Climbing parrots achieve pitch stability using forces and free moments produced by axial-appendicular couples.

During vertical climbing, the gravitational moment tends to pitch the animal's head away from the climbing surface and this may be countered by (1) applying a correcting torque at a discrete contact point, or (2) applying opposing horizontal forces at separate contact points to produce a free moment. We tested these potential strategies in small parrots with an experimental climbing apparatus imitating the fine branches and vines of their natural habitat. The birds climbed on a vertical ladder with four instrumented rungs that measured three-dimensional force and torque, representing the first measurements of multiple contacts from a climbing bird. The parrots ascend primarily by pulling themselves upward using the beak and feet. They resist the gravitational pitching moment with a free moment produced by horizontal force couples between the beak and feet during the first third of the stride and the tail and feet during the last third of the stride. The reaction torque from individual rungs did not counter, but exacerbated the gravitational pitching moment, which was countered entirely by the free moment. Possible climbing limitations were explored using two different rung radii, each with low and high friction surfaces. Rung torque was limited in the large-radius, low-friction condition; however, rung condition did not significantly influence the free moments produced. These findings have implications for our understanding of avian locomotor modules (i.e. coordinated actions of the head-neck, hindlimbs and tail), the use of force couples in vertical locomotion, and the evolution of associated structures.

Tunnel-tube and Fourier methods for measuring three-dimensional medium reaction force in burrowing animals.

Subterranean digging behaviors provide opportunities for protection, access to prey, and predator avoidance for a diverse array of vertebrates, yet studies of the biomechanics of burrowing have been limited by the technical challenges of measuring kinetics and kinematics of animals moving within a medium. We describe a new system for measuring 3D reaction forces during burrowing, called a 'tunnel-tube', which is composed of two, separately instrumented plastic tubes: an 'entry tube' with no medium, in series with a 'digging tube' filled with medium. Mean reaction forces are measured for a digging bout and Fourier analysis is used to quantify the amplitude of oscillatory digging force as a function of frequency. In sample data from pocket gophers digging in artificial and natural media, the mean ground reaction force is constant, whereas Fourier analysis resolves a reduced amplitude of oscillatory force in the artificial medium with lower compaction strength.

Linking Gait Dynamics to Mechanical Cost of Legged Locomotion.

For millenia, legged locomotion has been of central importance to humans for hunting, agriculture, transportation, sport, and warfare. Today, the same principal considerations of locomotor performance and economy apply to legged systems designed to serve, assist, or be worn by humans in urban and natural environments. Energy comes at a premium not only for animals, wherein suitably fast and economical gaits are selected through organic evolution, but also for legged robots that must carry sufficient energy in their batteries. Although a robot's energy is spent at many levels, from control systems to actuators, we suggest that the mechanical cost of transport is an integral energy expenditure for any legged system-and measuring this cost permits the most direct comparison between gaits of legged animals and robots. Although legged robots have matched or even improved upon total cost of transport of animals, this is typically achieved by choosing extremely slow speeds or by using regenerative mechanisms. Legged robots have not yet reached the low mechanical cost of transport achieved at speeds used by bipedal and quadrupedal animals. Here we consider approaches used to analyze gaits and discuss a framework, termed mechanical cost analysis, that can be used to evaluate the economy of legged systems. This method uses a point mass perspective to evaluate the entire stride as well as to identify individual events that accrue mechanical cost. The analysis of gait began at the turn of the last century with spatiotemporal analysis facilitated by the advent of cine film. These advances gave rise to the "gait diagram," which plots duty factors and phase separations between footfalls. This approach was supplanted in the following decades by methods using force platforms to determine forces and motions of the center of mass (CoM)-and analytical models that characterize gait according to fluctuations in potential and kinetic energy. Mechanical cost analysis draws from these approaches and provides a unified framework that interprets the spatiotemporal sequencing of leg contacts within the context of CoM dynamics to determine mechanical cost in every instance of the stride. Diverse gaits can be evaluated and compared in biological and engineered systems using mechanical cost analysis.

Scaling of the spring in the leg during bouncing gaits of mammals.

