Stability in legged locomotion

Stability is a key element in a gait synthesis. Static stability margins are widely adopted in crawlers, while no similar approach has been developed for dynamically stable systems. Utilizing an analytical approach, we developed a set of easy-to-calculate stability indices to describe instantaneous static and dynamic (In)stability for a certain group of walking systems. The analysis is based on a thorough analysis of the interaction between ground reaction forces and the walking system. The indices are applicable to walking systems regardless of the number of legs or mechanical/biological design. We show that static and dynamic stability are independent of each other. We suggest a possible categorization of gait modes based on stability. Stability characteristics are analyzed in a healthy and highly pathological human gait. Finally, we discuss the applicability of the proposed methods.     

Stability versus maneuverability in aquatic locomotion

The dictionary definition of stability as "Firmly established,not easily to be changed" immediately indicates the conflictbetween stability and maneuverability in aquatic locomotion.The present paper addresses several issues resulting from theseopposing requirements. Classical stability theory for bodiesmoving in fluids is based on developments in submarine and airshipmotions. These have lateral symmetry, in common with most animals.This enables the separation of the equations of motion intotwo sets of 3 each. The vertical (longitudinal) set, which includesmotions in the axial (surge), normal (heave) and pitching directions,can thus be separated from the lateral-horizontal plane whichincludes yaw, roll and sideslip motions. This has been founduseful in the past for longitudinal stability studies basedon coasting configurations but is not applicable to the analysisof turning, fast starts and vigorous swimming, where the lateralsymmetry of the fish body is broken by bending motions. Thepresent paper will also examine some of the aspects of the stabilityvs. maneuverability tradeoff for these asymmetric motions. Ananalysis of the conditions under which the separation of equationsof motions into vertical and horizontal planes is justified,and a definition of the equations to be used in cases wherethis separation is not accurate enough is presented.     

Mechanical aspects of legged locomotion control

We review the mechanical components of an approach to motion science that enlists recent progress in neurophysiology, biomechanics, control systems engineering, and non-linear dynamical systems to explore the integration of muscular, skeletal, and neural mechanics that creates effective locomotor behavior. We use rapid arthropod terrestrial locomotion as the model system because of the wealth of experimental data available. With this foundation, we list a set of hypotheses for the control of movement, outline their mathematical underpinning and show how they have inspired the design of the hexapedal robot, RHex.     

Dynamics and stability of legged locomotion in the horizontal plane: a test case using insects

 Motivated by experimental studies of insects, we propose a model for legged locomotion in the horizontal plane. A three-degree-of freedom, energetically conservative, rigid-body model with a pair of compliant virtual legs in intermittent contact with the ground allows us to study how dynamics depends on parameters such as mass, moment of inertia, leg stiffness, and length. We find periodic gaits, and show that mechanics alone can confer asymptotic stability of relative heading and body angular velocity. We discuss the relevance of our idealized models to experiments and simulations on insect running, showing that their gait and force characteristics match observations reasonably well. We perform parameter studies and suggest that our model is relevant to the understanding of locomotion dynamics across species. Received: 17 April 2001 / Accepted in revised form: 20 November 2001     

Quantifying dynamic stability of genetic memory circuits

Bistability/Multistability has been found in many biological systems including genetic memory circuits. Proper characterization of system stability helps to understand biological functions and has potential applications in fields such as synthetic biology. Existing methods of analyzing bistability are either qualitative or in a static way. Assuming the circuit is in a steady state, the latter can only reveal the susceptibility of the stability to injected DC noises. However, this can be inappropriate and inadequate as dynamics are crucial for many biological networks. In this paper, we quantitatively characterize the dynamic stability of a genetic conditional memory circuit by developing new dynamic noise margin (DNM) concepts and associated algorithms based on system theory. Taking into account the duration of the noisy perturbation, the DNMs are more general cases of their static counterparts. Using our techniques, we analyze the noise immunity of the memory circuit and derive insights on dynamic hold and write operations. Considering cell-to-cell variations, our parametric analysis reveals that the dynamic stability of the memory circuit has significantly varying sensitivities to underlying biochemical reactions attributable to differences in structure, time scales, and nonlinear interactions between reactions. With proper extensions, our techniques are broadly applicable to other multistable biological systems.     

