Repetitive gait of passive bipedal mechanisms in a three-dimensional environment

The inherent dynamics of bipedal, kneed mechanisms are studied with emphasis on the existence and stability, of repetitive gait in a three-dimensional environment, in the absence of external, active control. The investigation is motivated by observations that sustained anthropomorphic locomotion is largely a consequence of geometric and inertial properties of the mechanism. While the modeling excludes active control, the energy dissipated in ground and knee collisions is continuously re-injected by considering gait down slight inclines. The paper describes the dependence of the resulting passive gait in vertically constrained and unconstrained mechanisms on model parameters, such as ground compliance and ground slope. We also show the possibility of achieving statically unstable gait with appropriate parameter choices.     

Limb morphology,bipedal gait,and the energetics of hominid locomotion

How viable is the argument that increased locomotor efficiency was an important agent in the origin of hominid bipedalism? This study reviews data from the literature on the cost of human bipedal walking and running and compares it to data on quadrupedal mammals including several non-human primate species. Literature data comparing the cost of bipedal and quadrupedal locomotion in trained capuchin monkeys and chimpanzees are also considered. It is concluded that increased energetic efficiency would not have accrued to early bipeds. Presumably, however, selection for improved efficiency in the bipedal stance would have occurred once the transition was made. Would such a process have included selection for increased limb length? Data on the cost of locomotion vs. limb length reveal no significant relationship between these variables in 21 species of mammals or in human walking or running. © 1996 Wiley-Liss, Inc.     

A bipedal compliant walking model generates periodic gait cycles with realistic swing dynamics

A simple spring mechanics model can capture the dynamics of the center of mass (CoM) during human walking, which is coordinated by multiple joints. This simple spring model, however, only describes the CoM during the stance phase, and the mechanics involved in the bipedality of the human gait are limited. In this study, a bipedal spring walking model was proposed to demonstrate the dynamics of bipedal walking, including swing dynamics followed by the step-to-step transition. The model consists of two springs with different stiffnesses and rest lengths representing the stance leg and swing leg. One end of each spring has a foot mass, and the other end is attached to the body mass. To induce a forward swing that matches the gait phase, a torsional hip joint spring was introduced at each leg. To reflect the active knee flexion for foot clearance, the rest length of the swing leg was set shorter than that of the stance leg, generating a discrete elastic restoring force. The number of model parameters was reduced by introducing dependencies among stiffness parameters. The proposed model generates periodic gaits with dynamics-driven step-to-step transitions and realistic swing dynamics. While preserving the mimicry of the CoM and ground reaction force (GRF) data at various gait speeds, the proposed model emulated the kinematics of the swing leg. This result implies that the dynamics of human walking generated by the actuations of multiple body segments is describable by a simple spring mechanics.     

Humans exploit the biomechanics of bipedal gait during visually guided walking over complex terrain

How do humans achieve such remarkable energetic efficiency when walking over complex terrain such as a rocky trail? Recent research in biomechanics suggests that the efficiency of human walking over flat, obstacle-free terrain derives from the ability to exploit the physical dynamics of our bodies. In this study, we investigated whether this principle also applies to visually guided walking over complex terrain. We found that when humans can see the immediate foreground as little as two step lengths ahead, they are able to choose footholds that allow them to exploit their biomechanical structure as efficiently as they can with unlimited visual information. We conclude that when humans walk over complex terrain, they use visual information from two step lengths ahead to choose footholds that allow them to approximate the energetic efficiency of walking in flat, obstacle-free environments.     

Theories of bipedal walking: an odyssey

In this paper six theories of bipedal walking, and the evidence in support of the theories, are reviewed. They include: evolution, minimising energy consumption, maturation in children, central pattern generators, linking control and effect, and robots on two legs. Specifically, the six theories posit that: (1) bipedalism is the fundamental evolutionary adaptation that sets hominids--and therefore humans--apart from other primates; (2) locomotion is the translation of the centre of gravity along a pathway requiring the least expenditure of energy; (3) when a young child takes its first few halting steps, his or her biomechanical strategy is to minimise the risk of falling; (4) a dedicated network of interneurons in the spinal cord generates the rhythm and cyclic pattern of electromyographic signals that give rise to bipedal gait; (5) bipedal locomotion is generated through global entrainment of the neural system on the one hand, and the musculoskeletal system plus environment on the other; and (6) powered dynamic gait in a bipedal robot can be realised only through a strategy which is based on stability and real-time feedback control. The published record suggests that each of the theories has some measure of support. However, it is important to note that there are other important theories of locomotion which have not been covered in this review. Despite such omissions, this odyssey has explored the wide spectrum of bipedal walking, from its origins through to the integration of the nervous, muscular and skeletal systems.     

