Vol. 37 No.
4, July/August 2000

A Technical Note

*Harvard Medical School, Department of Orthopaedics, Massachusetts
General Hospital Biomotion Laboratory, Boston, MA 02114; MGH Institute of Health
Professions, Boston, MA 02114; Harvard Medical School, Department of Otolaryngology,
Jenks Vestibular Laboratory, Boston, MA 02114; Massachusetts Eye and Ear Infirmary,
243 Charles Street, Boston, MA 02114*

**Abstract — **We describe a quantitative method
to assess repeated stair stepping stability. In both the mediolateral (ML) and
anterio-posterior (AP) directions, the trajectory of the subject's center of
mass (COM) was compared to an ideal sinusoid. The two identified sinusoids were
unique in each direction but coupled. Two dimensionless numbers--the mediolateral
instability index (*I _{ML}*) and AP instability index (

**Key words:** *ataxia, balance assessment, body
sway, stability, vestibular subject.*

Address all correspondence and requests for reprints to: Michael D. McPartland, PhD, Harvard Medical School, Department of Orthopaedics, Massachusetts General Hospital Biomotion Laboratory, Boston, MA 02114; email: michael@space.mit.edu.

Reduced stability during standing and gait, dizziness, and other symptoms are common among patients with vestibular labyrinth defects (LD; 1-4). The loss or reduced function of this important sensory mechanism can significantly reduce a patient's ability to perform functional everyday tasks such as walking and stair climbing. There are a number of tests designed to quantify ataxia as a consequence of such diseases; however, many rely on ordinal scaling or an "all-or-none" quality (5-7). These qualities make such tests less than ideal for monitoring progressive changes. We report a method that produces unbounded scalar values in quantitatively assessing a patient's ability to perform a simple task, repeated stair stepping.

To simplify some of the complexities in studying the dynamics of human locomotion, fluidal and semirigid components of the body are often mapped together as rigid segments (8,9). The aggregate behavior of these body segments is often investigated by calculating and studying their resulting center of mass (COM) kinematics and dynamics (e.g., momentum). Dynamic stability, in relation to ataxia, is thus the ability to control the position and momentum of one's COM (4).

In normal human walking and stair climbing, an individual's
COM oscillates side to side sinusoidally as the body's weight is transferred
from one foot to the other as one advances forward (10). In repeated stair stepping,
the process of ascending and descending a single step while continuously facing
the same direction, the body's COM also oscillates side to side. The COM also
moves anteriorly and posteriorly in a similarly sinusoidal fashion, albeit at
half the frequency (**Figure 1**).

**Figure 1.** *
*Sketch indicating the order of a subject's footfalls onto step and floor
and how the COM oscillates ideally during the stepping protocol.

The oscillations of an individual's COM during repeated
stepping can be thought of as having a periodic component that is related to
the frequency of stepping and a residual sway component, which does not occur
at the frequency of stepping. Averaged power spectral estimates of mediolateral
(ML) COM trajectories for groups of nonimpaired controls and LD patients immediately
reveal that there are notable differences in the spectral signatures between
the groups (**Figure 2**). It can be observed from the ML power
spectrum of LD patients in **Figure 2A** that additional power
exists at frequencies outside and principally below that of the stepping component.
It is the energy content in these nonstepping-related frequencies that are the
focus of the present study.

**Figure 2.** *
*Averaged power spectra of COM ML (A) and AP (B) displacements. Displacements
were normalized by static standing COM heights prior to calculation of individual
spectra (n

The goal of the method described is to assess the functional ambulatory stability of patients. This is considerably different from applying tests to screen individuals for labyrinth defects; however, it does not preclude the method from being incorporated into a vestibular diagnostic test battery. A typical test battery for diagnosing both unilateral and bilateral vestibular hypofunction may include such examinations as caloric testing and electronystagmography. The results of such screenings, however, cannot predict the success of performing ambulatory tasks, in part because they do not account for developed compensatory strategies.

In this method we identify an ideal displacement
trajectory of an individual's COM while executing the repeated stepping task
*a posteriori*, that is, a trajectory that accounts for sway only related
to the stepping frequency. Once the ideal trajectory is identified, it is used
to develop two dimensionless numbers that describe the dynamic instability of
the individual in his or her respective ML and anterio-posterior (AP) directions.
We hypothesize that the observed displacement trajectory of controls will more
closely match their individual ideal trajectories than do those of LD subjects.
Because a considerable difference has been observed in the power spectra between
the control and LD groups in the ML direction in pilot data, and not necessarily
so in the AP direction, we further predict that analysis of ML translations
alone will prove most effective in exemplifying instability levels between groups.

