SUBMITTED TO INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH, BIOROB 2006 SPECIAL ISSUE 100
Broadcast Feedback of Stochastic Cellular
Actuators Inspired by Biological Muscle Control
Jun Ueda, Lael Odhner, and H. Harry Asada
Abstract— This paper presents a broadcast feedback approach selectively activated to accommodate the aggregate output of
to the distributed stochastic control of an actuator system the cellular units. It is known that the activation of sarcomeres
consisting of many cellular units. This control architecture is not governed by a deterministic control, but it contains
was inspired by skeletal muscles comprising a vast number
of tiny functional units, called sarcomeres. The output of the a stochastic process due to the diffusion of calcium ions
actuator system is an aggregate effect of numerous cellular [2]. Other references argue that the actomyosin contraction
units, each taking a bistable ON-OFF state. A central controller process, the essential process of actuation, is a Brownian
“broadcasts” the error between the aggregate output and a process [3]. It is also notable that a muscle can function
reference input. Rather than dictating the individual units to properly although a significant fraction of the cellular units
take specific states, the central controller merely broadcasts the
overall error signal to all the cellular units uniformly. In turn each are not functional. The system is quite robust and stable.
cellular unit makes a stochastic decision with a state transition The exact mechanism of skeletal muscle control is still
probability, which is modulated in relation to the broadcasted unknown. However, from the reported muscle behavior in
error. Stochastic properties of both open-loop and closed-loop those references we can gain some insights as to how a vast
control systems are analyzed. Stability conditions of the broadcast number of sarcomeres can be controlled with much fewer
feedback system are obtained by using a stochastic Lyapunov
function. The proposed method is simulated for an artificial sensors and motor neurons. The authors have presented new
cellular actuator, consisting of many segments of smart actuator control architecture inspired by the muscle behavior, which in
material. Theoretical results are verified through simulation. turn has the potential to be a novel approach to the control of
It is demonstrated that, even in the absence of deterministic a vast number of cellular units [4]. The proposed architecture,
coordination, the ensemble of the cellular units can track a given called “Broadcast Feedback”, elucidates the stochastic nature
trajectory stably and robustly.
of the cellular units as well as the relationship between
keywords: Micro Actuation, Broadcast, Stochastic Stability, many sarcomeres and few sensors and motor neurons. In
Markov Chain, Cellular Control System, Distributed Control, the broadcast feedback architecture, a central control unit
Muscle. simply broadcasts the error between the reference input and
the aggregate output of the cellular units. In turn individual
cellular units make independent stochastic decisions based on
I. INTRODUCTION the broadcasted signal of overall error. Our initial simulation
Muscles are dynamic systems with very many degrees of experiment has shown promising results. Although no individ-
internal freedom and relatively few inputs and outputs. A mus- ual commands are sent to the individual cells, the ensemble
cle is composed of small functional units called sarcomeres of the cells can track a desired trajectory when their state
which contract to provide varying levels of displacement and transition probabilities are modulated in proportion to the
stiffness [1]. These sarcomeres are far more numerous than broadcasted error signal. No addressing scheme is necessary
Golgi tendons and muscle spindles, the internal receptors used for broadcast control, since information is sent to all the cells
to measure force, velocity, and displacement in the muscle. rather than to a specific cell. Hence the method is highly
Furthermore, the number of motor neurons entering a muscle scalable to a vast number of cellular control systems.
is also much fewer than the number of sarcomeres. Clearly, the The idea of introducing randomness to the modeling and
central nervous system is not aware of the full internal state control of distributed, complex systems has been proposed [7]
of the muscles, nor can it specify the individual contractions and become an emerging research field . Not onlt theoretical
of the sarcomeres. However, a smooth and accurate gradation aspects but also applications have been studied actively, such
of response can be obtained from a muscle. This implies as to air traffic management and biochemical process mod-
that there is a certain mechanism coordinating a vast number eling [8]. However, to the authors’ knowledge, the idea of
of sarcomeres in such a way that a fraction of them are controlling a large array of stochastic actuator units by means
of broadcast control has never been considered before.
