An inelastic quadrupedal model discovers four-beat walking, two-beat running, and pseudo-elastic actuation as energetically optimal.

It is widely held that quadrupeds choose steady gaits that minimize their energetic cost of transport, but it is difficult to explore the entire range of possible footfall sequences empirically. We present a simple model of a quadruped that can spontaneously produce any of the thousands of planar footfall sequences available to quadrupeds. The inelastic, planar model consists of two point masses connected with a rigid trunk on massless legs. It requires only center of mass position, hind and forelimb proportions and a stride-length to speed relationship as input. Through trajectory optimization of a work and force-rate cost, and a large sample of random initial guesses, we provide evidence for the global optimality of symmetrical four-beat walking at low speeds and two beat running (trotting) at intermediate speeds. Using input parameters based on measurements in dogs (Canis lupus familiaris), the model predicts the correct phase offset in walking and a realistic walk-trot transition speed. It also spontaneously reproduces the double-hump ground reaction force profile observed in walking, and the smooth single-hump profile observed in trotting. Actuation appears elastic, despite the model's lack of springs, suggesting that spring-like locomotory behaviour emerges as an optimal tradeoff between work minimization and force-rate penalties.

Form in the context of function: Fundamentals of an energy effective striding walk, the role of the plantigrade foot and its expected size.

What morphological and functional factors allow for the unique and characteristic upright striding walk of the hominin lineage? Predictive models of locomotion that arise from considering mechanisms of energy loss indicate that collision-like losses at the transition between stance limbs are important determinants of bipedal gait. Theoretical predictions argue that these collisional losses can be reduced by having "functional extra legs" which are physically the heel and the toe part of a single anatomical foot. The ideal spacing for these "functional legs" are up to a quarter of a stride length, depending on the model employed. We evaluate the foot in the context of the dynamics of a bipedal system and compare predictions of optimal foot size against empirical data from modern humans, the Laetoli footprint trackways, and chimpanzees walking bipedally. The dynamics-based modeling approach provides substantial insight into how, and why, walking works as it does, even though current models are too simple to make predictions at a level adequate to anticipate specific morphology except at the most general level.

Determinants of optimal leg use strategy: horizontal to vertical transition in the parkour wall climb.

This study examined the mechanics of the horizontal to vertical transition used by parkour athletes in wall climbing. We used this task as an alternative to normal running - where the functional options differ substantially - exposing the movement control priorities required to successfully complete the task. Ground reaction forces were measured in several expert parkour athletes and centre of mass trajectory was calculated from force plates embedded in the ground and the wall. Empirical measures were compared with movements predicted by a work-based control optimization model. The model captured the fundamental dynamics of the transition and therefore allowed an exploration of parameter sensitivity for success at the manoeuvre (run-up speed, foot placement, etc.). The optimal transition of both the model and the parkour athletes used a common intermediate run-up speed and appears determined largely by a trade-off between positive and negative leg work that accomplishes the task with minimum overall work.

Load carrying with flexible bamboo poles: optimization of a coupled oscillator system.

In Asia, flexible bamboo poles are routinely used to carry substantial loads on the shoulder. Various advantages have been attributed to this load-carrying strategy (e.g. reduced energy consumption), but experimental evidence remains inconsistent - possibly because carriers in previous studies were inexperienced. Theoretical models typically neglect the individual's capacity to optimize interactions with the oscillating load, leaving the complete dynamics underexplored. This study used a trajectory optimization model to predict gait adaptations that minimize work-based costs associated with carrying compliant loads and compared the outcomes with naturally selected gait adaptations of experienced pole carriers. Gait parameters and load interactions (e.g. relative amplitude and frequency, phase) were measured in rural farmworkers in Vietnam. Participants carried a range of loads with compliant and rigid poles and the energetic consequences of step frequency adjustments were evaluated using the model. When carrying large loads, the empirical step frequency changes associated with pole type (compliant versus rigid) were largely consistent with model predictions, in terms of direction (increase or decrease) and magnitude (by how much). Work-minimizing strategies explain changes in leg compliance, harmonic frequency oscillations and fluctuations in energetic cost associated with carrying loads on a compliant bamboo pole.

Reducing gravity takes the bounce out of running.

