Human balancing of an inverted pendulum: is sway size controlled by ankle impedance?

Abstract

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. Two main questions were addressed. Can subjects systematically change sway size in response to instruction and availability of visual feedback? If so, do subjects decrease sway size by increasing ankle impedance or by some alternative mechanism? The position of the pendulum, the torque generated at each ankle and the soleus and tibialis anterior EMG were recorded. Results showed that subjects could significantly reduce the mean sway size of the pendulum by giving full attention to that goal. With visual feedback sway size could be minimised significantly more than without visual feedback. In changing sway size, the frequency of the sways was not changed. Results also revealed that ankle impedance and muscle co-contraction were not significantly changed when the sway size was decreased. As the ankle impedance and sway frequency do not change when the sway size is decreased, this implies no change in ankle stiffness or viscosity. Increasing ankle impedance, stiffness or viscosity are not the only methods by which sway size could be reduced. A reduction in torque noise or torque inaccuracy via a predictive process which provides active damping could reduce sway size without changing ankle impedance and is plausible given the data. Such a strategy involving motion recognition and generation of an accurate motor response may require higher levels of control than changing ankle impedance by altering reflex or feedforward gain.

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 axis of rotation of the pendulum, platform and footplates was co-linear with the subject's ankles. Force exerted by the subject's ankle musculature onto each footplate was transmitted by horizontally mounted load cells. These measured the torque that each leg applied to the pendulum via the rigidly attached platform. The footplates and the platform were independently mounted on precision ball races. A precision potentiometer measured sway of the pendulum. Absolute angle of the pendulum (Θ) and angular velocity were measured by an electronic inclinometer and a solid state gyroscope (not shown).

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 are identified by interpolating between the pairs of acceleration data points that cross zero. From these equilibrium times are selected those that occur while the acceleration is passing from positive to negative and while the velocity is positive (i.e. the pendulum is falling). These equilibrium times are shown as an asterisk. Ankle torque and pendulum position records are sampled at 0.04 s intervals for up to 5 s 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, Tg=KttsinΘ (load stiffness). The selected equilibria represent falling (increasing angle), spring-like (positive gradient) line crossings with an ankle impedance (Δtorque/Δangle) greater than the load stiffness. The four 0.96 s records shown in B are averaged to produce the record shown in C. The rising, positive gradient line crossings are selected and averaged in an analogous manner.

Figure 3. Representative sway of one subject

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, ⋄, and finishing point, □, are indicated. The line of equilibrium, load stiffness (Tg=KttsinΘ), is shown as a continuous straight line. Dashed lines parallel to this represent lines of constant torque error. Torque error produces a directly proportional acceleration of the pendulum in the direction indicated by the arrows. The inertia of the pendulum effectively ‘absorbs’ the torque; even with the largest torque error shown it will take 0.6 s for the deflection of the pendulum to reach 0.3 deg (this trial's average sway size) from rest. The smaller and smallest torque errors are associated with times of 1.0 and 1.4 s, respectively.

Figure 4. Effect of trial conditions on sway size

A, distributions of sway sizes for one subject under each of four trial conditions labelled 1, 2, 3 and 4 as described in Methods. Each trial lasted 200 s. A sway size was the angular displacement between successive turning points of the pendulum. Trial conditions were (1) stand still with visual feedback, (2) stand easy with visual feedback, (3) stand still with no visual feedback and (4) stand easy with no visual feedback. For each trial condition, B shows the mean, trial sway size and C shows the mean sway frequency. For both panels, values were averaged over 10 subjects for each of the four trial conditions. Error bars show 95 % confidence intervals for the mean values.

Figure 5. Effect of trial conditions on ankle impedance

A shows data averaged from 1.3 s before to 1.3 s after each positive gradient, equilibrium line crossing while the pendulum was falling. This is for one representative subject under each of four trial conditions labelled 1, 2, 3 and 4. Combined ankle torque is plotted against pendulum position with the same scaling for each graph. Trial conditions are the same as Fig. 3. Data points are at 40 ms intervals and proceed from label ‘a’ to ‘b’. The line of equilibrium (ignoring pendulum friction) is shown as a dashed line. The asterisk marks the point of equilibrium and maximum velocity. For each trial condition, B shows the mean, positive gradient, line crossing impedance (left bar, pendulum falling; right bar, pendulum rising) and C shows the mean EMG activity summed over both legs (left bar, tibialis anterior; right bar, soleus). For both panels, values were averaged over 10 subjects for each of the four trial conditions. Error bars show 95 % confidence intervals for the mean values.

