The condition for dynamic stability

The well-known condition for standing stability in static situations is that the vertical projection of the centre of mass (CoM) should be within the base of support (BoS). On the basis of a simple inverted pendulum model, an extension of this rule is proposed for dynamical situations: the position of (the vertical projection of) the CoM plus its velocity times a factor (square root l/g) should be within the BoS, l being leg length and g the acceleration of gravity. It is proposed to name this vector quantity 'extrapolated centre of mass position' (XcoM). The definition suggests as a measure of stability the 'margin of stability' b, the minimum distance from XcoM to the boundaries of the BoS. An alternative measure is the temporal stability margin tau, the time in which the boundary of the BoS would be reached without intervention. Some experimental data of subjects standing on one or two feet, flatfoot and tiptoe, are presented to give an idea of the usual ranges of these margins of stability. Example data on walking are also presented.     

Evaluation of time-to-contact measures for assessing postural stability

Postural stability has traditionally been examined through spatial measures of the center of mass (CoM) or center of pressure (CoP), where larger amounts of CoM or CoP movements are considered signs of postural instability. However, for stabilization, the postural control system may utilize additional information about the CoM or CoP such as velocity, acceleration, and the temporal margin to a stability boundary. Postural time-to-contact (TtC) is a variable that can take into account this additional information about the CoM or CoP. Postural TtC is the time it would take the CoM or CoP, given its instantaneous trajectory, to contact a stability boundary. This is essentially the time the system has to reverse any perturbation before stance is threatened. Although this measure shows promise in assessing postural stability, the TtC values derived between studies are highly ambiguous due to major differences in how they are calculated. In this study, various methodologies used to assess postural TtC were compared during quiet stance and induced-sway conditions. The effects of the different methodologies on TtC values will be assessed, and issues regarding the interpretation of TtC data will also be discussed.     

An Artificial Neural Network Estimation of Gait Balance Control in the Elderly Using Clinical Evaluations

The use of motion analysis to assess balance is essential for determining the underlying mechanisms of falls during dynamic activities. Clinicians evaluate patients using clinical examinations of static balance control, gait performance, cognition, and neuromuscular ability. Mapping these data to measures of dynamic balance control, and the subsequent categorization and identification of community dwelling elderly fallers at risk of falls in a quick and inexpensive manner is needed. The purpose of this study was to demonstrate that given clinical measures, an artificial neural network (ANN) could determine dynamic balance control, as defined by the interaction of the center of mass (CoM) with the base of support (BoS), during gait. Fifty-six elderly adults were included in this study. Using a feed-forward neural network with back propagation, combinations of five functional domains, the number of hidden layers and error goals were evaluated to determine the best parameters to assess dynamic balance control. Functional domain input parameters included subject characteristics, clinical examinations, cognitive performance, muscle strength, and clinical balance performance. The use of these functional domains demonstrated the ability to quickly converge to a solution, with the network learning the mapping within 5 epochs, when using up to 30 hidden nodes and an error goal of 0.001. The ability to correctly identify the interaction of the CoM with BoS demonstrated correlation values up to 0.89 (P<.001). On average, using all clinical measures, the ANN was able to estimate the dynamic CoM to BoS distance to within 1 cm and BoS area to within 75 cm². Our results demonstrated that an ANN could be trained to map clinical variables to biomechanical measures of gait balance control. A neural network could provide physicians and patients with a cost effective means to identify dynamic balance issues and possible risk of falls from routinely collected clinical examinations.     

