Leg force interference in polypedal locomotion

When leg number increases, COM oscillations decline more quickly with increasing deviations from alternating leg coordination. The examination of gaits and gait changes has been the focus of movement physiology and legged robot engineering since the first emergence of the fields. While most examinations have focused on bipedal and quadrupedal designs, many robotic implementations rely on the higher static stability of three or more pairs of legs. Thus far, however, the effect of number of pairs of legs on locomotion dynamics has not been examined. Accordingly, the present approach aims to extend available theory to polypedal designs and examines how the number of active walking legs affects body dynamics when combined with changing duty factors and phase relations. The model shows that ground force interference of higher numbers of active pairs of walking legs can prevent effective use of bouncing gaits, such as trot, and their associated advantages, such as energy efficiency, because significantly higher degrees of leg synchronization are required. It also shows that small changes in the leg coordination pattern have a much higher impact on the center-of-mass dynamics in locomotor systems with many legs than in those with fewer legs. In this way, the model reveals coordinative constraints for specific gaits facilitating the assessment of animal locomotion and economization of robotic locomotion.

Related papers:

title: The Co-Ordination of Insect Movements I The Walking Movements of Insects
abstract: 1. An account is given of the functional morphology of the coxal articulations in the cockroach. The greater range of movements executed by the prothoracic legs is made possible by the mobility of the trochantinal condyle relative to the episternal condyle. 2. The movements of the individual legs were studied by means of films taken at speeds of 16-32 frames/sec. These show that more than three legs are in contact with the ground throughout the cycle of movements at normal speeds. Suggestions are also made concerning the mechanics of walking, but these require experimental verification. 3. The rhythm of walking movements in Periplaneta, Blatta, Dytiscus, Hydropkilus, Carabus, Blaps and Chrysomela obeys two rules: (i) no foreleg or middle leg is protracted until the leg behind has taken up its supporting position; (ii) each leg alternates with the contralateral one of the same segment. Other pterygote insects appeared not to differ from the species studied in any essential feature. 4. An increase in speed is accompanied by a decrease in the times of both protraction ( p ) and retraction ( r ), a shortening of the stride length, and an increase in the distance between the successive points d9appui . The range of speeds is continuous and no distinction could be recognized between walking and running. 5. Several gaits have been observed, the most common order of protraction being R1, L2, R3, L1 , R2, L3, R1, etc., but these grade into one another if the ratio p/r is altered and the two rules obeyed. A system of rigidly alternating tripods would result if the ratio p/r was unity, but this is never quite realized as there is always a delay, sometimes extremely short, between the protraction of the three legs of a triangle. At very slow speeds the rhythm R3, R2, R1, L3, L2, L1 , R3, etc., may be present. 6. It is concluded that insects are the end-product of a process of limb reduction among terrestrial Arthropoda in which p/r may be increased to nearly one and yet the animal retains static stability throughout the whole cycle. This is impossible with fewer than six legs.

title: Crawling at High Speeds: Steady Level Locomotion in the Spider Cupiennius salei—Global Kinematics and Implications for Centre of Mass Dynamics
abstract: Spiders are an old yet very successful predatory group of arthropods. Their locomotor system differs from those of most other arthropods by the lack of extensor muscles in two major leg joints. Though specific functional characteristics can be expected regarding the locomotion dynamics of spiders, this aspect of movement physiology has been only scarcely examined so far. This study presents extensive analyses of a large dataset concerning global kinematics and the implications for dynamics of adult female specimens of the large Central American spider Cupiennius salei (Keyserling). The experiments covered the entire speed-range of straight runs at constant speeds. The analyses revealed specific characteristics of velocity dependent changes in the movements of the individual legs, as well as in the translational and rotational degrees of freedom of both the centre of mass and the body. In contrast to many other fast moving arthropods, C. salei avoid vertical fluctuations of their centre of mass during fast locomotion. Accordingly, aerial phases were not observed here. This behaviour is most likely a consequence of optimising energy expenditure with regard to the specific requirements of spiders' leg anatomy. A strong synchronisation of two alternating sets of legs appears to play only a minor role in the locomotion of large spiders. Reduced frequency and low centre of mass amplitudes as well as low angular changes of the body axes, in turn, seems to be the result of relatively low leg coordination.

title: Requirements and limits of anatomy-based predictions of locomotion in terrestrial arthropods with emphasis on arachnids
abstract: Modern computer-aided techniques foster the availability and quality of 3D visualization and reconstruction of extinct and extant species. Moreover, animated sequences of locomotion and other movements find their way into motion pictures and documentary films, but also gain attraction in science. While movement analysis is well advanced in vertebrates, particularly in mammals and birds, analyses in arthropods, with their much higher variability regarding general anatomy and size, are still in their infancies and restricted to a few laboratory species. These restrictions and deficient understanding of terrestrial arthropod locomotion in general impedes sensible reconstruction of movements in those species that are not directly observable (e.g., extinct and cryptic species). Since shortcomings like over-simplified approaches to simulate arthropod locomotion became obvious recently, in this review we provide insight into physical, morphological, physiological, behavioral, and ecological constraints, which are essential for sensible reconstructions of terrestrial arthropod locomotion. Such concerted consideration along with sensible evaluations of stability and efficiency requirements can pave the way to realistic assessment of leg coordination and body dynamics.

