Dynamic Stability of Passive Bipedal Walking on Rough Terrain:A Preliminary Simulation Study

A simplified 2D passive dynamic model was simulated to walk down on a rough slope surface defined by deterministic profiles to investigate how the walking stability changes with increasing surface roughness.Our results show that the passive walker can walk on rough surfaces subject to surface roughness up to approximately 0.1％ of its leg length.This indicates that bipedal walkers based on passive dynamics may possess some intrinsic stability to adapt to rough terrains although the maximum roughness they can tolerate is small.Orbital stability method was used to quantify the walking stability before the walker started to fall over.It was found that the average maximum Floquet multiplier increases with surface roughness in a non-linear form.Although the passive walker remained orbitally stable for all the simulation cases,the results suggest that the possibility of the bipedal model moving away from its limit cycle increases with the surface roughness if subjected to additional perturbations.The number of consecutive steps before falling was used to measure the walking stability after the passive walker started to fall over.The results show that the number of steps before falling decreases exponentially with the increase in surface roughness.When the roughness magnitude approached to 0.73％ of the walker's leg length,it fell down to the ground as soon as it entered into the uneven terrain.It was also found that shifting the phase angle of the surface profile has apparent affect on the system stability.This is probably because point contact was used to simulate the heel strikes and the resulted variations in system states at heel strikes may have pronounced impact on the passive gaits,which have narrow basins of attraction.These results would provide insight into how the dynamic stability of passive bipedal walkers evolves with increasing surface roughness.     

Virtual slope control of a forward dynamic bipedal walker

Active joint torques are the primary source of power and control in dynamic walking motion. However the amplitude, rate, timing and phasic behavior of the joint torques necessary to achieve a natural and stable performance are difficult to establish. The goal of this study was to demonstrate the feasibility and stable behavior of an actively controlled bipedal walking simulation wherein the natural system dynamics were preserved by an active, nonlinear, state-feedback controller patterned after passive downhill walking. A two degree-of-freedom, forward-dynamic simulation was implemented with active joint torques applied at the hip joints and stance leg ankle. Kinematic trajectories produced by the active walker were similar to passive dynamic walking with active joint torques influenced by prescribed walking velocity. The control resulted in stable steady-state gait patterns, i.e. eigenvalue magnitudes of the stride function were less than one. The controller coefficient analogous to the virtual slope was modified to successfully control average walking velocity. Furture developments are necessary to expand the range of walking velocities.     

The validity of stability measures: A modelling approach

Measures calculated from unperturbed walking patterns, such as variability measures and maximum Floquet multipliers, are often used to study the stability of walking. However, it is unknown if, and to what extent, these measures correlate to the probability of falling.We studied whether in a simple model of human walking, i.e., a passive dynamic walker, the probability of falling could be predicted from maximum Floquet multipliers, kinematic state variability, and step time variability. We used an extended version of the basic passive dynamic walker with arced feet and a hip spring. The probability of falling was manipulated by varying the foot radius and hip spring stiffness, or varying these factors while co-varying the slope to keep step length constant.The simulation data indicated that Floquet multipliers and kinematic state variability correlated inconsistently with probability of falling. Step time variability correlated well with probability of falling, but a more consistent correlation with the probability of falling was found by calculating the variability of the log transform of the step time. Our findings speak against the use of maximum Floquet multipliers and suggest instead that variability of critical variables may be a good predictor of the probability to fall.     

Effects of walking speed, strength and range of motion on gait stability in healthy older adults