Trotting, bipedal running, and especially hopping have long been considered the principal bouncing gaits of legged animals. We use the radial-leg spring constant [Formula: see text] to quantify the stiffness of the physical leg during bouncing gaits. The radial-leg is modeled as an extensible strut between the hip and the ground and [Formula: see text] is determined from the force and deflection of this strut in each instance of stance. A Hookean spring is modeled in-series with a linear actuator and the stiffness of this spring [Formula: see text] is determined by minimizing the work of the actuator while reproducing the measured force-deflection dynamics of an individual leg during trotting or running, and of the paired legs during hopping. Prior studies have estimated leg stiffness using [Formula: see text], a metric that imagines a virtual-leg connected to the center of mass. While [Formula: see text] has been applied extensively in human and comparative biomechanics, we show that [Formula: see text] more accurately models the spring in the leg when actuation is allowed, as is the case in biological and robotic systems. Our allometric analysis of [Formula: see text] in the kangaroo rat, tammar wallaby, dog, goat, and human during hopping, trotting, or running show that [Formula: see text] scales as body mass to the two-third power, which is consistent with the predictions of dynamic similarity and with the scaling of [Formula: see text]. Hence, two-third scaling of locomotor spring constants among mammals is supported by both the radial-leg and virtual-leg models, yet the scaling of [Formula: see text] emerges from work-minimization in the radial-leg model instead of being a defacto result of the ratio of force to length used to compute [Formula: see text]. Another key distinction between the virtual-leg and radial-leg is that [Formula: see text] is substantially greater than [Formula: see text], as indicated by a 30-37% greater scaling coefficient for [Formula: see text]. We also show that the legs of goats are on average twice as stiff as those of dogs of the same mass and that goats increase the stiffness of their legs, in part, by more nearly aligning their distal limb-joints with the ground reaction force vector. This study is the first allometric analysis of leg spring constants in two decades. By means of an independent model, our findings reinforce the two-third scaling of spring constants with body mass, while showing that springs in-series with actuators are stiffer than those predicted by energy-conservative models of the leg.

A comparative collision-based analysis of human gait.

This study compares human walking and running, and places them within the context of other mammalian gaits. We use a collision-based approach to analyse the fundamental dynamics of the centre of mass (CoM) according to three angles derived from the instantaneous force and velocity vectors. These dimensionless angles permit comparisons across gait, species and size. The collision angle Φ, which is equivalent to the dimensionless mechanical cost of transport CoTmech, is found to be three times greater during running than walking of humans. This threefold difference is consistent with previous studies of walking versus trotting of quadrupeds, albeit tends to be greater in the gaits of humans and hopping bipeds than in quadrupeds. Plotting the collision angle Φ together with the angles of the CoM force vector Θ and velocity vector Λ results in the functional grouping of bipedal and quadrupedal gaits according to their CoM dynamics-walking, galloping and ambling are distinguished as separate gaits that employ collision reduction, whereas trotting, running and hopping employ little collision reduction and represent more of a continuum that is influenced by dimensionless speed. Comparable with quadrupedal mammals, collision fraction (the ratio of actual to potential collision) is 0.51 during walking and 0.89 during running, indicating substantial collision reduction during walking, but not running, of humans.

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.

Modulation of joint moments and work in the goat hindlimb with locomotor speed and surface grade.

Goats and other quadrupeds must modulate the work output of their muscles to accommodate the changing mechanical demands associated with locomotion in their natural environments. This study examined which hindlimb joint moments goats use to generate and absorb mechanical energy on level and sloped surfaces over a range of locomotor speeds. Ground reaction forces and the three-dimensional locations of joint markers were recorded as goats walked, trotted and galloped over 0, +15 and -15 deg sloped surfaces. Net joint moments, powers and work were estimated at the goats' hip, knee, ankle and metatarsophalangeal joints throughout the stance phase via inverse dynamics calculations. Differences in locomotor speed on the level, inclined and declined surfaces were characterized and accounted for by fitting regression equations to the joint moment, power and work data plotted versus non-dimensionalized speed. During level locomotion, the net work generated by moments at each of the hindlimb joints was small (less than 0.1 J kg(-1) body mass) and did not vary substantially with gait or locomotor speed. During uphill running, by contrast, mechanical energy was generated at the hip, knee and ankle, and the net work at each of these joints increased dramatically with speed (P<0.05). The greatest increases in positive joint work occurred at the hip and ankle. During downhill running, mechanical energy was decreased in two main ways: goats generated larger knee extension moments in the first half of stance, absorbing energy as the knee flexed, and goats generated smaller ankle extension moments in the second half of stance, delivering less energy. The goats' hip extension moment in mid-stance was also diminished, contributing to the decrease in energy. These analyses offer new insight into quadrupedal locomotion, clarifying how the moments generated by hindlimb muscles modulate mechanical energy at different locomotor speeds and grades, as needed to accommodate the demands of variable terrain.

