Task-level regulation enhances global stability of the simplest dynamic walker

A model of task-level human stepping regulation yields semistable walking

A simple lateral dynamic walker, with swing leg dynamics and three adjustable input parameters, is used to study how motor regulation affects frontal plane stepping. Motivated by experimental observations and phenomenological models, we imposed task-level multiobjective regulation targeting the walker's optimal lateral foot placement at each step. The regulator prioritizes achieving step width and lateral body position goals to varying degrees by choosing a mixture parameter. Our model thus integrates a lateral mechanical template , which captures fundamental mechanics of frontal-plane walking, with a lateral motor regulation template , an empirically verified model of how humans manipulate lateral foot placements in a goal-directed manner. The model captures experimentally observed stepping fluctuation statistics and demonstrates how linear empirical models of stepping dynamics can emerge from first-principles nonlinear mechanics. We find that task-level regulation gives rise to a goal equivalent manifold in the system's extended state space of mechanical states and inputs, a subset of which contains a continuum of period-1 gaits forming a semistable set: perturbations off of any of its gaits result in transients that return to the set, though typically to different gaits.

The condition for dynamic stability in humans walking with feedback control

The walking human body is mechanically unstable. Loss of stability and falling is more likely in certain groups of people, such as older adults or people with neuromotor impairments, as well as in certain situations, such as when experiencing conflicting or distracting sensory inputs. Stability during walking is often characterized biomechanically, by measures based on body dynamics and the base of support. Neural control of upright stability, on the other hand, does not factor into commonly used stability measures. Here we analyze stability of human walking accounting for both biomechanics and neural control, using a modeling approach. We define a walking system as a combination of biomechanics, using the well known inverted pendulum model, and neural control, using a proportional-derivative controller for foot placement based on the state of the center of mass at midstance. We analyze this system formally and show that for any choice of system parameters there is always one periodic orbit. We then determine when this periodic orbit is stable, i.e. how the neural control gain values have to be chosen for stable walking. Following the formal analysis, we use this model to make predictions about neural control gains and compare these predictions with the literature and existing experimental data. The model predicts that control gains should increase with decreasing cadence. This finding appears in agreement with literature showing stronger effects of visual or vestibular manipulations at different walking speeds.

Periodic gaits and flip bifurcation of a biped robot walking on level ground with two feasible switching patterns of motion

In this article, a biped robot walking on horizontal ground with two feasible switching patterns of motion (two-phase gait and three-phase gait) is presented. By using the first-order Taylor approximate at the equilibrium point, a simplified linear continuous dynamic equation is obtained to discuss the walking dynamics of the biped robot. Conditions for the existence and stability of period-1 gaits ( P ( 1 , 2 ) , P ( 1 , 3 ) ) and period-2 gaits ( P ( 2 , 2 , 2 ) , P ( 2 , 2 , 3 ) , P ( 2 , 3 , 3 ) ) are obtained by using a discrete map. Among the periodic gaits, the P ( 2 , 2 , 3 ) type gait has never been reported in previous studies. Flip bifurcation of periodic gait is investigated. Numerical results for periodic gaits and bifurcation diagram are in good agreement with the theoretical analysis.

Adaptive multi-objective control explains how humans make lateral maneuvers while walking

To successfully traverse their environment, humans often perform maneuvers to achieve desired task goals while simultaneously maintaining balance. Humans accomplish these tasks primarily by modulating their foot placements. As humans are more unstable laterally, we must better understand how humans modulate lateral foot placement. We previously developed a theoretical framework and corresponding computational models to describe how humans regulate lateral stepping during straight-ahead continuous walking. We identified goal functions for step width and lateral body position that define the walking task and determine the set of all possible task solutions as Goal Equivalent Manifolds (GEMs). Here, we used this framework to determine if humans can regulate lateral stepping during non-steady-state lateral maneuvers by minimizing errors consistent with these goal functions. Twenty young healthy adults each performed four lateral lane-change maneuvers in a virtual reality environment. Extending our general lateral stepping regulation framework, we first re-examined the requirements of such transient walking tasks. Doing so yielded new theoretical predictions regarding how steps during any such maneuver should be regulated to minimize error costs, consistent with the goals required at each step and with how these costs are adapted at each step during the maneuver. Humans performed the experimental lateral maneuvers in a manner consistent with our theoretical predictions. Furthermore, their stepping behavior was well modeled by allowing the parameters of our previous lateral stepping models to adapt from step to step. To our knowledge, our results are the first to demonstrate humans might use evolving cost landscapes in real time to perform such an adaptive motor task and, furthermore, that such adaptation can occur quickly-over only one step. Thus, the predictive capabilities of our general stepping regulation framework extend to a much greater range of walking tasks beyond just normal, straight-ahead walking.

