in vivo estimation of the glenohumeral joint rotation center from scapular bony landmarks by linear regression in this paper a method is described for in vivo prediction of the glenohumeral joint rotation center gh r necessary for the construction of a humerus local coordinate system in shoulder kinematic studies the three dimensional positions of five scapula bony landmarks as well as a large number of data points on the surface of the glenoid and humeral head were collected at 36 sets of cadaver scapulae and adjacent humeri the position of gh r in each scapula was estimated by mathematically fitting spheres to the glenoid and humeral head gh r prediction from scapula geometry parameters by linear regression resulted in a rmse between measured and predicted gh r of 2 32 mm for the x coordinate 2 69 mm for the y coordinate and 3 04 mm for the z coordinate application in vivo revealed a random humerus orientation error due to measurement inaccuracies of 1 35 0 29 and 1 26 degrees standard deviation per rotation angle the estimated total humerus orientation error including the offset error due to the regression model inaccuracy was 2 86 0 84 and 2 69 degrees standard deviation as these errors were about 15 and 20 of respectively the intra and inter subject variability of the humerus orientations measured it is concluded that the method described in this paper allows for an adequate construction of a humerus local coordinate system

on the estimation of an average rigid body motion this paper investigates the definition and the estimation of the frechet mean of a random rigid body motion in super p the sample space se p contains objects me r t where r is a p p rotation matrix and t is a p 1 translation vector this work is motivated by applications in biomechanics where the posture of a joint at a given time is expressed as mise 3 the rigid body displacement needed to map a system of axes on one segment of the joint to a similar system on the other segment this posture can also be reported as m super 1e r super t r super tt by interchanging the role of the two segments several definitions of a frechet mean for a random motion are proposed using weighted least squares distances a special emphasis is given to a frechet mean that is equivariant with respect to the inverse transform this means that if p is the frechet mean for m then p super 1 is the frechet mean for m super 1 where m is a random se p object the sampling properties of moment estimators of the frechet means are studied in a large concentration setting where the scatter of the random ms around their mean value p is small and as the sample size goes to some simple exponential family models for se p data that generalize downs 1972 fisher von mises matrix distribution for rotation matrices are introduced and the least squares mean values for these distributions are calculated asymptotic comparisons between the estimators presented in this work are carried out for a particular model when pe2 a numerical example involving the motion of the ankle is presented to illustrate the methodology copyright 2012 oxford university press

compact isospectral sets of surfaces on etudie des ensembles de surfaces qui sont isospectrales par rapport a l operateur de laplace beltrami

isospectral deformations i riemannian structures on two step nilspaces on montre comment construire des exemples de deformations isospectrales de metriques de riemann sur des varietes sur lesquelles des recouvrements de certains groupes de lie nilpotents agissent

whole body inverse dynamics over a complete gait cycle based only on measured kinematics this paper presents a three dimensional 3d whole body multi segment model for inverse dynamics analysis over a complete gait cycle based only on measured kinematic data the sequence of inverse dynamics calculations differs significantly from the conventional application of inverse dynamics using force plate data a new validated smooth transition assumption was used to solve the indeterminacy problem in the double support phase kinematic data is required for all major body segments and hence a whole body gait measurement protocol is presented finally sensitivity analyses were conducted to evaluate the effects of digital filtering and body segment parameters on the accuracy of the prediction results the model gave reasonably good estimates of sagittal plane ground forces and moment however the estimates in the other planes were less good which we believe is largely due to their small magnitudes in comparison to the sagittal forces and moment the errors observed are most likely caused by errors in the kinematic data resulting from skin movement artefact and by errors in the estimated body segment parameters a digital filtering cut off frequency of 4 5 hz was found to produce the best results it was also shown that errors in the mass properties of body segments can play a crucial role with changes in properties sometimes having a disproportionate effect on the calculated ground reactions the implication of these results is that even when force plate data is available the estimated joint forces are likely to suffer from similar errors

nontrivial solitary waves to the generalized kadomtsev petviashvili equations abstract in this note we study the existence of nontrivial solitary waves for the generalized kadomtsev petviashvili equation in multi dimensional spaces our main tool is a recent linking theorem stated in a paper by szulkin and zou a szulkin w zou homoclinic orbit for asymptotically linear hamiltonian systems j funct anal 187 2001 25 41

on isospectral periodic potentials on a discrete lattice i on considere une version discretisee et generalisee de l equation de schrodinger et on etudie le probleme aux valeurs propres

estimation of the centre of rotation a methodological contribution the location of the centre of rotation of human joints that can be modelled as a spherical hinge can be estimated using kinematics information about the two adjacent bony segments involved recorded while the subject makes them move one relative to the other functional method in order to solve the relevant analytical problem several algorithms have been proposed most recently two methods one based on a spherical best fit approach and another based on the reuleaux construction have been presented as being different and submitted to comparative evaluation this paper modifies the second method taking all information in the data set into account and shows that having done this the two methods coincide analytically

line bundle laplacians over isospectral nilmanifolds we show that nontrivial isospectral deformations of a big class of compact riemannian two step nilmanifolds can be distinguished from trivial deformations by the behaviour of bundle laplacians on certain non flat hermitian line bundles over these manifolds

