small
number of these models have been
Figure 1: Mammalian Muscle Spindle. Strain
published describing all or part of the
applied across organ is transduced into Ia output. muscle spindle9,24,29. One such model, the
γmn input contracts intrafusal fiber tissue at
Schaafsma model29, was based upon the
distal ends, modulating Ia response.
widely held theory that complex spindle
behavior arises from mechanical interaction between the intrafusal muscle tissue and the
sensory region. It models the primary (Ia) response of a dynamic (bag1) fiber and static
(bag2 and nuclear chain) fiber. Each fiber consists of a linear elastic sensory region in
series with a contractile region. The primary afferent output is computed as a function of
sensory region length and its first derivative. Our robotic muscle spindle model
incorporates parts of the Schaafsma model for aspects of spindle behavior not intrinsic to
the mechatronic design.
1.1.3 Robotics
Biorobotic devices are being developed to replicate a variety of aspects of the peripheral
motor control system. Projects include a robotic replica of the upper arm and pneumatic
artificial muscles8,16. The robotic muscle spindle project, initiated by Marbot and
Hannaford18, presents the first biorobotic model of a muscle spindle. This device offers
IN PRESS: JAAX & HANNAFORD, ANNALS OF BIOMEDICAL ENGINEERING, 2002
the precision engineering and validation required for using it both as a platform for
further biological research and as a sensor in prosthetic and robotic applications.
1.2
Approach
This article describes the design and performance of a muscle spindle model in which
biological behavior is captured through both the performance characteristics of
mechatronic hardware and the modeling algorithms of the control software. In the
Methods section we describe the design and implementation process by which we
integrated three robotic subsystems into a structural model of the muscle spindle.
Technical engineering details of the robotic subsystems are described elsewhere15.
Tuning and validation were performed in two independent stages. The first stage, tuning
the parameters against five data sets obtained from the literature, is presented in the first
half of the Results. The second stage, validating the fully tuned model against five
additional experiments from the literature, comprises the remainder of the Results.
In the Discussion section we evaluate the model’s successes and limitations as revealed
by the tuning and validation studies. We also comment on the significance of the model
including use of the biorobotic modeling technique and potential contributions to
biological theory raised through the modeling process.
2 METHODS
2.1 Design
2.1.1 Conceptual Design
2.1.1.1 Modeling Approach
In conceptualizing the robotic muscle spindle, we abstracted three essential muscle
spindle functions for hardware implementation: (a) the mechanical filtering produced by
intrafusal muscle contractile tissue, (b) the neural transduction from strain to receptor
potential, and (c) the encoding of receptor potential as an action potential spike train.
The medium for implementing each of these functions was selected from the repertoire of
available engineering technology using the selection criteria that it must (a) meet
performance specifications derived from biological studies on the analogous
physiological system, and (b) be miniature enough to mount the full robotic muscle
spindle in parallel to a human biceps muscle. Once the technologies were selected,
specific robotic systems were designed and implemented to capture as much of the
physiological functionality as possible in the mechanical and electrical behavior of the
hardware itself. Aspects of muscle spindle’s behavior not intrinsic to the electrical and
mechanical design were implemented in control software using an adaptation of the
Schaafsma model29.
IN PRESS: JAAX & HANNAFORD, ANNALS OF BIOMEDICAL ENGINEERING, 2002
Figure 2: Robotic Muscle Spindle. Contractile element hardware replicates movement of intrafusal
muscle. Sensory element transduces its strain to millivolt potential. Encoder printed circuit board
(PCB) converts millivolt potential to frequency modulated square wave.
The physical length of the resulting muscle spindle model is 23mm when the intrafusal
muscle is stretched to optimal length, where force generation is maximal, and 20 mm at
the zero firing length, where there is no Ia output. This is approximately twice the length
of a typical cat muscle spindle. Ancillary engineering hardware increases the dimensions
of the physical device, shown in Figure 2, to 10x1x1cm.
2.1.1.2 Model Framework
The conceptual framework for the model consists of a contractile element in series with a
linear elastic sensory element. External position inputs are applied as strain across the
whole system. The strain is then unequally distributed between the contractile element
and the linear elastic sensory region. The contractile element’s force production is a
complex function of its length, velocity and contraction level, while the sensory
element’s force production is a simple linear function of length. The resulting
instantaneous variations in the mechanical properties of the two elements result in the
mechanical filtering behavior of the muscle spindle in which the strain across the sensory
region is different from that applied across the whole muscle spindle.
The model output is generated as a function of sensory region strain. First, the receptor
potential is calculated as a linear function of sensory element strain, reproducing the
transducer function. Second, the model’s output signal, Ia firing frequency, is calculated
as a function of receptor potential and its first derivative, reproducing the encoder
function.
The robotic model is comprised of a single physical fiber that, depending on the software
controlling it, models one of two fiber types: dynamic or static. The control software’s
parameter values model the analogous intrafusal fiber: the dynamic fiber models the
dynamic nuclear bag, while the static fiber models a hybrid of the static nuclear bag and
nuclear chain fiber. Further, these fibers receive their sole efferent input from the
dynamic γmn or static γmn, respectively.
IN PRESS: JAAX & HANNAFORD, ANNALS OF BIOMEDICAL ENGINEERING, 2002
2.1.2 Design Implementation
2.1.2.1 Sensory Element Model
We used data from the experimental muscle spindle literature to create performance
specifications for the sensory element. These specifications include: (a) absolute
deflection amplitudes greater than 0.24 mm, suitable for a maximum 3:1 scale model, (b)
resolution better than 2 µm6, and (c) a linear response region at small deflections,
followed by stiffening and decreasing sensitivity at increasing amplitudes6.
The resulting design, Fig. 3, is a pair of
CANTILEVER
strain-gaged cantilevers. The base of
the cantilevers is rigidly mounted to a
nut that defines the interface between
the contractile element and the sensory
element, Fig. 2. The cantilevers are
connected directly to a pair of cables
that provide external strain inputs
across the full length of the robotic
STRAIN GAGE
muscle spindle. Strain between the
cable insertion and the cantilever base
5 MM
is transduced by electronic circuitry
into a millivolt potential. This millivolt
Figure 3: CAD Drawing of Sensory Element Design.
potential, representing the strain across
Displacement of cable with respect to cantilever base
the sensory region, is then converted
causes bending. Strain gages mounted to cantilevers
into a frequency-modulated spike train
transduce bending into millivolt potential change
and transmitted to the computer as the
analogous to Ia receptor potential.
output of the sensory element.
