of the Ankle in Downhill Walking
Jonathan K. Holm[1], Sang-Wook Lee[2], John Jang[3], & Jonas Contakos[4]
The biomechanics literature is rich with data on level walking, but comparatively few studies have explored locomotion on slopes. This study investigated the energetics of the ankle during the stance phase of downhill walking and tested the hypothesis that the behavior of the ankle may be effectively modeled by passive mechanical components, i.e. a revolute spring and damper. Eight healthy, male, college-age participants were instructed to walk down an instrumented ramp at each of seven angles (0º,2º,3º,4º,5º,6º,8º). The planar ankle joint moment in the sagittal view was calculated using inverse dynamics methods. Mechanical energy gained or lost through the ankle joint was computed by integrating the ankle mechanical power across the duration of the stance phase. A linear function of joint angle and velocity was employed to model the behavior of the ankle during stance phase, and its parameters were estimated using linear regression.
Net energy at the ankle was approximately zero for level walking and decreased monotonically as downward slope angle increased, indicating the ankle’s role in absorbing energy on steeper walking slopes. Energy and power maxima during push-off remained relatively constant across the range of slopes. Measures of energy and power maxima during impact absorption and dorsiflexion became increasing negative with steeper slopes.
The passive ankle model exhibited relatively small discrepancy with the experimental data during dorsiflexion and larger discrepancy during impact absorption and push-off. Overall, the model-generated ankle moment demonstrated good agreement with experimental data as indicated by low root-mean-square (RMS) error values. RMS error demonstrated a local minimum between 2º and 4º, indicating the passive model was more accurate on those slopes than on level ground. All subjects showed a negative correlation between the spring parameter and slope angle and a positive correlation between the linear offset and slope angle. The damping parameter was observed to be negligible.
The energetics of human walking has long been an area of interest for biomechanists; more recently, the energy efficiency of human locomotion has drawn the interest of engineers designing bipedal robots and lower-limb prosthetics. Existing literature indicates the ankle plays a critical role in supplying much of the energy required for level walking [1, 2, 3], but exactly how the energetics of the ankle are affected by slope walking has not yet been fully explored. The primary purpose of this investigation was to study the energetics of the ankle during walking on a range of downhill slopes.
The second motivation for this study originated from recent developments in walking robots. Roboticists have demonstrated anthropomorphic slope walking in simple bipedal robots that require little or no energy input [4, 5, 6]. To date, however, ankle designs for these passive robots have been ad-hoc or inflexible. Recent work in biomechanics [7, 8] has suggested that some or all of the dynamics of the human ankle can be duplicated by a passive mechanism, i.e. a pin joint coupled with appropriately chosen revolute spring and damper. These results have inspired the secondary purpose of this investigation: to test the hypothesis that the human ankle can be effectively modeled as such a passive mechanism. Such a model will assist in the development of energy-efficient, biomimetic ankles for robots and below-knee human prostheses.
Ankle Energetics. The first aim of this investigation was to study the mechanical power and energy of the ankle during stance and observe how these measures are affected by walking slope. Energetics of the stance ankle play a key role in the overall energetics of locomotion. Winter [1] showed that the ankle of the stance leg contributes more energy to the forward motion of the body than do the knee and hip, concluding the ankle is the dominant source of the energy necessary for level walking. Gottshall and Kram [3] used electromyography (EMG) data and measured horizontal tugs to demonstrate the connection between activity of the ankle plantarflexors and forward motion of the body. Further EMG data reported by Radcliffe [2] and Stern [9] reveal the energy supplied via the ankle comes primarily through a burst of activity in the ankle plantarflexors (gastrocnemius and soleus) between heel-off (HO) and toe-off (TO), the push-off or propulsive part of stance phase [1, 7].