analytical study of the effects of soft tissue artefacts on functional techniques to define axes of rotation abstract the accurate location of the main axes of rotation aor is a crucial step in many applications of human movement analysis there are different formal methods to determine the direction and position of the aor whose performance varies across studies depending on the pose and the source of errors most methods are based on minimizing squared differences between observed and modelled marker positions or rigid motion parameters implicitly assuming independent and uncorrelated errors but the largest error usually results from soft tissue artefacts sta which do not have such statistical properties and are not effectively cancelled out by such methods however with adequate methods it is possible to assume that sta only account for a small fraction of the observed motion and to obtain explicit formulas through differential analysis that relate sta components to the resulting errors in aor parameters in this paper such formulas are derived for three different functional calibration techniques geometric fitting mean finite helical axis and sara to explain why each technique behaves differently from the others and to propose strategies to compensate for those errors these techniques were tested with published data from a sit to stand activity where the true axis was defined using bi planar fluoroscopy all the methods were able to estimate the direction of the aor with an error of less than 5 whereas there were errors in the location of the axis of 30 40 mm such location errors could be reduced to less than 17 mm by the methods based on equations that use rigid motion parameters mean finite helical axis sara when the translation component was calculated using the three markers nearest to the axis

calcul du spectre d une nilvariete de rang deux et applications on calcule en utilisant la theorie des orbites de kirillov le spectre d une nilvariete compacte de rang deux puis on utilise ce calcul pour etudier et caracteriser les deformations isospectrales de ces varietes

a comparison of two non invasive methods for measuring scapular orientation in functional positions identification of scapular dyskinesis and evaluation of interventions depend on the ability to properly measure scapulothoracic st motion the most widely used measurement approach is the acromion marker cluster amc which can yield large errors in extreme humeral elevation and can be inaccurate in children and patient populations recently an individualized regression approach has been proposed as an alternative to the amc this technique utilizes the relationship between st orientation humerothoracic orientation and acromion process position derived from calibration positions to predict dynamic st orientations from humerothoracic and acromion process measures during motion these individualized regressions demonstrated promising results for healthy adults however this method had not yet been compared to the more conventional amc this study compared st orientation estimates by the amc and regression approaches to static st angles determined by surface markers placed on palpated landmarks in typically developing adolescents performing functional tasks both approaches produced errors within the range reported in the literature for skin based scapular measurement techniques the performance of the regression approach suffered when applied to positions outside of the range of motion in the set of calibration positions the amc significantly underestimated st internal rotation across all positions and overestimated posterior tilt in some positions overall root mean square errors for the regression approach were smaller than the amc for every position across all axes of st motion accordingly we recommend the regression approach as a suitable technique for measuring st kinematics in functional motion

estimation of the axis of a screw motion from noisy data a new method based on plucker lines the problems of estimating the motion and orientation parameters of a body segment from two n point set patterns are analyzed using the plucker coordinates of a line plucker lines the aim is to find algorithms less complex than those in conventional use and thus facilitating more accurate computation of the unknown parameters all conventional techniques use point transformation to calculate the screw axis in this paper we present a novel technique that directly estimates the axis of a screw motion as a plucker line the plucker line can be transformed via the dual number coordinate transformation matrix this method is compared with schwartz and rozumalski 2005 a new method for estimating joint parameters from motion data journal of biomechanics 38 107 116 in simulations of random measurement errors and systematic skin movements simulation results indicate that the methods based on plucker lines plucker line method are superior in terms of extremely good results in the determination of the screw axis direction and position as well as a concise derivation of mathematical statements this investigation yielded practical results which can be used to locate the axis of a screw motion in a noisy environment developing the dual transformation matrix dtm from noisy data and determining the screw axis from a given dtm is done in a manner analogous to that for handling simple rotations a more robust approach to solve for the dual vector associated with dtm is also addressed by using the eigenvector and the singular value decomposition

a survey of formal methods for determining the centre of rotation of ball joints abstract the determination of an accurate centre of rotation cor from segment marker positions is of interest across a wide range of applications but particularly for clinical gait analysis and for estimating the hip joint centre during surgical intervention of the knee for limb alignment purposes for the first time in this survey of formal methods we classify analyse and compare different methods geometric algebraic bias compensated algebraic and pratt sphere fit methods as well as the centre transformation technique the holzreiter approach the helical pivot technique the schwartz transformation techniques the minimal amplitude point method and the stoddart approach for the determination of spherical joint centres from marker position data in addition we propose a new method the symmetrical cor estimation or score in which the coordinates of the joint centre must only remain constant relative to each segment thus not requiring the assumption that one segment should remain at rest for each method 1000 cor estimations were analysed with the application of isotropic independent and identically distributed gaussian noise standard deviation 0 1 cm to each of the marker positions to all markers on the segment simultaneously and the two in combination for the test conditions used here most techniques were capable of determining the cor to within 0 3 cm as long as the spherical range of motion rom of the joint was 45 or more under the most stringent conditions tested however the score was capable of best determining the cor to within approximately 1 2 mm with a rom of 20 the correct selection and application of these methodologies should help improve the accuracy of surgical navigation and clinical kinematic measurement

solitary waves for a class of generalized kadomtsev petviashvili equation in rn with positive and zero mass abstract in this paper we use variational methods to establish existence of solitary waves for a class of generalized kadomtsev petviashvili g k p equation in r n the positive and zero mass cases are considered the main argument is to find a palais smale sequence satisfying a property related to pohozaev identity as in 21 which was used for the first time by 23

new least squares solutions for estimating the average centre of rotation and the axis of rotation a new method is proposed for estimating the parameters of ball joints also known as spherical or revolute joints and hinge joints with a fixed axis of rotation the method does not require manual adjustment of any optimisation parameters and produces closed form solutions it is a least squares solution using the whole 3d motion data set we do not assume strict rigidity but only that the markers maintain a constant distance from the centre or axis of rotation this method is compared with other methods that use similar assumptions in the cases of random measurement errors systematic skin movements and skin movements with random measurement noise simulation results indicate that the new method is superior in terms of the algorithm used the closure of the solution consistency and minimal manual parameter adjustment the method can also be adapted to joints with translational movements

