## Abstract

Research on unperturbed stance is largely based on a one-segment inverted pendulum model. Recently, an increasing number of studies report a contribution of other major joints to postural control. Therefore this study evaluates whether the conclusions originating from the research based on a one-segment model adequately capture postural sway during unperturbed stance. High-pass filtered kinematic data (cutoff frequency 1/30 Hz) obtained over 3 min of unperturbed stance were analyzed in different ways. Variance of joint angles was analyzed. Principal-component analysis (PCA) was performed on the variance of lower leg, upper leg, and head–arms–trunk (HAT) angles, as well as on lower leg and COM angle (the orientation of the line from ankle joint to center of mass). It was found that the variance in knee and hip joint angles did not differ from the variance found in the ankle angle. The first PCA component indicated that, generally, the upper leg and HAT segments move in the same direction as the lower leg with a somewhat larger amplitude. The first PCA component relating ankle angle variance and COM angle variance indicated that the ankle joint angle displacement gives a good estimate of the COM angle displacement. The second PCA component on the segment angles partly explains the apparent discrepancy between these findings because this component points to a countermovement of the HAT relative to the ankle joint angle. It is concluded that postural control during unperturbed stance should be analyzed in terms of a multiple inverted pendulum model.

## INTRODUCTION

Bipedal stance is an important prerequisite for human functional movement. The ease with which we maintain our vertical posture is surprising when it is considered that, due to the destabilizing effect of gravity, the open-loop–controlled musculoskeletal system is unstable (Loram and Lakie 2002; Morasso and Sanguineti 2002; van Soest et al. 2003). Despite the fact that the postural control problem is widely studied, there is still little agreement on how the CNS makes bipedal standing into an almost effortless task in our daily activities.

An important part of the research investigating human balance control in unperturbed stance is based on a one-segment inverted pendulum model (Jeka et al. 2004; Loram and Lakie 2002; Loram et al. 2005; Morasso and Sanguineti 2002; Peterka 2002; Winter et al. 1998, 2001). In such a model, the human body is represented as a rigid segment with the body center of mass (COM) positioned approximately 1 m above and slightly in front of the ankle. This model is based on the assumption that postural control is performed at the ankle and that other joints do not contribute to both postural sway and postural control. The advantage of this model is that it reduces the system to a single degree-of-freedom (df) system while focusing on the joint for which the destabilizing effect of gravity is largest. This destabilizing effect of gravity is commonly expressed as the destabilizing gravitational stiffness dM_{G}/dφ_{ankle}, where M_{G} represents the moment of the force of gravity relative to the ankle joint axis and φ_{ankle} is the ankle joint angle. In the context of this 1 df model, a prerequisite for local stability at the equilibrium is that the net joint stiffness at the ankle, which arises from both intrinsic muscle properties and neural feedback, is larger than the destabilizing gravitational stiffness (Morasso and Schieppati 1999). In recent years, the questions of how large is the net ankle joint stiffness during standing and how is it regulated have received much attention (Casadio et al. 2005; Lakie et al. 2003; Loram and Lakie 2002; Loram et al. 2005; Morasso and Sanguineti 2002; van der Kooij et al. 2005; van Soest et al. 2003; Winter et al. 2001).

It has been shown that the stiffness of the Achilles tendon during standing is lower than the destabilizing gravitational stiffness (Loram and Lakie 2002; van Soest and Rozendaal 2008). This finding led Loram and Lakie (2002) to conclude that standing cannot be stable under direct proprioceptive feedback. Holding on to a single-segment inverted pendulum, they postulated that anticipatory control is likely used (Loram et al. 2005). However, it has recently become clear that the local stiffness at the ankle joint that is required for stability is much lower than the gravitational stiffness when a multisegment model is assumed (Rozendaal and van Soest 2008). In other words, modeling assumptions may have implications for discussions regarding the control strategy underlying standing.