Studies of downhill walking report simultaneous decreases in both the overall energetic cost of walking and the propulsive energy supplied though the ankle. Margaria’s [10] investigation of O2 consumption, recently duplicated and confirmed by Minetti et al. [11], demonstrated that the metabolic cost of walking diminishes on small downhill slopes and is minimized on a downhill slope of about 5.7º. Mitsui et al. [12] reported downhill walking on slopes between 0º and 26.5º resulted in decreased EMG activity in gastrocnemius while the soleus showed relatively little change. The observed a minimum in gastrocnemius EMG activity occurred on slopes of about 5.7º, the same slope corresponding to minimum energetic cost of walking according to Margaria and Minetti. The dynamics study of Kuster et al. [7] reported that ankle power maxima between HO and TO have diminished magnitude for walking on downhill slopes of about 10.7º, indicating a decrease in energy supplied by the ankle during push-off.
With this investigation, we sought to examine the propulsive power and energy of the ankle on a range of downhill slopes. We hypothesized that the minimum ankle plantarflexor activity at 5.7º would correspond to a minimum in energy supplied by the ankle during push-off on the same slope.
Passive Ankle Dynamics. The second aim of this investigation was to study whether the dynamics of the ankle could be effectively modeled by an entirely passive joint, i.e. a pin joint coupled with a revolute spring and damper.
The passive dynamics of the ankle have been characterized in stationary conditions when the ankle is bearing no weight. Weiss et al. [13] modeled the ankle of reclined participants over the full range of motion of the ankle, and reported spring and damper coefficients that were relatively constant for mid-range motions. This may indicate that locomotion at self-selected speeds can be modeled with a spring and a damper that maintain constant coefficients throughout stance. Hansen et al. [14] investigated the dynamics of the ankle of the stance leg during level walking. At self-selected walking speeds, they reported no net gain or loss of energy through the ankle and concluded the overall stance phase dynamics of the ankle are energetically passive and could be modeled as a simple pin joint with a revolute spring and damper. Palmer [8] documented passive spring behavior during portions of the stance phase of level walking from heel-strike (HS) to foot-flat (FF) and from FF to HO. Palmer concluded that damper behavior is negligible and spring behavior dominates these two stages of stance. However, Palmer treated the dynamics of each stage separately, did not use a passive model for push-off, and did not consider slope walking.
We proposed a passive model for the ankle that maintains constant spring and damper coefficients throughout the stance phase and sought to explore how these parameters and the accuracy the model vary with slope. We hypothesized that such a model would hold for level walking but would lose accuracy on downhill slopes.
2.1 Human participants
Eight healthy men (age 22-27, mean mass 74.2±4.7 kg, mean height 178.5±4.7 cm) with no known walking impairments participated in this study. Participants were carefully selected to be of the same gender and similar body dimensions in order to minimize gait variation due to gender, age, mass, and stature. Each subject was informed of the experimental protocol and provided consent in compliance with the University of Illinois Institutional Review Board. Each participant wore rigid bicycling shoes (U.S. size 10.5, which was within half a size of the shoes normally worn by each participant) that restricted the motion of the foot to that of a single segment.
2.2 Experimental setup
A variable slope apparatus was constructed consisting of three walking surfaces joined by hinges (Fig. 1). All walking surfaces were painted with nonslip paint. An AMTI force plate (model 02172; Advanced Medical Technology Inc., Watertown, MA) was embedded in the center section as shown in the figure. The section of the walkway where participants began walking was mounted on four hydraulic jacks (model F-2365; Pro-lift, Kansas City, MO). As the jacks were raised, hinges between the sections allowed the starting and ending sections to remain horizontal while the center section assumed various angles (Fig. 1). Care was taken in the design and construction of the walkway to minimize deflection of the walking surfaces.
Fig 1: Ramp apparatus used to test participants.
Fig 2: Model of the foot and shank segments showing how ankle angle theta(t) was defined. The foot segment was modeled as a triangle defined by the LM marker and the CAL and TO1 marker locations projected to the bottom of the shoe. The ankle angle theta(t) was defined as the angle between the shank segment and the line perpendicular to the bottom of the foot segment.
