@misc{carlisle2022OptimizationEnergyTime,
  title = {Optimization of Energy and Time Predicts Dynamic Speeds for Human Walking},
  author = {Carlisle, R. Elizabeth and Kuo, Arthur D.},
  year = {2022},
  month = jul,
  publisher = {{bioRxiv}},
  doi = {10.1101/2022.07.15.500158},
  abstract = {Humans make a number of choices when they walk, such as how fast and for how long. The preferred steady walking speed seems chosen to minimize energy expenditure per distance traveled. But the speed of actual walking bouts is not only steady, but rather a time-varying trajectory, which can also be modulated by task urgency or an individual's movement vigor. Here we show that speed trajectories and durations of human walking bouts are explained better by an objective to minimize Energy and Time, meaning the total work or energy to reach destination, plus a cost proportional to bout duration. Applied to a computational model of walking dynamics, this objective predicts speed vs. time trajectories with inverted U shapes. Model and human experiment (N = 10) show that shorter bouts are unsteady and dominated by the time and effort of accelerating, and longer ones are steadier and faster due to energy-per-distance. Individual-dependent vigor is characterized by the energy one is willing to spend to save a unit of time, which explains why some may walk faster than others, but everyone has similar-shaped trajectories due to similar walking dynamics. Tradeoffs between energy and time costs predict transient, steady, and vigor-related aspects of walking.},
  copyright = {\textcopyright{} 2022, Posted by Cold Spring Harbor Laboratory. This pre-print is available under a Creative Commons License (Attribution-NonCommercial 4.0 International), CC BY-NC 4.0, as described at http://creativecommons.org/licenses/by-nc/4.0/},
  langid = {english}
}

@article{carlisle2023OptimizationEnergyTime,
  title = {Optimization of Energy and Time Predicts Dynamic Speeds for Human Walking},
  author = {Carlisle, Rebecca Elizabeth and Kuo, Arthur D},
  editor = {Berman, Gordon J},
  year = {2023},
  month = feb,
  journal = {eLife},
  volume = {12},
  pages = {e81939},
  publisher = {{eLife Sciences Publications, Ltd}},
  issn = {2050-084X},
  doi = {10.7554/eLife.81939},
  abstract = {Humans make a number of choices when they walk, such as how fast and for how long. The preferred steady walking speed seems chosen to minimize energy expenditure per distance traveled. But the speed of actual walking bouts is not only steady, but rather a time-varying trajectory, which can also be modulated by task urgency or an individual's movement vigor. Here we show that speed trajectories and durations of human walking bouts are explained better by an objective to minimize Energy and Time, meaning the total work or energy to reach destination, plus a cost proportional to bout duration. Applied to a computational model of walking dynamics, this objective predicts dynamic speed vs. time trajectories with inverted U shapes. Model and human experiment (\&\#x1D441; = 10) show that shorter bouts are unsteady and dominated by the time and effort of accelerating, and longer ones are steadier and faster and dominated by steady-state time and effort. Individual-dependent vigor may be characterized by the energy one is willing to spend to save a unit of time, which explains why some may walk faster than others, but everyone may have similar-shaped trajectories due to similar walking dynamics. Tradeoffs between energy and time costs can predict transient, steady, and vigor-related aspects of walking.}
}

@article{darici2018OptimalRegulationBipedal,
  title = {Optimal Regulation of Bipedal Walking Speed despite an Unexpected Bump in the Road},
  author = {Darici, Osman and Temeltas, Hakan and Kuo, Arthur D.},
  year = {2018},
  month = sep,
  journal = {PLOS ONE},
  volume = {13},
  number = {9},
  pages = {e0204205},
  publisher = {{Public Library of Science}},
  issn = {1932-6203},
  doi = {10.1371/journal.pone.0204205},
  abstract = {Bipedal locomotion may occur over imperfect surfaces with bumps or other features that disrupt steady gait. An unexpected bump in the road is generally expected to slow down most types of locomotion. On wheels, speed may be regained quite readily with ``cruise control'' performed in continuous time. But legged locomotion is less straightforward, because the stance leg may be under-actuated, and the continuous-time dynamics are periodically disrupted by discrete ground contact events. Those events may also afford good control opportunities, albeit subject to the delay between discrete opportunities. The regulation of walking speed should ideally use these opportunities to compensate for lost time, and with good economy if possible. However, the appropriate control strategy is unknown. Here we present how to restore speed and make up for time lost going over a bump in the road, through discrete, once-per-step control. We use a simple dynamic walking model to determine the optimal sequence of control actions\textemdash pushing off from the leg at the end of each stance phase\textemdash for fast response or best economy. A two-step, deadbeat sequence is the fastest possible response, and reasonably economical. Slower responses over more steps are more economical overall, but a bigger difference is that they demand considerably less peak power. A simple, reactive control strategy can thus compensate for an unexpected bump, with explicit trade-offs in time and work. Control of legged locomotion is not as straightforward as with wheels, but discrete control actions also allow for effective and economical reactions to imperfect terrain.},
  langid = {english},
  keywords = {Acceleration,Biological locomotion,Legs,Pendulums,Robots,Velocity,Walking,Wheels},
  annotation = {00009}
}

