Criterion validity of 3D trunk accelerations to assess external work and power in able-bodied gait

1. Introduction 

In addition to the detailed instrumented analysis of gait, the global body function, i.e. the overall mechanical quality of the gait pattern, represents an important outcome in rehabilitation settings. Such large population study methods should be unsophisticated and performed in real-life, ambulatory conditions. They should be designed for application in large cohorts, in terms of time consumption and costs. One possible alternative to laboratory evaluation is gait analysis by means of accelerometry. This is limited to the measurement of linear kinematics of particular body locations, but permits the measurement of a “natural” walking over many repeated and uninterrupted cycles. In the past, accelerometers were used to assess spatial and temporal parameters [1], repeatability and variability [2], [3] and postural control [4], [5], [6]. Moe-Nilssen reported acceptable reliability of accelerometers with respect to acceleration amplitude during level walking [7], [8], [9]. Henriksen et al. showed that accelerometry was reliable for the determination of step length, stride length and cadence [10].

Comparisons have been made between the kinematics of the lower trunk and the body centre of mass (CoM) during walking. Saini et al. compared the sacral marker method with other methods of estimation of the vertical displacement of the CoM and reported moderate to moderately high correlations [11]. Similarly, Gard et al. reported overestimation of the vertical motion of the CoM by using the sacral marker method in comparison to a body-segment model and to a method using the force plate [12]. Using an accelerometer, Cavagna [13] showed that the CoM moves within and lags behind the trunk during the gait cycle. Whittle [14] quantified this anterior–posterior phase difference between the CoM and the pelvis excursions to 4–5% of the gait cycle. This phase difference between the CoM and the trunk probably introduces some systematic error in all predictions of CoM mechanics performed by simple trunk mechanics.

Further to kinematics, variables related to mechanical energy changes are essential for the understanding of the dynamic behavior of the walking body, particularly in relation to energy preservation [15]. External power in walking has been defined as the rate of change in mechanical energy of the CoM [13]. The concept of reducing the mechanical cost of walking to external power has been criticized: assessment of external power disregards multiple facets of mechanical energy changes: rotational kinetic energy fluctuations of body segments, simultaneous energy generation and absorption at different joints, co-contractions and work against gravity [13], [15], [16], [17], [18]. Finally, elastic energy storage remains unaccounted. Nevertheless, in global outcome assessments, external power measurement may be more adequate and sufficient. Internal forces do not directly influence CoM displacement. In comparison to the total and “internal” mechanical power, the understanding and interpretation of external power is straightforward. The pendulum mechanism arising from the potential-to-kinetic energy exchanges within the body and vice versa is accepted [19], [20], [21], [22].

Using outdoor measurement of trunk kinematics with a satellite positioning system, Terrier et al. [23] calculated external power and work. However, they failed to report correlations with a gold-standard. External work was also assessed by trunk accelerometry [13]. The authors are not aware of any work that studied the criterion validity of trunk accelerometry in predicting the external power and work as measured by means of force-platforms, which is considered as the current gold-standard [24].

Therefore, the aim of this work was to study the criterion validity of three-dimensional trunk acceleration measurement to assess external power and work in able-bodied gait. We hypothesized that in able-bodied level walking, there is a high correlation between external power and work derived from trunk movement and the “real” external power and work as measured by the force plate. In addition, we aimed at comparing the global kinematics and symmetry derived from the accelerometer measurements with those of the CoM movement.

2. Methods 

2.3. Data analysis 

3D GRF and accelerometer time series were synchronized by determination of the lags of maximal unbiased cross-correlation between the VT acceleration measured with the Physilog™ and the VT acceleration of the VICON system [3]. All time series were low-pass filtered at 30Hz (fourth-order zero-lag low-pass Butterworth). Mean walking speed was calculated from λ, the AP displacement of the left ankle during one stride period following left heel-strike. Stride period τ was determined by AP acceleration peak detection method [1]. For the determination of τ, Zijlstra and Hof reported time differences between the GRF method and the AP acceleration peak detection method that were smaller than 0.01s for normal gait speeds.