Simulating mechanical consequences of voluntary movement upon whole-body equilibrium: the arm-raising paradigm revisited

Voluntary arm-raising movement performed during the upright human stance position imposes a perturbation to an already unstable bipedal posture characterised by a high body centre of mass (CoM). Inertial forces due to arm acceleration and displacement of the CoM of the arm which alters the CoM position of the whole body represent the two sources of disequilibrium. A current model of postural control explains equilibrium maintenance through the action of anticipatory postural adjustments (APAs) that would offset any destabilising effect of the voluntary movement. The purpose of this paper was to quantify, using computer simulation, the postural perturbation due to arm raising movement. The model incorporated four links, with shoulder, hip, knee and ankle joints constrained by linear viscoelastic elements. The input of the model was a torque applied at the shoulder joint. The simulation described mechanical consequences of the arm-raising movement for different initial conditions. The variables tested were arm inertia, the presence or not of gravity field, the initial standing position and arm movement direction. Simulations showed that the mechanical effect of arm-raising movement was mainly local, that is to say at the level of trunk and lower limbs and produced a slight forward displacement of the CoM (1.5 mm). Backward arm-raising movement had the same effect on the CoM displacement as the forward arm-raising movement. When the mass of the arm was increased, trunk rotation increased producing a CoM displacement in the opposite direction when compared to arm movement performed without load. Postural disturbance was minimised for an initial standing posture with the CoM vertical projection corresponding to the ankle joint axis of rotation. When the model was reduced to two degrees of freedom (ankle and shoulder joints only) the postural perturbation due to arm-raising movement increased compared to the four-joints model. On the basis of these results the classical assumption that APAs stabilise the CoM is challenged.     

Computational stability of human knee joint at early stance in Gait: Effects of muscle coactivity and anterior cruciate ligament deficiency

As one of the most complex and vulnerable structures of body, the human knee joint should maintain dynamic equilibrium and stability in occupational and recreational activities. The evaluation of its stability and factors affecting it is vital in performance evaluation/enhancement, injury prevention and treatment managements. Knee stability often manifests itself by pain, hypermobility and giving-way sensations and is usually assessed by the passive joint laxity tests. Mechanical stability of both the human knee joint and the lower extremity at early stance periods of gait (0% and 5%) were quantified here for the first time using a hybrid musculoskeletal model of the lower extremity. The roles of muscle coactivity, simulated by setting minimum muscle activation at 0–10% levels and ACL deficiency, simulated by reducing ACL resistance by up to 85%, on the stability margin as well as joint biomechanics (contact/muscle/ligament forces) were investigated. Dynamic stability was analyzed using both linear buckling and perturbation approaches at the final deformed configurations in gait. The knee joint was much more stable at 0% stance than at 5% due to smaller ground reaction and contact forces. Muscle coactivity, when at lower intensities (<3% of its maximum active force), increased dynamic stability margin. Greater minimum activation levels, however, acted as an ineffective strategy to enhance stability. Coactivation also substantially increased muscle forces, joint loads and ACL force and hence the risk of further injury and degeneration. A deficiency in ACL decreases total ACL force (by 31% at 85% reduced stiffness) and the stability margin of the knee joint at the heel strike. It also markedly diminishes forces in lateral hamstrings (by up to 39%) and contact forces on the lateral plateau (by up to 17%). Current work emphasizes the need for quantification of the lower extremity stability margin in gait.     

Mechanics of Undulatory Swimming in a Frictional Fluid

The sandfish lizard (Scincus scincus) swims within granular media (sand) using axial body undulations to propel itself without the use of limbs. In previous work we predicted average swimming speed by developing a numerical simulation that incorporated experimentally measured biological kinematics into a multibody sandfish model. The model was coupled to an experimentally validated soft sphere discrete element method simulation of the granular medium. In this paper, we use the simulation to study the detailed mechanics of undulatory swimming in a “granular frictional fluid” and compare the predictions to our previously developed resistive force theory (RFT) which models sand-swimming using empirically determined granular drag laws. The simulation reveals that the forward speed of the center of mass (CoM) oscillates about its average speed in antiphase with head drag. The coupling between overall body motion and body deformation results in a non-trivial pattern in the magnitude of lateral displacement of the segments along the body. The actuator torque and segment power are maximal near the center of the body and decrease to zero toward the head and the tail. Approximately 30% of the net swimming power is dissipated in head drag. The power consumption is proportional to the frequency in the biologically relevant range, which confirms that frictional forces dominate during sand-swimming by the sandfish. Comparison of the segmental forces measured in simulation with the force on a laterally oscillating rod reveals that a granular hysteresis effect causes the overestimation of the body thrust forces in the RFT. Our models provide detailed testable predictions for biological locomotion in a granular environment.     