a hip joint kinematics driven model for the generation of realistic thigh soft tissue artefacts in human movement analysis accuracy and robustness of the algorithms used to determine the location of centres of rotation from stereophotogrametric data depend mainly on their capacity to deal with the artefacts due to soft tissue deformation sta while evaluating these algorithms using a mathematical simulation approach effectual realizations of stas are needed this study aimed at accomplishing this objective by modelling stas in twelve selected areas of the thigh as a linear function of the hip angles assuming no knee joint movement the proposed model was calibrated and assessed using ex vivo experiments this entailed that only the component of the sta due to skin stretching was accounted for photogrammetric data of markers placed on the skin and on hip bone and femur pins were recorded during passive flexion extension ab adduction rotation and circumduction of the hip joint artefact skin marker displacements were represented in a femur embedded anatomical frame model parameters were estimated by minimizing the least squares difference between measured and modelled stas the sta affecting a skin marker placed in a given thigh location of a given subject could be modelled with a high accuracy median root mean square difference over 4 subjects 3 trials 12 markers 3 coordinates 0 8 mm inter quartile range 1 0 mm this was also true for a hip joint movement different from the one used to calibrate the model high inter subject variability of the model parameters confirmed the subject dependency of the phenomenon

determining finger segmental centers of rotation in flexion extension based on surface marker measurement this paper describes the development of a novel algorithm for deriving finger segmental center of rotation cor locations during flexion extension from measured surface marker motions in vivo the algorithm employs an optimization routine minimizing the time variance of the internal link lengths and incorporates an empirically quantifiable relationship between the local movement of a surface marker around a joint termed surface marker excursion and the joint flexion extension the latter relationship constrains and simplifies the optimization routine to make it computationally tractable to empirically investigate this relationship and test the proposed algorithm an experiment was conducted in which hand cylinder grasping movements were performed by 24 subjects 12 males and 12 females spherical retro reflective markers were placed at various surface landmarks on the dorsal aspect of each subject s right grasping hand and were measured during the movements by an opto electronic system analysis of experimental data revealed a highly linear relationship between the surface marker excursion and the marker defined flexion extension angle the average r 2 in linear regression ranged from 0 89 to 0 97 the algorithm successfully determined the cors of the distal interphalangeal proximal interphalangeal and metacarpophalangeal joints of digits 2 5 during measured motions the derived cors appeared plausible as examined in terms of the physical locations relative to surface marker trajectories and the congruency across different joints and individuals

h type groups and iwasawa decompositions since their introduction by a kaplan kpl some ten years ago generalised heisenberg groups also known as groups of heisenberg type or h type groups have provided a framework in which to construct interesting examples in geometry and analysis see for instance c2 kp2 kp3 kpr k2 rl r2 tl tv the iwasawa n groups associated to all the real rank one simple lie groups are h type so one has a convenient vehicle for studying these in a unified way many problems on these simple lie groups can be reduced to a problem on h type groups via the so called noncompact picture and often problems on h type groups can be solved on all the groups of the family in one fell swoop as in for example ch ck dr out of this approach to studying simple lie groups several problems arise such as why only some h type groups correspond to simple lie groups of real rank one in this paper we discuss various features of iwasawa n groups which distinguish them in the class of all h type groups we shall show that all h type groups which possess certain geometric properties clearly possessed by iwasawa n groups satisfy a lie algebraic condition implicit in the work of b kostant kt2 that we shall call the j condition we shall also use elementary clifford algebra to classify the

robust estimation of dominant axis of rotation a simple method is developed for robustly estimating a fixed dominant axis of rotation aor of anatomical joints from surface marker data previous approaches which assume a model of circular marker trajectories use plane fitting to estimate the direction of the aor however when there is limited joint range of motion and rotation due to a second degree of freedom minimizing only the planar error can give poor estimates of the aor direction optimizing a cost function which includes the error component within a plane instead of only the component orthogonal to a plane leads to improved estimates of the aor direction for joints which exhibit additional rotational motion from a second degree of freedom results from synthetic data validation show the ranges of motion where the new method has lower estimation error compared to plane fitting techniques estimates of the flexion extension aor from empirical motion capture data of the knee and index finger joints were also more anatomically plausible

constrained least squares optimization for robust estimation of center of rotation this paper presents a new direct method for estimating the average center of rotation cor an existing least squares ls solution has been shown by previous works to have reduced accuracy for data with small range of motion rom alternative methods proposed to improve the cor estimation use iterative algorithms however in this paper we show that with a carefully chosen normalization scheme constrained least squares solutions can perform as well as iterative approaches even for challenging problems with significant noise and small rom in particular enforcing the normalization constraint avoids poor fits near plane singularities that can affect the existing ls method our formulation has an exact solution accounts for multiple markers simultaneously and does not depend on manually adjusted parameters simulation tests compare the method to four published cor estimation techniques the results show that the new approach has the accuracy of the iterative methods as well as the short computation time and repeatability of a least squares solution in addition application of the new method to experimental motion capture data of the thumb carpometacarpal cmc joint yielded a more plausible cor location compared to the previously reported ls solution and required less time than all four alternative techniques