To illustrate the efficacy of this objective technique,
we compared the calculated numerical results of the trajectories studied, to
their phase plane portraits. Phase plane portraits are plots of the velocity
of a trajectory *versus* its displacement in a single degree of freedom.
These plots are often used as a subjective approach to evaluate nonlinear systems
and graphically provide information on the stability and organization of human
locomotion (2,10-15).

**Subjects**

Six subjects comprised the LD group (three males, three females).
The mean age of these subjects was 55 years with a standard deviation (SD) in
age of 24 years (**Table 1**). Three had bilateral vestibular hypofunction
(BVH) and the remaining three had unilateral vestibular hypofunction (UVH) as
diagnosed by an otoneurologist. Diagnostic tests to evaluate the subjects included
electronystagmography, visual-vestibular interaction rotation and calculation
of vestibular-ocular reflex (VOR) gains, phase, and asymmetry on a computerized
sinusoidal vertical axis rotation (SVAR) device. Reduced VOR gains, at least
three SDs below normal values, during SVAR rotations between 0.01 and 0.10 Hz,
were the principle diagnostic criterion of the BVH patients. The UVH patients
had unilateral reduced caloric responses and confirmatory SVAR abnormalities,
including asymmetries.

Nine nonimpaired subjects (three males, six females),
with a mean age of 30 years and an SD of 10 years, were incorporated in the
study as a control group. All were free from vestibular or otoneurological pathologies
by history and physical examination (**Table 1**). All LD and control
subjects were capable of performing the stepping task and ambulating without
assistance.

Table 1.Subject characteristics. No Cond Sex Age Hgt 1 H F 30 92.1 2 H F 28 92.0 3 H F 25 90.2 4 H F 52 91.7 5 H M 26 95.8 6 H F 25 91.0 7 H F 25 90.1 8 H M 39 92.7 9 H M 25 88.2 10 UVH F 73 87.2 11 BVH M 25 94.7 12 UVH F 72 91.6 13 BVH F 70 90.5 14 BVH M 25 100.0 15 UVH M 69 102.0

No=subject number; Cond=condition; Age in years; Hgt=static center of mass height, in cm; H=healthy; UVH=unilateral vestibular hypofunction; BVH=bilateral vestibular hypofunction.

**Instrumentation**

Data collection instrumentation, determination of kinematic data,
and COM position estimates are described in detail elsewhere (9,16) and are
briefly presented here. Whole body kinematic data were collected using a four-camera
motion-capturing system (Selspot II). The kinematics of eleven body segments,
incorporating the head, thorax, pelvis, upper arms, thighs, legs, and feet,
were measured by collecting three-dimensional (3-D) position data from arrays
of strobed infrared light emitting diodes (LEDs) affixed to each segment. At
least three LEDs were mounted to each rigid disk to form an array. The arrays
were firmly attached to each body segment. Thus, translations and orientations
of each body segment can also be estimated. The displacements and rotations
of each segment were sampled at 150 Hz.

Measuring anthropomorphic details of each subject allows us to estimate the relative COM location of individual segments (17,18). We can then calculate the position of the whole-body COM knowing the COMs of the individual body segments and their locations in space.

**Procedure**

The stair-stepping task involved having the subject step repeatedly
on and off a single 7.6-cm-high platform. The width of the platform was 57.6
cm with a forward depth of 23.0 cm. Four consecutive foot placements constituted
a stepping cycle (**Figure 1**). Subjects were asked to step at
a cadence of 120 beats per minute (BPM) kept by a metronome. In this protocol,
subjects step forward up onto the platform with the right foot and then with
the left. Continuing with the cadence, the subject then steps backward off the
platform with the right foot and then with the left, and repeats this pattern
until instructed to stop. Data collection was initiated after the subjects had
completed at least two stepping cycles and proceeded thereafter for 10 s. Subjects
were barefoot and allowed to swing their arms freely at their sides. No instruction
was given to maintain any visual fixation during the trial.

