associations between trunk postural control in walking and unstable sitting at various levels of task demand abstract trunk postural control tpc has been investigated in several populations and tasks previous work observed targeted training of tpc via isolated trunk control tasks may improve performance in other activities e g walking however the nature of this relationship remains unknown we therefore investigated the relationship between tpc at both the global i e response to finite perturbations and local i e resistance to continuous perturbations levels during walking and unstable sitting both at varying levels of task demand thirteen individuals 11 male 2 female with no recent history past 12 months of illness injury or musculoskeletal disorders walked on a dual belt treadmill at four speeds 20 10 10 and 20 of self selected walking speed and completed an unstable sitting task at four levels of chair instability 100 75 60 and 45 of an individual s neutral stability as defined by the gravitational gradient three dimensional trunk and pelvic kinematics were collected tri planar lyapunov exponents and sample entropy characterized local tpc global tpc was characterized by ranges of motion and for seated trials metrics derived from center of pressure time series i e path length 95 confidence ellipse area mean velocity and rms position no strong or significant correlations 0 057

computing two dimensional poincar maps for hyperchaotic dynamics poincar map pm is one of the felicitous discrete approximation of the continuous dynamics to compute pm the discrete relation s between the successive point of interactions of the trajectories on the suitable poincar section ps are found out these discrete relations act as an amanuensis of the nature of the continuous dynamics in this article we propose a computational scheme to find a hyperchaotic pm hpm from an equivalent three dimensional 3d subsystem of a 4d or higher hyperchaotic model for the experimental purpose a standard four dimensional 4d hyperchaotic lorenz stenflo system hlss and a five dimensional 5d hyperchaotic laser model hlm is considered equivalent 3d subsystem is obtained by comparing the movements of the trajectories of the original hyperchaotic systems with all of their 3d subsystems the quantitative measurement of this comparison is made promising by recurrence quantification analysis rqa various two dimensional 2d poincar mas are computed for several suitable poincar sections for both the systems but only some of them are hyperchaotic in nature the hyperchaotic behavior is verified by positive values of both one dimensional 1d lyapunov exponent le i and 2d lyapunov exponent le ii at the end similarity of the dynamics between the hyperchaotic systems and their 2d hyperchaotic poincar maps hpm has been established through mean recurrence time mrt statistics for both of 4d hlss and 5d hlm and the best approximated discrete dynamics for both the hyperchaotic systems are found out

a 3d fem bem rolling contact formulation for unstructured meshes abstract this work presents a new formulation for solving 3d steady state rolling contact problems the convective terms for computing the tangential slip velocities involved in the rolling problem are evaluated using a new approximation inspired in numerical fluid dynamics techniques for unstructured meshes moreover the elastic influence coefficients of the surface points in contact are approached by means of the finite element method fem and or the boundary element method bem the contact problem is based on an augmented lagrangian formulation and the use of projection functions to establish the contact restrictions finally the resulting nonlinear equations set is solved using the generalized newton method with line search gnmls presenting some acceleration strategies as a new and more simplified projection operator which makes it possible to obtain a quasi complementarity of the contact variables reducing the number of contact problem unknowns and using iterative solvers the presented methodology is validated solving some rolling contact problems and analyzed for some unstructured mesh examples

is slow walking more stable several efforts have been made to study gait stability using measures derived from nonlinear timeseries analysis the maximum finite time lyapunov exponent lmax quantifies how a system responds to an infinitesimally small perturbation recent studies suggested that slow walking leads to lower lmax values and thus is more stable than fast walking but these studies suffer from methodological limitations we studied the effects of walking speed on the amount of kinematic variability and stability in human walking trunk motions of 15 healthy volunteers were recorded in 3d during 2 min of treadmill walking at different speeds from those time series maximum lyapunov exponents indicating short term and long term divergence ls stride and ll stride and mean standard deviation meansd were calculated ls stride showed a linear decrease with increasing speed for forward backward ap movements and quadratic effects inverted u shaped for medio lateral ml and up down vt movements ll stride showed a quadratic effect inverted u shaped of walking speed for ap movements a linear decrease for ml movements and a linear increase for vt movements moreover positive correlations between ls and meansd were found for all directions while ll stride and meansd were correlated negatively in the ap direction the different effects of walking speed on ls stride and ll stride for the different planes suggest that slow walking is not necessarily more stable than fast walking the absence of a consistent pattern of correlations between ll stride and meansd over the three directions suggests that variability and stability reflect at least to a degree different properties of the dynamics of walking

dynamic margins of stability during human walking in destabilizing environments understanding how humans maintain stability when walking particularly when exposed to perturbations is key to preventing falls here we quantified how imposing continuous pseudorandom anterior posterior ap and mediolateral ml oscillations affected the control of dynamic walking stability twelve subjects completed five 3 minute walking trials in the computer assisted rehabilitation environment caren system under each of 5 conditions no perturbation nop ap platform app or visual apv or ml platform mlp or visual mlv oscillations we computed ap and ml margins of stability mos for each trial mean mosml were consistently slightly larger during all perturbation conditions than during nop p 0 038 mean mosap for the app mlp and mlv oscillations were significantly smaller than during nop p 0 0005 variability of both mosap and mosml was significantly greater during the mlp and mlv oscillations than during nop p 0 0005 we also directly quantified how the mos on any given step affected the mos on the following step using first return plots there were significant changes in step to step mosml dynamics between experimental conditions p 0 0005 these changes suggested that subjects may have been trying to control foot placement and consequently stability during the perturbation conditions quantifying step to step changes in margins of dynamic stability may be more useful than mean mos in assessing how individuals control walking stability

dynamical grobner bases over dedekind rings abstract in this paper we extend the notion of dynamical grobner bases introduced by the second author to dedekind rings with zero divisors as an application we dynamically solve the ideal membership problem and compute a generating set for the syzygy module over multivariate polynomial rings with coefficients in dedekind rings we also give a partial positive answer to a conjecture about grobner rings

