The formation of trajectories during goal-oriented locomotion in humans. II. A maximum smoothness model

Decentralized policy learning with partial observation and mechanical constraints for multiperson modeling

Extracting the rules of real-world multi-agent behaviors is a current challenge in various scientific and engineering fields. Biological agents independently have limited observation and mechanical constraints; however, most of the conventional data-driven models ignore such assumptions, resulting in lack of biological plausibility and model interpretability for behavioral analyses. Here we propose sequential generative models with partial observation and mechanical constraints in a decentralized manner, which can model agents' cognition and body dynamics, and predict biologically plausible behaviors. We formulate this as a decentralized multi-agent imitation-learning problem, leveraging binary partial observation and decentralized policy models based on hierarchical variational recurrent neural networks with physical and biomechanical penalties. Using real-world basketball and soccer datasets, we show the effectiveness of our method in terms of the constraint violations, long-term trajectory prediction, and partial observation. Our approach can be used as a multi-agent simulator to generate realistic trajectories using real-world data.

Generalizing stepping concepts to non-straight walking

People rarely walk in straight lines. Instead, we make frequent turns or other maneuvers. Spatiotemporal parameters fundamentally characterize gait. For straight walking, these parameters are well-defined for the task of walking on a straight path. Generalizing these concepts to non-straight walking, however, is not straightforward. People follow non-straight paths imposed by their environment (sidewalk, windy hiking trail, etc.) or choose readily-predictable, stereotypical paths of their own. People actively maintain lateral position to stay on their path and readily adapt their stepping when their path changes. We therefore propose a conceptually coherent convention that defines step lengths and widths relative to predefined walking paths. Our convention simply re-aligns lab-based coordinates to be tangent to a walker's path at the mid-point between the two footsteps that define each step. We hypothesized this would yield results both more correct and more consistent with notions from straight walking. We defined several common non-straight walking tasks: single turns, lateral lane changes, walking on circular paths, and walking on arbitrary curvilinear paths. For each, we simulated idealized step sequences denoting "perfect" performance with known constant step lengths and widths. We compared results to path-independent alternatives. For each, we directly quantified accuracy relative to known true values. Results strongly confirmed our hypothesis. Our convention returned vastly smaller errors and introduced no artificial stepping asymmetries across all tasks. All results for our convention rationally generalized concepts from straight walking. Taking walking paths explicitly into account as important task goals themselves thus resolves conceptual ambiguities of prior approaches.

Powered Knee and Ankle Prosthesis Control for Adaptive Ambulation at Variable Speeds, Inclines, and Uneven Terrains

Ambulation in everyday life requires walking at variable speeds, variable inclines, and variable terrains. Powered prostheses aim to provide this adaptability through control of the actuated joints. Some powered prosthesis controllers can adapt to discrete changes in speed and incline but require manual tuning to determine the control parameters, leading to poor clinical viability. Other data-driven controllers can continuously adapt to changes in speed and incline but do so by imposing the same non-amputee gait patterns for all amputee subjects, which does not consider subjective preferences and differing clinical needs of users. Here, we present a controller for powered knee and ankle prostheses that can continuously adapt to different walking speeds, inclines, and uneven terrains without enforcing a specific prosthesis position, impedance, or torque. A virtual biarticular muscle connection determines the knee flexion torque, which changes with both speed and slope. Adaptation to inclines and uneven terrains is based solely on the global shank orientation. Continuously variable damping allows for speed adaptation. Minimum-jerk programming defines the prosthesis swing trajectory at variable cadences. Experiments with one individual with an above-knee amputation suggest that the proposed controller can effectively adapt to different walking speeds, inclines, and rough terrains.

People adapt a consistent center-of-mass trajectory in a novel force field

During human walking the whole body center-of-mass (COM) trajectory may be a control objective, a goal the central nervous system uses to plan and regulate movement. Our previous observation, that after practice walking in a novel laterally directed force field people adapt a COM trajectory similar to their normal trajectory, supports this idea. However, our prior work only presented data demonstrating changes in COM trajectory in response to a single force field. To evaluate whether this phenomena is robust, in the present study we present new data demonstrating that people adapt their COM trajectory in a similar manner when the direction of the external force field is changed resulting in drastically different lower limb joint dynamics. Specifically, we applied a continuous, left-directed force field (in the previous experiment the force field was applied to the right) to the COM as participants performed repeated trials of a discrete walking task. We again hypothesized that with practice walking in the force field people would adapt a COM trajectory that was similar to their baseline performance and exhibit aftereffects, deviation of their COM trajectory in the opposite direction of force field, when the field was unexpectedly removed. These hypotheses were supported and suggest that participants formed an internal model to control their COM trajectory. Collectively these findings demonstrate that people adapt their gait patterns to anticipate consistent aspects of the external environment. These findings suggest that this response is robust to force fields applied in multiple directions that may require substantially different neural control.NEW & NOTEWORTHY With experience people adapted a predictive internal model to control their whole body center-of-mass walking trajectory that anticipated the disruptive laterally directed forces of a novel and consistent external environment. Collectively these findings demonstrate that adaptation of gait to anticipate consistent aspects of the external environment is a response that is robust to force fields in multiple directions that require substantially different lower limb dynamics and neural control.