Engineering aspects of this design are described in Jaax et al.15. By meeting all of the
biologically-derived specifications, this robotic sensor reproduces the spindle’s strain-tomillivolt-potential transduction behavior directly in mechatronic hardware.
CABLE
Functionally, the sensory element output serves a dual role in the muscle spindle model.
First, it is the receptor potential analog and is used to calculate the muscle spindle output.
Second, it provides sensory information for the feedback control algorithm that drives the
contractile element.
2.1.2.2 Intrafusal Muscle Model
The muscle-like behavior of the contractile element is produced via a linear actuator
controlled by a muscle model-based software algorithm. Hence, the performance
requirement for the linear actuator is to respond to the controller’s commands rapidly
enough to reproduce the experimentally measured dynamics of intrafusal muscle. We
used published experimental data6,27 to identify the following biologically-motivated
performance specifications: (a) a rise time, defined as the time for actual position to
IN PRESS: JAAX & HANNAFORD, ANNALS OF BIOMEDICAL ENGINEERING, 2002
travel from 10% to 90% of desired position, of 22 msec for a 30mm/sec ramp and (b) a
maximum position error, defined as the maximum observed distance between the actual
and desired position during a ramp, of 0.3 mm during a 30mm/sec ramp and hold. The
latter specification ensures that the sensory element does not exceed its maximum
deflection. Using these specifications we designed and implemented a linear actuator and
controller. The resulting device successfully met the specifications with: (a) a rise time
on a 30 mm/sec ramp of 21 msec and (b) a maximum position error of 0.15 mm on a 30
mm/sec ramp. Engineering aspects of this design, Fig. 4, are described in Jaax et al.15
A software control algorithm supplies
muscle-like behavior to the lead screw
NUT
linear actuator. The muscle model
algorithm is adapted from an extrafusal
muscle fiber model developed by
LEAD
10 mm
Otten23. It calculates force as a function
SCREW
of velocity, length, and γmn input level.
In developing their muscle spindle
Figure 4: CAD Drawing of Contractile Element
model, Schaafsma et al.29 retuned the
Design. Motor rotates threaded rod, driving linear
travel of nut. Muscle model in the control algorithm
10 parameters of Otten’s extrafusal
(see text) generates muscle-like response to length
fiber model to mimic intrafusal fiber
and γmn inputs. (top housing removed for visibility)
dynamics. Our muscle model is an
adaptation of this intrafusal model. The equation for intrafusal force is:
MOTOR
k , F , v > 0
(1)
Fd = k a ,i Fa ,i Fv ,i Fq ,i + k p ,i Fp ,i + bi vi +  a i e i
 0, vi ≤ 0
where Fd is desired force, i is fiber type (1=dynamic fiber, 2= static fiber ), ka,i and kp,i
are maximum active and passive isometric force, respectively, Fa,i is active force
generated at current length (normalized), Fv,i is active force generated at current velocity
(normalized), Fq,i is active force generated by γmn stimulation rate (normalized), Fp,i is
passive force generated at current length (normalized), bi is passive damping, vi is
velocity of contractile region, and Fe is force enhancement, a term introduced by
Schaafsma et al.29 that creates a fixed positive offset in muscle force during lengthening.
Equations defining Fa, Fv, Fq, Fp are in Otten’s muscle model23. In tuning our model,
parameters were freed and tuned when justifiable on either biological grounds or due to
subsumption of the behavior into the mechatronic device, as described below.
Fig. 5 is a block diagram describing the software algorithm used to control the linear
actuator. The sensory element measures the actual force across the muscle spindle
model, Fa. The difference, Fd - Fa, is used as a force error signal, E, to control the
position of the linear actuator. Force errors arise from: (a) updates to Fd, the desired
force, calculated by the muscle model, (b) updates to C, the external position input, and
(c) control loop dynamics. A linear scaling factor, j, was implemented in the
“Mathematical Muscle Model” block of Fig. 5 to tune the muscle model force output, Fd,
to the stiffness of the sensory region. Actual forces generated across the biorobotic
muscle spindle never exceed 1 N, which is negligible compared to the 100-700 N
maximum force of the artificial extrinsic muscle it is designed to accompany8,16.
IN PRESS: JAAX & HANNAFORD, ANNALS OF BIOMEDICAL ENGINEERING, 2002
γ-mn input
Mathematical
Muscle
Model
External position
input, C
Nut position, B
Desired
Force,
Fd
+
-
Desired
Force
Position,
Error,
+
1/k
x
E
Convert force to
displacement
Physical
Plant
Controller
Position Controller
Sensory Region
k
Force, Fa
Convert displacement
to force
H
Feedback
Linearization
Nut
Position,
B
-
+
TRANSDUCTION
Compute Ia
Ia output
ENCODING
Sensory element
strain, ε
Figure 5: Block Diagram of Linear Actuator Controller. Algorithm compares actual force, FA, to
force predicted by muscle model, FD. The difference, E, is used as error signal to drive linear
actuator position. E arises from three sources: updates to FD from muscle model, external position
∆C), and dynamics of control loop.
inputs (∆
2.1.2.3 Encoder Model
The function of the encoder, translating receptor potential into a Ia action potential
frequency, is accomplished in two stages. The first stage, conversion from millivolt
receptor potential to a frequency modulated spike train, is done on a custom printed
circuit board (PCB) mounted directly to the spindle to minimize signal distortion15, Fig.
2. This signal is transmitted in the 1k–11k impulses/second (imp/sec) range to maximize
resolution and then rescaled in the computer. The second stage uses the following
algorithm adapted from Schaafsma et al.29 to convert the raw sensory element output into
a Ia signal:
(2)
P = ltp × d
i
i
i
Ia i = ptr × Pi + h × Pi
(3)
where Pi is receptor potential, ltpi is the conversion from sensory region length to
potential, di is the displacement of the sensory region beyond the zero firing length, ptr is
the conversion from receptor potential to Ia firing rate, h is rate sensitivity of encoding
from receptor potential to firing rate, and Iai is the Ia output firing rate. In the fully tuned
model, h is unidirectional, i.e. h=0 when dP/dt≤0, h=15 when dP/dt>0. This equation is
empirically derived from experimental data12 suggesting unidirectional rate sensitivity
across a broad frequency spectrum in the receptor potential to Ia output transfer function.