title: Speed dependent phase shifts and gait changes in cockroaches running on substrates of different slipperiness
abstract: BackgroundMany legged animals change gaits when increasing speed. In insects, only one gait change has been documented so far, from slow walking to fast running, which is characterised by an alternating tripod. Studies on some fast-running insects suggested a further gait change at higher running speeds. Apart from speed, insect gaits and leg co-ordination have been shown to be influenced by substrate properties, but the detailed effects of speed and substrate on gait changes are still unclear. Here we investigate high-speed locomotion and gait changes of the cockroach Nauphoeta cinerea, on two substrates of different slipperiness.ResultsAnalyses of leg co-ordination and body oscillations for straight and steady escape runs revealed that at high speeds, blaberid cockroaches changed from an alternating tripod to a rather metachronal gait, which to our knowledge, has not been described before for terrestrial arthropods. Despite low duty factors, this new gait is characterised by low vertical amplitudes of the centre of mass (COM), low vertical accelerations and presumably reduced total vertical peak forces. However, lateral amplitudes and accelerations were higher in the faster gait with reduced leg synchronisation than in the tripod gait with distinct leg synchronisation.ConclusionsTemporally distributed leg force application as resulting from metachronal leg coordination at high running speeds may be particularly useful in animals with limited capabilities for elastic energy storage within the legs, as energy efficiency can be increased without the need for elasticity in the legs. It may also facilitate locomotion on slippery surfaces, which usually reduce leg force transmission to the ground. Moreover, increased temporal overlap of the stance phases of the legs likely improves locomotion control, which might result in a higher dynamic stability.

title: Similarity in multilegged locomotion: Bouncing like a monopode
abstract: Despite impressive variation in leg number, length, position and type of skeleton, similarities of legged, pedestrian locomotion exist in energetics, gait, stride frequency and ground-reaction force. Analysis of data available in the literature showed that a bouncing, spring-mass, monopode model can approximate the energetics and dynamics of trotting, running, and hopping in animals as diverse as cockroaches, quail and kangaroos. From an animal's mechanical-energy fluctuation and ground-reaction force, we calculated the compression of a virtual monopode's leg and its stiffness. Comparison of dimensionless parameters revealed that locomotor dynamics depend on gait and leg number and not on body mass. Relative stiffness per leg was similar for all animals and appears to be a very conservative quantity in the design of legged locomotor systems. Differences in the general dynamics of gait are based largely on the number of legs acting simultaneously to determine the total stiffness of the system. Four- and six-legged trotters had a greater whole body stiffness than two-legged runners operating their systems at about the same relative speed. The greater whole body stiffness in trotters resulted in a smaller compression of the virtual leg and a higher natural frequency and stride frequency.

title: Leg design in hexapedal runners.
abstract: Many-legged animals, such as crabs and cockroaches, utilize whole-body mechanics similar to that observed for running bipeds and trotting quadrupedal mammals. Despite the diversity in morphology, two legs in a quadrupedal mammal, three legs in an insect and four legs in a crab can function in the same way as one leg of a biped during ground contact. To explain how diverse leg designs can result in common whole-body dynamics, we used a miniature force platform to measure the ground reaction forces produced by individual legs of the cockroach Blaberus discoidalis. Hexapedal runners were not like quadrupeds with an additional set of legs. In trotting quadrupedal mammals each leg develops a similar ground reaction force pattern that sums to produce the whole-body pattern. At a constant average velocity, each leg pair of the cockroach was characterized by a unique ground reaction force pattern. The first leg decelerated the center of mass in the horizontal direction, whereas the third leg was used to accelerate the body. The second leg did both, much like legs in bipedal runners and quadrupedal trotters. Vertical force peaks for each leg were equal in magnitude. In general, peak ground reaction force vectors minimized joint moments and muscle forces by being oriented towards the coxal joints, which articulate with the body. Locomotion with a sprawled posture does not necessarily result in large moments around joints. Calculations on B. discoidalis showed that deviations from the minimum moments may be explained by considering the minimization of the summed muscle forces in more than one leg. Production of horizontal forces that account for most of the mechanical energy generated during locomotion can actually reduce total muscle force by directing the ground reaction forces through the leg joints. Whole-body dynamics common to two-, four-, six- and eight-legged runners is produced in six-legged runners by three pairs of legs that differ in orientation with respect to the body, generate unique ground reaction force patterns, but combine to function in the same way as one leg of a biped.

title: Dynamics and kinematics of ant locomotion: do wood ants climb on level surfaces?
abstract: The biomechanics of running in small animals have remained poorly characterized because of the difficulty of recording three-dimensional ground reaction forces. Available techniques limit investigations to animals with a body mass above 1 g. Here we present, for the first time, single-leg ground reaction forces of ants (body mass 10 mg), measured with a custom-built miniature force plate. We investigated forces and high-speed kinematics for straight level runs (average speed: 8.4 cm s–1) of Formica polyctena workers. The major finding was that the time course of ground reaction forces strongly differed from previous observations of larger insects. Maximum vertical force was reached during the first third of the tripod contact phase. During this period the body was decelerated predominantly by the front legs. Subsequently, the front legs pulled and accelerated the body. This `climbing' type of stride may be useful on the bumpy and unstable substrates that the animals face in their natural habitats, and may therefore also occur on level ground. Propulsive forces were generated predominantly by the front and hind legs. Dragging of the gaster on the substrate resulted in a breaking momentum, which was compensated by the legs. Future investigations will reveal, whether the identified pattern is due to specialization.