Falls pose a tremendous risk to those over 65 and most falls occur during locomotion. Older adults commonly walk slower, which many believe helps improve walking stability. While increased gait variability predicts future fall risk, increased variability is also caused by walking slower. Thus, we need to better understand how differences in age and walking speed independently affect dynamic stability during walking. We investigated if older adults improved their dynamic stability by walking slower, and how leg strength and flexibility (passive range of motion (ROM)) affected this relationship. Eighteen active healthy older and 17 healthy younger adults walked on a treadmill for 5min each at each of 5 speeds (80-120% of preferred). Local divergence exponents and maximum Floquet multipliers (FM) were calculated to quantify each subject's inherent local dynamic stability. The older subjects walked with the same preferred walking speeds as the younger subjects (p=0.860). However, these older adults still exhibited greater local divergence exponents (p<0.0001) and higher maximum FM (p<0.007) than the younger adults at all walking speeds. These older adults remained more locally unstable (p<0.04) even after adjusting for declines in both strength and ROM. In both age groups, local divergence exponents decreased at slower speeds and increased at faster speeds (p<0.0001). Maximum FM showed similar changes with speed (p<0.02). Both younger and older adults exhibited decreased instability by walking slower, in spite of increased variability. These increases in dynamic instability might be more sensitive indicators of future fall risk than changes in gait variability.     

New assistive walker improved local dynamic stability in young healthy adults

In this study, we investigated the effect of walker type on gait pattern characteristics comparing normal gait (NG), gait with a regular walker (RW), and gait with a newly developed walker with vertical moveable handlebars, the Crosswalker (CW).Partial weight bearing (PWB) of the feet, peak joint angles and largest Lyapunov exponent (λ_max) of the lower extremities (hip, knee, ankle) in the sagittal plane, and gait parameters (gait velocity, stride length, cadence, stride duration) were determined for 18 healthy young adults performing 10 walking trials for each walking condition. Assistive gait with the CW improved local dynamic stability in the lower extremities (hip, knee, ankle) compared with RW and was not significantly different from NG. However, peak joint angles and stride characteristics in CW were different from NG. The PWB on the feet was lower with the RW (70.3%) compared to NG (82.8%) and CW (80.9%). This improved stability may be beneficial for the elderly and patients with impaired gait. However, increased PWB is not beneficial for patients during the early stages of rehabilitation.     

Optimal foot shape for a passive dynamic biped

Passive walking dynamics describe the motion of a biped that is able to "walk" down a shallow slope without any actuation or control. Instead, the walker relies on gravitational and inertial effects to propel itself forward, exhibiting a gait quite similar to that of humans. These purely passive models depend on potential energy to overcome the energy lost when the foot impacts the ground. Previous research has demonstrated that energy loss at heel-strike can vary widely for a given speed, depending on the nature of the collision. The point of foot contact with the ground (relative to the hip) can have a significant effect: semi-circular (round) feet soften the impact, resulting in much smaller losses than point-foot walkers. Collisional losses are also lower if a single impulse is broken up into a series of smaller impulses that gradually redirect the velocity of the center of mass rather than a single abrupt impulse. Using this principle, a model was created where foot-strike occurs over two impulses, "heel-strike" and "toe-strike," representative of the initial impact of the heel and the following impact as the ball of the foot strikes the ground. Having two collisions with the flat-foot model did improve efficiency over the point-foot model. Representation of the flat-foot walker as a rimless wheel helped to explain the optimal flat-foot shape, driven by symmetry of the virtual spoke angles. The optimal long period foot shape of the simple passive walking model was not very representative of the human foot shape, although a reasonably anthropometric foot shape was predicted by the short period solution.     

Gait modification when decreasing double support percentage

Much is still unknown about walking stability, including which aspects of gait contribute to higher stability. Walking stability appears to be related to walking speed, although the exact relationship is unclear. As walking speed decreases, the double support (DS) period of gait increases both in time and as a percentage of the gait cycle. Because humans have more control over their center of mass movement during DS, increasing DS duration may alter stability. This study examined how human gait is affected by changing DS percentage independent of walking speed. Sixteen young, healthy adults walked on a treadmill at a single speed for six one-minute trials. These trials included normal gait as well as longer- and shorter-than-normal DS percentage gaits. Subjects were consistently able to decrease DS percentage but had difficulty increasing DS percentage. In some cases, subjects altered their cadence when changing DS percentage, particularly when attempting to increase DS percentage. The changes to gait when decreasing DS percentage were similar to changes when increasing walking speed but occurred mainly during the swing period. These changes include increased hip and knee flexion during the swing period, increased swing foot height, and larger magnitude peaks in ground reaction forces. The changes in gait when attempting to increase DS percentage trended toward changes when decreasing walking speed. Altering DS percentage induced gait changes that were similar to, yet clearly distinct from, gait changes due to walking speed. Further, the difficulty of increasing DS percentage when walking at a constant speed suggests that people walk more slowly when they want to increase time spent in DS.     