A 2nd order low pass filter with a cutoff frequency of 20 Hz was implemented on the first
derivative of sensory element strain to minimize propagation of noise extraneous to the
experimental protocol30. The 20 Hz cutoff frequency was selected based on Fourier
analysis of the Ia signal that revealed a significant noise source in the motion of the linear
actuator mechanism at frequencies just above 20 Hz. This choice is in agreement with
the opinion stated by PBC Matthews that “frequencies above 20 Hz were not really
relevant for motor control21.” Given that we are not examining external vibration
protocols, frequencies in excess of 20 Hz are unlikely to be due to the physiology we are
examining.
IN PRESS: JAAX & HANNAFORD, ANNALS OF BIOMEDICAL ENGINEERING, 2002
2.2 Experimental Methods
2.2.1 Linear Positioning Device
A linear positioning device (LPD) was designed and built to apply position inputs to the
robotic muscle spindle in a manner analogous to that used in experimental muscle spindle
studies15. The LPD has a 19mm stroke length, allowing 3:1 scaling of typical amplitudes
used in the muscle spindle literature4,25. The resolution of the LPD’s length sensor is
0.33µm, within 0.1µm of the highest resolution length data in the muscle spindle
literature25,27,4.
2.2.2 Experimental Protocols
In experiments where we reproduced biological experiments, close attention was paid to
accurately implementing position trajectories. For example, physiologists often report
stretch amplitudes in terms of extrafusal muscle displacement. In such cases, we assume
this stretch is proportionally transmitted to an 11.5mm long biological muscle spindle
without distortion, and thus apply a linear scaling factor to calculate the spindle’s stretch
amplitude. The spindle length offset for each experiment was selected by experimentally
identifying the robotic spindle length at which there was optimal correspondence between
the magnitude of the biological and robotic Ia response across multiple γmn activation
levels. These lengths are reported along with the length offset used in the biological
experiment, if available. Since the robotic muscle spindle is a 2:1 scale model, all
displacements from biological experimental protocols are doubled before being applied to
the biorobotic model.
3
RESULTS
Tuning and validation of the model against data from the muscle spindle literature were
performed in two independent stages. In the first stage we tuned model parameters to
five metrics describing the muscle spindle’s ramp and hold response. In the second stage
we validated the fully tuned model against five additional experiments including
experimental protocols and results not used in tuning studies.
3.1 Model Tuning Studies
This section shows the degree of similarity achieved between robotic and biological
results by tuning the parameters to replicate specific sets of biological data. Most model
parameters retained the values identified by Schaafsma et al.29. Changes from these
parameter values, Table I, were justified by one of two reasons: (a) the behavior was
subsumed by the mechatronics of the robotic muscle spindle or (b) there is a biologicallybased reason for the new value. Specific changes are described in the Discussion.
IN PRESS: JAAX & HANNAFORD, ANNALS OF BIOMEDICAL ENGINEERING, 2002
Table I: Parameter values changed during tuning
Name
New Value
Biologically Motivated
K2
.4
h
15pps(mV/s) -1 , Pi > 0

-1  0pps(mV/s) , Pi ≤ 0
Mechatronically Motivated
Fx
0 FU
Fe
0
b1
8.6x10-4 FU(mm/s)-1
b2
4.6x10-4 FU(mm/s)-1
Function
static fiber F-v slope
encoder rate sensitivity
cross-bridge rupture
force enhancement
dynamic fiber passive damping
static fiber passive damping
Ia Output (Hz)
Optimization of the model parameters focused primarily on reproducing three metrics,
mean, peak and dynamic index, from a ramp and hold experiment performed by Crowe
and Matthews3. Dynamic index (DI) is defined as the change in the Ia output between
the end of the ramp and 0.5 seconds after the ramp3. Fig. 6 a, b, and c show the results
for mean, peak and DI, respectively, overlaid on original biological data. The plots
present the metrics as a function of ramp velocity and γmn activation level. The
biological metrics were originally reported as the “approximate average for several
spikes3.” In an effort to reproduce this methodology, we applied a 7 Hz low pass filter to
the robotic Ia data before calculating
a.)
b.)
c.) Dynamic Index
Mean
Peak
the metrics. The mean difference
between the robotic and biological
300
300
300
metrics is –4.7±12.3 (S.D.) imp/s at
5mm/s and 1.1±19.7 imp/s across all
200
200
200
ramp speeds. The few notable
discrepancies occur at high velocities
including high robotic peak and DI
100
100
100
metrics under static γmn input, and
low robotic mean response in the
0
0
0
0 10 20 30
0 10 20 30
0 10 20 30
passive cases.
Velocity (m m /sec)
Time domain plots of the muscle
spindle’s Ia response allow its
characteristic morphology to be
observed and tuned. The robotic
(gray) and biological3 (black)
responses to an identical 5mm/s ramp
and hold were overlaid (Fig. 7) to
show the success of the tuning process.
In the original biological data the xsweep rate of the recording
Figure 6: Model Parameter Tuning Study. Ia output
metrics during 6mm amplitude ramp and hold
experiment at varying velocities: (a) mean response
during ramp input, (b) peak response, (c) dynamic
index (see text). Robotic muscle spindle response
(markers with lines) closely matched cat soleus data
(markers without lines, Crowe et al.3) for different
levels of γmn stimulation (‘+,’ 100 Hz dynamic, “*,”
100 Hz static, “o,” none). Displacements refer to
biological host muscle. Final length in biological tissue
(max. physiologic length) similar to robotic muscle
spindle (24.5 mm).