2.3.1. 3D CoM acceleration 

The GRF data recorded at the end of the second force plate, that is, following one τ after left heel-strike on the first plate, were moved to the beginning of the cycle in order to obtain complete data for both double-stance phases [14]. Ignoring external frictional forces, the instantaneous 3D linear accelerations of the CoM were determined according to Newton:

where is the instantaneous 3D acceleration of the CoM, the instantaneous vector sum of the forces formed by the GRF components of both force plates, m the body mass, and is the terrestrial acceleration vector (0, 0, −g). This analysis yields only small errors in the estimation of the real CoM [24].

2.3.2. 3D trunk acceleration 

A coordinate transformation was performed to remove the static gravity effect generated by the incline of the sensing axes in the sagittal plane. This was performed by using the accelerometer as an inclinometer. Therefore, we assumed a steady state condition over the gait cycle and estimated the mean VT and AP acceleration to be 1 and 0g, respectively, and applied a trigonometric algorithm to correct for the mean incline angle, as presented by Moe-Nilssen [8]. Fluctuations of this tilt within the cycle were assumed to be less than ±2° [25], and upper limits of possible AP and VT acceleration errors generated by these fluctuations were calculated with basic trigonometry:

Finally, the gravity offset of the VT accelerations was removed to obtain .

2.3.3. Curve reliability, acceleration amplitude, variability and smoothness 

The mean trunk and the mean CoM acceleration cycle was determined by computing the mean cycle of at least seven reliable time-normalized trials. A trial was accepted if the gait curve reliability of the time-normalized trunk AP acceleration cycles including that trial was superior to 0.95. This cut-off value takes into account that sufficient reliability (>0.8 for single numerical measures) must be reconsidered for curve data [26]. Corresponding relative reliability indexes, intraclass correlation coefficients (ICC), were calculated from a one-way ANOVA as (BMS−WMS)/(BMS+(r−1)WMS), where BMS and WMS are the between-time and the within-time mean square, respectively, and r the number of trials [26], [27]. For the AP and VT direction, root-mean-square (RMS) accelerations, , and amplitude variability, , were calculated for both methods. Indexes of smoothness of the gait pattern, harmonic ratios, as described by Yack and Berger [28], were determined from the AP and VT mean acceleration cycles. Higher values of harmonic ratios indicate a smoother gait pattern.

2.3.4. Velocity and displacement of CoM and trunk 

Using trapezoidal double integration of the mean CoM and trunk acceleration cycles, we computed the respective velocity and displacement vector relative to a uniform rectilinear movement in AP direction: and . The integration constant was determined by requiring the average relative velocity to be 0 in all directions. Phase differences between trunk and CoM kinematics in percent of the gait cycle were determined by means of maximal cross-correlation between multiple repetitions of the corresponding time series. Right and left “functional stance phases” were defined arbitrarily as the periods between the temporal boundaries determined as the mean times between maximal AP velocity and minimal VT displacement. Fig. 1 shows a normalized cycle from a pilot trial with the two moments at the beginning and the middle of the cycle where AP velocity and VT displacement are stationary for the trunk and the CoM, respectively. Right and left “functional progression” was computed as .

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Fig. 1. The problem of determination of “functional stance phases” from kinematics of the lower trunk. Anterior–posterior (AP) velocity, solid line, and vertical (VT) displacement, dotted line, of the trunk and the centre of mass, respectively. Note that in AP direction, the centre of mass lags behind the trunk. As a result, the temporal relation of the AP velocity maximum to the respective VT displacement minimum is reversed. Therefore, start and end of functional phases were defined as the mean times between minimal VT positions and maximal AP velocities.

2.3.5. Energies of CoM and trunk 

We calculated mass normalized instantaneous external power P of the CoM and the trunk model as the sum of the respective translational kinetic (Pcin) and potential (Ppot) powers:

where . The lateral kinetic power component is negligible [13], [29] and was therefore discarded. The contribution of the kinetic vertical power is marginal only at low and intermediate speeds; therefore, this component was included in Pcin [21]. Mean absolute and RMS external power with respect to the mean gait cycle, the right and the left functional phases were calculated. External work was computed as the time-integral of absolute external power: . The fraction of mechanical energy recovered through within body transforms of potential and kinetic energy was computed as recovery=1−W/(Wcin+Wpot). Respective positive and negative powers of the right and left phases were integrated to compute positive and negative work for each phase. Standardized external work as the work performed per unit distance was calculated as Wstand=W/λ.