BigDog-inspired studies in the locomotion of goats and dogs.

Collision-based expenditure of mechanical energy and the compliance and geometry of the leg are fundamental, interrelated considerations in the mechanical design of legged runners. This article provides a basic context and rationale for experiments designed to inform each of these key areas in Boston Dynamic's BigDog robot. Although these principles have been investigated throughout the past few decades within different academic disciplines, BigDog required that they be considered together and in concert with an impressive set of control algorithms that are not discussed here. Although collision reduction is an important strategy for reducing mechanical cost of transport in the slowest and fastest quadrupedal gaits, walking and galloping, BigDog employed an intermediate-speed trotting gait without collision reduction. Trotting, instead, uses a spring-loaded inverted pendulum mechanism with potential for storage and return of elastic strain energy in appropriately compliant structures. Rather than tuning BigDog's built-in leg springs according to a spring-mass model-based virtual leg-spring constant , a much stiffer distal leg spring together with actuation of the adjacent joint provided good trotting dynamics and avoided functional limitations that might have been imposed by too much compliance in real-world terrain. Adjusting the directional compliance of the legs by adopting a knee-forward, elbow-back geometry led to more robust trotting dynamics by reducing perturbations about the pitch axis of the robot's center of mass (CoM). BigDog is the most successful large-scale, all-terrain trotting machine built to date and it continues to stimulate our understanding of legged locomotion in comparative biomechanics as well as in robotics.

A collisional perspective on quadrupedal gait dynamics.

The analysis of terrestrial locomotion over the past half century has focused largely on strategies of mechanical energy recovery used during walking and running. In contrast, we describe the underlying mechanics of legged locomotion as a collision-like interaction that redirects the centre of mass (CoM). We introduce the collision angle, determined by the angle between the CoM force and velocity vectors, and show by computing the collision fraction, a ratio of actual to potential collision, that the quadrupedal walk and gallop employ collision-reduction strategies while the trot permits greater collisions. We provide the first experimental evidence that a collision-based approach can differentiate quadrupedal gaits and quantify interspecific differences. Furthermore, we show that this approach explains the physical basis of a commonly used locomotion metric, the mechanical cost of transport. Collision angle and collision fraction provide a unifying analysis of legged locomotion which can be applied broadly across animal size, leg number and gait.

Effects of grade and mass distribution on the mechanics of trotting in dogs.

Quadrupedal running on grades requires balancing of pitch moments about the center of mass (COM) while supplying sufficient impulse to maintain a steady uphill or downhill velocity. Here, trotting mechanics on a 15 deg grade were characterized by the distribution of impulse between the limbs and the angle of resultant impulse at each limb. Anterior-posterior manipulation of COM position has previously been shown to influence limb mechanics during level trotting of dogs; hence, the combined effects of grade and COM manipulations were explored by adding 10% body mass at the COM, shoulder or pelvis. Whole body and individual limb ground reaction forces, as well as spatiotemporal step parameters, were measured during downhill and uphill trotting. Deviations from steady-speed locomotion were determined by the net impulse angle and accounted for in the statistical model. The limbs exerted only propulsive force during uphill trotting and, with the exception of slight hindlimb propulsion in late stance, only braking force during downhill trotting. Ratios of forelimb impulse to total impulse were computed for normal and shear components. Normal impulse ratios were more different from level values during uphill than downhill trotting, indicating that the limbs act more as levers on the incline. Differential limb function was evident in the extreme divergence of forelimb and hindlimb impulse angles, amplifying forelimb braking and hindlimb propulsive biases observed during level trotting. In both downhill and uphill trotting, added mass at the up-slope limb resulted in fore-hind distributions of normal impulse more similar to those of level trotting and more equal fore-hind distributions of shear impulse. The latter result suggests a functional trade-off in quadruped design: a COM closer to the hindlimbs would distribute downhill braking more equally, whereas a COM closer to the forelimbs would distribute uphill propulsion more equally. Because muscles exert less force when actively shortening than when lengthening, it would be advantageous for the forelimb and hindlimb muscles to share the propulsive burden more equally during uphill trotting. This functional advantage is consistent with the anterior COM position of most terrestrial quadrupeds.

Dynamics of goat distal hind limb muscle-tendon function in response to locomotor grade.

