61
IEEE TRANSACTIONS ON MAN-MACHINE SYSTEMS, VOL. MMS-9, NO. 3, SEPTEMBER 1968
A
Neuromuscular Actuation
DUANE T. McRUER,
System Model
RAYMOND E. MAGDALENO, MEMBER, IEEE, AND GEORGE P. MOORE
FELLOW, IEEE,
Abstract-Recently, both high quality physiological data and human-operator-describing function data of low variability and large dynamic range have become available. These data lead to control engineering descriptions for neuromuscular actuation systems that are compatible with the available data and that provide insight into the overall human control structure (e.g., the types of feedback systems used for various inputs). In this paper, some of these physiological and human-operator data are briefly reviewed, and a simple neuromuscular actuation system model is presented. The physiological data of interest include recent anatomical and physiological data for the muscle spindle and input-output studies of the muscle. These data indicate that simple linear models can describe the basic behavior of these two elements in tracking tasks. This paper contains two key developments: 1) the variation in system parameters as a function of average muscle tension or operating point; and 2) the role of the muscle spindle both as an equaliza-
tion element and in its effects on muscle tone or average tension. The simplest neuromuscular model suggested by and compatible with the data is one in which muscle spindles provide four functions in one entity: 1) the feedback of limb position; 2) lead/lag series equalization; 3) the source of at least one command signal to the system; and 4) a signal for adjustment of the spindle gain, equalization, and steady-state spindle output which produces the average muscle tension. The phase lag of the neuromuscular system's closed-loop describing function exhibits a variation with average muscle tension that is strikingly similar to that observed for the overall human operator in complex tracking tasks. The pertinent human-operatordescribing function data include the covariation of high- and lowfrequency phase data and the variation of high-frequency phase with set tension changes interpreted from force disturbance experiments.
INTRODUCTION nI pHE BASIC dynamics of the human operator and the precision of manual control are ultimately limited by the properties of the neuromuscular system. A basic understanding of this system has important practical ramifications in better appreciating these limitations, as well as in determining the likely effects of control system nonlinearities (e.g., hysteresis, backlash,
Coulomb friction, preload) on pilot/vehicle system stability and performance. Moreover, the neuromuscular system is an archetypical adaptive actuation system which, if understood operationally, might serve as the inspiration for analogous inanimate systems with similarly useful properties. Manuscript received May 17, 1968; revised July 1, 1968. This paper includes research efforts supported by the Ames Research Center, Man-Machine Integration Branch, Moffett Field, Calif., under NASA Contract NAS2-2824 and under Contract AF 33(657)10835 by the AF Flight Dynamics Laboratory (FDCC), Research and Technology Division, Wright-Patterson AFB, Ohio. D. T. McRuer and R. E. Magdaleno are with Systems Technology, Inc., Hawthorne, Calif. G. P. Moore is with the Department of Electrical Engineering and Physiology, University of Southern California, Los Angeles, Calif.
