Introduction

Aging is associated with reduced functional mobility, social, and physiological impairments, and an increased risk of falls.7 Falls and gait instabilities are among the most common and serious problems faced by older adults,24 with considerable medical cost and suffering.31 Older adults exhibit more conservative gait patterns, which are indications of greater cautiousness,46 and a large proportion of falls occur during locomotion.2,39 In everyday life, people commonly face various inclined surfaces, such as ramps to access buildings, or more abrupt hills on street sidewalks, which impose different demands on walking balance and energy expenditure. Thus, gait kinematic changes have been studied on multi-surface,26 irregular surfaces,28 and on a variety of sloped surfaces.12,27,41,48

Older adults exhibit different kinematic characteristics when walking on inclines, compared to level ground, to properly advance the center of mass forward and upward (or downward) against increased gravitational demands.12,41 They reduce speed and step length compared to young adults, adjusting to a more conservative gait (slow speed, shorter steps, increased double support, and increased cadence),12,41 suggesting a more stable gait pattern on inclines.12 Compared to level ground walking, the ability to walk on inclines requires a different lower extremity motor pattern, which includes increased range of motion in the ankle,48 increased/decreased hip and knee flexion at initial foot contact when walking up/down,21 and increased force production.27 Therefore, kinematic changes, including increased/decreased step length when walking up/down and decreased/increased gait cadence when walking up/down,27 are observed.

Furthermore, the relationship between energy expenditure and slope has a U-shape, so that energy requirements increase when walking up and down, with the optimum inclination for minimizing energy expenditure being −10%.29,30 These results can be explained by the additional loads imposed on muscles due to braking forces when walking down and gravitational demands when walking up that corresponds to eccentric and concentric muscle contraction, respectively.30

While older adults appear to prioritize stability, young adults optimize forward progression, mobility, and efficiency while walking.40 The gait patterns of young adults, characterized by phases of instability that lead to forward progression and lateral shifting of the body center of mass at each step, are detrimental in terms of balance. Since forward progression changes are likely when walking on an inclined surface, center of mass shifting and balance would also be affected. Therefore, previous studies’ results for level walking may not be generalized to inclined surfaces,46 particularly regarding nonlinear and linear features. To the best of our knowledge, no previous studies have assessed linear and nonlinear gait features for older adults when walking on inclined surfaces.

Thus, in addition to spatiotemporal gait features, it is important to assess gait variability, which is influenced by the ability to control gait optimally from one stride to the next.15 In biomechanics, variability refers to the ability of the motor system to perform a wide variety of tasks under environmental constraints.44 Variability is quantified using measures derived from linear statistics, such as the standard deviation of a time series, or using measures derived from nonlinear dynamics, such as maximum Lyapunov exponent and entropy. The variations in human movement are distinguishable from noise; they have a deterministic origin, being neither random nor independent. The variations display a temporal structure, captured by nonlinear tools. In this approach, the manner in which gait evolves over time becomes important.44 Therefore, linear and nonlinear variability, express different aspects of a time series: the magnitude of the variations presented in a time series, and the structure/organization of these variations, respectively. Thus, linear and nonlinear gait features were extracted in the present study.

For optimal operation of the motor system, a moderate amount of variability is expected as necessity for adaptability, since a system with a total absence of variability lacks the repertoire to accommodate even subtle perturbations.49 In contrast, a large amount of variability may be detrimental to the motor control system; greater variability can mean that the system is incapable of accurately achieving the desired goal. Therefore, the average standard deviation of trunk acceleration between strides (VAR), a common linear measure of variability in gait studies,9 and the sample entropy (SEn) of trunk velocity, a nonlinear measure, which refers to the regularity and the presence of nonrandom fluctuations in time within the apparently irregular dynamics of a system,25 were computed.

Gait stability was assessed by an inverted pendulum-model-based descriptor, margin of stability (MoS), which is a purely biomechanical concept. The MoS is calculated as the distance between the extrapolated center of mass and the border of the base of support, in which extrapolated center of mass accounts for not only the position of the center of mass but also its velocity.16 Decreased MoS is detrimental to stability, being affected by step length and step frequency,13 which are two analyzed spatiotemporal gait parameters. Local dynamic stability (LDS), which is derived from nonlinear dynamic system theory, was assessed by the maximum Lyapunov exponent (λ).3 LDS assumes that the motor control system ensures a dynamically stable gait if the divergence between trajectories in a reconstructed state-space remains low. Thus, an increase in λ implies a decreased in LDS.

