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J Physiol (2003), 551.1, pp. 357-370
© Copyright 2003 The Physiological Society
DOI: 10.1113/jphysiol.2002.036939
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ABSTRACT |
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These experiments were prompted by the recent discovery that the intrinsic stiffness of the ankle is inadequate to stabilise passively the body in standing. Our hope was that showing how a large inverted pendulum was manually balanced with low intrinsic stiffness would elucidate the active control of human standing. The results show that the pendulum can be satisfactorily stabilised when intrinsic stiffness is low. Analysis of sway size shows that intrinsic stiffness actually plays little part in stabilisation. The sway duration is also substantially independent of intrinsic stiffness. This suggests that the characteristic sway of the pendulum, rather than being dictated by stiffness and inertia, may result from the control pattern of hand movements. The key points revealed by these experiments are that with low intrinsic stiffness the hand provides pendulum stability by intermittently altering the bias of the spring and, on average, the hand moves in opposition to the load. The results lead to a new and testable hypothesis; namely that in standing, the calf muscle shortens as the body sways forward and lengthens as it sways backwards. These findings are difficult to reconcile with stretch reflex control of the pendulum and are of particular relevance to standing. They may also be relevant to postural maintenance in general whenever the CNS controls muscles which operate through compliant linkages. The results also suggest that in standing, rather than providing passive stability, the intrinsic stiffness acts as an energy efficient buffer which provides decoupling between muscle and body.
(Received 3 December 2002; accepted after revision 27 May 2003; first published online 27 June 2003)
Corresponding author M. Lakie: Applied Physiology Research Group, School of Sport and Exercise Sciences, University of Birmingham, Birmingham B15 2TT, UK. Email: m.d.lakie{at}bham.ac.uk
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INTRODUCTION |
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In this paper we describe some simple experiments where, in a balancing task analogous to standing, the intrinsic stiffness of the springs involved can be experimentally changed. The experiments were undertaken in an attempt to resolve some of the confusion that has recently arisen about the role of the intrinsic stiffness of the ankle in standing. The extent to which the intrinsic spring-like properties of the calf muscles can stabilise the body in standing has recently become the subject of much debate (Winter et al. 1998, 2001, 2003; Morasso & Schieppati, 1999; Loram & Lakie, 2002a,b; Morasso & Sanguineti, 2002).
In standing, it has become common to regard the body as an inverted pendulum pivoted at the ankles. This ponderous load is balanced in the anterior/posterior plane by the ankle musculature which is under the control of the nervous system. The muscles involved are primarily the calf muscles which are coupled to the ground by the Achilles' tendons and feet. This combination of foot, tendon and muscle determines the intrinsic stiffness of the ankle. If the intrinsic stiffness of the two ankles is greater than the torque-angle relationship produced by the influence of gravity on the body, then passive stability will result. This minimal stiffness required for stability is often expressed as total intrinsic ankle stiffness > mgh where m is the mass, g is the acceleration due to gravity and h is the height of the centre of mass above the pivot. In this case, any deflection from the body's equilibrium position will generate a restoring torque that will cause it to regain that position after a decrementing series of oscillations. As the body sways, energy can be stored and released in the stretching and shortening of the spring. This is the view which has been vigorously maintained for some years by Winter and his colleagues (see for example, Winter et al. 2001). In their opinion, the role of the CNS is seen as setting the tonic activity of the calf muscles at an appropriate level and the intrinsic ankle stiffness that results is adequate to produce stability.
However, a major problem with considering intrinsic stiffness to be the source of stability is that the muscles must operate in series with the tendon and the tissues of the foot. In any series arrangement the stiffness is limited by the most compliant element and the intrinsic ankle stiffness will be limited to a value smaller than its softest spring. If the tissues of the tendon and foot are insufficiently stiff then the intrinsic ankle stiffness will be inadequate to provide stability on its own regardless of the stiffness inherent in the active muscles. There has been a general tendency to regard the tendon and connective tissue as inevitably stiffer than the muscles. This may be true at high force levels, but at the modest levels encountered in standing the converse is true. Measurements have revealed considerable non-linearity in the stiffness of the tendon which is substantially less at low force levels (Ker, 1981). A number of recent relevant investigations on the ankles of human subjects are summarised in Table 1 (Hunter & Kearney, 1982; Kearney & Hunter, 1982; Hof, 1998; Mirbagheri et al. 2000; de Zee & Voigt, 2001; Loram & Lakie, 2002b; Maganaris, 2002).
Taken together, for the modest values of tension prevailing in quiet standing, these values for intrinsic ankle stiffness are too low for stabilisation of the body. Moreover, in very recent work in this laboratory we have directly measured intrinsic ankle stiffness during standing (value in Table 1) (Loram & Lakie, 2002b). The value that we obtain for intrinsic ankle stiffness is approximately 90 % of mgh and therefore too low to provide minimal effective stiffness for stability. Our measurements suggest that the limiting factor is the low stiffness of the series combination of the foot and the tendon. Our measurements, and those of Gurfinkel et al. (1994), suggest that the tendon and the foot both make a substantial contribution to this compliance.
Naturally, if the intrinsic ankle stiffness is inadequate to provide passive stability, an alternative active control scheme must be sought. Based on the results from our previous experiments we have recently suggested a possible mechanism (Loram & Lakie, 2002a). In this paper we study the proposed mechanism in action in a task analogous to standing. The force produced by the series combination of the foot and tendon will be a function of their elongation. In standing, it is often forgotten that there are two factors producing tendon-foot elongation. Stretch is produced passively by movement of the body (sway-dependent) but stretch can also be produced by an active length change of the muscle which need not be associated at any instant with body movement (i.e. not sway dependent). The proximal end (origin) of the calf muscles moves passively as the body sways and the sole of the foot remains stationary on the ground. The muscle transmits this movement to its distal end (insertion), producing a reaction force on the ground by stretching the Achilles' tendon and the arch of the foot. Active changes in muscle length will create extra stretch of the tendon-foot spring and thus generate extra force. This is not an intrinsic stiffness of the ankle - it is produced by appropriately timed neural intervention. The neural modulation may, in principle, result from feedback or predictive processes. The nervous system exerts control by dynamically altering the stretch of the tendon-foot spring.