title: Passive mechanical properties of legs from running insects
abstract: While the dynamics of running arthropods have been modeled as a spring-mass system, no such structures have been discovered that store and return energy during bouncing. The hindleg of the cockroach Blaberus discoidalis is a good candidate for a passive, vertical leg spring because its vertically oriented joint axes of rotation limit the possibility of active movements and contributions of muscle properties. We oscillated passive legs while measuring force to determine the leg's dynamic, mechanical properties. The relative dimensionless stiffness of an individual cockroach leg was equal to that estimated for a single leg of a biped or quadruped. Leg resilience ranged from 60 to 75%, affording the possibility that the leg could function as a spring to store and return the mechanical energy required to lift and accelerate the center of mass. Because hysteresis was independent of oscillation frequency, we rejected the use of a Voigt model – a simple spring in parallel with a viscous damper. A hysteretic damping model fit the cockroach leg force–displacement data over a wide range of frequencies and displacement using just two parameters. Rather than simply acting as a spring to minimize energy, we hypothesize that legs must manage both energy storage and absorption for rapid running to be most effective.

title: Running springs: speed and animal size.
abstract: Trotting and hopping animals use muscles, tendons and ligaments to store and return elastic energy as they bounce along the ground. We examine how the musculoskeletal spring system operates at different speeds and in animals of different sizes. We model trotting and hopping as a simple spring-mass system which consists of a leg spring and a mass. We find that the stiffness of the leg spring (k(leg)) is nearly independent of speed in dogs, goats, horses and red kangaroos. As these animals trot or hop faster, the leg spring sweeps a greater angle during the stance phase, and the vertical excursion of the center of mass during the ground contact phase decreases. The combination of these changes to the spring system causes animals to bounce off the ground more quickly at higher speeds. Analysis of a wide size range of animals (0.1-140 kg) at equivalent speeds reveals that larger animals have stiffer leg springs (k(leg) [symbol: see text] M0.67, where M is body mass), but that the angle swept by the leg spring is nearly independent of body mass. As a result, the resonant period of vertical vibration of the spring-mass system is longer in larger animals. The length of time that the feet are in contact with the ground increases with body mass in nearly the same way as the resonant period of vertical vibration.

title: The spring-mass model for running and hopping.
abstract: A simple spring-mass model consisting of a massless spring attached to a point mass describes the interdependency of mechanical parameters characterizing running and hopping of humans as a function of speed. The bouncing mechanism itself results in a confinement of the free parameter space where solutions can be found. In particular, bouncing frequency and vertical displacement are closely related. Only a few parameters, such as the vector of the specific landing velocity and the specific leg length, are sufficient to determine the point of operation of the system. There are more physiological constraints than independent parameters. As constraints limit the parameter space where hopping is possible, they must be tuned to each other in order to allow for hopping at all. Within the range of physiologically possible hopping frequencies, a human hopper selects a frequency where the largest amount of energy can be delivered and still be stored elastically. During running and hopping animals use flat angles of the landing velocity resulting in maximum contact length. In this situation ground reaction force is proportional to specific contact time and total displacement is proportional to the square of the step duration. Contact time and hopping frequency are not simply determined by the natural frequency of the spring-mass system, but are influenced largely by the vector of the landing velocity. Differences in the aerial phase or in the angle of the landing velocity result in the different kinematic and dynamic patterns observed during running and hopping. Despite these differences, the model predicts the mass specific energy fluctuations of the center of mass per distance to be similar for runners and hoppers and similar to empirical data obtained for animals of various size.

title: The mechanics of running: how does stiffness couple with speed?
abstract: A mathematical model for terrestrial running is presented, based on a leg with the properties of a simple spring. Experimental force-platform evidence is reviewed justifying the formulation of the model. The governing differential equations are given in dimensionless form to make the results representative of animals of all body sizes. The dimensionless input parameters are: U, a horizontal Froude number based on forward speed and leg length; V, a vertical Froude number based on vertical landing velocity and leg length, and KLEG, a dimensionless stiffness for the leg-spring. Results show that at high forward speed, KLEG is a nearly linear function of both U and V, while the effective vertical stiffness is a quadratic function of U. For each U, V pair, the simulation shows that the vertical force at mid-step may be minimized by the choice of a particular step length. A particularly useful specification of the theory occurs when both KLEG and V are assumed fixed. When KLEG = 15 and V = 0.18, the model makes predictions of relative stride length S and initial leg angle theta o that are in good agreement with experimental data obtained from the literature.

title: Invited Editorial on “Interaction of leg stiffness and surface stiffness during human hopping”
abstract: when a human runs or a kangaroo hops, the center of mass of the body rises and falls like a bouncing ball. The analogy has proved helpful in several studies of running and hopping (e.g., Ref.  and is used again by Ferris and Farley in an article ([6][2]) in this issue of the Journal. They ask

title: Time of contact and step length: the effect of limb length, running speed, load carrying and incline.
abstract: Using published values for twelve species of birds and mammals, we investigated the effects of limb length and running speed on time of contact and step length. In addition, we measured the time of contact in horses trotting up a 10 % incline and when carrying a load averaging 19 % of their body mass. From these values, we calculated stride period and step length. Our analysis of the interspecific data yielded the following relationship between time of contact (t(c) in s) and leg length (L in m) and running speed (v in m s(-)(1)): t(c)=0.80L(0.84)/v(0.87) (r(2)=0.97). Both exponents in this relationship are significantly different from 1.0, indicating that step length increases with speed and that small species use a step length that, relative to their leg length, is longer than the relative step length used by larger species. Time of contact increased when a horse carried a load but not when it trotted up an incline.