Since trunk kinematic data are most sensitive to differences between young and older adults,19 as maintaining stability of the upper body is critical for human locomotion,32 we computed gait features from trunk kinematics. Therefore, the present study evaluated nonlinear and linear gait features on inclined surfaces for older adults walking at different speeds, since older adults alter their speed on inclines. We compared the gait of young and older adults on surfaces inclined by −8, 0, and 8%, at 100% and ±20% of their preferred walking speed. Considering the kinematic and kinetic changes observed for both young and older adults when walking on inclines,27,41 we hypothesized that inclines decrease gait stability and LDS, and increase linear gait variability and nonlinear measures, pronouncedly for older adults when compared to young adults, and that these effects are more evident in walking down an incline, since some previous studies have shown that this is biomechanically more challenging than walking up an incline.27,41

Methods

Subjects

A total of 37 healthy older adults (15 males, 22 females, 69.3 ± 6.6 years old, 66.5 ± 12.0 kg, 1.58 ± 0.08 m) and 50 healthy young adults (26 males, 24 females, 25.2 ± 5.1 years old, 68.9 ± 10.6 kg, 1.71 ± 0.07 m) participated in this study. All participants were healthy, could ambulate independently without an assistive device, were not diagnosed with a neurologic condition, without any history of musculoskeletal surgery and without any injury or pain at the time of data collection. The older adults were in good health, and they had active lifestyles: some of them practiced regular physical activity; others practiced sports informally, while others were still employed, as gardeners, drivers, teachers, and farmers. They reported no falls in the past 12 months. All participants agreed to sign an informed consent form. Next, they submitted to protocols previously approved by the Local Research Ethics Committee.

Equipment

Reflective markers were attached to the lateral malleoli, the heels, the head of the second and fifth metatarsals, and the T1 vertebrae of all participants. A cluster of four reflective markers was attached between the left and right posterior superior iliac spines for MoS calculations. The markers were used for movement registration with a 3-D motion capture system that used ten infrared cameras operating at 100 samples/s (Vicon Nexus, Oxford Metrics, Oxford, UK).

Protocol

The participants walked on a level and inclined treadmill at their preferred walking speed (PWS) on each inclination and ±20% PWS. First, the PWS was evaluated, following a previously reported protocol17 on each inclination, since there may be significant differences in PWS across inclinations for older adults.41 However, we did not find any differences in PWS on the tested inclinations for both young and older adults.

The participants then performed nine trials of 4 min at treadmill inclinations of −8% (DOWN—walking down), 0% (HOR—horizontal), and 8% (UP—walking up) in the corresponding PWS, +20% PWS (120PWS), and −20% PWS (80PWS) presented in a randomized order. The participants rested for 2 min between trials. An inclination of 8% (1:12) was chosen because it is a standard for inclines in civil engineering and architecture.34,42

Data Analysis

Prior to data analysis, with the exception of the calculation of λ and SEn, kinematic data were low-pass filtered through a fourth-order, zero-lag Butterworth filter with a cut-off frequency of 6 Hz. All parameters were calculated for 150 strides. First, the initial and final 15 s of each trial were discarded,13 after which all steps were detected as the zero-cross of the heel-marker velocity.50 The intermediate 150 strides were then selected, discarding the initial and final strides exceeding 150. A customized Matlab (R2014a, MathWorks, Natick, MA) code was used for data analysis.

Spatiotemporal Features

Step frequency was determined as the inverse of the average duration between two subsequent heel-strikes, and step length was calculated as the average duration between two subsequent heel-strikes multiplied by average treadmill speed. For fixed walking speed conditions, step length variations imply inverse variations in step frequency.