**Data Analysis**

We were concerned only with horizontal translations of the COM in
the plane of the floor of the testing facility. The horizontal trajectory was
separated into two orthogonal components and AP displacements of the COM were
delimited as COM_{AP} and ML displacements (frontal plane) as COM_{ML}.
To separate a subject's extraneous COM translations from their ideal translations
for a given experiment the data were analyzed as follows.

This analysis assumes that the COM trajectories
were sinusoidal (**Figure 3**) with the AP sway frequency equal
to one half the ML sway frequency.

**Figure 3.** *
*Example of COM horizontal trajectories for a control subject from a ten-second
stepping trial. Left: Orthogonal components of COM displacements along ML (upper)
and AP directions. Right: Combined COM displacements depicting actual path over
floor.

The expected frequency for ƒ* _{ML}*, 1 Hz, was the stepping frequency (120 BPM,
i.e., 2 Hz) divided by the number of steps (two) required for a single oscillation
of COM

We consider that an ideal trajectory exists for
both COM_{ML} and for COM_{AP}. Determination of ideal trajectories
for a given trial was a multistage process encompassing several optimization
steps. The first procedure was to remove the majority of the low-frequency trends
in the time-series data. This was done by using a fourth-order infinite-impulse
response high-pass filter implemented bidirectionally with a cut-off frequency
set at one half the trajectory's expected frequency of oscillation.

The goal of the second procedure was to simultaneously
identify the frequency and phase shifts ƒ* _{ML}*, F

The cost function *J* to be minimized in this
procedure was a function of the Pearson's coefficient of correlation of the
filtered trajectories (*t*) to their respective basis function (Equation 3). Because
we wanted to maximize the sum of two correlations, we defined the cost function
as the negative of the sum and employed a quasi-Newton optimization approach
to minimize Equation 4 (19).

Using the identified parameters from the minimization
of Equation 4, a simple least-squares fit was performed separately on each *x*(*t*)
to calculate a gain *A* and an offset *b* as used in Equation 5 for
each basis function to produce the ideal trajectories *X*(*t*).

*X _{ML}*(

Taking the root-mean-square (RMS) of the error vectors produces two performance values with units of centimeters. To transform these values to dimensionless numbers useful for inter-subject comparison as well as to normalize as a function of anthropomorphic characteristics, we divide the RMS values by the subject's nominal COM height in centimeters. The COM height was estimated from data collected during quasi-static standing with feet set parallel and 30 cm apart.

Though the analysis procedure may appear complex, it was easily implemented in the high-level programming language, MATLAB®.

Low numerical values for the instability indices will denote that a subject differed little from an ideal execution of the repeated stepping task. Large values will indicate that the subject differed significantly from his or her potential ideal. Thus high instability indices, which represent poor performance, also signify ataxia or a lack of uncoordinated power in movement.

To determine whether the method distinguished between
controls and LD patients, an analysis of variance (ANOVA) for unequal sample
sizes was applied to the resulting directional stability parameters *I _{ML}*
and

The identified ideal trajectory parameters, mediolateral
sway frequencies ƒ* _{ML}*), trajectory amplitudes (

The correlation values *r _{ML}* and

**Table 4** lists the instability index
parameters, *I*, for each subject. On average, *I _{ML}* and

Table 4.Instability indices of healthy subjects (Nos 1-9) and subjects with labyrinth defects (Nos 10-15). No I_{ML} I_{AP} No I_{ML} I_{AP} 1 1.21 1.58 10 1.11 1.55 2 1.12 1.10 11 1.55 1.60 3 0.87 1.47 12 2.20 4.21 4 0.98 1.43 13 4.51 2.47 5 1.27 1.77 14 4.51 3.91 6 1.28 1.71 15 6.76 6.61 7 3.18 1.61 8 1.32 1.36 9 2.41 1.90 Mean 1.51 1.55 3.44 3.39SD0.760.242.181.94

No=subject number; ML=medio-lateral; AP=antero-posterior; all values ×100; SD=standard deviation.

The LD subjects, on average, have significantly
more ML and AP sway (*p*_{ML}<0.014, *p*_{AP}<0.006)
at frequencies that are not related to the stepping frequency compared to controls
(**Table 5**). An example of an ideal trajectory as fit to COM_{ML}
for a control subject is given in **Figure 4A**. The error trajectory
for this case is shown in **Figure 4B**. In contrast, **Figures
4C** and **D** show similar plots produced for an LD subject.