local dynamic stability in turning and straight line gait successful community and household ambulation require the ability to navigate corners and maneuver around obstacles posing unique challenges compared to straight line walking the challenges associated with turning may contribute to an increased incidence of falling and the occurrence of fall related injuries a measure of stability applied to turning gait may be able to quantify a system s response to naturally occurring disturbances associated with turning and identify subjects at greater risk of falling an index of stability has been used previously to assess the rate of kinematic separation local dynamic stability during straight line gait the purpose of this study was to determine if local dynamic stability during constant speed turning is reduced compared to straight line treadmill walking maximum finite time lyapunov exponents lambda were used to estimate the local stability of able bodied subjects n 19 sagittal plane hip knee and ankle trajectories for turning compared to straight line walking at two different walking speeds turning lambda was greater than straight lambda for the hip right knee and ankle p 0 05 turning lambda for the left knee angle was similar to straight lambda there were no differences in lambda between left and right limbs for the hip and ankle and also no differences between the inside and outside limbs during turning for all joints these findings indicate able bodied subjects hip right knee and ankle kinematics are less locally stable while turning than walking in a straight line and may be used as a comparative tool for determining the efficacy of therapeutic interventions for mobility impaired populations

a zero dimensional valuation ring is 1 grobner abstract in this study we consider the following conjecture and open problem 1 a valuation ring r is 1 grobner if and only if r is an archimedean ring 2 is a 1 grobner valuation ring coherent first we prove that a zero dimensional valuation ring is 1 grobner furthermore we give a positive answer to 1 and a negative answer to 2 finally we present an example of a valuation ring that is 1 grobner but not coherent or noetherian

calculation of lyapunov exponents avoiding spurious elements abstract a new approach to evaluate lyapunov exponents from time series is discussed with the help of this approach the number of calculated lyapunov exponents can be reduced to the physically relevant ones calculations in high dimensional embedding spaces are performed on smaller data bases with improved speed more direct information is obtained from the dynamics of the underlying dynamical system

computing syzygies over v x1 xk v a valuation domain abstract we give an algorithm for computing the v saturation of any finitely generated submodule of v x 1 x k m k n m n where v is a valuation domain our algorithm is based on a notion of echelon form which ensures its correctness the proposed algorithm terminates when two hilbert series on the quotient field and the residue field of v coincide as application our algorithm computes syzygies over v x 1 x k

effects of fatigue on inter cycle variability in cross country skiing the aim of the study was to examine the inter cycle variability in cross country skiing gait and its evolution with fatigue both issues were investigated to understand the flexibility capabilities of the neuromuscular system four women and four men skied on a treadmill up to exhaustion the angular displacements of the arms and legs movements were obtained for 40s period at the beginning and end of the skiing test mean inter cycle standard deviation sd c largest lyapunov exponent lambda 1 and correlation dimension d c were computed for each time series and surrogate counterpart to evaluate the magnitude and nature of the variability for any experimental time series lambda 1 was positive d c greater than 1 and both were found to be different from their surrogate counterparts confirming that the temporal variations of the data had a deterministic origin more larger sd c d c and lambda 1 values were observed at the end of the test indicating more variability noise and local dynamic instability in the data with fatigue hence the fluctuations of limb angular displacements displayed a chaotic behavior which reflected flexibility of the neuromuscular system to adapt to possible perturbations during skiing however such chaotic behavior degraded with fatigue making the neuromuscular system less adaptable and more unstable

comparison of different state space definitions for local dynamic stability analyses abstract measures of local dynamic stability such as the local divergence exponent s quantify how quickly small perturbations deviate from an attractor that defines the motion when the governing equations of motion are unknown an attractor can be reconstructed by defining an appropriate state space however state space definitions are not unique and accepted methods for defining state spaces have not been established for biomechanical studies this study first determined how different state space definitions affected s for the lorenz attractor since exact theoretical values were known a priori values of s exhibited errors s decreased post fatigue for all 6 state spaces differences were statistically significant for 3 of these state spaces for state spaces defined using delay embedding increasing the embedding dimension decreased s in both the lorenz and experimental data overall our findings suggest that direct numerical comparisons between studies that use different state space definitions should be made with caution however trends across experimental comparisons appear to persist biomechanical state spaces constructed using positions and velocities or delay reconstruction of individual states are likely to provide consistent results

complexity and synchronization in chaotic fiber optic systems abstract in this work we investigate experimentally the complexity of chaotic attractors generated by a semiconductor laser subjected to optical feedback and associate their dimensionality with the synchronization efficiency of the corresponding chaotic transmitter receiver configuration the complexity is characterized by calculating the correlation dimension d 2 of experimental chaotic time series for different values of the optical feedback we present the effect of d 2 on the synchronization efficiency and determine the optimal operating condition that leads to the most complex chaotic carrier and simultaneously to the most successful synchronization lastly we associate and explain our experimental results with theoretical predictions in the research literature

local dynamic stability of treadmill walking intrasession and week to week repeatability abstract repetitive falls degrade the quality of life of elderly people and of patients suffering of various neurological disorders in order to prevent falls while walking one should rely on relevant early indicators of impaired dynamic balance the local dynamic stability lds represents the sensitivity of gait to small perturbations divergence exponents maximal lyapunov exponents assess how fast a dynamical system diverges from neighbor points although numerous findings attest the validity of lds as a fall risk index reliability results are still sparse the present study explores the intrasession and intersession repeatability of gait lds using intraclass correlation coefficients icc and standard error of measurement sem ninety five healthy individuals performed 5 min treadmill walking in two sessions separated by 9 days trunk acceleration was measured with a 3d accelerometer three time scales were used to estimate lds over 4 10 strides 4 10 over one stride 1 and over one step 0 5 the intrasession repeatability was assessed from three repetitions of either 35 strides or 70 strides taken within the 5 min tests the intersession repeatability compared the two sessions which totalized 210 strides the intrasession iccs 70 strides estimates 35 strides estimates were 0 52 0 18 for 4 10 and 0 84 0 77 for 1 and 0 5 the intersession iccs were around 0 60 the sem results revealed that 0 5 measured in medio lateral direction exhibited the best reliability sufficient to detect moderate changes at individual level 20 however due to the low intersession repeatability one should average several measurements taken on different days in order to better approximate the true lds

noether normalization theorem and dynamical grobner bases over bezout domains of krull dimension 1 abstract we propose a new version of noether normalization theorem over bezout domains of krull dimension one the ring z of integers as main example we also show that one can avoid branching when computing dynamical grobner bases over a bezout domain of krull dimension one