Goal-oriented locomotion in children with spastic diplegia: Anticipatory orienting strategies and trajectory formation

Goal-oriented locomotion (GOL) is a complex task integrating navigation and gait control. To our knowledge, this is the first study of GOL in subjects with Cerebral Palsy (CP). Thirteen subjects with spastic diplegia and 26 with typical development were enrolled in the study. Subjects performed a GOL task to reach luminous targets. Within-subject trajectory variability, maximal head deviation from trajectory and mean head anticipation over trajectory were analyzed. While all subjects showed gait impairment, only 8 of 13 subjects also showed navigation abnormalities as revealed by either: a) abnormal head orientation and trajectory formation, or b) abnormal head orientation with normal trajectory formation. Abnormal gait patterns do not account for and can be distinguished from navigation disorders in spastic diplegic CP. This distinction has important implications for novel rehabilitation methods that should specifically address navigation, not only gait.

Variable Impedance Control Based on Target Position and Tracking Error for Rehabilitation Robots During a Reaching Task

To obtain an anthropomorphic performance in physical human-robot interaction during a reaching task, a variable impedance control (vIC) algorithm with human-like characteristics is proposed in this article. The damping value of the proposed method is varied with the target position as well as through the tracking error. The proposed control algorithm is compared with the impedance control algorithm with constant parameters (IC) and another vIC algorithm, which is only changed with the tracking error (vIC-e). The different control algorithms are validated through the simulation study, and are experimentally implemented on a cable-driven rehabilitation robot. The results show that the proposed vIC can improve the tracking accuracy and trajectory smoothness, and reduce the interaction force at the same time.

Control of Mammalian Locomotion by Somatosensory Feedback

When animals walk overground, mechanical stimuli activate various receptors located in muscles, joints, and skin. Afferents from these mechanoreceptors project to neuronal networks controlling locomotion in the spinal cord and brain. The dynamic interactions between the control systems at different levels of the neuraxis ensure that locomotion adjusts to its environment and meets task demands. In this article, we describe and discuss the essential contribution of somatosensory feedback to locomotion. We start with a discussion of how biomechanical properties of the body affect somatosensory feedback. We follow with the different types of mechanoreceptors and somatosensory afferents and their activity during locomotion. We then describe central projections to locomotor networks and the modulation of somatosensory feedback during locomotion and its mechanisms. We then discuss experimental approaches and animal models used to investigate the control of locomotion by somatosensory feedback before providing an overview of the different functional roles of somatosensory feedback for locomotion. Lastly, we briefly describe the role of somatosensory feedback in the recovery of locomotion after neurological injury. We highlight the fact that somatosensory feedback is an essential component of a highly integrated system for locomotor control. © 2021 American Physiological Society. Compr Physiol 11:1-71, 2021.

A unified energy-optimality criterion predicts human navigation paths and speeds

Navigating our physical environment requires changing directions and turning. Despite its ecological importance, we do not have a unified theoretical account of non-straight-line human movement. Here, we present a unified optimality criterion that predicts disparate non-straight-line walking phenomena, with straight-line walking as a special case. We first characterized the metabolic cost of turning, deriving the cost landscape as a function of turning radius and rate. We then generalized this cost landscape to arbitrarily complex trajectories, allowing the velocity direction to deviate from body orientation (holonomic walking). We used this generalized optimality criterion to mathematically predict movement patterns in multiple contexts of varying complexity: walking on prescribed paths, turning in place, navigating an angled corridor, navigating freely with end-point constraints, walking through doors, and navigating around obstacles. In these tasks, humans moved at speeds and paths predicted by our optimality criterion, slowing down to turn and never using sharp turns. We show that the shortest path between two points is, counterintuitively, often not energy-optimal, and, indeed, humans do not use the shortest path in such cases. Thus, we have obtained a unified theoretical account that predicts human walking paths and speeds in diverse contexts. Our model focuses on walking in healthy adults; future work could generalize this model to other human populations, other animals, and other locomotor tasks.

Extraction of Vehicle Turning Trajectories at Signalized Intersections Using Convolutional Neural Networks

This paper aims at developing a convolutional neural network (CNN)-based tool that can automatically detect the left-turning vehicles (right-hand traffic rule) at signalized intersections and extract their trajectories from a recorded video. The proposed tool uses a region-based CNN trained over a limited number of video frames to detect moving vehicles. Kalman filters are then used to track the detected vehicles and extract their trajectories. The proposed tool achieved an acceptable accuracy level when verified against the manually extracted trajectories, with an average error of 16.5 cm. Furthermore, the trajectories extracted using the proposed vehicle tracking method were used to demonstrate the applicability of the minimum-jerk principle to reproduce variations in the vehicles’ paths. The effort presented in this paper can be regarded as a way forward toward maximizing the potential use of deep learning in traffic safety applications.