Jun Ueda (corresponding author) is with d’Arbeloff Laboratory for In- In this paper a formal description and theoretical analysis
formation Systems and Technology, Department of Mechanical Engineering, of broadcast feedback are presented. Based on stochastic
Massachusetts Institute of Technology, Cambridge, MA 02139. Tel: 617-253-
3772, Fax: 617-258-6575, and also with the Graduate School of Information Lyapunov analysis [5] [6], this paper rigorously shows that (1)
Science, Nara Institute of Science and Technology, Nara, 630-0192, Japan. merely broadcasting the aggregate output error can guarantee
E-mail: uedajun@mit.edu. that an ensemble of cellular units asymptotically converge to
Lael Odhner and H. Harry Asada are with d’Arbeloff Laboratory for
Information Systems and Technology, Department of Mechanical Engineering, a reference input with probability one, that (2) robustness
Massachusetts Institute of Technology, Cambridge, MA 02139. against failure is guaranteed although the number of dead
SUBMITTED TO INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH, BIOROB 2006 SPECIAL ISSUE 101
cells or nonfunctional cells is unknown, and that (3) as the Figure 2 shows an artificial muscle control system having a
number of cells tends to infinity, the ensemble behavior of the binary cellular structure. Instead of driving the whole actuator
cells approaches that of deterministic control. These theoretical material as a bulk, the actuator material is divided into many
results will be applicable to diverse systems consisting of a small segments, each controlled as a bistable ON-OFF finite
vast number of cellular units. However, the formulation of state machine [15]. The displacement of the actuator is given
the problem as well as the simulation of the results will be by the aggregate sum of the binary outputs of all the cellular
provided in the context of cellular actuator systems. actuators. As the size of each cell decreases, the speed of
response increases and the resolution improves.
II. INSPIRATION FROM BIOLOGICAL MUSCLE Broadcast control. Increasing the number of cellular units
CONTROL and reducing the size of each cell can bring about improved
A skeletal muscle consists of five layers of hierarchical resolution and faster response. However, as the number of
structure, starting with sarcomeres as the lowest functional cellular units increases, it is infeasible, or at least difficult, to
units. At the molecular level, recent studies have reported control all the cellular units directly with a central controller.
that stochastic behavior is essential in explaining intracellular Figure 3 (a) illustrates a central controller directly controlling a
calcium transport [2] and actomyosin contraction itself [3]. number of individual cellular units through a communication
At the macroscopic level, a skeletal muscle shows smooth line, e.g., a bus line. This traditional control architecture is
motion although the muscle fibers are known to have either not likely the case for the biological muscle for two reasons.
“ON” (producing tension) or “OFF” (relaxed) state [1], and One is that, to each motor neuron, more than 1,500,000,000
they exhibit prominent hysteresis [9]. Today’s artificial muscle sarcomeres (150 fibers/neuron[18], 1,000 myofibrils/fiber[19],
actuators, although similar in some aspects, are significantly and 10,000 sarcomeres/myofibril[1]) are connected, which are
different in structure from a biological muscle. Assimilating too many to communicate and control individually. Addressing
the anatomical structure and motor control architecture of all the cellular units for sending individual control commands
a skeletal muscle, we can gain some insights as to how entails long addressing bits, which eat up the channel capacity.
an artificial muscle can be built and controlled. This leads Second, each motor neuron transmits a control signal from the
to an alternative to the design of today’s artificial muscle central nervous system to a target muscle fiber. The control
actuators, which is worth investigation for long-term research signal is then disseminated through a network of T tubules to
interests. The following are three major aspects inspired by a number of sarcoplasmic reticula, which activate a bundle of
the biological muscle. sarcomeres.
Binary Cellular Structure. Functional units lower in the It is natural to consider that the same information is deliv-
muscle hierarchy take a binary state, which can be modeled ered to a vast number of low-level units, at least to the level
as ON-OFF finite state machines. Bistable ON-OFF control of sarcoplasmic reticula. This anatomical fact implies that a
has salient features in coping with complex nonlinearities of signal from the central nervous system is broadcasted over a
actuator materials. Muscle fibers have prominent hysteresis vast number of cellular units, rather than different information
as addressed by [9]. Most materials for artificial muscle is delivered to individual units.
actuators, too, have prominent hysteresis and state-dependent Fig. 3 (b) illustrates the broadcast nature of communications
complex nonlinearities [10][11][12][13][14]. As shown in Fig. between the motor neuron and the cellular units.