Mechanisms of adaptation from a multiple to a single step recovery strategy following repeated exposure to forward loss of balance in older adults

When released from an initial, static, forward lean angle and instructed to recover with a single step, some older adults are able to meet the task requirements, whereas others either stumble or fall. The purpose of the present study was to use the concept of margin of stability (MoS) to investigate balance recovery responses in the anterior-posterior direction exhibited by older single steppers, multiple steppers and those that are able to adapt from multiple to single steps following exposure to repeated forward loss of balance. One hundred and fifty-one healthy, community dwelling, older adults, aged 65-80 years, participated in the study. Participants performed four trials of the balance recovery task from each of three initial lean angles. Balance recovery responses in the anterior-posterior direction were quantified at three events; cable release (CR), toe-off (TO) and foot contact (FC), for trials performed at the intermediate lean angle. MoS was computed as the anterior-posterior distance between the forward boundary of the Base of Support (BoS) and the vertical projection of the velocity adjusted centre of mass position (XCoM). Approximately one-third of participants adapted from a multiple to a single step recovery strategy following repeated exposure to the task. MoS at FC for the single and multiple step trials in the adaptation group were intermediate between the exclusively single step group and the exclusively multiple step group, with the single step trials having a significant, 3.7 times higher MoS at FC than the multiple step trials. Consistent with differences between single and multiple steppers, adaptation from multiple to single steps was attributed to an increased BoS at FC, a reduced XCoM at FC and an increased rate of BoS displacement from TO to FC. Adaptations occurred within a single test session and suggest older adults that are close to the threshold of successful recovery can rapidly improve dynamic stability following repeated exposure to a forward loss of balance.     

Retention of improvement in gait stability over 14 weeks due to trip-perturbation training is dependent on perturbation dose

Perturbation training is an emerging approach to reduce fall risk in the elderly. This study examined potential differences in retention of improvements in reactive gait stability over 14 weeks resulting from unexpected trip-like gait perturbations. Twenty-four healthy middle-aged adults (41–62 years) were assigned randomly to either a single perturbation group (SINGLE, n = 9) or a group subjected to eight trip-like gait perturbations (MULTIPLE, n = 15). While participants walked on a treadmill a custom-built brake-and-release system was used to unexpectedly apply resistance during swing phase to the lower right limb via an ankle strap. The anteroposterior margin of stability (MoS) was calculated as the difference between the anterior boundary of the base of support and the extrapolated centre of mass at foot touchdown for the perturbed step and the first recovery step during the first and second (MULTIPLE group only) perturbation trials for the initial walking session and retention-test walking 14 weeks later. Group MULTIPLE retained the improvements in reactive gait stability to the perturbations (increased MoS at touchdown for perturbed and first recovery steps; p < 0.01). However, in group SINGLE no differences in MoS were detected after 14 weeks compared to the initial walking session. These findings provide evidence for the requirement of a threshold trip-perturbation dose if adaptive changes in the human neuromotor system over several months, aimed at the improvement in fall-resisting skills, are to occur.     

Inertial artifact in viscoelastic measurements of striated muscle: Modeling and experimental results