IN PRESS: JAAX & HANNAFORD, ANNALS OF BIOMEDICAL ENGINEERING, 2002
oscilloscope was a linear function of the muscle spindle position input3. Accordingly, the
x-axis time scale only applies to the hold region. We plotted the robotic data with a
similar x-axis distortion during the ramp (solid bar) to allow direct comparison of the
results. Note that all time domain Ia response plots are filtered with a 2nd order 60 Hz
low pass filter. At all γmn activation levels, the robotic model replicates the major
elements of the muscle spindle Ia response. First, the accuracy of the length gain is
evident in (a) the Ia output at ramp onset, (b) the Ia slope during the ramp, and (c) the Ia
magnitude during the hold. Second, the accuracy of the velocity gain is demonstrated by
the offset of the Ia response during the ramp at all three γmn activation levels.
In building a structural physical
model, we sought to accurately
reproduce the mechanical
deformations of the sensory and
contractile regions. Fig. 8 depicts the
displacement of the sensory region of
the robotic muscle spindle and the
displacement of a comparable point on
a biological muscle spindle4, 0.3 mm
from the spindle equator, during
identical ramp and hold experiments.
Figure 7: Model Parameter Tuning Study.
Note that as a 2:1 scale model, the
Comparison of Ia responses (top graph) during ramp
actual robot displacements are 2x the
and hold input (bottom graph). Robotic muscle
values presented here. Both the
spindle response (black) closely reproduces cat soleus
3
robotic and biological data exhibit a
muscle spindle response (gray, Crowe et al. ) under
varying γmn stimulation levels ((a) none (b) 70 Hz
peak displacement of 21 µm. Further,
dynamic, (c) 70 Hz static). Solid bar indicates region
the relative magnitude of the passive
where x axis is a function of position input, not time.
vs. dynamic γmn stimulated response
See text for details. Lengths refer to displacements of
is similar, with the robotic muscle
host muscle. Final length in biological tissue (max
physiological length) similar to robotic muscle spindle spindle exhibiting a slightly larger
(24.5 mm).
passive response. Finally, between 0
and 150 msec, the displacements of
both the robotic and biological sensory regions show an initial burst spike typical of
short-range stiffness.
3.2 Model Validation Studies
Once the robotic muscle spindle was tuned, we validated its performance by comparing
its behavior to a different set of five experiments obtained from the muscle spindle
literature. No parameter values were adjusted while performing these studies.
IN PRESS: JAAX & HANNAFORD, ANNALS OF BIOMEDICAL ENGINEERING, 2002
Figure 9 compares the robotic muscle
spindle’s ramp and hold response to
biological data from Boyd et al.2.
Under both dynamic and static γmn
stimulation the robotic response
replicates well the morphology of the
biological response. In the passive
case the morphology is similar,
although an unusually large initial gain
in the biological data produces a 45%
offset not found in the robotic data.
The morphological similarities
between the robotic and biological
data include length gain, velocity gain
and initial burst. The length gain
similarity can be seen both in the Ia
slope during the ramp and the steady
state Ia value during the hold. The
velocity gain similarity is most evident
in the Ia offsets during the ramp. Both
the time course and magnitude of the
initial burst are mimicked well under
Figure 8: Model Parameter Tuning Study. Sensory
region stretch during ramp and hold stretch applied
both static and dynamic γmn
across whole muscle spindle. (a) Robotic muscle
stimulation, although under passive
spindle sensory region stretch, (b) Input displacement
conditions the robotic time course is
applied across whole muscle spindle, (c) Displacement
too fast. Note that the data are
of cat tenuissimus muscle spindle tissue 0.3 mm from
normalized to the full depth of
spindle equator, just beyond sensory region (Dickson
et al.4). For all graphs, Left column: no γmn
modulation of that muscle spindle’s
stimulation, Right column: 100 Hz dynamic γmn
response under dynamic γmn
stimulation. Range and shape of sensory region
stimulation, robotic or biological, with
displacement closely matches biological data. Lengths
zero set as the minimum Ia value in
refer to displacements applied directly to biological
each individual response. This allows muscle spindle. Final length in biological tissue not
comparison of the morphology despite available to compare to robotic spindle length (24
mm).
substantial differences in Ia
amplitudes. In this experiment the
robotic data exhibits a range of 200 imp/s while the biological range is only 48 imp/s.
The robotic ramp and hold response also matched data from P.B.C. Matthews20, but the
normalization of Fig. 9 was not required. The only major discrepancy was a small
velocity gain in the robotic response under both dynamic γmn input and no γmn input
(passive), resulting in a smaller offset in the robotic Ia response during the ramp.
During a 2 mm peak-to-peak amplitude, 1 Hz sinusoidal input, the robotic muscle
spindle’s time domain Ia response closely matched biological data from Hulliger et al.11
under passive, maximal dynamic and maximal static γmn activation. Similarities
included a phase lead of approximately 80° across all γmn activation levels, dynamic γmn
IN PRESS: JAAX & HANNAFORD, ANNALS OF BIOMEDICAL ENGINEERING, 2002
input generating the maximum Ia depth of modulation, and zero Ia output in the passive
muscle spindle at lengths less than the “zero length.” Scaling of the robotic passive
response, though, was notably smaller than the biological response.
The effect of sinusoid amplitude on Ia
depth of modulation is shown in Fig.
10. The robotic response under both
static and dynamic γmn stimulation is
very similar to the biological behavior
reported by Hulliger et al.11, exhibiting
slopes indicative of the gain
compression phenomenon. These
include a steep linear slope indicating
high length gains at small amplitudes,
and a shallower slope indicating low
length gains at larger sinusoid
amplitudes. In the passive case,
however, the robotic response is much
smaller than its biological counterpart.
To test the origin of this, a sensitivity
analysis was done on the passive
damping parameter, b1, which had been
reduced from 9.91x10-3 to 8.60x10-4
FU (mm/s)-1 due to the intrinsic
damping of the mechatronics. Note
that 1 FU is the force required to
stretch the sensory region 1 mm.
Restoring b1 to its original value only
increased the passive response
amplitude by 5-8 imp/s.
Figure 9: Completed Model Validation Study (cf.