Research on postural responses following platform or visual perturbations has already described the contribution of hip and knee to balance control (e.g., Alexandrov et al. 2005; Bardy et al. 1999, 2002; Nashner and Mccollum 1985). Only recently, publications on unperturbed stance bring into focus the importance of rotations at joints other than the ankle (Aramaki et al. 2001; Creath et al. 2005). Aramaki et al. (2001) found the angular displacement at the hip to be significantly greater than the angular displacement at the ankle and further found that angular acceleration of the ankle was compensated for by oppositely directed acceleration of the hip joint. More recently, Creath et al. (2005) found a simultaneous coexistence of an in-phase and antiphase pattern between trunk and leg angles. Furthermore, notable movements of other major joints have been reported (Gage et al. 2004; Hsu et al. 2007) and were found to contribute to the minimization of the movement of the body COM. A complicating factor in comparing published research results and conclusions concerning the nature of postural control mechanisms is the variety of experimental conditions used in the different studies and the lack of a general definition on what constitutes perturbed and unperturbed stance (Bardy et al. 2007). Aramaki et al. (2001) restricted movements in knee and head–neck–trunk with stiff wooden splints. Creath et al. (2005) restricted visual information by using an eyes-closed condition and Gage et al. (2004) restricted visual information by a fixed-gaze condition. An evaluation of unperturbed stance, defined in this study as *a situation in which neither mechanical nor perceptual variables are intentionally perturbed by the experimenter*, is in our view warranted. In this study we investigate the kinematics at ankle, knee, and hip joints during unperturbed stance as just defined.

As recently argued by Rozendaal and van Soest (2008), nonrigidity of knee and hip joints during unperturbed stance would have important implications for the ongoing discussion on local stability of unperturbed stance. In particular, Rozendaal and van Soest (2008) showed that the net ankle joint stiffness required for local stability depends on the net joint stiffness at the hip and knee joints and on the assumed structure of neural and/or mechanical interconnections between joints. For example, it was shown that in the absence of such interconnections, the net ankle joint stiffness required for local stability increases substantially when net joint stiffness at knee and hip is reduced from infinite (representing the rigid knee and hip joints of a single-segment model) to more reasonable values (Rozendaal and van Soest 2005). This conclusion was recently confirmed (Edwards 2007). Yet it seems more likely to assume that neural and mechanical interconnections between joints do exist and contribute to postural control. For instance an angular displacement in the ankle leading away from equilibrium can lead to a “restoring” torque not only in the ankle but also in the knee joint by means of biarticular muscles or neural interconnections. Analysis of this type of control is only starting to emerge (e.g., Rozendaal and van Soest 2008).

Given the implications that nonrigidity at the knee and hip may have for the discussions on the control of unperturbed stance, a comprehensive analysis of the ankle, knee, and hip movement is warranted. This analysis should reveal whether the one-segment inverted pendulum model is indeed an oversimplification of reality. If this is indeed the case, the resulting description of the multiple-segment postural sway may provide a starting point for future model-based evaluation of stability. First of all, we will examine the amplitude of the movement in the ankle, knee, and hip after removal of low-frequency drift, and determine whether the amplitude at the ankle is substantially larger than that at knee and hip. Only if this is not the case—i.e., only if knee and hip are far from infinitely stiff—we will continue to describe the coordination between ankle, knee, and hip movement using principal-component analyses (PCAs). PCA was chosen because it can reveal how lower leg, upper leg, and trunk angle are interrelated and how ankle, knee, and hip joint movements contribute to the movement of the body COM. Using these results we will reexamine whether the body movements during unperturbed stance are adequately captured by a one-segment model.

## METHODS

### Participants

Ten healthy persons, four male and six female [mean age 32 yr (range 23–52 yr)], participated in this study. Additional information about their body characteristics can be found in Table 1. The experiment was approved by the local ethical committee and all participants signed an informed consent statement.

### Equipment

Kinematic data were obtained with two Optotrak 3020 position sensors (Northern Digital, Waterloo, Ontario, Canada), operating at 100 Hz. Ground reaction forces were acquired with a Kistler Forceplate (Kistler Instruments, Amherst, NY), sampled at 100 Hz. Synchronization of force plate and kinematics was realized by means of a SCXi 1100 module (National Instruments). Data processing was done with Matlab 6.5 (The MathWorks, Natick, MA) and statistical analysis with SPSS 11.5 (LEAD Technologies).

### Measurements

Optotrak markers were placed on the spine (two on thoracic and one on a cervical processus spinosus vertebrae) and bilaterally at the greater trochanter of the femur, the epicondylus lateralis of the femur (2 cm above the knee line), the malleolus lateralis of the fibula, and the fifth metatarsophalangeal joint. The participants were asked to stand on the force plate with the sensors positioned as shown in Fig. 1. In this setup, the Optotrak markers remained within range of the same camera during the measurements. For all trials kinematic data were obtained. In a preliminary trial, force plate data were acquired for the computation of the COM of the head–arm–trunk (HAT) segment relative to the Optotrak markers on that segment (see next section).