Viscoelastic properties of striated muscle are often measured using length perturbation analysis and quantified as a complex modulus, whose elastic and viscous components reflect the energy-storage and energy-absorbing properties of the tissue, respectively. The energy stored as inertia is commonly ignored due to the small size of samples examined, typically <1 mm. Considering recent advances in tissue engineering to generate muscle tissues of larger sizes, we questioned whether ignoring the inertial artifact was still reasonable in these samples. To answer this question, we derived and solved the one-dimensional wave equation that describes the propagation of strain along the length of a sample. The inertial artifact was predicted to contaminate the elastic modulus with (2πf)²L₀²ρ/6, where f is perturbation frequency, L₀ is muscle length, and ρ is muscle density. We then measured viscoelastic properties up to 500 Hz in mouse skeletal muscle fibers at long (4.8 mm) and short (<1 mm) lengths and up to 100 Hz in rat cardiac slices at long (10–12 mm) and short (<2 mm) lengths. We found the elastic modulus of long preparations was elevated as frequency increased and was about half the magnitude of that predicted by the model. While the prediction tended to overestimate the measured inertial artifact, these results provided some validity to the model. We used the predicted artifact as an overly conservative estimate of error that might arise in a mechanics assay of mammalian striated muscle, whose nominal resting stiffness is on the order 100 kN m^?2. We found that muscle lengths of <1 mm resulted in negligible inertial artifact (<0.5% error) for perturbation frequencies under 250 Hz. Muscle samples longer than 5 mm, on the other hand, would result in >5% error at frequencies of 200 Hz and higher.     

On Heels and Toes: How Ants Climb with Adhesive Pads and Tarsal Friction Hair Arrays

Ants are able to climb effortlessly on vertical and inverted smooth surfaces. When climbing, their feet touch the substrate not only with their pretarsal adhesive pads but also with dense arrays of fine hairs on the ventral side of the 3^rd and 4^th tarsal segments. To understand what role these different attachment structures play during locomotion, we analysed leg kinematics and recorded single-leg ground reaction forces in Weaver ants (Oecophylla smaragdina) climbing vertically on a smooth glass substrate. We found that the ants engaged different attachment structures depending on whether their feet were above or below their Centre of Mass (CoM). Legs above the CoM pulled and engaged the arolia (‘toes’), whereas legs below the CoM pushed with the 3^rd and 4^th tarsomeres (‘heels’) in surface contact. Legs above the CoM carried a significantly larger proportion of the body weight than legs below the CoM. Force measurements on individual ant tarsi showed that friction increased with normal load as a result of the bending and increasing side contact of the tarsal hairs. On a rough sandpaper substrate, the tarsal hairs generated higher friction forces in the pushing than in the pulling direction, whereas the reverse effect was found on the smooth substrate. When the tarsal hairs were pushed, buckling was observed for forces exceeding the shear forces found in climbing ants. Adhesion forces were small but not negligible, and higher on the smooth substrate. Our results indicate that the dense tarsal hair arrays produce friction forces when pressed against the substrate, and help the ants to push outwards during horizontal and vertical walking.     

Controlling propulsive forces in gait initiation in transfemoral amputees

During prosthetic gait initiation, transfemoral (TF) amputees control the spatial and temporal parameters that modulate the propulsive forces, the positions of the center of pressure (CoP), and the center of mass (CoM). Whether their sound leg or the prosthetic leg is leading, the TF amputees reach the same end velocity. We wondered how the CoM velocity build up is influenced by the differences in propulsive components in the legs and how the trajectory of the CoP differs from the CoP trajectory in able bodied (AB) subjects. Seven TF subjects and eight AB subjects were tested on a force plate and on an 8 m long walkway. On the force plate, they initiated gait two times with their sound leg and two times with their prosthetic leg. Force measurement data were used to calculate the CoM velocity curves in horizontal and vertical directions. Gait initiated on the walkway was used to determine the leg preference. We hypothesized that because of the differences in propulsive components, the motions of the CoP and the CoM have to be different, as ankle muscles are used to help generate horizontal ground reaction force components. Also, due to the absence of an active ankle function in the prosthetic leg, the vertical CoM velocity during gait initiation may be different when leading with the prosthetic leg compared to when leading with the sound leg. The data showed that whether the TF subjects initiated a gait with their prosthetic leg or with their sound leg, their horizontal end velocity was equal. The subjects compensated the loss of propulsive force under the prosthesis with the sound leg, both when the prosthetic leg was leading and when the sound leg was leading. In the vertical CoM velocity, a tendency for differences between the two conditions was found. When initiating gait with the sound leg, the downward vertical CoM velocity at the end of the gait initiation was higher compared to when leading with the prosthetic leg. Our subjects used a gait initiation strategy that depended mainly on the active ankle function of the sound leg; therefore, they changed the relative durations of the gait initiation anticipatory postural adjustment phase and the step execution phase. Both legs were controlled in one single system of gait propulsion. The shape of the CoP trajectories, the applied forces, and the CoM velocity curves are described in this paper.     