title: Passive Joint Forces Are Tuned to Limb Use in Insects and Drive Movements without Motor Activity
abstract: Limb movements are generally driven by active muscular contractions working with and against passive forces arising in muscles and other structures. In relatively heavy limbs, the effects of gravity and inertia predominate, whereas in lighter limbs, passive forces intrinsic to the limb are of greater consequence. The roles of passive forces generated by muscles and tendons are well understood, but there has been little recognition that forces originating within joints themselves may also be important, and less still that these joint forces may be adapted through evolution to complement active muscle forces acting at the same joint.  We examined the roles of passive joint forces in insect legs with different arrangements of antagonist muscles. We first show that passive forces modify actively generated movements of a joint across its working range, and that they can be sufficiently strong to generate completely passive movements that are faster than active movements observed in natural behaviors. We further demonstrate that some of these forces originate within the joint itself. In legs of different species adapted to different uses (walking, jumping), these passive joint forces complement the balance of strength of the antagonist muscles acting on the joint. We show that passive joint forces are stronger where they assist the weaker of two antagonist muscles.  In limbs where the dictates of a key behavior produce asymmetry in muscle forces, passive joint forces can be coadapted to provide the balance needed for the effective generation of other behaviors.

title: Neural Control of Unloaded Leg Posture and of Leg Swing in Stick Insect, Cockroach, and Mouse Differs from That in Larger Animals
abstract: Stick insect (Carausius morosus) leg muscles contract and relax slowly. Control of stick insect leg posture and movement could therefore differ from that in animals with faster muscles. Consistent with this possibility, stick insect legs maintained constant posture without leg motor nerve activity when the animals were rotated in air. That unloaded leg posture was an intrinsic property of the legs was confirmed by showing that isolated legs had constant, gravity-independent postures. Muscle ablation experiments, experiments showing that leg muscle passive forces were large compared with gravitational forces, and experiments showing that, at the rest postures, agonist and antagonist muscles generated equal forces indicated that these postures depended in part on leg muscles. Leg muscle recordings showed that stick insect swing motor neurons fired throughout the entirety of swing. To test whether these results were specific to stick insect, we repeated some of these experiments in cockroach (Periplaneta americana) and mouse. Isolated cockroach legs also had gravity-independent rest positions and mouse swing motor neurons also fired throughout the entirety of swing. These data differ from those in human and horse but not cat. These size-dependent variations in whether legs have constant, gravity-independent postures, in whether swing motor neurons fire throughout the entirety of swing, and calculations of how quickly passive muscle force would slow limb movement as limb size varies suggest that these differences may be caused by scaling. Limb size may thus be as great a determinant as phylogenetic position of unloaded limb motor control strategy.

title: Integrating the Physiology, Mechanics and Behavior of Rapid Running Ghost Crabs: Slow and Steady Doesn't Always Win the Race
abstract: In 1979 Bliss predicted that, “land crabs are and will undoubtedly continue to be promising objects of scientific research.” Studies of rapid running ghost crabs support her contention and have resulted in several general findings relating to locomotion and activity. 1) Energy exchange mechanisms during walking are general and not restricted to quadrupedal and bipedal morphologies. 2) “Equivalent gaits,” such as trots and gallops, may exist in 4-, 6- and 8-legged animals that differ greatly in leg and skeletal ( i.e. , exo- vs . endoskeletal) design. These findings support the hypothesis that terrestrial locomotion in many species can modeled by an inverted pendulum or spring-mass system. 3) An open circulatory system and chitin-covered gills do not necessarily limit the rate at which oxygen consumption can be increased or the factorial increase oxygen consumption over resting rates. 4) Interspecific and intraspecific i.e. , ontogenetic) scaling of sub-maximal oxygen consumption and maximal aerobic speed can differ significantly. 5) Locomotion at speeds above the maximal aerobic speed requiring non-aerobic contributions may be far more costly than can be predicted from aerobic costs alone. The cost transport may attain a minimum at less than maximum speed. 6) The speed which elicits maximal oxygen consumption during continuous exercise is attained at moderate walking speeds in crabs and probably other ectotherms. Speeds 15- to 20-fold faster are possible, but cannot be sustained. 7) The low endurance associated with the low maximal oxygen consumption and maximal aerobic speed of ectotherms moving continuously can be increased or decreased by altering locomotor behavior and moving intermittently. Ectotherms can locomote at high speeds and travel for considerable distances or remain active for long periods by including rest pauses. Alternatively, intense activity with extended exercise periods with short pause periods may actually reduce behavioral capacity or work accomplished relative to continuous activity during which the behavior is carried out at a lower intensity level without pauses.

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

title: Mechanics of six-legged runners.
abstract: Six-legged pedestrians, cockroaches, use a running gait during locomotion. The gait was defined by measuring ground reaction forces and mechanical energy fluctuations of the center of mass in Blaberus discoidalis (Serville) as they travelled over a miniature force platform. These six-legged animals produce horizontal and vertical ground-reaction patterns of force similar to those found in two-, four- and eight-legged runners. Lateral forces were less than half the vertical force fluctuations. At speeds between 0.08 and 0.66 ms-1, horizontal kinetic and gravitational potential energy changes were in phase. This pattern of energy fluctuation characterizes the bouncing gaits used by other animals that run. Blaberus discoidalis attained a maximum sustainable stride frequency of 13 Hz at 0.35 ms-1, the same speed and frequency predicted for a mammal of the same mass. Despite differences in body form, the mass-specific energy used to move the center of mass a given distance (0.9 J kg-1m-1) was the same for cockroaches, ghost crabs, mammals, and birds. Similarities in force production, stride frequency and mechanical energy production during locomotion suggest that there may be common design constraints in terrestrial locomotion which scale with body mass and are relatively independent of body form, leg number and skeletal type.