### Protocol

In a preliminary trial participants were asked to stand as motionless as possible for about 20 s in three consecutive postures: with the trunk 0, 20, and 40° flexed in the hips (and the arms folded in front of the chest). These trials were used to calculate the position of the HAT COM relative to the Optotrak markers on that segment.

In the main trial (unperturbed stance trial), the participants were asked to stand quietly and relaxed for 3 min with their feet side by side and slightly apart and their arms folded in front of their chest. In daily life balance control is not perceived consciously and it exists to facilitate other performances. We therefore chose to impose a suprapostural task: the participants had to respond to questions taken from the party game *Trivial Pursuit*. As was known to the participants, these questions are of a trivial nature and are not indicative of a person's knowledge or intelligence. The participants were aware of the fact that their performance on this task was not registered and that the task was primarily aimed to distract them from the process of standing during the long trials.

### Data processing

For the unperturbed stance trials the first minute was eliminated for all participants. Several participants made sudden movements with head or arms in this period as they answered the questions that were posed to them. They were asked not to do so and after the first minute, all participants stood quietly and relaxedly while they responded to the questions. For participant 9 the last 20 s were eliminated as well due to sudden knee movement.

The three Optotrak markers that were placed on the spine were averaged to one virtual marker. The kinematic data from the left and right sides of the body were averaged, yielding one set of marker coordinates in the sagittal plane. This procedure suppresses the effects of torsion at the level of knee or hip on the sagittal sway.

Segment angles of lower leg, upper leg, and HAT (respectively φ_{LL}, φ_{UL}, and φ_{HAT}) were defined as the angle between the horizontal and the line from the distal to the proximal segment marker, as shown in Fig. 2. Joint angles (φ_{ankle}, φ_{knee}, and φ_{hip}) were defined as the upper segment angle minus the lower segment angle, as a result of which a joint angle of zero represents a fully extended joint. For the upper and lower leg, the mass and the location of the COM were set according to parameters found in the literature (Winter 1979), using individual anthropometric data and total body mass. For the HAT segment, the location of the COM was calculated according to Kingma et al. (1995) using the data from the preliminary trial. As indicated in Fig. 2, the COM angle (φ_{COM}) was defined as the angle between the horizontal axis and the line connecting the ankle with the COM of the whole body.

A baseline measurement was performed to quantify the measurement error in the segment angle. In the same experimental setup as used in the main measurement, Optotrak markers were placed in pairs at 0, 0.2, and 1 m above the ground. These Optotrak marker pairs were averaged to three virtual markers as was done in the data processing of the present study. The error in the measurement for the horizontal marker displacement was 0.018 mm; the corresponding estimated error in the segment and joint angles was 0.003°. Relative to the measured movement amplitude (see Table 2) this error is acceptable (<4%).

Statistical analysis of sway data is complicated by the nonstationary character of these data (Riley et al. 1999). To remove the low-frequency drift of the equilibrium position around which the postural sway occurred, all angles were filtered with a second-order high-pass Butterworth filter with a very low cutoff frequency of 1/30 Hz. The filtering was done bidirectionally, leading to a fourth-order filter with an effective cutoff frequency of 1/24 Hz. By filtering the data the segment angles were reduced to angular displacements eliminating the mean segment angle. It should be noted that, notwithstanding high-pass filtering, the data still contain low-frequency components due to the very low cutoff frequency.

### Is the amplitude of the movement found in knee and hip negligible?

To assess the movement amplitude of ankle, knee, and hip angles the SD of the high-pass filtered joint angles was calculated (SD_{ankle}, SD_{knee}, SD_{hip}). A one-way ANOVA was used to compare these SDs. In addition, the SD of φ_{COM} (SD_{COM}) was calculated and compared with SD_{ankle} using a paired-samples *t*-test.

### What is the contribution of the knee and hip rotations to the COM angle variation?

The contribution of the knee and hip joint movement to the variance found in φ_{COM} was investigated by analyzing the (co)variances found in the high-pass filtered (cutoff 1/30 Hz) segment angles (φ_{LL}, φ_{UL}, and φ_{HAT}). This was done using a principal component analysis (PCA) (Daffertshofer et al. 2004; Sokal and Rohlf 1969). The PCA decomposes the variances in linear relations or patterns between φ_{LL}, φ_{UL}, and φ_{HAT}. The first pattern is determined as the linear relation between the segment angles that explains the largest possible fraction of the total variances. This linear relation can be plotted as a first principal axis in the three-dimensional (3D) scatter diagram of the segment angles (see Fig. 3). Likewise, for the remaining variance a second linear relation is determined as well as a third (second and third principal axes not shown in Fig. 3). In a PCA-based decomposition, all principal axes are by definition orthogonal.