Effect of intervertebral translational flexibilities on estimations of trunk muscle forces,kinematics, loads,and stability

Due to the complexity of the human spinal motion segments, the intervertebral joints are often simulated in the musculoskeletal trunk models as pivots thus allowing no translational degrees of freedom (DOFs). This work aims to investigate, for the first time, the effect of such widely used assumption on trunk muscle forces, spinal loads, kinematics, and stability during a number of static activities. To address this, the shear deformable beam elements used in our nonlinear finite element (OFE) musculoskeletal model of the trunk were either substantially stiffened in translational directions (SFE model) or replaced by hinge joints interconnected through rotational springs (HFE model). Results indicated that ignoring intervertebral translational DOFs had in general low to moderate impact on model predictions. Compared with the OFE model, the SFE and HFE models predicted generally larger L4–L5 and L5–S1 compression and shear loads, especially for tasks with greater trunk angles; differences reached ～15% for the L4–L5 compression, ～36% for the L4–L5 shear and ～18% for the L5–S1 shear loads. Such differences increased, as location of the hinge joints in the HFE model moved from the mid-disc height to either the lower or upper endplates. Stability analyses of these models for some select activities revealed small changes in predicted margin of stability. Model studies dealing exclusively with the estimation of spinal loads and/or stability may, hence with small loss of accuracy, neglect intervertebral translational DOFs at smaller trunk flexion angles for the sake of computational simplicity.     

Kinematics of lower limbs during walking are emulated by springy walking model with a compliantly connected,off-centered curvy foot

The dynamics of the center of mass (CoM) during walking and running at various gait conditions are well described by the mechanics of a simple passive spring loaded inverted pendulum (SLIP). Due to its simplicity, however, the current form of the SLIP model is limited at providing any further information about multi-segmental lower limbs that generate oscillatory CoM behaviors and their corresponding ground reaction forces. Considering that the dynamics of the CoM are simply achieved by mass-spring mechanics, we wondered whether any of the multi-joint motions could be demonstrated by simple mechanics. In this study, we expand a SLIP model of human locomotion with an off-centered curvy foot connected to the leg by a springy segment that emulates the asymmetric kinematics and kinetics of the ankle joint. The passive dynamics of the proposed expansion of the SLIP model demonstrated the empirical data of ground reaction forces, center of mass trajectories, ankle joint kinematics and corresponding ankle joint torque at various gait speeds. From the mechanically simulated trajectories of the ankle joint and CoM, the motion of lower-limb segments, such as thigh and shank angles, could be estimated from inverse kinematics. The estimation of lower limb kinematics showed a qualitative match with empirical data of walking at various speeds. The representability of passive compliant mechanics for the kinetics of the CoM and ankle joint and lower limb joint kinematics implies that the coordination of multi-joint lower limbs during gait can be understood with a mechanical framework.     

Determining the location of the body׳s center of mass for different groups of physically active people

