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The concept of the 'extrapolated center of mass (XcoM)', introduced by Hof et al., (2005, J. Biomechanics 38 (1), p. 1–8), extends the classical inverted pendulum model to dynamic situations. The vector quantity XcoM combines the center of mass position plus its velocity divided by the pendulum eigenfrequency. In this concept, the margin of stability (MoS), i.e., the minimum signed distance from the XcoM to the boundaries of the base of support was proposed as a measure of dynamic stability. Here we describe the conceptual evolution of the XcoM, discuss key considerations in the estimation of the XcoM and MoS, and provide a critical perspective on the interpretation of the MoS as a measure of instantaneous mechanical stability.

Split-belt treadmill walking has contributed to our understanding of human locomotor adaptation and motor learning. On the split-belt treadmill, each limb walks on a belt moving at a different speed. An obvious question is: how fast is a person walking on a split-belt treadmill? This question has implications for metabolic and mechanical energy consumption, stability, and neuromotor adaptation. There are two interpretations of this question that provide different answers and implications. First, how fast a person is walking can be interpreted as the center of mass velocity, which is a vector of the rate of displacement in the three directions of an inertial reference frame. The center of mass velocity is zero on average in a global reference frame and oscillates throughout the gait cycle, providing information about mechanical and stability demands. Second, how fast a person is walking can be interpreted as stride speed, which is a scalar defined as stride length divided by stride time. Stride speed is constrained by the split-belt treadmill to the average belt speed, subtracted by the interaction between belt speed difference and step time asymmetry. Thus, stride speed is slower than the average speed of the belts due to modifications in step time asymmetry. Here, we provide a framework for calculating and contextualizing these two quantities, highlighting their implications for recent work in split-belt adaptation. We suggest that the design and interpretation of future split-belt adaptation studies can benefit from careful consideration of the analyses presented here.

In a mixture of two kinds of identical bosons, there are two types of pairs: identical bosons’ pairs, of either species, and pairs of distinguishable bosons. In the present work, the fragmentation of pairs in a trapped mixture of Bose–Einstein condensates is investigated using a solvable model, the symmetric harmonic-interaction model for mixtures. The natural geminals for pairs made of identical or distinguishable bosons are explicitly contracted by diagonalizing the intra-species and inter-species reduced two-particle density matrices, respectively. Properties of pairs’ fragmentation in the mixture are discussed, the role of the mixture’s center-of-mass and relative center-of-mass coordinates is elucidated, and a generalization to higher-order reduced density matrices is made. As a complementary result, the exact Schmidt decomposition of the wave function of the bosonic mixture is constructed. The entanglement between the two species is governed by the coupling of their individual center-of-mass coordinates, and it does not vanish at the limit of an infinite number of particles where any finite-order intra-species and inter-species reduced density matrix per particle is 100% condensed. Implications are briefly discussed.

Accurate estimation of the whole-body center of mass (CoM) is essential for assessing human stability and postural control. However, selecting a suitable estimation method considering the nature of the activity, availability of equipment such as force plates measuring ground reaction forces (GRFs), and the time needed for data collection and processing is challenging. This study compares four methods for estimation of the 3D CoM position and velocity — “Pelvis Markerset” (PM), “Pelvis Markerset & GRFs” (PMG), “Whole-Body Markerset” (WM), and “Whole-Body Markerset & GRFs” (WMG) — across 4 activities classified as Static (e.g., standing with eyes closed), and 10 as Dynamic (e.g., picking up an object from the ground). Using the root mean square (RMS) of “external force residual” as a performance metric, we found that in the Static group, all the methods performed similarly for both position and velocity estimation. During the Dynamic activities, for position estimation, the pelvis-based estimates showed higher residuals compared to the whole-body methods (p<0.001, Cohen’s d=3.03). For velocity estimation, the residual of WMG was similar to WM, and both outperformed PM (p<0.001, d=3.17); meanwhile, PMG achieved lower residuals than PM (p<0.001, d=2.26). Given our results, we recommend the WM method as it performed well and did not require fusion with the GRFs. The PM method can be used in activities similar to the Static group, during which markers on the pelvis reflect whole-body kinematics. When the GRFs are also measured, it is possible to improve the velocities estimates of this method using the Kalman filter.