when human walking becomes random walking fractal analysis and modeling of gait rhythm fluctuations abstract we present a random walk fractal analysis of the stride to stride fluctuations in the human gait rhythm the gait of healthy young adults is scale free with long range correlations extending over hundreds of strides this fractal scaling changes characteristically with maturation in children and older adults and becomes almost completely uncorrelated with certain neurologic diseases stochastic modeling of the gait rhythm dynamics based on transitions between different neural centers reproduces distinctive statistical properties of the gait pattern by tuning one model parameter the hopping transition range the model can describe alterations in gait dynamics from childhood to adulthood including a decrease in the correlation and volatility exponents with maturation

step width variability but not step length variability or step time variability discriminates gait of healthy young and older adults during treadmill locomotion the variability of spatial and temporal step kinematics have separately been shown to prospectively predict falls by older adults however the published literature has not addressed the relative importance of the information related to locomotion control contributed by variability of spatial variables temporal variables or both we conducted a post hoc analysis to determine the extent to which the variability of spatial and temporal step kinematics are independent descriptors of locomotion control in healthy young and older adults a second purpose of the analysis was to establish the extent to which treadmill walking mimics overground walking using a benchmark for step kinematics eighteen young adults and 12 older adults walked at a self selected velocity on an instrumented treadmill from which step length variability step width variability and step time variability were computed stepwise discriminant analysis correctly classified group membership of 70 percent of the 30 subjects i e young or older adult using only step width variability p 0 037 wilk s lamda 0 854 step width variability was 70 57 percent larger than step length variability p 0 001 this relationship was similar to that of the benchmark established for overground locomotion the results suggest that for healthy younger and older adults step width variability is a more meaningful descriptor of locomotion control than step length variability and step time variability given the potential clinical impact further systematic study and improvement of the methods and technology for acquiring step kinematic data are warranted

gait stability variability and complexity on inclined surfaces this study evaluated the gait stability variability and complexity of healthy young adults on inclined surfaces a total of 49 individuals walked on a treadmill at their preferred speed for 4min at inclinations of 6 8 and 10 in upward up and downward down conditions and in horizontal 0 condition gait variability was assessed using average standard deviation trunk acceleration between strides var gait stability was assessed using margin of stability mos and maximum lyapunov exponent s and gait complexity was assessed using sample entropy sen trunk variability var increased in the medial lateral ml anterior posterior and vertical directions for all inclined conditions the sen values indicated that movement complexity decreased almost linearly from down to up conditions reflecting changes in gait pattern with longer and slower steps as inclination increased the down conditions were associated with the highest variability and lowest stability in the mos ml but not in s stability was lower in up conditions which exhibited the largest s values the overall results support the hypothesis that inclined surfaces decrease gait stability and alter gait variability particularly in up conditions

a constructive comparison of the rings r x and r x and application to the lequain simis induction theorem we constructively prove that for any ring r with krull dimension d the ring r x locally behaves like the ring r x or a localization of a polynomial ring of type s 1r x with s a multiplicative subset of r such that the krull dimension of s 1r is d 1 as an application we give a simple and constructive proof of the lequain simis induction theorem which is an important variation of the quillen induction theorem

effect of noise and filtering on largest lyapunov exponent of time series associated with human walking this study aimed to determine the effect of added noise filtering and time series length on the largest lyapunov exponent lye value calculated for time series obtained from a passive dynamic walker the simplest passive dynamic walker model comprising of two massless legs connected by a frictionless hinge joint at the hip was adopted to generate walking time series the generated time series was used to construct a state space with the embedding dimension of 3 and time delay of 100 samples the lye was calculated as the exponential rate of divergence of neighboring trajectories of the state space using rosenstein s algorithm to determine the effect of noise on lye values seven levels of gaussian white noise snr 55 25db with 5db steps were added to the time series in addition the filtering was performed using a range of cutoff frequencies from 3hz to 19hz with 2hz steps the lye was calculated for both noise free and noisy time series with different lengths of 6 50 100 and 150 strides results demonstrated a high percent error in the presence of noise for lye therefore these observations suggest that rosenstein s algorithm might not perform well in the presence of added experimental noise furthermore findings indicated that at least 50 walking strides are required to calculate lye to account for the effect of noise finally observations support that a conservative filtering of the time series with a high cutoff frequency might be more appropriate prior to calculating lye

fundamental limitations for estimating dimensions and lyapunov exponents in dynamical systems we show that values of the correlation dimension estimated over a decade from the grassberger procaccia algorithm cannot exceed the value 2 log10n if n is the number of points in the time series when this bound is saturated it is thus not legitimate to conclude that low dimensional dynamics is present the estimation of lyapunov exponents is also discussed

phase dependent changes in local dynamic stability during walking in elderly with and without knee osteoarthritis previously we reported reduced time averaged knee local stability in the unaffected but not the affected leg of elderly with knee osteoarthritis oa compared to controls since stability may show phase related changes we reanalyzed the dataset reported previously using time dependent local stability t and also calculated time averaged local stability s for comparison we studied treadmill walking at increasing speeds focusing on sagittal plane knee movements 16 patients 12 healthy peers and 15 young subjects were measured we found a clear maximum in t i e minimum in stability at around 60 of the stride cycle stancemax t a second clear maximum swingmax t at around 95 followed by a minimum between 70 and 100 swingmin t stancemax t of both legs was significantly higher in the oa than the young control group values for healthy elderly fell between those of the other groups were significantly higher than in young adults but there was only a trend towards a significant difference with the stancemax t of the oa group s affected side time averaged and time dependent stability measures within one leg were uncorrelated while time dependent stability measures at the affected side were inversely correlated with s at the unaffected side the results indicate that time dependent local dynamic stability might provide a more detailed insight into the problems of gait stability in oa than conventional averaged local dynamic stability measures and support the notion that the paradoxical decline in unaffected side time averaged local stability may be caused by a trade off between affected and unaffected side stability

recurrence plot analysis of irregularly sampled data irregularly sampled time series usually require data preprocessing before a desired time series analysis can be applied we propose an approach for distance measuring of pairs of data points which is directly applicable to irregularly sampled time series in order to apply recurrence plot analysis to irregularly sampled time series we use this approach and detect regime transitions in prototypical models and for an application from palaeoclimatatology this approach might be useful for any method that is based on distance measuring e g correlation sum or lyapunov exponent estimation

