Human balancing of an inverted pendulum: position control by small, ballistic-like, throw and catch movements - PubMed (original) (raw)
Human balancing of an inverted pendulum: position control by small, ballistic-like, throw and catch movements
Ian D Loram et al. J Physiol. 2002.
Abstract
In standing, there are small sways of the body. Our interest is to use an artificial task to illuminate the mechanisms underlying the sways and to account for changes in their size. Using the ankle musculature, subjects balanced a large inverted pendulum. The equilibrium of the pendulum is unstable and quasi-regular sway was observed like that in quiet standing. By giving full attention to minimising sway subjects could systematically reduce pendulum movement. The pendulum position, the torque generated at each ankle and the soleus and tibialis anterior EMGs were recorded. Explanations about how the human inverted pendulum is balanced usually ignore the fact that balance is maintained over a range of angles and not just at one angle. Any resting equilibrium position of the pendulum is unstable and in practice temporary; movement to a different resting equilibrium position can only be accomplished by a biphasic 'throw and catch' pattern of torque and not by an elastic mechanism. Results showed that balance was achieved by the constant repetition of a neurally generated ballistic-like biphasic pattern of torque which can control both position and sway size. A decomposition technique revealed that there was a substantial contribution to changes in torque from intrinsic mechanical ankle stiffness; however, by itself this was insufficient to maintain balance or to control position. Minimisation of sway size was caused by improvement in the accuracy of the anticipatory torque impulses. We hypothesise that examination of centre of mass and centre of pressure data for quiet standing will duplicate these results.
Figures
Figure 1. Inverted pendulum apparatus
Subjects balanced a backward-leaning, real inverted pendulum of mass and inertia equivalent to a medium sized woman. The subjects were unable to sway since they were strapped round the pelvis to a fixed vertical support. The rotation of the pendulum, platform and footplates was co-axial with the subject's ankles. Force exerted by the subject's ankle musculature onto each footplate was transmitted by horizontally mounted load cells. A contactless precision potentiometer measured sway of the pendulum.
Figure 2. Illustration of the line crossing averaging process
A, a 6 s record of angular velocity and angular acceleration against time for a representative subject. Equilibrium times were identified by interpolating between the pairs of acceleration data points that cross zero. From these equilibrium times, those that occurred while the acceleration was passing from positive to negative and while the velocity was positive (i.e. the pendulum is falling) were selected. These equilibrium times are shown as an asterisk. Ankle torque and pendulum position records were sampled at 0.04 s intervals before and after these selected equilibrium times. The four selected equilibrium times in A are shown in B, together with ± 0.48 s of surrounding data, plotted as ankle torque against pendulum position. The straight dashed lines represent the line of equilibrium defined by the gravitational torque acting on the pendulum. The selected equilibria represent falling (increasing angle), spring-like (positive-gradient) line crossings as indicated by the arrow in C. The four 0.96 s records shown in B are averaged to produce the record shown in C. In a 200 s trial, over 100 examples would be averaged. The rising, positive-gradient line crossings were selected and averaged in an analogous manner.
Figure 4. Effect of intention on averaged spring-like line crossings
Averaged data are shown from 1 s before to 1 s after all positive-gradient, equilibrium line crossings while the pendulum was falling. These data were averaged over all 10 subjects in the ‘still’ (1) and ‘easy’ (3) conditions while visual feedback was available. A, combined torque from both legs vs. pendulum position. B, combined soleus EMG from both legs vs. pendulum position. EMG data were normalised between subjects. Data points are at 40 ms intervals and proceed from reversal points a to b via the arrow. The line of equilibrium (ignoring pendulum friction) is shown as the continuous straight line in A. The asterisk marks the instant of equilibrium and maximum velocity.
Figure 3. Pendulum sway
A, a 12 s record from one subject is plotted as combined ankle torque against pendulum position. Data points are at 40 ms intervals. The starting point (diamond) and finishing point (square) are indicated. The line of equilibrium (gravitational torque on the pendulum) is shown as a continuous straight line. For each trial condition, B shows the mean sway size and C shows the mean sway duration. For both panels, values were averaged over 10 subjects for each of the four trial conditions. Error bars show 95 % simultaneous confidence intervals for the mean values. As described in Methods, a sway is the angular movement between successive reversal points of the pendulum. Trial conditions were (1) stand still with visual feedback, (2) stand still with no visual feedback, (3) stand easy with visual feedback, and (4) stand easy with no visual feedback.