2.3.6. Symmetry 

For all relevant variables and with respect to the functional phases, we calculated two types of symmetry indexes: ratios and asymmetry indexes (AI):

Data analysis was performed with Matlab 6.0, Signal Processing Toolbox (The MathWorks Inc., Natick, MA).

3. Results 

The mean sagittal tilt angle of the accelerometer VT measurement axis relative to the real vertical was 7.5° with a S.D. of 4.8°. Maximal possible errors from assumed 2° tilt fluctuations were smaller than 0.05g for AP and smaller than 0.01g for VT accelerations. The mean walking speed of the 12 subjects was 1.41±0.13m/s. Stride frequency was 59.42±4.11cycles/min, stride length 1.43±0.08m, stride time 1.01±0.07s. Table 1 shows the time lags between the trunk and the CoM for the AP and VT kinematics. For velocity and displacement, the respective lags were significant and larger than 3% in AP direction (p<0.001) and smaller than 0.4% in VT direction (p<0.05). For all subjects, the trunk leaded the CoM in AP direction.

Table 1. Lags (percent of the gait cycle) and peak correlations (mean±S.D.) between trunk and centre of mass kinematics

		VT	AP
	Lag, acceleration (% cycle)	0.64±0.52a	−2.75±0.85b
	Lag, velocity (% cycle)	0.38±0.34a	−3.11±0.79b
	Lag, displacement (% cycle)	0.25±0.32a	−3.54±1.02b
	Peak correlation acceleration	0.95±0.02	0.90±0.04
	Peak correlation velocity	0.98±0.01	0.95±0.02
	Peak correlation displacement	0.96±0.04	0.93±0.06

Negative signs indicate that the centre of mass lags behind the trunk. Note the important lag between trunk and centre of mass kinematics in anterior–posterior direction. VT: vertical; AP: anterior–posterior.

a p<0.05. b p<0.001 for H0: lag=0%.

Fig. 2 illustrates representative cycles of AP and VT accelerations of the trunk and the CoM, respectively. The acceleration amplitudes of the CoM were smaller, and the CoM accelerations demonstrated a smoother pattern in comparison to the trunk. RMS VT and AP accelerations correlated highly and significantly between the CoM and the trunk model (r=0.97 for the VT, r=0.81 for the AP accelerations).

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Fig. 2. Representative vertical (VT) and anterior–posterior (AP) accelerations of the lower trunk and the centre of mass (CoM), respectively.

Table 2 shows the mean and S.D. of the kinematics resulting from trunk and CoM movement, their correlation and the explained variance.

Table 2. Trunk and center of mass kinematics (mean±S.D.) and corresponding Pearson and squared correlations

	Variable	Trunk model	CoM	Pearson	Explained
	Period R (s)	0.51±0.04	0.51±0.03	0.98	0.97
	Period L (s)	0.50±0.03	0.51±0.04	0.99	0.97
	Period ratio	0.98±0.03	1.00±0.03	0.57	0.33
	Period Al (%)	2.34±2.55	0.22±2.67	0.57	0.33
	Lambda R (m)	0.72±0.04	0.72±0.05	0.98	0.97
	Lambda L (m)	0.70±0.04	0.71±0.04	0.98	0.96
	Lambda ratio	0.97±0.03	1.00±0.03	0.67	0.44
	Lambda Al (%)	2.70±3.20	0.38±2.94	0.67	0.44
	RMSVT (g)	0.31±0.07	0.22±0.05	0.97	0.93
	RMSAP (g)	0.19±0.04	0.13±0.02	0.81	0.65
	Harmonic ratio VT	5.29±1.22	5.23±1.81	0.67	0.44
	Harmonic ratio AP	4.64±0.96	8.35±2.70	0.19a	0.04
	Amp variability VT (g)	0.07±0.02	0.05±0.01	0.84	0.70
	Amp variability AP (g)	0.04±0.01	0.02±0.01	0.76	0.57
	Range VT (cm)	5.39±0.91	4.14±0.57	0.91	0.83
	Range VT R (cm)	5.18±0.98	3.99±0.60	0.91	0.83
	Range VT L (cm)	5.35±0.91	4.05±0.51	0.92	0.86
	Range VT ratio	1.04±0.05	1.02±0.07	0.80	0.64
	Range AP (cm)	3.33±0.95	2.30±0.42	0.85	0.72
	Range AP R (cm)	2.63±0.43	1.90±0.30	−0.19a	0.04
	Range AP L (cm)	2.80±1.24	1.98±0.57	0.79	0.62
	Range AP ratio	1.11±0.56	1.06±0.29	0.46a	0.21