Linear Gait Variability

Linear gait variability was assessed using the average standard deviation of medial–lateral (ML), anterior–posterior (AP), and vertical (V) trunk acceleration along normalized strides (VAR).9,47 To compute VAR, each stride was time-normalized (0–100%). At each of the 101 normalized time points, the standard deviation of the ML, AP, and V trunk acceleration over the 150 strides was calculated, after which the average of these 101 standard deviations was determined.9 As maintaining stability of the upper body is critical for human locomotion,32 higher levels of trunk variability are evidence of poor task performance.49

Gait Stability

Gait stability was assessed using MoS. The MoS was calculated using the adapted method proposed by Hak et al.13 The center of mass was estimated as the average of the four cluster markers attached between the left and right posterior superior iliac spines, using the length of the inverted pendulum (l) as the maximum height of the estimated center of mass. The extrapolated center of mass was calculated as center of mass plus its velocity multiplied by the factor \(\sqrt {l/g}\),16 where g is gravity acceleration. The MoS was calculated as the distance between extrapolated center of mass and the fifth metatarsal head marker of the leading foot for ML MoS, and the heel marker of the leading foot for backward (BW) MoS. The minimum value of the MoS within each step was averaged over the 150 strides. The closer the extrapolated center of mass to the border of the base of support, the more likely to occur a loss of balance. Thus, a decrease in MoS is detrimental to stability.16

Local Dynamic Stability

The LDS evaluation was based on the maximum Lyapunov exponent (λ) computation, using Rosenstein’s algorithm.13,23,46,47 Briefly, the ML, AP, and V trunk velocities were calculated using the three points method from the T1 marker raw data.18 Next, the velocity signal was time-normalized to 15,000 samples, preserving differences in stride time between strides.4 A high-dimension attractor was constructed, using the normalized T1 marker velocity and its delayed copies. A delay of 10 samples was used, based on the mean value of the minimum of the mutual information function across all data, and a dimensionality of five was found to be appropriate, based on a global false nearest neighbor analysis in which the percentage of false neighbors is nearly zero.9,10 For each point in the state-space, a nearest neighbor was found, and the Euclidean distance between these points was tracked for 10 strides, resulting in as many time distance curves as time points in the state-space. The divergence curve was calculated as the mean of the natural log of the time–distance curves (for a more detailed algorithm’s description see Appendix A in Ref. 45). The λ was calculated as the slope of a linear fit to the first 50 samples (time needed for one step) of the divergence curve.35,46,47 Thus, λ indicated the relative rate of divergence over one step, resulting from a small difference in initial conditions. Unlike traditional measures of gait variability that treat each gait cycle as independent, the LDS evaluates the evolution of stability over the course of several strides.49 LDS was smaller for older fall-prone adults when compared to young and older healthy adults,23 it was smaller for patients with neurological disorders when compared to healthy individuals,37 and it decreases from 40 to 50 years old.46

Gait Regularity

SEn was calculated to quantify the degree of regularity and predictability of variations of ML, AP, and V trunk velocities.38 SEn is the negative natural logarithm of the conditional probability that two m-dimensional delayed vectors that are close within a tolerance r, remain close in the (m + 1)-dimensional state-space, without allowing self-matches.33 Parameters values of m = 2 and r = 0.2 (times the standard deviation of the data) were selected based on a previously reported practical solution (Fig. 1).33 SEn reflects the likelihood that similar patterns of observations will not be followed by additional, similar observations. A time series containing numerous repetitive patterns—one that is predictable—has a relatively small SEn and greater regularity, whereas a less predictable process has a higher SEn and smaller regularity.

Figure 1
figure 1

(a) Median sample entropy for ML trunk velocity computed for several r (0.05–1.0) and m (1–4) values. The curves are superimposed for m ≥ 2; thus m = 2 was adopted in this study. (b) Median of maximum relative error of sample entropy estimation computed for several r (0.05–1.0) and m (2–3) values. The curves show a minimum at r = 0.2; thus r = 0.2 was adopted in this study.

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A movement with high regularity (low entropy) or a very random movement (high entropy) would reveal poor performance, inferring the motor control process as either unyielding or too imprecise, respectively.44

Statistical Analysis

A mixed repeated measures analysis of variance (ANOVA) was used to assess the main effects of age, treadmill inclinations, and speed, and two-way interaction effects, since all descriptors presented a normal distribution (Shapiro–Wilk test, p > 0.05). Greenhouse–Geisser correction was applied when sphericity was violated. The statistical analysis was performed using SPSS software, version 17 (SPSS Inc., Chicago, IL, US), and the significance level was set at p < 0.05.