Contributions of muscles and passive dynamics to swing initiation over a range of walking speeds

Stiff-knee gait is a common walking problem in cerebral palsy characterized by insufficient knee flexion during swing. To identify factors that may limit knee flexion in swing, it is necessary to understand how unimpaired subjects successfully coordinate muscles and passive dynamics (gravity and velocity-related forces) to accelerate the knee into flexion during double support, a critical phase just prior to swing that establishes the conditions for achieving sufficient knee flexion during swing. It is also necessary to understand how contributions to swing initiation change with walking speed, since patients with stiff-knee gait often walk slowly. We analyzed muscle-driven dynamic simulations of eight unimpaired subjects walking at four speeds to quantify the contributions of muscles, gravity, and velocity-related forces (i.e. Coriolis and centrifugal forces) to preswing knee flexion acceleration during double support at each speed. Analysis of the simulations revealed contributions from muscles and passive dynamics varied systematically with walking speed. Preswing knee flexion acceleration was achieved primarily by hip flexor muscles on the preswing leg with assistance from biceps femoris short head. Hip flexors on the preswing leg were primarily responsible for the increase in preswing knee flexion acceleration during double support with faster walking speed. The hip extensors and abductors on the contralateral leg and velocity-related forces opposed preswing knee flexion acceleration during double support.     

Energetics of actively powered locomotion using the simplest walking model

We modified an irreducibly simple model of passive dynamic walking to walk on level ground, and used it to study the energetics of walking and the preferred relationship between speed and step length in humans. Powered walking was explored using an impulse applied at toe-off immediately before heel strike, and a torque applied on the stance leg. Although both methods can supply energy through mechanical work on the center of mass, the toe-off impulse is four times less costly because it decreases the collision loss at heel strike. We also studied the use of a hip torque on the swing leg that tunes its frequency but adds no propulsive energy to gait. This spring-like actuation can further reduce the collision loss at heel strike, improving walking energetics. An idealized model yields a set of simple power laws relating the toe-off impulses and effective spring constant to the speed and step length of the corresponding gait. Simulations incorporating nonlinear equations of motion and more realistic inertial parameters show that these power laws apply to more complex models as well.     

Stable bipedal walking with a swing-leg protraction strategy

In bipedal locomotion, swing-leg protraction and retraction refer to the forward and backward motion, respectively, of the swing-leg before touchdown. Past studies have shown that swing-leg retraction strategy can lead to stable walking. We show that swing-leg protraction can also lead to stable walking. We use a simple 2D model of passive dynamic walking but with the addition of an actuator between the legs. We use the actuator to do full correction of the disturbance in a single step (a one-step dead-beat control). Specifically, for a given limit cycle we perturb the velocity at mid-stance. Then, we determine the foot placement strategy that allows the walker to return to the limit cycle in a single step. For a given limit cycle, we find that there is swing-leg protraction at shallow slopes and swing-leg retraction at steep slopes. As the limit cycle speed increases, the swing-leg protraction region increases. On close examination, we observe that the choice of swing-leg strategy is based on two opposing effects that determine the time from mid-stance to touchdown: the walker speed at mid-stance and the adjustment in the step length for one-step dead-beat control. When the walker speed dominates, the swing-leg retracts but when the step length dominates, the swing-leg protracts. This result suggests that swing-leg strategy for stable walking depends on the model parameters, the terrain, and the stability measure used for control. This novel finding has a clear implication in the development of controllers for robots, exoskeletons, and prosthetics and to understand stability in human gaits.     