Shifts of the center of pressure (CoP) through modulation of foot placement and ankle moments (CoP-mechanism) cause accelerations of the center of mass (CoM) that can be used to stabilize gait. An additional mechanism that can be used to stabilize gait, is the counter-rotation mechanism, i.e., changing the angular momentum of segments around the CoM to change the direction of the ground reaction force. The relative contribution of these mechanisms to the control of the CoM is unknown. Therefore, we aimed to determine the relative contribution of these mechanisms to control the CoM in the anteroposterior (AP) direction during a normal step and the first recovery step after perturbation in healthy adults. Nineteen healthy subjects walked on a split-belt treadmill and received unexpected belt acceleration perturbations of various magnitudes applied immediately after right heel-strike. Full-body kinematic and force plate data were obtained to calculate the contributions of the CoP-mechanism and the counter-rotation mechanism to control the CoM. We found that the CoP-mechanism contributed to corrections of the CoM acceleration after the AP perturbations, while the counter-rotation mechanism actually counteracted the CoM acceleration after perturbation, but only in the initial phases of the first step after the perturbation. The counter-rotation mechanism appeared to prevent interference with the gait pattern, rather than using it to control the CoM after the perturbation. Understanding the mechanisms used to stabilize gait may have implications for the design of therapeutic interventions that aim to decrease fall incidence.

Accurate estimation of the center of mass is necessary for evaluating balance control during quiet standing. However, no practical center of mass estimation method exists because of problems with estimation accuracy and theoretical validity in previous studies that used force platforms or inertial sensors. This study aimed to develop a method for estimating the center of mass displacement and velocity based on equations of motion describing the standing human body. This method uses a force platform under the feet and an inertial sensor on the head and is applicable when the support surface moves horizontally. We compared the center of mass estimation accuracy of the proposed method with those of other methods in previous studies using estimates from the optical motion capture system as the true value. The results indicate that the present method has high accuracy in quiet standing, ankle motion, hip motion, and support surface swaying in anteroposterior and mediolateral directions. The present method could help researchers and clinicians to develop more accurate and effective balance evaluation methods.

Estimating the position of the whole-body centre of mass (CoM) based on skin markers and anthropometric tables requires tracking the pelvis and lower body, which is impossible for wheelchair users due to occlusion. In this work, we present a method to track the user’s whole-body CoM using visible markers affixed to the user and wheelchair where the user remains seated in their wheelchair, by expressing the pelvis and lower body segments in wheelchair coordinates. The accuracy of this method was evaluated on the anterior-posterior (AP) and medial–lateral (ML) axes by comparing the projected CoM to the centre of pressure measured by four force plates, for 11 able-bodied participants adopting 9 static postures that include extreme reaching postures. The estimation accuracy was within 33 mm (AP) and 9 mm (ML), with a precision within 23 mm (AP) and 12 mm (ML). Tracking the whole-body CoM during wheelchair propulsion will allow researchers to better understand the dynamics of propulsion, which may help devise new approaches to increase the energy transfer from the arms to the ground and reduce the risks of developing musculoskeletal disorders.