Table 5.Results of ANOVA between healthy and labyrinth defect groups. Parameterp<r_{ML} 0.007r_{AP} 0.060I_{ML} 0.014I_{AP} 0.006

r=maximized correlation values for each direction;I=instability indices for each direction; ML=medio-lateral; AP=antero-posterior.

Figure 4.*
* COM

The scatter plot of all AP and ML instability indices
in **Figure 5** shows the moderate correlation (*r*=0.84,
*p*<0.001) between the two indices. Two control subjects clearly stand
out as outliers, with notably higher values of *I _{ML}*, from the
otherwise tight grouping. However, the values of

Figure 5. Scatter plot and regression line of instability indices for control and LD subjects. |
_{ML} or
I_{AP} (×100) for each trajectory are displayed as bold
numbers inside each graph. Subjects are sorted row-wise by value of I_{ML}
with the least stable subjects towards the top. Plotting scales are constant
along columns. |

We also found that the ML and AP displacement trajectories of each subject's COM and estimate of their pelvic center were highly correlated in our experiments. The average Pearson correlation between the COM and pelvic center trajectories among all subjects was greater than 0.98 in each axis.

A subject's "ideal" trajectory is dependent on his or her anthropomorphic characteristics as well as on the protocol of the assigned stepping task. Body mass distribution and segment lengths contribute to the height of the COM which in turn contributes to its 3-D trajectory. Additionally, the stepping rate and height of the platform also affect the ideal trajectory. The first by affecting the COM velocity along the path, the later by affecting the general gait pattern by defining the required range of motion for the limbs. Once the ideal trajectory was determined, the extraneous translations were readily separated and used to evaluate the performance. Scalar values were calculated, normalized, and transformed into dimensionless numbers for easy comparison with other subjects.

The ideal trajectory technique discriminates between control and pathologic groups in ML and AP whole-body COM displacements. Because control subjects have less variation in their displacement trajectories than the LD subjects in this time-paced task, we can assert that they also had superior control of their center of mass velocity. Therefore, we conclude that subjects with lower instability indices also had superior control of their whole-body momentum and kinetic energy during the task.

Due to the frequency coupling stated in Equation
1, the phase plane portraits (**Figure 6**) of COM_{ML}
show twice as many orbits as that for COM_{AP}. This can give the potentially
misleading impression that the COM_{ML} trajectories are inherently
more complex.

The first assumption of this analysis technique
was that ideal trajectories in both the ML and AP directions are sinusoidal.
Therefore, a phase plane portrait of an ideal trajectory is effectively a plot
of the derivative of a basis function in Equation 2, a cosine, *versus*
that basis function. Because these functions will have the same frequency and
phase, the portrait will be of a single ellipse with its major and minor axes
aligned with the plot axes. The less elliptical and, more importantly, the less
repeatable a measured trajectory is, the less stable the subject tends to be
(11). Instability indices *I _{ML}* and

Assuming that all individuals naturally attempt to perform a given physical activity such that they minimize their metabolic energy cost for that activity, it is reasonable to assert that a subject's ideal displacement trajectory is the minimum energy time series path to complete the stepping task (11,12,22). Deviation from the ideal displacement trajectory also means deviation from an ideal velocity trajectory, because this is a time-based task. Such deviations produced by LD subjects indicate that their absolute COM trajectories are greater than their ideal trajectories. Therefore, under the assumption that stair stepping is an energy-dissipating process (23), LD subjects are expending additional energy to complete the same task as that of control subjects.

The excessive energy use by LD subjects is further
substantiated by review of the power spectra in **Figure 2**. The
plots represent the normalized (for COM height) and averaged power spectra for
both the control and LD groups. The area between the two curves is proportional
to the relative difference in the magnitude of movement between the two groups.
Because such movement is an energy-dissipative process, this also means that
the area must be proportional to the relative difference in energy expended
by each group to complete the same task. Consider this analogous to the power
spectra of an alternating current through an ideal resistor, where the time
integral of the spectra is proportional to the total energy dissipated.