Energy efficient walking with central pattern generators: from passive dynamic walking to biologically inspired control

Like human walking, passive dynamic walking—i.e. walking down a slope with no actuation except gravity—is energy efficient by exploiting the natural dynamics. In the animal world, neural oscillators termed central pattern generators (CPGs) provide the basic rhythm for muscular activity in locomotion. We present a CPG model, which automatically tunes into the resonance frequency of the passive dynamics of a bipedal walker, i.e. the CPG model exhibits resonance tuning behavior. Each leg is coupled to its own CPG, controlling the hip moment of force. Resonance tuning above the endogenous frequency of the CPG—i.e. the CPG’s eigenfrequency—is achieved by feedback of both limb angles to their corresponding CPG, while integration of the limb angles provides resonance tuning at and below the endogenous frequency of the CPG. Feedback of the angular velocity of both limbs to their corresponding CPG compensates for the time delay in the loop coupling each limb to its CPG. The resonance tuning behavior of the CPG model allows the gait velocity to be controlled by a single parameter, while retaining the energy efficiency of passive dynamic walking.     

Kinematic variability and local dynamic stability of upper body motions when walking at different speeds

A ubiquitous characteristic of elderly and patients with gait disabilities is that they walk slower than healthy controls. Many clinicians assume these patients walk slower to improve their stability, just as healthy people slow down when walking across ice. However, walking slower also leads to greater variability, which is often assumed to imply deteriorated stability. If this were true, then slowing down would be completely antithetical to the goal of maintaining stability. This study sought to resolve this paradox by directly quantifying the sensitivity of the locomotor system to local perturbations that are manifested as natural kinematic variability. Eleven young healthy subjects walked on a motorized treadmill at five different speeds. Three-dimensional movements of a single marker placed over the first thoracic vertebra were recorded during continuous walking. Mean stride-to-stride standard deviations and maximum finite-time Lyapunov exponents were computed for each time series to quantify the variability and local dynamic stability, respectively, of these movements. Quadratic regression analyses of the dependent measures vs. walking speed revealed highly significant U shaped trends for all three mean standard deviations, but highly significant linear trends, with significant or nearly significant quadratic terms, for five of the six finite-time Lyapunov exponents. Subjects exhibited consistently better local dynamic stability at slower speeds for these five measures. These results support the clinically based intuition that people who are at increased risk of falling walk slower to improve their stability, even at the cost of increased variability.     

A passive dynamic walking robot that has a deterministic nonlinear gait

There is a growing body of evidence that the step-to-step variations present in human walking are related to the biomechanics of the locomotive system. However, we still have limited understanding of what biomechanical variables influence the observed nonlinear gait variations. It is necessary to develop reliable models that closely resemble the nonlinear gait dynamics in order to advance our knowledge in this scientific field. Previously, Goswami et al. [1998. A study of the passive gait of a compass-like biped robot: symmetry and chaos. International Journal of Robotic Research 17(12)] and Garcia et al. [1998. The simplest walking model: stability, complexity, and scaling. Journal of Biomechanical Engineering 120(2), 281-288] have demonstrated that passive dynamic walking computer models can exhibit a cascade of bifurcations in their gait pattern that lead to a deterministic nonlinear gait pattern. These computer models suggest that the intrinsic mechanical dynamics may be at least partially responsible for the deterministic nonlinear gait pattern; however, this has not been shown for a physical walking robot. Here we use the largest Laypunov exponent and a surrogation analysis method to confirm and extend Garcia et al.'s and Goswami et al.'s original results to a physical passive dynamic walking robot. Experimental outcomes from our walking robot further support the notion that the deterministic nonlinear step-to-step variations present in gait may be partly governed by the intrinsic mechanical dynamics of the locomotive system. Furthermore the nonlinear analysis techniques used in this investigation offer novel methods for quantifying the nature of the step-to-step variations found in human and robotic gait.     

