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(a) A convex function. For the convex function, the function value (solid line) at some velocity v is less than or equal to any weighted average (dotted line) of the function values at two or more velocities (say v 1 and v 2 ) whose weighted average is v. (b) A non-convex function. For the non-convex function, there exist velocities v 1 and v 2 such that the aforementioned inequality is violated. (c) The objective is to keep the mass at a given position on average, with zero average velocity. (d ) The objective is to exert a given average force on the wall, with possibly some periodic motion of the point P.
Source publication
A popular hypothesis regarding legged locomotion is that humans and other large animals walk and run in a manner that minimizes the metabolic energy expenditure for locomotion. Here, using numerical optimization and supporting analytical arguments, I obtain the energy-minimizing gaits of many different simple biped models. I consider bipeds with po...
Contexts in source publication
Context 1
... an average force. Consider the situ- ation shown in figure 3c. The goal is to use the muscle to support the mass (m ¼ 1) at some vertical position on average, in the presence of gravity, while minimizing the cost C g with a strictly convex g(v), over some long time period (or per unit time). ...
Context 2
... equation of motion is v ˙ ¼ F 2 1 (gravity constant ¼1). The periodicity equations imply figure 3a,b, if some point on the solid line represents the left-hand side of equation (6.3), the point on the dotted straight line with the same abscissa (v-value) represents the right-hand side of equation (6.3). ...
Context 3
... of non-convexity. If g(v) were not convex (concave) around v ¼ 0, for instance, as in figure 3b, there exists a time-varying m(t), and therefore F(t), for which the above inequalities are reversed; that is, C g g(0)t p and a time-varying F(t) ('tremor') would be better than holding still. In the non-convex example of figure 3b, one would be able to reduce the effective g(v) to the dotted line, from the solid line, by operating between v 1 and v 2 . ...
Citations
... In this expression, the absolute kinetic energy change from rest to speed v or from speed v to rest is mv 2 /2. This kinetic energy change is scaled by the reciprocal of approximate muscle efficiencies for performing positive and negative mechanical work (η pos = 0.25 and η neg = 1.2 [86]) to obtain, respectively, an estimate of the metabolic cost of positive work of acceleration mv 2 /(2η pos ) and the metabolic cost of negative work during deceleration mv 2 /(2η neg ). We then multiply this estimate by an empirically derived scaling factor a change , previously suggested by Seethapathi and Srinivasan [17], who denoted it λ instead of a change and estimated the metabolic cost of changing speeds by making subjects explicitly change walking speeds on an inertial frame and performing indirect calorimetry. ...
Preferred walking speed is a widely-used performance measure for people with mobility issues, but is usually measured in straight line walking for fixed distances or durations, and without explicitly accounting for turning. However, daily walking involves walking for bouts of different distances and walking with turning, with prior studies showing that short bouts with at most 10 steps could be 40% of all bouts and turning steps could be 8-50% of all steps. Here, we studied walking in a straight line for short distances (4 m to 23 m) and walking in circles (1 m to 3 m turning radii) in people with transtibial amputation or transfemoral amputation using a passive ankle-foot prosthesis (Jaipur Foot). We found that the study participants’ preferred walking speeds are lower for shorter straight-line walking distances and lower for circles of smaller radii, which is analogous to earlier results in subjects without amputation. Using inverse optimization, we estimated the cost of changing speeds and turning such that the observed preferred walking speeds in our experiments minimizes the total cost of walking. The inferred costs of changing speeds and turning were larger for subjects with amputation compared to subjects without amputation in a previous study, specifically, being 4x to 8x larger for the turning cost and being highest for subjects with transfemoral amputation. Such high costs inferred by inverse optimization could potentially include non-energetic costs such as due to joint or interfacial stress or stability concerns, as inverse optimization cannot distinguish such terms from true metabolic cost. These experimental findings and models capturing the experimental trends could inform prosthesis design and rehabilitation therapy to better assist changing speeds and turning tasks. Further, measuring the preferred speed for a range of distances and radii could be a more comprehensive subject-specific measure of walking performance than commonly used straight line walking metrics.
... Cavagna et al. [26] first proposed a one-degree-of-freedom (DOF) inverted pendulum model. The 1-DOF inverted pendulum is the simplest mechanical model and has been expanded in many versions (by adding springs, dampers, and telescopic actuators) [27][28][29][30][31][32][33][34][35][36] as the ...