In the present investigation we studied the ability of subjects to balance an inverted pendulum using an equivalent model where an extension spring represents the tendon-foot spring. One end of the spring is attached to the pendulum and the other to the subject's hand which represents the muscle. The force at any instant is a function of movement of both ends of the spring, i.e. the spring's overall length. Sway of the pendulum produces movement of one end (stretch) and the hand produces movement of the other end (offset). With appropriate timing between stretch and offset the torque applied at any pendulum angle can be varied over a wide range. The torque produced is a summation of a mechanical component and a neurally generated element. The spring stiffness sets the intrinsic stiffness of the system; it remains constant until the spring is changed. By using different springs, the intrinsic stiffness could be experimentally varied over a wide range, from a value which is much greater than that needed to produce passive stability of the chosen pendulum ('stiff springs') down to a value which is completely inadequate ('soft springs').
The subjects balanced the pendulum manually and we examined the hand movement pattern that they employed to control the pendulum. Although it is not the same thing, this may be a reflection of the neural strategy used to control the offset of the tendon-foot spring; hand movement mimics the movement of the myotendinous junction in the calf. We have three aims. First, to show that it is possible to balance an inverted pendulum in a well controlled way when the intrinsic stiffness is too small to permit passive stability. Second, to examine the nature of the hand movements that are employed, and see how this is related to movement of the pendulum. By inference, the movement of the hand may illuminate the way in which the calf muscles are controlled. Third, we describe the implications of this model in particular for the control of human balance and more generally for postural maintenance. These simple experiments provide a dramatic test of our assertion that a pendulum can be satisfactorily stabilised when the intrinsic stiffness of the control system is inadequate and they reveal the mechanism responsible for generating the extra torque.
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METHODS |
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Subjects
Seven subjects (age 20-48 years, 4 male) carried out these simple experiments which conformed to the Declaration of Helsinki. They gave their informed, written consent and permission was given by the local ethical committee.
Apparatus - the inverted pendulum
The apparatus which is illustrated in Fig. 1 comprised a large inverted pendulum which was pivoted on precision ball races so that it moved in one plane. The pendulum consisted of a mass (m) of 51.01 kg at a height (h) of 1.03 m from the axis of rotation. These characteristics were similar to those of a medium-sized adult. Gravitational acceleration (g) operating on the pendulum generates a toppling torque (
) which is very nearly directly proportional to its inclination (
) when it is within a few degrees of vertical. Using the equation for the pendulum, = mgh, the calculated toppling torque per unit angle for this pendulum was 9.15 N m deg-1. The toppling torque per unit angle (Loram et al. 2001) has been called the load stiffness (Fitzpatrick et al. 1992) and also the gravitational spring (Winter et al. 2001). We refer to it in this paper as the load stiffness. The actual load stiffness of the pendulum was calibrated directly by measurement at a number of angles and was found to be greater than the calculated value due to the mass of the rod and other minor unbalanced parts; it was 11.03 N m deg-1. The moment of inertia of the pendulum and other rotating parts was determined by attaching a spring of known stiffness and recording the damped oscillations following a small push; it was 64.1 kg m2. We have earlier calculated the friction constant (0.045 ± 0.1 N m) and the viscous drag (0.061 ± 0.02 N m s deg-1); both are very small (Loram et al. 2001). The same pendulum was used for every experiment and end stops restricted the range of its movement from vertical to approximately 8 deg backwards. The pendulum was controlled by a steel cable (1.5 mm diameter) attached to an extension spring of adjustable stiffness. It was made up of a series combination of identical springs (T32 090, Springmasters, UK) with the number determining the overall stiffness (Table 2). Adjustable links were used to keep the overall length of the spring plus cable the same (approximately 0.8 m). The point of attachment of the spring to the rod was always 0.87 m above the axis. As the line of pull was very nearly normal to this, and because only small deflections of the pendulum were involved, it was appropriate to express the pendulum characteristics in linear rather than angular terms. At the point of attachment the load stiffness was 0.727 N mm-1. In Table 2, spring stiffness is expressed in the same units and the relationship to the load stiffness is given by: relative stiffness (%) = spring stiffness/load stiffness.
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Figure 1. The apparatus used A large inverted pendulum (P) was supported on ball races mounted on a very substantial trunnion (Tr). Pendulum inclination was measured by a Hall-effect potentiometer attached to its base. End-stops (not shown) limited its range of movement. The subject controlled the pendulum by flexion-extension movements of the forearm. The upper arm was secured in a cradle (C). The forearm was coupled to the pendulum by an armlet (A) which was linked by an adjustable spring (K) to a load cell (LC) mounted on the pendulum rod. Arm movement was recorded by an infra-red optical rangefinder (R) which was aimed at a lightweight reflective target (Ta). A positive signal was produced by a rise in force and by movement towards the right in this figure, of the hand and pendulum. The approximate site of the EMG electrodes over the flexor and extensor compartments is indicated. Further details are given in the text. | ||
In standing, the pull of the Achilles' tendon has a moment arm of approximately 5.0 cm. The moment arm in our model is much larger at 87 cm. This has the effect of scaling up the size of any movement that is required by a factor of 17.4. We chose to do this as it seemed reasonable to expect that the wrist position would be controlled with very much less precision than the controlling muscles themselves. Because of the length of the lever that forms the human arm, the ratio of movement of the biceps muscle to the pendulum might be expected to be broadly comparable to the ratio of the movement of the calf muscles to the body in standing. The force that is required, measured at the wrist, is only about one seventeenth of the force that would be generated by the calf muscle, but once again the force in the biceps muscle itself may be expected to be broadly comparable with that in the calf. As both the force and the displacement are similarly scaled, the stiffness of our springs (in N mm-1 as in Table 2) would have to be multiplied by 17.42 (302) if it is desired to make a direct comparison with linear tendon stiffness in standing. The angular stiffness of our model is comparable to that of a standing subject, but because its moment arm is much longer, the linear stiffness of our springs is much less.
Instrumentation
Force. At the point of attachment to the pendulum rod, a force transducer consisting of a very stiff cantilever strain gauge measured the force that was applied.
Pendulum angle. Pendulum inclination was measured by a precision Hall effect contactless potentiometer (CP2UTX, Midori Precision Co. Ltd, Japan) attached to its base. An oscilloscope positioned approximately one metre in front of the subject's head provided visual feedback about pendulum angle. Its screen sensitivity was 1 cm of deflection per degree of pendulum inclination.