the centre of rotation of the shoulder complex and the effect of normalisation shoulder motions consist of a composite movement of three joints and one pseudo joint which together dictate the humerothoracic motion the purpose of this work was to quantify the location of the centre of rotation cor of the shoulder complex as a whole dynamic motion of 12 participants was recorded using optical motion tracking during coronal scapular and sagittal plane elevation the instantaneous cor was found for each angle of elevation using helical axes projected onto the three planes of motion the location of an average cor for each plane was evaluated using digitised and anthropometric measures for normalisation when conducting motion in the coronal scapular and sagittal planes respectively the coefficients for locating the cors of the shoulder complex are 61 61 and 65 of the anterior posterior dimension the vector between the midpoint of the incisura jugularis and the xiphoid process and the midpoint of the seventh cervical vertebra and the eighth thoracic vertebra 0 1 and 2 of the superior inferior dimension the vector between the midpoint of the acromioclavicular joints and the midpoint of the anterior superior iliac spines and 57 57 and 78 of the medial lateral dimension 0 129 times the height of the participant knowing the location of the cor of the shoulder complex as a whole enables improved participant positioning for evaluation and rehabilitation activities that involve movement of the hand with a fixed radius such as those that employ isokinetic dynamometers

optimal average path of the instantaneous helical axis in planar motions with one functional degree of freedom this paper presents a model for determining the path of the instantaneous helical axis iha that optimally represents human planar motions with one functional degree of freedom fdof a human movement is said to have one fdof when all degrees of freedom dofs are coordinated such that all the kinematic variables can be expressed across movement repetitions as functions of only one independent dof except for a small natural intercycle variability quantified as lower than a prespecified value the concept of fdof allows taking into account that due to motor coordination human movements are executed in a repeatable manner our method uses the measurement of several repetitions of a given movement to obtain the optimal average iha path the starting point is a change of variables from time to a joint position magnitude generally an angle in this way instead of operating with the time dependent single valued trajectory of the successive cycles our model permits the representation of any motion variable e g positions and their time derivatives as a cloud of points dependent on the joint angle this allows the averaging to be performed over the displacements and their derivatives before determining the mean iha path we thus avoid the nonlinear magnification of errors and variability inherent in the iha computation moreover the iha path can be considered as a geometric attribute of the joint and the type of motion rather than of each single movement execution an experiment was performed that show the accuracy and usefulness of the method

the inaudible geometry of nilmanifolds we show that isospectral deformations of compact riemannian two step nilmanifolds can be systematically detected by simple changes in the behavior of their geodesics in spite of the fact that the length spectrum which measures the lengths of all closed geodesics remains constant

continuous families of isospectral metrics on simply connected manifolds we construct continuous families of riemannian metrics on certain simply connected manifolds with the property that the resulting riemannian manifolds are pairwise isospectral for the laplace operator acting on functions these are the first examples of simply connected riemannian manifolds without boundary which are isospectral but not isometric for example we construct continuous isospectral families of metrics on the product of spheres s4 x s3 x s3 the metrics considered are not locally homogeneous for a big class of such families the set of critical values of the scalar curvature function changes during the deformation moreover the manifolds are in general not isospectral for the laplace operator acting on 1 forms

the effect on conventional gait model kinematics and kinetics of hip joint centre equations in adult healthy gait abstract the conventional gait model cgm needs to benefit from large investigations on localization of the hip joint centre hjc incorrect positions from the native equations were demonstrated sangeux et al 2014 harrington et al 2007 more accurate equations were proposed but their impact on kinematics and kinetic cgm outputs was never evaluated this short communication aims at examining if adoption of new hjc equations would alter standard cgm outputs sixteen able bodied participants underwent a full 3 d optoelectronic gait analysis followed by a 3 d ultrasound localization of their hips data were processed through the open source python package pycgm2 replicating kinematic and kinetic processing of the native cgm compared with 3d ultrasound location hara equations improved the accuracy of sagittal plane kinematics 0 6 and kinetics 0 02 n m kg 1 for the hip the worst case participant exhibited harrington s equations reached a deviation of 3 for the sagittal kinematics in the coronal plane hara and harrington equations presented similar differences 1 for the hip whilst davis equations had the largest deviation for hip abduction 2 7 and hip abductor moment 0 10 n m kg 1 both harrington and hara equations improved the cgm location of the hjc hara equations improved results in the sagittal plane plus utilise a single anthropometrics measurement leg length that may be more robust however neither set of equations had significant effect on kinematics we reported some effects on kinetics particularly in the coronal plane which warrant caution in interpreting outputs using different sets of equations

a new method for estimating the axis of rotation and the center of rotation a new method is presented for estimating the parameters of two different models of a joint the two models are 1 a rotational joint with a fixed axis of rotation also referred to as a hinge joint and 2 a ball and socket model corresponding to a spherical joint given the motion of a set of markers it is shown how the parameters can be estimated utilizing the whole data set the parameters are estimated from motion data by minimizing two objective functions the method does not assume a rigid body motion but only that each marker rotates around the same fixed axis of rotation or center of rotation simulation results indicate that in situations where the rigid body assumption is valid and when measurement noise is present the proposed method is inferior to methods that utilize the rigid body assumption however when there are large skin movement artefacts simulation results show the proposed method to be more robust

a new method for estimating joint parameters from motion data joint centers and axes of rotation joint parameters are central to all branches of movement analysis in gait analysis the standard protocol used to determine hip and knee joint parameters is prone to errors arising from palpation anthropometric regression equations and misplaced alignment devices several alternative methods have been proposed but to date none have been shown to be accurate and reliable enough for use in the clinical setting this article describes a new method for joint parameter estimation the new method can be summarized as follows i the motions of two adjacent segments spanning a single joint are tracked ii the axis of rotation between every pair of observed segment configurations is computed iii the most likely intersection of all axes effective joint center and most likely orientation of the axes effective joint axis is found initial validation of the method was conducted on a hinged mechanical analog and a single healthy adult subject for the analog the center was found to be within 3 8 mm of the geometric center and 2 0 degrees of the geometric axis standard deviation for the adult subject hip centers varied on the order of 1 3 mm knee centers by 3 9 mm and knee axes by 2 0 degrees the results suggest that the new method is an objective precise and practical alternative to the standard clinical approach