Gait models are important for the design and control of lower limb exoskeletons. The inverted pendulum model has advantages in simplicity and computational efficiency, but it also has the limitations of oversimplification and lack of realism. This paper proposes a two-degrees-of-freedom (DOF) inverted pendulum walking model by considering the knee joints for describing the characteristics of human gait. A new parameter, roll factor, is defined to express foot function in the model, and the relationships between the roll factor and gait parameters are investigated. Experiments were conducted to verify the model by testing seven healthy adults at different walking speeds. The results demonstrate that the roll factor has a strong relationship with other gait kinematics parameters, so it can be used as a simple parameter for expressing gait kinematics. In addition, the roll factor can be used to identify walking styles with high accuracy, including small broken step walking at 99.57%, inefficient walking at 98.14%, and normal walking at 99.43%.
... Dynamic balance control is closely associated with energy expenditure (Donelan et al., 2001), and in healthy humans, symmetric walking is believed to be energetically optimal (Srinivasan, 2011;Ellis et al., 2013). According to a simple model, the metabolic cost of leg movement is predicted to depend more heavily on step frequency compared to step length (Kuo, 2001), meaning greater changes in temporal features have a greater impact on energy cost compared to spatial ones. ...
Introduction:
During locomotion, cutaneous reflexes play an essential role in rapidly responding to an external perturbation, for example, to prevent a fall when the foot contacts an obstacle. In cats and humans, cutaneous reflexes involve all four limbs and are task- and phase modulated to generate functionally appropriate whole-body responses.
Methods:
To assess task-dependent modulation of cutaneous interlimb reflexes, we electrically stimulated the superficial radial or superficial peroneal nerves in adult cats and recorded muscle activity in the four limbs during tied-belt (equal left-right speeds) and split-belt (different left-right speeds) locomotion.
Results:
We show that the pattern of intra- and interlimb cutaneous reflexes in fore- and hindlimbs muscles and their phase-dependent modulation were conserved during tied-belt and split-belt locomotion. Short-latency cutaneous reflex responses to muscles of the stimulated limb were more likely to be evoked and phase-modulated when compared to muscles in the other limbs. In some muscles, the degree of reflex modulation was significantly reduced during split-belt locomotion compared to tied-belt conditions. Split-belt locomotion increased the step-by-step variability of left-right symmetry, particularly spatially.
Discussion:
These results suggest that sensory signals related to left-right symmetry reduce cutaneous reflex modulation, potentially to avoid destabilizing an unstable pattern.
... Consideration of animal locomotor mechanics can be pursued at a number of levels, from the highly simplifying point-mass models [3][4][5][6][7][8][9][10] right through to the detailed musculoskeletal analyses (e.g. [11][12][13][14][15][16][17]). ...
Human walking appears complicated, with many muscles and joints performing rapidly varying roles over the stride. However, the function of walking is simple: to support body weight as it translates economically. Here, a scenario is proposed for the sequence of joint and muscle actions that achieves this function, with the timing of muscle loading and unloading driven by simple changes in geometry over stance. In the scenario, joints of the legs and feet are sequentially locked, resulting in a vaulting stance phase and three or five rapid 'mini-vaults' over a series of 'virtual legs' during the step-to-step transition. Collision mechanics indicate that the mechanical work demand is minimized if the changes in the centre-of-mass trajectory over the step-to-step transition are evenly spaced, predicting an even spacing of the virtual legs. The scenario provides a simple account for the work-minimizing mechanisms of joints and muscles in walking, and collision geometry allows leg and foot proportions to be predicted, accounting for the location of the knee halfway down the leg, and the relatively stiff, plantigrade, asymmetric, short-toed human foot.
... It is known that humans tend to prefer walking in metabolically economical ways [22]; hence, metabolic energy expenditure (specifically, the cost of transport or the energy consumed per distance traveled) often appears as a term in the cost function for predictive simulations of gait [2,[18][19][20][23][24][25][26][27][28][29][30]. Other commonly used terms include the minimization of muscle effort or work [24,26,28,[30][31][32][33][34][35][36][37], fatigue [35,36,38], ligament strain [29,30], joint torque [28,39], joint load [25], head motion [19,29,30], asymmetry [34,35], and the derivative of ground reaction force [30,33]-each of which can be justified biologically. Many combinations and relative weightings of these cost function terms have been used in the literature to generate predictive simulations of gait with realistic joint kinematics and other features. ...