The purpose of the present study was to compare the location of the body center of mass (CoM) determined by using a high accuracy reaction board (RB) and two different segment parameter models for motion analysis (Dempster, 1955, DEM and de Leva, 1996 adjusted from Zatsiorsky and Seluyanov, ZAT). The body CoM (expressed as percentage of the total body height) was determined from several subjects including athletes as well as physically active students and sedentary people. Some significant differences were found in the location of the body CoM between the used segment models and the reaction board method for all male subjects (n=58, 57.03±0.79%, 56.20±0.76% and 57.60±0.76% for RB, ZAT and DEM, respectively) and separately for male (n=12, RB 57.02±0.41%, ZAT 56.74±0.62%, DEM 58.19±0.60%) and female (n=12, RB 55.91±0.88%, ZAT 57.24±0.77%) students of physical activity. The ZAT model was a good match with RB for high jumpers (56.26±0.94% and 56.63±0.56%) whereas the DEM model was better for gymnasts (57.38±0.46% and 57.89±0.49%) and throwers (58.19±0.69% and 57.79±0.45%). For ice hockey players (IH) and ski jumpers (SJ) both segment models, ZAT and DEM, differed significantly from the reaction board results. The results of the present study showed that careful attention should be paid while selecting the proper model for motion analysis of different type of athletes.     

Neuronal firing sensitivity to morphologic and active membrane parameters

Both the excitability of a neuron's membrane, driven by active ion channels, and dendritic morphology contribute to neuronal firing dynamics, but the relative importance and interactions between these features remain poorly understood. Recent modeling studies have shown that different combinations of active conductances can evoke similar firing patterns, but have neglected how morphology might contribute to homeostasis. Parameterizing the morphology of a cylindrical dendrite, we introduce a novel application of mathematical sensitivity analysis that quantifies how dendritic length, diameter, and surface area influence neuronal firing, and compares these effects directly against those of active parameters. The method was applied to a model of neurons from goldfish Area II. These neurons exhibit, and likely contribute to, persistent activity in eye velocity storage, a simple model of working memory. We introduce sensitivity landscapes, defined by local sensitivity analyses of firing rate and gain to each parameter, performed globally across the parameter space. Principal directions over which sensitivity to all parameters varied most revealed intrinsic currents that most controlled model output. We found domains where different groups of parameters had the highest sensitivities, suggesting that interactions within each group shaped firing behaviors within each specific domain. Application of our method, and its characterization of which models were sensitive to general morphologic features, will lead to advances in understanding how realistic morphology participates in functional homeostasis. Significantly, we can predict which active conductances, and how many of them, will compensate for a given age- or development-related structural change, or will offset a morphologic perturbation resulting from trauma or neurodegenerative disorder, to restore normal function. Our method can be adapted to analyze any computational model. Thus, sensitivity landscapes, and the quantitative predictions they provide, can give new insight into mechanisms of homeostasis in any biological system.     

Adaptational responses in dynamic stability during disturbed walking in the elderly

The purpose of this study was to examine the age-related predictive and feedback adaptive locomotor improvements in the components of dynamic stability control during disturbed walking. Thirteen old (62–74 yrs) and ten young (23–30 yrs) male subjects performed a gait protocol on a gangway, which included one covered element. By exchanging this element, the subjects walked either solely over hard surface or experienced a perturbation of the gait on the soft surface element. The gait protocol consisted of a baseline on hard surface and an adaptation phase with 19 trials on soft or hard (2nd, 8th and 19th) surface. The investigation of dynamic stability was made by using the margin of stability (MS), which was calculated as the difference between the base of support and the extrapolated center of mass (CM). Horizontal velocity of CM and its vertical projection in anterior–posterior direction as well as the eigenfrequency of an inverted pendulum generate the extrapolated CM. As a result of the first unexpected disturbance, MS was decreased in the step following the perturbation compared to baselines in both age-groups. This decrease was higher for the old participants compared to the young ones, indicating a more unstable position in the step after the perturbation for the elderly. In the following adaptation phase, MS returned to baseline values in both age-groups. In the hard trial after the first unexpected perturbation, both age-groups increased MS at touchdown of the disturbed leg compared to baseline, reflecting fast predictive adjustments. Our findings show that the well-known age-related biological impairments did not inhibit the adaptive improvements in the components of dynamic stability in the elderly. However, the feedback corrections after the first unexpected perturbation were less effective for the elderly. This may increase the risk of falling.     