The effect of uphill and downhill walking on gait parameters: A self-paced treadmill study

It has been shown that gait parameters vary systematically with the slope of the surface when walking uphill (UH) or downhill (DH) (Andriacchi et al., 1977; Crowe et al., 1996; Kawamura et al., 1991; Kirtley et al., 1985; McIntosh et al., 2006; Sun et al., 1996). However, gait trials performed on inclined surfaces have been subject to certain technical limitations including using fixed speed treadmills (TMs) or, alternatively, sampling only a few gait cycles on inclined ramps. Further, prior work has not analyzed upper body kinematics. This study aims to investigate effects of slope on gait parameters using a self-paced TM (SPTM) which facilitates more natural walking, including measuring upper body kinematics and gait coordination parameters.Gait of 11 young healthy participants was sampled during walking in steady state speed. Measurements were made at slopes of +10°, 0° and −10°. Force plates and a motion capture system were used to reconstruct twenty spatiotemporal gait parameters. For validation, previously described parameters were compared with the literature, and novel parameters measuring upper body kinematics and bilateral gait coordination were also analyzed.Results showed that most lower and upper body gait parameters were affected by walking slope angle. Specifically, UH walking had a higher impact on gait kinematics than DH walking. However, gait coordination parameters were not affected by walking slope, suggesting that gait asymmetry, left-right coordination and gait variability are robust characteristics of walking. The findings of the study are discussed in reference to a potential combined effect of slope and gait speed. Follow-up studies are needed to explore the relative effects of each of these factors.     

Dynamic gait stability of treadmill versus overground walking in young adults

Treadmill has been broadly used in laboratory and rehabilitation settings for the purpose of facilitating human locomotion analysis and gait training. The objective of this study was to determine whether dynamic gait stability differs or resembles between the two walking conditions (overground vs. treadmill) among young adults. Fifty-four healthy young adults (age: 23.9 ± 4.7 years) participated in this study. Each participant completed five trials of overground walking followed by five trials of treadmill walking at a self-selected speed while their full body kinematics were gathered by a motion capture system. The spatiotemporal gait parameters and dynamic gait stability were compared between the two walking conditions. The results revealed that participants adopted a “cautious gait” on the treadmill compared with over ground in response to the possible inherent challenges to balance imposed by treadmill walking. The cautious gait, which was achieved by walking slower with a shorter step length, less backward leaning trunk, shortened single stance phase, prolonged double stance phase, and more flatfoot landing, ensures the comparable dynamic stability between the two walking conditions. This study could provide insightful information about dynamic gait stability control during treadmill ambulation in young adults.     

Slower speeds in patients with diabetic neuropathy lead to improved local dynamic stability of continuous overground walking

Patients with diabetic peripheral neuropathy are significantly more likely to fall while walking than subjects with intact sensation. While it has been suggested that these patients walk slower to improve locomotor stability, slower speeds are also associated with increased locomotor variability, and increased variability has traditionally been equated with loss of stability. If the latter were true, this would suggest that slowing down, as a locomotor control strategy, should be completely antithetical to the goal of maintaining stability. The present study resolves these seemingly paradoxical findings by using methods from nonlinear time series analysis to directly quantify the sensitivity of the locomotor system to local perturbations that are manifested as natural kinematic variability. Fourteen patients with severe peripheral neuropathy and 12 gender-, age-, height-, and weight-matched non-diabetic controls participated. Sagittal plane angles of the right hip, knee, and ankle joints and tri-axial accelerations of the trunk were measured during 10 min of continuous overground walking at self-selected speeds. Maximum finite-time Lyapunov exponents were computed for each time series to quantify the local dynamic stability of these movements. Neuropathic patients exhibited slower walking speeds and better local dynamic stability of upper body movements in the horizontal plane than did control subjects. The differences in local dynamic stability were significantly predicted by differences in walking speed, but not by differences in sensory status. These results support the hypothesis that reductions in walking speed are a compensatory strategy used by neuropathic patients to maintain dynamic stability of the upper body during level walking.     