CoM: centre of mass; R: right; L: left; Period R, L: right, left functional phase period; Lambda R, L: progression during R and L functional stance phase; g: 9.81m/s2; VT: vertical; AP: anterior–posterior; RMS VT, AP: root-mean-square of the VT and AP acceleration amplitude of the trunk and the center of mass, respectively; amplitude variability: square root of the within-time mean square from one-way ANOVA; harmonic ratio: ratio of the sum of amplitudes of the even to the sum of amplitudes of the odd harmonics; range: difference between maximal and minimal displacement; Al: asymmetry index (negative values indicating value L>value R); ratio: value L/value R.

a Absence of significant product moment correlation.

Similarly, Table 3 shows the results for variables associated to external power/work of the two models. Fig. 3 shows the respective energy time series for one subject.

Table 3. External work and power (mean±S.D.) determined from trunk movement and from the center of mass movement

	Variable	Trunk model	CoM	Pearson	Explained
	Work (J/kg)	1.41±0.37	1.09±0.23	0.77	0.59
	Work R (J/kg)	0.67±0.14	0.51±0.11	0.76	0.58
	Work L (J/kg)	0.75±0.25	0.58±0.13	0.77	0.60
	Work ratio	1.11±0.25	1.13±0.13	0.66	0.43
	Work Al (%)	−7.85±24.37	−11.17±12.11	0.64	0.40
	Stand work (J/kgm)	0.98±0.21	0.76±0.13	0.62	0.39
	Stand work R (J/kgm)	0.92±0.16	0.72±0.13	0.65	0.42
	Stand work L (J/kgm)	1.05±0.31	0.81±0.15	0.67	0.45
	Stand work ratio	1.15±0.28	1.13±0.15	0.70	0.49
	Stand work Al (%)	−10.52±26.24	−11.53±12.86	0.67	0.46
	Mean power (W/kg)	1.41±0.41	1.09±0.26	0.83	0.70
	Mean power R (W/kg)	1.32±0.32	1.02±0.24	0.83	0.68
	Mean power L (W/kg)	1.51±0.55	1.15±0.29	0.84	0.70
	Power ratio	1.14±0.27	1.13±0.14	0.69	0.48
	Mean power Al (%)	−12.86±23.08	−11.21±12.40	0.82	0.66
	RMS power (W/kg)	1.87±0.54	1.37±0.36	0.84	0.71
	RMS power R (W/kg)	1.73±0.39	1.32±0.33	0.82	0.67
	RMS power L (W/kg)	1.99±0.71	1.43±0.40	0.83	0.70
	RMS power ratio	1.13±0.24	1.09±0.12	0.72	0.52
	RMS power Al (%)	−10.21±22.57	−7.97±11.38	0.69	0.48
	Recovery	0.67±0.04	0.64±0.06	−0.26a	0.07
	Recovery R	0.68±0.03	0.66±0.06	0.11a	0.01
	Recovery L	0.65±0.07	0.62±0.06	−0.12a	0.01
	Recovery ratio	0.96±0.11	0.94±0.06	0.60	0.36
	Recovery Al (%)	4.74±11.32	6.35±6.37	0.62	0.38

Corresponding Pearson and squared correlations. CoM: center of mass; R: right; L: left; work: external work per cycle/phases; stand work: external work per unit distance; mean power: mean absolute external power; RMS power: root-mean-square external power; recovery: fraction of energy conservation; Al: asymmetry index (negative value indicating value L>value R); ratio: value L/value R.

a Absence of significant product moment correlation.

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Fig. 3. Mechanical energy time series for one subject: centre of mass (upper panel), approximation from trunk accelerations (lower panel).