To clarify the contribution of the knee and hip joint rotations to the COM angle variation, the linear relationship between φ_{LL}–φ_{UL}, φ_{UL}–φ_{HAT}, and φ_{LL}–φ_{HAT} was analyzed by projecting the principal axis on the plane formed by φ_{LL} and φ_{UL}, the plane formed by φ_{UL} and φ_{HAT}, and the plane formed by φ_{LL} and φ_{HAT} (see Fig. 3). When for instance the slope of the projected first principal axis on the φ_{LL} and φ_{UL} plane (s_{UL,LL}) equals 1, this implies that in the first component, any change in φ_{LL} is accompanied by an equal change in φ_{UL}, which means that the knee joint angle is constant. Similarly, an s_{UL,LL} >1 would indicate that in the first component, any change in φ_{LL} is accompanied by a larger change in φ_{UL}; this would imply that the change in knee joint angle amplifies the contribution of the change in lower leg angle to the change in COM angle.

In this PCA of the segment angles, PCA^{seg}, the segment angle covariance matrix was (1) The eigenvectors and eigenvalues of this covariance matrix were determined. The fraction of the variance that is accounted for by the *i*th principal axis was calculated as the corresponding eigenvalue divided by the sum of all eigenvalues (2)

The slope of the projected principal axis on the planes formed by φ_{LL} and φ_{UL}, by φ_{UL} and φ_{HAT}, and by φ_{LL} and φ_{HAT} was calculated by dividing the two associated coefficients in the eigenvector. For each PCA component this leads to (3)

A second (two-dimensional) PCA was carried out on the relation between the high-pass filtered φ_{LL} and φ_{COM} (PCA^{COM}). This PCA starts from the covariance matrix (4) In both PCAs, the component corresponding to the smallest eigenvalue accounted for <12% of the total variance; therefore these components will not be discussed.

Circular statistics were applied to calculate the mean direction angle of the projected principal axes and their SDs over the 10 participants (Batschelet 1981). For this purpose the slopes (s_{UL,LL}, s_{HAT,UL}, and s_{HAT,LL}) were transformed to direction angles in degrees. It was tested whether the direction angles were uniformly distributed using a Rao's spacing test for uniformity as described in Batchelet (1981). For this test α was set on 0.05, which is comparable to a Rao's spacing test parameter (*L*) exceeding 172.1 to reject randomness or uniform distribution. In cases where the distribution was nonuniform, the 95% confidence region of the mean direction angle was calculated using the bootstrap method described by Fisher (1993), based on 1,000 bootstraps.

Using the PCA components the segment angle displacements can be decomposed into a contribution of the first, second, and third PCA components. Every measured sample *n* can be thought of as a position vector in the 3D space formed by the three orthogonal axes φ_{LL}, φ_{UL}, and φ_{HAT}. This same position vector can be built from any basis for this 3D space (i.e., of three independent vectors in this space). The three eigenvectors constitute such an independent set. This means that every measured sample *n* can be decomposed in terms of eigenvectors as (5) where *p*_{1}, *p*_{2}, and *p*_{3} are the new set of coordinates in the coordinate system formed by the eigenvectors and the three terms on the right-hand side of the equation represent the contribution to the segment angle displacements of the first, second, and third PCA components, respectively. For the first and the second components of the PCA, the segment angle displacements in time were reconstructed this way for the 10 individual participants.

To compare our results with other studies in which conclusions are based on analyses performed in the frequency domain power spectral density (PSD) plots were calculated for the first two components. This was done in Matlab with function “PSD” and a Welch averaging method with a 20-s Hamming window. To allow comparison with Creath et al. 2005 the percentage of power for frequencies <1 Hz was calculated as the surface below the PSD curve and normalized to the total surface below the PSD curve.

## RESULTS

### Is the amplitude of the movement found in knee and hip negligible?

An example of high-pass filtered (1/30 Hz) joint angles as a function of time is given in Fig. 4. Note that due to the very low cutoff frequency the signal still contains low-frequency components.