Kinematics of soft-bodied, legged locomotion in Manduca sexta larvae

Caterpillar crawling is distinct from that of worms and molluscs; it consists of a series of steps in different body segments that can be compared to walking and running in animals with stiff skeletons. Using a three-dimensional kinematic analysis of horizontal crawling in Manduca sexta, the tobacco hornworm, we found that the phase of vertical displacement in the posterior segments substantially led changes in horizontal velocity and the segments appeared to pivot around the attached claspers. Both of the motions occur during vertebrate walking. In contrast, vertical displacement and horizontal velocity in the anterior proleg-bearing segments were in phase, as expected for running gaits coupled by elastic storage. We propose that this kinematic similarity to running results from the muscular compression and release of elastic tissues. As evidence in support of this proposal, the compression and extension of each segment were similar to harmonic oscillations in a spring, although changes in velocity were 70 degrees out of phase with displacement, suggesting that the spring was damped. Measurements of segment length within, and across, intersegmental boundaries show that some of these movements were caused by folding of the body wall between segments. These findings demonstrate that caterpillar crawling is not simply the forward progression of a peristaltic wave but has kinetic components that vary between segments. Although these movements can be compared to legged locomotion in animals with stiff skeletons, the underlying mechanisms of caterpillar propulsion, and in particular the contribution of elastic tissues, remain to be discovered.     

A unified theory for the energy cost of legged locomotion

Small animals are remarkably efficient climbers but comparatively poor runners, a well-established phenomenon in locomotor energetics that drives size-related differences in locomotor ecology yet remains poorly understood. Here, I derive the energy cost of legged locomotion from two complementary components of muscle metabolism, Activation–Relaxation and Cross-bridge cycling. A mathematical model incorporating these costs explains observed patterns of locomotor cost both within and between species, across a broad range of animals (insects to ungulates), for a wide range of substrate slopes including level running and vertical climbing. This ARC model unifies work- and force-based models for locomotor cost and integrates whole-organism locomotor cost with cellular muscle physiology, creating a predictive framework for investigating evolutionary and ecological pressures shaping limb design and ranging behaviour.     

An analytical estimation of the energy cost for legged locomotion

Legged locomotion requires the determination of a number of parameters such as stride period, stride length, order of leg movements, leg trajectory, etc. How are these parameters determined? It has been reported that the locomotor patterns of many legged animals exhibit common characteristics, which suggests that there exists a basic strategy for legged locomotion. In this study we derive an equation to estimate the cost of transport for legged locomotion and examine a criterion of the minimization of the transport cost as a candidate of the strategy. The obtained optimal locomotor pattern that minimizes the cost suitably represents many characteristics of the pattern observed in legged animals. This suggests that the locomotor pattern of legged animals is well optimized with regard to the energetic cost. The result also suggests that the existence of specific gait patterns and the phase transition between them could be the result due to optimization; they are induced by the change in the distribution of ground reaction forces for each leg during locomotion.     

Balancing requirements for stability and maneuverability in cetaceans

The morphological designs of animals represent a balance betweenstability for efficient locomotion and instability associatedwith maneuverability. Morphologies that deviate from designsassociated with stability are highly maneuverable. Major featuresaffecting maneuverability are positions of control surfacesand flexibility of the body. Within odontocete cetaceans (i.e.,toothed whales), variation in body design affects stabilityand turning performance. Position of control surfaces (i.e.,flippers, fin, flukes, peduncle) provides a generally stabledesign with respect to an arrow model. Destabilizing forcesgenerated during swimming are balanced by dynamic stabilizationdue to the phase relationships of various body components. Cetaceanswith flexible bodies and mobile flippers are able to turn tightlyat low turning rates, whereas fast-swimming cetaceans with lessflexibility and relatively immobile flippers sacrifice smallturn radii for higher turning rates. In cetaceans, body andcontrol surface mobility and placement appear to be associatedwith prey type and habitat. Flexibility and slow, precise maneuveringare found in cetaceans that inhabit more complex habitats, whereashigh-speed maneuvers are used by cetaceans in the pelagic environment.     

Tradeoff between stability and maneuverability during whole-body movements

Background

Understanding how stability and/or maneuverability affects motor control strategies can provide insight on moving about safely in an unpredictable world. Stability in human movement has been well-studied while maneuverability has not. Further, a tradeoff between stability and maneuverability during movement seems apparent, yet has not been quantified. We proposed that greater maneuverability, the ability to rapidly and purposefully change movement direction and speed, is beneficial in uncertain environments. We also hypothesized that gaining maneuverability comes at the expense of stability and perhaps also corresponds with decreased muscle coactivation.

Materials and Methods

We used a goal-directed forward lean movement task that integrated both stability and maneuverability. Subjects (n = 11) used their center of pressure to control a cursor on a computer monitor to reach a target. We added task uncertainty by shifting the target anterior-posterior position mid-movement. We used a balance board with a narrow beam that reduced the base of support in the medio-lateral direction and defined stability as the probability that subjects could keep the balance board level during the task.