Boyd et al. 19772). Parameters tuned with data from
Crowe et al.3 (Fig. 5 and Fig. 6) and Dickson et al.4
(Fig. 7) applied to data from Boyd et al.. Comparison
of Ia response to ramp and hold position input
(bottom row) under different γmn stimulation levels:
Left column: none (passive), Center column: 100 Hz
dynamic (dynamic), Right Column: 100 Hz static
(static). Normalized robotic muscle spindle response
(top row) very closely matches normalized response of
cat tenuissimus muscle spindle under dynamic and
static γmn input (middle row), although amplitude of
passive is small. All Ia responses normalized to
maximum depth of modulation of response of
respective spindle, robotic or biological, under 100 Hz
dynamic γmn input. Positions refer to deformations
applied to host muscle. Final length data for
biological muscle spindle not available to compare to
robotic muscle spindle (24.4mm).
Figure 11 shows the effect of varying
γmn firing rate on the mean Ia
response. The robotic Ia data matches
well biological data reported by
Hulliger10 under both static and
dynamic γmn stimulation, with all
robotic data falling within the standard
deviation bars reported in the biological experiment. The robotic Ia slopes are very
similar to their biological counterparts, exhibiting only a 10 imp/s offset. Finally, the
saturation point to γmn input corresponds well at approximately 100 imp/s.
IN PRESS: JAAX & HANNAFORD, ANNALS OF BIOMEDICAL ENGINEERING, 2002
Typical noise in the robotic spindle
exhibits a gaussian distribution with a
standard deviation of ~10.1 imp/s.
This compares favorably with the
active (γmn stimulated) biological
spindle’s gaussian distributed noise
which exhibits a standard deviation of
~8 imp/s. The robotic spindle’s noise
is substantially larger than that of
passive biological spindles, though,
which exhibit gaussian noise with a
standard deviation of only 1 imp/s 22.
Figure 10: Completed Model Validation Study (cf.
Hulliger et al. 197711). Comparison of depth of
modulation of Ia output in response to varying
amplitude of sinusoidal stretch input. Robotic muscle
spindle data (dashed lines) closely match cat soleus
muscle spindle data (solid lines) during dynamic γmn
(“+”, 100 Hz dynamic) and static γmn (“o”, 100 Hz
static) stimulation, while the passive response (“*”, 0
γmn input) is about 25% of experimental amplitude.
Amplitudes refer to displacement of the host muscle.
Mean length of biological spindle (1-2 mm less than
physiological max) similar to robotic spindle (22 mm).
4
DISCUSSION
This article describes the design and
performance of a robotic muscle
spindle model intended for
applications ranging from basic
science to prosthetics and robotics. In
this discussion, we first examine the
model tuning, including which
parameters were tuned and why, as
well as its successes and limitations.
We then evaluate the validation studies
for the model’s ability to capture key elements of muscle spindle behavior in a more
general context. Finally, we conclude by presenting hypotheses about muscle spindle
function generated through the development and validation of this model.
4.1 Model Tuning
The initial parameters of the model included six determined by the mechatronics15 and
twelve intrafusal muscle model parameters29,23. Using these parameters, we compared
the model’s performance against five biological metrics characterizing the ramp and hold
response. When discrepancies arose, the responsible parameter was identified and
evaluated using the following criteria: (a) did evidence regarding the muscle spindle’s
physiology or anatomy support changing the parameter value, and (b) was this parameter
duplicated between the mechatronics and the software controller? If either criterion was
met, the parameter was freed and tuned accordingly.
4.1.1 Mechatronically Motivated Parameter Changes
The first modification resulting from the mechatronics was elimination of the software
algorithm modeling short-range stiffness29. We instead modeled it with a physically
analogous mechanism: stiction. In the biological muscle spindle, short-range stiffness is
thought to arise from temporary persistence of bound cross-bridges13. In our linear
actuator, short-range stiffness arises from the temporary persistence of a surface bond
IN PRESS: JAAX & HANNAFORD, ANNALS OF BIOMEDICAL ENGINEERING, 2002
between the nut and lead screw. The success of this physical model in producing an
initial burst by transmitting initial displacements directly to the sensory region is
demonstrated by the sensory region displacement, Fig 8, and the Ia response, Fig 9.
Figure 11: Completed Model Validation Study (cf.
Hulliger 197910). Comparison of effect of varying γmn
stimulation level on Ia response. Robotic muscle
spindle data (dotted lines) match slope and saturation
point of cat soleus muscle spindle response (solid lines,
error bars and shading indicate std. dev.) under two
different types of γmn stimulation (dynamic “+” and
static “*”). Muscle spindle held at constant length
throughout all experiments. Biological muscle spindle
length (2 mm less than physiological max) similar to
robotic muscle spindle (22.5 mm). Note that robotic
data exactly overlie biological data if inequality
allowed between length under static (23mm) and
dynamic (22mm) γmn input.
The second mechatronically motivated
change was force enhancement, Fe,
which had been added by Schaafsma et
al.29 to the original Otten muscle
model23 as a discontinuous force offset
term: a positive constant in
lengthening and zero in shortening
(Eq. 1). We again removed Fe
because its effect was to increase the
eccentric force-velocity term, which in
the dynamic fiber is already near
maximum. Further, the discontinuity
introduces instability in the feedback
control system for the muscle model.
Finally, since the mechanical plant has
intrinsic damping, the passive damping
term, b1, was redundant and we
reduced its value accordingly.
4.1.2 Biologically Motivated
Parameter Changes
The first change was to h, the encoder
rate sensitivity term, which was
increased and made unidirectional
12
based on biological evidence , allowing it to occupy the functional role of the absent Fe
term. In the past, bi-directional rate sensitivity has been incorporated into both ion
channel level transducer models17 and high level encoder models9,29. We observed,
though, that a large bi-directional rate sensitivity led to large sustained non-physiological
Ia undershoots during falling receptor potentials, e.g. ramp cessation. Experimentation
with our model revealed that eliminating h just during falling receptor potentials allowed
the Ia output to maintain its velocity-dependent offset during the ramp, while eliminating
the undershoot on ramp cessation.