Results

Spatiotemporal Features

A significant main effect of the treadmill inclination existed for both step length and step frequency (Table 1), with increasing and decreasing values from DOWN to UP conditions, respectively. There was also a significant interaction effect between the treadmill inclination and age for both step length and step frequency, although the effect size was small.

Table 1 Statistical results: spatiotemporal features for inclined conditions.
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A significant main effect of speed was found for both step length and step frequency (Fig. 2), with increasing values from 80PWS to 120PWS. A significant interaction effect also occurred between speed and age for both step length and step frequency, and the effect of speed was greater for young adults. Additionally, there was a significant interaction effect between the treadmill inclination and speed, but only for step frequency, although the effect size was small.

Figure 2
figure 2

Mean values for walking against treadmill inclination for each speed condition: (a) step length, and (b) step frequency.

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The results showed a significant effect of age for both step length and step frequency, with smaller step length values and greater step frequency values for older adults (Fig. 2).

Linear Gait Variability

A significant main effect of inclination was found for VAR in all directions (Table 2), with increased values in both UP and DOWN conditions. A significant interaction effect also occurred between inclination and age in all directions, and the inclinations’ effects were greater for young adults.

Table 2 Statistical results: linear gait variability for inclined conditions.
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There was a significant main effect of speed for VAR in all directions (Fig. 3), with the greatest value at 120PWS. A significant interaction effect occurred between speed and age for VAR ML, and VAR AP, but not for VAR V. Interestingly, the effect of speed was even greater for young adults.

Figure 3
figure 3

Mean values for walking against treadmill inclination for each speed condition: (a) VAR ML, (b) VAR AP, and (c) VAR V.

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A significant interaction effect between treadmill inclination and speed was also demonstrated in all directions.

However, there was no significant effect of age for VAR (Fig. 3).

Gait Stability

There was a significant main effect of treadmill inclination for MoS ML and MoS BW (Table 3), with increasing and decreasing values from DOWN to UP conditions, respectively. In addition, a significant interaction effect was found between treadmill inclination and age but only for MoS BW. The effect of inclination was greater for young adults.

Table 3 Statistical results: margin of Stability for inclined conditions.
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There was a significant main effect of speed for MoS BW, with increasing values from 80 to 120 PWS, but not for MoS ML (Fig. 4). A significant interaction effect was found between speed and age but only for MoS BW, and the effect of speed was greater for young adults. Significant interaction effect between inclination and speed was not found.

Figure 4
figure 4

Mean values for walking against treadmill inclination for each speed condition: (a) MoS ML and (b) MoS BW.

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There was no significant effect of age for MoS ML, or MoS BW (Fig. 4).

Local Dynamic Stability

Treadmill inclination exerted a significant main effect for λ in all directions (Table 4), with increased values in the DOWN and UP conditions (except in ML direction where λ decreased in the DOWN condition). A significant interaction effect was found between the inclination and age but only for λ V. The effects of inclination were greater for young adults.

Table 4 Statistical results: maximum Lyapunov exponent for inclined conditions.
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Speed showed to have a significant main effect for λ in all directions (Fig. 5), with increased values in 80PWS for ML and AP directions, but increased values in 120PWS for V direction. A significant interaction effect occurred between speed and age but only for λ V. The effect of speed was greater for young adults.

Figure 5
figure 5

Mean values for walking against treadmill inclination for each speed condition: (a) λ ML, (b) λ AP, and (c) λ V.

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Treadmill inclination and speed exhibited a significant interaction effect only for λs V, although the effect size was small.

There were no significant age effects for λ (Fig. 5).

Gait Regularity

Significant main effects were found with the treadmill inclination for SEn in all directions (Table 5), with decreasing values from DOWN to UP conditions. There was also a significant interaction effect between the treadmill inclination and age for SEn in all directions. The effect of inclination was greater for young adults in the ML and AP directions, but not in the V direction.

Table 5 Statistical results: sample entropy for inclined conditions.
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Speed also showed significant main effects in all directions (Fig. 6), with increased values in 120PWS, PWS and 80PWS for ML, AP and V directions, respectively. There was no significant interaction effect between speed and age for SEn in any direction.

