David V. Lee

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.

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

SUMMARY 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.

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 Dynamics 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 BigDogs 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 robots 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 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.

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.

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.

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.

Linking gait dynamics to the 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 robots 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.

Conclusion: Related research projects and future directions

As the conclusion of the book, state-of-the-art research regarding legged locomotion and the application to daily activities are presented in this chapter. By comparing engineered legged systems with animals we address how far we are from nature. The recent technologies on assistive devices demonstrate the applicability of the scientific methods to fill this gap by developing products usable in human life. An overview of current research on legged locomotion presented in this chapter depicts the roadmap for future research.

Chapter 2 – Fundamental Subfunctions of Locomotion

Abstract Legged locomotion is a complex hybrid, nonlinear and highly dynamic problem. Animals have solved this complex problem as they are able to generate energy efficient and robust locomotion resulted from million years of evolution. However, different aspects of locomotion in biological legged systems such as mechanical design, actuation and control are still not fully understood. Splitting such a complicated problem to simpler subproblems may facilitate understanding and control of legged locomotion. Inspired from template models explaining biological locomotory systems and legged robots, we define three basic locomotor subfunctions: stance leg function, leg swinging and balancing. Combinations of these three subfunctions can generate different gaits with diverse properties. Basic analysis on human locomotion using conceptual models can result in developing new methods in design and control of legged systems like humanoid robots and assistive devices.