Evaluating Human Movement Coordination During Immersive Walking in a Virtual Crowd

This paper describes our investigation on how participants coordinate movement behavior in relation to a virtual crowd that surrounds them while immersed in a virtual environment. The participants were immersed in a virtual metropolitan city and were instructed to cross the road and reach the opposite sidewalk. The participants performed the task ten times. The virtual crowd that surrounded them was scripted to move in the same direction. During the experiment, several measurements were obtained to evaluate human movement coordination. Moreover, the time and direction in which the participants started moving toward the opposite sidewalk were also captured. These data were later used to initialize the parameters of simulated characters that were scripted to become part of the virtual crowd. Measurements were extracted from the simulated characters and used as a baseline to evaluate the movement coordination of the participants. By analyzing the data, significant differences between the movement behaviors of the participants and the simulated characters were found. However, simple linear regression analyses indicated that the movement behavior of participants was moderately associated with the simulated characters' movements when performing a locomotive task within a virtual crowd population. This study can be considered as a baseline for further research that evaluates the movement coordination of participants during human-virtual-crowd interactions using measurements obtained by the simulated characters.

Modeling and Simulation of Pedestrian Movement Planning Around Corners

: Owing to the complexity of behavioral dynamics and mechanisms associated with turning maneuvers, capturing pedestrian movements around corners in a mathematical model is a challenging task. In this study, minimum jerk and one-thirds power law concepts, which have been initially applied in neurosciences and brain research domains, were utilized in combination to model pedestrian movement planning around bends. Simulation outputs explained that the proposed model could realistically represent the behavioral characteristics of pedestrians walking through bends. Comparison of modeled trajectories with empirical data demonstrated that the accuracy of the model could further be improved by using appropriate parameters in the one-thirds power law equation. Sensitivity analysis explained that, although the paths were not sensitive to the boundary conditions, speed and acceleration profiles could be remarkably varied depending on boundary conditions. Further, the applicability of the proposed model to estimate trajectories of pedestrians negotiating bends under different entry, intermediate, and exit conditions was also identified. The proposed model can be applied in microscopic simulation platforms, virtual reality, and driving simulator applications to provide realistic and accurate maneuvers around corners.

American Society of Biomechanics Journal of Biomechanics Award 2018: Adaptive motor planning of center-of-mass trajectory during goal-directed walking in novel environments

To aid in the successful execution of goal-directed walking (discrete movement from a start location to an end target) the central nervous system forms a predictive motor plan. For the motor plan to be effective, it must be adapted in response to environmental changes. Despite motor planning being inherent to goal-directed walking, it is not understood how the nervous system adapts these plans to interact with changing environments. Our objective was to understand how people adapt motor plans of center of mass (COM) trajectory during goal-directed walking in response to a consistent change in environmental dynamics. Participants preformed a series of goal-directed walking trials in a novel environment created by a cable robot that applied a lateral force field to their COM. We hypothesized that participants would adapt to the environment by forming an internal model of their COM trajectory within the force field. Our findings support this hypothesis. Initially, we found COM trajectory significantly deviated in the same direction as the applied field, relative to baseline (no field) (p = 0.002). However, with practice in the field, COM trajectory adapted back to the baseline (p = 0.6). When we unexpectedly removed the field, participants demonstrated after-effects, COM trajectory deviated in the direction opposite of the field relative to baseline (p < 0.001). Our findings suggest that when performing a goal-directed walking task, people adapt a motor plan that predicts the COM trajectory that will emerge from the interaction between a specific set of motor commands and the external environment.

Gaussian Process Autoregression for Simultaneous Proportional Multi-Modal Prosthetic Control With Natural Hand Kinematics

Matching the dexterity, versatility, and robustness of the human hand is still an unachieved goal in bionics, robotics, and neural engineering. A major limitation for hand prosthetics lies in the challenges of reliably decoding user intention from muscle signals when controlling complex robotic hands. Most of the commercially available prosthetic hands use muscle-related signals to decode a finite number of predefined motions and some offer proportional control of open/close movements of the whole hand. Here, in contrast, we aim to offer users flexible control of individual joints of their artificial hand. We propose a novel framework for decoding neural information that enables a user to independently control 11 joints of the hand in a continuous manner-much like we control our natural hands. Toward this end, we instructed six able-bodied subjects to perform everyday object manipulation tasks combining both dynamic, free movements (e.g., grasping) and isometric force tasks (e.g., squeezing). We recorded the electromyographic and mechanomyographic activities of five extrinsic muscles of the hand in the forearm, while simultaneously monitoring 11 joints of hand and fingers using a sensorized data glove that tracked the joints of the hand. Instead of learning just a direct mapping from current muscle activity to intended hand movement, we formulated a novel autoregressive approach that combines the context of previous hand movements with instantaneous muscle activity to predict future hand movements. Specifically, we evaluated a linear vector autoregressive moving average model with exogenous inputs and a novel Gaussian process ( ) autoregressive framework to learn the continuous mapping from hand joint dynamics and muscle activity to decode intended hand movement. Our approach achieves high levels of performance (RMSE of 8°/s and ). Crucially, we use a small set of sensors that allows us to control a larger set of independently actuated degrees of freedom of a hand. This novel undersensored control is enabled through the combination of nonlinear autoregressive continuous mapping between muscle activity and joint angles. The system evaluates the muscle signals in the context of previous natural hand movements. This enables us to resolve ambiguities in situations, where muscle signals alone cannot determine the correct action as we evaluate the muscle signals in their context of natural hand movements. autoregression is a particularly powerful approach which makes not only a prediction based on the context but also represents the associated uncertainty of its predictions, thus enabling the novel notion of risk-based control in neuroprosthetics. Our results suggest that autoregressive approaches with exogenous inputs lend themselves for natural, intuitive, and continuous control in neurotechnology, with the particular focus on prosthetic restoration of natural limb function, where high dexterity is required for complex movements.