A comparison between a new model and current models for estimating trunk segment inertial parameters

Modeling of the body segments to estimate segment inertial parameters is required in the kinetic analysis of human motion. A new geometric model for the trunk has been developed that uses various cross-sectional shapes to estimate segment volume and adopts a non-uniform density function that is gender-specific. The goal of this study was to test the accuracy of the new model for estimating the trunk's inertial parameters by comparing it to the more current models used in biomechanical research. Trunk inertial parameters estimated from dual X-ray absorptiometry (DXA) were used as the standard. Twenty-five female and 24 male college-aged participants were recruited for the study. Comparisons of the new model to the accepted models were accomplished by determining the error between the models’ trunk inertial estimates and that from DXA. Results showed that the new model was more accurate across all inertial estimates than the other models. The new model had errors within 6.0% for both genders, whereas the other models had higher average errors ranging from 10% to over 50% and were much more inconsistent between the genders. In addition, there was little consistency in the level of accuracy for the other models when estimating the different inertial parameters. These results suggest that the new model provides more accurate and consistent trunk inertial estimates than the other models for both female and male college-aged individuals. However, similar studies need to be performed using other populations, such as elderly or individuals from a distinct morphology (e.g. obese). In addition, the effect of using different models on the outcome of kinetic parameters, such as joint moments and forces needs to be assessed.     

Perturbation analysis of 6DoF flight dynamics and passive dynamic stability of hovering fruit fly Drosophila melanogaster

Insects exhibit exquisite control of their flapping flight, capable of performing precise stability and steering maneuverability. Here we develop an integrated computational model to investigate flight dynamics of insect hovering based on coupling the equations of 6 degree of freedom (6DoF) motion with the Navier-Stokes (NS) equations. Unsteady aerodynamics is resolved by using a biology-inspired dynamic flight simulator that integrates models of realistic wing-body morphology and kinematics, and a NS solver. We further develop a dynamic model to solve the rigid body equations of 6DoF motion by using a 4th-order Runge-Kutta method. In this model, instantaneous forces and moments based on the NS-solutions are represented in terms of Fourier series. With this model, we perform a systematic simulation-based analysis on the passive dynamic stability of a hovering fruit fly, Drosophila melanogaster, with a specific focus on responses of state variables to six one-directional perturbation conditions during latency period. Our results reveal that the flight dynamics of fruit fly hovering does not have a straightforward dynamic stability in a conventional sense that perturbations damp out in a manner of monotonous convergence. However, it is found to exist a transient interval containing an initial converging response observed for all the six perturbation variables and a terminal instability that at least one state variable subsequently tends to diverge after several wing beat cycles. Furthermore, our results illustrate that a fruit fly does have sufficient time to apply some active mediation to sustain a steady hovering before losing body attitudes.     

Stepping boundary of external force-controlled perturbations of varying durations: Comparison of experimental data and model simulations

This study investigated the stepping boundary – the force that can be resisted without stepping – for force-controlled perturbations of different durations. Twenty-two healthy young adults (19–37 years old) were instructed to try not to step in response to 86 different force/time combinations of forward waist-pulls. The forces at which 50% of subjects stepped (F₅₀) were identified for each tested perturbation durations. Results showed that F₅₀ decreased hyperbolically when the perturbation’s duration increased and converged toward a constant value (about 10% BW) for longer perturbations (over 1500 ms). The effect of perturbation duration was critical for the shortest perturbations (less than 1 s).In parallel, a simple function was proposed to estimate this stepping boundary. Considering the dynamics of a linear inverted pendulum + foot model and simple balance recovery reactions, we could express the maximum pulling force that can be withstood without stepping as a simple function of the perturbation duration. When used with values of the main model parameters determined experimentally, this function replicated adequately the experimental results.This study demonstrates for the first time that perturbation duration has a major influence on the outcomes of compliant perturbations such as force-controlled pulls. The stepping boundary corresponds to a constant perturbation force-duration product and is largely explained by only two parameters: the reaction time and the displacement of the center of pressure within the functional base of support. Future work should investigate pathological populations and additional parameters characterizing the perturbation time-profile such as the time derivative of the perturbation.