The SDs of φ_{ankle}, φ_{knee}, and φ_{hip} and φ_{COM} are presented for each of the 10 participants in Table 2. A one-way ANOVA indicated that the joint angle SDs were not significantly different (*F* = 0.214, *P* = 0.809). In fact, the difference between movement amplitude at the ankle and that at the knee/hip was close to zero. The mean value of SD_{knee} − SD_{ankle} was −0.019° (95% confidence interval: −0.127 to 0.089°); the mean value of SD_{hip} − SD_{ankle} was +0.004° (95% confidence interval −0.201 to 0.210°). Thus sway amplitude at the ankle was not clearly larger than sway amplitude at either the knee or the hip.

Regarding COM movement, a paired-samples *t*-test revealed that the SD of the φ_{COM} did not differ significantly from the SD of φ_{ankle} (*t* = 0.271, *P* = 0.793); thus COM angle displacement was not significantly larger (or smaller) than ankle angle displacement. At first sight, this finding seems to be in conflict with the finding that the movement in both knee and hip was not negligible. Therefore the SD of the φ_{COM} that is to be expected on the basis of the SD in ankle, knee, and hip joint angles, was derived (SD_{COM}^{exp}). This derivation rests on the mechanical relation between joint motions and motion of the COM (see the appendix, *Model II* for a description of the underlying method). In this model, it was assumed that joint angles were uncorrelated. The last column of Table 2 presents the SD_{COM} to be expected based on the measured SD of the joint angles (SD_{COM}^{exp}). The difference between this SD_{COM}^{exp} and the actually measured SD_{COM} was not statistically significant (*t* = 1.141, *P* = 0.284). Thus the relatively small mean SD_{COM} compared with the mean SD of the joint angles can be explained by the small contribution of φ_{knee} and φ_{hip} to the φ_{COM} due to the fact that hip and knee joints are located at a smaller distance from the COM.

### What is the contribution of the knee and hip rotations to the COM angle variation?

An example of scatterplots of the three high-pass filtered segment angles is provided in Fig. 5 as well as an example of a φ_{ankle} − φ_{COM} scatterplot. In Table 3, for each participant the Pearson's correlation coefficients are reported with regard to each pair of segment angles (CORR^{seg}) and φ_{LL} and φ_{COM} (CORR^{COM}). Intersubject variability in the correlation coefficients between segment angles is substantial. The correlation between ankle angle and COM angle is quite high and quite consistent over participants: *r* = 0.90 (range 0.79–0.98).

In Table 4, the results of the PCA^{seg} analysis are summarized. The largest part of the variance in segment angles is captured by the first principal component: the variance fraction captured by the first component is 0.78 (range 0.62–0.92) (left part of Table 4). A clear pattern over participants is evident in this first component: φ_{UL} and φ_{HAT} show an enlargement compared with the angular displacement of the segment beneath. This is indicated by the slopes of the first component >1. These results are visualized in Fig. 6, *A* and *C*. An augmentation of φ_{HAT} compared with φ_{LL} is evident over 9 of the 10 participants.

The first component of the PCA^{COM} accounts for 0.95% (range 0.92–0.99) of the total variance (Table 5). The mean orientation of the first principal axis on the φ_{LL} − φ_{COM} is not significantly different from 45°, indicating that for this component, the COM angle variance was about as large as the lower leg angle variance. The latter result is in agreement with the reported SDs in Table 2 but surprising, given the results of the first component of the PCA^{seg}. In this component both knee and hip show a clear amplification of the sway of the lower leg. To relate the *s*_{COM,LL} to the slopes found in the PCA^{seg} we calculated the slope between COM and lower leg that should be expected (*s*_{COM,LL}^{exp}) on the basis of *s*_{UL,LL} and *s*_{HAT,LL} found in the first component if this component accounted for all the variance found in the segment angles. (See *Eq.* A*6* in the appendix.) This *s*_{COM,LL}^{exp} is reported in Table 5 (right column). A paired-samples *t*-test revealed that the observed *s*_{COM,LL} was significantly smaller than *s*_{COM,LL}^{exp} (*t* = −4.883, *P* = 0.001). This indicates that the variance not accounted for by the first component of the PCA^{seg} must be structured in such a way that it counteracts the amplification of segment angle displacement observed in the first component of PCA^{seg}.

Analysis of the second principal component, accounting for a fraction of 0.18 (range 0.07–0.31) of the variance (see Table 4), is indeed in line with this indication. In this component, the only direction angles which are clustered around the mean value is the *s*_{HAT,LL}. The corresponding slopes are <1 for all participants (see Fig. 6) and the average direction angle corresponds to a slope of −0.8. This indicates that in this component, a countermovement of φ_{HAT} relative to φ_{LL} is present. This countermovement is shown in Fig. 7 (*right column*) where the variance in φ_{LL}, φ_{UL}, and φ_{HAT}, based on the second principal component, is plotted over time. Thus this component indeed provides a reduction of the angular deviation in φ_{COM} relative to the angular deviation in φ_{LL}.

