The rotary component of leg force during walking and running.

The component of ground reaction force (GRF) acting perpendicular to the leg in the sagittal plane during human locomotion (acting in a rotary direction) has not been systematically investigated and is not well understood. In this paper, we investigate this rotary component of the GRF of 11 human subjects (mean age ± s.d.: 26.6 ± 2.9 years) while walking and speed walking on a treadmill, along with eight human subjects (mean age ± s.d.: 26.3 ± 3.1) running on a treadmill. The GRF on both legs was measured, along with estimates of the subject's mass centre and the centre of pressure of each foot to yield total leg lengths and leg angle. Across all steady walking and running speeds, we find that the rotary component of the GRF has significant magnitude (peak values from 5% to 38% of body weight, from slow walking to moderate running, respectively) and implies leg propulsion of the mass centre in the rotary direction. Furthermore, peak rotary force magnitude over stance increases with locomotion speed for both walking and running ( p < 0.05), and the time-averaged (mean) rotary force shows a slight increase with walking speed (though the mean force trend is uncertain for running). Also, an estimate of average power input from the rotary force of the leg acting at the mass centre shows moderate and strong positive correlation with locomotion speed for running and walking respectively ( p < 0.05). This study also shows that the rotary force acts differently in walking versus running: rotary force is predominantly positive during running, but during walking it exhibits both positive and negative phases with net positive force found over the whole stride.

Effective leg stiffness of animal running and the co-optimization of energetic cost and stability.

The relative leg stiffness of most running animals falls in a small range between 7 and 27. Here we present a theoretical study of an established running model, an actuated Spring Loaded Inverted Pendulum model, to determine if the energetic cost and stability of running might be co-optimized over this range of leg stiffness values. The energetic cost of the model is quantified as the energy spent to move a unit mass a unit distance. The stability of the model is based on the system response to perturbations with respect to periodic locomotion solutions, and uses the linearized dynamics of Poincaré return maps and the resulting maximum eigenvalue and singular value decomposition in order to analyze asymptotic stability and the overall system response to perturbations, respectively. We find that there exists a tradeoff between stability and energetic cost in the model with respect to variation in forcing (actuation) level: For a given leg stiffness, the energetic cost tends to be more optimal with smaller forcing, and the opposite for stability. We find that intermediate levels of forcing can achieve near asymptotic stability or complete asymptotic stability while remaining small enough to yield a relatively low energetic cost consistent with human-like values. We demonstrate that this outcome can be achieved in the model with a simple optimization function that balances stability and energetic cost. We then investigate the stability and energetic cost when both leg stiffness and forcing are varied. Overall, the analysis shows that leg stiffness values in or near the biological range offers a good chance of simultaneously achieving both reasonable energetic cost and stability in the model. The results of this study suggest that stability and energetic cost may be interacting factors that have a combined influence on the effective leg stiffness and actuation (forcing) used by running animals.

A modelling approach to the dynamics of gait initiation.

Gait initiation is an integral and complex part of human locomotion. In this paper, we present a novel compliant-leg model-based approach to understanding the key phases of initiation, the nature of the effective forces involved in initiation, and the importance of the anticipatory postural adjustments (APAs). The results demonstrate that in the presence of APAs, we observe a change in the characteristic of forcing required for initiation, and the energetic cost of gait initiation is also reduced by approximately 58%. APAs also result in biologically relevant leg landing angles and trajectories of motion. Furthermore, we find that a sublinear functional relationship with the velocity error from steady state predicts the required force, consistent with an open loop control law basis for gait initiation.

Suspending loads decreases load stability but may slightly improve body stability.

Here, we seek to determine how compliantly suspended loads could affect the dynamic stability of legged locomotion. We theoretically model the dynamic stability of a human carrying a load using a coupled spring-mass-damper model and an actuated spring-loaded inverted pendulum model, as these models have demonstrated the ability to correctly predict other aspects of locomotion with a load in prior work, such as body forces and energetic cost. We report that minimizing the load suspension natural frequency and damping ratio significantly reduces the stability of the load mass but may slightly improve the body stability of locomotion when compared to a rigidly attached load. These results imply that a highly-compliant load suspension could help stabilize body motion during human, animal, or robot load carriage, but at the cost of a more awkward (less stable) load.