For all correlations, the lower limits of the bootstrap 95% CI were higher than the lower limits of the Fisher 95% CI. Fig. 4 shows a scatter plot of the regression of CoM RMS power on trunk RMS power and the corresponding distribution of bootstrapped correlation coefficients.

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Fig. 4. Linear regression of the centre of mass (CoM) RMS external power on trunk model RMS external power. Lower panel: corresponding bootstrapped correlation coefficients.

Mean absolute and RMS power, right and left mean absolute and RMS power showed high and significant correlations (0.82<r<0.84). The correlation for work, right and left work amounted to 0.77. Standardized external work showed also a significant, but moderate correlation (r=0.62 for the cycle, 0.65 for the right phase and 0.67 for the left phase). The correlations for right phase, left phase and global recovery were not significant. Symmetry indexes for power showed correlations of 0.69 for mean absolute power ratio and RMS power AI, 0.82 for mean absolute power AI and 0.72 for RMS power ratio. Work ratio correlation was 0.66; work AI 0.64, standardized work ratio and AI showed correlations of 0.70 and 0.68, respectively. Recovery ratio and AI showed moderate and significant correlations. The correlations for the ratios of positive to negative work for the two functional phases were not significant.

For all variables associated with energy cost, the trunk model overestimated corresponding CoM variables. The slopes of the linear regressions of the CoM on the corresponding trunk model power and work were smaller than 1.

Agreement was substantially lower than the correlation between the two methods. The best agreement with absence of heteroscedasticity was given for right and left period and right and left functional progression (ICC(3,1)=0.98, ICC(1,1)=0.97 for the right and left period, ICC(3,1)=0.98, ICC(1,1)=0.97 for right, ICC(3,1)=0.97, ICC(1,1)=0.95 for left functional progression). All lower limits of the corresponding 95% CI were larger than 0.9. For mean absolute power, ICCs amounted to 0.75 and 0.44 for the (3,1) and the (1,1) model, respectively. For RMS power, the corresponding values were 0.78 and 0.38. However, only ICC(3,1) lower confidence limit was larger than 0 and there was evidence of heteroscedasticity for the bivariate distributions of these power variables. The symmetry indexes mean absolute power ratio, mean absolute power AI, RMS power ratio and AI, standardized work ratio and AI showed all significant ICCs(1,1) and ICCs(3,1). These ICCs ranged from 0.57 to 0.62 and there was absence of evidence of heteroscedasticity.

4. Discussion 

Global gait quality is an important outcome in rehabilitation settings. Face validity of the modeling of the CoM by a single-segment model or by the trunk is frequently stated, even for activities representing smaller CoM within-body stability than walking. We compared the global mechanics of the sinusoidal trunk movement determined with accelerometry to the mechanics of the sinusoidal CoM movement determined with force-platforms. In particular, we compared external power and work for the two methods. We confirmed a significant AP phase difference between the trunk and the CoM. In normal level walking, the CoM moved within and lagged behind the trunk. We quantified this phase difference to 3.5% of the gait cycle for AP displacement, little less than reported by Whittle [14]. This time lag influences the correlation between the CoM and the trunk model recovery, since the energy conservation is a function of the relation of VT displacement (potential energy) to AP velocity (kinetic energy).

In our sample, amplitude variability correlated highly and significantly between the methods. This is an interesting finding, as variability of gait has been described as an important parameter for balance control [2]. VT harmonic ratio as a measure of smoothness of the gait pattern showed also a significant correlation and agreement. Conversely, there was no significant correlation for AP harmonic ratio. The trunk AP acceleration pattern showed harmonic ratios that were much smaller than those for CoM, indicating an important increase of the odd harmonics in the trunk AP movement. To describe symmetry, we defined right and left functional stance phases of the gait cycle. Functional stance phases have been defined as intervals between maximal AP velocities during the gait cycle [20], [31]. The phase difference between the trunk and the CoM AP velocity resulted in an inversion of the temporal relation of AP velocity maximum to the VT displacement minimum. Our compromise was to choose the mean times of AP velocity maximum and VT displacement minimum to determine these intervals, for trunk and CoM movement, respectively. The AP leading of the trunk influences the correlation and agreement between the accelerometer and the force-platform method relative to symmetry. A posteriori, we also calculated all variables and correlations by defining the respective boundaries of functional phases only by highest CoM and trunk AP velocities. This resulted in slightly higher correlations for energy-related variables, but in lower correlations for kinematic symmetries.