We identified two studies in the biological literature with data to support the theory of
unidirectional encoder rate sensitivity. Hunt et al.12 overlaid on top of an actual Ia
response a theoretical Ia response predicted as a linear function of receptor potential. The
actual Ia response was much greater than predicted during rising receptor potentials, but
corresponded well to the predicted value during falling receptor potentials. Fukami’s
data showed similar results for snake muscle spindles5. Hunt and Gladden also observed
in reviews that Ia output during stretch is proportionally greater than receptor potential
IN PRESS: JAAX & HANNAFORD, ANNALS OF BIOMEDICAL ENGINEERING, 2002
predicts6,13, although neither explicitly addressed Ia output during shortening. Based on
this evidence, we implemented the following amendment to Eq. 3:
15, Pi > 0
Ia = ptr × Pi + h × Pi , h 
(4)
 0, Pi ≤ 0
The second parameter changed was K2, the slope of the static fiber force-velocity curve
in Eq. 5 below. The parameter changes above, notably to Fe and h, were tuned to the
dynamic fiber, resulting in an excessive velocity gain in the static fiber. Accordingly, we
sought a parameter to selectively decrease static fiber sensitivity during stretch. The
optimal choice was the static fiber force-velocity (F-v) relationship23:
1 − v / V max 2

,v ≥ 0

1 + v /( K 2 × V max 2 )
(5)
Fv 
1 + v / V max 2
e2 − (e2 − 1)
,v < 0

1 − 7.56 v / (K 2 × V max 2 )
where: Fv is the force due to velocity, v is velocity, Vmax2 is the maximum static fiber
velocity, e2 is maximum static fiber force due to velocity, and K2 is the slope of the static
fiber force-velocity curve. K2 was selected because its value for intrafusal muscle has
not been measured and available evidence suggests extremely low viscosity in the static
fiber, e.g. fast myosin isoforms6, driving in the nuclear chain fiber13, and extremely small
dynamic indices3. Based on this biological support we increased K2 from 0.25 to 0.4.
4.1.3 Quality of Fit
The goal of the tuning process was to match the model’s output to five different measures
of the biological muscle spindle’s ramp and hold response. The results for these five
measures, which span quantitative metrics (Fig. 6), time domain morphology (Fig. 7),
and physical displacements of the sensory region (Fig. 8), show a strong fit between
robotic and biological data, particularly on 5 mm/s ramps. All responses under both
dynamic γmn input and no γmn input (passive) are quite accurate at all speeds, reflecting
a high quality of fit for the dynamic fiber. At higher velocities, though, under static γmn
input the static fiber exhibits an excessive velocity gain. This is because the robotic
spindle’s intrinsic damping makes it difficult to replicate the static fiber’s extremely low
velocity gain at high velocities. Sources of damping in the static muscle model, b2 and
K2, were tuned to minimize damping. A sensitivity analysis on b2, K2, and e2, the static
force-velocity curve’s maximum value, showed that further changes would not
appreciably lower the peak and dynamic index metrics.
The physical displacement tuning study (Fig. 8) was included to (a) ensure that the major
Ia response features are present in the dynamics of the intrafusal muscle model’s physical
displacement2,4,19 and (b) tune the range of the sensory region displacement to match the
biological data. The similarity between the robotic and biological displacements (Fig. 8)
confirms the structurally analogous origin of our muscle spindle model behavior.
4.1.4 Muscle Length
IN PRESS: JAAX & HANNAFORD, ANNALS OF BIOMEDICAL ENGINEERING, 2002
In tuning the robotic muscle spindle, it quickly became apparent that accurately
reproducing the baseline offset in spindle length is an important factor in replicating the
biological Ia response. This phenomenon arises from several factors. First, the
nonlinearity of the muscle force-length relationship across different γmn activation levels
causes the relative initial values of the Ia response to change with length. Second, the
relationship between spindle position sensitivity and length varies with γmn stimulation,
with the passive spindle alone exhibiting a substantial increase in sensitivity with length.
Finally, the passive muscle spindle has zero Ia response below its zero firing length. To
accommodate this, for each experiment we repeated the test at 5 different baseline length
offsets throughout the robotic muscle spindle’s working range. We then used the relative
Ia amplitudes across the various γmn activation levels to determine which length offset
best corresponded to the length offset of the biological muscle spindle when the data
were collected.
4.2 Validation
To validate our robotic muscle spindle we tested the fully tuned model under novel
circumstances, e.g. fusimotor input and sinsusoidal position input studies from the cat
muscle spindle literature, to examine its general applicability. No modifications were
made while performing these studies. The only variable adjusted to get the best match
was the length offset at which each experimental protocol was applied.
The ramp and hold validation studies demonstrated that, under active γmn stimulation,
the robotic muscle spindle exhibited strong similarity to the results of Boyd et al.2 and
Matthews20, while under passive conditions the robotic Ia amplitude was small. This
indicates that, for the case of active γmn input, in a generalized ramp and hold experiment
the model is able to replicate both the position and velocity gains as well as the initial
burst of the biological Ia response.
The data in Fig. 9 were normalized due to range differences that we suspect result from
tuning our model to muscle spindles with larger depths of modulation than the muscle
spindles used in Boyd et al.2. When we compare the robotic muscle spindle’s behavior to
data20 from P. B. C. Matthews, the author who published the data used for tuning3, we
find that the robotic muscle spindle’s range is quite accurate.
Sinusoidal experiments test whether the robotic muscle spindle model is complete
enough to reproduce a range of muscle spindle behaviors beyond its tuning studies. The
model’s time domain sinusoidal response test was successful. It showed good
correspondence to the biological response, including sinusoidal phase lead and the
passive biological spindle’s zero firing length. The scale of the robotic passive response,
though, is smaller than the biological response.
The second sinusoidal study (Fig. 10) was included to test the gain compression
phenomenon. Biological muscle spindles exhibit a linear range with high position gains
at small amplitude stretches while the cross-bridges are still bound. At larger stretches
the cross-bridges rupture and there is a lowering of the position gain. This test of the
IN PRESS: JAAX & HANNAFORD, ANNALS OF BIOMEDICAL ENGINEERING, 2002
robotic muscle spindle’s ability to predict new results was successful. It produced this
behavior very nicely under both dynamic and static γmn stimulation, not only matching
the depth of Ia modulation very well, but also exhibiting a distinct linear range. The
passive robotic data, however, are again much smaller than the biological data.