Walking Ahead: The Headed Social Force Model

Human motion models are finding an increasing number of novel applications in many different fields, such as building design, computer graphics and robot motion planning. The Social Force Model is one of the most popular alternatives to describe the motion of pedestrians. By resorting to a physical analogy, individuals are assimilated to point-wise particles subject to social forces which drive their dynamics. Such a model implicitly assumes that humans move isotropically. On the contrary, empirical evidence shows that people do have a preferred direction of motion, walking forward most of the time. Lateral motions are observed only in specific circumstances, such as when navigating in overcrowded environments or avoiding unexpected obstacles. In this paper, the Headed Social Force Model is introduced in order to improve the realism of the trajectories generated by the classical Social Force Model. The key feature of the proposed approach is the inclusion of the pedestrians’ heading into the dynamic model used to describe the motion of each individual. The force and torque representing the model inputs are computed as suitable functions of the force terms resulting from the traditional Social Force Model. Moreover, a new force contribution is introduced in order to model the behavior of people walking together as a single group. The proposed model features high versatility, being able to reproduce both the unicycle-like trajectories typical of people moving in open spaces and the point-wise motion patterns occurring in high density scenarios. Extensive numerical simulations show an increased regularity of the resulting trajectories and confirm a general improvement of the model realism.

Locomotor deficits in recently concussed athletes and matched controls during single and dual-task turning gait: preliminary results

Background

There is growing evidence that mild traumatic brain injury (concussion) can affect locomotor characteristics for prolonged periods of time even when physical signs and symptoms are absent. While most locomotor deficits post-concussion have involved straight walking, turning gait has received little attention despite its pervasiveness in everyday locomotion and athletic competition.

Methods

This study longitudinally examined kinematic characteristics during preplanned turning in a small sample of recently concussed athletes (n = 4) and healthy matched control athletes (n = 4) to examine potential deficits during single and dual-task turning gait over the initial 6 weeks post-injury, with a one-year follow-up. Turning path kinematics (curvature, obstacle clearance, path length), stride kinematics (stride length, stride width, stride time), and inclination angles were calculated from motion capture of participants walking around an obstacle.

Results

Concussed athletes had larger dual-task costs in turning speed and stride time compared to healthy controls. After controlling for speed and turn curvature, recently concussed athletes increased their inclination towards the inside of the turn over time and decreased their stride time compared to controls indicating a prolonged recovery. Kinematic differences between groups were estimated to recover to healthy levels between 100 and 300 days post-injury, suggesting future prospective longitudinal studies should span 6–12 months post-injury.

Conclusion

Turning gait should be included in future studies of concussion and may be a clinically useful tool. Future longitudinal studies should consider examining gait changes for up to 6–12 months post-injury.

Electronic supplementary material

The online version of this article (doi:10.1186/s12984-016-0177-y) contains supplementary material, which is available to authorized users.

Socially compliant mobile robot navigation via inverse reinforcement learning

Mobile robots are increasingly populating our human environments. To interact with humans in a socially compliant way, these robots need to understand and comply with mutually accepted rules. In this paper, we present a novel approach to model the cooperative navigation behavior of humans. We model their behavior in terms of a mixture distribution that captures both the discrete navigation decisions, such as going left or going right, as well as the natural variance of human trajectories. Our approach learns the model parameters of this distribution that match, in expectation, the observed behavior in terms of user-defined features. To compute the feature expectations over the resulting high-dimensional continuous distributions, we use Hamiltonian Markov chain Monte Carlo sampling. Furthermore, we rely on a Voronoi graph of the environment to efficiently explore the space of trajectories from the robot’s current position to its target position. Using the proposed model, our method is able to imitate the behavior of pedestrians or, alternatively, to replicate a specific behavior that was taught by tele-operation in the target environment of the robot. We implemented our approach on a real mobile robot and demonstrated that it is able to successfully navigate in an office environment in the presence of humans. An extensive set of experiments suggests that our technique outperforms state-of-the-art methods to model the behavior of pedestrians, which also makes it applicable to fields such as behavioral science or computer graphics.