The moderate or poor correlation and agreement for some variables can be attributed to the limited range of our sample, which included able-bodied individuals only. However, external power showed correlations that were larger than 0.8, while correlation for work was 0.77. Recovery, although not correlated between methods, showed similar values (trunk model: 67%, CoM: 64%). Correlations for symmetry outcomes of functional progression, period and vertical range were significant. Correlation for standardized external work was 0.62. This correlation was smaller than hypothesized.

Mechanics determined from trunk movement overestimated CoM mechanics (30% for power and work), and in comparison to CoM values, the variability of trunk values was larger for these variables; all slopes of CoM on trunk regressions were smaller than 1. The reasons for this overestimation were larger 3D displacements of the accelerometer region relative to the CoM. The mean VT and AP range was 5.39 and 3.33cm for the trunk, 4.14 and 2.30cm for the CoM, respectively. Vertical excursions and related symmetry correlated highly between the methods (r>0.9). Saini et al. [11] reported a smaller correlation (0.43) between the force plate and sacral marker method for vertical range. Given that we measured the L3 region, this supports the assumption that the level of sensor fixation plays a role. Fixation of the accelerometer at higher levels than L3 would, possibly yield higher agreement between the two methods.

There are limitations in the measurement of unbiased CoM movement with two force-platforms, since the sequence between stance phases is always the same (left–right in our case). We observed substantial asymmetries. Participants were aware of the force-platforms and displayed consistent asymmetry in their gait pattern. Left power and work was generally larger than the right. Linear ankle kinematics showed that the right step length prior to heel-strike of the right foot on the second (right) force plate was in general smaller than the left step length prior to heel-strike of left foot on the first plate. This asymmetry was also detected by the accelerometer. Laboratory measurements with force plates carry the disadvantage of a biased gait pattern.

Correlations for power and work symmetry were significantly larger than 0.65. In the presence of gait abnormality, the interpretation of the right-phase and left-phase ratios of positive versus negative external work, representing the loss and gain of energy during both functional stance phases, could be helpful [31]. However, we did not detect significant correlations for these ratios on either side; we stipulate that in pathological gait patterns, these correlations would be higher. Finally, the critical assumption of a steady state gait pattern and the manipulation of the GRF time series necessary to construct the first double-stance period probably introduced another bias in the CoM and the trunk movement. This may have resulted in under- or overestimations of the correlation between the methods.

Our results were also affected by a systematic bias related to the sensor attachment and to the uncorrected instantaneous sensor orientation. In contrast to the mean sensor orientation, orientation fluctuations cannot be corrected without a 3D gyroscope. Therefore, accelerometers can produce biased and even lagged estimates. AP kinematics, in particular, can be very sensitive to axis rotation, since they depend on the sine of the fluctuation angle. VT acceleration depends on the cosine of this angle and is therefore more robust. The resulting malalignment with reference systems should be removed by estimation of instantaneous absolute accelerometer orientation. This requires the integration of a 3D gyroscope and an appropriate calibration procedure.

Other studies have assessed the reliability of the global mechanics and kinematics in level walking determined from trunk accelerations [9], [10]. The relative and absolute reliability of external power/work and kinematics assessed by accelerometry was amounted to ICCs(1,1) larger than 0.96, with standard errors of measurement smaller than 0.04J/kgm for standardized external work and smaller than 1.2mm for VT displacement [32]. Further research should investigate the criterion validity of trunk acceleration measures in disabled people with disturbed gait pattern and at different walking speeds. Larger sample sizes, the influence of different positions and methods of sensor fixation and the integration of gyroscopes should also be considered. Validity should be examined with respect to different pathologies, as accelerometry may be an option for a straightforward cost-effective outcome measure in various conditions.

5. Conflict of interest statement 

The authors had no financial support for this work. This work was carried out by the corresponding author for his master-thesis at the study course “Sciences in Physiotherapy” at the universities of Maastricht/NL and Zurich/CH.