We have chosen to use systems engineering descriptions since these are a natural language for integrating the rapidly expanding knowledge of biological servomechanisms, and serve to implement the interpretation of physiological data, construct and validate models of basic physiological processes, and suggest further experimentation. By neuromuscular actuation system we mean the output servomechanism portion of the human-that portion often called the "motor" system, which includes sensory and motor neurons at the spinal cord level and their associated muscles, joints, and receptors in the periphery. In physical reality the effective connections or block diagram topography of this adaptive system can have many structural forms, each peculiar to a specific control situation. The control situation discussed in this paper involves neuromuscular system operations in which the command inputs arriving at the spinal level are the consequences of random-appearing visual inputs and the outputs are small motions of the hand exerted against spring-restrained lowinertia manipulators. For this situation an appropriate neuromuscular system model can be synthesized by connecting ensembles of sensory and equalization components with associated ensembles of muscle and manipulator elements into an equivalent single-loop feedback system. In this paper we shall present a summary of the data now available concerning the dynamic properties of those neuronal and muscular elements which are relevant to the problem of neuromuscular control at the spinal level. We shall summarize certain experimental findings about these elements, introduce some simplified mathematical models for them, and consider the overall open- and closedloop properties of a system composed of these elements and organized according to well-established neurological principles. Having considered some features of the operating characteristics of the overall system, we shall compare the behavior of the model to existing data obtained from the intact human operator in order to account for certain findings in terms of our model. DESCRIPTION OF THE COMPONENTS This section describes the muscle actuation elements and the muscle-spindle feedback and actuation elements. Muscle Actuation Elements We are interested in a simple muscle model appropriate for a tracking task involving relatively small movements. Involved here will be an agonist/antagonist muscle pair (Fig. 1) where each has an average tension P0 (which of course would not appear across the load). For motion to occur, one muscle of the pair must generate a force greater
62
IEEE TRANSACTIONS ON MAN--MACHINE Limb
Control Position
Position -
x
H
Tendons Muscle Pair
I,
MUSCLE
:
_
2 75 _
(Interface) n~~~~~~~~~~~~~~~~~~~~~~~~~~~~ , Linrnb
Bone
[ Control Stick
lI//Feel System
a)
50 [
LIMB
70
MANIPULATOR
-
f =-O
85 100 115 L (Percent of Rest Length)
Fig. 2. Isometric tension-length curve [1]-[3].
Pa,(L,, f) + - (f -fo) + dL (L- L,) P = Po, + Cf Af
I/
C_
0
(1)
Km AL
where
Po = tension at the operating point Af = change in average firing rate or average electrical activity
AL
0X
o 25
Joirnt
than that of the other, and thus the tension levels must fluctuate about the average. The relationship between tension and length in response to motor nerve commands can be found from the tension-length and force-velocity curves for muscles[1]-[6]. A family of isometric tension-length curves are shown in Fig. 2 for several values of motor nerve firing frequency f. Here P is the tension in the muscle as a function of its length L when stimulated by the motor nerve at a firing frequency f. The solid curves (maximum and minimum stimulation frequencies) are typical of skeletal muscle [1]. The dashed curves are derived from experiments on isolated muscles in which the slope of the curves increased as the firing frequency increased [2] and are consistent with other results based on electromyographic measurements of muscle in vivo [3], [7]. Thus the change in isometric tension due to small perturbations about the operating point (La,, f,) can be described by the first order terms in a Taylor series expansion as
P(L, f)
C
O
(L
Fig. 1. Agonist/antagonist muscle pair.
P
SYSTEMS, SEPTEMBER 1968
Increasing
,
Mean
L2
S \\ Absolute
EMG
0
VM
0
Shortening Velocity
Fig. 3. Force-velocity curves at various mean absolute EMG levels [7].
