Ten healthy people, of whom six were male, aged between 18 and 45 years took part in this study. The subjects gave written informed consent, and the study was approved by the local human ethics committee and conformed to the principles of the Declaration of Helsinki.
Subjects were strapped round the pelvis to a vertical support that effectively eliminated their actual sway (Fig. 1
). The subject stood on two footplates with his/her ankles positioned to be co-axial with the axis of rotation of an inverted pendulum. The footplates were exactly horizontal when the backward lean of the pendulum was 3 deg, thus approximating typical forward lean in standing. The pendulum had a mass of 61.65 kg with a centre supported 0.937 m from the axis of rotation (distance h
). The subject balanced the inverted pendulum, which was free to move forwards and backwards, in a parasagittal plane while always tending to topple backwards. The same mass and distance h
were used for all subjects. The constant static and dynamic properties of the pendulum presented each subject with an identical task. This allowed results from all subjects to be pooled. The toppling torque of the pendulum was measured to be 10.2 ± 0.4 N m deg−1
). Using a spring of known stiffness and by recording the damped oscillations of the pendulum, the moment of inertia was determined to be 62.6 ± 2 kg m2
(which included the contribution of the rod and other rotating parts), the viscous damping was 0.061 ± 0.02 N m s deg−1
and friction was 0.045 ± 0.1 N m (means ±s.d.
The relative angular position of the pendulum was measured using a Hall effect precision potentiometer (with an effective range of 15 deg) (CP 2UTX, Midori Precision Company Ltd, Japan) and fixed gain amplifier. Absolute angular position was measured using an electronic inclinometer (Cline R1, Cline Labs Inc., USA), of resolution 0.001 deg, attached to the base of the pendulum. The angular velocity of the pendulum was measured using a piezoelectric vibrating gyroscope (range ±90 deg s−1) (ENV-05 A+C, Murata Co. Ltd, Japan) in conjunction with an instrumentation amplifier. The subject exerted torque on the pendulum via each footplate. The left and right torque signals were recorded using horizontally mounted miniature load cells (Sensotec model 31, Sensotec Inc., USA) followed by a two-channel bridge amplifier and low-pass filter (Sensotec UBP). The load cells were mounted in compression in the horizontal plane. One end of the transducer was rigidly bolted and the other made contact with a polished surface. This method of mounting effectively decoupled the load cells from off-axis loads and prevented the slight deflection of the structure caused by the subject's weight from producing a signal which would be falsely registered as a torque (Kelly, 1998). Electromyographic (EMG) activity from the right and left tibialis anterior and soleus muscles was recorded using home-constructed bipolar surface electrodes with encapsulated preamplifiers (Johnson et al. 1977). These signals containing the entire bandwidth were then amplified and passed through an analogue full-wave rectifier and r.m.s. averaging filter with a time constant of 100 ms. Data from all sensors were recorded by computer, and sampled at 25 Hz via an analog-to-digital converter (CED 1401, Cambridge Electronic Design, UK). The resolution of the recorded data was limited by input noise levels of less than 0.002 deg, 0.02 deg s−1, 0.03 N m, 60 μV for relative angular position, angular velocity, right or left torque and surface EMG, respectively.
The inverted pendulum apparatus has been designed to study the effect of limiting the kinds of sensory inputs influencing ankle mechanisms used to control upright balance (Fitzpatrick et al. 1992b
). Since the subjects themselves were prevented from swaying, vestibular feedback was not available to them. The pendulum mass and rod were screened from view, though an oscilloscope was available providing the option of visual feedback regarding the position of the pendulum. The oscilloscope was 1 m away from the subject and had a gain of 1 cm deflection per degree change in angular position. When visual feedback was not used, the oscilloscope was turned off though subjects still had their eyes open. Proprioceptive information from the legs was available to all subjects. Tactile information from the areas of the trunk in contact with the support was also available, but is likely to be inconsequential (Fitzpatrick et al. 1992b
In four separate trials, subjects were asked to balance the inverted pendulum under differing instructions and visual conditions. The four trial conditions were (1) stand still using visual feedback, (2) stand easy using visual feedback, (3) stand still with no visual feedback and (4) stand easy with no visual feedback. The order in which the four conditions were carried out was randomised. The duration of each trial was 200 s. In all cases the subjects were asked to keep the pendulum between 0.5 and 5.5 deg from the vertical so as to approximate standing sway.