## DISCUSSION

### Main findings

Movements at the knee and hip joints in the high-pass filtered data were not found to be different in amplitude compared with the ankle joint movement; the hip and knee do not behave as joints that are infinitely stiff. In the PCA on the segment angles two patterns became evident. The first PCA component revealed that displacements in upper leg and HAT angle were larger than those in the distal adjoining segment. This might lead one to expect that the COM angle displacements are also larger than the displacements in the ankle joint angle. This is not the case; the first component of the PCA relating ankle joint angle to COM angle revealed that displacements in these angles are of similar magnitude for all participants. An explanation for this apparent inconsistency can be found in the second principal component of the PCA on the segment angles, in which the HAT segment consistently showed a countermovement relative to the lower leg.

### Limitations

The angular displacements and consequently the marker displacements that occur during unperturbed stance are very small. This raises the question as to whether these displacements can be measured reliably. For this reason, a baseline measurement was done to estimate the measurement error of the segment angles. This measurement was described in methods. It was found that this error was <4% compared with the measured movement amplitude as shown in Table 2. Furthermore, the effects on the PCA results would not have changed the main outcomes. Segment angles were calculated using one single marker for each combining joint (knee and hip). When adjoining segments share a marker, inaccurate measurement of the horizontal marker displacement leads to a misrepresentation in the appearance of the upper segment rotating in the opposite direction from the lower adjoining segment. This misrepresentation can only have reduced the pattern found in the first PCA^{seg} component where angular displacements in the upper segments were larger than those in the distal adjoining segment. The second PCA^{seg} component did show a countermovement, but the main result concerned the HAT angle compared with the lower leg angle. These are not adjoining segments and, consequently, this finding cannot be due to marker placement.

The present study reveals compensatory movement in both knee and hip with a high intersubject variability. All participants have undergone one trial of unperturbed stance. Whether a participant has a preference to use either knee or hip or changed between them cannot be determined and should be addressed in future studies.

Additionally it should be noted that in the analyses of unperturbed stance carried out in this study, the very low frequency components of the behavior were disregarded by considering only high-pass filtered data (cutoff frequency 1/30 Hz).

### Movement amplitude of the ankle, knee, and hip

In unperturbed stance, the body COM shows a low-frequency drift making the signal nonstationary (Riley et al. 1999). When measuring for long periods (>1 min) this can complicate the data analysis. Gage et al. (2004) broke their 2-min trial in 30-s blocks to reduce this drift. In the present study, a high-pass filter with a cutoff at 1/30 Hz was used. This filtering process with very low cutoff frequency successfully removed the low-frequency drift while still leaving in a large part of the low-frequency components, as can be seen in Fig. 4. Clearly, this filtering process decreased the amplitude of the angular displacement in the ankle, knee, and hip joints (see Table 6). The SDs in the joint angles reported in this study are lower than those reported by Gage et al. (2004). As shown in Table 6, this difference is largely due to a difference in filtering. Application of only a low-pass filter (cutoff 1.5 Hz) to our data, as used by Gage et al. (2004), resulted in joint angle SDs that were comparable to those reported by these authors. The results found by Hsu et al. (2007) show higher SDs for ankle, knee, and hip displacements than both Gage et al. (2004) and the low-pass filter data of the present study. We suspect that the high variability measures reported by Hsu et al. (2007) are the result of calculating variability measures over a long time series for a nonstationary signal; if this suspicion is correct, then this illustrates the complications that arise due to nonstationarity caused by setpoint drift.