@article{darici2020AnticipatoryControlMomentum,
  title = {Anticipatory Control of Momentum for Bipedal Walking on Uneven Terrain},
  author = {Darici, Osman and Temeltas, Hakan and Kuo, Arthur D.},
  year = {2020},
  month = jan,
  journal = {Scientific Reports},
  volume = {10},
  number = {1},
  pages = {540},
  publisher = {{Nature Publishing Group}},
  issn = {2045-2322},
  doi = {10.1038/s41598-019-57156-6},
  abstract = {Humans and other walking bipeds often encounter and compensate for uneven terrain. They might, for example, regulate the body's momentum when stepping on stones to cross a stream. We examined what to do and how far to look, as a simple optimal control problem, where forward momentum is controlled to compensate for a step change in terrain height, and steady gait regained with no loss of time relative to nominal walking. We modeled planar, human-like walking with pendulum-like legs, and found the most economical control to be quite stereotypical. It starts by gaining momentum several footfalls ahead of an upward step, in anticipation of the momentum lost atop that step, and then ends with another speed-up to regain momentum thereafter. A similar pattern can be scaled to a variety of conditions, including both upward or downward steps, yet allow for considerably reduced overall energy and peak power demands, compared to compensation without anticipation. We define a ``persistence time'' metric from the transient decay response after a disturbance, to describe how momentum is retained between steps, and how far ahead a disturbance should be planned for. Anticipatory control of momentum can help to economically negotiate uneven terrain.},
  copyright = {2020 The Author(s)},
  langid = {english},
  annotation = {00000  Bandiera\_abtest: a Cc\_license\_type: cc\_by Cg\_type: Nature Research Journals Primary\_atype: Research Subject\_term: Biomedical engineering;Mechanical engineering Subject\_term\_id: biomedical-engineering;mechanical-engineering}
}

@article{darici2022HumansOptimallyAnticipate,
  title = {Humans Optimally Anticipate and Compensate for an Uneven Step during Walking},
  author = {Darici, Osman and Kuo, Arthur D.},
  editor = {Ting, Lena H},
  year = {2022},
  month = jan,
  journal = {eLife},
  volume = {11},
  pages = {e65402},
  publisher = {{eLife Sciences Publications, Ltd}},
  issn = {2050-084X},
  doi = {10.7554/eLife.65402},
  abstract = {The simple task of walking up a sidewalk curb is actually a dynamic prediction task. The curb is a disturbance that could cause a loss of momentum if not anticipated and compensated for. It might be possible to adjust momentum sufficiently to ensure undisturbed time of arrival, but there are infinite possible ways to do so. Much of steady, level gait is determined by energy economy, which should be at least as important with terrain disturbances. It is, however, unknown whether economy also governs walking up a curb, and whether anticipation helps. Here we show that humans compensate with an anticipatory pattern of forward speed adjustments, predicted by a criterion of minimizing mechanical energy input. The strategy is mechanistically predicted by optimal control for a simple model of bipedal walking dynamics, with each leg's push-off work as input. Optimization predicts a tri-phasic trajectory of speed (and thus momentum) adjustments, including an anticipatory phase. In experiment, human subjects ascend an artificial curb with the predicted tri-phasic trajectory, which approximately conserves overall walking speed relative to undisturbed flat ground. The trajectory involves speeding up in a few steps before the curb, losing considerable momentum from ascending it, and then regaining speed in a few steps thereafter. Descending the curb entails a nearly opposite, but still anticipatory, speed fluctuation trajectory, in agreement with model predictions that speed fluctuation amplitudes should scale linearly with curb height. The fluctuation amplitudes also decrease slightly with faster average speeds, also as predicted by model. Humans can reason about the dynamics of walking to plan anticipatory and economical control, even with a sidewalk curb in the way.},
  annotation = {00000}
}

@article{darici2022HumansPlanFuture,
  title = {Humans Plan for the near Future to Walk Economically on Uneven Terrain},
  author = {Darici, Osman and Kuo, Arthur D.},
  year = {2022},
  month = jul,
  journal = {arXiv},
  volume = {2207.11224},
  eprint = {2207.11224},
  eprinttype = {arxiv},
  primaryclass = {cs, eess, math, q-bio},
  doi = {10.48550/arXiv.2207.11224},
  abstract = {Humans experience small fluctuations in their gait when walking on uneven terrain. The fluctuations deviate from the steady, energy-minimizing pattern for level walking, and have no obvious organization. But humans often look ahead when they walk, and could potentially plan anticipatory fluctuations for the terrain. Such planning is only sensible if it serves some an objective purpose, such as maintaining constant speed or reducing energy expenditure, that is also attainable within finite planning capacity. Here we show that humans do plan and perform optimal control strategies on uneven terrain. Rather than maintain constant speed, they make purposeful, anticipatory speed adjustments that are consistent with minimizing energy expenditure. A simple optimal control model predicts economical speed fluctuations that agree well with experiments with humans (N = 12) walking on seven different terrain profiles (correlated with model r = 0.517 std. 0.109, P {$<$} 0.05 all terrains). Participants made repeatable speed fluctuations starting about seven to eight steps ahead of each terrain feature (up to 7.5 cm height difference each step, up to 16 consecutive features). They need not plan farther ahead, because each leg collision with ground dissipates energy, preventing momentum from persisting indefinitely. About seven to eight steps of continuous look-ahead and working memory thus suffice to practically optimize for any length of terrain. Humans reason about walking in the near future to plan complex optimal control sequences.},
  archiveprefix = {arXiv},
  keywords = {Computer Science - Robotics,Electrical Engineering and Systems Science - Systems and Control,Mathematics - Optimization and Control,Quantitative Biology - Neurons and Cognition}
}

