Neural-Body Coupling for Emergent Locomotion: A Musculoskeletal

2011 IEEE/RSJ International Conference on
Intelligent Robots and Systems
September 25-30, 2011. San Francisco, CA, USA
Neural-Body Coupling for Emergent Locomotion:
a Musculoskeletal Quadruped Robot with Spinobulbar Model
Yasunori Yamada, Satoshi Nishikawa, Kazuya Shida, Ryuma Niiyama, and Yasuo Kuniyoshi
Abstract— To gain a synthetic understanding of how the body
and nervous system co-create animal locomotion, we propose an
investigation into a quadruped musculoskeletal robot with biologically realistic morphology and a nervous system. The muscle
configuration and sensory feedback of our robot are compatible
with the mono- and bi-articular muscles of a quadruped animal
and with its muscle spindles and Golgi tendon organs. The
nervous system is designed with a biologically plausible model
of the spinobulbar system with no pre-defined gait patterns such
that mutual entrainment is dynamically created by exploiting
the physics of the body. In computer simulations, we found that
designing the body and the nervous system of the robot with
the characteristics of biological systems increases information
regularities in sensorimotor flows by generating complex and
coordinated motor patterns. Furthermore, we found similar
results in robot experiments with the generation of various
coordinated locomotion patterns created in a self-organized
manner. Our results demonstrate that the dynamical interaction
between the physics of the body with the neural dynamics
can shape behavioral patterns for adaptive locomotion in an
autonomous fashion.
I. I NTRODUCTION
Through evolutionary processes, the animal morphology
and nervous system have mutually adapted themselves in
order to achieve efficient sensorimotor integration within
the environment. As a result, various complex behaviors
marked by efficiency in energy consumption as well as
self-organization can emerge from dynamical interactions
between the body, the neural system, and the environment.
These skills are possible because the neural system exploits
the physics of the body on the one hand, while on the other
hand, the body dynamics structure the neural dynamics via
sensory stimuli. This constitutes a fundamental property of
embodied intelligence [1][2][3].
Recently, many researchers have followed this line of
investigation to better understand the mechanisms underlying animal locomotor skills in order to apply them into
robots [4][5]. Particular attention has been focused on the
central pattern generator (CPG) to replicate animal locomotion in biologically-inspired robots [6][7]. For instance,
the dog-like Tekken series [8] can accomplish stable locomotion pattern using sensory feedback whereas the insectlike AMOS-WD06 [9] can generate various complex behaviors by exploiting the chaotic properties of CPG models.
Y. Yamada, K. Shida, and Y. Kuniyoshi are with Department of
Mechano-Informatics, Graduate School of Information Science and
Technology, The University of Tokyo, S. Nishikawa is Graduate School
of Interdisciplinary Information Studies, The University of Tokyo, R.
Niiyama is Robot Locomotion Group, Computer Science and Artificial
Intelligence Lab, MIT, Cambridge, USA. {y-yamada, nisikawa,
shida, kuniyosh}@isi.imi.i.u-tokyo.ac.jp and
[email protected]
978-1-61284-455-8/11/$26.00 ©2011 IEEE
However, these robots are either too rigid or linearly joint
controlled by electromagnetic motors. In contrast, an animal’s musculoskeletal system is comprised of a complex and
redundant structural morphology with nonlinear materials for
visco-elastic muscle-tendon tissues, and it has no sensors or
actuators to directly sense and control its joints [10]. Thus
these robots are difficult to explain the role of the body as
musculoskeletal system for locomotion behaviors.
On the other hand, several researchers have focused
on the animal body as musculoskeletal system. Tsujita et
al. [11] achieved stable locomotion with a musculoskeletal
quadruped robot, while Verrelst et al. [12] developed a
bipedal musculoskeletal robot that had a walk-driven antagonistic mechanism. Although these robots were driven by
an antagonistic mechanism with mono-articular pneumatic
muscles, various kinds of animals very commonly have biarticular muscles, which are two cross joints that add torque
to both joints simultaneously [13]. Therefore, these robots
do not replicate this important aspect of biological musculoskeletal systems. In contrast, bipedal musculoskeletal
robots with both mono- and bi-articular muscles based on
the animals’ muscle configurations have been developed and
have performed dynamic motions [14][15]. However, the role
of the nervous system has never been considered in controlling these robots, and how locomotion is achieved through
their body and nervous system has not been identified.