The integral of the spectra in **Figure 2A**
(5.8 dB-Hz) indicates that the LD group will dissipate notably more energy than
the control group in the ML direction. There is only a negligible difference
in area between the spectra of **Figure 2B** (-0.2 dB-Hz), indicating
that both groups will expend similar amounts of energy to ambulate in the AP
direction. We believe there is a greater difference between the ML spectra because
subjects are not physically constrained in their frontal plane and can progress
to the left and right on the step as much as necessary as long as their foot
placements stay within the step width. Some LD subjects have been observed traveling
nearly the full extent of the step width during a single trial, though all trials
are initiated in the middle of the step. Motion in the AP direction has a biomechanical
constraint, in that in order to continue stepping on and off the step, the median
location of the COM_{AP} must be near the edge of the step.

We had anticipated that COM_{ML} trajectories,
by our original inspection of their power spectra and phase plane portraits,
as well by anecdotal evidence, would be more telling about a subject's stability
and provide a better discriminator between the groups than COM_{AP}.
We found conflicting evidence relating to this prediction when comparing both
the correlation values and the instability indices. Although *I _{AP}*
has higher discriminating power, due to the small sample sizes this finding
should not be considered definitive. Moreover, for these data,

The instability indices, as expected, are superior
to the correlation values for intersubject comparison for two reasons. First,
*I* can discriminate between groups in both directions investigated. Second,
the scales of the values are tremendously different from those of *r*.
The between-group differences for the averaged values of *r*_{ML}
and *r*_{AP} were only nine and two percent, respectively. For
*I _{ML}* and

The resultants of Equations 2 and 7 are reasonably
sensitive to the values of ƒ* _{ML}*. Therefore, an iterative optimization approach
was employed to obtain this parameter and not a more common numerical approach
such as the discrete Fourier transform (DFT). The DFT proved to be too coarse
in its frequency intervals for application in this technique, given the sampling
characteristics of the data.

We examined two degrees of freedom (DOFs) of a three-DOF
trajectory. Omitting the vertical trajectory and limiting the analysis to the
two remaining DOFs allowed for easy comparison to phase plane portraits. As
stated above, a greater difference can be seen between the two groups by analysis
of COM_{AP} alone. However, *I _{AP}* only describes the
performance in a single DOF and does not necessarily describe the comprehensive
functional performance of an individual ambulating in 2-D space. A potential
way to address this would be to produce the magnitude from the two values

The number of LD subjects used in this study was limited. This shortcoming was in part due to the inherent difficulty of stepping at 120 BPM. Not all LD subjects tested were included in the analysis as several were unable to perform the task at the indicated stepping frequency of 120 BPM. Future work with this method should test control subjects and patients at slower stepping rates. A procedure that would allow the quantification and normalization of a task's relative difficulty would be useful to compare data from different subjects tested at different stepping frequencies.

The age distribution of control and LD subjects
was limited but does not affect the proof-of-concept for ideal trajectory analysis.
Our future work will include precisely age-matched samples as well as an increased
number of subjects. Future work to corroborate or contradict this study's finding
that there is no correlation of *I* with age would be useful.

High correlations between COM and pelvic center trajectories during the stepping task imply that the instrumentation and some data reduction used in our analysis can be simplified or eliminated if we elect to analyze the pelvis center instead of the COM. For instance, to analyze the pelvic center, one would not need to calculate segmental mass properties. It is expected that measurement of displacements of a single point on a subject, e.g., a tracked marker or the end of a mechanical tracker placed near the L4 or L5 vertebra, would likely be sufficient to estimate the relative displacement of the pelvic center. Saini et al. have shown that tracking a marker place on the sacrum can accurately estimate the vertical displacement of the COM (24). A more practical approach, in particular if employing passive reflective markers, may be to track two markers, one on each anterior superior iliac crest, to describe the relative displacement. Thus, a whole-body motion tracking system could be substituted by motion tracking devices that are more suitable for clinical or field use. Acoustical, mechanical boom, or electromagnetic motion tracking systems are examples of such devices.

We found that the repeated stepping task is a simple and efficient task to implement, and that it models a quasi-functional activity. The approach is sufficient to excite the dynamics of interest while requiring little testing area. The analysis technique provides useful numerical values that will be advantageous for intersession comparison of groups and individuals (3). The instability indices coincide well visually with the ordering of performance by examination of subjects' phase plane portraits.

We conclude that ideal trajectory COM analysis during repeated stair stepping is useful and practical, and should be included in balance assessment profiles. Additionally, this technique can be employed for quantitative gait assessment of astronauts to evaluate the postflight influence of neurovestibular adaptation to microgravity.

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