title: Climbing in hexapods: a plain model for heavy slopes.
abstract: Usually, a climbing cockroach attaches with three legs to a substrate. According to a recent model study, pulling forces underneath the front leg are required at some critical slope angle in upward locomotion. This critical angle depends on the animal's anatomy and leg positioning. In this study, we asked especially how this critical angle can be biased by one parameter that may be controlled during climbing: the body height above the substrate. We found that the typical ratio between body height and length (0.2) adopted by cockroaches is slightly higher than the very ratio (0.15) at which the critical slope angle can be increased most strongly for a given decrease in body height. In other words, it is likely that a geometrical body design of cockroaches evolved, which enables a delicate reduction in body height perfectly suitable for preventing the danger of slipping or even falling over rearwards at steepening slopes (approaching the vertical). In that sense, our model predicts, not just for hexapods but rather for any three-point climber, that taking up a low ratio of body height to the distance between the foremost and the hindmost attachment point (very crouched posture) makes body height a good parameter for climbing control.

title: Gait characterisation and classification in horses
abstract: Although a large number of foot-fall sequences are possible in quadrupeds, few sequences are routinely used. The aim of this paper is to characterise, by foot-fall pattern, the gaits used by horses and develop a novel technique to classify symmetric and asymmetric gaits using one common criterion. To achieve this speed and relative foot-fall, timings of all four limbs of eight Icelandic horses were measured using accelerometers. Linear discriminant analysis (LDA) was performed to find criteria that are optimal for discriminating between the different gaits. This also allowed us to evaluate whether gaits should be considered a continuum or as discrete entities. Foot-fall timings (stance times, swing times, duty factors and stride frequencies) for walk, tolt, trot, pace, left canter, right canter, left gallop and right gallop during over-ground locomotion at a range of speeds are presented. In the gaits of walk, tolt, trot and pace, foot-fall timings were equal between left and right hindlimbs and forelimbs so these gaits can be considered as symmetrical. Differences in stance times and duty factors were observed between gaits but are unlikely to be of biological significance due to their similar magnitude and inconsistent relative trends. This implies that metabolics or peak limb forces derived from contact times are unlikely to be the principal driving factors in gait transition between walk, trot, pace, canters and gallops, although these factors may influence the use of tolt at the lower and higher speeds. Gaits did cluster in the LDA space and the running gaits (tolt, trot, pace, left and right canters and gallops) could be considered a kinematic continuum but the relative relationship with walk may be more complex. Thus, LDA analysis has enabled common criteria to be discovered to accurately classify equine gaits on the basis of foot-fall timings on a stride-by-stride basis.

title: Gait selection in the ostrich: mechanical and metabolic characteristics of walking and running with and without an aerial phase
abstract: It has been argued that minimization of metabolic–energy costs is a primary determinant of gait selection in terrestrial animals. This view is based predominantly on data from humans and horses, which have been shown to choose the most economical gait (walking, running, galloping) for any given speed. It is not certain whether a minimization of metabolic costs is associated with the selection of other prevalent forms of terrestrial gaits, such as grounded running (a widespread gait in birds). Using biomechanical and metabolic measurements of four ostriches moving on a treadmill over a range of speeds from 0.8 to 6.7 m s−1, we reveal here that the selection of walking or grounded running at intermediate speeds also favours a reduction in the metabolic cost of locomotion. This gait transition is characterized by a shift in locomotor kinetics from an inverted–pendulum gait to a bouncing gait that lacks an aerial phase. By contrast, when the ostrich adopts an aerial–running gait at faster speeds, there are no abrupt transitions in mechanical parameters or in the metabolic cost of locomotion. These data suggest a continuum between grounded and aerial running, indicating that they belong to the same locomotor paradigm.

title: Shape optimization in exoskeletons and endoskeletons: a biomechanics analysis
abstract: This paper addresses the question of strength and mechanical failure in exoskeletons and endoskeletons. We developed a new, more sophisticated model to predict failure in bones and other limb segments, modelled as hollow tubes of radius r and thickness t. Five failure modes were considered: transverse fracture; buckling (of three different kinds) and longitudinal splitting. We also considered interactions between failure modes. We tested the hypothesis that evolutionary adaptation tends towards an optimum value of r/t, this being the value which gives the highest strength (i.e. load-carrying capacity) for a given weight. We analysed two examples of arthropod exoskeletons: the crab merus and the locust tibia, using data from the literature and estimating the stresses during typical activities. In both cases, the optimum r/t value for bending was found to be different from that for axial compression. We found that the crab merus experiences similar levels of bending and compression in vivo and that its r/t value represents an ideal compromise to resist these two types of loading. The locust tibia, however, is loaded almost exclusively in bending and was found to be optimized for this loading mode. Vertebrate long bones were found to be far from optimal, having much lower r/t values than predicted, and in this respect our conclusions differ from those of previous workers. We conclude that our theoretical model, though it has some limitations, is useful for investigating evolutionary development of skeletal form in exoskeletons and endoskeletons.