Effects of independently altering body weight and mass on the energetic cost of a human running model.

The mechanisms underlying the metabolic cost of running, and legged locomotion in general, remain to be well understood. Prior experimental studies show that the metabolic cost of human running correlates well with the vertical force generated to support body weight, the mechanical work done, and changes in the effective leg stiffness. Further, previous work shows that the metabolic cost of running decreases with decreasing body weight, increases with increasing body weight and mass, and does not significantly change with changing body mass alone. In the present study, we seek to uncover the basic mechanism underlying this existing experimental data. We find that an actuated spring-mass mechanism representing the effective mechanics of human running provides a mechanistic explanation for the previously reported changes in the metabolic cost of human running if the dimensionless relative leg stiffness (effective stiffness normalized by body weight and leg length) is regulated to be constant. The model presented in this paper provides a mechanical explanation for the changes in metabolic cost due to changing body weight and mass which have been previously measured experimentally and highlights the importance of active leg stiffness regulation during human running.

The leg stiffnesses animals use may improve the stability of locomotion.

Despite a wide diversity of running animals, their leg stiffness normalized by animal size and weight (a relative leg stiffness) resides in a narrow range between 7 and 27. Here we determine if the stability of locomotion could be a driving factor for the tight distribution of animal leg stiffness. We simulated an established physics-based model (the actuated Spring-Loaded Inverted Pendulum model) of animal running and found that, with the same energetic cost, perturbations to locomotion are optimally corrected when relative leg stiffness is within the biologically observed range. Here we show that the stability of locomotion, in combination with energetic cost, could be a significant factor influencing the nearly universally observed animal relative leg stiffness range. The energetic cost of locomotion has been widely acknowledged as influencing the evolution of physiology and locomotion behaviors. Specifically, its potential importance for relative leg stiffness has been demonstrated. Here, we demonstrate that stability of locomotion may also be a significant factor influencing relative leg stiffness.

Dynamics of carrying a load with a handle suspension.

Carrying loads with a compliant pole or backpack suspension can reduce the peak forces of the load acting on the body when the suspension natural frequency is tuned below the stepping frequency. Here we investigate a novel application for a load suspension that could be used to carry a load by hand, which is a common yet difficult method of load carriage and results in inherently asymmetric dynamics during load carriage. We hypothesize that the asymmetric dynamics of carrying a load in one hand will result in multiple locomotion frequency modes which can affect the forces of carrying a load with a handle suspension. We tested an adjustable-stiffness hand-held load suspension with four different natural frequency values while walking and running compared to a rigid handle. As expected, the peak forces acting on the body decrease compared to a rigid handle as the effective suspension stiffness decreases below the stepping frequency. However, the asymmetric dynamics of carrying a load with one hand introduce another frequency mode at half the stepping frequency which increases the peak forces acting on the body when the natural frequency of the handle is tuned near this frequency. We conclude that hand-held load suspensions should be designed to have a natural frequency below the half-stepping frequency of walking to minimize the peak forces and the musculoskeletal stress on the human body while carrying loads with one hand.

Animals prefer leg stiffness values that may reduce the energetic cost of locomotion.

Despite the neuromechanical complexity and wide diversity of running animals, most run with a center-of-mass motion that is similar to a simple mass bouncing on a spring. Further, when animals׳ effective leg stiffness is measured and normalized for size and weight, the resulting relative leg stiffness that most animals prefer lies in a narrow range between 7 and 27. Understanding why this nearly universal preference exists could shed light on how whole animal behaviors are organized. Here we show that the biologically preferred values of relative leg stiffness coincide with a theoretical minimal energetic cost of locomotion. This result strongly implies that animals select and regulate leg stiffness in order to reduce the energy required to move, thus providing animals an energetic advantage. This result also helps explain how high level control targets such as energy efficiency might influence overall physiological parameters and the underlying neuromechanics that produce it. Overall, the theory presented here provides an explanation for the existence of a nearly universal preferred leg stiffness. Also, the results of this work are beneficial for understanding the principles underlying human and animal locomotion, as well as for the development of prosthetic, orthotic and robotic devices.

A model of human walking energetics with an elastically-suspended load.