1, bistable ON-OFF control does not depend on these complex Distributed stochastic control. If the same information is
nonlinearities, as long as the state of the material is pushed broadcasted to all the cellular units and each unit can take
towards either ON or OFF state. In Fig. 1, the input is only ON or OFF state, the consequence is that all the units
temperature and the output is displacement if we take shape- turn ON or OFF at the same time. This contradicts to the
memory alloy (SMA) as an example. Dynamic transition may fact in muscle physiology that an ensemble of sarcomeres
be influenced by the varying nonlinearities. Nonetheless, the can take multiple levels of excitation. This contradiction can
control problem becomes much simpler for ON-OFF control, be resolved if each cellular unit makes a stochastic decision
as demonstrated by [15] for SMA and by [16] for dielectric in response to the broadcasted information. Each sarcomere
elastomers. is activated with calcium ions through a diffusion process,
The cellular architecture has another important feature with which is a stochastic process. In other words, the sarcomere
respect to speed of response. As the size of cellular units activation is stochastic, and the probability with which each
reduces, the speed of response increases for those actuator sarcomere is activated depends on the ion density and diffusion
materials that entail transport of matter. Activating sarcomeres characteristics. In the literature a number of groups have
entails diffusion of calcium ions, activating SMA needs heat reported the stochastic nature of calcium release and recapture
transfer, and conducting polymers need ion migration. Com- processes. Moreover, stochastic behavior can be observed at
mon to all these actuator materials is the fact that speed of various motor control processes, ranging from motor unit
response increases when the actuator materials are segmented firing[20] to actomyosin motors[3]. Especially, molecular-level
into many small units or thin films, and the reservoir of ions processes, such as calcium release, breakdown of ATP, etc., are
or heat is closely located to the cellular units. For example, influenced by thermal noise resulting in stochastic behavior.
thin film SMA for a micro actuator [17] has a small amount This implies that even though the control command, or nerve
of thermal capacitance, thus the response time is substantially impulse, is sent uniformly to all units, the response of all
reduced. the units may not be the same. Stochastic decision-making at
SUBMITTED TO INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH, BIOROB 2006 SPECIAL ISSUE 102
Output Central Nervous System
ON Control
(Motor neuron)
Broadcast signal
OFF
Cellular Units
Input
Noise (Sarcomeres)
uOFF uON OFF
OFF
ON
ON
OFF
OFF
ON
OFF
ON
ON
OFF
OFF
ON
OFF
ON
ON
OFF ON
Fig. 1. Bi-stable ON-OFF Control
Sensors
+ (Muscle spindle)
Control Local on/off
control
-
Fig. 4. Schematic Diagram of Muscle Control
Segmented material
Fig. 2. Segmented Binary Control p
1-p OFF ON 1-q
Data
Address + Data
q
Fig. 5. Single Cell
(a) Bus line communication (b) Broadcast communication
III. STOCHASTIC CELLULAR CONTROL SYSTEMS
Fig. 3. Communication between Controller and Cellular Units
A. Single Cells
A cell is defined to be the smallest functional unit having its
own state and producing an output. Each individual cell takes
bistable ON-OFF states as shown in Fig. 5. Each cell has a
local units regulates the aggregate output of the ensemble units decision-making unit that changes the transition probability
without deterministic coordination. from one state to the other by receiving a broadcast signal.
Combining the above three aspects inspired by a skeletal Let p (0 ≤ p ≤ 1) be the transition probability from OFF
muscle lead to the model depicted in Fig. 4. A control signal to ON, and q (0 ≤ q ≤ 1) be the transition probability from
generated at a central control, i.e. the central nervous system, ON to OFF. We assume that the transition is performed in
is broadcasted from a broadcast station, i.e. the motor neuron. discrete time step, hence the behavior of the cell is modeled
Each cellular unit, i.e. the sarcomere, makes a stochastic as a discrete-time, non-homogeneous Markov process. We also
decision. This results in a probabilistic distribution of ON- assume that all cells are uniform in size, i.e., providing an
state units and OFF-state units. The aggregate outputs, i.e. uniform displacement:
muscle displacement, force, etc., are detected by sensors, i.e. η, ON
muscle spindles and golgi tendon organs. Note that this system yt =
i
, (1)
0, OF F
architecture is not for fully explaining the true biological mus-
cle. Highly complex neurological and biochemical processes where yt is the displacement of the ith cell at time t. In this
i
involved in the five-layer muscle hierarchy are ignored, and paper, we focus on a position control of the cellular actuators.