Stability of bipedal stance: the contribution of cocontraction and spindle feedback

 The aim of this study is to assess the contribution of cocontraction and spindle feedback to local stability during bipedal stance. To that aim, an existing nonlinear state space model of the human musculoskeletal system is linearized in a reference equilibrium state. The maximal real part of the eigenvalues of the linearized system matrix A and the low-frequency joint stiffness are used as a measure of local stability. Muscle properties, as represented in a Hill-type muscle model, are shown to improve the behavior, the improvement being larger at high cocontraction. However, even at maximal cocontraction the low-frequency joint stiffness generated by the muscle properties is insufficient to yield a locally stable system. It follows that feedback is necessary to ensure local stability. In this study, the potential contribution of spindle feedback is investigated by optimizing the feedback gains for contractile element length and velocity for each muscle. It is found that in the case of time-delayed negative feedback, it is impossible to stabilize the system on the basis of spindle feedback. When positive time-delayed feedback is allowed, a barely stable system is obtained. When the time delays are removed, the feedback gains can be chosen such that a locally stable system is obtained, indicating the limitations imposed by the presence of time delays. Finally, it is shown that for small perturbations the response of the linear system to an arbitrary perturbation is similar to that of the nonlinear system, indicating the validity of the approach used. It is concluded that the combination of muscle properties and time-delayed spindle feedback is insufficient to obtain a system with reasonable local stability. Received: 10 October 2002 / Accepted: 3 December 2002 / Published online: 3 April 2003 Correspondence to: A. J. van Soest (e-mail: a_j_van_soest@fbw.vu.nl) Acknowledgements. We acknowledge Richard Casius for his help in carrying out the optimizations.     

Adaptation to bipedal gait and fifth metatarsal structural properties in Australopithecus,Paranthropus, and Homo

Humans, unlike African apes, have relatively robust fifth metatarsals (Mt5) presumably reflecting substantial weight-bearing and stability function in the lateral column of the former. When this morphological difference emerged during hominin evolution is debated. Here we investigate internal diaphyseal structure of Mt5s attributed to Australopithecus (from Sterkfontein), Paranthropus (from Swartkrans), and Homo (from Olduvai, Dmanisi, and Dinaledi) placed in the context of human and African ape Mt5 internal diaphyseal structure. ‘Whole-shaft’ properties were evaluated from 17 cross sections sampling 25% to 75% diaphyseal length using computed tomography. To assess structural patterns, scaled cortical bone thicknesses (sCBT) and scaled second moments of area (sSMA) were visualized and evaluated through penalized discriminant analyses. While the majority of fossil hominin Mt5s exhibited ape-like sCBT, their sSMA were comparatively more human-like. Human-like functional loading of the lateral column existed in at least some fossil hominins, although perhaps surprisingly not in hominins from Dmanisi or Dinaledi.     

Spatio-temporal gait characteristics of the hind-limb cycles during voluntary bipedal and quadrupedal walking in bonobos (Pan paniscus)

Spatio-temporal gait characteristics (step and stride length, stride frequency, duty factor) were determined for the hind-limb cycles of nine bonobos (Pan paniscus) walking quadrupedally and bipedally at a range of speeds. The data were recalculated to dimensionless quantities according to the principle of dynamic similarity. Lower leg length was used as the reference length. Interindividual variability in speed modulation strategy of bonobos appears to be low. Compared to quadrupedal walking, bipedal bonobos use smaller steps to attain a given speed (differences increase with speed), resulting in shorter strides at a higher frequency. In the context of the ("hybrid") dynamic pattern approach to locomotion (Latach, 1998) we argue that, despite these absolute differences, intended walking speed is the basic control variable which elicits both quadrupedal and bipedal walking kinematics in a similar way. Differences in the initial status of the dynamic system may be responsible for the differences in step length between both gaits. Comparison with data deduced from the literature shows that the effects of walking speed on stride length and frequency are similar in bonobos, common chimpanzees, and humans. This suggests that (at least) within extant homininae, spatio-temporal gait characteristics are highly comparable, and this in spite of obvious differences in mass distribution and bipedal posture.     

Collision-based mechanics of bipedal hopping

The muscle work required to sustain steady-speed locomotion depends largely upon the mechanical energy needed to redirect the centre of mass and the degree to which this energy can be stored and returned elastically. Previous studies have found that large bipedal hoppers can elastically store and return a large fraction of the energy required to hop, whereas small bipedal hoppers can only elastically store and return a relatively small fraction. Here, we consider the extent to which large and small bipedal hoppers (tammar wallabies, approx. 7 kg, and desert kangaroo rats, approx. 0.1 kg) reduce the mechanical energy needed to redirect the centre of mass by reducing collisions. We hypothesize that kangaroo rats will reduce collisions to a greater extent than wallabies since kangaroo rats cannot elastically store and return as high a fraction of the mechanical energy of hopping as wallabies. We find that kangaroo rats use a significantly smaller collision angle than wallabies by employing ground reaction force vectors that are more vertical and center of mass velocity vectors that are more horizontal and thereby reduce their mechanical cost of transport. A collision-based approach paired with tendon morphometry may reveal this effect more generally among bipedal runners and quadrupedal trotters.     