Hand position. Hand position was recorded by an infra-red analog reflex rangefinder (HT66 MGV 80, Wenglor Sensoric GmbH, Germany) which recorded the position of a lightweight cardboard target attached to the subject's armlet. The transducer used recorded hand movement relative to a fixed reference point as our main interest was in the way that the hand and pendulum moved relative to each other.
EMG. EMG activity was recorded from the biceps brachii and triceps brachii muscles of the right arm using bipolar silver cutaneous electrodes and an integrated preamplifier (gain 1000). Each electrode had a diameter of 5 mm and the electrode centres were 10 mm apart. The signals were rectified and integrated with a leaky integrator with a time constant of 44 ms. The EMG records were used only to see if at any relative stiffness co-contraction was employed to maintain a stationary hand position. The EMG records are not directly comparable with those of the calf muscles. In our simple model, the pendulum represents the body, the spring represents the intrinsic tendon and foot tissue stiffness and the hand position represents the movement of the insertion of the calf muscles that pull on the tendon. The biceps and triceps muscles which control the hand position themselves operate through tendons and levers and these additional series elements and the inertia of the arm itself mean that the EMG of the arm and calf muscles cannot be directly compared.
Resolution. The resolution of the signals was mainly determined by the noise which was present; it was 0.002 deg for pendulum angle, 0.0005 m for hand position, 0.16 N for force and 60 µV for surface EMGs.
Protocol
Subjects sat in a comfortable chair with the right shoulder facing towards the pendulum so that it was out of their direct line of sight. The upper arm was abducted until it was approximately horizontal and the elbow was supported at shoulder height in a padded cradle. The supinated forearm was inclined vertically. The subject controlled the pendulum by arm flexion/extension movements at the elbow. An aluminium armlet encircled the wrist and was attached to the pendulum by the steel cable and extension spring. Subjects were asked to attempt to maintain the angle of the pendulum so that it had a backward lean of 3 deg. At this angle the static force required to balance was 38 N, the line of pull was perpendicular to the forearm and the oscilloscope display was centred. Subjects were informed only that the linkage connecting them to the pendulum was going to be changed and they would have to find a strategy for balancing it. The same randomised order of spring stiffness was used for each subject. Only two subjects performed the test with the 106, 124 and 746 % relative stiffness springs. After each change they were given a few minutes to practice and when they declared themselves to be confident of their ability a recording was made. Three recordings (40 s duration each) were made for each spring stiffness value.
Subsequent to making recordings with subjects, the passive resonant properties of the pendulum and spring combinations were determined. The pendulum was balanced at a mean backward inclination of 3 deg using springs of different intrinsic stiffness (K). The other end of the spring was fixed to a rigid support. Naturally, balance could only be achieved with springs that were stiffer than the load stiffness. A small (~1 deg) push was imparted to the pendulum and the resulting prolonged transient oscillation recorded. Fast Fourier transform (FFT) analysis of the transient oscillation allowed the damped resonant frequency of the pendulum-spring combination to be determined.
Analysis
Pendulum angle, force, hand position and EMG were recorded with 12 bit resolution by an a-d interface (CED 1401) and a PC. Their values were computed in degrees, newtons, centimetres and voltage, respectively. Pendulum angle and hand position were low-pass filtered (REMEZ FIR filter; cut-off frequency 2.25 Hz). Pendulum sway and hand movement were analysed by measuring the duration and size of the unidirectional sway (defined as the time and displacement from one reversal to the next). We used this technique in preference to a frequency domain method as hand and pendulum movements are irregular and aperiodic. Simple visual inspection of a sway record shows that it consists of a series of unidirectional sways with brief but different durations. These sways are not well represented in a FFT analysis as the coherence time of the record is short and is only of the same order of magnitude as the sways themselves.
Cross-correlation analysis was used to study how the hand and pendulum moved with respect to each other. Cross-correlation was used because it measures the three main features that we wish to show. These features are (1) the extent to which hand movement can predict pendulum position, (2) whether, on average, the hand moves in the same direction or opposite direction to the pendulum and (3) whether the hand movement lags, leads, or is in time with the pendulum movement. Cross-correlation is an appropriate technique when it is desired to compare two signals in the time domain. It provided two values. The peak value of the cross-correlation function (r) indicates the extent to which pendulum position can be predicted from the hand position. Strictly, it is the degree to which the two signals are linearly related. A correlation value of 1.0 indicates that the two are identical and moving in the same direction. Conversely, a correlation value of -1.0 indicates that they are complete opposites (identical but moving in opposite directions - a mirror image). A correlation of zero indicates that hand position provides no information concerning pendulum position. The time offset of the peak of the cross-correlation function shows the average lag or lead between movement of the hand and movement of the pendulum at which they are most highly correlated.
Statistics
Significance was tested by ANOVA and Tukey's post hoc multiple pairwise comparison tests were used. The level of significance chosen was P 
0.05.
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RESULTS |
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Passive pendulum characteristics
The characteristics of the pendulum when balanced passively by springs are shown in Fig. 2. As would be expected for a very lightly damped pendulum stabilised by a spring, the (resonant frequency)2 is proportional to the spring stiffness (K). The value of K for which the natural frequency becomes zero is where the intrinsic stiffness of the spring becomes equal to the load stiffness of the pendulum. From the equation for the regression line, the predicted critical value is 0.721 N mm-1 which is in very close agreement with the calculated value (0.727 N mm-1). Values of intrinsic stiffness that are less than this will not passively stabilise the pendulum and consequently cannot produce a resonance; there is no natural frequency.
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Figure 2. The passive behaviour of the pendulum The resonant frequency of oscillation of the pendulum was determined when stabilised by springs of different stiffness. The (resonant frequency)2 is linearly related to stiffness and the value of stiffness at which the natural frequency would become zero is the point where the stiffness of the spring becomes equal to the load stiffness of the pendulum. The load stiffness of the pendulum is shown on the figure. There is no natural frequency with springs of lesser stiffness than this. | ||
Active balancing
Subjects were told that they should maintain the position of the pendulum as near to a 3 deg of backward inclination as they could. The recording range of our apparatus extended from 0.5 to 5.5 deg and we accepted a trial if these boundaries were not exceeded. All subjects found this task easy to do with the stiffest springs, although for some the prolonged, almost isometric force, (~40 N) at the wrist was quite fatiguing. On gross examination of the results under these conditions the movement of the pendulum was small and subjects quickly realised that it slavishly followed the hand position. However, even with the stiffest springs, the position of the pendulum and hand were not static. With the softer springs, subjects quickly realised that active movement of the hand was required to maintain balance of the pendulum. Learning this ability took variable amounts of time and some subjects continued to find the task difficult with the softest spring. With the intermediate springs all subjects found that the task quickly became semi-automatic and they were able to perform it easily with some finesse while talking or subjected to other minor distractions. All subjects reported that the secret of success lay in anticipation of pendulum movement. Four of the subjects who became very proficient at the task (including two of the authors) attempted to balance the pendulum with the oscilloscope turned off (no visual information about pendulum angle). This was possible with the very stiffest springs (where hand position unambiguously dictated pendulum angle) but none of these subjects succeeded for more than a few seconds with the springs of a relative stiffness of < 186 %.