In gravity below Earth-normal, a person should be able to take higher leaps in running. We asked 10 subjects to run on a treadmill in five levels of simulated reduced gravity and optically tracked centre-of-mass kinematics. Subjects consistently reduced ballistic height compared with running in normal gravity. We explain this trend by considering the vertical take-off velocity (defined as maximum vertical velocity). Energetically optimal gaits should balance the energetic costs of ground-contact collisions (favouring lower take-off velocity), and step frequency penalties such as leg swing work (favouring higher take-off velocity, but less so in reduced gravity). Measured vertical take-off velocity scaled with the square root of gravitational acceleration, following energetic optimality predictions and explaining why ballistic height decreases in lower gravity. The success of work-based costs in predicting this behaviour challenges the notion that gait adaptation in reduced gravity results from an unloading of the stance phase. Only the relationship between take-off velocity and swing cost changes in reduced gravity; the energetic cost of the down-to-up transition for a given vertical take-off velocity does not change with gravity. Because lower gravity allows an elongated swing phase for a given take-off velocity, the motor control system can relax the vertical momentum change in the stance phase, thus reducing ballistic height, without great energetic penalty to leg swing work. Although it may seem counterintuitive, using less 'bouncy' gaits in reduced gravity is a strategy to reduce energetic costs, to which humans seem extremely sensitive.

Metabolic cost of human hopping.

To interpret the movement strategies employed in locomotion, it is necessary to understand the source of metabolic cost. Muscles must consume metabolic energy to do work, but also must consume energy to generate force. The energy lost during steady locomotion and, hence, the amount of mechanical work muscles need to perform to replace it can be reduced and, in theory, even eliminated by elastically storing and returning some portion of this energy via the tendons. However, even if muscles do not need to perform any mechanical work, they still must generate sufficient force to tension tendons and support body weight. This study shows that the metabolic cost per hop of human hopping can largely be explained by the cost of producing force over the duration of a hop. Metabolic cost determined via oxygen consumption is compared with theoretical predictions made using a number of different cost functions that include terms for average muscle work, force, force rate and impulse (time integral of muscle force). Muscle impulse alone predicts metabolic cost per hop as well as more complex functions that include terms for muscle work, force and force rate, and explains a large portion (92%) of the variation in metabolic cost per hop. This is equivalent to 1/effective mechanical advantage, explaining a large portion (66%) of the variation in metabolic cost per time per unit body weight. This result contrasts with studies that suggest that muscle force rate or muscle force rate per time determines the metabolic cost per time of force production in other bouncing gaits such as running.

The apparently contradictory energetics of hopping and running: the counter-intuitive effect of constraints resolves the paradox.

Metabolic rate appears to increase with the rate of force application for running. Leg function during ground contact is similar in hopping and running, so one might expect that this relationship would hold for hopping as well. Surprisingly, metabolic rate appeared to decrease with increasing force rate for hopping. However, this paradox is the result of comparing different cross-sections of the metabolic cost landscapes for hopping and running. The apparent relationship between metabolic rate and force rate observed in treadmill running is likely not a fundamental characteristic of muscle physiology, but a result of runners responding to speed constraints, i.e. runners selecting step frequencies that minimize metabolic cost per distance for a series of treadmill-specified speeds. Evaluating hopping metabolic rate over a narrow range of hop frequencies similar to that selected by treadmill runners yields energy use trends similar to those of running.

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.

Constrained optimization of metabolic cost in human hopping.

Constrained optimization of metabolic cost/distance travelled largely predicts the gait parameters selected by humans during walking and running. This study evaluates whether this is also the case for human hopping. Hop frequency (f), height (h) and metabolic energy expenditure were measured in partly constrained (f, h or hop speed, s ≡ fh, specified), fully constrained (both f and h specified) and unconstrained conditions (neither f nor h specified) for 4 min trials. Hop frequency and height were also measured in frequency-constrained (f specified), fully constrained (maximal height and f specified) and unconstrained conditions for 15 s trials. Metabolic cost surfaces were constructed from experimental data from the 4 min trials, and the least costly behaviour for each constraint was calculated. Subjects selected the same height-frequency pattern for all three partly constrained conditions because the metabolic cost/height surface for hopping was a slope with no observed minimum. The heights selected for the 15 s frequency-constrained trials were only slightly lower than maximal, the optimal behaviour predicted by constrained optimization of metabolic cost/height. This supports the hypothesis that constrained optimization of metabolic cost largely predicts movement selection during hopping. However, subjects often chose noticeably lower than optimal heights and higher than optimal frequencies during partly constrained and unconstrained conditions for the 4 min trials. It appears that they selected heights and frequencies that incurred a slightly greater metabolic cost/height in order to reduce metabolic cost/time to a level they could comfortably sustain for 4 min.