using higher order correlations to define an embedding window abstract correlation functions of up to fourth order are investigated numerically in search of a reliable window for embedding or reconstructing chaotic flows by the method of time delays a number of these higher order correlations display coincident extrema the time at which this coincidence occurs is shown to be a reasonable embedding window for calculating the correlation dimension of a number of well known chaotic attractors

measurement of the lyapunov spectrum from a chaotic time series the exponential divergence or convergence of nearby trajectories lyapunov exponents is conceptually the most basic indicator of deterministic chaos we propose a new method to determine the spectrum of several lyapunov exponents including positive zero and even negative ones from the observed time series of a single variable we have applied the method to various known model systems and also to the rayleigh b enard experiment and have elucidated the dependence of the lyapunov exponents on the rayleigh number

a wavelet based method for surrogate data generation hypothesis testing based on surrogate data has emerged as a popular way to test the null hypothesis that a signal is a realisation of a linear gaussian stochastic process if these surrogates are constrained to the values and power spectrum of the original data there is no need to formulate a pivotal test statistic in this paper a method is presented for generating constrained surrogates using a wavelet transform introducing a threshold above which wavelet detail coefficients are pinned to their original values such surrogates avoid problems of nonstationarity for pseudo periodic data and appear to be more robust than conventional approaches for situations where period modulation is corrupting a gaussian stochastic process when used for generating ensemble realisations of a process the approach used here avoids some of the difficulties of methods based on simple randomisation of wavelet coefficients

the effects of general fatigue induced by incremental exercise test and active recovery modes on energy cost gait variability and stability in male soccer players abstract the aerobic endurance is considered an important physiological capacity of soccer players which is examined by incremental exercise test iet however it is not clear how general fatigue induced by iet influences physiological and biomechanical gait features in soccer players and how players recover optimally at post iet here the effect of general fatigue induced by iet on energy cost gait variability and stability in soccer players was investigated to identify an optimal recovery mode the effect of walking at preferred walking speed pws running at individual ventilation threshold ivt two active recovery modes and rest a passive recovery mode on aforementioned features were studied nine male players walked 4 min at pws on a treadmill prior iet pret which was followed by four 4 min walking trials post 0 1 2 and 3 with three 4 min recovery intervals pws ivt or rest between them in three sessions one for each recovery mode in a random order energy cost gait variability and stability were examined at pret baseline and at post 0 1 2 and 3 intervals of respectively 0 4 8 12 16 20 24 28 min at post iet gait variability was assessed by the standard deviation of trunk angle and gait stability was assessed by the local dynamic stability of trunk angular velocity gait stability was not affected by iet despite increases in gait variability and energy cost different from ivt pws and rest recovery modes reduced energy cost at post iet gait variability and energy cost recovered at post 1 and post 2 suggesting that 8 12 and 16 20 min recovery intervals respectively were required for returning to their baselines no preference for active over passive recovery was found in terms of gait variability and energy cost

energy cost of balance control during walking decreases with external stabilizer stiffness independent of walking speed human walking requires active neuromuscular control to ensure stability in the lateral direction which inflicts a certain metabolic load the magnitude of this metabolic load has previously been investigated by means of passive external lateral stabilization via spring like cords in the present study we applied this method to test two hypotheses 1 the effect of external stabilization on energy cost depends on the stiffness of the stabilizing springs and 2 the energy cost for balance control and consequently the effect of external stabilization on energy cost depends on walking speed fourteen healthy young adults walked on a motor driven treadmill without stabilization and with stabilization with four different spring stiffnesses between 760 and 1820 nm 1 at three walking speeds 70 100 and 130 of preferred speed energy cost was calculated from breath by breath oxygen consumption gait parameters mean and variability of step width and stride length and variability of trunk accelerations were calculated from kinematic data on average external stabilization led to a decrease in energy cost of 6 p 0 005 as well as a decrease in step width 24 p 0 001 step width variability 41 p 0 001 and variability of medio lateral trunk acceleration 12 5 p 0 005 increasing stabilizer stiffness increased the effects on both energy cost and medio lateral gait parameters up to a stiffness of 1260 nm 1 contrary to expectations the effect of stabilization was independent of walking speed p 0 111 these results show that active lateral stabilization during walking involves an energetic cost which is independent of walking speed

a boundary element formulation for wear modeling on 3d contact and rolling contact problems the present work shows a new numerical treatment for wear simulation on 3d contact and rolling contact problems this formulation is based on the boundary element method bem for computing the elastic influence coefficients and on projection functions over the augmented lagrangian for contact restrictions fulfillment the constitutive equations of the potential contact zone are signorini s contact conditions coulomb s law of friction and holm archard s law of wear the proposed methodology is applied to predict wear on different contact and rolling contact problems results are validated with numerical solutions and semi analytical models presented in the literature the bem considers only the degrees of freedom involved on these kind of problems those on the solids surfaces reducing the number of unknowns and obtaining a very good approximation on contact tractions using a low number of elements together with the formulation an acceleration strategy is presented allowing to reduce the times of resolution

a benchmark test of accuracy and precision in estimating dynamical systems characteristics from a time series characteristics of dynamical systems are often estimated to describe physiological processes for instance lyapunov exponents have been determined to assess the stability of the cardio vascular system respiration and more recently human gait and posture however the systematic evaluation of the accuracy and precision of these estimates is problematic because the proper values of the characteristics are typically unknown we fill this void with a set of standardized time series with well defined dynamical characteristics that serve as a benchmark estimates ought to match these characteristics at least to good approximation we outline a procedure to employ this generic benchmark test and illustrate its capacity by examining methods for estimating the maximum lyapunov exponent in particular we discuss algorithms by wolf and co workers and by rosenstein and co workers and evaluate their performances as a function of signal length and signal to noise ratio in all scenarios the precision of rosenstein s algorithm was found to be equal to or greater than wolf s algorithm the latter however appeared more accurate if reasonably large signal lengths are available and noise levels are sufficiently low due to its modularity the presented benchmark test can be used to evaluate and tune any estimation method to perform optimally for arbitrary experimental data