Compliant bipedal model with the center of pressure excursion associated with oscillatory behavior of the center of mass reproduces the human gait dynamics

Although the compliant bipedal model could reproduce qualitative ground reaction force (GRF) of human walking, the model with a fixed pivot showed overestimations in stance leg rotation and the ratio of horizontal to vertical GRF. The human walking data showed a continuous forward progression of the center of pressure (CoP) during the stance phase and the suspension of the CoP near the forefoot before the onset of step transition. To better describe human gait dynamics with a minimal expense of model complexity, we proposed a compliant bipedal model with the accelerated pivot which associated the CoP excursion with the oscillatory behavior of the center of mass (CoM) with the existing simulation parameter and leg stiffness. Owing to the pivot acceleration defined to emulate human CoP profile, the arrival of the CoP at the limit of the stance foot over the single stance duration initiated the step-to-step transition. The proposed model showed an improved match of walking data. As the forward motion of CoM during single stance was partly accounted by forward pivot translation, the previously overestimated rotation of the stance leg was reduced and the corresponding horizontal GRF became closer to human data. The walking solutions of the model ranged over higher speed ranges (~1.7 m/s) than those of the fixed pivoted compliant bipedal model (~1.5 m/s) and exhibited other gait parameters, such as touchdown angle, step length and step frequency, comparable to the experimental observations. The good matches between the model and experimental GRF data imply that the continuous pivot acceleration associated with CoM oscillatory behavior could serve as a useful framework of bipedal model.     

Voluntary bipedal walking of infant chimpanzees

The voluntary bipedal walking of infant chimpanzees was studied by the analysis of foot force and by motion analysis. The infants were trained to locomote on a level platform without any restrictions on the locomotor pattern. The voluntary bipedal walking was compared with the other types of locomotion at the same age and with the trained bipedal walking performed by other chimpanzees, including adult chimpanzees. The characteristics of voluntary bipedal walking in the infant until one year of age were: (1) high-speed walking with short cycle duration; (2) short stance phase duration; (3) small braking component of the preceding leg and large acceleration of the following leg; (4) one downward peak in the vertical component; and (5) a relatively small transverse component. Bipedal walking usually continued for less than one second and ended in quadrupedal locomotion. During walking, the preceding foot touched the floor, heel first, as in the case of older chimpanzees and humans. At this age, bipedal walking was similar to high-speed locomotion. The voluntary bipedal walking of the two-year-old and frour-yearold chimpanzees was characterized as follows: (1) slower speed than during quadrupedal locomotion, (2) relatively long periods and distances; (3) well balanced accelerating and braking components; and (4) a vertical component showing two downward peaks and a trough in between during numerous trials. The last characteristic means that the body center of gravity is higher in the single stance phase, just as in the bipedal walkinbg of the adult chimpanzees and humans. The bipedal walking of infant chimpanzees was discussed in comparison with the walking of humans, including infants.     

Metatarsal robusticity in bipedal rats

All metatarsals have a significantly greater robusticity in the male than in the female rat. The robusticity formula of the rat's foot is 1 > 5 > 2 > 3 > 4. In bipedal rats that formula remains unchanged, but the robusticity of the metatarsals is increased especially in females. The tripod arrangement of the human foot with its particular robustness of the marginal metatarsals 1 and 5 and a strong calcaneum has been related to upright posture. The similar robusticity pattern in the rat's marginal metatarsals 1 and 5 raises the question of whether that part of the formula might not represent a more general plantigrade pattern.     

Central pattern generators for bipedal locomotion

Golubitsky, Stewart, Buono and Collins proposed two models for the achitecture of central pattern generators (CPGs): one for bipeds (which we call leg) and one for quadrupeds (which we call quad). In this paper we use symmetry techniques to classify the possible spatiotemporal symmetries of periodic solutions that can exist in leg (there are 10 nontrivial types) and we explore the possibility that coordinated arm/leg rhythms can be understood, on the CPG level, by a small breaking of the symmetry in quad, which leads to a third CPG architecture arm. Rhythms produced by leg correspond to the bipedal gaits of walk, run, two-legged hop, two-legged jump, skip, gallop, asymmetric hop, and one-legged hop. We show that breaking the symmetry between fore and hind limbs in quad, which yields the CPG arm, leads to periodic solution types whose associated leg rhythms correspond to seven of the eight leg gaits found in leg; the missing biped gait is the asymmetric hop. However, when arm/leg coordination rhythms are considered, we find the correct rhythms only for the biped gaits of two-legged hop, run, and gallop. In particular, the biped gait walk, along with its arm rhythms, cannot be obtained by a small breaking of symmetry of any quadruped gait supported by quad.