Neglected losses and key costs: tracking the energetics of walking and running.

As one of the most energetically demanding daily activities, locomotion has attracted substantial investigative attention. Although legged locomotion has been well described, it is currently not well understood. Looking at energy accounting might be a good pathway with which to solve this problem. One relatively simple way of analyzing energy management is to look directly at the flow of mechanical energy into and out of the system, in terms of costs and losses (with some attention to the mechanisms responsible for this flow). In this commentary we argue that a key source of energetic loss has largely been neglected: the redirection of body motion from downward to upward at each step. We discuss the role of this loss and the compensating energetic costs, identifying some of the general features of the trade-offs that determine gait optimization strategies. We find that even at a conceptual level, a focus on the main mechanism of loss and the strategies available to the organism to effectively compensate for losses can yield substantial insight into observations as diverse as the functional limits of a playground swing through to the strikingly different effect of reduced gravity on human walking and running. Such insight changes the interpretation of fundamental features of leg function, such as push-off timing and the role of elastic deflection during stance.

Competing Models of Work in Quadrupedal Walking: Center of Mass Work is Insufficient to Explain Stereotypical Gait.

The walking gaits of cursorial quadrupedal mammals tend to be highly stereotyped as a four-beat pattern with interspersed periods of double and triple stance, often with double-hump ground reaction force profiles. This pattern has long been associated with high energetic economy, due to low apparent work. However, there are differing ways of approximating the work performed during walking and, consequently, different interpretations of the primary mechanism leading to high economy. A focus on Net Center of Mass (COM) Work led to the claim that quadrupedal walking is efficient because it effectively trades potential and kinetic energy of the COM. Individual Limbs COM Work instead focuses on the ability of the limbs to manage the trajectory of the COM to limit energetic losses to the ground ("collisions"). By focusing on the COM, both these metrics effectively dismiss the importance of rotation of the elongate quadrupedal body. Limb Extension Work considers work required to extend and contract each limb like a strut, and accounts for the work of body pitching. We tested the prescriptive ability of these approximations of work by optimizing them within a quadrupedal model with two approximations of the body as a point-mass or a rigid distributed mass. Perfect potential-kinetic energy exchange of the COM was possible when optimizing Net COM Work, resulting in highly compliant gaits with duty factors close to one, far different than observed mammalian gaits. Optimizing Individual Limbs COM Work resulted in alternating periods of single limb stance. Only the distributed mass model, with Limb Extension Work as the cost, resulted in a solution similar to the stereotypical mammalian gait. These results suggest that maintaining a near-constant limb length, with distributed contacts, are more important mechanisms of economy than either transduction of potential-kinetic energy or COM collision mitigation for quadrupedal walking.

Evaluating the energetics of entrainment in a human-machine coupled oscillator system.

During locomotion, humans sometimes entrain (i.e. synchronize) their steps to external oscillations: e.g. swaying bridges, tandem walking, bouncy harnesses, vibrating treadmills, exoskeletons. Previous studies have discussed the role of nonlinear oscillators (e.g. central pattern generators) in facilitating entrainment. However, the energetics of such interactions are unknown. Given substantial evidence that humans prioritize economy during locomotion, we tested whether reduced metabolic expenditure is associated with human entrainment to vertical force oscillations, where frequency and amplitude were prescribed via a custom mechatronics system during walking. Although metabolic cost was not significantly reduced during entrainment, individuals expended less energy when the oscillation forces did net positive work on the body and roughly selected phase relationships that maximize positive work. It is possible that individuals use mechanical cues to infer energy cost and inform effective gait strategies. If so, an accurate prediction may rely on the relative stability of interactions with the environment. Our results suggest that entrainment occurs over a wide range of oscillation parameters, though not as a direct priority for minimizing metabolic cost. Instead, entrainment may act to stabilize interactions with the environment, thus increasing predictability for the effective implementation of internal models that guide energy minimization.

'Mini-interval gait' switching: understanding the positive implications of a novel training regime.