We propose to investigate this issue in a quadruped robot
with biologically-realistic body and nervous system, both in
computer simulations and robot experiments. We make three
contributions. First, we developed a biologically realistic
musculoskeletal quadruped robot in terms of actuator and
its configuration, sensory feedback, and nervous system. This
robot allows us to investigate how the body and nervous system co-create locomotion behaviors. Second, we quantified
the contribution of the morphology in structuring sensorimotor information via embodied interaction. We discovered
through computer simulations and robot experiments that the
sensory information structure shaped by the biologicallyrealistic body can produce complex and coordinate motor
patterns via neural-body coupling. We also suggested the
functional role of bi-articular muscles from the perspective
of structuring informational regularity. Third, we showed that
various and coordinated locomotor patterns could emerge
from dynamic interaction with body and nervous system with
no pre-defined coordination circuit.
Our results suggest that the proposed biological strategy
has a more general nature to enable a wide range of dynamical locomotor patterns emerging from neural-body coupled
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TABLE I
S PECIFICATION OF THE ROBOT.
actuators
passive elements
valve
pressure sensor
potentiometer
CPU board
OS
mass of the robot
materials
Fig. 1.
Quadruped musculoskeletal robot.
length of body
shoulder - elbow
elbow - forefoot
hip - knee
knee - ankle
ankle - hind foot
Fig. 2.
Muscle configuration. The symbols are, LD: latissimus dorsi
muscle, DEL: deltoid muscle, TB: triceps brachii muscle, BRA: brachialis
muscle, GMAX: gluteus maximus muscle, IL: iliopsoas muscle, RF: rectus
femoris muscle, BF: biceps femoris muscle, GAS: gastrocnemius muscle,
TA: tibialis anterior muscle. The symbols with underline are bi-articular
muscles.
dynamics to be spontaneously and autonomously explored.
It could aid in a further investigation of how a wide range
of dynamical behaviors emerge from neural-body coupled
dynamics as a key synthetic approach for understanding
embodied intelligence.
II. M ATERIALS AND M ETHODS
A. Musculoskeletal Quadruped Robot
We designed a simplistic quadruped robot that was sufficiently realistic to capture important features of animal
musculoskeletal system to achieve an embodiment of the
neural system [16][17] (Fig. 1, Table I).
We employed McKibben-type pneumatic artificial muscles
that reproduced some of the non-linear properties of biological muscles in terms of damping and elasticity [18][19][20].
The muscle configuration of the robot was carefully chosen
based on that of quadruped animals [21](Fig. 2). The artificial muscles were supplied with air from an external air
compressor and we used proportional pressure-control valves
to control the inner pressure of the artificial muscles.
Sensory feedback in real muscles is done by muscle
spindles that sense the muscle length and by Golgi tendon
organs that sense muscle tension. We replicated this feature
by computing the length and the tension of the artificial
muscles using pressure sensors and potentiometers. The
muscle lengths were calculated from each joint angle with
geometric calculations (see Appendix I). Muscle tension
10 artificial pneumatic muscles
(McKibben type)
10 springs (Misumi, AWY12-70)
10 proportional pressure valves
(Hoerbiger, tecno basic)
Fujikura, XFGM-6001MPGSR
Murata, PVS1A103A01
General Robotix, Lepracaun CPU board
linux-2.6.21.1-ARTLinux
4.0 kg
ABS resin, CFRP pipes
and aluminum boards
350 mm
length of each part
140 mm
180 mm
150 mm
150 mm
105 mm
F [N] is theoretically estimated as a function of muscle
pressure p [Pa] and muscle length lm [m] as [22][23]:
(
)
lm 2
F = p A(1 −
) −B ,
(1)
Lmax
1
3
where A = πD02 cot2 θ0 , B = πD02 cosec2 θ0 .
4
4
Here, Lmax [m] is the maximum length of the muscle,
D0 [m] is the initial diameter of the rubber tube, and θ0
is the initial angle between a braided thread and the axis
along the rubber tube. We used the following parameters of
D0 = 0.008 and θ0 = 16.
B. Nervous System
As the nervous system concerns locomotion, many researchers have focused on the CPG and proposed various
types of CPG models depending on what phenomena were
being studied, and applied these models to robots as new
control technologies [6][7][8][9]. Most of these CPG bioinspirations in robotics have used a CPG network composed
of oscillators implemented with basic limit cycle behaviors
and neural modules that determine the relationship between
each element and then generate a specific gait pattern. While
these types of CPG controls have proved highly successful in
terms of generating stability and robustness in locomotion,
these techniques are not suitable as models to investigate the
underlying mechanisms for generating adaptive and various
animal locomotion as a result of neural-body coupling.