Finally, the fusimotor validation study showed that the robotic muscle spindle is able to
predict Ia output at various frequencies of γmn stimulation (Fig. 11). The model’s
response matches the slope, magnitude and saturation point of the biological response
under both types of γmn stimulation, static and dynamic.
We feel that the ability of the γmn stimulated robotic muscle spindle to predict the
biological Ia response across these five validation studies is indicative of its general
accuracy in replicating the behavior of the active biological muscle spindle under these
types of experimental protocols.
4.2.1 Limitations
Although in 5 of the 9 measures the robotic muscle spindle’s passive response was
accurate, in the remaining four cases its amplitude was much smaller than the biological
response, representing the only major limitation of the model’s general applicability. We
identified three possible sources for this behavior: failure to correctly identify the length
offset, a missing term in the passive model and stretch activation.
The length offset theory comes from the fact that, unlike the γmn stimulated spindle, the
passive spindle Ia position sensitivity rises as a function of spindle length. Performing
these studies at a longer length would likely restore the relative amplitudes under the
three γmn input cases: passive, dynamic and static. The absolute magnitude of the Ia
response would then exceed the biological data, but such variability in scaling is observed
in biological data1. This explanation is appealing since only some of the passive
experiments exhibited low output amplitudes.
An absent term in the passive muscle spindle model is the second possibility. Careful
examination of the passive sinusoidal time domain response suggests it has insufficient
phase lead, indicative of a missing damping term. Sensitivity analysis calculations,
confirmed by experimentation, showed that increasing passive damping by a factor of 10
only increases the passive Ia depth of modulation by 5-8 Hz during a 1 mm sinusoid.
Since this change is so slight and would have equal effect in the active spindle, we
concluded that the passive damping term was not contributing substantially to the small
passive response.
Stretch activation is the final possibility. If indeed the act of stretching a passive
intrafusal fiber can lead to contraction4, as some data suggests27, this could account for
the four biological experiments whose passive Ia response amplitude we were unable to
replicate.
Other limitations to the model’s fidelity include absence of creep, which is the
IN PRESS: JAAX & HANNAFORD, ANNALS OF BIOMEDICAL ENGINEERING, 2002
phenomenon of slow Ia firing rate decay on ramp cessation, and excessive noise in the
passive response.
4.3 Summary of Contributions
4.3.1 First Biorobotic Muscle Spindle Model
Our device and its prototype18 are the first muscle spindle models to be built using the
biorobotic modeling technique. This technique offers several unique advantages over
traditional software modeling including (a) rigorous adherence to all physical laws, (b)
insights gained through implementing concepts in physical hardware, (c) the ability to
apply physical inputs directly to the model, (d) educational advantages of having students
physically interact with the model and (e) creating a working device that is then available
for other applications.
The biorobotic modeling technique enhanced the results of this project in several
respects. First, we recognized that the discontinuity introduced by intrafusal muscle force
enhancement made feedback control of our muscle model extremely difficult. From this,
we postulate that similar difficulties in a motor control stretch reflex loop might be
created by a discontinuity in the muscle spindle’s velocity gain. From this we
hypothesize that the ideal muscle spindle design would not include a discontinuous force
enhancement term. Second, by building in-house a Linear Positioning Device to apply
position inputs, we gained insight into the bandwidth of our model as well as the
technology with which the biological data were collected. Third, since our model is
physically realized in robust robotic hardware, we can install it on a robot or prosthetic.
This feature is especially significant for researchers developing biologically accurate
biorobotic models of the stretch reflex.
4.3.2 Potential Applications to Biological Theory
Ideally, the modeling process is closely coupled with experimentation. We have drawn
extensively upon the work of experimenters to develop and validate this model and in this
final section we hope to offer something in return. While developing this model, two
issues arose from which we wish to postulate new hypotheses about muscle spindle
mechanisms. The first issue is force enhancement, implemented in the Schaafsma model
as a discontinuous term that produces a constant positive force offset during lengthening
that is absent during shortening. We suspect that this term would produce a discontinuity
in the muscle spindle’s velocity-Ia output transfer function. Such a discontinuity can
introduce instability in closed loop control systems. We therefore hypothesize that, if
force enhancement does occur in the intrafusal fiber, it has a more continuous form, e.g.
sigmoidal.
The second hypothesis we propose is unidirectional rate sensitivity in the encoding
process. Symmetrical rate sensitivity between receptor potential and Ia frequency led to
non-physiological large undershoots on ramp cessation. Investigation of biological data
on the encoding process12 supports the hypothesis that this rate sensitivity is indeed only
IN PRESS: JAAX & HANNAFORD, ANNALS OF BIOMEDICAL ENGINEERING, 2002
present during increasing receptor potentials, not decreasing. We speculate that the
mechanism underlying this might be depression of action potential firing thresholds12
which occurs only during positive rates of change in receptor potentials. Unidirectional
rate sensitivity was implemented in our model and successfully eliminated undershoots
on ramp cessation. Hence, we hypothesize that the encoding function exhibits only
unidirectional rate sensitivity and encourage further experimentation to test this theory.
The final element we wish to comment on is a functional implication of the relative
length sensitivities of the muscle spindle. In both the robotic muscle spindle model and
biological muscle spindles7, passive position sensitivity increases substantially as a
function of length while active position sensitivity increases only slightly (biological with
dynamic γmn input), remains constant (biological with static γmn input), or decreases
slightly (robotic) as a function of length. These relative effects in which the γmn input
stabilized the position sensitivity7 made it important to replicate the length offset of the
biological muscle spindle when attempting to match the relative responses of the passive
and active spindles. Such effects may also contribute to biological phenomenon such as
the dependence of ankle joint motion sensitivity on extensor muscle length, observed in
the passive limb28.
ACKNOWLEDGEMENTS
We wish to thank Pierre-Henry Marbot for his intellectual contributions to the
development of this model. This study was supported by a Whitaker Foundation
Graduate Fellowship to K.N. Jaax.
REFERENCES
1
Botterman, R. B. and E. Eldred. Static stretch sensitivity of Ia and ii afferents in the cat's
gastrocnemius. Pflügers Archive. 395:204-211, 1982.