Anticipatory control and spatial cognition in locomotion and navigation through typical development and in cerebral palsy

Behavioural evidence, summarized in this narrative review, supports a developmental model of locomotor control based on increasing neural integration of spatial reference frames. Two consistent adult locomotor behaviours are head stabilization and head anticipation: the head is stabilized to gravity and leads walking direction. This cephalocaudal orienting organization aligns gaze and vestibula with a reference frame centred on the upcoming walking direction, allowing anticipatory control on body kinematics, but is not fully developed until adolescence. Walking trajectories and those of hand movements share many aspects, including power laws coupling velocity to curvature, and minimized spatial variability. In fact, the adult brain can code trajectory geometry in an allocentric reference frame, irrespective of the end effector, regulating body kinematics thereafter. Locomotor trajectory formation, like head anticipation, matures in early adolescence, indicating common neurocomputational substrates. These late-developing control mechanisms can be distinguished from biomechanical problems in children with cerebral palsy (CP). Children's performance on a novel navigation test, the Magic Carpet, indicates that typical navigation development consists of the increasing integration of egocentric and allocentric reference frames. In CP, right-brain impairment seems to reduce navigation performance due to a maladaptive left-brain sequential egocentric strategy. Spatial integration should be considered more in rehabilitation.

Drawing ellipses in water: evidence for dynamic constraints in the relation between velocity and path curvature

Several types of continuous human movements comply with the so-called Two-Thirds Power Law (2/3-PL) stating that velocity (V) is a power function of the radius of curvature (R) of the endpoint trajectory. The origin of the 2/3-PL has been the object of much debate. An experiment investigated further this issue by comparing two-dimensional drawing movements performed in air and water. In both conditions, participants traced continuously quasi-elliptic trajectories (period T = 1.5 s). Other experimental factors were the movement plane (horizontal/vertical), and whether the movement was performed free-hand, or by following the edge of a template. In all cases a power function provided a good approximation to the V-R relation. The main result was that the exponent of the power function in water was significantly smaller than in air. This appears incompatible with the idea that the power relationship depends only on kinematic constraints and suggests a significant contribution of dynamic factors. We argue that a satisfactory explanation of the observed behavior must take into account the interplay between the properties of the central motor commands and the visco-elastic nature of the mechanical plant that implements the commands.

Walking paths to and from a goal differ: on the role of bearing angle in the formation of human locomotion paths

The path that humans take while walking to a goal is the result of a cognitive process modulated by the perception of the environment and physiological constraints. The path shape and timing implicitly embeds aspects of the architecture behind this process. Here, locomotion paths were investigated during a simple task of walking to and from a goal, by looking at the evolution of the position of the human on a horizontal (x,y) plane. We found that the path while walking to a goal was not the same as that while returning from it. Forward-return paths were systematically separated by 0.5-1.9m, or about 5% of the goal distance. We show that this path separation occurs as a consequence of anticipating the desired body orientation at the goal while keeping the target in view. The magnitude of this separation was strongly influenced by the bearing angle (difference between body orientation and angle to goal) and the final orientation imposed at the goal. This phenomenon highlights the impact of a trade-off between a directional perceptual apparatus—eyes in the head on the shoulders—and and physiological limitations, in the formation of human locomotion paths. Our results give an insight into the influence of environmental and perceptual variables on human locomotion and provide a basis for further mathematical study of these mechanisms.

Corner height influences center of mass kinematics and path trajectory during turning

Despite the prevalence of directional changes during every-day gait, relatively little is known about turning compared to straight gait. While the center of mass (COM) movement during straight gait is well characterized, the COM trajectory and the factors that influence it are less established for turning. This study investigated the influence of a corner׳s height on the COM trajectory as participants walked around the corner. Ten participants (25.3±3.74 years) performed both 90° step and spin turns to the left at self-selected slow, normal, and fast speeds while walking inside a marked path. A pylon was placed on the inside corner of the path. Four different pylon heights were used to correspond to heights of everyday objects: 0 cm (no object), 63 cm (box, crate), 104 cm (desk, table, counter), 167 cm (shelf, cabinet). Obstacle height was found to significantly affect the COM trajectory. Taller obstacles resulted in more distance between the corner and the COM, and between the corner and the COP. Taller obstacles also were associated with greater curvature in the COM trajectory, indicating a smaller turning radius despite the constant 90° corner. Taller obstacles correlated to an increased required coefficient of friction (RCOF) due to the smaller turning radii. Taller obstacles also tended towards greater mediolateral (ML) COM-COP angles, contrary to the initial hypothesis. Additionally, the COM was found to remain outside the base of support (BOS) for the entire first half of stance phase for all conditions indicating a high risk of falls resulting from slips.

Adjustments of speed and path when avoiding collisions with another pedestrian

When walking in open space, collision avoidance with other pedestrians is a process that successfully takes place many times. To pass another pedestrian (an interferer) walking direction, walking speed or both can be adjusted. Currently, the literature is not yet conclusive of how humans adjust these two parameters in the presence of an interferer. This impedes the development of models predicting general obstacle avoidance strategies in humans’ walking behavior. The aim of this study was to investigate the adjustments of path and speed when a pedestrian is crossing a non-reactive human interferer at different angles and speeds, and to compare the results to general model predictions. To do so, we designed an experiment where a pedestrian walked a 12 m distance to reach a goal position. The task was designed in such a way that collision with an interferer would always occur if the pedestrian would not apply a correction of movement path or speed. Results revealed a strong dependence of path and speed adjustments on crossing angle and walking speed, suggesting local planning of the collision avoidance strategy. Crossing at acute angles (i.e. 45° and 90°) seems to require more complex collision avoidance strategies involving both path and speed adjustments than crossing at obtuse angles, where only path adjustments were observed. Overall, the results were incompatible with predictions from existing models of locomotor collision avoidance. The observed initiations of both adjustments suggest a collision avoidance strategy that is temporally controlled. The present study provides a comprehensive picture of human collision avoidance strategies in walking, which can be used to evaluate and adjust existing pedestrian dynamics models, or serve as an empirical basis to develop new models.