amplitude effects of medio lateral mechanical and visual perturbations on gait falls during walking are a major contributor to accidental deaths and injuries that can result in debilitating hospitalization costs lost productivity and diminished quality of life to reduce these losses we must develop a more profound understanding of the characteristic responses to perturbations similar to those encountered in daily life this study addresses this issue by building on our earlier studies that examined mechanical and visual perturbations in the same environment by applying the same continuous pseudo random perturbations at multiple 3 mechanical 5 visual amplitudes walking variability during mechanical perturbations increased significantly with amplitude for all subjects and differences as measured by variabilities of step width com position and com velocity these parameters were the only ones sensitive to the presence of visual perturbations but none of them changed significantly with perturbation amplitude additionally visual perturbation effects were far less consistent across participants with several who were essentially unaffected by visual perturbations at any level the homogeneity of the mechanical perturbation effects demonstrates that human responses to mechanical perturbations are similar because they are driven by kinetics that require similar corrections that must be made in order to maintain balance conversely responses to visual perturbations are driven by the perceived need to make corrections and this perception is not accurate enough to produce amplitude related corrections even for a single participant nor is this perception consistent across individuals this latter finding is likely to be relevant to future visual perturbation studies and the diagnosis and rehabilitation of gait and balance disorders

an algorithm to solve coupled 2d rolling contact problems this work presents a new approach to the steady state rolling contact problem for two dimensional elastic bodies with and without force transmission the problem solution is achieved by minimizing a general function representing the equilibrium equation and the contact restrictions the boundary element method is used to compute the elastic influence coefficients of the surface points involved in the contact equilibrium equations while the contact conditions are represented with the help of variational inequalities and projection functions finally the minimization problem is solved by the generalized newton s method with line search four classic rolling problems are also solved and commented on copyright 2000 john wiley sons ltd

a scaling exponent based detector of chaos in oscillatory circuits abstract in this paper a scaling exponent based approach is proposed to determine the state of chaotic circuits and the scaling exponent is calculated using detrended fluctuation analysis dfa the corresponding detector is designed using the fact that the scaling exponent changes for various states of chaotic circuits simulation examples in this paper are performed for chua s circuit and other chaotic systems and compared with the state of the art in the field the proposed detector outperforms existing techniques in ability to distinguish the chaotic and periodic states in the circuits for relatively high noise

extracting qualitative dynamics from experimental data abstract we consider the notion of qualitative information and the practicalities of extracting it from experimental data our approach based on a theorem of takens draws on ideas from the generalized theory of information known as singular system analysis due to bertero pike and co workers we illustrate our technique with numerical data from the chaotic regime of the lorenz model

effects of perturbation magnitude on dynamic stability when walking in destabilizing environments external perturbations applied to the walking surface or visual field can challenge an individual s ability to maintain stability during walking accurately quantifying and predicting changes in stability during walking will further our understanding of how individuals respond to challenges encountered during daily life and guide the development of assessments and rehabilitation interventions for individuals at increased risk of falling this study is the first to determine how orbital and local dynamic stability metrics including maximum floquet multipliers and local divergence exponents change in response to continuous mediolateral visual and surface perturbations of different amplitudes eleven healthy individuals walked in a fully immersive virtual environment participants completed two 3 min walking trials each under the following nine conditions no perturbations surface perturbations at each of 3 amplitudes and visual perturbations at each of 5 amplitudes all perturbations were applied as continuous pseudo random oscillations during both surface and visual perturbations individuals were significantly more orbitally and locally unstable compared to un perturbed walking as walking surface perturbation amplitudes increased individuals were more orbitally but not locally unstable as visual perturbation amplitudes increased individuals were more locally but not orbitally unstable between lower and higher amplitudes overall these dynamic stability metrics were much less sensitive to changes in perturbation amplitudes than to differences between un perturbed and perturbed walking or to differences between mechanical and visual perturbations this suggests that the type of perturbation s applied has a far greater impact than the magnitude of those perturbations in determining the response that will be elicited

formulation and comparison of algorithms for frictional contact problems this paper presents two algorithms for solving the discrete quasi static small displacement linear elastic contact problem with coulomb friction the algorithms are adoptions of a newton method for solving b differentiable equations and an interior point method for solving smooth constrained equations for the application of the former method the contact problem is formulated as a system of b differentiable equations involving the projection operator onto sets with simple structure for the application of the latter method the contact problem is formulated as a system of smooth equations involving complementarity conditions and with the non negativity of variables treated as constraints the two algorithms are numerically tested for two dimensional problems containing up to 100 contact nodes and up to 100 time increments results show that at the present stage of development the newton method is superior both in robustness and speed additional comparison is made with a commercial finite element code

walking variability during continuous pseudo random oscillations of the support surface and visual field abstract walking on uneven surfaces or while undergoing perturbations has been associated with increased gait variability in both modeling and human studies previous gait research involving continuous perturbations has focused on sinusoidal oscillations which can result in individuals predicting the perturbation and or entraining to it therefore we examined the effects of continuous pseudo random support surface and visual field oscillations on 12 healthy young participants participants walked in a virtual reality environment under no perturbation nop anterior posterior ap walking surface and visual oscillation and mediolateral ml walking surface and visual oscillation conditions participants exhibited shorter p 0 005 wider p

a practical method for calculating largest lyapunov exponents from small data sets detecting the presence of chaos in a dynamical system is an important problem that is solved by measuring the largest lyapunov exponent lyapunov exponents quantify the exponential divergence of initially close state space trajectories and estimate the amount of chaos in a system we present a new method for calculating the largest lyapunov exponent from an experimental time series the method follows directly from the definition of the largest lyapunov exponent and is accurate because it takes advantage of all the available data we show that the algorithm is fast easy to implement and robust to changes in the following quantities embedding dimension size of data set reconstruction delay and noise level furthermore one may use the algorithm to calculate simultaneously the correlation dimension thus one sequence of computations will yield an estimate of both the level of chaos and the system complexity

the problem of spurious lyapunov exponents in time series analysis and its solution by covariant lyapunov vectors we briefly recall the methods for the determination of lyapunov exponents from time series data and their limitations one particular problem is given by the fact that reconstructed phase spaces have usually extra dimensions compared to the true phase space of a dynamical system leading to extra so called spurious lyapunov exponents several methods to identify the true ones have been proposed which do not give satisfactory results we show that the geometric information contained in covariant lyapunov vectors can be used to identify the true exponents we illustrate its use and its limitations by applying it to experimental nmr laser data this article is part of a special issue of journal of physics a mathematical and theoretical devoted to lyapunov analysis from dynamical systems theory to applications