The functional roles of the lateral gastrocnemius (LG), medial gastrocnemius (MG) and superficial digital flexor (SDF) muscle-tendon units (MTUs) in domestic goats (N=6) were studied as a function of locomotor grade, testing the hypothesis that changes in distal limb muscle work would reflect changes in mechanical work requirements while goats walked or trotted on the level, 15 deg. decline and 15 deg. incline. As steep terrain-adapted animals, changes in muscle work output are expected to be particularly important for goats. In vivo muscle-tendon forces, fascicle length changes and muscle activation were recorded via tendon force buckles, sonomicrometry and electromyography to evaluate the work performance and elastic energy recovery of the three distal MTUs. These recordings confirmed that fascicle strain and force within goat distal hind limb muscles are adjusted in response to changes in mechanical work demand associated with locomotor grade. In general, muscle work was modulated most consistently by changes in fascicle strain, with increased net shortening (P<0.001) observed as goats switched from decline to level to incline locomotion. Peak muscle stresses increased as goats increased speed from a walk to a trot within each grade condition (P<0.05), and also increased significantly with grade (P<0.05 to P<0.01). Due to the increase in net fascicle shortening and muscle force, net muscle work per cycle also increased significantly (P<0.05 to P<0.005) as goats switched from decline to level to incline conditions (LG work: 20 mJ to 56 mJ to 209 mJ; MG work: -7 mJ to 34 mJ to 179 mJ; SDF work: -42 mJ to 14 mJ to 71 mJ, at a 2.5 ms(-1) trot). Although muscle work was modulated in response to changes in grade, the amount of work produced by these three distal pennate muscles was small (being <3%) in comparison with the change in mechanical energy required of the limb as a whole. Elastic energy recovery in the SDF and gastrocnemius (GA) tendons was substantial across all three grades, with the SDF tendon recovering 2.4 times more energy, on average, than the GA tendon. In parallel with the increase in muscle-tendon force, tendon energy recovery also increased as goats increased speed and changed gait, reaching the highest levels when goats trotted on an incline at 2.5 ms(-1) (GA: 173 mJ; SDF: 316 mJ). In general, tendon elastic energy exceeded net muscle work across all grade and gait conditions. These results demonstrate, for the first time in a quadruped, similar findings to those observed in ankle extensor muscles in humans, wallabies, turkeys and guinea fowl, suggesting that distal muscle-tendon architecture more generally favors a design for economic force production and tendon elastic energy recovery, with the majority of limb work during incline or decline running performed by larger proximal muscles.

Differential muscle function between muscle synergists: long and lateral heads of the triceps in jumping and landing goats (Capra hircus).

We investigate how the biarticular long head and monoarticular lateral head of the triceps brachii function in goats (Capra hircus) during jumping and landing. Elbow moment and work were measured from high-speed video and ground reaction force (GRF) recordings. Muscle activation and strain were measured via electromyography and sonomicrometry, and muscle stress was estimated from elbow moment and by partitioning stress based on its relative strain rate. Elbow joint and muscle function were compared among three types of limb usage: jump take-off (lead limb), the step prior to jump take-off (lag limb), and landing. We predicted that the strain and work patterns in the monoarticular lateral head would follow the kinematics and work of the elbow more closely than would those of the biarticular long head. In general this prediction was supported. For instance, the lateral head stretched (5 +/- 2%; mean +/- SE) in the lead and lag limbs to absorb work during elbow flexion and joint work absorption, while the long head shortened (-7 +/- 1%) to produce work. During elbow extension, both muscles shortened by similar amounts (-10 +/- 2% long; -13 +/- 4% lateral) in the lead limb to produce work. Both triceps heads functioned similarly in landing, stretching (13 +/- 3% in the long head and 19 +/- 5% in the lateral) to absorb energy. In general, the long head functioned to produce power at the shoulder and elbow, while the lateral head functioned to resist elbow flexion and absorb work, demonstrating that functional diversification can arise between mono- and biarticular muscle agonists operating at the same joint.

Compliance, actuation, and work characteristics of the goat foreleg and hindleg during level, uphill, and downhill running.

We model the action of muscle-tendon system(s) about a given joint as a serial actuator and spring. By this technique, the experimental joint moment is imposed while the combined angular deflection of the actuator and spring are constrained to match the experimental joint angle throughout the stance duration. The same technique is applied to the radial leg (i.e., shoulder/hip-to-foot). The spring constant that minimizes total actuator work is considered optimal, and this minimum work is expressed as a fraction of total joint/radial leg work, yielding an actuation ratio (AR; 1 = pure actuation and 0 = pure compliance). To address work modulation, we determined the specific net work (SNW), the absolute value of net divided by total work. This ratio is unity when only positive or negative work is done and zero when equal energy is absorbed and returned. Our proximodistal predictions of joint function are supported during level and 15 degrees grade running. The greatest AR and SNW are found in the proximal leg joints (elbow and knee). The ankle joint is the principal spring of the hindleg and shows no significant change in SNW with grade, reflecting the true compliance of the common calcaneal tendon. The principal foreleg spring is the metacarpophalangeal joint. The observed pattern of proximal actuation and distal compliance, as well as the substantial SNW at proximal joints, minimal SNW at intermediate joints, and variable energy absorption at distal joints, may emerge as general principles in quadruped limb mechanics and help to inform the leg designs of highly capable running robots.