finite centroid and helical axis estimation from noisy landmark measurements in the study of human joint kinematics abstract recent work on joint kinematics indicates that the finite centroid centre of rotation and the finite helical axis axis of rotation screw axis twist axis are highly susceptible to measurement errors when they are experimentally determined from landmark position data this paper presents an analytical model to describe these effects under isotropic conditions for the measurement errors and for the spatial landmark distribution it appears that the position and direction errors are inversely proportional to the rotation magnitude and that they are much more error prone than the relatively well determined rotation and translation magnitudes furthermore the direction and rotation magnitude errors are inversely proportional to the landmark distribution radius and the position and translation magnitude errors are minimal if the mean position of the landmarks coincides with the centroid or helical axis for the planar centroid the use of rigid body constraints results in considerable precision improvement relative to the classical finite reuleaux method for centroid reconstruction these analytical results can be used to define suitable measurement configurations and they are used in this paper to explain experimental results on rontgenphotogrammetrically acquired in vitro wrist joint movement

bayesian inverse kinematics vs least squares inverse kinematics in estimates of planar postures and rotations in the absence of soft tissue artifact a variety of inverse kinematics ik algorithms exist for estimating postures and displacements from a set of noisy marker positions typically aiming to minimize ik errors by distributing errors amongst all markers in a least squares ls sense this paper describes how bayesian inference can contrastingly be used to maximize the probability that a given stochastic kinematic model would produce the observed marker positions we developed bayesian ik for two planar ik applications 1 kinematic chain posture estimates using an explicit forward kinematics model and 2 rigid body rotation estimates using implicit kinematic modeling through marker displacements we then tested and compared bayesian ik results to ls results in monte carlo simulations in which random marker error was introduced using gaussian noise amplitudes ranging uniformly between 0 2 mm and 2 0 mm results showed that bayesian ik was more accurate than ls ik in over 92 of simulations with the exception of one center of rotation coordinate planar rotation for which bayesian ik was more accurate in only 68 of simulations moreover while ls errors increased with marker noise bayesian errors were comparatively unaffected by noise amplitude nevertheless whereas the ls solutions required average computational durations of less than 0 5 s average bayesian ik durations ranged from 11 6 s for planar rotation to over 2000 s for kinematic chain postures these results suggest that bayesian ik can yield order of magnitude ik improvements for simple planar ik but also that its computational demands may make it impractical for some applications

generalized kadomtsev petviashvili equation with an infinite dimensional symmetry algebra abstract a generalized kadomtsev petviashvili equation describing water waves in oceans of varying depth density and vorticity is discussed a priori it involves 9 arbitrary functions of one or two variables the conditions are determined under which the equation allows an infinite dimensional symmetry algebra this algebra can involve up to three arbitrary functions of time it depends on precisely three such functions if and only if it is completely integrable

homoclinic orbits for asymptotically linear hamiltonian systems we study the existence of homoclinic orbits for first order time dependent hamiltonian systems z jhz z t where h z t depends periodically on t and hz z t is asymptotically linear in z as z we also consider an asymptotically linear schrodinger equation in rn

validation of a functional method for the estimation of hip joint centre location the present study assesses the accuracy with which the subject specific coordinates of the hip joint centre hjc in a pelvic anatomical frame can be estimated using different methods the functional method was applied by calculating the centre of the best sphere described by the trajectory of markers placed on the thigh during several trials of hip rotations different prediction methods proposed in the literature and in the present investigation which estimate the hjc of adult subjects using regression equations and anthropometric measurements were also assessed the accuracy of each of the above mentioned methods was investigated by comparing their predictions with measurements obtained on a sample of 11 male adult able bodied volunteers using roentgen stereophotogrammetric analysis rsa assumed to provide the true hjc locations prediction methods estimated the hjc location at an average rms distance of 25 30 mm the functional method performed significantly better and estimated hjcs within a rms distance of 13 mm on average this result may be confidently generalised if the photogrammetric experiment is carefully conducted and an optimal analytical approach used the method is therefore suggested for use in motion analysis when the subject s hip range of motion is not limited in addition the facts that it is not an invasive technique and that it has relatively small and un biased errors make it suitable for regression equations identification with no limit to sample size and population typology

comparative evaluation of scapular and humeral coordinate systems based on biomedical images of the glenohumeral joint bio imaging techniques represent a powerful tool for shoulder joint biomechanical analysis however the restricted field of view may prevent the acquisition of complete scapula and humerus bone models and hence limiting the applicability of standardized anatomical coordinate system acs definitions the aim of this study was to propose acs definitions for both scapula and humerus which can be implemented when limited portions of the relevant bones are available magnetic resonance mr images of twenty right humeri and scapulae were acquired the proposed acss were assessed in terms of 1 sensitivity to bone morphological variation 2 intra and inter operator repeatability and 3 consistency with the anatomical cardinal directions a comparison with alternative acs definitions was also performed overall our acs scapular proposal and that presented in kedgley and dunning 2010 were found to be the least sensitive to the morphometric variability mean angular absolute deviation lower than 8 3 deg and they were characterized by a high intra and inter operator repeatability mean angular absolute deviation lower than 1 5 deg the humeral acs proposal showed a morphometric variability similar to amadi et al 2009b mean angular absolute deviation lower than 8 3 deg but a higher reproducibility the scapular and humeral acs mean angular deviation from the reference anatomical cardinal directions were smaller than 15 deg and 8 6 deg respectively the proposed scapular and humeral acs definitions are therefore suitable to be applied when a limited portion of the glenohumeral joint is available as it may occur in standard shoulder clinical exams