2
Boyd, I. A., M. H. Gladden, P. N. McWilliam and J. Ward. Control of dynamic and
static nuclear bag fibres and nuclear chain fibres by gamma and beta axons in isolated cat
muscle spindles. J. Physiol. (Lond.). 265:133-162, 1977.
3
Crowe, A. and P. B. C. Matthews. The effects of stimulation of static and dynamic
fusimotor fibres on the response to stretching of the primary endings of muscle spindles.
J. Physiol. (Lond.). 174:109-131, 1964.
4
Dickson, M., M. H. Gladden, D. M. Halliday and J. Ward. Fusimotor mechanisms
determining the afferent output of muscle spindles. Prog. Brain Res. 80:9-17, 1989.
5
Fukami, Y. Receptor potential and spike initiation in two varieties of snake muscle
spindles. J. Neurophysiol. 41:1546-1556, 1978.
6
Gladden, M. H. Mechanical factors affecting the sensitivity of mammalian muscle
spindles. Trends Neurosci. 9:295-297, 1986.
IN PRESS: JAAX & HANNAFORD, ANNALS OF BIOMEDICAL ENGINEERING, 2002
7
Goodwin, G. M., M. Hulliger and P. B. C. Matthews. The effects of fusimotor
stimulation during small amplitude stretching on the frequency-response of the primary
ending of the mammalian muscle spindle. J. Physiol. (Lond.). 253:175-206, 1975.
8
Hannaford, B., J. M. Winters, C. P. Chou and P. H. Marbot. The anthroform biorobotic
arm: A system for the study of spinal circuits. Ann. Biomed. Eng. 23:399-408, 1995.
9
Hasan, Z. A model of spindle afferent response to muscle stretch. J. Neurophysiol.
49:989-1006, 1983.
10
Hulliger, M. The responses of primary spindle afferents to fusimotor stimulation at
constant and abruptly changing rates. J. Physiol. (Lond.). 294:461-482, 1979.
11
Hulliger, M., P. B. C. Matthews and J. Noth. Static and dynamic fusimotor action on
the response of Ia fibers to low frequency sinusoidal stretching of widely ranging
amplitude. J. Physiol. (Lond.). 267:811-838, 1977.
12
Hunt, C. and D. Ottoson. Impulse activity and receptor potential of primary and
secondary endings of isolated mammalian muscle spindles. J. Physiol. (Lond.). 252:259281, 1975.
13
Hunt, C. C. Mammalian muscle spindle: Peripheral mechanisms. Physiol. Rev. 70:643663, 1990.
14
Hunt, C. C. and R. S. Wilkinson. An analysis of receptor potential and tension of
isolated cat muscle spindles in response to sinusoidal stretch. J. Physiol. (Lond.).
302:241-262, 1980.
15
Jaax, K. N., P. H. Marbot and B. Hannaford. "Development of a biomimetic position
sensor for robotic kinesthesia." In Proceedings of the 2000 IEEE/RSJ International
Conference on Intelligent Robots and Systems. Takamatsu, Japan: IEEE. 2000. pp. 12551260.
16
Klute, G. K., J. M. Czerniecki and B. Hannaford. "Mckibbon artificial muscles:
Pneumatic actuators with biomechanical intelligence." In Proceedings of the
IEEE/ASME 1999 Intl. Conf. on Advanced Intelligent Mechatronics. Atlanta, GA: 1999.
pp.
17
Kruse, M. N. and R. E. Poppele. Components of the dynamic response of mammalian
muscle spindles that originate in the sensory terminals. Exp. Brain Res. 56:359-366,
1991.
18
Marbot, P. H. and B. Hannaford. "The mechanical spindle: A replica of the mammalian
muscle spindle." In Proceedings of the IEEE Conference on Engineering in Medicine and
Biology. San Diego, CA: IEEE. 1993.
IN PRESS: JAAX & HANNAFORD, ANNALS OF BIOMEDICAL ENGINEERING, 2002
19
Matthews, B. H. C. Nerve endings in mammalian muscle. J. Physiol. (Lond.). 78:1-53,
1933.
20
Matthews, P. B. C. The differentiation of two types of fusimotor fibre by their effects
on the dynamic response of muscle spindle primary endings. Q. J. Exp. Physiol. 47:324333, 1962.
21
Matthews, P. B. C. Review lecture: Evolving views on the internal operation and
functional role of the muscle spindle. J. Physiol. (Lond.). 320:1-30, 1981.
22
Matthews, P. B. C. and R. B. Stein. The regularity of primary and secondary muscle
spindle afferent discharges. J. Physiol. (Lond.). 202:59-82, 1969.
23
Otten, E. A myocybernetic model of the jaw system of the rat. J. Neurosci. Methods.
21:287-302, 1987.
24
Otten, E., K. A. Scheepstra and M. Hulliger. "An integrated model of the mammalian
muscle spindle." In Proceedings of the Symposium on Alpha and Gamma Motor
Systems. London: Plenum Press. 1995. pp. 294-301.
25
Ottoson, D. and G. M. Shepherd. Length changes within isolated frog muscle spindle
during and after stretching. J. Physiol. (Lond.). 207:747-759, 1970.
26
Poppele, R. E. and R. J. Bowman. Quantitative description of linear behavior of
mammalian muscle spindles. J. Neurophysiol. 33:59-72, 1970.
27
Poppele, R. E. and D. C. Quick. Stretch-induced contraction of intrafusal muscle in the
muscle spindle. Journal of Neuroscience. 1:1069-1074, 1981.
28
Refshauge, K. M. and R. C. Fitzpatrick. Perception of movement at the human ankle:
Effects of leg position. J. Physiol. (Lond.). 488:243-248, 1995.
29
Schaafsma, A., E. Otten and J. D. van Willigen. A muscle spindle model for primary
afferent firing based on a simulation of intrafusal mechanical events. J. Neurophysiol.
65:1297-1312, 1991.
30
Usui, S. and I. Amidror. Digital low-pass differentiation for biological signal
processing. IEEE Trans. Biomed. Eng. 29:686-693, 1982.
31
Wallace, K. R. and G. K. Kerr. A numerical simulation of muscle spindle ensemble
encoding during planar movement of the human arm. Biol. Cybern. 75:339-350, 1996.