muscle under maximum stimulation frequency [1], [4]-[6]. An exception is that the relationship for lengthening muscle (V negative) is somewhat steeper than an extrapolation of the shortening curves would predict [6], [7]. This results in a change in slope in the region of most interest (small positive and negative velocities). However, in the context of an agonist/antagonist muscle pair, one muscle will be lengthening while the other is shortening, resulting in a continuous overall force/velocity relationship. Each of the force/velocity curves in Fig. 3 can be identified by the activity level, or alternatively, by their intersection with the zero velocity line, since there the load that can just be supported is equal to the isometric tension. The curves in Fig. 3 can be approximated by F
=
P(1
-
V/ Vm)
(2)
(1 + V/b)
change in muscle length (chosen positive in the where direction of muscle shortening) Vm = the maximum velocity of shortening Km = slope of the tension-length curves for constant f b = a constant Cf = slope of the tension-length curves for constant P = the isometric tension pertinent to the operating length. This tends to increase as f, increases. point length and muscle activity F = the muscle output force Note that Cf and Km are evaluated at the operating point V = velocity of shortening. defined by f, and L, A family of human force-velocity data curves [7] is relationship is directly analogous to the force/speed given in Fig. 3. Here F is the force exerted on the load by This characteristic of an actuator. Again a Taylor's series can muscle during a shortening contraction at a constant be used to the small perturbation motions. To describe velocity V. The family parameter here is motor nerve first order terms input frequency, estimated by electromyographic recordings. (At V = 0 then F = P, the isometric tension.) The F = F(P, V) -F. + d- (P Pa) + a-V( Vo,). load velocity increases as the load force decreases. The shape of the curves is typical of that obtained in isolated (3) =
-
63
MCRUER et al.: NEUROMUSCULAR ACTUATION SYSTEM MODEL
Either actual data or equations fitted to the actual data can be used to evaluate the partials. In this case the second possibility is far more convenient, since (2) is directly pertinent. The partial derivatives evaluated at PO, VO, are
aOF
aP
-
P
-
Fb +
1-)
x
BBBowo); M m
Kc
IBc
(4)
aF + l/Vm) Po(1/b (1 + V10/b)2 aV For steady-state tracking the operating point of interest is V1 = 0, F. = P,. Thus, substituting (4) into (3) yields F
0
DFriving Force C~A fmR)
-
CHARACTERISTICS
Limb Position
m
(1 Vo/ Vm) (1 + V0/b)
MANIPULATOR
MUSCLE ACTIVE AND PASSIVE CHARACTERISTICS
AV
Fig. 4. Schematic of limb/manipulator dynamics for neuromuscular actuation system. Low Tension
(5)
where
___
_
N
o
_~~~dEOdB
_
AV = V - VO.
Substituting (1) for P into (5) yields F -P0 + Cf Af Bm AV - Km AL
-40 dB/decade
(6)
°M
where
Bm = Po(1/b + 1/Vm), a direct function of PO Km = function of PO also (see Fig. 2). There is some evidence to indicate that Km is a linear function of P. [2]. In terms of an analogous physical system, the linearized equation for a muscle given above corresponds to a force source, PO + Cf Af, coupled to a parallel spring/viscousdamper-combination, The damper element has a damping coefficient which is linearly related to the operating point tension. This behavior is typical of both individual muscle fibers and of whole muscle and is a key feature of our succeeding discussion. Skeletal muscles can only contract actively, so movements involving high-grade skill, such as tracking, generally require coordinated groups of muscles. The simplest of these is an agonist/antagonist pair connected at opposite ends of a first-class lever to provide rotary motion. For rotation to occur, one muscle must contract while the other extends. If the opposing muscles each have a steadystate tension PO in the static situation caused by some steady-state or average firing rate f0, rotation can be accomplished by increasing the firing rate for the contracting muscle by the increment Af while simultaneously decreasing the firing rate in the antagonist by about the same increment. The actual muscle system involved in almost any complex limb motion is seldom, if ever, as simple as that described above. We shall assume the same principles hold for each agonist/antagonist pair involved, and thus the model represents the average behavior of all the pairs contributing to the actual limb motion of interest. Fig. 4 shows the limb/manipulator system schematic for a simple spring-mass-damper manipulator and the small signal level model, (6), for the limb system. Since the limb and manipulator inertias are in parallel, the two are
( C(K,.A - KC)/Mv) -, scal,
Tl
TM2 TMI
Fig. 5. jw-Bode diagram for limb/manipulator dynamics.