Cognitive loading affects motor awareness and movement kinematics but not locomotor trajectories during goal-directed walking in a virtual reality environment

The primary purpose of this study was to investigate the effects of cognitive loading on movement kinematics and trajectory formation during goal-directed walking in a virtual reality (VR) environment. The secondary objective was to measure how participants corrected their trajectories for perturbed feedback and how participants' awareness of such perturbations changed under cognitive loading. We asked 14 healthy young adults to walk towards four different target locations in a VR environment while their movements were tracked and played back in real-time on a large projection screen. In 75% of all trials we introduced angular deviations of ±5° to ±30° between the veridical walking trajectory and the visual feedback. Participants performed a second experimental block under cognitive load (serial-7 subtraction, counter-balanced across participants). We measured walking kinematics (joint-angles, velocity profiles) and motor performance (end-point-compensation, trajectory-deviations). Motor awareness was determined by asking participants to rate the veracity of the feedback after every trial. In-line with previous findings in natural settings, participants displayed stereotypical walking trajectories in a VR environment. Our results extend these findings as they demonstrate that taxing cognitive resources did not affect trajectory formation and deviations although it interfered with the participants' movement kinematics, in particular walking velocity. Additionally, we report that motor awareness was selectively impaired by the secondary task in trials with high perceptual uncertainty. Compared with data on eye and arm movements our findings lend support to the hypothesis that the central nervous system (CNS) uses common mechanisms to govern goal-directed movements, including locomotion. We discuss our results with respect to the use of VR methods in gait control and rehabilitation.

Spatial constant equi-affine speed and motion perception

The two-thirds power law, postulating an inverse local relation between the velocity and cubed root of curvature of planar trajectories, is a long-established simplifying principle of human hand movements. In perception, the motion of a dot along a planar elliptical path appears most uniform for speed profiles closer to those predicted by the power law than to constant Euclidean speed, a kinetic-visual illusion. Mathematically, complying with this law is equivalent to moving at constant planar equi-affine speed, while unconstrained three-dimensional drawing movements generally follow constant spatial equi-affine speed. Here we test the generalization of this illusion to visual perception of spatial motion for a dot moving along five differently shaped paths, using stereoscopic projection. The movements appeared most uniform for speed profiles closer to constant spatial equi-affine speed than to constant Euclidean speed, with path torsion (i.e., local deviation from planarity) directly affecting the speed profiles perceived as most uniform, as predicted for constant spatial equi-affine speed. This demonstrates the dominance of equi-affine geometry in spatial motion perception. However, constant equi-affine speed did not fully account for the variability among the speed profiles selected as most uniform for different shapes. Moreover, in a followup experiment, we found that viewing distance affected the speed profile reported as most uniform for the extensively studied planar elliptical motion paths. These findings provide evidence for the critical role of equi-affine geometry in spatial motion perception and contribute to the mounting evidence for the role of non-Euclidean geometries in motion perception and production.

Kinematic evaluation of virtual walking trajectories

Virtual walking, a fundamental task in Virtual Reality (VR), is greatly influenced by the locomotion interface being used, by the specificities of input and output devices, and by the way the virtual environment is represented. No matter how virtual walking is controlled, the generation of realistic virtual trajectories is absolutely required for some applications, especially those dedicated to the study of walking behaviors in VR, navigation through virtual places for architecture, rehabilitation and training. Previous studies focused on evaluating the realism of locomotion trajectories have mostly considered the result of the locomotion task (efficiency, accuracy) and its subjective perception (presence, cybersickness). Few focused on the locomotion trajectory itself, but in situation of geometrically constrained task. In this paper, we study the realism of unconstrained trajectories produced during virtual walking by addressing the following question: did the user reach his destination by virtually walking along a trajectory he would have followed in similar real conditions? To this end, we propose a comprehensive evaluation framework consisting on a set of trajectographical criteria and a locomotion model to generate reference trajectories. We consider a simple locomotion task where users walk between two oriented points in space. The travel path is analyzed both geometrically and temporally in comparison to simulated reference trajectories. In addition, we demonstrate the framework over a user study which considered an initial set of common and frequent virtual walking conditions, namely different input devices, output display devices, control laws, and visualization modalities. The study provides insight into the relative contributions of each condition to the overall realism of the resulting virtual trajectories.