Behavioural and physical reactions of the Colorado potato beetle, Leptinotarsa decemlineata (Say) (Coleoptera: Chrysomelidae) walking on a slanted surface

 A natural reaction of the adult Colorado potato beetle [Leptinotarsa decemlineata (Say)] is to walk uphill on sloped surfaces. The geotaxis reaction of the beetle was observed on slopes of 20° or steeper. It was demonstrated that the uphill orientation behaviour was not a consequence of physical limitation for across-slope locomotion. The walking speed of insects deviating from the fall line did not change within the range of slope angles tested. The speed of adult beetles decreased with an increase in the slope of the substrate as a reaction to the increased gravitational force vector opposing uphill movement. The larger size of the hind legs might make uphill locomotion more efficient than traversing a sloped surface. As the angle of the slope increased, the gait changed from a 3/3 to a 5/1, as did the posterior and anterior extreme position of the legs. This behaviour might be triggered by the need to maintain balance on slanted surfaces as the vertical projection of the centre of mass on the substrate moved outside the support base pattern at the steeper angles. In one experiment beetles were made to pull a load when walking over a horizontal surface. The loads pulled were equivalent to the gravitational loads opposing forward motion when walking up a slope. No differences in forward speed or gait were observed at the lower-angle equivalent compared to beetles walking on slopes. Differences in speed were noted at slope angles higher than 40° indicating that adaptation of the walking strategy might be needed on steeply slanted surfaces. Received: 2 May 2000 / Accepted in revised form: 11 September 2000     

Complexity,symmetry and variability of forward and backward walking at different speeds and transfer effects on forward walking: Implications for neural control

This study aimed to investigate effects of walking direction and speed on gait complexity, symmetry and variability as indicators of neural control mechanisms, and if a period of backward walking has acute effects on forward walking. Twenty-two young adults attended 2 visits. In each visit participants walked forwards at preferred walking speed (PWS) for 3-minutes (pre) followed by 5-minutes walking each at 80%, 100% and 120% of PWS of either forward or backward walking then a further 3-minutes walking forward at PWS (post). The order of walking speed in each visit was randomised and walking direction of each visit was randomised. An inertial measurement unit was placed over L5 vertebra to record tri-axial accelerations. From the trunk accelerations multiscale entropy, harmonic ratio and stride time variability were calculated to measure complexity, symmetry and variability for each walk. Complexity increased with increasing walking speed for all axes in forward and backward walking, and backward walking was less complex than forward walking. Stride time variability was also greater in backward than forward walking. Anterio-posterior and medio-lateral complexity increased following forward and backward walking but there was no difference between forward and backward walking post effects. No effects were found for harmonic ratio. These results suggest during backward walking trunk motion is rigidly controlled but central pattern generators responsible for temporal gait patterns are less refined for backward walking. However, in both directions complexity increased as speed increased suggesting additional constraint of trunk motion, normally characterised by reduced complexity, is not applied as speed increases.     

Influence of walking speed on gait parameters

Modern three-dimensional gait analysis systems give information on joint angles and moments in the sagittal and coronal planes, for which normal ranges may not be readily available in the literature. Since patients with joint disease tend to walk slowly and with a short stride, it is essential that normal ranges for gait parameters should be defined with reference to speed of walking. This we have done using a population of 10 normal male subjects agea from 18 to 63 years, walking at speeds which range from very slow to very fast. The ranges of knee angle and moment are given, together with the changes in these parameters with walking speed. Peak knee flexion moment is strongly related to walking speed, whereas coronal plane knee angle is virtually independent of it. The stride length is probably the best basis for deciding the normal range for a particular measurement.     

Trot Gait Design and CPG Method for a Quadruped Robot

Developing efficient walking gaits for quadruped robots has intrigued investigators for years. Trot gait, as a fast locomotion gait, has been widely used in robot control. This paper follows the idea of the six determinants of gait and designs a trot gait for a parallel-leg quadruped robot, Baby Elephant. The walking period and step length are set as constants to maintain a relatively fast speed while changing different foot trajectories to test walking quality. Experiments show that kicking leg back improves body stability. Then, a steady and smooth trot gait is designed. Furthermore, inspired by Central Pattern Generators （CPG）, a series CPG model is proposed to achieve robust and dynamic trot gait. It is generally believed that CPG is capable of producing rhythmic movements, such as swimming, walking, and flying, even when isolated from brain and sensory inputs. The proposed CPG model, inspired by the series concept, can automatically learn the previous well-designed trot gait and reproduce it, and has the ability to change its walking frequency online as well. Experiments are done in real world to verify this method.