on intrinsic equivalences of the finite helical axis the instantaneous helical axis and the sara approach a mathematical perspective accurate determination of joint axes is essential for understanding musculoskeletal function whilst numerous algorithms to compute such axes exist the conditions under which each of the methods performs best remain largely unknown typically algorithms are evaluated for specific conditions only limiting the external validity of conclusions regarding their performance we derive exact mathematical relationships between three commonly used algorithms for computing joint axes from motion data finite helical axes fha instantaneous helical axes iha and sara symmetrical axis of rotation approach including relationships for an extension to the mean helical axes methods that facilitate determining joint centres and axes through the derivation of a sound mathematical framework to objectively compare the algorithms we demonstrate that the fha and sara approach are equivalent for the analysis of two time frames moreover we show that the position of a helical axis derived from the iha using positional data is affected by a systematic error perpendicular to the true axis direction whereas the axis direction is identical to those computed with either the fha or sara approach true direction finally with an appropriate choice of weighting factors the mean fha mfha method is equivalent to the symmetrical centre of rotation estimation score algorithm for determination of a centre of rotation cor and similarly equivalent to the sara algorithm for determination of an axis of rotation aor the deep understanding of the equivalences between methods presented here enables readers to choose numerically efficient robust methods for determining aors and cors with confidence

spinal shape changes resulting from scoliotic spine surgical instrumentation expressed as intervertebral rotations and centers of rotation this paper reports the changes in spinal shape resulting from scoliotic spine surgical instrumentation expressed as intervertebral rotations and centers of rotation the objective is to test the hypothesis that the type of spinal instrumentation system cotrel dubousset versus colorado does not influence these motion parameters intervertebral rotations and centers of rotation of the scoliotic spines were computed from the pre and post operative radiographs of 82 patients undergoing spinal correction the three dimensional 3d reconstruction of six anatomical landmarks was achieved for each of the thoracic and lumbar vertebrae a least squares approach based on singular value decomposition was used to calculate the rigid body transformation parameters average centers of rotation for all intervertebral levels are located in the neural canal at the mid sagittal plane and approximately at the superior endplate level of the inferior vertebra intervertebral rotations have components in all planes 6 7 degrees frontal 5 5 degrees sagittal and 4 5 degrees transverse rms for all intervertebral levels nearly all intervertebral rotations and centers of rotation are not significantly different for the two instrumentation systems various intervertebral rotations and 3d reconstruction errors were simulated on a theoretical model of a lumbar functional unit to assess the proposed method intervertebral rotation errors were 1 7 degrees when simulating 3d errors of 3mm on the position of the landmarks maximum errors for the position of centers of rotation were below 1cm in the case of intervertebral rotations larger than 2 5 degrees most cases but were larger 38 mm for small intervertebral rotations 1 degrees the type of instrumentation system did not influence intervertebral rotations and centers of rotation these results provide valuable data for the development and validation of simulation models for surgical instrumentation of idiopathic scoliosis

a regression based 3 d shoulder rhythm abstract in biomechanical modeling of the shoulder it is important to know the orientation of each bone in the shoulder girdle when estimating the loads on each musculoskeletal element however because of the soft tissue overlying the bones it is difficult to accurately derive the orientation of the clavicle and scapula using surface markers during dynamic movement the purpose of this study is to develop two regression models which predict the orientation of the clavicle and the scapula the first regression model uses humerus orientation and individual factors such as age gender and anthropometry data as the predictors the second regression model includes only the humerus orientation as the predictor thirty eight participants performed 118 static postures covering the volume of the right hand reach the orientation of the thorax clavicle scapula and humerus were measured with a motion tracking system regression analysis was performed on the euler angles decomposed from the orientation of each bone from 26 randomly selected participants the regression models were then validated with the remaining 12 participants the results indicate that for the first model the r2 of the predicted orientation of the clavicle and the scapula ranged between 0 31 and 0 65 and the rmse obtained from the validation dataset ranged from 6 92 to 10 39 for the second model the r2 ranged between 0 19 and 0 57 and the rmse obtained from the validation dataset ranged from 6 62 and 11 13 the derived regression based shoulder rhythm could be useful in future biomechanical modeling of the shoulder

comparison of the score and ha methods for locating in vivo the glenohumeral joint centre a recent paper has described a new functional method the symmetrical centre of rotation score for locating joint centre position ehrig r m taylor w r duda g n heller m o 2006 a survey of formal methods for determining the centre of rotation of ball joints journal of biomechanics 39 15 2798 2809 for in vitro analyses the score method showed better precision than helical axis ha or sphere fitting methods despites ha determination is very sensitive to small angular velocity the international society of biomechanics has recommended to use ha for locating the glenohumeral joint centre this paper aims at comparing the score method with the ha method for locating in vivo the glenohumeral joint centre according to the movement characteristics nine subjects performed 10 cycles of three different movements at two different velocities for each test combination of movements the location of the centre of rotation was estimated with both methods score and ha analyses focused on the 3d location of the glenohumeral joint centre and on the repeatability of location standard deviation this study showed that score and ha methods yielded the same gh location nevertheless with score method the location of the glenohumeral joint centre was different according to the test this study evidenced that the score method was more precise than ha method error of 3 mm versus 4 6 mm and that the gh location with the score method was not affected by movements with slow velocities