Figure 6. Relationship between ankle impedance and frequency

The pendulum falling and pendulum rising, positive gradient, line crossing equilibria were separately grouped into five frequency bins of 0.10-0.29, 0.30-0.49, 0.50-0.69, 0.70-0.99 and 1.00-1.50 Hz. The frequency for each line crossing was calculated from the duration of the associated sway. For each frequency bin for each trial, the mean falling and rising line crossing impedances, and the mean frequency associated with the line crossings, were calculated. The impedances were averaged over 10 subjects for each of the trial conditions. A, the mean line crossing impedance for each of the four trial conditions plotted against mean binned frequency. (For this plot the rising and falling impedances have been combined and the points have been plotted at the mean frequencies rather than the central bin frequencies.) The stand still conditions are plotted as continuous lines and the stand easy conditions are plotted as dashed lines. The lowest dashed line is the load impedance. The load impedance was calculated using the formula Z = complex modulus of (I(jw)2 - Ktt+bjw) where I is the pendulum moment of inertia, Ktt is the load stiffness, b is the viscous drag of the pendulum, w is the angular frequency and j is the square root of -1 (Schwarzenbach & Gill, 1992). For each frequency bin and for each trial condition, B shows the mean, positive gradient, line crossing impedance (left bar, pendulum falling; right bar, pendulum rising). Two-way ANOVA, N = 40, P = 0.80, 0.20, 0.54, 0.42, 0.48 for the falling impedances in order of increasing bin frequency and P = 0.52, 0.2, 0.1, 0.51, 0.75 for the rising impedances in order of increasing bin frequency. C, the fraction of occurrences populating each bin for each trial condition. A group of four trial conditions are shown (order 1, 2, 3, 4 from left to right) with the group centred at the mean frequency for each bin. Two-way ANOVA, N = 80, P = 0.11, 0.0004, 0.24, 0.13, 0.81 for the five bins in order of increasing frequency. For all three panels, the error bars show the 95 % simultaneous confidence intervals in the mean values.

Figure 7. Effect of ankle stiffness, viscosity and torque noise on sway

These results were generated using the model described in Appendix. A-D, data averaged from 1 s before to 1 s after each falling, positive gradient, equilibrium line crossing. The same scaling is used for each graph. Data points are at 40 ms intervals and proceed from label ‘a’ to ‘b’. The line of equilibrium (ignoring pendulum friction) is shown as a dashed line. The asterisk marks the point of equilibrium and maximum velocity. A has normal values for ankle stiffness and viscosity for A/P sway taken from Winter et al. (1998) (K = 1440 N m rad−1= 25.1 N m deg−1, B = 350 N m s rad−1= 6.11 N m s deg−1). B, reduced stiffness (K = 600 N m rad−1= 10.5 N m deg−1) compared to A. C, reduced viscosity (B = 100 N m s rad−1= 1.75 N m s deg−1) compared to A. D shows four times the torque noise power as A. E, the measured ‘line crossing gradients’ for the simulated trials shown in A-D. These line crossing gradients are inevitably higher than the stiffness as the impedance includes a viscous and noise component. F, the mean sway size; G, the mean sway frequency for the same simulated trials A-D.

Figure 8. Model of inverted pendulum and ankle torque

The Simulink model used to represent the pendulum and the ankle torque is shown. I, pendulum moment of inertia; b, viscous damping; Ktt, gravitational toppling torque per unit angle of the pendulum. Values used were those for our own pendulum. I = 62.6 kg m2, b = 0.061 N m s deg−1, Ktt= 10.3 N m deg−1. K and B are the ankle stiffness and viscosity, respectively. th0 is the offset angle for the ankle stiffness. Typical values of K and B for A/P standing sway are taken from Winter et al. (1998). K = 850 +Ktt= 1440 N m rad−1= 25.1 N m deg−1; B = 350 N m s rad−1= 6.11 N m s deg−1. Band-limited white noise was used in conjunction with a first-order low-pass filter, 1/(1 +τs) (where s is the Laplace variable) to reduce the step-like nature of the noise. A noise sampling time tsample= 0.4 s, a noise power of 6.4, and a filter time constant τ= 0.5 s were used.