Figure 8. Comparison of soleus, tibialis anterior and gastrocnemius EMG
A and B, averaged data from 1 s before to 1 s after all positive-gradient, equilibrium line crossings while the pendulum was falling. Data points are at 40 ms intervals and proceed from reversal points a to b via the arrow. The asterisk marks the instant of equilibrium and maximum velocity. A, soleus (S) and tibialis anterior (TA) EMG from both legs vs. pendulum position. These data were averaged over all 10 subjects and over all 40 trials including ‘still’ and ‘easy’ conditions both with and without visual feedback. B (for one leg only), soleus (S), tibialis anterior (TA), gastrocnemius medialis (GM) and gastrocnemius lateralis (GL) EMG vs. pendulum position as well as ankle torque vs. pendulum position. This data set is different from the one reported in the body of this paper although the methods used were identical. Twelve subjects were asked to balance the pendulum for 200 s.
Figure 5. Decomposition of ankle torque according to our model
The decomposition of ankle torque into intrinsic elastic and neurally modulated components is shown. The model is described in Appendix A. A-C, illustrative data averaged from all subjects for falling, positive-gradient equilibrium line crossings in condition 3. A, the changes in torque arising from stretching and releasing of the elastic components. B, the variation in torque resulting from neural modulation (continuous line) and the preceding variation in soleus EMG (dotted line). C, the actual variation in torque as well as the modelled variation (dotted line). _A_-C, the same position, EMG and torque data as Fig. 4 (‘easy’). Data points are at 40 ms intervals and the reversal points a and b correspond to those in Fig. 4. The model was applied to averaged, positive-gradient, line crossing data from each trial. Values of parameters for falling and rising line crossings were averaged. For each trial condition, D shows the mean intrinsic mechanical stiffness, E shows the mean neural gain and F shows the mean electromechanical delay between changes in soleus EMG and changes in torque. The neural gain is expressed relative to the isometric neural gain (N m V−1). Parameter values were averaged over 10 subjects for each of the four trial conditions. Error bars show 95 % simultaneous confidence intervals for the mean values.
Figure 6. Effect of intention on binned sways
A, the effect of intention on sways sampled and grouped according to their velocity at the first positive-gradient line crossing. A, the acceleration at the end of the sway. B and C, the effect of intention on sways sampled and grouped according to their initial acceleration. B, the sway size to the first positive-gradient line crossing equilibrium and to the reversal point at the end of the sway. C, the duration to the first line crossing and to the end of the sway. For all panels the ‘still’ results (crosses on continuous line) and the ‘easy’ results (dots on dashed line) were averaged over falling and rising sways and over with and without visual feedback for all subjects. The abscissa values are the mean binned values. The error bars represent 95 % confidence intervals in the mean ordinate values for each bin. Two-way ANOVA for ascending bins in A gives n = 80, F = 1.0, 8.7, 16.1, 9.7, 11.7, 9.1, P = 0.3, 0.004, 0.0001, 0.003, 0.001, 0.003.
Figure 7. Effect of intention on the current sway and subsequent return sway
For every sway the pendulum starts from transient rest, passes a positive-gradient line crossing (a speed maximum) and ultimately comes to a reversal point where it changes direction. For each bin, A shows the mean size of a sway vs. the velocity at the first positive-gradient line crossing. After the reversal point, the pendulum executes a return sway in the opposite direction to the current sway. B (continuous and dashed line) shows the mean size of the subsequent return sway vs. the velocity at the first positive-gradient line crossing of the current sway. The dotted line shows the size of the return sway calculated by interpolation from Fig. 6_A_ and B. The ‘still’ results (crosses on continuous line) and the ‘easy’ results (dots on dashed line) were averaged for all subjects over falling and rising sways including with and without visual feedback conditions. The abscissa values are the mean binned values. The error bars represent 95 % confidence intervals in the mean ordinate values for each bin. Two-way ANOVA for ascending bins in B gives n = 80, F = 7.3, 7.3, 11.5, 8.9, 11.2, 11.0, P = 0.008, 0.009, 0.001, 0.004, 0.001, 0.001. C, for a range of current line crossing speeds the continuous line (tot.) shows the total difference in the size of the subsequent return sway caused by the intention of the subject. The dashed (i. a.) and dotted (oth.) lines show respectively the component differences caused by reducing the initial acceleration of the subsequent sway and by other minimisations occurring during the subsequent sway. All three lines were calculated by interpolation from Fig. 6_A_ and B, not Fig. 7_B_.
Figure 9. An ideal, perfect drop and catch pattern
A-C a perfect biphasic drop and catch pattern that will move the pendulum from rest and equilibrium at one position to rest and equilibrium at a new position. A, pendulum position vs. time. B, torque vs. time. C, torque vs. position. D, a double drop and catch pattern. Points proceed at 40 ms intervals in the direction of the arrow. The dashed line is the line of equilibrium. The asterisk indicates the positive-gradient line crossing equilibrium which is the instant of maximum speed. The derivation is given in Appendix B.
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