lumped together in the single effective inertia M. The transfer function between limb rotation and differential firing rate will then be Ct x
Af
M
s2
+
[Bm(P) +
B,]
Km(Po)
+
K,
(7)
We shall consider cases where the manipulator spring K, is much greater than Km. Furthermore, for precision movements in tracking, there is usually some average tension acting, and ordinarily this is sufficient to make the damping ratio considerably greater than unity. For this case the appropriate form for (6) is X
Af
_
Cf (Km + K) (TM.s + 1) (TM2s + 1)
(8)
The dynamics of this equivalent system are illustrated for two cases of tension by the jw-Bode diagram of Fig. 5. From this it is apparent that the effect of changing the tension of the muscle group is to decrease the low-frequency pole and increase the high-frequency pole. As will be seen later, these changes, due to an increase in steadystate tension, have most important consequences on the neuromuscular system dynamics. Muscle Spindle Physiology Much of the control of neuromuscular behavior in the periphery is dependent on a complex organ located in
64
IEEE TRANSACTIONS ON MAN-MACHINE SYSTEMS, SEPTEMBER 1968
most, muscles of the body, the muscle spindle. It is in itself a complex neuromuscular integrative system receiving a continuous barrage of motor control or command signals from the central nervous system and sending a constant stream of sensory signals via its several paths back to the central nervous system. There are other sensory organs involved in motor tasks (Golgi tendon organs, joint receptors, Pacinian corpuscles), however, these will not be included in the present discussion (largely due to an inadequate data base for them). Since the influence of the spindle in neuromuscular control is more far reaching than previously suspected, it is important to review some of its basic anatomical and physiological features. These have been documented in several recent publications [8]-[12], and in what follows we shall present a brief summary of this work, in which the complexities of spindle structure and function not relevant to the present discussion are either simplified or omitted. A typical muscle may have 50 to 80 spindles embedded at various points among the tension-producing ("extrafusal") muscle fibers of the main muscle mass [12]. A typical spindle is elongated in shape, may be several millimeters in length, and is oriented parallel with the extrafusal fibers. A highly simplified diagrammatic view of a spindle is shown in Fig. 6. It has a central swelling, the nuclear bag region, which is the main sensory portion of this organ. The bag is suspended between two so-called "intrafusal muscle fibers"' attached in parallel with the main muscle fibers. The intrafusal fiber properties are similar to those of ordinary muscle (discussed in the previous section), except that they have their own independent motor pathways, the so-called "gamma axons". These cause length or tension changes in the intrafusal fibers independent of length or tension changes in the surrounding muscle. Thus activity in the gamma pathways, as well as length changes of the surrounding muscle, will influence the sensory bag region. In a typical spindle, there will usually arise one large axon whose principal termination winds around the nuclear bag region (a region of the spindle without muscle or contractile elements). These are the primary or annulospiral endings. The type la axons serving these do not have terminations on other spindles.2 Mechanical deformation of these sensory endings leads to the development, at the terminals, of electrical potential fields that are directly proportional to the strength of the deformation [22]. These generator potentials are accurate mappings of the forces operating on the terminals and can follow rather high frequencies of change in the deforming stimulus. Nerve impulses are generated at a rate directly proportional to the magnitude of the generator potential and thus of the strength of the deformation. In the absence of any motor signals from the spinal cord, the primary ending
MUSCLE
Alpha Motor Axon Extrafusal Muscle Fibers
MUSCLE SPINDLE
Intrafusal
Muscle Fiber
lendon
Fig. 6. Simplified diagrammatic view of a muscle spindle and its orientation with muscle.