title: Bone strain: a determinant of gait and speed?
abstract: Principal strains were recorded in vivo from the radial and tibial midshafts of three goats as they increased speed and changed gait. These data were compared with strain data measured for the radius and tibia of the dog (Rubin & Lanyon, 1982) and the horse (Biewener, Thomason & Lanyon, 1983b) in order to test the hypothesis that similar peak bone strains (stresses) occur at functionally equivalent points in the gaits of different species. Multiple recordings of in vivo strain along the caudal diaphyses of the radius and tibia of one goat were made to test the validity of this technique for measuring peak locomotor stress. Measured strains were extremely consistent over the animal's full range of speed (coefficient of variation for the radius 0.05-0.08, and for the tibia 0.06-0.11). The data from the three gauges, which were spaced 15 mm apart, demonstrated that maximal strains act at the midshaft, substantiating the use of this technique to measure peak locomotor bone strains. Strain levels recorded at the trot-gallop transition and top galloping speeds of the goat were similar to the values reported for the dog and horse, despite large differences in absolute speed (goat, 4.3 ms-1; dog, 6.9 ms-1; horse, 7.5 ms-1 at maximum gallop). The second moments of area of the tibia and radius (+ ulna) of the dog are 29% and 113% greater than for goats of equal size, explaining how similar strains are achieved in the dog at higher speeds than the goat. Furthermore, peak bone strains recorded at the fastest trotting speed were similar to those recorded at the fastest galloping speed for each species. Peak strains recorded for the goat at a maximum gallop correspond to stresses of +37.9 MPa (cranial) and -47.7 MPa (caudal) in the radius and +36.3 MPa (cranial) and -50.3 MPa (caudal) in the tibia, representing a safety factor to yield failure of three.

title: Gait and the energetics of locomotion in horses
abstract: It seems reasonable that quadrupeds should change gait from a walk to a trot to a gallop in such a way as to minimize their energy consumption, as human beings are known1 to change from a walk to a run at a particular speed (2.4 m s−1) below which walking requires less energy than running and above which the opposite is true. Thus by changing gait, human beings keep the energy cost of locomotion to a minimum as their speed increases. One reason this relation holds is that in humans, metabolic rate increases curvilinearly with walking speed1. If metabolism were a curvilinear function of speed within each of the gaits used by quadrupeds, it would support the hypothesis that they also change gait to minimize energetic cost. There is an old controversy about whether metabolic rate increases linearly or curvilinearly in running humans1,2 but all previous reports have suggested that metabolic rate increases linearly with speed in quadrupeds. Extended gaits were an important experimental tool in the study of human gait changes; thus we have trained three small horses (110–170 kg) to walk, trot and gallop on a motorized treadmill, and to extend their gaits on command. We report here that, using measurements of rates of oxygen consumption as an indicator of rates of energy consumption, we have confirmed that the natural gait at any speed indeed entails the smallest possible energy expenditure.

title: EXOSKELETAL STRAIN: EVIDENCE FOR A TROT-GALLOP TRANSITION IN RAPIDLY RUNNING GHOST CRABS
abstract: Equivalent gaits may be present in pedestrians that differ greatly in leg number, leg design and skeletal type. Previous studies on ghost crabs found that the transition from a slow to a fast run may resemble the change from a trot to a gallop in quadrupedal mammals. One indication of the trot‐gallop gait change in quadrupedal mammals is a distinct alteration in bone strain. To test the hypothesis that ghost crabs ( Ocypode quadrata) change from a trot to a gallop, we measured in vivo strains of the meropodite of the second trailing leg with miniature strain gauges. Exoskeletal strains changed significantly (increased fivefold) during treadmill locomotion at the proposed trot‐gallop transition. Maximum strains attained during galloping and jumping (100031026‐300031026) were similar to the values reported for mammals. Comparison of the maximum load possible on the leg segment (caused by muscular tension) with the strength of the segment under axial loading revealed a safety factor of 2.7, which is similar to values measured for jumping and running mammals. Equivalent gaits may result from similarities in the operation of pedestrian locomotory systems.

title: Partitioning the Energetics of Walking and Running: Swinging the Limbs Is Expensive
abstract: Explaining the energetics of walking and running has been difficult because the distribution of energy use among individual muscles has not been known. We estimated energy use by measuring blood flow to the hindlimb muscles in guinea fowl. Blood flow to skeletal muscles is controlled locally and varies directly with metabolic rate. We estimate that the swing-phase muscles consume 26% of the energy used by the limbs and the stance-phase muscles consume the remaining 74%, independent of speed. Thus, contrary to some previous suggestions, swinging the limbs requires an appreciable fraction of the energy used during terrestrial legged locomotion. Models integrating the energetics and mechanics of running will benefit from more detailed information on the distribution of energy use by the muscles.

title: An analytical estimation of the energy cost for legged locomotion.
abstract: 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.

title: Energetics of locomotion and patterns of respiration in tenebrionid beetles from the Namib Desert
abstract: Summary1.Rates of oxygen consumption
$$\left(  } \right)$$
 were measured in 4 species of flightless tenebrionids ranging in mass from 0.6 to 1.3 g. Three were fast-moving day-active species (Onymacris plana, Physadesmia globosa, Physosterna cribripes), and one was slow-moving and crepuscular (Epiphysa arenicola).2.For motionless animals at 22–24 °C during the inactive phase of their daily cycle, the regression of log
$$\dot V_{O_2 } $$
 (ml O2/h) on mass (g) is described by Y=0.105X0.796.3.In maleO. plana
$$\dot V_{O_2 } $$
 increased with running speed up to 13 cm/s. From 13 to 22 cm/s the slope of the regression of
$$\dot V_{O_2 } $$
 on speed did not differ significantly from zero. We hypothesize that the independence of
$$\dot V_{O_2 } $$
 and high running speeds inO. plana results from the generation of aerodynamic lift by its flattened and laterally expanded body which acts as an airfoil. InP. globosa andP. cribripes
$$\dot V_{O_2 } $$
 increased linearly with running speed at all speeds measured.4.
$$\dot V_{O_2 } $$
 was strongly cyclical. The periodicity was most conspicuous in motionless animals but was also apparent during locomotion. The cycles corresponded with periods of ventilation associated with episodes of pumping by the abdomen which lies within a space inclosed in a rigid chitinous capsule formed by the fused elytra and fused abdominal sternites.5.Factorial metabolic scope was greates inO. plana, the most cursorial of the species studied.