Results

During the uncertainty condition, subjects were able to change direction of their anterior-posterior center of pressure more rapidly, indicating that subjects were more maneuverable. Furthermore, medio-lateral center of pressure excursions also approached the edges of the beam and reduced stability margins, implying that subjects were less stable (i.e. less able to keep the board level). On the narrow beam board, subjects increased muscle coactivation of lateral muscle pairs and had greater muscle activity in the left leg. However, there were no statistically significant differences in muscle activity amplitudes or coactivation with uncertainty.

Conclusions/Significance

These results demonstrate that there is a tradeoff between stability and maneuverability during a goal-directed whole-body movement. Tasks with added uncertainty could help individuals learn to be more maneuverable yet sufficiently stable.     

Quantifying beta-sheet stability by phage display

Antigens I/II are large multifunctional adhesins from oral viridans streptococci that exert immunomodulatory effects on human cells and play important roles in inflammatory disorders. Among them, Streptococcus mutans plays a major role in the initiation of dental caries. The structure of the V-region (SrV+, residues 464-840) of the antigen I/II of S. mutans has been determined using the multiwavelength anomalous diffraction phasing technique with seleno-methionine-substituted recombinant protein and subsequently refined at 2.4 A resolution. The crystal structure of SrV+ revealed a lectin-like fold that displays a putative preformed carbohydrate-binding site stabilized by a metal ion. Inhibition of this binding site may confer to humans a protection against dental caries and dissemination of the bacteria to extra-oral sites involved in life-threatening inflammatory diseases. This crystal structure constitutes a first step in understanding the structure-function relationship of antigens I/II and may help in delineating new preventive or therapeutic strategies against colonization of the host by oral streptococci.     

Dynamics and stability of lateral plane locomotion on inclines

An actuated, lateral leg spring model is developed to investigate lateral plane locomotion dynamics and stability on inclines. A single actuation input, the force-free leg length, is varied in a feedforward fashion to explicitly and implicitly match prescribed lateral and fore-aft force profiles, respectively. Forward dynamic simulations incorporating the prescribed leg actuation are employed to identify periodic orbits for gaits in which the leg acts to either push the body away from or pull the body towards the foot placement point. Gait stability and robustness to external perturbation are found to vary significantly as a function of slope and velocity for each type of leg function. Results of these analyses suggest that the switch in leg function from pushing to pulling is governed by gait robustness, and occurs at increasing inclines for increasing velocities.     

Postural stability of the human mandible during locomotion

Movements of the head and of the mandible relative to the head were measured in human subjects walking and running on a treadmill at various speeds and inclinations. A miniature magnet and piezo-electric accelerometer assembly was mounted on the mandibular incisors, and a Hall-effect sensor along with a second accelerometer mounted on a maxillary incisor along a common vertical axis. Signals from these sensors provided continuous records of vertical head and mandible acceleration, and relative jaw position. Landing on the heel or on the toe in different forms of locomotion was followed by rapid deceleration of the downward movement of the head and slightly less rapid deceleration of the downward movement of the mandible, i.e., the mandible moved downwards relative to the maxilla, then upwards again to near its normal posture within 200 ms. No tooth contact occurred in any forms of gait at any inclination. The movement of the mandible relative to the maxilla depended on the nature and velocity of the locomotion and their effects on head deceleration. The least deceleration and hence mandibular displacement occurred during toe-landing, for example, during "uphill" running. The maximum displacement of the mandible relative to the head was less than 1mm, even at the fastest running speed. The mechanisms that limit the vertical movements of the jaw within such a narrow range are not known, but are likely to include passive soft-tissue visco-elasticity and stretch reflexes in the jaw-closing muscles.     

Torque around the center of mass: dynamic stability during quadrupedal arboreal locomotion in the Siberian chipmunk (Tamias sibiricus)

When animals travel on tree branches, avoiding falls is of paramount importance. Animals swiftly running on a narrow branch must rely on movement to create stability rather than on static methods. We examined how Siberian chipmunks (Tamias sibiricus) remain stable while running on a narrow tree branch trackway. We examined the pitch, yaw, and rolling torques around the center of mass, and hypothesized that within a stride, any angular impulse (torque during step time) acting on the center of mass would be canceled out by an equal and opposite angular impulse. Three chipmunks were videotaped while running on a 2 cm diameter branch trackway. We digitized the videos to estimate center of mass and center of pressure positions throughout the stride. A short region of the trackway was instrumented to measure components of the substrate reaction force. We found that positive and negative pitch angular impulse was by far the greatest in magnitude. The anterior body was pushed dorsally (upward) when the forelimbs landed simultaneously, and then the body pitched in the opposite direction as both hindlimbs simultaneously made contact. There was no considerable difference between yaw and rolling angular impulses, both of which were small and equal between fore- and hindlimbs. Net angular impulses around all three axes were usually greater than or less than zero (not balanced). We conclude that the chipmunks may balance out the torques acting on the center of mass over the course of two or more strides, rather than one stride as we hypothesized.     