the whole system is reduced to just a two-layer distributed In order to simplify the analysis, we assume that each cell
stochastic control system. Rather, the objective of this model provides an uniform displacement η when ON regardless of
is to manifest how the aggregate output of vast cellular units the stress applied to the cell.
can be controlled, although the number of independent units is
numerous than the number of feedback control loops. To verify
that this control architecture functions properly and stably, B. Cellular Control System
a precise mathematical description and rigorous analysis are Consider a cellular control system in which N cells are
needed, which are the focus of the following sections. connected in series. The output yt of the system is given by
SUBMITTED TO INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH, BIOROB 2006 SPECIAL ISSUE 103
N OFF (= N − N ON ) N ON
q(e)
1-p(e) OFF ON 1-q(e)
OFF ON
N dead N dead
p(e)
inactive cells inactive cells
0 r L 0 r L 0 r L
Fig. 6. Aggregate Markov Model N; small N; medium N; large
Fig. 7. Probability Distribution for different N
the aggregate output of all the cells. Suppose that, at time t,
NtON cells are ON and NtOF F cells are OFF. From (1):
IV. BROADCAST FEEDBACK FOR CELLULAR
N
CONTROL SYSTEM
yt = yt = η · NtON ,
i
(2)
i=1
A. Open-Loop Characteristics
The gross stroke (output range of the system) is then given by Consider a feedforward (open-loop) control problem at first.
L = η · N . The number of OFF cells, NtOF F (= N − NtON ), Without loss of generality, we can assume that all the N cells
does not contribute to the aggregate output. are OFF at time t in the following analysis. Let r = ηNd (0 ≤
r ≤ L) be a reference input, where Nd is the number of
For example, the probability density function for the number
ON cells for r. By setting the transition probability p of the
of activated cells at time t + 1 when all the N cells are OFF
individual cells as
at time t is given by
r
p= (0 ≤ p ≤ 1), (6)
L
N
Pr{Nt+1 = x|NtOF F = N } =
ON
px (1 − p)N −x , (3) the expected aggregate output is given by
x
E[yt+1 |yt = 0] = η · E[Nt+1 |NtON = 0] = η · p · N = r, (7)
ON
N N!
where = (N −x)!x! .
x which agrees with the reference output. More importantly, the
variance of the output is given by
C. Markov Model V ar[yt+1 |yt = 0] = E[(yt+1 − E[yt+1 |yt = 0])2 ]
The ensemble behavior of N cells, each having the single = η 2 · E[(Nt+1 − E[Nt+1 |NtON ])2 ]
ON ON
cell state transition probabilities, p and q, can be represented L2
as a Markov process [21] shown in Fig. 6. The conditional = η 2 · N · p(1 − p) = p(1 − p). (8)
N
mean of the number of ON cells and that of OFF cells are
Note that the variance reduces as the number of cells
given by the state transition equation:
increases. In other words, the variance reduces as the given
gross stroke length L is divided into more cellular units, each
E[Nt+1 |NtON , NtOF F ]
ON
1−q p NtON producing a finer output η. Figure 7 shows plots of the output
= . (4)
E[Nt+1 |NtON , NtOF F ]
OF F
q 1−p NtOF F probability distribution for different N . The standard deviation
√
σt+1 = V ar[yt+1 |yt = 0] reduces in proportion to 1/ N .
In practice, however, some fractions of the cells are non-
This property implies that the cellular control system having
functional, i.e. dead cells. Suppose that Ndead cells are dead
ON
more cells turns out to be more predictable. As N tends to
and stay in the ON state for the next transition and that
infinity it can be driven to produce a desired output with an
Ndead are dead, taking the same OFF state. For simplicity,
OF F
arbitrary accuracy in the mean square sense by broadcasting
we consider the case where dead cells stay either in the ON
the error or state transition probability.
or OFF state but in the intermediate states. The above state
transition equation is then modified to
B. Closed-Loop Control by Broadcast Feedback
E[Nt+1 |NtON , NtOF F ]
ON
E[Nt+1 |NtON , NtOF F ]
OF F The drawback of the above broadcast open-loop control are
• For a finite number of cells, N , the output inevitably de-
1−q p NtON − Ndead
ON
Ndead
ON
= OF F + . (5) viates from the reference having a probability distribution
q 1 − p Nt OF F
− Ndead Ndead
OF F
with a finite variance.
