Platform-Based Methods for Posturographic Analysis

Various methods for estimating CM kinematics, based principally on force platform measurements, have been applied to posturographic analyses. Some of these rely on inverted pendulum models (e.g. Barbier et al., 2003; Karlsson and Lanshammar, 1997; King and Zatsiorsky, 1997; Morasso et al., 1999) that are

this approach is also applicable to gait analysis (e.g. Benda et al., 1994; Caron et al., 1997).

Lenzi et al. (2003) compared the method of Caron et al. (1997) with the IA methods of Shimba (1984) and Zatsiorsky and King (1998), which are described in more detail on pages 14 to 24. The latter two IA methods were rated more favourably than the low-pass filter method, in terms of CM trajectory estimation performance, when these methods were compared with a computer-simulated segmental kinematic (SK) model and in terms of sensitivity to BSP errors. Lafond et al. (2004) compared both the low-pass filter method of Caron et al. (1997) and an IA method developed by Zatsiorsky and Duarte (2000) with an SK determination of CM kinematics. The Zatsiorsky and Duarte (2000) method is described in more detail on pages 17-24. Lafond et al. (2004) found that the low-pass filter method produced significantly different antero-posterior CM trajectories to the SK and Zatsiorsky and Duarte (2000) methods for several posturographic activities (viz. quiet standing, one-legged stance, voluntary oscillation about the ankles, and voluntary oscillation about the hips and ankles), whereas, there was no significant difference between the latter two methods. They also provided graphical evidence that the low-pass filter method produced unrealistic results for quiet stance and one-legged stance, insofar as the CM trajectory was not always confined within the dynamic range of the COP trajectory, as expected during quiet standing (Winter et al., 1996a). For example, Fig. 1 is a reproduction of figure 2 from Lafond et al. (2004), showing plots of CM trajectory for the SK, ZPZP and low-pass filter methods, relative to COP

trajectory for quiet stance. The graph indicates that the method of Caron et al. (1997) produced results that were clearly unrealistic.

Figure 1. The quiet stance plots from figure 2 of Lafond et al. (2004), reprinted with permission of Elsevier, showing the good agreement between the CM

trajectory plots derived from the SK approach (labelled COM) and the ZPZP

method of Zatsiorsky and Duarte (2000) (labelled GLP), and less agreement with the CM trajectory plot derived from the low-pass filter method of Caron et al.

(1997) (labelled LPF). More significantly, the relationship between COP

trajectory (labelled COP) and LPF is clearly unrealistic (e.g. for the first three seconds, LPF CM trajectory changes direction several times while COP remains on one side of LPF).

Caron et al. (1997) reported that they did not apply their method to quiet stance, “because of the ‘noise’ of the instrumentation” affecting their determination of the

likely from the results of Lafond et al. (2004) that such a demonstration may have been difficult for Caron et al. (1997) to achieve, even if they had tried. Caron (2005) questioned how Lafond et al. (2004) had implemented the low-pass filter method, and claimed the method of Caron et al. (1997) was “more precise” than the ZPZP method. In reply, Prince et al. (2005) confirmed they had followed the procedure of Caron et al. (1997) and also highlighted the shortcomings of the claims by Caron et al. (1997) and Caron (2005) regarding the suitability and accuracy, respectively, of their low-pass filter method for quiet stance.

The validity of low-pass filtering approaches was also questioned by Zatsiorsky and King (1998) on the fundamental grounds that such approaches do not account for phase differences between transverse plane CM and COP trajectories and that results are dependent on the choice of low-pass filter cut-off frequency. Benda et al. (1994), who themselves used a low-pass filter method, mentioned the out-of-phase nature of CM and COP trajectories but only urged for cautious interpretation of the results when low-pass filtering methods are applied to activities that are more dynamic than quiet stance. Considering the phase differences between CM and COP trajectories and the results published by Lafond et al. (2004), low-pass filtering of COP data appears to be an invalid approach for estimating CM trajectory for quiet stance activities.

Several researchers have applied the integration approach (IA) to estimate CM kinematics for posturographic analysis (Eng and Winter, 1993; King and Zatsiorsky, 1997; Levin and Mizrahi, 1996; Shimba, 1984; Zatsiorsky and King, 1998; Zatsiorsky and Duarte, 2000). Note that the basic procedure underpinning

the IA for posturographic analyses applies equally to gait and more generic movement pattern analyses because it is based on Newton’s 2nd Law. If the only external forces acting on the body are gravitational force and the ground reaction forces acting on the feet measured by one or two force platforms (implying aerodynamic forces are considered negligible), then the acceleration of the CM can be determined by dividing the net force acting on the body by whole body mass: g m t F t M C WB + = ′′( ) ( ) (1)

where CM′′(t) is the acceleration of the CM at time t, F(t) is the net ground

reaction force (GRF) of the reaction forces applied to both feet at time t, mWB is

the whole body mass and g is gravitational acceleration. Subsequently, integration of CM′′(t) with respect to time allows determination of CM velocity:

0 ) ( 1 ) ( F t dt gt V m t M C WB + + = ′

_∫

where CM′(t) is the velocity of the CM at time t and V0 is the first integration

constant representing the initial velocity of the CM at t = t0. Similarly, integration

0 0 2 2 1 ) ( 1 ) ( F t dt gt V t S m t CM WB + + + =

_∫∫

where CM(t) is the displacement-time history of the CM and S0 is the second

integration constant representing the initial displacement of the CM at t = t0.