effect of various upper limb multibody models on soft tissue artefact correction a case study soft tissue artefacts sta introduce errors in joint kinematics when using cutaneous markers especially on the scapula both segmental optimisation and multibody kinematics optimisation mko algorithms have been developed to improve kinematics estimates mko based on a chain model with joint constraints avoids apparent joint dislocation but is sensitive to the biofidelity of chosen joint constraints since no recommendation exists for the scapula our objective was to determine the best models to accurately estimate its kinematics one participant was equipped with skin markers and with an intracortical pin screwed in the scapula segmental optimisation and mko for 24 chain models including four variations of the scapulothoracic joint were compared against the pin derived kinematics using root mean square error rmse on cardan angles segmental optimisation led to an accurate scapula kinematics 1 1 rmse 3 3 even for high arm elevation angles when mko was applied no clinically significant difference was found between the different scapulothoracic models 0 9 rmse 4 1 except when a free scapulothoracic joint was modelled 1 9 rmse 9 6 to conclude using mko as a sta correction method was not more accurate than segmental optimisation for estimating scapula kinematics

existence regularity and concentration phenomenon of nontrivial solitary waves for a class of generalized variable coefficient kadomtsev petviashvili equation in this paper we establish some results concerning the existence regularity and concentration phenomenon of nontrivial solitary waves for a class of generalized variable coefficient kadomtsev petviashvili equation variational methods are used to get an existence result as well as to study the concentration phenomenon while the regularity is more delicate because we are leading with functions in an anisotropic sobolev space

model guided derivation of lumbar vertebral kinematics in vivo reveals the difference between external marker defined and internal segmental rotations this study investigated whether the external marker defined spine inter segmental rotation is different from the internal vertebral rotation and explored how to estimate the latter from limited surface measurement a kinematic model was first created to elucidate analytically the relation between the external and internal rotations a novel approach guided by the model was proposed for deriving vertebral centers of rotation cors from measured planar trajectories of skin surface markers the approach involved a recursive procedure for establishing local anatomical coordinate systems and an optimization routine that identified the maximum likelihood circles best fitting the marker trajectories in local coordinate systems an experiment with 10 subjects 5 males and 5 females was conducted to test the approach along with the model skin surface markers were strategically placed over individual spinous processes and other body landmarks and recorded by an opto electronic system while sagittally symmetric load lifting movements were being performed for the majority 89 of measured motions the cor locations for lumbar vertebrae l2 l5 were derived successfully solutions resulting from the optimization routine met a convergence criterion governed by the model and were in agreement with existing data from radiographic or cadaveric studies empirical results confirmed the differences between the external marker defined inter segmental motions and corresponding internal vertebral rotations 1 1 5 8 degrees on average all statistically significant the study demonstrated the necessity and viability of quantifying internal vertebral kinematics when utilizing non invasive marker based measurement for spine related clinical diagnosis and biomechanical modeling

the glenohumeral joint rotation centre in vivo within the framework of the current call for standardization in upper extremity research three methods to determine the glenohumeral joint rotation centre in vivo were tested therefore subjects performed humeral movements while a 3d electromagnetic tracking device registered the motion of the humerus with respect to the scapula for the first method to estimate the glenohumeral joint rotation centre five scapular bony landmarks served as input to regression equations the second method fitted a sphere through the humeral position data and the third method calculated the rotation centre determining an optimal helical axis the experiment consisted of two parts at first one subject was measured 10 times subsequently one observer measured 10 subjects twice and another observer measured these subjects once the first part of the experiment demonstrated that all methods are capable to reproduce the rotation centre within 4mm but the location of the centre differed significantly between methods p 0 001 the second part showed that inter and intra observer reliability was sufficiently for the sphere fitting method and for the helical axes method the two observations of one observer differed significantly p 0 008 using the regression method the authors prefer the helical axes method it is a reliable and valid method which can be applied in movement registration of healthy subjects and patients with a shoulder endoprosthesis it can be applied in hinge joints to determine a rotation axis instead of a rotation centre which is desirable in standardized upper extremity research and calculation time is short

solutions of a variable coefficient kadomtsev petviashvili equation via computer algebra we apply the truncated painleve expansion and computer algebra to a type of the variable coefficient kadomtsev petviashvilli equations and find a class of the analytical solutions along with the corresponding constraints on the variable coefficients shown in this class are the physically meaningful solutions which possess the usual soliton profile

functional knee axis based on isokinetic dynamometry data comparison of two methods mri validation and effect on knee joint kinematics this paper compares geometry based knee axes of rotation transepicondylar axis and geometric center axis and motion based functional knee axes of rotation faor two algorithms are evaluated to calculate faors gamage and lasenby s sphere fitting algorithm gl and ehrig et al s axis transformation algorithm sara calculations are based on 3d motion data acquired during isokinetic dynamometry aors are validated with the equivalent axis based on static mr images we quantified the difference in orientation between two knee axes of rotation as the angle between the projection of the axes in the transversal and frontal planes and the difference in location as the distance between the intersection points of the axes with the sagittal plane maximum differences between faors resulting from gl and sara were 5 7 and 15 4mm respectively maximum differences between faors resulting from gl or sara and the equivalent axis were 5 4 11 5mm and 8 6 12 8mm respectively differences between geometry based axes and ea are larger than differences between faor and ea both in orientation maximum 10 6 and location maximum 20 8mm knee joint angle trajectories and the corresponding accelerations for the different knee axes of rotation were estimated using kalman smoothing for the joint angles the maximum rms difference with the mri based equivalent axis which was used as a reference was 3 for the knee joint accelerations the maximum rms difference with the equivalent axis was 20 s 2 functional knee axes of rotation describe knee motion better than geometry based axes gl performs better than sara for calculations based on experimental dynamometry

a correction for axis misalignment in the joint angle curves representing knee movement in gait analysis this paper derives a simple mathematical model relating changes in the orientations of the two cartesian coordinate systems involved in recording knee movement and the varus valgus and the internal external rotation angles for describing the knee s motion rotation matrices are given for changing the orientations of the two cartesian coordinate systems in such a way that the quadratic variations in the varus valgus and in the external internal angles are minimal these estimated rotation matrices are used to correct for axis misalignment the correction is calibrated by considering the impact of the new orientation of the thigh cartesian coordinate system on the hip joint angles the procedure is applied to kinematic data collected on normal subjects the uncertainty about the specification of the thigh cartesian coordinate system is shown to explain some of the between subject variability in the varus valgus and in the internal external rotation angles curves