Figure 3 is a representative trace from one subject. It shows four records, each of 10 s with springs having a stiffness: much greater than(A), greater than(B), approximately the same as (C) and less than (D) the load stiffness of the pendulum. In general, the sway size of the pendulum increased as spring stiffness decreased. With high spring stiffness the controlling hand and pendulum were clearly generally in phase and they moved in the same direction. With intermediate stiffness the pendulum and hand were partly in phase and partly out of phase. With the soft spring the movement of the pendulum and hand was very clearly predominantly in antiphase - at the same instant they were generally moving in opposite directions. Movement of the controlling hand was small with high stiffness and became greater as the stiffness decreased. The force reflected the position of the pendulum (angle) and also the position of the hand (offset). At any instant its size was proportional to the elongation of the spring which connects the two together. At each level of intrinsic stiffness, pendulum position and force can be seen to be predominantly in phase.
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Figure 3. Representative traces from one subject Each record is 10 s long. The records are of pendulum position, hand position and force with springs of four different relative stiffnesses: A, 746 %; B, 249 %; C, 94 %; and D, 58 %. The gain is identical in all records to facilitate direct comparison. The movement of the pendulum is generally less with the greatest intrinsic stiffness. The pendulum and hand move mainly in phase in A and in antiphase in D. The force record reflects both pendulum position and hand position. Force is clearly predominantly in phase with pendulum position. Movements of the hand are more frequent than movements of the pendulum. | ||
In Fig. 4 the relationship between hand position and pendulum position is shown as an x-y plot for one subject balancing the pendulum using springs of different relative stiffness. With the stiffest spring (Fig. 4A) the relationship between hand position and pendulum position was necessarily very close. This is because the hand and pendulum were, in effect, tightly coupled together and on average moved as a unit in the same direction. The small approximately horizontal oscillations resulted from the hand rapidly stretching and releasing the spring. With a somewhat softer spring (Fig. 4B) the situation was similar but overall the pendulum moved appreciably further than the hand because of the greater stretch in the spring. Consequently, the gradient of pendulum position vs. hand position was increased. The rapid excursions of the hand were also larger. With a spring of 94 % relative stiffness (Fig. 4C) it can be seen that a particular pendulum angle could co-exist with many hand positions. Hand position is not a predictor of pendulum position. It can be shown that for a spring of exactly 100 % relative stiffness, the principal gradient would be vertical and all positions of the pendulum would be associated with a mean hand position of zero. With a spring of 58 % relative stiffness (Fig. 4D) the principal gradient was dramatically reversed (negative). The slope was quite similar to that of Fig. 4A but in the opposite sense; thus overall the hand moved approximately the same distance as the pendulum but in the opposite direction. Because of the soft spring, the rapid hand excursions were very large.
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Figure 4. Relationship between hand and pendulum position The values are for linear displacement of the hand and of the pendulum at the point of attachment. All graphs are on the same scale and have been centred on zero. A positive displacement of the pendulum indicates that it is falling and a positive displacement of the hand indicates movement in the same direction (towards the pendulum). A, relative stiffness 746 %. Hand and pendulum move almost as a unit in the same direction. The small horizontal excursions represent small rapid movements of the hand which compress and extend the spring. B, relative stiffness 249 %. The pendulum moves overall more than the hand and the hand movement is larger causing greater compression and extension of the spring. C, relative stiffness 94 %. There is no clear relationship between hand movement and pendulum movement. The hand makes small movements of up to ~5 mm around its mean position and the pendulum drifts through a range of approximately 15 mm. D, relative stiffness 58 %. The gradient relating hand movement to pendulum movement has reversed. On average, hand and pendulum move in opposite directions. The rapid hand movements are large showing that there is considerable compression and extension of the spring. | ||
The interaction of hand position and pendulum position is shown more clearly in Fig. 5A where part of the data from Fig. 4C have been replotted on a larger scale to show more clearly the situation at a relative stiffness of 94 %. Hand positional changes of up to a few millimetres occurred but they were not related in any unambiguous way to pendulum position which, over this part of the record, changed by approximately 4 mm. In Fig. 5B and C, even smaller sections of the record from Fig. 4C are shown. These reveal a consistently observed feature which is that for most reversals of pendulum sway there was more than one reversal of hand movement. Thus, the hand made considerably more movements than the pendulum.
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Figure 5. Hand and pendulum position at 94 % relative stiffness A, some of the data from Fig. 4C replotted to show the relationship between pendulum position and hand position in more detail at a relative stiffness of 94 %. It is possible to make out alterations in the direction of sway of the pendulum (represented by turning points in the vertical axis) and alterations in the direction of hand movement (represented by turning points in the horizontal axis). There are more sways in the horizontal sense than in the vertical sense. This is shown more clearly for two single sways of the pendulum in B and C. In B there are three hand movements for the single pendulum sway and in C there are four. In B and C the dots are 40 ms apart. | ||
In order to examine in a more systematic way the relative movement of the hand and pendulum and the degree to which they were linearly related, cross-correlation analysis was performed between the pendulum position and the hand position for each subject at each relative stiffness. A negative value means that the hand movement is leading the pendulum and all values are in milliseconds.