active control of lateral balance in human walking we measured variability of foot placement during gait to test whether lateral balance must be actively controlled against dynamic instability the hypothesis was developed using a simple dynamical model that can walk down a slight incline with a periodic gait resembling that of humans this gait is entirely passive except that it requires active control for a single unstable mode con ned mainly to lateral motion an especially e cient means of controlling this instability is to adjust lateral foot placement we hypothesized that similar active feedback control is performed by humans with fore aft dynamics stabilized either passively or by very low level control the model predicts that uncertainty within the active feedback loop should result in variability in foot placement that is larger laterally than fore aft in addition loss of sensory information such as by closing the eyes should result in larger increases in lateral variability the control model also predicts a slight coupling between step width and length we tested 15 young normal human subjects and found that lateral variability was 79 larger than fore aft variability with eyes open and a larger increase in lateral variability 53 vs 21 with eyes closed consistent with the model s predictions we also found that the coupling between lateral and fore aft foot placements was consistent with a value of 0 13 predicted by the control model our results imply that humans may harness passive dynamic properties of the limbs in the sagittal plane but must provide signi cant active control in order to stabilize lateral motion 2000 elsevier science ltd all rights reserved

harmonic ratios a quantification of step to step symmetry the harmonic ratio hr derived from the fourier analysis of trunk accelerations has been described in various ways as a measure of walking smoothness walking rhythmicity or dynamic stability there is an increasing interest in applying the hr technique to investigate the impact of various pathologies on locomotion however explanation of the method has been limited the aim here is to present a clear description of the mathematical basis of hrs and an understanding of their interpretation we present harmonic theory the interpretation of the hr using sinusoidal signals and an example using actual trunk accelerations and harmonic analyses during limb loading conditions we suggest that the hr method may be better defined not as a measure of rhythmicity or stability but as a measure of step to step symmetry within a stride

a direct comparison of spine rotational stiffness and dynamic spine stability during repetitive lifting tasks stability of the spinal column is critical to bear loads allow movement and at the same time avoid injury and pain however there has been a debate in recent years as to how best to define and quantify spine stability with the outcome being that different methods are used without a clear understanding of how they relate to one another therefore the goal of the present study was to directly compare lumbar spine rotational stiffness calculated with an emg driven biomechanical model to local dynamic spine stability calculated using lyapunov analyses of kinematic data during a series of continuous dynamic lifting challenges twelve healthy male subjects performed 30 repetitive lifts under three varying load and three varying rate conditions with an increase in the load lifted constant rate there was a significant increase in mean maximum and minimum spine rotational stiffness po 0 001 and a significant increase in local dynamic stability po 0 05 both stability measures were moderately to strongly related to one another r 0 55 to 0 71 with an increase in lifting rate constant load there was also a significant increase in mean and maximum spine rotational stiffness po 0 01 however there was a non significant decrease in the minimum rotational stiffness and a non significant decrease in local dynamic stability p40 05 weak linear relationships were found for the varying rate conditions r 0 02 to 0 27 the results suggest that spine rotational stiffness and local dynamic stability are closely related to one another as they provided similar information when movement rate was controlled however based on the results from the changing lifting rate conditions it is evident that both models provide unique information and that future research is required to completely understand the relationship between the two models using both techniques concurrently may provide the best information regarding the true effects of in stability under different loading and movement scenarios and in comparing healthy and clinical populations

solving 2d transient rolling contact problems using the bem and mathematical programming techniques this work presents a new approach to the transient rolling contact of two dimensional elastic bodies a solution will be obtained by minimizing a general b differentiable function representing the equilibrium equations and the contact conditions at each time step inertial effects are not taken into account and the boundary element method is used to compute the elastic influence coefficients of the surface points involved in contact equilibrium equations the contact conditions are represented with the help of variational inequalities and projection functions finally the minimization problem is solved using the generalized newton s method with line search the results are compared with some example problems and the influence of discretization and integration time step on the results is discussed copyright 2001 john wiley sons ltd

the grobner ring conjecture in the lexicographic order case we prove that a valuation domain mathbf v has krull dimension le 1 if and only if for any n fixing the lexicographic order as monomial order in mathbf v x 1 ldots x n for every finitely generated ideal i of mathbf v x 1 ldots x n the ideal generated by the leading terms of the elements of i is also finitely generated this proves the grobner ring conjecture in the lexicographic order case the proof we give is both simple and constructive the same result is valid for prufer domains as a scoop contrary to the common idea that grobner bases can be computed exclusively on noetherian ground we prove that computing grobner bases over mathbf r x 1 ldots x n where mathbf r is a prufer domain has nothing to do with noetherianity it is only related to the fact that the krull dimension of mathbf r is le 1

quaternions and joint angles in an analysis of local stability of gait for different variants of walking speed and treadmill slope the authors describe an example of application of a nonlinear time series analysis directed at identifying the presence of deterministic chaos in human gait kinematic data by means of the largest lyapunov exponent lle a positive lle value is interpreted as an indicator of local instability the research was aimed at assessment of the influence of both walking speed and ground slope on the resilience of gait control to infinitesimally small perturbations that occur naturally during walking the analysis of treadmill gait data was carried out twofold 1 for the time series representing the following joint angles hip flexion extension knee flexion extension and dorsiflexion plantarflexion of the ankle and 2 for the time series representing rotations of foot tibia and femur segments through euler angles converted to a quaternion representation a comparison between both approaches as well as a dependency between treadmill inclination and lle values constitute the original aspects of this study the lle value was estimated threefold for every time series as the short term lle for both the first step and the first stride and as the long term lle over a fixed interval between the fourth and the tenth stride it was confirmed that all considered movements are characterized by positive lle values which quantify a local instability moreover a tendency to attenuate the perturbation consequences is evident in all variants of walking speed and treadmill slope

the effect of walking speed on local dynamic stability is sensitive to calculation methods local dynamic stability has been assessed by the short term local divergence exponent s which quantifies the average rate of logarithmic divergence of infinitesimally close trajectories in state space both increased and decreased local dynamic stability at faster walking speeds have been reported this might pertain to methodological differences in calculating s therefore the aim was to test if different calculation methods would induce different effects of walking speed on local dynamic stability ten young healthy participants walked on a treadmill at five speeds 60 80 100 120 and 140 of preferred walking speed for 3min each while upper body accelerations in three directions were sampled from these time series s was calculated by three different methods using a a fixed time interval and expressed as logarithmic divergence per stride time s a b a fixed number of strides and expressed as logarithmic divergence per time s b and c a fixed number of strides and expressed as logarithmic divergence per stride time s c mean preferred walking speed was 1 16 0 09m s there was only a minor effect of walking speed on s a s b increased with increasing walking speed indicating decreased local dynamic stability at faster walking speeds whereas s c decreased with increasing walking speed indicating increased local dynamic stability at faster walking speeds thus the effect of walking speed on calculated local dynamic stability was significantly different between methods used to calculate local dynamic stability therefore inferences and comparisons of studies employing s should be made with careful consideration of the calculation method language en