in vivo estimation of the glenohumeral joint centre by functional methods accuracy and repeatability assessment abstract several algorithms have been proposed for determining the centre of rotation of ball joints these algorithms are used rather to locate the hip joint centre few studies have focused on the determination of the glenohumeral joint centre however no studies have assessed the accuracy and repeatability of functional methods for glenohumeral joint centre this paper aims at evaluating the accuracy and the repeatability with which the glenohumeral joint rotation centre ghrc can be estimated in vivo by functional methods the reference joint centre is the glenohumeral anatomical centre obtained by medical imaging five functional methods were tested the algorithm of gamage and lasenby 2002 bias compensated halvorsen 2003 symmetrical centre of rotation estimation ehrig et al 2006 normalization method chang and pollard 2007 helical axis woltring et al 1985 the glenohumeral anatomical centre ghac was deduced from the fitting of the humeral head four subjects performed three cycles of three different movements flexion extension abduction adduction and circumduction for each test the location of the glenohumeral joint centre was estimated by the five methods analyses focused on the 3d location on the repeatability of location and on the accuracy by computing the euclidian distance between the estimated ghrc and the ghac for all the methods the error repeatability was inferior to 8 25 mm this study showed that there are significant differences between the five functional methods the smallest distance between the estimated joint centre and the centre of the humeral head was obtained with the method of gamage and lasenby 2002

existence of solution for two classes of elliptic problems in r n with zero mass in this paper we show the existence of a positive solution for the following class of elliptic equations u k x f u in rn where f is a continuous function with quasicritical growth and k is a nonnegative function verifying some conditions of two kinds we analyze the problem with k asymptotically periodic k periodic and k belonging to some lr

double calibration an accurate reliable and easy to use method for 3d scapular motion analysis the most recent non invasive methods for the recording of scapular motion are based on an acromion marker am set and a single calibration sc of the scapula in a resting position however this method fails to accurately measure scapular kinematics above 90 of arm elevation due to soft tissue artifacts of the skin and muscles covering the acromion the aim of this study was to evaluate the accuracy and inter trial and inter session repeatability of a double calibration method dc in comparison with sc the sc and dc data were measured with an optoelectronic system during arm flexion and abduction at different angles of elevation 0 180 they were compared with palpation of the scapula using a scapula locator dc data was not significantly different from palpation for 5 6 axes of rotation tested y x and z in abduction and flexion where as sc showed significant differences for 5 6 axes the root mean square errors ranged from 2 96 to 4 48 for dc and from 6 to 9 19 for sc the inter trial repeatability was good to excellent for sc and dc the inter session repeatability was moderate to excellent for sc and moderate to good for dc coupling am and dc is an easy to use method which yields accurate and reliable measurements of scapular kinematics for the complete range of arm motion it can be applied to the measurement of shoulder motion in many fields sports orthopaedics and rehabilitation especially when large ranges of arm motion are required

bias compensated least squares estimate of the center of rotation abstract it is shown that a recently published least squares method for the estimation of the average center of rotation is biased consequently a correction term is proposed and an iterative algorithm is derived for finding a bias compensated solution to the least squares problem the accuracy of the proposed bias compensated least squares method is compared to the previously proposed least squares method by monte carlo simulations the tests show that the new method gives a substantial improvement in accuracy

comparison between tripod and skin fixed recording of scapular motion non invasive dynamical measurements of 3d scapular motion can be performed easily by attachment of a 6 dof electromagnetic receiver onto the skin above the acromion to quantify the introduction of possible errors due to skin displacement we assessed 3d scapular positions on n 8 subjects by both tripod and skin fixed method error analysis included the variables method tripod skin fixed simultaneously with tripod separate skin fixed at 0 and 0 25hz of elevation speed plane of elevation 0 degrees and 90 degrees and observation receiver replacement n 3 inter individual group differences depended on elevation plane and showed an average underestimation of scapular rotation of 6 5 degrees worst case 13 degrees using the skin fixed method only the group rmse not the individual rmse could be successfully lowered using linear regression to about 2 degrees inter trial reliability rmse 0 94 and rmse between 0 and 0 25hz recordings about 2 5 degrees were satisfactory intra observer rmse after replacement of the skin fixed receiver was 5 degrees the skin fixed method is suitable for dynamic recordings of scapular rotations however measurements are precise only when the acromion receiver is not replaced combined with a relatively low accuracy we conclude that the skin fixed method should be used only in combination with tripod calibration

the structure of underlying lie algebras abstract every lie algebra over a field e gives rise to new lie algebras over any subfield f e by restricting the scalar multiplication this paper studies the structure of these underlying lie algebra in relation to the structure of the original lie algebra in particular the question how much of the original lie algebra can be recovered from its underlying lie algebra over subfields f by introducing the conjugate of a lie algebra we show that in some specific cases the lie algebra is completely determined by its underlying lie algebra furthermore we construct examples showing that these assumptions are necessary as an application we give for every positive n an example of a real 2 step nilpotent lie algebra which has exactly n different bi invariant complex structures this answers an open question by di scala lauret and vezzoni motivated by their work on quasi kahler chern flat manifolds in differential geometry the methods we develop work for general lie algebras and for general galois extensions f e in contrast to the original question which only considered nilpotent lie algebras of nilpotency class 2 and the field extension r c we demonstrate this increased generality by characterizing the complex lie algebras of dimension 4 which are defined over r and over q