of a spindle usually shows some discharge even when the extrafusal muscle fibers are at their normal resting length. The firing rate in the spindle will increase monotonically as a function of increasing muscle length. This results from the disposition within the muscle of the spindle, which serves to transmit length changes in the muscle to the bag region where the change is reflected as an increase in bag tension. Conversely, shortening of the muscle (either passively or in response to an alpha motor command signal) will reduce the tension on the bag and hence reduce the spindle Ia sensory fiber firing frequency. Recent studies [8], [9] of single gamma fibers ending on spindles whose primary endings were being monitored have shown that there are two different gamma fiber types. Stimulating these fibers at a constant rate produced different primary ending responses in response to muscle length as well as velocity. At a constant muscle length the primary ending firing rate was approximately a linear function of the stimulation rate of either gamma fiber type. Generally, the slope of this qurve increased for greater muscle lengths. At a constant stimulation rate of either gamma fiber type the primary ending firing rate was approximately a linear function of the length of the muscle. Generally, the slope of this curve increased as gamma fiber stimulation rate increased. While there were differences in the slopes of these curves for the two fiber types, the most dramatic difference between these fibers was their effect on the primary ending response to an abrupt change in the velocity of muscle stretching. The sudden application of a ramp input change in length (a step of velocity) produces in the primary ending firing rate a transient increase that can be related to the spindle's velocity sensitivity. The experimental sequence consisted of a step velocity input, then a wait for the 1 In this paper we will consider only one class of intrafusal primary ending response to reach a steady-state velocity, the so-called "bag fibers." fiber, 2 Other receptors are also associated with some muscle spindles, and then an abrupt reduction of the velocity of stretching but will not be considered here. to zero, after which muscle length remained constant. The
65
MCRUER et al.: NEUROMUSCULAR ACTUATION SYSTEM MODEL Primory
difference in output firing rate between that just prior to step velocity reduction and that when the system reaches a new steady state is a measure of the spindle's velocity sensitivity. Stimulation of one fiber type (called the "dynamic fiber") increased the spindle's velocity sensitivity as an approximately linear function of the input stimulation rate. Stimulation of the other fiber type (called the "static fiber") produced essentially no change in the spindle's velocity sensitivity. However, static fiber stimulation seemed to make the primary ending more sensitive to static length changes than did the same stimulation frequency when applied to a dynamic fiber.
Intrafusal Fiber
Afferert
Nuclear Bag
Introfusal Fer
Capsule
Iniroe uW
Intrafuscl Fiber Elements
Fiber Elements
Ydv
(a) Equivkaent Lumped Parometer Mec/tionial NlteAorkr
PSD+ CftAf5 r~ ,rIoI
Muscle Spindle Model A muscle spindle model compatible with the data just discussed is shown in Fig. 7. The output of the spindle is the primary ending firing rate. The inputs are the dynamic and static fiber firing rates as well as muscle length. In Fig. 7(a), the region of the nuclear bag from which the primary ending originates is represented by a relatively stiff spring. The primary ending firing rate is assumed to be proportional to nuclear bag deformation. Thus Ia firing rate is proportional to spring deflection. The intrafusal muscle fiber portions are shown with dynamic Yd and static -Y. fiber inputs. Each of these fiber elements are muscles, so each one can be represented as a muscle model similar to that shown in Fig. 4. This is the basis of the equivalent model shown in Fig. 7(b), which also takes advantage of the symmetry about the midpoint of the nuclear bag capsule spring. Note that Ia firing rate is proportional to X9. The details of the portion of the intrafusal muscle fiber innervated by the dynamic fibers Yd are assumed to be typical of muscle as modeled previously (6). These comprise a force generator, Pd = Pd. + Cfd Afd, plus an equivalent spring K,, and damper B,, both of which depend on the operating point conditions fd. and fiber length, which determine Pdo. The portion of the intrafusal muscle fiber innervated by the static fibers has a similar model (force generator plus spring Kh ) but lacking a viscosity element, since the data just reviewed indicates that it is either not present or is very small. Note that the effect of the static fiber force generator (shown dashed), P8 = P,0 + Cf. AJ'1, on nuclear bag deformation can alternatively be described by an equivalent length input 7c. This can be seen by applying a step increase in Af. (with muscle shortening x kept constant) that produces a step reduction in xm. This immediately increases x8 since the damper B,, cannot change length instantaneously. Subsequently, the damper smoothly extends, allowing the length change to distribute throughout all three springs as a new equilibrium is reached. Inspection of Fig. 7(b) reveals that this response pattern is the same as that produced by a step length change in x. Consequently, the static fiber input will henceforth be replaced by an equivalent length input y,,
Primary (la)
Ending
Kh(P50) "Stalic
Pdo, Cfd Afd
Kn(Pdc)
Ks
B (Pd,) no
Dynamic Fiber"
Fiber
Bag tNuclear Deformrotion -
x,
Effeciive Length
|Input Sigal
Intrafusol Fiber Choracterislics
(bi Smp//f/ed Mec/onicalf fietworA
Fig. 7. Schematic of muscle spindle model.