Six infrared cameras (model 460; Vicon Motion Systems Ltd., Oxford, UK) were placed around the walkway to collect kinematic data. Kinematic and force data were both sampled at 100 Hz. Markers were placed over the acromion, sacrum, and bilaterally over the anterior-superior iliac spine, lateral epicondyle of the femur (LEF), and lateral malleolus (LM). Four additional markers were placed on each shoe over the calcaneous (CAL) and 1st metatarsal of each foot (TO1) as shown in (Fig. 2).
2.3 Experimental protocol
Participants were allowed to walk around the lab until they felt comfortable walking in the rigid-soled shoes. For each slope angle, the participants were instructed to walk the length of the walkway at a self-selected pace. No mention was made of the force plate in the ramp section of the walkway, and participants were instructed to look straight ahead in order to prevent them from biasing their gait based on its location. A successful trial consisted of heel-strike (HS) and toe-off (TO) of either foot completely within the boundaries of the force plate without the contralateral foot contacting the force plate. If a subject’s foot placement did not yield a step entirely on the force plate, experimenters adjusted the subject’s starting position until a successful trial was achieved. The data from unsuccessful trials were discarded. Participants completed a minimum of two successful trials at each angle of ramp inclination and were provided breaks while the ramp angle was adjusted.
The angles of the ramp used in this study were selected based on the range of angles tested in previous energetics studies of human slope walking and slope walking by bipedal robots whose gait dynamics and energy requirements closely mimic that of humans. Margaria [10] and Minetti et al. [11] showed the energetic cost of human walking is minimized on downhill slopes of about 5º-6º; Misui et al. [12] reported minimal ankle plantarflexor EMG activity on slopes in the same range. In bipedal robotics, Goswami et al. [5] and Yamakita & Asano [15] demonstrated minimal energy requirement for human-like gait on slopes of 2º-4º. Therefore, the set of test angles was selected with the most resolution in a range between 2º and 6º. In particular, the participants walked on the ramp with angles of 0º, 2º, 3º, 4º, 5º, 6º, and 8º (approximately the legal maximum angle of a wheelchair ramp [16]) in that order. The order was chosen to allow the participants to acclimate to rigid-foot walking on gentle slopes first before tackling the steeper slopes.
2.4 Data analysis
Kinematic data were conditioned using a 2nd-order zero-phase Butterworth lowpass filter with 10 Hz cutoff using MATLAB (version 6.5; The MathWorks, Natick, MA). The marker locations were projected onto the sagittal plane for analysis (Fig. 2). Sagittal analysis was deemed sufficient since 93% of the work done at the ankle during walking occurs in this plane [17]. The ankle angle theta(t) was defined as shown in (Fig. 2), in agreement with [18]. The angular velocity thetadot(t) was computed using a numeric approximation of the derivative, the change in angle theta(t) divided by time between frames. Two-way analysis of variance (ANOVA) was performed to evaluate the statistical significance of participant difference and slope variation on all measures of power, energy, and passive model parameters.
For each subject and angle, measures from the two successful trials were averaged. Exceptions were made in two instances in which one of the successful trials for a given angle produced power and moment profiles starkly inconsistent with other data; in these instances, only data from the error-free trial were considered. These two exceptions occurred in different participants and on different angles, suggesting the errors were not correlated and were therefore not repeated.
Ankle Energetics. The ankle moment tau(t) was computed using the measured ground reaction force, center of pressure, and the motion data in a bottom-up inverse dynamics procedure. Mechanical power at the ankle was calculated using the formula P(t) = tau(t)thetadot(t). To compute the energy supplied or absorbed via the ankle joint, we integrated the mechanical power with respect to time
where limits of integration t0, t1 were chosen by specific gait phases. For example, to compute the energy during the propulsion, we integrated from HO to toe-off TO; to compute the total energy during stance, we integrated from heel-strike HS to TO.
Passive Ankle Dynamics. To test our hypothesis that the behavior of a spring and damper effectively models the dynamics of the stance ankle, the following model of the ankle torque due was used
where kp and kd represent the unknown spring and damper coefficients respectively. In the equation above, c is a constant offset term that accounts for the angle of spring relaxation as well as minor offsets in marker placement that affect the calculation of the angle theta. These parameters were estimated by fitting the model-generated ankle torque profile tau_sd(t) to the actual profile tau(t) using linear regression.