We employed the spinobulbar model developed by Kuniyoshi and his colleagues [24][25] based on a biological
perspective (Fig. 3) for these reasons. This model was
composed of independent elements. In other words, it had no
pre-defined motor coordination circuits and the only builtin reflex was a stretch reflex. However, due to a chaotic
property of the neural oscillator model, this spinobulbar
model can explore and get entrained into a variety of
embodied dynamics exploiting body dynamics. Although
the assumption that there are no internal connections is an
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entropy is formulated as:
∑
(l)
(k)
p(yt , yt−1 , xt−1 )×
TX→Y =
(l)
(k)
yt ,yt−1 ,xt−1
(l)
log2
Fig. 3.
Spinobulbar model. The arrows and filled circles respectively
represent excitatory and inhibitory connections. The symbols are, Neural
oscillator: neural oscillator neuron model, S0: afferent sensory interneuron
model, α: α motor neuron model, γ: γ motor neuron model, Spindle:
muscular sensory organ model, Tendon: Golgi tendon organ model.
extreme hypothesis that may deviate from biological reality,
we predicted this model allows us to investigate how various
locomotion patterns emerged through neural-body coupling
without using pre-defined coordination circuits.
One unitary element of the spinobulbar model consists of
a muscle, one α and γ motor neuron, one afferent sensory
interneuron, and one neural oscillator model. All modules are
integrated with time delay (transfer lag between modules)
and gain parameters. These parameters were determined
based on previous work by Kuniyoshi et al. [25]. Details
of each model including the definitions of the variables and
the parameters are given in the Appendix II.
C. Information Metrics
We adopted informational measures of mutual information, transfer entropy, integration, complexity, and phase
synchronization index to investigate the information structure
of the sensory and motor flows shaped by the morphology
and nervous system.
1) Mutual Information: We used mutual information to
measure the deviation from statistical independence between
two information flows. The mutual information of two discrete variables, X and Y , can be expressed as:
M I(X, Y ) = −
∑∑
p(x, y) log2
p(x)p(y)
.
p(x, y)
(2)
If X and Y are statistically independent, this is zero.
2) Transfer Entropy: To investigate how morphology
shapes the causality between sensor and motor flows in
neural-body coupling, we used transfer entropy, which allowed us to identify the directed exchange of information
(also referred to as ”causal dependency”) between time
series [26][27]. Given two time series X and Y , transfer
(k)
p(yt |yt−1 , xt−1 )
(l)
p(yt |yt−1 )
,
(3)
where p(·|·) denotes the transition probability, k and l are
the dimensions of the two delay vectors, and index TX→Y
indicates the influence of X on Y . Transfer entropy is clearly
non-symmetric under exchange of X and Y and can thus be
used to identify coupling and directional transport between
two systems.
3) Integration and Complexity: To capture the global aspects of the statistical dependence within motor patterns, we
employed two information theoretical measures: integration
and complexity.
Integration is generally the multivariate generalization
of mutual information and globally captures the amount
of statistical dependence within a given system or set of
elements X = (x1 , x2 , . . . , xN ). Integration is defined as
the difference between the individual entropy of elements
and their joint entropy [28]:
∑
H(xi ) − H(X),
(4)
I(X) =
i
where H(·) denotes entropy. As in mutual information, if all
elements xi are statistically independent, I(X) = 0.
Complexity allows us to capture the interplay between
segregation and integration in dynamic patterns in terms of
statistical dependencies within the system across all spatial
scales. This complexity is represented as [29]:
∑
H(xi |X − xi ),
(5)
C(X) = H(X) −
i
where H(xi |X − xi ) is the conditional entropy of one
element xi given complement X − xi that comprises the
rest of the system. Only systems combining local and global
structures generate high levels of complexity. In other words,
complex systems combine a certain level of randomness and
disorder with a certain level of regularity and order.
4) Phase Synchronization Index: We analyzed the relationship between a set of elements generating global information structures to better understand the information structure
of motor patterns. To achieve this, the additional theoretical measure we employed was the phase synchronization
index [30].
This measure captures the strength of phase locking between two elements using the instantaneous phase. This
instantaneous phase is calculated as:
1501
φ = arctan
xH (t)
,
x(t)
(6)
Fig. 4. Muscles configuration with only mono-articular muscles. TB’, RF’,
and GAS’ are the replacements for TB, RF, and GAS of bi-articular muscles.