Renewable fluid dynamic energy derived from aquatic animal locomotion

Aquatic animals swimming in isolation and in groups are known to extract energy from the vortices in environmental flows, significantly reducing muscle activity required for locomotion. A model for the vortex dynamics associated with this phenomenon is developed, showing that the energy extraction mechanism can be described by simple criteria governing the kinematics of the vortices relative to the body in the flow. In this way, we need not make direct appeal to the fluid dynamics, which can be more difficult to evaluate than the kinematics. Examples of these principles as exhibited in swimming fish and existing energy conversion devices are described. A benefit of the developed framework is that the potentially infinite-dimensional parameter space of the fluid-structure interaction is reduced to a maximum of eight combinations of three parameters. The model may potentially aid in the design and evaluation of unsteady aero- and hydrodynamic energy conversion systems that surpass the Betz efficiency limit of steady fluid dynamic energy conversion systems.     

Situating mortality: Quantifying crisis points and periods of stability

A wide range of stressors can cause a dramatic and sudden rise in the death rate in populations, typically resulting in what is referred to as crisis mortality. Here we present a method to standardize the assessment of identifying moments of crises. A modification of the mortality Z‐score methodology which is combined with time series analysis was used to investigate mortality events over the course of nearly two centuries for two populations: Gibraltar and Malta. A benefit of this method is that it situates the yearly death rate within the prevailing mortality pattern, and by doing so allows the researcher to assess the relative impact of that event against the norm for the period under investigation. A series of threshold values were established to develop levels of mortality to distinguish moments of lower mortality than expected, background mortality, a crisis, and a catastrophe. Our findings suggested that within defined periods, a limited number of events constituted moments of excessive mortality in the range of a crisis or higher. These included epidemics (yellow fever and influenza in Gibraltar only, and cholera) and casualties associated with World War II. Episodes of lower than expected mortality were only detected (although not significant) in the 20th century in Malta, and at the micro level, the harvesting effect appears to have occurred following cholera epidemics in both locations and influenza in Gibraltar. The analysis demonstrates clearly that the impact of epidemics can be highly variable across time and populations. Am J Phys Anthropol 152:459–470, 2013. © 2013 Wiley Periodicals, Inc.     

Quantifying dynamic resource allocation illuminates foraging strategy in Phanerochaete velutina

Saprotrophic woodland fungi forage for mineral nutrients and woody resources by extension of a mycelial network across the forest floor. Different species explore at different rates and establish networks with qualitatively differing architecture. However, detailed understanding of fungal foraging behaviour has been hampered by the absence of tools to quantify resource allocation and growth accurately and non-invasively. To solve this problem, we have used photon-counting scintillation imaging (PCSI) to map and quantify nutrient allocation and localised growth simultaneously in heterogeneous resource environments. We show that colonies spontaneously shift to an asymmetric growth pattern, even in the absence of added resources, often with a distinct transition between the two growth phases. However, the extent of polarisation was much more pronounced and focussed in the presence of an additional cellulose resource. In this case, there was highly localised growth, often at the expense of growth elsewhere in the colony, and marked accumulation of ¹⁴C-AIB in the sector of the colony with the added resource. The magnitude of the response was greatest when resource was added around the time of the endogenous developmental transition. The focussed response required a metabolisable resource, as only limited changes were seen with glass fibre discs used to mimic the osmotic and thigmotropic stimuli upon resource addition. Overall the behaviour is consistent with an adaptive foraging strategy, both to exploit new resources and also to redirect subsequent foraging effort to this region, presumably with an expectation that the probability of finding additional resources is increased.     

Quantifying the effect of brush layering on slope stability

Soil bioengineering techniques that use vegetation as a structural element gained popularity in the field of natural and man-made slope stabilisation due to their ability to combine safety and environmental conservation elements. In spite of such popularity, little research has been done to quantify their effect on slope stability. This work presents a simple scheme for the evaluation of the Factor of Safety for slopes reinforced by brush layering, which is one of the most common techniques adopted in slope stabilisation works. The proposed model is based on the limit equilibrium principle and accounts for geotechnical soil properties (cohesion, friction angle, unit weight of soil), soil saturation, slope steepness, and brush layer design parameters (number of stems per meter, length and diameter of stems, distance between brush layers). The model provides the value of the Factor of Safety for a given slope and soil depth. Laboratory pullout tests were carried out in order to estimate relevant parameters of cuttings of purple willow (Salix purpurea L.) and to perform a slope stability analysis via the model.