Note that 0 ≤ Ndead , Ndead and 0 ≤ Ndead + Ndead ≤ N
ON OF F ON OF F • The exact number of the usable cells must be known.
hold obviously. Ndead and Ndead may vary, but we treat them
ON OF F
The latter drawback will be an important issue particularly
as constant values assuming that the variation is slow. for a large scale cellular system, where it is difficult to
In the following, we modulate the transition probabilities maintain all the cells functional. Some of the cells may die
p and q as a function of the broadcasted error. Hence this or do not respond to the inputs. It is desired if the control
Markov process is not homogeneous. system works without knowing the exact number of functional
SUBMITTED TO INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH, BIOROB 2006 SPECIAL ISSUE 104
p Cell
problem and avoids zero dynamics as will be described in V-
1-p OFF ON 1-q
B. This changes the number of ON cells to
u q
r + e
Control yi Nt+1 = NtON + ΔNt+1 ,
ON ON
(9)
-
y + and the output and the error to
Dead cell
yt+1 = η · Nt+1
ON
(10)
Active cell
et+1 = r − yt+1 , (11)
respectively.
Fig. 8. Broadcast Feedback for a Cellular Control System with Distributed The expected value, i.e., conditional mean, of ΔNt+1 is
ON
Decision-making units
obtained from the binomial probability distribution:
¯ ON
ΔNt+1 = E[ΔNt+1 |NtOF F ] = pt+1 · NtOF F ,
ON
(12)
N
which brings the error to
N tON N tOFF
t =t
E[et+1 |et , r] = pt+1 (r − L) + (1 − pt+1 )et . (13)
From (9) to (12), the variance of ΔNt+1 is also obtained from
ON
ΔN tON
+1
the binomial probability distribution:
N tON N tOFF
t = t +1
+1 +1
V ar[ΔNt+1 |NtOF F ] = pt+1 (1 − pt+1 ) · NtOF F .
ON
(14)
The variance of error at t + 1 is then given by
(a) Change of the Number of ON cells
L =η ⋅ N V ar[et+1 |et , r] = η 2 NtOF F pt+1 (1 − pt+1 )
r 1
= L(L − r + et )pt+1 (1 − pt+1 ). (15)
N
yt = η ⋅ N tON et = r − yt Note that the variance gets smaller as the number of cells
t =t increases. Similar results can be obtained for et 0. If et > 0, more cells must be turned ON. scale cellular control system.
The updated probability pt+1 is calculated in all the cells, We assume that the sampling rate of the broadcast feedback
which independently make stochastic decisions. As a result, is sufficiently slow compared to the cell dynamics, so that
ΔNt+1 cells are turned on among NtOF F cells that were OFF
ON
each cell completes transition within the sampling period. The
at time t. Let us assume that the transition from ON to OFF basic motivation of cellular architecture, where the actuator
is prohibited when et > 0. This prohibition simplifies the material is divided into many small units, is to increase speed
SUBMITTED TO INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH, BIOROB 2006 SPECIAL ISSUE 105
of response as well as to overcome the complex hysteresis and B. Unilateral Transition Control
nonlinear properties of the material. The above assumption
is based on this design concept; the cell dynamics are much In order to simplify the problem, the transition from ON to
faster than the broadcast feedback using the aggregate output. OFF is prohibited when et > 0, and the transition from OFF
This design concept is achieved without difficulty, when micro to ON is prohibited when et 0)
state transition is known only in the stochastic sense. There-
fore, stability theory for deterministic systems is not applica- q(e) (e 0, x = 0. V S (x) Random walk models. By substituting (19) and (20) for (5),
has continuous first derivatives in the bounded set Qm = {x : the following discrete systems are obtained:
V S (x) 0) (21)
OF F
condition in Qm . If a non-negative, real, scalar function k(xt ) E[et+1 |et ] = et + ηq(et )(NtON − Ndead ) (et 0. This implies et−1 > 0 and Δet 0 and q = 0. Similarly, Δet > 0 if
is deterministic and, thereby, the variance is zero, the stability ∃i such that yt−1 = η and yt = 0. Therefore, the following
i i
condition has no difference from that of a deterministic Lya- proposition holds:
punov function. Due to the stochastic nature of the process, the
left hand side of the above inequality condition is larger with
the added variance term. Therefore, more strict (conservative) ∃i, s.t. Δyt = 0 ⇒ |Δet | = 0.