Dotted lines represent the original bi-articular muscles.
(a) body with only mono-articular (b) body with mono- and bi-articular
muscle
muscles
Fig. 5. Mutual information across sensory information with two types of
body in random movements.
where xH (t) is Hilbert transformation. Using this instantaneous phase, phase synchronization index Φ is defined as:
√
Φ = < cos ∆φ(t) >2T + < sin ∆φ(t) >2T ,
(7)
where < · >T denotes the temporal average. If phase
differences indicate a constant value, two elements are phase
synchronized, and Φ = 1. In contrast, for uniformly distributed phase differences (i.e., no synchronization) Φ = 0.
(a) body with only mono-articular (b) body with mono- and bi-articular
muscles
muscles
Fig. 6.
Transfer entropy between sensor to motor information.
III. S IMULATION E XPERIMENTS
An animal’s body dynamics plays a crucial role in shaping
neural dynamics via sensory information. Thus, a biologically plausible body is essential to understand how the neural
system interacts with body dynamics in generating various
and adaptive locomotion.
We investigated the information structure of sensory and
motor information shaped using a biologically correct body
to better understand the underlying mechanisms that generated animal behaviors. We especially focused on the biarticular muscles because we predicted that they would
enhance correlation and redundancies across sensory information and such correlation would help to capture the state
of the body with the environment on the morphological level
and generate coordinated and adaptive locomotor patterns.
To this achieve end, we conducted computer simulation
using dynamics simulator OpenHRP3 [31] and analyzed
the information structures. The model parameters (mass,
inertia, and geometry) were obtained from a CAD data of
the quadruped robot that we developed. We calculated a
force of each muscle model using the theoretical equation
of artificial muscle as can be seen in (1). We compared
two types of muscle configurations. The first had the same
muscle configuration as the robot with mono- and bi-articular
muscles based on biological knowledge. The second muscle
configuration was made by replacing the bi-articular muscles
with mono-articular muscles, i.e., they were composed of
only mono-articular muscles (Fig. 4). We simplified the
model using movement in four legs only, with the body
trunk held in a fixed position to produce more clearly
understandable results.
We analyzed three information structures: sensory information structures, informational flow from sensory to motor
information, and the information structure of motor patterns.
We used muscle length flows as sensory information because
motor patterns mainly depend on these flows by projecting
them to neural oscillator models. We set the time step
iteration of the simulation to 1 ms, and ran the simulations
for 200 s.
A. Sensory Information Structures
We moved both models with the same random motor
commands and analyzed mutual information from the time
series of muscle length to identify statistical dependencies
across the sensory information shaped by the morphology.
Sensory information had low statistical dependencies (Fig.
5(a)) in the model that was only composed of mono-articular
muscles. However, we observed statistical dependencies
across muscles (Fig. 5(b)) in the model with mono- and biarticular muscles.
B. Information Flow from Sensory to Motor
We predicted that although each element was not directly
connected in the neural circuit, if sensory information were
mutually coupled on the body level, there would be information flow from sensory to motor information without direct
connections. We employed transfer entropy between sensory
flows and motor commands to measure this information flow.
Causal dependencies between sensors and motors with no
direct connections increased more in the model with monoand bi-articular muscles than in the body without bi-articular
muscles (Fig. 6). Moreover, we observed information transfer
between GMAX and GAS, i.e., gluteus maximus and gastrocnemius muscles, in the model with mono- and bi-articular
muscles that had no correlation across sensory information
in random movements.
To measure the degree of informational flow from sensors
to motors across muscles via the body, or body coupling,
we calculated the rate of causal dependencies from sensors
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(a) Integration
Fig. 7.
(b) Complexity
Global information structure of motor pattern.
Fig. 8. Phase synchronization index between two muscle activations.
(a) Experiment A.
Fig. 9.
(b) Experiment B.
Snapshot of locomotion patterns in two experiments.
to motors with no direct connections in the nervous model.
This measure β can be calculated as:
∑
j6=i TSi →Mj
β = < ∑
>i ,
(8)
j TSi →Mj
where <>i denotes the average. We obtained the following
results: β = 0.325 in the body with only mono-articular
muscles, and β = 0.530 in the body with mono- and biarticular muscles. We found how the body with mono- and
bi-articular muscles led to an increase of 63% in body
coupling.
C. Information Structure of Motor Patterns
We calculated integration and complexity in both time series to identify the global nature of emergent motor patterns.