It was explained that ‘stand still’ meant to reduce the sway of the pendulum to an absolute minimum and to keep the pendulum at the same angle. Subjects were told that ‘stand easy’ meant to balance the pendulum while giving the least possible attention to the sway of the pendulum. When subjects were ‘standing still’ they were encouraged to give their full attention to the oscilloscope when that was turned on and to give full attention to what they could register through their legs when the oscilloscope trace was blanked. When subjects were ‘standing easy’ they were engaged in meaningful conversation to take their mind off the task as much as possible.
All subjects were given a preliminary experience of balancing the pendulum at different angles ranging from 1 to 5 deg using visual feedback from the oscilloscope. They also practised balancing the pendulum without the use of visual feedback. The subjects then reported the angle at which they preferred to balance the pendulum. This was around 3-4 deg for all subjects. For each trial, recording started with the pendulum at the preferred angle of the subject.
Principles and methods of data analysis
During balancing, the pendulum sways to and fro in a quasi-regular fashion. We identified the times at which the pendulum reversed direction by interpolating between the data points when the velocity changes sign. The unidirectional movement between one turning point and the next was categorised as a sway. For any trial, the mean sway size was the average magnitude of the sways. Mean sway frequency was calculated as the total number of sways (positive and negative) divided by 2 and divided by the trial duration.
Gravity exerts a torque on the inverted pendulum given by Tg=KttsinΘ≈KttΘ, where Ktt is the gravitational toppling torque per unit angle. At any angle Θ from the vertical, this formula defines the ankle torque that is required to balance exactly the pendulum. (Dynamic torques due to frictional and viscous damping of the pendulum are very small.) On a plot of torque vs. angle this formula defines a line of unstable equilibrium (which has also been called the load stiffness of the pendulum; Fitzpatrick et al. 1992b). To keep the sway size between certain limits, the ankle torque must be repeatedly alternated above and below this line. Line crossings represent repeated events around which data can be averaged and from which ankle impedance at equilibrium can be measured.
Using Savitzky-Golay filters, the position data were double differentiated to produce a record of acceleration (Press et al. 1999). From Newton's second law of motion the angular acceleration is zero at equilibrium. Equilibrium points represent moments at which the subject perfectly balances the static and dynamic torques exerted on them via the pendulum. We identified those equilibrium points when zero acceleration was crossed by interpolating between the data points when the acceleration changes sign.
In each unidirectional sway a spring-like equilibrium occurs at least once. This is represented by a positive gradient of torque vs. angle crossing the line of equilibrium. These equilibria were identified by an acceleration changing from positive to negative while the pendulum was falling, or negative to positive while the pendulum was rising. The data surrounding these equilibrium points were averaged to show the mean responses. The impedance at these averaged equilibrium points was calculated as the regression value for torque/angle encompassing one data point (40 ms) either side of the equilibrium. This method of identifying equilibrium points, sampling around these points and averaging is illustrated in Fig. 2.
Illustration of the line crossing averaging process
This line crossing impedance is a biased measure in that the ankle impedance of high frequency sways will be greater than that for low frequency sways. So, to investigate the effect of frequency, we have grouped the line crossing equilibria into frequency bins and then separately averaged the line crossings for each frequency category. The appropriate frequency bin for each line crossing was determined from the duration between the nearest reversal points surrounding the line crossing, using the formula: frequency = 1/(2 × inter-reversal duration).
The position and velocity records possess small, relatively high frequency variations of the same magnitude and frequency as the noise that is recorded when a subject is not standing on the footplate to balance the pendulum. Given the large inertia of the pendulum, these variations are taken to be a noise product of the measuring and recording process and were eliminated by smoothing as part of the differentiation process. A Savitzky-Golay filter algorithm was used, which assumes that the noise is normally distributed and independent of the slowly changing variable and that a moving polynomial can be fitted to the data (Gander & Hrebicek, 1997; Press et al. 1999). This algorithm was effectively used as a low-pass filter with a bandwidth of 3 Hz and zero phase shift. Data from the velocity and position sensors were cross-checked to corroborate the differentiation and smoothing process.
We wanted to know the effect of ankle stiffness, viscosity and noise on our line crossing measure of ankle impedance, on sway size and on sway frequency. A second-order model of the inverted pendulum was constructed for simulation purposes as described in Appendix
. Torque generated at the ankles was modelled as having a stiffness component, a viscous component and a noise component as described by Winter et al. (1998)
. Results generated from this model were subjected to the same analysis procedures used for real data.