title: The Mechanism of Rapid Running in the Ghost Crab, Ocypode Ceratophthalma
abstract: 1. Ocypode ceratophthalma has a maximum speed of 2.1 m/sec when running on a measured track with a base of hard-packed sand. Speed increases linearly with the width of the carapace up to a certain size, beyond which larger crabs run slower than smaller ones. 2. The crab does not run at these high speeds by making extremely rapid movements as these data seemed to require. The highest frequency of leg movements observed was 20 Hz. 3. Electromyographs of muscles used in running, made from the freely running, intact crab, showed asymmetry in the motoneurone discharges. Extensors and flexors in the meropodites of legs on the leading side frequently showed only a maintained tonus and could not have contributed to the running movements. Those on the trailing side showed alternation synchronous with stepping. 4. It is concluded that the crab mainly pushes itself along rather than using a push-pull combination. 5. Three pairs of legs are commonly used in running. Legs 2 and 4 of one side move together with leg 3 of the opposite side and provide a tripod of support. At the highest speeds only legs 2 and 3 of one side are used to provide thrust alternately. 6. The high speed is achieved by the crab leaping through the air as it steps, thereby increasing the effective length of the steps.

title: Walking and running in the desert ant Cataglyphis fortis
abstract: Path integration, although inherently error-prone, is a common navigation strategy in animals, particularly where environmental orientation cues are rare. The desert ant Cataglyphis fortis is a prominent example, covering large distances on foraging excursions. The stride integrator is probably the major source of path integration errors. A detailed analysis of walking behaviour in Cataglyphis is thus of importance for assessing possible sources of errors and potential compensation strategies. Zollikofer ( 1994a) demonstrated consistent use of the tripod gait in Cataglyphis, and suggested an unexpectedly constant stride length as a possible means of reducing navigation errors. Here, we extend these studies by more detailed analyses of walking behaviour across a large range of walking speeds. Stride length increases linearly and stride amplitude of the middle legs increases slightly linearly with walking speed. An initial decrease of swing phase duration is observed at lower velocities with increasing walking speed. Then it stays constant across the behaviourally relevant range of walking speeds. Walking speed is increased by shortening of the stance phase and of the stance phase overlap. At speeds larger than 370 mms−1, the stride frequency levels off, the duty factor falls below 0.5, and Cataglyphis transitions to running with aerial phases.

title: Locomotor mechanics of the tölt in Icelandic horses.
abstract: To evaluate the locomotor mechanics of the tölt in Icelandic horses.



10 adult Icelandic horses with no history of lameness.


PROCEDURES
Force platform data were captured for 27 trials for horses ridden at a tölt in a lateral sequence single-foot gait at a steady speed from 0.89 to 5.98 m/s. Simultaneous kinematic data were obtained by tracking retroflective markers overlying the right fore- and hind limbs. These kinetic and kinematic data were combined to evaluate 3 mechanical approaches, duty factor, Froude number, and center of mass (COM) mechanics, and to evaluate the capacity to recover mechanical energies during tölting via inverse pendulum and spring-mass (bouncing) mechanics.



Tölting horses had in-phase fluctuations of gravitational potential and kinetic energies of their COM and a capacity to recover mechanical energy through elastic recoil of spring elements in their limbs. These characteristics, along with Froude numbers exceeding values expected for the walk-run transition, are indicative of bouncing mechanics and, hence, most strongly ally tölting with running. Only the footfall pattern of a lateral sequence single-foot gait and low vertical excursions of the COM are more commonly associated with walking.



At the tölt, horses have unique mechanical characteristics that should be understood for veterinary care. Differences in interlimb coordination between tölting and trotting mask the overall similarities in most other aspects of their locomotor dynamics.

title: Controlling legs for locomotion-insights from robotics and neurobiology.
abstract: Walking is the most common terrestrial form of locomotion in animals. Its great versatility and flexibility has led to many attempts at building walking machines with similar capabilities. The control of walking is an active research area both in neurobiology and robotics, with a large and growing body of work. This paper gives an overview of the current knowledge on the control of legged locomotion in animals and machines and attempts to give walking control researchers from biology and robotics an overview of the current knowledge in both fields. We try to summarize the knowledge on the neurobiological basis of walking control in animals, emphasizing common principles seen in different species. In a section on walking robots, we review common approaches to walking controller design with a slight emphasis on biped walking control. We show where parallels between robotic and neurobiological walking controllers exist and how robotics and biology may benefit from each other. Finally, we discuss where research in the two fields diverges and suggest ways to bridge these gaps.