Directionally compliant legs influence the intrinsic pitch behaviour of a trotting quadruped.

Limb design is well conserved among quadrupeds, notably, the knees point forward (i.e. cranial inclination of femora) and the elbows point back (i.e. caudal inclination of humeri). This study was undertaken to examine the effects of joint orientation on individual leg forces and centre of mass dynamics. Steady-speed trotting was simulated in two quadrupedal models. Model I had the knee and elbow orientation of a quadruped and model II had a reversed leg configuration in which knees point back and elbows point forward. The model's legs showed directional compliance determined by the orientation of the knee/elbow. In both models, forward pointing knees/elbows produced a propulsive force bias, while rearward pointing knees/elbows produced a braking force bias. Hence, model I showed the same pattern of hind-leg propulsion and fore-leg braking observed in trotting animals. Simulations revealed minimal pitch oscillations during steady-speed trotting of model I, but substantially greater and more irregular pitch oscillations of model II. The reduced pitch oscillation of model I was a result of fore-leg and hind-leg forces that reduced pitching moments during early and late stance, respectively. This passive mechanism for reducing pitch oscillations was an emergent property of directionally compliant legs with the fore-hind configuration of model I. Such intrinsic stability resulting from mechanical design can simplify control tasks and lead to more robust running machines.

Effects of mass distribution on the mechanics of level trotting in dogs.

The antero-posterior mass distribution of quadrupeds varies substantially amongst species, yet the functional implications of this design characteristic remain poorly understood. During trotting, the forelimb exerts a net braking force while the hindlimb exerts a net propulsive force. Steady speed locomotion requires that braking and propulsion of the stance limbs be equal in magnitude. We predicted that changes in body mass distribution would alter individual limb braking-propulsive force patterns and we tested this hypothesis by adding 10% body mass near the center of mass, at the pectoral girdle, or at the pelvic girdle of trotting dogs. Two force platforms in series recorded fore- and hindlimb ground reaction forces independently. Vertical and fore-aft impulses were calculated by integrating individual force-time curves and Fourier analysis was used to quantify the braking-propulsive (b-p) bias of the fore-aft force curve. We predicted that experimental manipulation of antero-posterior mass distribution would (1) change the fore-hind distribution of vertical impulse when the limb girdles are loaded, (2) decrease the b-p bias of the experimentally loaded limb and (3) increase relative contact time of the experimentally loaded limb, while (4) the individual limb mean fore-aft forces (normalized to body weight + added weight) would be unaffected. All four of these results were observed when mass was added at the pelvic girdle, but only 1, 3 and 4 were observed when mass was added at the pectoral girdle. We propose that the observed relationship between antero-posterior mass distribution and individual limb function may be broadly applicable to quadrupeds with different body types. In addition to the predicted results, our data show that the mechanical effects of adding mass to the trunk are much more complex than would be predicted from mass distribution alone. Effects of trunk moments due to loading were evident when mass was added at the center of mass or at the pelvic girdle. These results suggest a functional link between appendicular and axial mechanics via action of the limbs as levers.

Pace length effects in human walking: "groucho" gaits revisited.

In the present study, the authors explored the effect of manipulating the distance between footfalls (pace length) in level walking in 3 men. The translational kinetic energy to gravitational potential energy (E(k)-E(p)) exchange was found to be maintained until pace length reached approximately 110% of single limb length. Beyond that pace length, the compliance of the support limb increased, apparently to allow the muscular absorption of excess kinetic energy. E(p) oscillations decreased with increasing limb compliance, and the participants could control that relationship with pace length alone. The authors compared the present results with previous descriptions of the relationship between limb compliance and pace length in running. An analog of the running relationship holds for walking as well. The trigger for those gait modifications has not yet been identified, however. The present results have implications for the exploration of the integrated control of mechanical consequences in human locomotion and for understanding the walk-run transition in bipeds other than humans, such as ground-dwelling birds.