We found the body with mono- and bi-articular muscles
increased information regularities in neural-body coupling
and integration and complexity increased about twice that in
the body without bi-articular muscles (Fig. 7).
Moreover, to capture the relationship between each element shaping the global dynamics of motor patterns, we
analyzed the phase synchronization index. We observed high
levels of synchronization in the model with bi-articular
muscles. The phase synchronization indexes of three pairs
increased by 2.5, 1.7, and 3.1 times, and only in the one pair,
corresponding to the gluteus maximus and rectus femoris
muscles in the model with bi-articular muscles; this index
slightly decreased by 11% (Fig. 8).
IV. ROBOT E XPERIMENTS
We conducted some experiments with the quadruped musculoskeletal robot to investigate emergent phenomena by
using the dynamic interactions between the body and nervous
system with the environment. The robot was mounted with
a CPU board running a real-time OS that sent the pressure
commands to the valves and received the sensor values from
the pressure sensors and potentiometers every about 7.5 ms.
One external PC communicated with the CPU board every
about 100 ms and computed the neural dynamics every 1
ms. During the experiments we used the same initial posture
and parameters for the nervous model.
We observed various types of emergent behaviors in the
experiments. For example, the robot generated dynamically
forward movements for several steps (Fig. 9(a): left) and it
then switched to another pattern by performing backward
movements for several steps (Fig. 9(a): middle). After a
period of time, it returned back to its previous dynamics and
re-generated forward movements (Fig. 9(a): right). In this
experiment, the robot generated a series of the movements
with the footfall pattern in Fig. 10, and a speed of 0.24 [m/s]
on average and 0.66 [m/s] maximum speed. We noted that
this type of behavior did not always occur throughout the
experiments, which demonstrated the dynamical nature of
the system. For instance, we observed that locomotion was
only backward in one experiment (Fig. 9(b)). Among other
behaviors, there were only forward movement and jumpinglike motion.
We analyzed the information structure of sensors and motors in the experiments to enable the underlying mechanisms
1503
Fig. 10. The footfall pattern in experiment in Fig. 9(a). The symbols are,
LF: left forelimb, LH: left hindlimb, RF: right forelimb, RH: right hindlimb.
Fig. 12.
Fig. 13.
Fig. 11. Transfer entropy from sensor to motor in experiment in Fig. 9(b).
to be more clearly understood.
First, we analyzed the information flow from sensors to
motor information in the experiment in Fig. 9(b) to identify
the degree of body-coupling. We observed a wide range of
causal dependencies between sensors and motors through
interaction with the environment compared with the results
from simulation (Fig. 11).
Next, we calculated the time change in phase differences
between the left/right latissimus dorsi and gluteus maximus
muscles to investigate the transition in motor patterns in the
experiment in Fig. 9(a). We found a large change in phase
differences before the transition times of about 6 and 8 s
(Fig. 12).
Furthermore, we investigated the phase synchronization
index between the left/right latissimus dorsi and gluteus
maximus muscles in these two experiments (Fig. 9(a) and
Fig. 9(b)) to identify the coordinate relationships between
each leg. We can see more than half the combinations were
highly synchronized, and these relationships changed across
the experiments and even during the same experiment (Fig.
13).
V. D ISCUSSION
In the same way animals exploit the physics of their
body, robots can perform dynamical behaviors if the neural
dynamics comply with the external stimuli. In this paper, we
explained how such mechanism occurs in a neural system
embodied in a robot and how it interacts with the body dynamics in order to generate various and adaptive locomotion
patterns. We designed for this a quadruped musculoskeletal
Time series of phase differences in experiment in Fig. 9(a).
Time series of phase synchronization index in two experiments.
robot that replicates important biological features, though in
a simplified form.
We found in our simulations that bi-articular muscles
produce correlations and redundancies across the body and
therefore in sensory feedback. Such structured information
can create causal relations in the motor patterns without
imposing fixed coupling between the neural circuits. We
found also that the produced motor patterns have globally
richer information regularities and better coordination than
for mono-articular robot.
Although few studies have underlined the importance of
bi-articular muscles for force control [32][33], their functions
are not precisely known. Thus, our results may shed some
new lights on the functions of bi-articular muscles from
the perspective of morphological computation. Bi-articular
muscles, or in general multi-articular muscles, enhance correlations and redundancies across sensory flows and enable
the nervous system to exploit the physics of the body in
dynamical neural-body coupling.