title: Static antennae act as locomotory guides that compensate for visual motion blur in a diurnal, keen-eyed predator
abstract: High visual acuity allows parallel processing of distant environmental features, but only when photons are abundant enough. Diurnal tiger beetles (Carabidae: Cicindelinae) have acute vision for insects and visually pursue prey in open, flat habitats. Their fast running speed causes motion blur that degrades visual contrast, forces stop-and-go pursuit and potentially impairs obstacle detection. We demonstrate here that vision is insufficient for obstacle detection during running, and show instead that antennal touch is both necessary and sufficient for obstacle detection. While running, tiger beetle vision appears to be photon-limited in a way reminiscent of animals in low-light habitats. Such animals often acquire wide-field spatial information through mechanosensation mediated by longer, more mobile appendages. We show that a nocturnal tiger beetle species waves its antennae in elliptical patterns typical of poorly sighted insects. While antennae of diurnal species are also used for mechanosensation, they are rigidly held forward with the tips close to the substrate. This enables timely detection of path obstructions followed by an increase in body pitch to avoid collision. Our results demonstrate adaptive mechanosensory augmentation of blurred visual information during fast locomotion, and suggest that future studies may reveal non-visual sensory compensation in other fast-moving animals.

title: A collisional model of the energetic cost of support work qualitatively explains leg sequencing in walking and galloping, pseudo-elastic leg behavior in running and the walk-to-run transition.
abstract: Terrestrial legged locomotion requires repeated support forces to redirect the body's vertical velocity component from down to up. We assume that the redirection is accomplished by impulsive leg forces that cause small-angle glancing collisions of a point-mass model of the animal. We estimate the energetic costs of these collisions by assuming a metabolic cost proportional to positive muscle work involved in generating the impulses. The cost of bipedal running estimated from this collisional model becomes less than that of walking at a Froude number (v2/gl) of about 0.7. Two strategies to reduce locomotion costs associated with the motion redirection are: (1) having legs simulate purely elastic springs, as is observed in human running; and (2) sequencing the leg forces during the redirection phase; examples of this sequencing are the ba-da-dump pattern of a horse gallop and having push-off followed by heel-strike in human walking.

title: The role of leaf structure in vibration propagation.
abstract: The leaf and its structural components play a key role in the propagation of short transient signals produced by insects. In this paper, it is shown how the complex structure of an apple leaf could be modeled by a much simpler one for the analysis of vibratory signal propagation. Waves were produced by impacts of small spheres and the propagation studied using two laser vibrometers, followed by a wavelets analysis. Three components of the leaf were investigated: the midvein, minor veins, and the interspaced homogeneous regions making up the leaf lamina. The loss of signal energy over the leaf lamina and across minor veins and midvein was studied. For the midvein, the loss of energy decreased from 80% at the leaf base to 40% at the apex. For minor veins, the loss of energy decreased from 70% at the leaf base to 31% at the apex. The loss in homogeneous regions was 40%. A signal decomposition into two frequency ranges, above and below 1.7 kHz, showed that the midvein acted as a low-pass filter. As energy loss was mainly a function of vein diameter and not vein type, veins smaller or equal to 0.2 mm were considered as equivalent to homogeneous regions. Hence, a model leaf reduced to the leaf lamina and veins with a diameter >0.2 mm is retained for the study of signal propagation in a leaf.

title: Force and flow at the onset of drag in plowed granular media.
abstract: We study the transient drag force FD on a localized intruder in a granular medium composed of spherical glass particles. A flat plate is translated horizontally from rest through the granular medium to observe how FD varies as a function of the medium's initial volume fraction, ϕ. The force response of the granular material differs above and below the granular critical state, ϕc, the volume fraction which corresponds to the onset of grain dilatancy. For ϕ<ϕc FD increases monotonically with displacement and is independent of drag velocity for the range of velocities examined (<10 cm/s). For ϕ>ϕc, FD rapidly rises to a maximum and then decreases over further displacement. The maximum force for ϕ>ϕc increases with increasing drag velocity. In quasi-two-dimensional drag experiments, we use granular particle image velocimetry (PIV) to measure time resolved strain fields associated with the horizontal motion of a plate started from rest. PIV experiments show that the maxima in FD for ϕ>ϕc are associated with maxima in the spatially averaged shear strain field. For ϕ>ϕc the shear strain occurs in a narrow region in front of the plate, a shear band. For ϕ<ϕc the shear strain is not localized, the shear band fluctuates in space and time, and the average shear increases monotonically with displacement. Laser speckle measurements made at the granular surface ahead of the plate reveal that for ϕ<ϕc particles are in motion far from the intruder and shearing region. For ϕ>ϕc, surface particles move only during the formation of the shear band, coincident with the maxima in FD, after which the particles remain immobile until the sheared region reaches the measurement region.

title: Comparative three-dimensional kinematics of the hindlimb for high-speed bipedal and quadrupedal locomotion of lizards
abstract: Although lizards have been model organisms for testing locomotor performance and in ecomorphological studies, the limb movements of lizards during high-speed locomotion are poorly understood. Thus, we quantified the three-dimensional kinematics of the hindlimb, body and tail for five morphologically distinct species of lizard during steady-speed locomotion near maximum sprinting speed (2-5 m s-1). The kinematics of different species had little multivariate overlap. More than half of the strides of all species had digitigrade foot posture, but the frequency of using digitigrade foot posture varied among species. The combination of digitigrade foot posture and large foot size of the lizards contributed substantially to the high values of hip height. For each species, different suites of kinematic variables distinguished bipedal from quadrupedal strides. Interspecific morphological variation did not correspond globally to variation in kinematics, although lizard species with elongated hindlimbs took longer strides than species with shorter hindlimbs. The Froude numbers and relative stride lengths of all lizards running near maximal speeds were large compared with those reported previously for other vertebrates.