Figure 6
figure 6

Mean values when walking against treadmill inclination for each speed condition: (a) SEn ML, (b) SEn AP, and (c) SEn V.

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However, a significant interaction effect occurred between inclination and speed for SEn V.

Additionally, a significant effect of age was found but only for SEn AP, with greater values for older adults.

Discussion

In the present study, we investigated linear and nonlinear gait features in older adults when walking on an inclined treadmill at different speeds. We hypothesized that inclined surfaces would decrease gait stability and alter linear gait variability and nonlinear measures. Further, we hypothesized that older adults would exhibit poorer results compared to young adults. For this purpose, we used different spatiotemporal and nonlinear and linear descriptors to investigate gait features on an −8, 0, and 8% inclined treadmill at 120% PWS, 100% PWS, and 80% PWS.

The results showed that spatiotemporal descriptors changed with inclination and speed, as did linear variability (VAR), stability (MoS), LDS (λ), and regularity (SEn), which partially supported the initial hypothesis. For example, DOWN conditions presented the greatest variability and the lowest stability in relation to MoS ML, but not in relation to LDS, which was higher in DOWN conditions. The largest λ values, indicating decreased LDS, and the lowest SEn values, indicating increased regularity, were observed for UP conditions.

Spatiotemporal Features

Our results were in accordance with those of previous studies of gait on inclined surfaces.21,27 Step length increased from DOWN to UP conditions, and step frequency decreased from DOWN to UP conditions. Moreover, both step length and step frequency increased with increased speed, as expected. Older adults walked with smaller step length and higher step frequency than young adults. This gait pattern occurred particularly in DOWN conditions, which can be interpreted as a strategy to maintain balance and improve stability, and are not necessarily indicative of a fall risk.13,41

Linear Gait Variability

The increased linear gait variability in inclined conditions, assessed by VAR, may have operated as compensation for the demands imposed on the motor system. A common assumption in many studies on locomotion has been that increased variability in spatiotemporal gait variables was detrimental to system performance. Additionally, variability has been associated with instability and increased fall risk.15 Nevertheless, in multiple degrees of freedom systems, variability is an inevitable and necessary condition for optimality and adaptability.22,25,49 Indeed, some studies have shown that a certain amount of variability appeared to be evidence of healthy and stable movement.49 Therefore, variability may indicate a normal range of coordination patterns, in which variability performs a functional role that expresses the range of possible patterns that a movement system can accomplish.49

Consequently, since older adults exhibited a slightly decreased linear variability compared to young adults (Fig. 3), they appear to be more cautious and less adaptable to inclines, prioritizing stability. In this regard, young adults exhibited greater main effects of both inclination and speed for variability, possibly an indication of more versatile movement repertoire, optimizing mobility, and efficiency while walking.40

Gait Stability

In this study, ML MoS decreased in DOWN conditions, indicating poorer dynamic stability for both groups (Fig. 4). However, this decrease, with a corresponding increase in step frequency, contrasted with the results obtained by Hak et al.13 for level walking, as these authors found a direct relationship between MoS ML and step frequency. A possible explanation for this may lie in the increased movement of the pelvis in DOWN conditions, due to the differences of foot contact heights. Such pelvic movement leads to greater and faster oscillation of the center of mass in the ML direction, which influences the calculation of the extrapolated center of mass. Consequently, this issue should be the subject of further studies.

Considering MoS BW, our results were in accordance with those of Hak et al.13 The increased MoS BW in DOWN conditions occurred likely due to a corresponding decrease in stride length. Our results supported previous observations that decreased step length was a strategy to increase MoS BW, and therefore to increase stability.13

However, there was significant interaction effect between MoS BW and both inclination and speed, and this effect was greater for young adults. MoS BW was greater in DOWN conditions and for higher speed, and these alterations appear to be greater for young adults, although the differences between the two groups were not significant.