(the static fiber force generator is shown in dashed lines, for this same reason). The equations of motion can be written by summing the forces at each node to zero. On rearranging, these become - (K, + B,s)
[(K, + X,, + Bns) - (K. + BRo)
II,
(Kh + K. + B.s)iLxmi = r j(-Y + x) - Pd LPd
(9) -I
where Pd = PdI + C1fd Afd, the force due to dynamic fiber stimulation. Solving for the nuclear bag deformation x8 = (y, + x) - xn,, which is proportional to the primary ending firing rate, yields K,(s + 1/TK) (Y. + X) + Pd/B, (10) (s + l/aTK)
where 1
Kn,
TK
B,,
1 aTK
K, +Kn Bn
1 _1
K,
Kh
1
K.
Thus the primary ending has a lead/lag response to muscle length and/or static fiber input. The response to dynamic fiber input is a simple lag. Note that the lead break frequency is dependent on the operating point dynamic fiber stimulation frequency, whereas the lag break frequency depends on both the dynamic and static fiber inputs. The overall position of this organ in our model can be
66
IEEE TRANSACTIONS ON MAN-MACHINE SYSTEMS, SEPTEMBER
1968
A Inhibitory Synapses A Excitatory Synapses
y Motorneuron a
Spindle
Fig. 8. Reflex arcs of muscle spindle. a
Motorneuron
Fig. 9. Elementary neuromuscular system model.
related by means of Fig. 8, which shows the relation of the spindle to the surrounding muscle and spinal cord command systems. The sensory fiber is seen to form a feedback path to alpha command cells in the cord, which supply the muscle on which the spindle lies. Thus an increase in muscle length or contraction of spindle, by increasing spindle firing, reflexly induces increased motor cell firing, thus tending to offset the increase in length. For the same reason, command signals to cause muscle contraction can be initiated in two ways: either directly via the alpha or indirectly via the gamma control to the
spindle and, by the feedback loop described, back to the alpha motor neuron. SIMPLIFIED NEUROMUSCULAR SYSTEM AND CLOSED-LOOP DYNAMICS
Block Diagram The single-loop feedback system which corresponds to the neuromuscular system just described is shown in Fig. 9. The signals in this block diagram represent perturbation operations about steady-state operating points.
MCRUER et
y~ O.x
al.: NEUROM-USCULAR
(aTK)' aTK
67
ACTUATION SYSTEM MODEL LC'i TENSION
Roof locl cf reol CAis rccf shown by infetsect/ion of heavy lines and 0 dP fine
HIGH TENSION 0
dB line
TK
log
w
(c) Amplitude Ratios
-A
_-
oe,
I0 dog'
@ '
Low Tension I og
, High Tension
\
w
(b) Approximate Closed-Loop Phases Fig. 10. Closed-loop neuromuscular subsystem dynamics for two levels of tension.
Consequently, all the signals indicated can be either positive or negative and the agonist/antagonist muscles and their spindles are implicit in the composite diagram. The spindle ensemble provides four functions in one entity: 1) sensory feedback of limb position; 2) some lead/lag series equalization whose parameters are controlled by the steady-state gamma bias signal Yb,; 3) the source of one command to the system y6; and 4) a means for adjustment of the steady-state bias signal to the muscle. The spindle output incremental firing rate Af,, is summed with an alpha motor neuron command input a, with the result, after conduction and synaptic delays, being an alpha motor neuron incremental firing rate Af. This in turn perturbs the muscles and manipulator, giving rise to limb movement, which is then sensed by the spindle ensemble. For our needs in this paper we have lumped all of the time delays distributed around the feedback loop into one location. (In a more precise model these should be redistributed to their proper locations.) In addition we shall assume that the alpha motor neuron command a, is not involved in the tracking tasks of interest in this paper. The effective damping in the limb/manipulator dynamics transfer function is set by the operating point muscle tension P0 which is due to the alpha motor neuron firing rate fo This is indicated in Fig. 9 by the total
gamma bias y, shown entering the GM block. This bias is due to the steady spindle model response to the steadystate firing rates of the static and dynamic gamma fibers represented by yb,. In the next section we shall find the closed-loop limb response to a gamma command.