The neuromechanical reorganization required to change gaits imposes an energetic cost 75% greater than either a walking or running step at the same speed. By combining walking and running with the requisite gait switching transition steps, an exercise protocol can be generated with virtually any desired metabolic output even at relatively slow treadmill speed. Gait switching increases metabolic demand through discrete events, which can be tolerated more easily by individuals recovering from health problems, just as interval training allows greater work production for healthy individuals. In addition to cardio-respiratory benefits, 'mini-intervals' with frequent gait switching also provides positive effects and attributes such as distributing muscle group activation, re-training neural coordination, and avoiding repetitive joint overloading. It has the added benefit of developing stability during transitions while a safety hand rail is present which can lead to greater stability in more complex natural environments. Finally, increased mental focus may help avoid the monotony of usual treadmill workouts, aiding adherence to an exercise program. We review evidence for the cost increase of the gait transition step and explain the mechanisms involved. We also discuss literature supporting the range of benefits for mini-interval gait switching as a training and rehabilitation tool.

The Landscape of Movement Control in Locomotion: Cost, Strategy, and Solution.

Features of gait are determined at multiple levels, from the selection of the gait itself (e.g., walk or run) through the specific parameters utilized (stride length, frequency, etc.) to the pattern of muscular excitation. The ultimate choices are determined neurally, but what is involved with deciding on the appropriate strategy? Human locomotion appears stereotyped not so much because the pattern is predetermined, but because these movement patterns are good solutions for providing movement utilizing the machinery available to the individual (the legs and their requisite components). Under different circumstances the appropriate solution may differ broadly (different gait) or subtly (different parameters). Interpretation of the neural decision making process would benefit from understanding the influences that are utilized in the selection of the appropriate solution in any set of circumstances, including normal conditions. In this review we survey an array of studies that point to energetic cost as a key input to the gait coordination system, and not just an outcome of the gait pattern implemented. We then use that information to rigorously define the construct proposed by Sparrow and Newell (1998) where the effects of environment, organism, and task act as constraints determining the solution set available, and the coordination pattern is then implemented under pressure for energetic economy. The fit between the environment and the organism define affordances that can be actualized. We rely on a novel conceptualization of task that recognizes that the task goal needs to be separated from the mechanisms that achieve it so that the selection of a particular implementation strategy can be exposed and understood. This reformulation of the Sparrow and Newell construct is then linked to the proposed pressure for economy by considering it as an optimization problem, where the most readily selected gait strategy will be the one that achieves the task goal at (or near) the energetic minimum.

Step length and required friction in walking.

The effect of step length on minimum required coefficient of friction (microR) during a walking step was isolated from other features that influence the mechanics of foot contact (such as speed). Ground reaction force (GRF) from defined step lengths at consistent forward speed was used to calculate (microR), required coefficent of friction. Some individuals walked in a manner that generated a (microR) that was 50% larger than others, in spite of being restricted to the same speed, step length and step frequency. Unshod subjects had greater (microR) compared to shod subjects except at the shortest step lengths. Understanding the dynamic interaction of applied vertical and horizontal forces is necessary to develop strategies to effectively evaluate unsafe circumstances, or modify behavior and develop safer equipment (at home and/or in the workplace) to deal with adverse footing environments.

Compliant walking appears metabolically advantageous at extreme step lengths.

BACKGROUND: Humans alter gait in response to unusual gait circumstances to accomplish the task of walking. For instance, subjects spontaneously increase leg compliance at a step length threshold as step length increases. Here we test the hypothesis that this transition occurs based on the level of energy expenditure, where compliant walking becomes less energetically demanding at long step lengths. RESEARCH QUESTION: To map and compare the metabolic cost of normal and compliant walking as step length increases. METHODS: 10 healthy individuals walked on a treadmill using progressively increasing step lengths (100%, 120%, 140% and 160% of preferred step length), in both normal and compliant leg walking as energy expenditure was recorded via indirect calorimetry. Leg compliance was controlled by lowering the center-of-mass trajectory during stance, forcing the leg to flex and extend as the body moved over the foot contact. RESULTS: For normal step lengths, compliant leg walking was more costly than normal walking gait, but compliant leg walking energetic cost did not increase as rapidly for longer step lengths. This led to an intersection between normal and compliant walking cost curves at 114% relative step length (regression analysis; r2 = 0.92 for normal walking; r2 = 0.65 for compliant walking). SIGNIFICANCE: Compliant leg walking is less energetically demanding at longer step lengths where a spontaneous shift to compliant walking has been observed, suggesting the human motor control system is sensitive to energetic requirements and will employ alternate movement patterns if advantageous strategies are available. The transition could be attributed to the interplay between (i) leg work controlling body travel during single stance and (ii) leg work to control energy loss in the step-to-step transition. Compliant leg walking requires more stance leg work at normal step lengths, but involves less energy loss at the step-to-step transition for very long steps.