can stability really predict an impending slip related fall among older adults the primary purpose of this study was to systematically evaluate and compare the predictive power of falls for a battery of stability indices obtained during normal walking among community dwelling older adults one hundred and eighty seven community dwelling older adults participated in the study after walking regularly for 20 strides on a walkway participants were subjected to an unannounced slip during gait under the protection of a safety harness full body kinematics and kinetics were monitored during walking using a motion capture system synchronized with force plates stability variables including feasible stability region measurement margin of stability the maximum floquet multiplier the lyapunov exponents short and long term and the variability of gait parameters including the step length step width and step time were calculated for each subject sensitivity of predicting slip outcome fall vs recovery was examined for each stability variable using logistic regression results showed that the feasible stability region measurement predicted fall incidence among these subjects with the highest sensitivity 68 4 except for the step width with an sensitivity of 60 2 no other stability variables could differentiate fallers from those who did not fall for the sample included in this study the findings from the present study could provide guidance to identify individuals at increased risk of falling using the feasible stability region measurement or variability of the step width

a perturbed lagrangian formulation for the finite element solution of contact problems making use of a perturbed lagrangian formulation a finite element procedure for contact problems is developed for the general case in which node to node contact no longer holds the proposed procedure leads naturally to a discretization of the contact interface into contact segments within the context of a bilinear interpolation for the displacement field a mixed finite element approximation is introduced by assuming discont naous contact pressure constant on the contact segment because of this piece wise constant approximation the gap function enters into the formulation in an average sense instead of through a point wise definition numerical examples are presented that illustrate the performance of the proposed procedure current finite element formulations for contact problems based on either the classical lagrange parameter procedure l 3 12 203 or the penalty function method 4 6 11 are characterized by a point wise enforcement of the contact constraint condition in the sense that penetration of the bodies is established on a nodal basis moreover in this methods the recovery of the contact pressure over the element from the contact nodal forces generally requires an additional procedure within the framework of classical lagrange multiplier methods the contact condition is exactly satisfied by transforming the constrained problem into an unconstrained one with the introduction of additional variables lagrange multipliers these extra variables add computational effort to the solution process which often requires special procedures to handle the presence of zero diagonal terms penalty methods on the other hand enable one to transform the constrained problem into an unconstrained one without introducing additional variables the constraint condition is now satisfied only approximately for finite values of the penalty parameter the main difhculty associated with these methods however lies in the poor conditioning of the problem as the penalty is increased to more accurately enforce the constraint condition this is a well understood phenomenon particularly in the context of the incompressible and nearly incompressible problem in solid and fluid mechanics e g see l 22 25 for a review recently augmented lagrangian procedures have been proposed as a promising way to partially overcome these difficulties and regularize the penalty formulation e g see the survey in 7 8 within the framework of zinearited e u c it is possible to restrict the finite element formulation of contact problems by assuming that node to node contact occurs this is in fact the case often considered in the literature l 2 4 6 12 20 211 in the general context of fully

kinematic variability and local dynamic stability of upper body motions when walking at different speeds a ubiquitous characteristic of elderly and patients with gait disabilities is that they walk slower than healthy controls many clinicians assume these patients walk slower to improve their stability just as healthy people slow down when walking across ice however walking slower also leads to greater variability which is often assumed to imply deteriorated stability if this were true then slowing down would be completely antithetical to the goal of maintaining stability this study sought to resolve this paradox by directly quantifying the sensitivity of the locomotor system to local perturbations that are manifested as natural kinematic variability eleven young healthy subjects walked on a motorized treadmill at five different speeds three dimensional movements of a single marker placed over the first thoracic vertebra were recorded during continuous walking mean stride to stride standard deviations and maximum finite time lyapunov exponents were computed for each time series to quantify the variability and local dynamic stability respectively of these movements quadratic regression analyses of the dependent measures vs walking speed revealed highly significant u shaped trends for all three mean standard deviations but highly significant linear trends with significant or nearly significant quadratic terms for five of the six finite time lyapunov exponents subjects exhibited consistently better local dynamic stability at slower speeds for these five measures these results support the clinically based intuition that people who are at increased risk of falling walk slower to improve their stability even at the cost of increased variability

application of bem and optimization technique to wear problems in this paper a new method for solving wear problems using the boundary element method bem is proposed the proposed method uses a shape optimization technique for rapid calculation of the maximum wear depth and the final geometry of the bodies in contact after a specified service period the optimization method solves the wear problem directly without using increments of the sliding distance as in the classical methods the bem is used for modelling both bodies in contact treating the problem as a multi region problem the material loss is presented in terms of the applied load and the sliding distance and is modelled using a linear wear model the proposed method is shown to be robust as it requires only few iterations to achieve a converged solution the reduction in computational effort as compared to the classical incremental method becomes more significant as the sliding distance increases

dangers of geometric filtering abstract we argue that there are unrecognised dangers in the popular and powerful geometric filtering method carelessly applied geometric filtering can produce apparent structure in data which is pure noise or can cause severe distortions in clean deterministic data the explanations for these effects are straightforward and the dangers are easily avoided by taking simple precautions in the filtering process

dynamic stability of human walking in visually and mechanically destabilizing environments abstract understanding how humans remain stable during challenging locomotor activities is critical to developing effective tests to diagnose patients with increased fall risk this study determined if different continuous low amplitude perturbations would induce specific measureable changes in measures of dynamic stability during walking we applied continuous pseudo random oscillations of either the visual scene or support surface in either the anterior posterior or mediolateral directions to subjects walking in a virtual environment with speed matched optic flow floquet multipliers and short term local divergence exponents both increased indicating greater instability during perturbed walking these responses were generally much stronger for body movements occurring in the same directions as the applied perturbations likewise subjects were more sensitive to both visual and mechanical perturbations applied in the mediolateral direction than to those applied in the anterior posterior direction consistent with previous experiments and theoretical predictions these responses were likewise consistent with subjects anecdotal perceptions of which perturbation conditions were most challenging contrary to the floquet multipliers and short term local divergence exponents which both increased long term local divergence exponenets decreased during perturbed walking however this was consistent with specific changes in the mean log divergence curves which indicated that subjects movements reached their maximum local divergence limits more quickly during perturbed walking overall the floquet multipliers were less sensitive but reflected greater specificity in their responses to the different perturbation conditions conversely the short term local divergence exponents exhibited less specificity in their responses but were more sensitive measures of instability in general