Another factor that may have influenced the movement in ankle, knee, and hip is the suprapostural task used in our experiment. We imposed a suprapostural task because in daily life it is uncommon that attention is focused on the process of standing. During the experiment, participants were asked to answer questions from the party game *Trivial Pursuit*. It is well known that suprapostural task characteristics affect postural control (e.g., Dault et al. 2001; Ramenzoni et al. 2007; Yardley et al. 1999). Dault et al. (2001) imposed a working-memory task and observed an increase in the mean power frequency and a decrease of the variability of the center of pressure and interpreted these findings as a tighter control of posture. In contrast, Yardley et al. (1999) found no effect of attentional load on sway path (i.e., the total length of the center of pressure trajectory). More recently, Ramenzoni et al. (2007) argued that the effects of suprapostural tasks are specific to the type of working-memory task imposed. It may be added to this that the mere requirement to speak has also been reported to affect postural control (Yardley 1999). These authors found an increase in sway path as a result of the requirement to speak as part of the suprapostural task. In the suprapostural task imposed in this study, the participant's performance was irrelevant, and this was known to the participants. The questions were of a trivial nature and participants indicated that they felt amused rather than under pressure. We expect that the cognitive demand of the suprapostural task imposed in our experiment was lower than that in Dault et al. (2001). However, participants did have to speak every now and then and this may have had an effect on the amplitude of their postural sway. It is an open question whether the necessity to speak also affects the coordination between hip, knee, and ankle motion during standing as observed in this study. Addressing this question is beyond the scope of this study.

### Results of the PCA^{seg} compared with previously reported analyses

The PCA^{seg} performed on the high-pass filtered segment angles of lower leg, upper leg, and HAT revealed two different patterns (see Fig. 7). The first PCA component revealed an augmentation of the segmental sway in knee and hip joints, leading to larger changes in upper leg and HAT angle than those in the distal adjoining segment. The second principal component of the PCA^{seg} showed a pattern in which the HAT segment rotates opposite compared with the lower leg angle (see Table 4). The origin of this countermovement, however, was not consistent over the participants; for some participants it originated at the knee joint, for others at the hip joint, and for one subject at both joints (see Fig. 6). Compensatory movement in knee and hip during unperturbed stance has been reported in recent literature (Aramaki et al. 2001; Creath et al. 2005; Gage et al. 2004). Aramaki et al. (2001) found reciprocal angular acceleration at the ankle and hip joints and suggested that ankle and hip rotations serve to minimize acceleration of the COM. Gage et al. (2004) reported the occurrence of compensatory knee joint angular displacements because they found that lower limb angular displacements correlated more closely to COM displacements than lower leg angular displacement.

When interpreting the components found in the PCA and comparing them to other publications it should be kept in mind that PCA results are not invariant under coordinate transformations, even in cases where different coordinate systems are linearly related. In other words, PCA results in terms of a description in joint angles may result in components that are quite different from those obtained in terms of a description in segment angles. Thus the choice of coordinates is decisive for the results of any PCA. In our view it is surprising that this issue is rarely discussed in studies using PCA. In the present study we found considered segment angles to be the most informative coordinates since they result in a tightly clustered variance in the scatterplot of lower leg, upper leg, and HAT angles, leading to a high proportion of variance that can be explained by the first component of the PCA.

Creath et al. (2005) performed a spectral analysis based on a two-segment model with an ankle and a hip joint. They reported an in-phase pattern of leg and trunk angle for frequencies <1 Hz and an antiphase pattern for frequencies >1 Hz. Cross-spectral density analysis of our segment angle data showed an in-phase pattern at frequencies <1 Hz and an antiphase pattern at frequencies >1 Hz, thus confirming the findings reported by Creath et al. (2005). The first and second components of the PCA^{seg} carried out in this study further confirm the presence of, respectively, an in-phase and an antiphase pattern between lower leg and trunk. To establish whether our PCA components differed in frequency content, a spectral analysis was performed as described in methods and the power ratio between frequencies below and above 1 Hz was calculated. There was a small but significant difference in power ratio between the in-phase and the antiphase components of the PCA^{seg}: the antiphase component contained 2% more power in the frequencies >1 Hz. Thus the PCA^{seg} revealed both the in-phase and antiphase patterns as well as the difference in frequency band as reported by Creath et al. (2005).