Local Dynamic Stability

With regard to LDS, there were no effects of age for these older adults in any direction, similar to previous results on level walking.23 However, the effects of the treadmill inclination on LDS, evaluated by λ, revealed a decrease in the ML direction for the UP condition, and in the AP and V directions for both UP and DOWN conditions. Several authors have previously highlighted the use of λ to assess dynamic balance.11,13,35,36,46 Theoretical and experimental studies have highlighted the importance of the frontal plane in the regulation of dynamic stability, since gait is considered to be stable in the AP and V directions, and active control is required to stabilize the body in the ML direction.1

LDS decreased in the ML direction for the UP condition, but not for the DOWN condition. The increased step length and decreased step frequency in UP conditions lead to an increased gait cycle duration, and, consequently, to longer swing and stance phases.21 The longer single-leg stance phase, associated with increased swing-leg hip flexion and foot clearance,21 and, hence, more pronounced body weight transfer from one side to the other during walking up, may have contributed to the decreased ML LDS in the UP condition. However, further studies are necessary to confirm this hypothesis.

Furthermore, previous studies have established a relationship between energy expenditure and inclination during walking at slopes between −15 and 15%.29 The curve of energy expenditure as a function of incline has a shape very similar to that of ML LDS (Fig. 5a), in which −8% presented the smallest λ value and, consequently, the greatest LDS. Indeed, −8% of inclination is also very close to the optimal value for minimizing energy expenditure.29,30 The minimal cost at about −8% originates from the relationship between down acceleration and braking effect on the body COM when walking down.29

However, we found a more significant increase for λ in the AP and V directions. These results indicated that inclines substantially increase divergence rate in the AP and V directions, a situation that equally required active control to stabilize the body. Inclines impose several factors that must be negotiated by the motor control system, including requirements to raise or lower the body’s center of mass during the associated mechanical work, the vertical displacement during each stride, and the altered friction demands and foot clearance.27 There are short phases in early and late stance phases when friction demands are high due to high horizontal forces. The pelvic tilt, which appears to be related to trunk motion, increases, influencing hip flexion, maintaining forward momentum and positioning the body’s center of mass more anteriorly.21,27 All these aspects of walking on inclines occur in the sagittal plane and may explain the changes observed in AP and V LDS.

The values of λ altered significantly with speed. Higher speed produced lower λ and increased LDS in the ML and AP directions, but speed produced the opposite effect in the V direction: higher speed produced higher λ and decreased LDS. Our results were in accordance with those of previous studies of level walking at different speeds.17,18 The concordance of results also supported the idea that slower speeds, decreased step length, and increased step frequency in older adults were compensatory strategies used to maintain the dynamic stability of the upper body.5,10,14

Gait Regularity

In this study, the values of SEn indicated that movement regularity increased almost linearly from the DOWN to the UP conditions. The higher movement regularity observed in UP conditions can be explained by the gait pattern adopted, with larger step length and smaller step frequency. Although previous studies analyzing gait regularity using entropy are based on stride interval times series, trunk kinematics is intrinsically influenced by the lower limb movement,18 so that our conclusions may be valid. In addition, the SEn in AP direction in the present study (Fig. 6) has the same behavior to that reported by Costa et al.6 computed from stride interval times series, in which normal speed gait presented greater entropy than fast and slow speed gait.

Furthermore, although linear variability increased in the UP condition, the increase was less than the increase in the DOWN condition. Together, these results may have contributed to the decreased ML LDS observed in the UP conditions: smaller SEn and linear variability in the UP conditions compared to the DOWN conditions can be interpreted as a decreased capacity of the system to recover from perturbations.

SEn exhibited effects of age only in the AP direction, but, surprisingly, the older adults showed greater values than young adults. This increased randomness can be viewed as a loss of movement precision.44

Conclusion

Our results showed that, although older adults present several functional alterations widely reported in literature,8,20 they demonstrated similar performance to young adults when walking on inclines at different speeds related to stability, and nonlinear and linear features. The kinematic alterations noted in older adults, in the face of the observed results, were a compensatory strategy used to maintain dynamic stability of the upper body. In conclusion, aging per se does not necessarily imply impaired balance control during walking, which is a point of view support by some authors,43 and the building standards for inclines are appropriate. Such conclusions reinforced the importance of adopting an active life in old age, and the need for public policies specifically addressed to the elderly. These needs are particularly acute in developing countries, such as Brazil, whose population undergoes an accelerated aging process.