Effect of Average Tension on the Closed-Loop Dynamics A representation of the open- and closed-loop dynamics of the neuromuscular system is indicated in Fig. 10 via a block diagram, root locus, and Bode plot. The open-loop poles and zeros are starting and end points on a conventional root locus; these correspond to the breakpoints of the asymptotic Bode plots for the open-loop amplitude ratio Ix/(y x) 1, i.e., the low-frequency muscle/manipulator root 1/TM,, the spindle lead 1/ TK, the high frequency muscle root 1/TM,,, and the spindle lag 1/aTK. (Note that the small pure time delay within the loop rTa is neglected since its effects are minor at the frequencies of interest here.) For a particular tension level these plots show that as gain is increased, the low-frequency muscle/manipulator root approaches the lead zero of the spindle, while the high-frequency muscle root and the spindle lag approach one another, rendezvous, and break into a second order -
pair.
For a particular loop gain, e.g., that shown by the zero-dB line for the plot labeled "Low Tension," the closed-
68
IEEE TRANSACTIONS ON MAN-MACHINE SYSTEMS, SEPTEMBER 1968
loop asymptotic plot of Ix7-y is indicated by the dashed asymptotes. The real-axis closed-loop poles (1/ T,K', 1/TM,2, i/aTK,) are given by the intersection of the zero-dB line and the locus of real-axis roots or Siggy plots [23], [24], while the closed-loop zero is the same as the open-loop, i.e., 1/TTK. The approximate closed-loop phase curve is close to zero degrees in the mid-frequency region (reflecting the long flat portion of lx/(-y - x)j) whereas it droops down at either end. The general shape in the mid-range resembles that of an umbrella. The low-frequency phase lag is due to the low-frequency closed-loop lag/lead, whereas the high-frequency phase lag is a mid-frequency view of the two high-frequency closed-loop lags. Consider, now, the effect of an operating point tension increase on the neuromuscular system dynamics. For simplicity we shall assume that changes in mean tension are consequences of changes in alpha motor neuron activity that do not involve concomitant changes in that type of gamma activity which would alter the spindle lead and lag values [9]. The change in the limb/manipulator dynamics as mean tension is increased (Fig. 5) is reflected in the plot labeled HIGH TENSION in Fig. 10, i.e., the low-frequency muscle root is decreased while the high-frequency root is increased. The zero-dB line in the HIGH TENSION case reflects the assumption that the gain reduction of the limb/manipulator dynamics in the mid-band region is offset by the increased gain of the spindle. Now, the closed-loop dynamics are modified significantly. The closed-loop low-frequency lag/lead is more widely spaced than for the low-tension case, giving rise to a larger low-frequency phase lag, as seen from midfrequencies; and the high-frequency lag is substantially reduced, yielding less high-frequency phase lag. The net effect of the tension increase is to shift the "phase umbrella" to the right. Having presented the basic elements of the peripheral neuromuscular model, and some of the dynamic properties of the closed-loop system associated with them, it is natural to ask whether the model is consistent with other tension data on neuromuscular control. Unfortunately, in a system constructed from physiological elements whose isolated properties alone are known, it is particularly difficult to formulate validation criteria from data derived from measurements on intact humans performing complex tasks. However, we shall now discuss some data trends for the overall human operator that reflect neuromuscular system effects. OVERALL HUMAN OPERATOR DATA REFLECTING NEUROMUSCULAR SYSTEM DYNAMICS There are now fairly large bodies of high precision data
(single-loop compensatory display) that describe the dynamics of the overall human when tracking a randomappearing command input [13]-[18] and those of the human neuromuscular system [19]-[21] in response to force disturbance inputs.