Although analytical integration notation is used above, in practice, digitised ground reaction force data can only be integrated numerically. V0 and S0 are

usually not known precisely. If they can be estimated, then:

ε V V V₀ = ˆ₀+ _{, and (4)} ε S S S₀ = ˆ₀+ (5)

where ₀ and ₀ are the estimates and S_ε an V_ε are the associated error terms. If V0 cannot be estimated, only relative CM′(t) can be determined. If V0 can be

estimated, absolute CM′(t) and relative CM(t) can be estimated. Further, if S0 can

be estimated, absolute CM(t) can also be estimated. However, accurate estimation of V0 and S0 is usually problematic. The error (Sε) in estimating the second integration constant will introduce an offset error in calculated CM(t). Even more critically, the error ( ) in estimating the first integration constant will introduce a cumulative error in calculated CM(t). The cumulative error increases linearly as time elapses. Thus, errors in estimating initial CM velocity and displacement values must be minimised if the IA is to be used successfully to determine CM displacement. Most importantly, initial CM velocity must be known accurately in order to avoid cumulative integration errors.

ˆ

V Sˆ d

ε

Eng and Winter (1993) evaluated the IA for calculating antero-posterior CM(t) for posturographic applications. They suggested that cumulative errors made the method only suitable for transient activities. An attempt to define what might be the maximum length of time applicable to the term ‘transient’ was not made. However, it is clear that the more accurately V0 is estimated, the longer the

analysis can continue before CM(t) errors become unacceptably large. Approaches for determining V0 and S0 for posturographic activities have ranged

from the oversimplified assumption that V0 = 0 (Eng and Winter, 1993), to more

complex approaches (e.g. King and Zatsiorsky, 1997; Levin and Mizrahi, 1996; Shimba, 1984; Zatsiorsky and King, 1998).

Using a least squares approach, Shimba (1984) estimated V0, S0 and an offset error

term in the GRF measurements, by fitting the IA CM(t) function with an alternative function that was also claimed to approximate CM(t). The alternative function was related to the Newtonian principle that the rate of change of angular momentum of a body about the body’s CM (H′CM) is equal to the net external

torque acting on the body about its CM. Shimba chose to regard the unknown value of H′CM as negligible, thus reducing the alternative CM(t) function, E(t), to

z x GF z x t₎ 0 ( = _p + in the x F E 0 direction, and z y G p F F z y t E 0 0 ) ( = + in the y direction (6) me knowledge or stimation of the anthropometry of the subject to determine zG.

ʹ

ta. Contrary

(1996), Lenzi et al. (2003) suggested that H′  was not always negligible for quiet

H′ becomes significant when these methods of determination of the initial where xp and yp were the coordinates of the transverse plane COP, and F0x, F0y

and F0z were the components of the GRF. The other parameter, zG, was the height

of the CM above the ground, so Shimba’s method required so e

Levin and Mizrahi (1996) agreed with Shimba’s assumption that the unknown value of HCM as negligible for posturographic applications. They extended the

work of Shimba by using bilateral force platforms and introducing an iterative process to evaluate H′CM and subsequently refine the estimate of CM trajectory.

However, this process required the provision of even more anthropometric data and a five-segment body model. Lenzi et al. (2003) evaluated Shimba’s method by comparison to benchmark simulation da to Levin and Mizrahi

CM

stance. Whether or not this is so, it is reasonable to suggest that the influence of

CM

conditions are applied to more dynamic activities. Levin and Mizrahi (1996) suggested their method would overcome this issue but conceded that it would require more iterations to converge (i.e. to arrive at the solution). However, this was not assessed in their work and the author is unaware of any subsequent

quires the foot (or feet) to be flat n the force platform throughout the analysis.

attempts by other researchers to apply Levin and Mizrahi’s method to more dynamic tasks. The application of the approach proposed by Levin and Mizrahi (1996) has similar limitations to that of Shimba (1984). Both require some anthropometric data and both assume the height of the CM to be constant. The latter point clearly prevents their use for activities such as lifting tasks and stair climbing. Levin and Mizrahi’s approach also re

o

King and Zatsiorsky (1997) proposed two IA methods with different algorithms for estimating the initial conditions when determining antero-posterior CM(t) for stance activities. They called the first algorithm the ‘trend-eradication’ technique. Only relative CM(t) was determined with this method as S0 was not known. It

involved assuming initially that V0 was zero when double-integrating the CM

antero-posterior acceleration. The resultant CM displacement-time history was then fitted with a linear regression line. The slope of this line was then accepted as the improved estimate of V0. King and Zatsiorsky (1997) ensured the analysis

commenced and finished when the COP trajectory was “at peak values of COP.” However, a source of error exists because the two peak COP displacements may not always correspond to two peak CM displacements; and even if they did, the two peak CM displacements will not necessarily be equivalent. Consider the case when the real initial and final CM displacements are not equivalent at the times corresponding to the chosen initial and final COP peaks and where true V0 is

Knowledge of the true initial and final CM positions would overcome this problem and permit analysis over shorter time spans.

The other IA method proposed by King and Zatsiorsky (1997) was the ‘zero-point-to-zero-point’ integration (ZPZP) method. This method aimed to determine the first instant (and all subsequent instants) in time when the CM and COP antero-posterior displacements coincided. The first such instant in time was then assigned as the starting point for the analysis, rather than attempting to determine the initial conditions at the very beginning of the trial. Their approach was based on the assertion that, during stance, antero-posterior CM displacement and COP coincide whenever the antero-posterior GRF is momentarily zero (King and Zatsiorsky, 1997; Zatsiorsky and King, 1998). Zatsiorsky and Duarte (1999, 2000) called these absolute antero-posterior COP positions the ‘zero-force points’ or ‘instant equilibrium points’ (IEPs). Fundamentally, the step-by-step algorithm applied by all of these researchers in the antero-posterior dimension was as follows:

Step 1: Find the first two IEPs (the experimentally recorded COP values at the