@article{donelan2002MechanicalWorkSteptostep,
  title = {Mechanical Work for Step-to-Step Transitions Is a Major Determinant of the Metabolic Cost of Human Walking},
  author = {Donelan, J Maxwell and Kram, Rodger and Kuo, Arthur D.},
  year = {2002},
  month = dec,
  journal = {The Journal of Experimental Biology},
  volume = {205},
  number = {Pt 23},
  pages = {3717--3727},
  issn = {0022-0949},
  abstract = {In the single stance phase of walking, center of mass motion resembles that of an inverted pendulum. Theoretically, mechanical work is not necessary for producing the pendular motion, but work is needed to redirect the center of mass velocity from one pendular arc to the next during the transition between steps. A collision model predicts a rate of negative work proportional to the fourth power of step length. Positive work is required to restore the energy lost, potentially exacting a proportional metabolic cost. We tested these predictions with humans (N=9) walking over a range of step lengths (0.4-1.1 m) while keeping step frequency fixed at 1.8 Hz. We measured individual limb external mechanical work using force plates, and metabolic rate using indirect calorimetry. As predicted, average negative and positive external mechanical work rates increased with the fourth power of step length (from 1 W to 38 W; r(2)=0.96). Metabolic rate also increased with the fourth power of step length (from 7 W to 379 W; r(2)=0.95), and linearly with mechanical work rate. Mechanical work for step-to-step transitions, rather than pendular motion itself, appears to be a major determinant of the metabolic cost of walking.},
  lccn = {0155},
  pmid = {12409498},
  keywords = {Calorimetry; Indirect,Linear Models,Mathematics,Oxygen Consumption,Physical Exertion,Regression Analysis},
  annotation = {00804}
}

@article{kuo2001SimpleModelBipedala,
  title = {A Simple Model of Bipedal Walking Predicts the Preferred Speed-Step Length Relationship},
  author = {Kuo, Arthur D.},
  year = {2001},
  month = jun,
  journal = {Journal of Biomechanical Engineering},
  volume = {123},
  number = {3},
  pages = {264--269},
  issn = {0148-0731},
  abstract = {We used a simple model of passive dynamic walking, with the addition of active powering on level ground, to study the preferred relationship between speed and step length in humans. We tested several hypothetical metabolic costs, with one component proportional to the mechanical work associated with pushing off with the stance leg at toe-off, and another component associated with several possible costs of forcing oscillations of the swing leg. For this second component, a cost based on the amount of force needed to oscillate the leg divided by the time duration of that force predicts the preferred speed-step length relationship much better than other costs, such as the amount of mechanical work done in swinging the leg. The cost of force/time models the need to recruit fast muscle fibers for large forces at short durations. The actual mechanical work performed by muscles on the swing leg appears to be of relatively less importance, although it appears to be minimized by the use of short bursts of muscle activity in near-isometric conditions. The combined minimization of toe-off mechanical work and force divided by time predicts the preferred speed-step length relationship.},
  lccn = {0113},
  pmid = {11476370}
}

@article{kuo2002EnergeticsActivelyPowereda,
  title = {Energetics of Actively Powered Locomotion Using the Simplest Walking Model},
  author = {Kuo, Arthur D.},
  year = {2002},
  month = feb,
  journal = {Journal of Biomechanical Engineering},
  volume = {124},
  number = {1},
  pages = {113--120},
  issn = {0148-0731},
  abstract = {We modified an irreducibly simple model of passive dynamic walking to walk on level ground, and used it to study the energetics of walking and the preferred relationship between speed and step length in humans. Powered walking was explored using an impulse applied at toe-off immediately before heel strike, and a torque applied on the stance leg. Although both methods can supply energy through mechanical work on the center of mass, the toe-off impulse is four times less costly because it decreases the collision loss at heel strike. We also studied the use of a hip torque on the swing leg that tunes its frequency but adds no propulsive energy to gait. This spring-like actuation can further reduce the collision loss at heel strike, improving walking energetics. An idealized model yields a set of simple power laws relating the toe-off impulses and effective spring constant to the speed and step length of the corresponding gait. Simulations incorporating nonlinear equations of motion and more realistic inertial parameters show that these power laws apply to more complex models as well.},
  lccn = {0248},
  pmid = {11871597},
  keywords = {Computer Simulation,Hip,Posture,Sensitivity and Specificity},
  annotation = {00876}
}