Figure 6A shows how the correlation between hand and pendulum movement changes with intrinsic stiffness. With a very stiff spring the hand and the pendulum were, in effect, strongly mechanically coupled together and constrained to move as a unit. This is reflected in the very high value of r (0.939) at 746 % relative stiffness. As stiffness was decreased, the coupling between hand and pendulum became weaker and the value of r reduced. The movement of the hand and pendulum had become more decoupled. When the relative stiffness fell below 100 % (soft springs), the value of r became negative, indicating that hand and pendulum movement were now generally moving in opposition as seen in Fig. 4D. The magnitude of the negative r value was largest (-0.847) with the lowest relative stiffness; the movement of the hand was almost a mirror image of the pendulum. The lowest correlation was found with the springs around 100 % relative stiffness. This confirms the impression arising from Fig. 4D and Fig. 5 which is that there is little correlation between hand and pendulum movement around this relative stiffness.
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Figure 6. Cross-correlation of pendulum and hand position Values are means and S.E.M. for three trials at each stiffness with seven subjects (2 at 106 %, 124 % and 746 %). A, the value of the correlation coefficient with springs of different intrinsic stiffness. The value is high and positive with stiff springs; hand and pendulum move generally in phase. With soft springs it is high and negative; hand and pendulum move generally in antiphase. B, the time lag between the pendulum position and hand position (time offset at which the correlation product is maximal). With stiff springs, pendulum position lags the hand and the delay can be considerable. With soft springs the situation is entirely different; there is almost no time interval between hand and pendulum movement. | ||
The time delay between hand and pendulum movement is shown in Fig. 6B. With the stiffest spring the lag between hand and subsequent pendulum movement was inevitably minimal. However the time lag increased rapidly as the spring stiffness was reduced and when the relative stiffness was 186 %, the lag of pendulum position on hand position was more than 500 ms. The lag increased yet further as stiffness fell, reaching a value of nearly 1.0 s at 106 %. When the relative stiffness was reduced below 100 % the relationship between hand and pendulum position changed abruptly. The time delay between hand movement and pendulum position became small. Taking into account the negative r value, this implies that, on average, the hand and pendulum are at any instant moving in opposite directions. The tendency for simultaneous antiphase movement became strong as relative stiffness was further reduced. Taking Fig. 6A and B together, a rigid link and any spring stiffness materially less than 100 % are similar in that they have essentially zero time lag between hand and pendulum movement. They are however completely different in that with a rigid link, hand and pendulum move on average in the same direction whereas with a soft spring they move on average in opposite directions.
Pendulum movement and hand movement were categorised by two simple procedures
The time and displacement associated with each reversal point were identified. The distance between each reversal point was used to give a measure of pendulum sway or hand-movement size and the time interval was used to give a measure of pendulum sway or hand-movement duration. The median value of both parameters was calculated for each trial. The median value was used as sway parameters are not normally distributed. Values for each trial and each subject could then be pooled and the mean and S.E.M. calculated. These values are plotted in Fig. 7 which shows data from all seven subjects and all trials (only two subjects at 106, 124 and 746 %).
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Figure 7. Pendulum sway and hand movement measures Most records show means ± S.E.M. of the parameters derived from three trials and seven subjects; however, only two subjects performed the test with the 106, 124 and 746 % relative stiffness. Movement of the hand is in millimetres and pendulum movement is measured both in degrees and millimetres at the point of attachment of the spring. A, pendulum median sway size. Sway size increases slightly as stiffness decreases. Post hoc comparison showed that the 58 % relative stiffness was significantly different from all the other relative stiffnesses and the 76 % relative stiffness was significantly different from 151, 189, 252 and 746 % relative stiffnesses. B, pendulum median sway duration. This is the time taken for a unidirectional sway. The | ||
Figure 7A shows the median sway size of the pendulum for the seven subjects. The tendency was for sway size to increase as stiffness decreased. The subjects performed in a generally similar way. One-way ANOVA for all subjects' data showed that the increase in sway size was significant (F = 16.0, d.f. = 8, 142, P < 0.001) and post hoc pairwise comparison showed that the 58 and 76 % relative stiffness were significantly different from the others.
Figure 7B (filled symbols) shows the median sway duration of the pendulum for all subjects. The duration is the time for the pendulum to complete a transit between reversals; it is therefore a unidirectional sway. Median unidirectional sway duration is remarkably constant across the whole range of stiffnesses which were used, ranging from ~0.8 to 1.2 s. In Fig. 7B the calculated resonant sway duration values for the pendulum and spring combination are also plotted for six values of stiffness greater than 100 % (open symbols). These were calculated from the data in Fig. 2. The continuous line shows the best fit. As would be expected if the pendulum were stabilised purely by a spring, the sway duration increased in inverse proportion to the square root of stiffness.
Figure 7C shows the median hand-movement size. The hand movement became greater as stiffness was reduced. The high value for the softest spring was partly attributable to some of the subjects whose hand movements were much bigger than the others. One-way ANOVA showed that hand movement was significantly affected by spring stiffness (F = 34.0, d.f. = 8, 142, P < 0.001) and post hoc pairwise analysis showed that this significant difference was confined to the 58 and 76 % relative stiffness. The median size of the hand movements is remarkably small, even with low relative stiffness.
Figure 7D shows the hand movement duration calculated in the same way as the pendulum sway duration. There is a suggestion that the duration was minimal at relative stiffness around 100 % and the duration was significantly longer at 746 % relative stiffness (F = 80.0, d.f. = 8, 142, P < 0.001).
In Fig. 8 force has been plotted against pendulum position for one subject at a relative stiffness of 94 %. Also shown on the figure is the static force-position line for the inverted pendulum. There are three noteworthy aspects of this figure. First, as would be expected, the principal gradient of the figure was very close to the pendulum's force-position line. Although at any instant it may be too high or too low, on average the force was appropriate at every angle. Second, there are many instances where there were relatively large changes in force associated with small changes in pendulum position. These correspond to stretch of the spring (offset) produced by a relatively rapid hand movement. Third, there were sometimes quite prolonged periods when the force changed with a gradient which was close to that of the pendulum's force-position line. At these times the hand was relatively stationary.
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Figure 8. The relationship of pendulum position and force at a relative stiffness of 94 % These are the same data as shown in Fig. 5A. The continuous line indicates the static force-position relationship of the pendulum. The force vs. pendulum position record is shown as a thin line with dots superimposed at 40 ms intervals. The record is made up in part of periods where the force changes with angle in much the same way as would be expected for the pendulum. In these periods, the hand is relatively stationary and the force change results from the pendulum stretching the spring. There are also many parts of the record where the force changes relatively suddenly with no clear relationship to pendulum position. These changes result from offset adjustments produced by hand movement. The overall force-position record is the 'effective stiffness' but it results from the combination of two different components. | ||
The EMG of the biceps and triceps muscles was rectified and integrated and the root mean square level determined for both muscles in each experiment. These values were analysed by ANOVA and no significant influence of spring stiffness was found (biceps, F = 1.238, d.f. = 8, 142, P = 0.28; triceps, F = 1.806, d.f. = 8, 142, P = 0.08). Biceps activity was remarkably constant at all stiffness levels except at the highest relative stiffness where it was increased. On average there was little activity in the triceps muscle throughout. There was no obvious evidence of appreciable co-contraction at any relative stiffness.