Energetics. A representative plot of ankle power against time during the stance phase is provided
(Fig. 3). We identified four power maxima (identified by numerals I-IV) during stance phase, averaged across participants for each angle, and plotted these against ramp angle (Fig. 4). The greatest variation occurred in maximum I, the power minimum during impact absorption, which ranged from -56.8W on level ground to -148.9W on 8º. Slope significantly affected I (p < 0.0001), as well as III (p < 0.001), the local minimum during dorsiflexion, and IV (p < 0.01), the maximum during push-off.
Ankle power was integrated across the entire stance phase (HS to TO) to compute the net mechanical energy produced or absorbed at the ankle (Fig. 5). Net energy was closest to zero on level ground, indicating very little net gain or loss of energy through the ankle for level walking, consistent with the analysis of [14]. Ramp angle significantly affected net ankle energy (p < 0.00001), resulting in a monotonic decrease in net energy as ramp angle increased.
Fig 3: Representative plot of ankle power versus time during the stance phase (subject AM walking on 5º slope). The stance phase is divided into three stages: impact absorption (plantarflexion) from HS to FF, dorsiflexion from FF to HO, and propulsion (plantarflexion) from HO to TO. Various power maxima are identified with numerals: I, minimum during impact absorption; II and III, local maximum and local minimum, respectively, during dorsiflexion; and IV, maximum during propulsion.
Fig 4: Trends in various ankle power maxima (mean ±1 standard deviation) as ramp angle varies.
Fig 5: Total mechanical energy of the ankle during stance versus ramp angle.
Fig 6: Trends in various measures of ankle energy as ramp angle varies: energy during impact absorption (from HS to FF), energy during dorsiflexion (FF to HO), and energy during propulsion (HO to TO).
Integrating power from HS to FF (impact absorption), FF to HO (dorsiflexion), and HO to TO
(push-off), we computed energy during these different stages of stance and observed trends as ramp angle varied (Fig. 6). The greatest variation occurred in dorsiflexion energy, which ranged from -9.03J on level ground to -19.8J on 8º. Slope significantly affected dorsiflexion energy (p < 0.0001) as well as impact absorption energy (p < 0.001). Push-off energy remained relatively constant and was not significantly affected by slope.
Passive Ankle Dynamics. A representative plot of ankle torque versus time during the stance phase is shown (Fig. 7). Using linear regression, the passive model was fit to this data, and the model torque has been overlaid for comparison. To judge the efficacy of the passive spring-damper model we computed the root-mean-square (RMS) error during impact absorption, dorsiflexion, push-off, as well as across all of stance and compared these error measures with the RMS ankle torque of the human participants (Fig. 8). Ramp angle significantly affected the RMS error of the model (p < 0.01).
Evolution of the passive parameters as slope varied revealed monotonic trends in all parameters
(Fig. 9). The spring coefficient kp decreased as ramp angle increased while the damping coefficient kd and linear offset c increased with increasing ramp angle. Ramp angle significantly affected all model parameters (p < 0.00001).
Fig 7: (a) Representative plot of ankle torque versus time during the stance phase (participant AM walking on 5º slope). The torque profile from the spring-damper model is overlaid for comparison and error residual is shown in the lower plot. (b) Residual error versus time during the stance phase, illustrating the error is greatest (model is least accurate) during absorption and propulsion stages (i.e., from HS to FF and from HO to TO).
Fig 8: RMS error of the passive ankle model versus ramp angle: (a) RMS error for various stages during stance phase, (b) overall RMS error for stance phase compared with RMS ankle torque value.
Fig 9: Parameters of the passive model versus ramp angle: (a) spring kp (b) damper kd (c) linear offset c.