Strategies of locomotor collision avoidance

Collision avoidance during locomotion can be achieved by a variety of strategies. While in some situations only a single trajectory will successfully avoid impact, in many cases several different strategies are possible. Locomotor experiments in the presence of static boundary conditions have suggested that the choice of an appropriate trajectory is based on a maximum-smoothness strategy. Here we analyzed locomotor trajectories of subjects avoiding collision with another human crossing their path orthogonally. In such a case, changing walking direction while keeping speed or keeping walking direction while changing speed would be two extremes of solving the problem. Our participants clearly favored changing their walking speed while keeping the path on a straight line between start and goal. To interpret this result, we calculated the costs of the chosen trajectories in terms of a smoothness-maximization criterion and simulated the trajectories with a computational model. Data analysis together with model simulation showed that the experimentally chosen trajectory to avoid collision with a moving human is not the optimally smooth solution. However, even though the trajectory is not globally smooth, it was still locally smooth. Modeling further confirmed that, in presence of the moving human, there is always a trajectory that would be smoother but would deviate from the straight line. We therefore conclude that the maximum smoothness strategy previously suggested for static environments no longer holds for locomotor path planning and execution in dynamically changing environments such as the one tested here.

Multi-finger coordination in healthy subjects and stroke patients: a mathematical modelling approach

Background

Approximately 60% of stroke survivors experience hand dysfunction limiting execution of daily activities. Several methods have been proposed to objectively quantify fingers' joints range of motion (ROM), while few studies exist about multi-finger coordination during hand movements. The present work analysed this aspect, by providing a complete characterization of spatial and temporal aspects of hand movement, through the mathematical modelling of multi-joint finger motion in healthy subjects and stroke patients.

Methods

Hand opening and closing movements were examined in 12 healthy volunteers and 14 hemiplegic stroke survivors by means of optoelectronic kinematic analysis. The flexion/extension angles of metacarpophalangeal (MCPJ) and proximal interphalangeal joints (IPJ) of all fingers were computed and mathematically characterized by a four-parameter hyperbolic tangent function. Accuracy of the selected model was analysed by means of coefficient of determination (R²) and root mean square error (RMSE). Test-retest reliability was quantified by intraclass correlation coefficient (ICC) and test-retest errors. Comparison between performances of healthy controls and stroke subjects were performed by analysing possible differences in parameters describing angular and temporal aspects of hand kinematics and inter-joint, inter-digit coordination.

Results

The angular profiles of hand opening and closing were accurately characterized by the selected model, both in healthy controls and in stroke subjects (R² > 0.973, RMSE < 2.0°). Test-retest reliability was found to be excellent, with ICC > 0.75 and remarking errors comparable to those obtained with other methods. Comparison with healthy controls revealed that hemiparetic hand movement was impaired not only in joints ROM but also in the temporal aspects of motion: peak velocities were significantly decreased, inter-digit coordination was reduced of more than 50% and inter-joint coordination patterns were highly disrupted. In particular, the stereotypical proximal-to-distal opening sequence (reversed during hand closing) found in healthy subjects, was altered in stroke subjects who showed abnormally high delay between IPJ and MCPJ movement or reversed moving sequences.

Conclusions

The proposed method has proven to be a promising tool for a complete objective characterization of spatial and temporal aspects of hand movement in stroke, providing further information for a more targeted planning of the rehabilitation treatment to each specific patient and for a quantitative assessment of therapy's outcome.

Invariance of locomotor trajectories across visual and gait direction conditions

We studied the influence of vision (walking with or without vision) and of gait direction (walking forward or backward) on goal-oriented locomotion in humans. Subjects had to walk, in a free environment, from a given position and orientation towards a distant arrow which constrained their final position and orientation. We found that the average trajectories were mostly similar across the tested conditions, which suggests that locomotor trajectories are generated at a high cognitive level and, to some extent, independently of the detailed sensory and motor implementation levels. The variability profiles around the average trajectories were similar across the gait direction conditions but differed greatly across the visual conditions, indicating the existence of motor-independent and vision-dependent control mechanisms. Taken together, our observations argue further in favour of a top-down implementation of goal-oriented locomotion, where the control of locomotion is specified at the level of whole-body trajectories and then implemented through specific motor strategies.

Analytical and numerical analysis of inverse optimization problems: conditions of uniqueness and computational methods

One of the key problems of motor control is the redundancy problem, in particular how the central nervous system (CNS) chooses an action out of infinitely many possible. A promising way to address this question is to assume that the choice is made based on optimization of a certain cost function. A number of cost functions have been proposed in the literature to explain performance in different motor tasks: from force sharing in grasping to path planning in walking. However, the problem of uniqueness of the cost function(s) was not addressed until recently. In this article, we analyze two methods of finding additive cost functions in inverse optimization problems with linear constraints, so-called linear-additive inverse optimization problems. These methods are based on the Uniqueness Theorem for inverse optimization problems that we proved recently (Terekhov et al., J Math Biol 61(3):423-453, 2010). Using synthetic data, we show that both methods allow for determining the cost function. We analyze the influence of noise on the both methods. Finally, we show how a violation of the conditions of the Uniqueness Theorem may lead to incorrect solutions of the inverse optimization problem.