Precision Model
Y~~ ~~~(1)(1
1)- 0
(
3
Approximate
)[(6. 5)
(
16.5
/ (---
2)I)e-
= (42) Yp >
°
2
2
}
Fig. 11. Typical pilot describing function data and models [151 [YC = KC(S- 2); COj = 4.0 rad/s].
Typical data for the overall human are provided in Fig. 11 which shows an average describing function for pilots engaged in tracking a so-called subcritical first order controlled -element [Y, = K,/(s - 2)], using a very light manipulator with a moderate spring restraint. These data have been fitted both by an approximate model and a precision model [13], [15]. In contrast with the precision model, the approximate model describes all but the very highest and lowest frequency data points. In the approximate model, the midfrequency effects of the high-frequency terms are approximated by either a first order lag (so-called neuromuscular lag) or a pure time delay. The latter can be summed with the basic latencies to give an overall time delay Te. Similarly, if the low-frequency effects are modeled by transfer characteristics containing m lags and leads, then the incremental phase shift due to these will be
Ajol.,
mz
m
=
tan-' (wTTleadi) E i=l
-
E tan' i-l
(wTIagi).
(11)
At frequencies well above the break frequencies of the lags and leads, i.e., w > 1/Tleadi, 1/TIagj, this can be approximated by jI
L2 1TleadIi
CO i=1
(7lead
j1
2
coTlag)i
(12)
Tlag)i.
a
Thus the net phase is approximated by a term of the form 6i(r+±/ ) which has the shape of a phase umbrella as in Fig. 11, for the approximate model. Empirically, from measurements on humans involved in tracking tasks, we know that the a and -r, parameters vary systematically together (Fig. 12), at least under
MCRUER et al.: NEUROMUSCULAR ACTUATION SYSTEM MODEL
69
0.6
G0.2 -
i40
4= 0.1
35
4.0[
0.5
Increasing \ Tension
(Inferred)
jw
X
\
130
(rad/s)
a- 0.4 c-
Wi
-120
T
2.51 a)
0
0.3 _
-5 o(rad/s) 0 Fig. 13. Upper root location of a complex pair fitted to some transient responses [18]. -10
U-
:a)
0.2 H
ILi
3:
0
0D Induced by wi changes ,Yc =Kc-s2
0.1~
0
0 Induced by T changes, Yc
_
0
1.0
2.0
=
Kc/s(s
0.5
T) . wi=1.5
I_ 3.0
4.0
Inverse Equivalent Time Deloy, I/Tn 1.0
-
0.33
5.0
s-1 0
0.25
Equivalent Time Delay
40
PO(Tension Inferred from Average
Te . s
Sphygmomanometer Reading in
Fig. 12. Connection between equivalent time delay and low-frequency phase lag [20].
circumstances in which the subject is forced to make adaptive responses as a result, for example, of changes in the effective bandwidth of the forcing function, or changes in the stability of the controlled element [15]. As can be seen in Fig. 12 there is a joint variation of and l/re such that increases in either ci or 1/T wi]l shift the phase umbrella to the right. Typically these adaptive response patterns are accompanied by a subjective impression of increased mean tension levels in the experimental subject. In other studies [20], [24] the response to torque impulse disturbances delivered to the arm of a human subject resembled the transient response of a dominant second order system with light damping. Systematic variations in the position of the upper pole with overall muscle tension levels were noted. In general, increasing mean tension levels increased the natural frequency of these roots while leaving the damping ratio relatively unchanged (Fig. 13). We can see the effect of this trend on the highfrequency phase by noting that for a second order system a
1-) .co 1(-) (JN
¢JN
I + (-) 2@ (J;oN
mm
Hg)
Fig. 14. Effective time constant of the closed-loop neuromuscular system as a function of inferred tension.
(13)
for frequencies cw