Energetics. The ankle power profiles for all participants were similar to each other and were in general agreement with other studies [1, 7, 17]. The monotonic decrease in net ankle energy suggests the ankle is responsible for dissipating energy during slope walking. The energy being removed from the gait is dissipated during the early stages of stance, as evidenced by decreases in both energy and power maxima during impact absorption (HS to FF) and dorsiflexion (FF to HO).
The results disproved our first hypothesis that the energy supplied by the ankle during push-off would demonstrate a local minimum on a slope of about 5.7º. Instead, our data showed a local minimum in dorsiflexion energy at 6º while push-off energy did not vary significantly with changes in slope. This suggests that the reduction in plantarflexor activity that results in minimum EMG activity on 5.7º may occur during the earlier stages in stance, while the activity during push-off may remain constant. We note here that Mitsui et al. [12] observed diminished EMG activity at 5.7º only in the gastrocnemius, while the activity of the soleus remained relatively constant. Perhaps the constant activity of the soleus is responsible for the steady supply of energy during push-off, while the work of the gastrocnemius is seen in other parts of stance and accounts for the minimum in EMG activity on 5.7º.
Passive Ankle Dynamics. The ankle torque profiles were relatively consistent among the eight participants and in general agreement with related studies [1, 7, 17, 18]. Error between the model and experimental torques was greatest during the absorption and propulsion stages for all but two of the angles tested. RMS error during dorsiflexion was less than that of absorption for all angles tested and less than that of propulsion for all angles except 0ºand 2º. Overall stance phase RMS error remained low for all angles, bounded below 14Nm and never surpassing 25% of the RMS torque value.
Our second hypothesis predicted the passive model would be effective on level ground but would
lose accuracy on downhill slopes. On the basis of the relatively low RMS error values we accept the first part of the hypothesis, that the passive model was effective on level ground. This conclusion confirms those of Hansen et al. [14]. However, we reject the second part of the hypothesis, that the passive model would lose accuracy on downhill slopes. Measures of RMS error during individual stages of stance as well as overall RMS error demonstrated a local minimum between 2º-4º, indicating the passive model was actually more accurate on those slopes than on level ground.
All parameters of the passive model varied significantly with ground slope, indicating that a single spring and damper ankle will be insufficient for effectively duplicating the behavior of the ankle on a variety of walking slopes. Linear offset parameter c was nonzero on all slopes, suggesting our specification of the line normal to the bottom of the foot as the angle of spring relaxation is incorrect. Moreover, the variation of the parameter c as slope changed implies the angle of spring relaxation changes with walking slope.
Values of spring parameter kp were one order of magnitude greater than spring coefficients reported by Weiss et al. [13] for mid-range motions when the ankle is bearing no weight but only slightly larger than the coefficients reported by Palmer [8] for the impact absorption phase alone of level walking at self-selected speed. Values of the damping parameter kd in our model were more than one order of magnitude smaller than values of the spring parameter, implying that damping at the ankle is negligible for walking at self-selected speed on the range of angles tested. This is consistent with the conclusions of other studies [8, 13, 14].
Limitations. In spite of the imposed restrictions on the mass, stature, shoe size, and gender of study participants, differences between participants significantly influenced all measures of power, energy, and passive model parameters (p < 0.001). This suggests that the energetics and passive dynamics of the ankle are particular to individual morphological characteristics other than the four we controlled.
The linear passive model we proposed and analyzed is only one of many possible passive models that could be tested. Indeed, there are a number of nonlinear passive models that could also be proposed and studied. We note that the effectiveness of our passive model in mimicking the behavior of the human ankle does not indicate how other, nonlinear passive models may behave.
The authors thank the faculty members of the interdisciplinary locomotion seminar at UIUC for providing invaluable instruction, advice, and criticism while this investigation was conducted. In particular, we are indebted to Karl S. Rosengren, Elizabeth T. Hsiao-Wecksler, and John D. Polk. We also thank the team of graduate students who assisted and contributed to this study: Richard Doyle, Eric Dudley, Timothy Filipiak, Kelly McHugh, Arun Ramachandran, and David Lim.
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