An analytical approach to the problem of inverse optimization with additive objective functions: an application to human prehension

We consider the problem of what is being optimized in human actions with respect to various aspects of human movements and different motor tasks. From the mathematical point of view this problem consists of finding an unknown objective function given the values at which it reaches its minimum. This problem is called the inverse optimization problem. Until now the main approach to this problems has been the cut-and-try method, which consists of introducing an objective function and checking how it reflects the experimental data. Using this approach, different objective functions have been proposed for the same motor action. In the current paper we focus on inverse optimization problems with additive objective functions and linear constraints. Such problems are typical in human movement science. The problem of muscle (or finger) force sharing is an example. For such problems we obtain sufficient conditions for uniqueness and propose a method for determining the objective functions. To illustrate our method we analyze the problem of force sharing among the fingers in a grasping task. We estimate the objective function from the experimental data and show that it can predict the force-sharing pattern for a vast range of external forces and torques applied to the grasped object. The resulting objective function is quadratic with essentially non-zero linear terms.

On the open-loop and feedback processes that underlie the formation of trajectories during visual and nonvisual locomotion in humans

We investigated the nature of the control mechanisms at work during goal-oriented locomotion. In particular, we tested the effects of vision, locomotor speed, and the presence of via points on the geometric and kinematic properties of locomotor trajectories. We first observed that the average trajectories recorded in visual and nonvisual locomotion were highly comparable, suggesting the existence of vision-independent processes underlying the formation of locomotor trajectories. Then by analyzing and comparing the variability around the average trajectories across different experimental conditions, we were able to demonstrate the existence of on-line feedback control in both visual and nonvisual locomotion and to clarify the relations between visual and nonvisual control strategies. Based on these insights, we designed a model in which maximum-smoothness and optimal feedback control principles account, respectively, for the open-loop and feedback processes. Taken together, the experimental and modeling findings provide a novel understanding of the nature of the motor, sensory, and "navigational" processes underlying goal-oriented locomotion.

Maneuvers during legged locomotion

Maneuverability is essential for locomotion. For animals in the environment, maneuverability is directly related to survival. For humans, maneuvers such as turning are associated with increased risk for injury, either directly through tissue loading or indirectly through destabilization. Consequently, understanding the mechanics and motor control of maneuverability is a critical part of locomotion research. We briefly review the literature on maneuvering during locomotion with a focus on turning in bipeds. Walking turns can use one of several different strategies. Anticipation can be important to adjust kinematics and dynamics for smooth and stable maneuvers. During running, turns may be substantially constrained by the requirement for body orientation to match movement direction at the end of a turn. A simple mathematical model based on the requirement for rotation to match direction can describe leg forces used by bipeds (humans and ostriches). During running turns, both humans and ostriches control body rotation by generating fore-aft forces. However, whereas humans must generate large braking forces to prevent body over-rotation, ostriches do not. For ostriches, generating the lateral forces necessary to change movement direction results in appropriate body rotation. Although ostriches required smaller braking forces due in part to increased rotational inertia relative to body mass, other movement parameters also played a role. Turning performance resulted from the coordinated behavior of an integrated biomechanical system. Results from preliminary experiments on horizontal-plane stabilization support the hypothesis that controlling body rotation is an important aspect of stable maneuvers. In humans, body orientation relative to movement direction is rapidly stabilized during running turns within the minimum of two steps theoretically required to complete analogous maneuvers. During straight running and cutting turns, humans exhibit spring-mass behavior in the horizontal plane. Changes in the horizontal projection of leg length were linearly related to changes in horizontal-plane leg forces. Consequently, the passive dynamic stabilization associated with spring-mass behavior may contribute to stability during maneuvers in bipeds. Understanding the mechanics of maneuverability will be important for understanding the motor control of maneuvers and also potentially be useful for understanding stability.

Complex Unconstrained Three-Dimensional Hand Movement and Constant Equi-Affine Speed

One long-established simplifying principle behind the large repertoire and high versatility of human hand movements is the two-thirds power law—an empirical law stating a relationship between local geometry and kinematics of human hand trajectories during planar curved movements. It was further generalized not only to various types of human movements, but also to motion perception and prediction, although it was unsuccessful in explaining unconstrained three-dimensional (3D) movements. Recently, movement obeying the power law was proved to be equivalent to moving with constant planar equi-affine speed. Generalizing such motion to 3D space—i.e., to movement at constant spatial equi-affine speed—predicts the emergence of a new power law, whose utility for describing spatial scribbling movements we have previously demonstrated. In this empirical investigation of the new power law, subjects repetitively traced six different 3D geometrical shapes with their hand. We show that the 3D power law explains the data consistently better than both the two-thirds power law and an additional power law that was previously suggested for spatial hand movements. We also found small yet systematic modifications of the power-law's exponents across the various shapes, which further scrutiny suggested to be correlated with global geometric factors of the traced shape. Nevertheless, averaging over all subjects and shapes, the power-law exponents are generally in accordance with constant spatial equi-affine speed. Taken together, our findings provide evidence for the potential role of non-Euclidean geometry in motion planning and control. Moreover, these results seem to imply a relationship between geometry and kinematics that is more complex than the simple local one stipulated by the two-thirds power law and similar models.