Together, the two components of the PCA^{seg} lead to an amplitude and direction of the COM angle displacement strongly resembling the displacement in lower leg angle. Thus although COM angle displacements can be accurately described using ankle angle displacements, the amplitude and direction of the displacements in upper leg and trunk angle do contribute to the COM angle displacement as well. Hsu et al. (2007), using the uncontrolled manifold concept, found that movements in all major joints along the longitudinal axis of the body were coordinated such that they resulted in a minimal variance in the position of the COM. Their analysis revealed that a major part of the joint angle variance resulted in a minimal displacement of the COM, whereas a small part of the joint angle variance led to the major part of the COM variance measured. Although the PCA in the present study was done on segment angles instead of joint angles, it is of interest to compare a task-related analysis (uncontrolled manifold [UCM]) and a descriptive analysis (PCA). In the PCA^{seg} a major part of the segment angle variance was explained by the first component (78%; range 62–92%; Table 4). This component showed an intersegmental pattern that resulted in a relatively high variance of the COM angle. An additional calculation showed this was 90.3% (range 67–99%) of the total COM angle variance measured. The second component of the PCA^{seg}, explaining a minor part of the segment angle variance (18%; range 7–31%; Table 4) showed an intersegmental pattern that resulted in small COM angle variance: 9.2% (range 0–33%) of the COM variance measured. Only a small part of the segmental sway seems to be coordinated in such a way that it reduces COM angle displacement. This finding is hard to reconcile with the results of Hsu et al. (2007) based on their UCM decomposition of joint angle variance. Therefore a UCM decomposition was performed on the joint angle data of the present study. A ratio of *V*_{ucm} to *V*_{orth} of 8.6 (SD 4.6) was found for our three-segment model. This value fits well between the ratios found by Hsu et al. (2007) for the two- and six-segment models they used. From this we conclude that the data in both studies are comparable. The divergence in the results is caused by the difference in coordinates used (segment angles vs. joint angles) and/or in the difference in character of the two methods of analysis used.

### Implications for studies on postural control

The results of this study indicate that, although COM angle displacements can accurately be described by ankle angle displacements, a one-segment inverted pendulum model cannot give a comprehensive description of postural sway data. The knee and hip joint rotations can be decomposed into both an amplifying and reducing pattern with regard to the position of the body COM. This finding—in combination with recent claims that requirements on joint stiffness depend on the number of degrees of freedom considered (Rozendaal and van Soest 2008)—leads us to conclude that unperturbed bipedal standing should be analyzed using the framework of multiple inverted pendulums. Conclusions concerning the nature of postural control based on research using a one-segment inverted pendulum model should be reevaluated.

## APPENDIX

Herein we determine how the displacements in center-of-mass (COM) angle (φ_{COM}) are related to the displacement in segment or joint angle to help interpret experimentally observed relations between these variables.

### Model I

The COM coordinates and φ_{COM} depend in a fairly complex manner on the constant length and mass of the body segments and on the variable segment angles (A1) where *m* is the segment mass, *d* is the distance from the lower end of the segment to the segment COM, and *l* is the segment length. However, the segment angle displacements as found in the sway of unperturbed stance are small. The Δφ_{COM} displacement relative to the nominal value of φ̄_{COM} can therefore be calculated from the segment angle displacements (Δφ_{seg}) relative to a nominal posture defined by φ̄_{seg} using a linear approximation (A2) The coefficients *c*_{1}, *c*_{2}, and *c*_{3} can be obtained by linearizing *Eq.* A*1* around a mean posture defined by φ̄_{LL}, φ̄φ_{UL}, and φ̄_{HAT}. Taking into consideration that a change in *y*_{COM} will not lead to a significant change in φ_{COM} for the small angular displacements around a vertical COM angle, the partial derivatives can be described as (A3) Since *x _{com}* ≪

*y*,

_{com}*Eq.*A

*3*can be further simplified (A4) The mean values and SDs found for

*c*

_{1},

*c*

_{2}, and

*c*

_{3}are 0.465 (SD 0.008), 0.357 (SD 0.024), and 0.178 (SD 0.027), respectively. With substitution of

*Eq.*A

*1*for

*y*

_{COM}in

*Eq.*A

*4*it can be proven that

*c*

_{1}+

*c*

_{2}+

*c*

_{3}= 1 because they should be in any COM calculation.

With *Eq.* A*2* and the slopes found in the PCA of the segment angles (PCA^{seg}), a prediction can be made concerning the relation between φ_{COM} and φ_{LL} (A5) Thus if the first component of the PCA^{seg} would account for all the variance found in the segment angles, the slope of principal axis found in the PCA of φ_{COM} and φ_{LL} is expected to be equal to (A6)

### Model II

Analogous to Model I, the displacement in φ_{COM} can be expressed in terms of the displacement in joint angles (again using a linear approximation) (A7) *Equation* A*7* can be used to express the variance found in φ_{COM} in terms of the variance found in φ_{ankle}, φ_{knee}, and φ_{hip} (A8) If it is assumed that the movements in ankle, knee, and hip are independent, the covariances in *Eq.* A*8* are expected to be zero. Then the variance in the φ_{COM} is expected to be equal to (A9)

## GRANTS

This work was supported by Netherlands Organization for Scientific Research NWO Grant 575-23-014 to L. A. Rozendaal.

## Footnotes

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- Copyright © 2008 by the American Physiological Society