walking at the preferred stride frequency maximizes local dynamic stability of knee motion healthy humans display a preference for walking at a stride frequency dependent on the inertial properties of their legs walking at preferred stride frequency psf is predicted to maximize local dynamic stability whereby sensitivity to intrinsic perturbations arising from natural variability inherent in biological motion is minimized previous studies testing this prediction have employed different variability measures but none have directly quantified local dynamic stability by computing maximum finite time lyapunov exponent max which quantifies the rate of divergence of nearby trajectories in state space here ten healthy adults walked 45 m overground while sagittal motion of both knees was recorded via electrogoniometers an auditory metronome prescribed 7 different frequencies relative to each individual s psf psf 5 10 15 strides min stride frequencies were performed under both freely adopted speed fs and controlled speed cs set at the speed of psf trials conditions local dynamic stability was maximal max was minimal at the psf becoming less stable for higher and lower stride frequencies this occurred under both fs and cs conditions although controlling speed further reduced local dynamic stability at non preferred stride frequencies in contrast measures of variability revealed effects of stride frequency and speed conditions that were distinct from max in particular movement regularity computed by approximate entropy apen increased for slower walking speeds appearing to depend on speed rather than stride frequency the cadence freely adopted by humans has the benefit of maximizing local dynamic stability which can be interpreted as humans tuning to their resonant frequency of walking

evaluation of the threshold of stability for the human spine the threshold of stability tos is introduced as a new tool for evaluating spinal stability current methods have evaluated the magnitude of spinal kinematic variability stability diffusion or lyapunov exponents this present method differs by modifying task difficulty to obtain a critical value delineating regions of stability and instability conceptually as task difficulty increases kinematic variability of the system increases and the basin of stability decreases when kinematic variability exceeds the basin of stability stability cannot be maintained over time this concept is first illustrated using a mathematical model from which an effective potential function was calculated to show the relationships among kinematic variability the basin of stability and task difficulty in addition an experiment was performed to evaluate the sensitivity of the tos to changes in postural control using visual feedback as a control variable the tos was found to be more sensitive than the lyapunov exponent to removal of visual feedback suggesting it may have use as a diagnostic indicator e g for low back pain furthermore this new method has an additional advantage in which minimal instrumentation is needed its simplicity sensitivity and low cost suggest that the tos has potential as a diagnostic or prognostic tool in a clinical setting

influence of simulated neuromuscular noise on movement variability and fall risk in a 3d dynamic walking model people at risk of falling exhibit increased gait variability which may predict future falls however the causal mechanisms underlying these correlations are not well known increased neuronal noise associated with aging likely leads to increased gait variability which could in turn lead to increased fall risk this paper presents a model of how changes in neuromuscular noise independently affect gait variability and probability of falling and aims to determine the extent to which changes in gait variability directly predict fall risk we used a dynamic walking model that incorporates a lateral step controller to maintain lateral stability noise was applied to this controller to approximate neuromuscular noise in humans noise amplitude was varied between low amplitudes that did not induce falls and high amplitudes for which the model always fell with increases in noise amplitude the model fell more often and after fewer steps gait variability increased with noise amplitude and predicted increased probability of falling importantly these relationships were not linear at either low gait variability or very high gait variability small increases in noise and variability affected probability of falling very little conversely at intermediate noise and or variability levels the same small increases resulted in large increases in probability of falling our results validate the idea that age related increases in neuromuscular noise likely play a direct contributing role in increasing fall risk however neuromuscular noise remains only one of many important factors that need to be considered these findings have important implications for fall prevention research and practice

characterization of strange attractors a new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable the relation of this new measure to fractal dimension and information theoretic entropy is discussed

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow

the grobner ring conjecture in one variable we prove that a valuation domain v has krull dimension 1 if and only if for every finitely generated ideal i of v x the ideal generated by the leading terms of elements of i is also finitely generated this proves the grobner ring conjecture in one variable the proof we give is both simple and constructive the same result is valid for semihereditary rings

does the personal lift assist device affect the local dynamic stability of the spine during lifting abstract the personal lift assist device plad is an on body ergonomic aid that reduces low back physical demands through the restorative moment of an external spring element which possesses a mechanical advantage over the erector spinae although the plad has proven effective at reducing low back muscular demand spinal moments and localized muscular fatigue during laboratory and industrial tasks the effects of the device on the neuromuscular control of spinal stability during lifting have yet to be assessed thirty healthy subjects 15m 15f performed repetitive lifting for three minutes at a rate of 10 lifts per minute with and without the plad maximum finite time lyapunov exponents representing short term max s and long term max l divergence were calculated from the measured trunk kinematics to estimate the local dynamic stability of the lumbar spine using a mixed design repeated measures anova it was determined that wearing the plad did not significantly change max s np 0 335 p 0 321 p 0 225 but did significantly reduce max l np 0 0024 p 0 0011 p 0 014 2 0 197 there were no between subject effects of sex or significant interactions p 0 720 the present results indicated that max s was not statistically different between the device conditions but that the plad significantly reduced max l to a negative stable value this shows that subjects neuromuscular systems were able to respond to local perturbations more effectively when wearing the device reflecting a more stable control of spinal movements these findings are important when recommending the plad for long term industrial or clinical use

a newton method for three dimensional fretting problems the present paper concerns the numerical treatment of fretting problems using a finite element analysis the governing equations resulting from a formal finite element discretization of an elastic body with a potential contact surface are considered in a quasi static setting the constitutive equations of the potential contact surface are signorini s contact conditions coulomb s law of friction and archard s law of wear using a backward euler time discretization and an approach based on projections the governing equations are written as an augmented lagrangian formulation which is implemented and solved using a newton algorithm for three dimensional fretting problems of didactic nature details concerning the implementation are provided