Properties of traditional bamboo carrying poles have implications for user interactions.

Compliant bamboo poles have long been used for load carriage in Asian cultures. Although this custom differs from Western conventions of rigid body attachments (e.g. backpack), potential benefits include reduced peak shoulder forces as well as metabolic transport cost savings. Evidence that carrying a flexible pole benefits locomotion remains mixed, perhaps in part because the properties of pole design (e.g. bamboo material, structural geometry, etc.) have largely been neglected. These properties influence vibrational forces and consequently, the energy required by the user to manage the oscillations. We collected authentic bamboo poles from northern Vietnam and characterized their design parameters. Four poles were extensively studied in the lab (load-deflection testing, resonance testing, and computed tomography scans of three-dimensional geometry), and 10 others were tested at a rural Vietnamese farm site (basic measures of form and resonance). A mass-spring-damper model was used to characterize a relationship between resonant frequency (which affects the energetics of the pole-carrier system) and pole properties concerning stiffness, damping, etc. Model predictions of resonant frequencies agreed well with empirical data. Although measured properties suggest the poles are not optimally designed to reduce peak oscillation forces, resonant frequencies are within range of a typical human walking cadence, and this is likely to have a consequence on locomotion energetics.

Affordance Boundaries Are Defined by Dynamic Capabilities of Parkour Athletes in Dropping from Various Heights.

Available behaviors are determined by the fit between features of the individual and reciprocal features of the environment. Beyond some critical boundary certain behaviors become impossible causing sudden transitions from one movement pattern to another. Parkour athletes have developed multiple movement patterns to deal with their momentum during landing. We were interested in whether drop distance would cause a sudden transition between a two-footed (precision) landing and a load-distributing roll and whether the transition height could be predicted by dynamic and geometric characteristics of individual subjects. Kinematics and ground reaction forces were measured as Parkour athletes stepped off a box from heights that were incrementally increased or decreased from 0.6 to 2.3 m. Individuals were more likely to roll from higher drops; those with greater body mass and less explosive leg power, were more likely to transition to a roll landing at a lower height. At some height a two-footed landing is no longer feasible but for some athletes this height was well within the maximum drop height used in this study. During low drops the primary task constraint of managing momentum could be achieved with either a precision landing or a roll. This meant that participants were free to select their preferred landing strategy, which was only partially influenced by the physical demands of the task. However, athletes with greater leg power appeared capable of managing impulse absorption through a leg mediated strategy up to a greater drop height.

The goal of locomotion: Separating the fundamental task from the mechanisms that accomplish it.

Human locomotion has been well described but is still not well understood. This is largely true because the observable aspects of locomotion-neuromuscular activity that generates forces and motions-relate to both the task solution and the problem being solved. Identifying the fundamental task achieved in locomotion makes it possible to critically evaluate the motor control strategy used to accomplish the task goal. We contend that the readily observed movements and activities of locomotion should be considered mechanism(s). Our proposal is that the fundamental task of walking and running is analogous to flight, and should be defined in terms of the interaction of the individual's mass with the medium in which it moves: a low-density fluid for flight, or the supporting substrate for legged locomotion. A rigorous definition of the fundamental task can help identify the constraints and opportunities that influence its solution and guide the selection of appropriate mechanisms to accomplish the task effectively. The results from robotics-based modeling studies have demonstrated how the interaction of the mass and substrate can be optimized, making the goal of movement a defined trajectory of the individual's mass. We assessed these concepts by evaluating the ground reaction forces generated by an optimization model that satisfies the task but uses none of the mechanisms that are available to the human leg. Then we compared this model to normal human walking. Although it is obvious that the specific task of locomotion changes with a variety of movement challenges, clearly identifying the fundamental task of locomotion puts all other features in an interpretable context.