Although reactive arm motions are important in recovering from a slip event, the biomechanical influences of upper extremity motions during slipping are not clear. The purpose of the current study was to determine whether reactive arm motions during slip recovery leads to increased margins of stability (MoS), and decreased center of mass (CoM) velocity and excursion. Thirty-two participants were randomized into 2 conditions: arms free and arms constrained. Participants traversed a 10-meter walkway and were exposed to an unexpected slip while wearing a protective harness. Anterior-posterior and medial–lateral MoS, as well as the CoM excursion and velocity during the slip perturbation was quantified using a three-dimensional motion capture system. In the frontal plane, individuals with their arms unconstrained demonstrated greater MoS (0.06 ± 0.03 vs −0.01 ± 0.02 m, p < 0.01), decreased CoM excursion (0.05 ± 0.02 vs 0.08 ± 0.01 m, p = 0.015), and a reduced CoM velocity (0.07 ± 0.03 vs. 0.14 ± 0.02 m/s, p < 0.01) compared to individuals with their arms constrained. In the sagittal plane, individuals with their arms unconstrained demonstrated, decreased CoM excursion (0.83 ± 0.13 vs 1.14 ± 0.20 m, p < 0.01) reduced CoM velocity (1.71 ± 0.08 vs. 1.79 ± 0.07 m/s, p = 0.02), but no differences in margins of stability (0.89 ± 0.13 vs 0.94 ± 0.10 m, p = 0.32). Our findings demonstrate that arm motions during a slip perturbation act to restore balance by minimizing displacement and velocity of the body CoM during a slip event in the frontal plane.

We study the problem of finding the global Riemannian center of mass of a set of data points on a Riemannian manifold. Specifically, we investigate the convergence of constant step-size gradient descent algorithms for solving this problem. The challenge is that often the underlying cost function is neither globally differentiable nor convex, and despite this one would like to have guaranteed convergence to the global minimizer. After some necessary preparations we state a conjecture which we argue is the best convergence condition (in a specific described sense) that one can hope for. The conjecture specifies conditions on the spread of the data points, step-size range, and the location of the initial condition (i.e., the region of convergence) of the algorithm. These conditions depend on the topology and the curvature of the manifold and can be conveniently described in terms of the injectivity radius and the sectional curvatures of the manifold. For $2$-dimensional manifolds of nonnegative curvature and manifolds of constant nonnegative curvature (e.g., the sphere in $\\mathbb{R}^{n}$ and the rotation group in $\\mathbb{R}^{3}$) we show that the conjecture holds true. For more general manifolds we prove convergence results which are weaker than the conjectured one (but still superior to the available results). We also briefly study the effect of the configuration of the data points on the speed of convergence. Finally, we study the global behavior of the algorithm on certain manifolds proving (generic) convergence of the algorithm to a local center of mass with an arbitrary initial condition. An important aspect of our presentation is our emphasis on the effect of curvature and topology of the manifold on the behavior of the algorithm. [PUBLICATION ABSTRACT]

Center of mass (COM) state, specifically in a local reference frame (i.e., relative to center of pressure), is an important variable for controlling and quantifying bipedal locomotion. However, this metric is not easily attainable in real time during human locomotion experiments. This information could be valuable when controlling wearable robotic exoskeletons, specifically for stability augmentation where knowledge of COM state could enable step placement planners similar to bipedal robots. Here, we explored the ability of simulated wearable sensor-driven models to rapidly estimate COM state during steady state and perturbed walking, spanning delayed estimates (i.e., estimating past state) to anticipated estimates (i.e., estimating future state). We used various simulated inertial measurement unit (IMU) sensor configurations typically found on lower limb exoskeletons and a temporal convolutional network (TCN) model throughout this analysis. We found comparable COM estimation capabilities across hip, knee, and ankle exoskeleton sensor configurations, where device type did not significantly influence error. We also found that anticipating COM state during perturbations induced a significant increase in error proportional to anticipation time. Delaying COM state estimates significantly increased accuracy for velocity estimates but not position estimates. All tested conditions resulted in models with R 2  > 0.85, with a majority resulting in R 2  > 0.95, emphasizing the viability of this approach. Broadly, this preliminary work using simulated IMUs supports the efficacy of wearable sensor-driven deep learning approaches to provide real-time COM state estimates for lower limb exoskeleton control or other wearable sensor-based applications, such as mobile data collection or use in real-time biofeedback.