i
(25)
stability condition must be met for the stochastic process. It is
obvious that V ar[et+1 ] → 0 (see (15)) and et+1 → E[et+1 ] The contraposition of this proposition gives Lemma 1.2
if N → ∞, resulting in deterministic analysis shown in Since there is no zero dynamics, a Lyapunov function based
Appendix. on only the aggregate output e is enough to analyze the
When the inequality condition, (18), is satisfied, the pro- stability of the entire cellular system. Unlike the above state
cess is called a nonnegative supermartingale, for which the transition law, if bilateral state transitions are allowed, there is
Lyapunov function is guaranteed to converge to a nonnegative a chance that prolonged oscillations may occur. Furthermore,
limit with probability one. See [5] [6] [23] for more detail and proof of stability including internal stability becomes more
proof. complicated.
SUBMITTED TO INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH, BIOROB 2006 SPECIAL ISSUE 106
p(e) − 2e −η 2e −η
q= q p p=
η ( N − 1) η ( N − 1)
e>0 1-p(e) OFF ON 1 1 1
η η
e=− e=
2 2
e 0 also implies that the internal state of every cell converges
ΔVtS = . (26) with probability one, i.e., Δy i → 0 ∀i. The proof is a direct
⎪ {η(1 − q) + 2et + q(ηNd − ηNdead − et )}
⎪
ON
t
⎩
×q(ηNd − ηNdead − et )
ON
et η/2 any feasible r if the system is stable for N (100% cells are
2e−η
d dead
(27) active).
−2e−η
0 0
is less than the resolution of the output η.
0 0. The following theorem provides the addition, only the nominal gross stroke L is required for the
OF F
solution: design of the transition probabilities.
Theorem 2: Error Broadcast for the Cellular Control Remark 3: No overshoot response. No overshoot response
System. Suppose that a broadcast feedback controller performs of E[et ] for any negative initial error can be obtained if 0 0) for (32). This convergence is irrelevant to the design
p(e) = 2e−η (29) of q(e).
0 η/2
−2e−η
0 0. 2
OF F OF F
when fully contracted is 0.2604[m]. The feedback controller
SUBMITTED TO INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH, BIOROB 2006 SPECIAL ISSUE 107
Full extension Output Reference
broadcasts the error between the reference and the current
displacement: et = r − yt . 0.28 0.28
The broadcast signal et is updated in every 0.01[sec] so
Length [m]
Length [m]
0.27
that decision-making in each individual cell is performed 0.27
in sync with this update. Since the response of the PZT 0.26 0.26 Full contraction (with dead cells)
actuator in excess of 5 kHz is fast enough, the dynamics of 0 0.5 1 1.5 2 0 0.5 1 1.5 2
PZT actuator is negligible compared to the dynamics of the Time [sec] Full contraction (nominal) Time [sec]
decision-making. Sampling delay T = 0.01[sec] and white (a) Healthy (gp=gq=0.8) (b) 30% dead (gp=gq=0.8)
noise are added to the observation of e. 0.28
0.28
Length [m]
Length [m]
0.27
B. Transition Probability Design 0.27
Assume that the number of the cells (N = 1000) is large 0.26 0.26
enough. From (31) and (32), a practical design for p(e) and 0 0.5 1 1.5 2 0 0.5 1 1.5 2
q(e) that satisfies the condition of stability is given as follows: Time [sec]
(c) Healthy (gp=gq=1.5)
Time [sec]
(d) 30% dead (gp=gq=1.5)
0 (e ≤ 0) 0.28 0.28
p(e) = (33)
Length [m]
Length [m]
min(gp e/L, 1) (e > 0) 0.27 0.27
min(−gq e/L, 1) (e 0
=
performance. Consideration of a time-varying reference as ⎪ q(ηNd − ηNdead − et )
⎪
ON
⎩
well as time-varying number of nonfunctional cells is neces- ×{2et + q(ηNd − ηNdead − et )}
ON
et 0 [20] C. T. Moritz, B. K. Barry, M. A. Pascoe and R. M. Enoka, “ Discharge
dead Rate Variability Influences the Variation in Force Fluctuations Across
(36) the Working Range of a Hand Muscle,” J. Neurophysiology 93: pp.
2449–2459, 2005.
−2e
0 0
0 < q(e) < min(−2e/L, 1) e < 0
q(e) = . (39)
0 e≥0
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