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DISCUSSION |
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We have three main aims. First, we wish to show that it is possible to balance an inverted pendulum in a well-controlled way when the intrinsic stiffness is too low to permit passive stability. Our second aim is to examine the hand movement that occurs and to show how the central nervous system may control this in relation to the pendulum movement to produce stability. Third, we make a testable prediction of how the calf muscles might be controlled in standing when the intrinsic stiffness is low and we also identify a role for the intrinsic stiffness as an energy efficient passive buffer in an active balancing control system.
Balance of an inverted pendulum using springs of different intrinsic stiffness
With stiff springs (greater intrinsic stiffness than the pendulum load stiffness) the system is passively stable. A disturbance of pendulum angle will generate a restoring force which will bring it back to the equilibrium position. With soft springs (lower stiffness than the pendulum load stiffness) this is not so, and active hand movement is required to generate extra force to restore the pendulum position. Balance of the pendulum could be satisfactorily achieved with all the springs that we employed. Subjects generally found that balancing was most difficult with the softest spring (58 %). Pendulum sway became larger as relative stiffness was reduced and this was significant for the two lowest stiffnesses (Fig. 7A). As the relative stiffness was increased above 94 % (Fig. 7A), there was a further reduction in sway size, but this effect appeared to progressively decrease, and median sway size reached a minimum value of ~0.1 deg. The size of the necessary hand movements was also affected by the relative stiffness and much larger movements were made when this was very low (Figs 3D, 4D and 7C). The implication of this is that intrinsic stiffness plays a small but positive role in minimising the sway of the pendulum. With all the intermediate springs it was easy to preserve a state of balance and the transition from a stiff spring to a soft one was not particularly remarked on by any of the subjects. The range of median sway sizes of the pendulum (Fig. 7A) was similar to the mean sway size of the human body in standing and to the sway of a similar pendulum balanced by the feet in an equivalent task (Loram et al. 2001; Loram & Lakie, 2002b). This suggests that the task was genuinely comparable to standing. Even with the highest relative stiffness, movement of the hand and pendulum occurred and there was no evidence that position was maintained by a strategy of co-contraction of the arm muscles to increase its rigidity. Maintenance of balance when intrinsic stiffness was low required a distinctive strategy for control where the inadequate force change arising from stretch of the spring was augmented by hand movement (see below). All the subjects were able to develop this strategy by trial and error without necessarily understanding exactly what they were doing. It is quite probable that the performance of subjects would have improved if they had been given longer to practice. In order to perform the task with low intrinsic stiffness, subjects required some visual knowledge of pendulum position. In these experiments pendulum position was shown to the subjects on an oscilloscope, but in preliminary experiments the subjects watched the pendulum directly. The source of the visual information did not make any obvious difference to the results. It was also noticeable that subjects did not need to give their undivided attention to the pendulum or oscilloscope.
How is the pendulum stabilised?
In Fig. 7B the median unidirectional sway duration of the pendulum is shown when it oscillated while passively stabilised by stiff springs. The period of the resonance under these conditions (time for a to-and-fro sway) ranged from > 7.0 s at 106 % relative stiffness to approximately 0.8 s at 373 % relative stiffness. In contrast, when the pendulum was actively balanced it was clear that it did not resonate. Figure 7B shows that its median sway duration (approximately 1.0 s) was remarkably constant at all levels of intrinsic stiffness. If a 1 s sway in one direction were followed by a 1 s sway in the other, this could be regarded as a period of 2.0 s which corresponds to a frequency of ~0.5 Hz (but as will be seen below the movement is not a regular oscillation of this type).
The relationship between pendulum sway and hand movement. Inspection of Fig. 4 and Fig. 5 provides an important insight into the relationship of movement of the hand and pendulum. With the stiffest spring there is a tight mechanical coupling between the hand and the pendulum (Fig. 4A) and on average the two move in the same direction. As stiffness is reduced, the two still move, on average, in the same direction but the pendulum moves further than the hand because of weaker coupling ('give' in the spring; Fig. 4B). The situation is very different for the lowest intrinsic stiffness. Here, on average the hand has to move in the opposite direction to the pendulum in order to generate sufficient torque (Fig. 4D). With relative stiffness around 100 % (Fig. 4C and Fig. 5) the hand 'dances' to and fro by a few millimetres or less and the pendulum position varies apparently randomly by approximately 15 mm. The pendulum is being actively controlled and balanced but its position is not predictable from hand position. The cross-correlation analysis confirms this impression (Fig. 6A). With stiff springs the correlation between hand and pendulum position is positive and strong; with weak springs it is negative and strong, and at around 100 % relative stiffness it is much lower. The inference is clear: when the relative stiffness is close to 100 % the hand does not control the pendulum purely by its position. Another factor must be involved and this is the time for which the position of the hand is maintained. Thus, control of the pendulum position depends on hand offset and duration - an impulsive controller.
It is also clear that the pendulum and hand move at different rates. Inspection of Fig. 5 shows that for a reversal of the pendulum to occur there are usually several reversals of hand movement (two in Fig. 5B and three in Fig. 5C). This is confirmed by a comparison of Fig. 7B and D which show that the duration of a pendulum sway is on average two to three times longer than a hand movement. Thus for each unidirectional pendulum sway there are on average two to three unidirectional hand movements. These hand movements represent anticipatory efforts to control the pendulum. We have previously described this discontinuous behaviour as a drop and catch mechanism (Loram & Lakie, 2002a). In our view, the pendulum sway duration is determined by the time that the CNS takes to plan and execute an anticipatory offset adjustment to catch the next sway, and by the fact that on average it takes two to three attempts to do so.