Elastically-suspended loads have been shown to reduce the peak forces acting on the body while walking with a load when the suspension stiffness and damping are minimized. However, it is not well understood how elastically-suspended loads can affect the energetic cost of walking. Prior work shows that elastically suspending a load can yield either an increase or decrease in the energetic cost of human walking, depending primarily on the suspension stiffness, load, and walking speed. It would be useful to have a simple explanation that reconciles apparent differences in existing data. The objective of this paper is to help explain different energetic outcomes found with experimental load suspension backpacks and to systematically investigate the effect of load suspension parameters on the energetic cost of human walking. A simple two-degree-of-freedom model is used to approximate the energetic cost of human walking with a suspended load. The energetic predictions of the model are consistent with existing experimental data and show how the suspension parameters, load mass, and walking speed can affect the energetic cost of walking. In general, the energetic cost of walking with a load is decreased compared to that of a stiffly-attached load when the natural frequency of a load suspension is tuned significantly below the resonant walking frequency. The model also shows that a compliant load suspension is more effective in reducing the energetic cost of walking with low suspension damping, high load mass, and fast walking speed. This simple model could improve our understanding of how elastic load-carrying devices affect the energetic cost of walking with a load.

Insects running on elastic surfaces.

In nature, cockroaches run rapidly over complex terrain such as leaf litter. These substrates are rarely rigid, and are frequently very compliant. Whether and how compliant surfaces change the dynamics of rapid insect locomotion has not been investigated to date largely due to experimental limitations. We tested the hypothesis that a running insect can maintain average forward speed over an extremely soft elastic surface (10 N m(-1)) equal to 2/3 of its virtual leg stiffness (15 N m(-1)). Cockroaches Blaberus discoidalis were able to maintain forward speed (mean +/- s.e.m., 37.2+/-0.6 cm s(-1) rigid surface versus 38.0+/-0.7 cm s(-1) elastic surface; repeated-measures ANOVA, P=0.45). Step frequency was unchanged (24.5+/-0.6 steps s(-1) rigid surface versus 24.7+/-0.4 steps s(-1) elastic surface; P=0.54). To uncover the mechanism, we measured the animal's centre of mass (COM) dynamics using a novel accelerometer backpack, attached very near the COM. Vertical acceleration of the COM on the elastic surface had a smaller peak-to-peak amplitude (11.50+/-0.33 m s(-2), rigid versus 7.7+/-0.14 m s(-2), elastic; P=0.04). The observed change in COM acceleration over an elastic surface required no change in effective stiffness when duty factor and ground stiffness were taken into account. Lowering of the COM towards the elastic surface caused the swing legs to land earlier, increasing the period of double support. A feedforward control model was consistent with the experimental results and provided one plausible, simple explanation of the mechanism.

Dynamics and stability of insect locomotion: a hexapedal model for horizontal plane motions.

We develop a simple hexapedal model for the dynamics of insect locomotion in the horizontal plane. Each leg is a linear spring endowed with two inputs, controlling force-free length and "hip" position, in a stereotypical feedforward pattern. These represent, in a simplified manner, the effects of neurally activated muscles in the animal and are determined from measured foot force and kinematic body data for cockroaches. We solve the three-degree-of-freedom Newtonian equations for coupled translation-yawing motions in response to the inputs and determine branches of periodic gaits over the animal's typical speed range. We demonstrate a close quantitative match to experiments and find both stable and unstable motions, depending upon input protocols. Our hexapedal model highlights the importance of stability in evaluating effective locomotor performance and in particular suggests that sprawled-posture runners with large lateral and opposing leg forces can be stable in the horizontal plane over a range of speeds, with minimal sensory feedback from the environment. Fore-aft force patterns characteristic of upright-posture runners can cause instability in the model. We find that stability can constrain fundamental gait parameters: our model is stable only when stride length and frequency match the patterns measured in the animal. Stability is not compromised by large joint moments during running because ground reaction forces tend to align along the leg and be directed toward the center of mass. Legs radiating in all directions and capable of generating large moments may allow very rapid turning and extraordinary maneuvers. Our results further weaken the hypothesis that polypedal, sprawled-posture locomotion with large lateral and opposing leg forces is less effective than upright posture running with fewer legs.