Software packages that use optimization to predict the motion of dynamic systems are powerful tools for studying human movement. These “predictive simulations” are gaining popularity in parameter optimization studies for designing assistive devices such as exoskeletons. The cost function is a critical component of the optimization problem and can dramatically affect the solution. Many cost functions have been proposed that are biologically inspired and that produce reasonable solutions, but which may lead to different conclusions in some contexts. We used OpenSim Moco to generate predictive simulations of human walking using several cost functions, each of which produced a reasonable trajectory of the human model. We then augmented the model with motors that generated hip flexion, knee flexion, or ankle plantarflexion torques, and repeated the predictive simulations to determine the optimal motor torques. The model was assumed to be planar and bilaterally symmetric to reduce computation time. Peak torques varied from 41.3 to 79.0 N·m for the hip flexion motors, from 48.0 to 94.2 N·m for the knee flexion motors, and from 42.6 to 79.8 N·m for the ankle plantarflexion motors, which could have important design consequences. This study highlights the importance of evaluating the robustness of results from predictive simulations.
... In more recent years, a great variety of two-dimensional kinetic models based on the IP model have been developed in order to better mimic and then understand the pattern of human walking. These models include springs [11][12][13][14][15][16], dampers [17], and/or additional segments and joints [18,19] (for other works based on the IP model see, e.g., [20][21][22][23][24][25][26][27][28][29][30]). ...
... At the theoretical limit for vaulting in the hindlimbs (U ′ H 1), the hindlimbs also shift to bouncing ( Figure 6C) (see Supplementary Video S1 for animations of these solutions). This same general shift in gait is also seen at the lowest force-rate penalty ( Figures 6D-F), though the peak forces approach impulsivity, as is expected from work-minimizing optimization (Srinivasan, 2010). At walking speeds, the solutions can exhibit periods during stance where the hindlimbs are completely offloaded ( Figure 6D). ...
Fossil trackways provide a glimpse into the behavior of extinct animals. However, while providing information of the trackmaker size, stride, and even speed, the actual gait of the organism can be ambiguous. This is especially true of quadrupedal animals, where disparate gaits can have similar trackway patterns. Here, predictive simulation using trajectory optimization can help distinguish gaits used by trackmakers. First, we demonstrated that a planar, five-link quadrupedal biomechanical model can generate the qualitative trackway patterns made by domestic dogs, although a systematic error emerges in the track phase (relative distance between ipsilateral pes and manus prints). Next, we used trackway dimensions as inputs to a model of Batrachotomus kupferzellensis, a long-limbed, crocodile-line archosaur (clade Pseudosuchia) from the Middle Triassic of Germany. We found energetically optimal gaits and compared their predicted track phases to those of fossil trackways of Isochirotherium and Brachychirotherium. The optimal results agree with trackways at slow speeds but differ at faster speeds. However, all simulations point to a gait transition around a non-dimensional speed of 0.4 and another at 1.0. The trackways likewise exhibit stark differences in the track phase at these speeds. In all cases, including when simulations are constrained to the fossil track phase, the optimal simulations after the first gait transition do not correspond to a trot, as often used by living crocodiles. Instead, they are a diagonal sequence gait similar to the slow tölt of Icelandic horses. This is the first evidence that extinct pseudosuchians may have exhibited different gaits than their modern relatives and of a gait transition in an extinct pseudosuchian. The results of this analysis highlight areas where the models can be improved to generate more reliable predictions for fossil data while also showcasing how simple models can generate insights about the behavior of extinct animals.
... SLIP models can demonstrate steady state locomotion through equilibrium gaits, which can be produced by the correct choice of touchdown angle [12]. In legged locomotion, when a gait has the same height and velocity at the start and end of a cycle (Fig. 1), a limit cycle called an equilibrium gait has been achieved [13]. To apply these models to legged robots, damping and actuation can be added so that leg actuation strategies can be investigated. ...