The drop and catch mechanism. It has been known for many years (Craik, 1947, 1948; Vince, 1948) and confirmed more recently (Neilson et al. 1988) that discrete neural outputs can be made at only a modest rate, with most estimates being two to three events per second. The movements of the hand which we observed were made at this rate. The pendulum falls and is caught by an anticipatory offset adjustment which is planned and executed by the CNS. However the anticipatory adjustment does not always arrest the pendulum (as it is often inadequate in size) and we suggest based on Fig. 7B and D that there are, on average, approximately two to three anticipatory hand adjustments per unidirectional pendulum sway. The same mechanism operates for movements of the pendulum in the opposite (rising) direction with the adjustments being reversed. This mechanism also provides a simple explanation for how the pendulum can be driven to any desired new mean position. Pendulum position is not controlled by hand position but rather by the impulse which acts on it. Impulse represents the product of the size of the offset and its duration. The drop and catch mechanism represents the output of a neurally controlled impulse generator. As we have previously pointed out, a drop and catch (or its converse a throw and catch) is the only method for moving an inverted pendulum from one stabilised position to another (Loram & Lakie, 2002a). This mechanism has the advantage of parsimony; it can provide stability and control movement. It also provides a simple explanation for the pendulum sway durations that were observed. The sway durations are a statistical property of the number of impulses that are necessary to arrest the pendulum. Thus, on this view, control of balance of the inverted pendulum is a discrete, rather than a continuous process and pendulum sway is a consequence of this discontinuous process.
The relationship of the drop and catch mechanism to 'effective stiffness'. In an alternative interpretation of our data the controller could be thought of as an active and anticipatory effective stiffness-generation mechanism. Effective stiffness of the system is defined as the change in torque applied to the pendulum divided by the change in pendulum angle. Movement of the offset of the spring permits the torque at any pendulum angle to be altered at will so that effective stiffness can be altered by appropriate hand movements. Intrinsic stiffness is the stiffness of the spring. With low values of intrinsic stiffness the hand is moved in opposition to the pendulum to produce additional length modulation of the spring. This produces a value of effective stiffness greater than the intrinsic stiffness. The 'goal' of the CNS would be to produce an adequate effective stiffness to balance the pendulum.
As stated above, the median frequency of the pendulum could be considered to be consistently ~0.5 Hz. This frequency is the resonant frequency of the pendulum passively stabilised by a spring of ~200 % relative stiffness. On this basis the effective stiffness of our controller could be said to be 200 %. This value (200 %) is certainly consistent with many calculations of stiffness which have previously been made for antero-posterior body sway on the basis of sway characteristics (Winter et al. 1998; Carpenter et al. 1999). However, Winter et al. (1998, 2001) wrongly identify this effective stiffness as intrinsic stiffness because they have omitted to include the element of torque produced by EMG modulation within the cycle (Loram & Lakie, 2002a; Morasso & Sanguineti, 2002). The regulation of effective stiffness in this way could, in principle, stabilise the pendulum. Notwithstanding this, inspection of Fig. 8 shows that that the actual torque-angle relationship does not resemble a stiffness. An effective stiffness could be computed from the trace, but it is clear from Fig. 8 that the change in torque associated with a change in pendulum angle is in reality an amalgam of two different processes. The concept of effective stiffness is a useful abstraction but it does not explain the impulsive intermittent control that is observed. Also there is no obvious reason why the effective stiffness should be set at 200 %. Nor does it explain the characteristics of the sway that are observed. Unlike the conventional pendulum of a clock, the inverted pendulum in these experiments does not sway in a periodic manner. The motion is aperiodic so that the duration of one unidirectional sway does not predict the next (I. D. Loram & M. Lakie, unpublished observations). It is for this reason that we are uneasy about assigning a 'frequency' to the sway. This in itself militates against the view that the sway results from the inertial properties of the pendulum stabilised by a stiff spring. The intermittent movements of the hand are not periodic. The non-periodic nature of these neurally controlled anticipatory events can explain the non-periodic sway of the pendulum. In summary, the pendulum and controller have both an intrinsic stiffness (the spring in our experiments) and an effective stiffness (which can be calculated from the sway characteristics). However, we do not think that effective stiffness is the variable that is regulated by the CNS.
Relationship of hand movement size and duration to intrinsic stiffness. Figure 7D shows that hand movement duration was long with the highest stiffness and was very similar to pendulum sway duration (Fig. 7B). This similarity is presumably due to the tight coupling - the hand and pendulum move almost as a unit and the hand movement takes on the characteristics of the pendulum sway. The duration of the hand movements and the size of the pendulum sway also increased somewhat as the intrinsic stiffness decreased below 200 % (Fig. 7D). This may be because the size of the hand movements also increased as the intrinsic stiffness decreased (Fig. 7C). If the process is limited by the velocity with which the offset can be adjusted, the bigger adjustments will inevitably take more time and must proceed at a lower rate. In this experiment the biceps muscle had to contend with the inertia of the arm, armlet and spring, which were accelerated and decelerated. The velocity of offset adjustment will reflect the mechanical properties of the muscle and the load and the capability of the neural controller. Balance can be maintained by this intermittent biasing method regardless of the intrinsic stiffness of the system. However, the intrinsic stiffness does have a useful role in that it can partly compensate for the gravitational torque and allows the CNS to concentrate on the dynamic offset adjustments. It will also act as an energy-storing buffer (see below).
Implications of a low intrinsic ankle stiffness for standing
We have recently shown that in quiet standing the intrinsic ankle stiffness is 91 % (± 23 %) of the body's toppling torque per unit angle (Loram & Lakie, 2002a,b). Also, Morasso & Sanguineti (2002) have recently theoretically calculated that ankle stiffness alone is insufficient for stability. This conclusion is supported by a number of other calculations of foot-tendon compliance which will act as a limiting factor for intrinsic stiffness (Table 1). In the light of the present results we can make a number of suggestions about the implications for standing of an intrinsic ankle stiffness of approximately 100 %.
First, triceps surae length will be poorly (or, at exactly 100 % relative stiffness, not at all) correlated with body position. The advantage of an intrinsic stiffness of 100 % is that the mean length of the muscle will be the same over the whole range of standing positions and for any angle of body inclination the muscle will be able to operate within its optimal length region. The result does, however, have serious implications for some theories of reflex control of standing. In our experiments the counter-intuitive finding was that at around this value of relative stiffness, the position of the hand is not in itself sufficient to control the pendulum. Conversely, knowledge of the position of the hand cannot provide information about the position of the pendulum. By implication, the myotatic reflex will be of limited value in standing.