Viability, task switching, and fall avoidance of the simplest dynamic walker

Walking humans display great versatility when achieving task goals, like avoiding obstacles or walking alongside others, but the relevance of this to fall avoidance remains unknown. We recently demonstrated a functional connection between the motor regulation needed to achieve task goals (e.g., maintaining walking speed) and a simple walker's ability to reject large disturbances. Here, for the same model, we identify the viability kernel-the largest state-space region where the walker can step forever via at least one sequence of push-off inputs per state. We further find that only a few basins of attraction of the speed-regulated walker's steady-state gaits can fully cover the viability kernel. This highlights a potentially important role of task-level motor regulation in fall avoidance. Therefore, we posit an adaptive hierarchical control/regulation strategy that switches between different task-level regulators to avoid falls. Our task switching controller only requires a target value of the regulated observable-a "task switch"-at every walking step, each chosen from a small, predetermined collection. Because humans have typically already learned to perform such goal-directed tasks during nominal walking conditions, this suggests that the "information cost" of biologically implementing such controllers for the nervous system, including cognitive demands in humans, could be quite low.

Comparing system identification techniques for identifying human-like walking controllers

While human walking has been well studied, the exact controller is unknown. This paper used human experimental walking data and system identification techniques to infer a human-like controller for a spring-loaded inverted pendulum (SLIP) model. Because the best system identification technique is unknown, three methods were used and compared. First, a linear system was found using ordinary least squares. A second linear system was found that both encoded the linearized SLIP model and matched the first linear system as closely as possible. A third nonlinear system used sparse identification of nonlinear dynamics (SINDY). When directly mapping states from the start to the end of a step, all three methods were accurate, with errors below 10% of the mean experimental values in most cases. When using the controllers in simulation, the errors were significantly higher but remained below 10% for all but one state. Thus, all three system identification methods generated accurate system models. Somewhat surprisingly, the linearized system was the most accurate, followed closely by SINDY. This suggests that nonlinear system identification techniques are not needed when finding a discrete human gait controller, at least for unperturbed walking. It may also suggest that human control of normal, unperturbed walking is approximately linear.

Implementation and acceleration of optimal control for systems biology

Optimal control theory provides insight into complex resource allocation decisions. The forward–backward sweep method (FBSM) is an iterative technique commonly implemented to solve two-point boundary value problems arising from the application of Pontryagin’s maximum principle (PMP) in optimal control. The FBSM is popular in systems biology as it scales well with system size and is straightforward to implement. In this review, we discuss the PMP approach to optimal control and the implementation of the FBSM. By conceptualizing the FBSM as a fixed point iteration process, we leverage and adapt existing acceleration techniques to improve its rate of convergence. We show that convergence improvement is attainable without prohibitively costly tuning of the acceleration techniques. Furthermore, we demonstrate that these methods can induce convergence where the underlying FBSM fails to converge. All code used in this work to implement the FBSM and acceleration techniques is available on GitHub at https://github.com/Jesse-Sharp/Sharp2021.

Walking humans trade off different task goals to regulate lateral stepping

People walk in complex environments where they must adapt their steps to maintain balance and satisfy changing task goals. How people do this is not well understood. We recently developed computational models of lateral stepping, based on Goal Equivalent Manifolds that serve as motor regulation templates, to identify how people regulate walking movements from step-to-step. In normal walking, healthy adults strongly maintain step width, but also lateral position on their path. Here, we used this framework to pose empirically-testable hypotheses about how humans might adapt their lateral stepping dynamics when asked to prioritize different stepping goals. Participants walked on a treadmill in a virtual-reality environment under 4 conditions: normal walking and, while given direct feedback at each step, walking while trying to maintain constant step width, constant absolute lateral position, or constant heading (direction). Time series of lateral stepping variables were extracted, and variability and statistical persistence (reflecting step-to-step regulation) quantified. Participants exhibited less variability of the prescribed stepping variable compared to normal walking during each feedback condition. Stepping regulation results supported our models' predictions: to maintain constant step width or position, people either maintained or increased regulation of the prescribed variable, but also decreased regulation of its complement. Thus, people regulated lateral foot placements in predictable and systematic ways determined by specific task goals. Humans regulate stepping movements to not only "just walk" (step without falling), but also to achieve specific goal-directed tasks within a specific environment. The framework and motor regulation templates presented here capture these important interactions.

Task-level regulation enhances global stability of the simplest dynamic walker

Much remains unknown about how considerations such as stability and energy minimization shape the way humans walk. While active neuromotor control keeps humans upright, they also need to choose from multiple stepping regulation strategies to achieve one or more task goals, such as maintaining a desired speed or direction. Experiments on human treadmill walking motivate an important question: why do humans prefer one task-level regulation strategy over another-perhaps to enhance their ability to reject large disturbances? Here, we study the relationship between task-level regulation and global stability in a powered compass walker on a treadmill, with added step-to-step speed and position regulators. For treadmill walking, we find that speed regulation greatly enlarges and regularizes the unregulated walker's stability region, i.e. its basin of attraction, much more than position regulation. Thus, our results suggest a possible explanation for the experimental finding that humans strongly prioritize regulating speed from one stride to the next, even as they walk economically on average. Furthermore, our work suggests a functional connection between task-level motor regulation and global stability-and, thus, perhaps even fall risk.