In this paper, we investigate whether applying ankle torques during mid-stance can be a more effective way to reduce energetic cost of locomotion than actuating leg length alone. Ankles are useful in human gaits for many reasons including static balancing. In this work, we specifically avoid the heel-strike and toe-off benefits to investigate whether the progression of the center of pressure from heel-to-toe during mid-stance, or some other approach, is beneficial in and of itself. We use an "Ankle Actuated Spring Loaded Inverted Pendulum" model to simulate the shifting center of pressure dynamics, and trajectory optimization is applied to find limit cycles that minimize cost of transport. The results show that, for the vast majority of gaits, ankle torques do not affect cost of transport. Ankles reduce the cost of transport during a narrow band of gaits at the transition from grounded running to aerial running. This suggests that applying ankle torque during mid-stance of a steady gait is not a directly beneficial strategy, but is most likely a path between beneficial heel-strikes and toe-offs.
... There are also different types of human locomotion models, including simple dynamic models and data-driven mathematical models. These models have provided great insights into the dynamic principles of walking and running [167][168][169][170], the stability and optimality of steady and non-steady gaits [171][172][173][174][175][176][177], and the control and adaptation of legged locomotion [166,[178][179][180][181]. As these models often account for representative characteristics, such as the center of mass movement and foot placement, they could be used in modeling the higher layer of hierarchical controllers. ...
Modeling human motor control and predicting how humans will move in novel environments is a grand scientific challenge. Researchers in the fields of biomechanics and motor control have proposed and evaluated motor control models via neuromechanical simulations, which produce physically correct motions of a musculoskeletal model. Typically, researchers have developed control models that encode physiologically plausible motor control hypotheses and compared the resulting simulation behaviors to measurable human motion data. While such plausible control models were able to simulate and explain many basic locomotion behaviors (e.g. walking, running, and climbing stairs), modeling higher layer controls (e.g. processing environment cues, planning long-term motion strategies, and coordinating basic motor skills to navigate in dynamic and complex environments) remains a challenge. Recent advances in deep reinforcement learning lay a foundation for modeling these complex control processes and controlling a diverse repertoire of human movement; however, reinforcement learning has been rarely applied in neuromechanical simulation to model human control. In this paper, we review the current state of neuromechanical simulations, along with the fundamentals of reinforcement learning, as it applies to human locomotion. We also present a scientific competition and accompanying software platform, which we have organized to accelerate the use of reinforcement learning in neuromechanical simulations. This “Learn to Move” competition was an official competition at the NeurIPS conference from 2017 to 2019 and attracted over 1300 teams from around the world. Top teams adapted state-of-the-art deep reinforcement learning techniques and produced motions, such as quick turning and walk-to-stand transitions, that have not been demonstrated before in neuromechanical simulations without utilizing reference motion data. We close with a discussion of future opportunities at the intersection of human movement simulation and reinforcement learning and our plans to extend the Learn to Move competition to further facilitate interdisciplinary collaboration in modeling human motor control for biomechanics and rehabilitation research
... Despite the importance of turning in ecological settings, we do not have a coherent theoretical account of the paths and speeds observed in such locomotion. Human-subject experiments (2-7) and mathematical models (5,(8)(9)(10)(11)(12)(13)(14)(15) have suggested that energy optimality explains many aspects of straight-line locomotion, at least approximately. However, we do not know if such energy optimality generalizes to walking while navigating a more complex environment. ...
... This small force increase cannot directly account for the nearly 50% increases in metabolic costs we have measured. Similarly, if we conceptualize the additional centripetal acceleration as increasing the "effective gravity" (resulting in the increased leg forces) by 1 to 2%, then the leg-work requirements would also increase by a similar small percentage, and not by over 50% (12,42,43). One might speculate that the larger increases in metabolic rate may be partly due to recruitment of additional muscles that may not be at their optimal operating regimes (37). ...
... For more complex walking tasks, where the walking path is not predetermined (Figs. 5-7), we solved for the total time duration, body position, body orientation, and their derivatives as functions of time using numerical trajectory-optimization methods (12). For this trajectory optimization, we used the metabolic-cost function described in the previous paragraph. ...
Significance
Why do humans move the way they do? Here, we obtain a physiologically based theory of the speeds and paths with which humans navigate their environment. We measure the metabolic energy cost of walking with turning and show that minimizing this cost explains diverse phenomena involving navigating around obstacles, walking in complex paths, and turning. We explain why humans slow down while turning, avoid sharp turns, do not always use the shortest path, and other naturalistic locomotor phenomena.