In our experiments with robots, the sensorimotor interactions between body dynamics and the nervous system
modified dynamically the legs coordination to various behavioral patterns activated sequentially, even during the same
experiment, in a self-organized fashion. We discovered that
dynamical interaction between the body and nervous system
with the environment generated a wide range of information transfer from sensors to motors and within the body
coupling. Furthermore, the dynamical coupling determined
the coordination of each leg and their transitions to different
locomotion patterns.
Our results suggest that this biologically-inspired strategy
can generate a wide range of dynamical behavioral patterns
in a self-organized fashion. It can help us to understand how
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the body and neural system in biological systems shape their
locomotion patterns.
In future works, we will investigate other body morphologies in order to develop novel and more complex
locomotion patterns. We believe that such approach, which
emphasizes the ecological balance between the body and
the nervous system, is essential for understanding embodied
intelligence [3].
ACKNOWLEDGEMENTS
We thank Tatsuya Harada for his valuable comments on
the draft of the manuscript. We are grateful to Alexandre Pitti
for many fruitful discussions and his advice on this paper.
We also would like to thank Yuya Yamashita for his advice
about informational analysis.
Fig. 14. Definition of parameters for calculating muscle length. The length
of a, b, c, r and l are constant values, θ is sensor value of potentiometer,
and φ, x, and y are functions of θ.
The Golgi tendon organ model is modeled as [34]:
(1 + s/0.15)(1 + s/1.5)(1 + s/16)
Ib (s)
,
=
(1 + s/0.2)(1 + s/2)(1 + s/37)
F̄ (s)
A PPENDIX I
C ALCULATION OF MUSCLE LENGTHS
When we get each joint angle [· · · , θi , θj , · · · ] as sensory
feedback of potentiometer, we can estimate muscle length
L as following equations (see symbol definition of Fig. 14).
When muscle is mono-articular muscle,
L(θi ) = L(θi,0 ) + (x(θi ) + y(θi )) − (x(θi,0 ) + y(θi,0 )) ,
(9)
and when muscle is bi-articular muscle,
L(θi , θj ) = L(θi,0 , θj,0 )+
(x(θi ) + y(θi )) − (x(θi,0 ) + y(θi,0 ))
+ (x(θj ) + y(θj )) − (x(θj,0 ) + y(θj,0 )) ,
(10)
where θi and θj denote joint angles that the muscle adds
torque to, and θi,0 , θj,0 , L(θi,0 ) and L(θi,0 , θj,0 ) constitute
the specific values of angles and lengths. We calculated x
and y as:
x(θ) = l(π/2 − φ),
y(θ) = (b + c sin θ − r cos θ + l sin φ)/ cos φ.
(11)
(12)
Here, φ is calculated as follows:
√
φ = − arctan(B/A) + arccos(l/ A2 + B 2 ),
where A = −a + r sin θ + c cos θ,
B = b − r cos θ + c sin θ.
(13)
(15)
where F̄ is the normalized muscles contraction force, and
Ib [nA] is the output of tendon organ.
The spinal neurons in Fig. 3, α, γ, and S0, are modeled
by the following transfer function as [34]:
(
)
2
1.5
1
+
s/33
+
(s/33)
o(s)
=
(16)
2 ,
i(s)
1 + 2(s/58) + (s/58)
where i [nA] denotes an input signal, and o [pulse/s] the
output.
Neural oscillator model is calculated by the following BVP
equation:
(
)
(
)
dx
x3
τ
=c x−
− y + Iconst + δ IS0 − x ,
dt
3
(17)
1
dy
= (x − by + a) + ²IS0 .
τ
dt
c
Here, the constants are set as a = 0.7, b = 0.675, c = 1.75,
δ = 0.013, ² = 0.022, Iconst = 0.7, and τ = 0.077. IS0 is
calculated as:
IS0 = −40OS0 + 20,
(18)
where, OS0 is the output of S0. The output of neural
oscillator model ONO , input to α motor neuron, is defined
as:
ONO = 0.25x + 0.34.
(19)
The initial value of x was set to a random value within the
range −0.1 ≤ x ≤ 0.0.
R EFERENCES
A PPENDIX II
E ACH MODEL IN NERVOUS SYSTEM
The spindle model is represented by the following transfer
function [34]:
(1 + s/7.23)(1 + s/74.07)
Ia (s)
= 3.2×105
,
z(s)
(1 + s/12.46)(1 + s/123.28)(1 + s/250)
(14)
where z [m] denotes the length of sensor part of the spindle,
and Ia [pulse/s] denotes the output of the spindle.
1505
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