Second, as the average intrinsic stiffness is a little less than 100 % such small alteration of mean triceps surae muscle length that does occur will be in opposition to the body shift. Thus in forward sway the triceps surae muscle shortens and in backward sway it lengthens (Loram, 2003).
Third, as the intrinsic stiffness is inadequate for stability, we suggest that the force deficit is supplied by a neurally generated, intermittent pattern of impulses. Neural control of balance and position involves the generation of extra force by dynamically pulling on or releasing the tendon- foot spring to shift its offset. In standing, as the body sways forward the foot, tendon and muscle are all stretched. In order to generate sufficient extra tension the muscle must shorten, producing extra force by lengthening the foot- tendon spring. As shown by Fig. 6B, with low intrinsic stiffness there is no time lag between offset and pendulum position. To provide compensation for the inevitable biomechanical delays the neural controller has to act in an anticipatory manner. Accordingly, force resulting from tendon-foot stretch and muscle bias will be, on average, in phase with angle of the body. The phase relationship between force and angle is a consequence of active anticipatory control, not a passive elastic mechanism as has been claimed (Winter et al. 1998, 2001). Subtle alterations of impulse timing can be used to drive the body to any desired new position. This is the drop and catch mechanism as mentioned above and described in Loram & Lakie (2002b). The body is in effect nudged to a new mean position by an appropriate pattern of impulses. The necessary neural pattern can only be produced as the result of a process which estimates in anticipation the force required which is necessary to arrest and reverse the body. What we observe as postural sway is the result of these anticipatory steps, endlessly repeated. Sway duration is not dictated by spring stiffness or body inertia but by the neural controller's decision-making and execution time, and by mechanical limitations to the rate of change and accuracy of the offset. The anticipatory steps are made more precisely when sway is to be minimised (Loram & Lakie, 2002b) so that sway size is a function of the accuracy of an estimation of bias.
Fourth, it is possible to estimate the extent of the bias adjustment required in standing. As explained in Methods, the lever arm in our system is approximately 17.4 times longer than in the leg. We have previously suggested that the intrinsic stiffness in standing is about 90 % of the toppling torque per unit angle. With the 94 % stiffness spring that we employed, median hand movement was ~2 mm. This suggests that the size of the bias adjustment required in the triceps surae is approximately 120 µm. This small movement must be broadly similar to that which the biceps muscle itself was making in our present experiments.
The low intrinsic stiffness is caused by the tendon and foot which together act as an energy buffer
Although it cannot provide passive stability, we suggest that the low intrinsic stiffness does serve a very important role as a decoupler and energy store. When force at the ankle is changed, the intrinsic elastic element will be stretched and energy can be stored in it. Our recent work (Loram & Lakie, 2002a) suggests that the greatest stretch will occur in the tendon and the foot and will be considerable in both. As they are not very stiff, the tendon and foot act as a buffer between the muscles and the body. During standing, the triceps surae muscle is not constrained to move by the same amount, at the same rate or in the same direction as the body; they are decoupled and the muscle is cushioned from the very large inertial load of the body. Rather, the muscle acts indirectly on the body by rapidly and anticipatorily altering the elastic energy in a spring. Control of position is not dictated by muscle length but by adjustment of impulse which is the product of spring length and time. At any given torque, the energy that is stored is proportional to the extension that is generated - i.e. the energy storage is greatest in the weakest spring. The weak spring permits the contracting muscle to do more external work as it generates its force over a much longer distance. The weak spring is the tendon and foot stiffness which effectively stores the energy (integrates the impulses) and imparts it to the body over a longer time period. For storage of energy imparted by offset adjustments, a low stiffness of tendon and foot is best. The lower limit of stiffness is probably dictated by three factors. These are; first, the range and rapidity of the offset that the muscle can generate; second, the predictive accuracy with which changes in offset can be made and third, the need for the tendon and foot to be ultimately sufficiently stiff to transmit the highest torques that the muscle can generate. So far we have considered only the situation where the muscle acts on the foot, but the decoupling is also an advantage when the foot acts on the muscle. In this situation, sudden deflections of the foot (e.g. stumbling) do not produce high shock loadings in the muscles to which it is connected.
Conclusion
We have shown that a large inverted pendulum can be satisfactorily balanced when it is coupled to a limb by a soft spring. This demands a mode of control where the actuator and the pendulum move on average in opposite directions. As the two move with virtually no delay between them, anticipatory control seems inevitable. This does not resemble a linear proportional controller, mainly because there are always more movements of the hand than there are of the pendulum. The movement of the hand is consistent with intermittent control by the nervous system and we suggest that it is the aperiodic nature of these movements and the low intrinsic stiffness which are responsible for the unpredictable sway of the pendulum. We suggest that it is this process that may underlie human standing. Standing, or balancing a pendulum, is a particularly demanding task as the load is unstable and errors grow by positive feedback. Myotatic reflex control of this task would be difficult as with a relative stiffness close to 100 %, movement of the muscle and load will be on average poorly correlated. This may mean that the role of reflexes in standing will need to be reconsidered. It is an open question whether similar constraints will also apply in controlling a stable load through compliant tendons.
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symbols show the experimentally derived values when the pendulum is actively stabilised. There is little variation in sway duration as the stiffness is changed and the duration is generally close to 1.0 s. Post hoc pairwise comparisons showed that the sway duration at 252 % relative stiffness was significantly shorter than the values at 76, 94, 108 and 151 % relative stiffness. The 
symbols show how the pendulum behaves when passively stabilised by springs of stiffness greater than 100 % of the load stiffness. The unidirectional sway durations have been calculated from Fig. 2. The continuous line is a line of best fit. It shows that the sway duration, when passively stabilised in this way, increased in inverse proportion to the square root of stiffness. The lack of similarity between active and passive stabilisation at low levels of intrinsic stiffness is striking. C, hand movement median size. Hand movement increases as spring stiffness decreases. Post hoc pairwise comparisons showed that at this stiffness (58 %), hand movement was significantly larger than that at all the other relative stiffnesses. The movement size at 76 % relative stiffness was also significantly larger than that at the 151, 189 and 252 % relative stiffnesses. D, median duration of unidirectional hand movements. Generally, hand movements are rapid events with a duration of ~400 ms. Post hoc pairwise comparisons showed that the duration at 746 % was significantly greater than that at all the other relative stiffnesses.