A Lagrange-based generalised formulation for the equations of motion of simple walking models

Simple 2D models of walking often approximate the human body to multi-link dynamic systems, where body segments are represented by rigid links connected by frictionless hinge joints. Performing forward dynamics on the equations of motion (EOM) of these systems can be used to simulate their movement. However, deriving these equations can be time consuming. Using Lagrangian mechanics, a generalised formulation for the EOM of n-link open-loop chains is derived. This can be used for single support walking models. This has an advantage over Newton–Euler mechanics in that it is independent of coordinate system and prior knowledge of the ground reaction force (GRF) is not required. Alternative strategies, such as optimisation algorithms, can be used to estimate joint activation and simulate motion. The application of Lagrange multipliers, to enforce motion constraints, is used to adapt this general formulation for application to closed-loop chains. This can be used for double support walking models. Finally, inverse dynamics are used to calculate the GRF for these general n-link chains. The necessary constraint forces to maintain a closed-loop chain, calculated from the Lagrange multipliers, are one solution to the indeterminate problem of GRF distribution in double support models. An example of this method’s application is given, whereby an optimiser estimates the joint moments by tracking kinematic data.     

Experimental/analytical analysis of human locomotion using bondgraphs

A new vectorial bondgraph approach for modeling and simulation of human locomotion is introduced. The vectorial bondgraph is applied to an eight-segment gait model to derive the equations of motion for studying ground reaction forces (GRFs) and centers of pressure (COPs) in single and double support phases of ground and treadmill walking. A phase detection technique and accompanying transition equation is proposed with which the GRFs and COPs may be calculated for the transitions from double-to-single and single-to-double support phases. Good agreement is found between model predictions and experimental data obtained from force plate measurements. The bondgraph modeling approach is shown to be both informative and adaptable, in the sense that the model resembles the human body structure, and that modeled body segments can be easily added or removed.     

Prediction of ground reaction forces during gait based on kinematics and a neural network model

Kinetic information during human gait can be estimated with inverse dynamics, which is based on anthropometric, kinematic, and ground reaction data. While collecting ground reaction data with a force plate is useful, it is costly and requires regulated space. The goal of this study was to propose a new, accurate methodology for predicting ground reaction forces (GRFs) during level walking without the help of a force plate. To predict GRFs without a force plate, the traditional method of Newtonian mechanics was used for the single support phase. In addition, an artificial neural network (ANN) model was applied for the double support phase to solve statically indeterminate structure problems. The input variables of the ANN model, which were selected to have both dependency and independency, were limited to the trajectory, velocity, and acceleration of the whole segment's mass centre to minimise errors. The predicted GRFs were validated with actual GRFs through a ten-fold cross-validation method, and the correlation coefficients (R) for the ground forces were 0.918 in the medial–lateral axis, 0.985 in the anterior–posterior axis, and 0.991 in the vertical axis during gait. The ground moments were 0.987 in the sagittal plane, 0.841 in the frontal plane, and 0.868 in the transverse plane during gait. The high correlation coefficients(R) are due to the improvement of the prediction rate in the double support phase. This study also proved the possibility of calculating joint forces and moments based on the GRFs predicted with the proposed new hybrid method. Data generated with the proposed method may thus be used instead of raw GRF data in gait analysis and in calculating joint dynamic data using inverse dynamics.     

Spatial reconstruction of the human motion based on images of a single camera

The inverse dynamic analysis procedures used in the study of the human gait require that the kinematics of the supporting biomechanical model is known beforehand. The first step to obtain the kinematic data is the reconstruction of human spatial motion, i.e., the evaluation of the anatomic points positions that enables to uniquely define the position of all anatomical segments. In photogrammetry, the projection of each anatomical point is described by two linear equations relating its three spatial coordinates with the two coordinates of the projected point. The need for the image of two cameras arises from the fact that three equations are necessary to find the original spatial position of the anatomical point. It is shown here that the kinematic constraint equations associated with a biomechanical model can be used as the extra set of equations required for the reconstruction process, instead of the equations associated with the second camera. With this methodology, the system of equations arising from the point projections and biomechanical model kinematic constraints are solved simultaneously. Since the system of equations has multiple solutions for each image, a strategy based on the minimization of the cost function associated to the smoothness of the reconstructed motion is devised, leading to an automated computer procedure enabling a unique reconstruction.     

Prediction of ground reaction forces and moments during various activities of daily living

Inverse dynamics based simulations on musculoskeletal models is a commonly used method for the analysis of human movement. Due to inaccuracies in the kinematic and force plate data, and a mismatch between the model and the subject, the equations of motion are violated when solving the inverse dynamics problem. As a result, dynamic inconsistency will exist and lead to residual forces and moments. In this study, we present and evaluate a computational method to perform inverse dynamics-based simulations without force plates, which both improves the dynamic consistency as well as removes the model?s dependency on measured external forces. Using the equations of motion and a scaled musculoskeletal model, the ground reaction forces and moments (GRF&Ms) are derived from three-dimensional full-body motion. The method entails a dynamic contact model and optimization techniques to solve the indeterminacy problem during a double contact phase and, in contrast to previously proposed techniques, does not require training or empirical data. The method was applied to nine healthy subjects performing several Activities of Daily Living (ADLs) and evaluated with simultaneously measured force plate data. Except for the transverse ground reaction moment, no significant differences (P>0.05) were found between the mean predicted and measured GRF&Ms for almost all ADLs. The mean residual forces and moments, however, were significantly reduced (P>0.05) in almost all ADLs using our method compared to conventional inverse dynamic simulations. Hence, the proposed method may be used instead of raw force plate data in human movement analysis using inverse dynamics.     

Standing sway: iterative estimation of the kinematics and dynamics of the lower extremities from force-plate measurements

In this study, a model for the estimation of the dynamics of the lower extremities in standing sway from force plate data only is presented. A three-dimensional, five-segment, four-joint model of the human body was used to describe postural standing sway dynamics. Force-plate data of the reactive forces and centers of pressure were measured bilaterally. By applying the equations of motion to these data, the transversal trajectory of the center of gravity (CG) of the body was resolved in the sagittal and coronal planes. An inverse kinematics algorithm was used to evaluate the kinematics of the body segments. The dynamics of the segments was then resolved by using the Newton-Euler equations, and the model's estimated dynamic quantities of the distal segments were compared with those actually measured. Differences between model and measured dynamics were calculated and minimized, using an iterative algorithm to re-estimate joint positioning and anthropometric properties. The above method was tested with a group of 11 able-bodied subjects, and the results indicated that the relative errors obtained in the final iteration were of the same order of magnitude as those reported for closed loop problems involved in direct kinematics measurements of human gait. Received: 22 July 1997 / Accepted in revised form: 29 January 1998     

Validation protocol of models for centre of mass estimation.

The estimation of the body centre-of-mass (COM) position requires the modelling of the human body as a system of rigid segments and the measurement of the position of related external anatomical landmarks. Many models for COM position estimation have been proposed with different levels of complexity and, in some cases, specific protocols have been used for model accuracy evaluation. In this paper, we propose a general method for the quantitative assessment of any COM model in relation to a determined set of movements. It consists of an experimental protocol and of a set of comparative indices, which quantify the congruence among the estimated kinematic variables and their expected values. The general applicability of the method is specifically addressed to models' comparison, aiming to support the user in the process of choice and validation of the most suitable model for her/his purposes. In this frame, the results of the analytical comparison among two kinematic models with different levels of complexity are reported.     

Coupled oscillators utilised as gait rhythm generators of a two-legged walking machine

The gait of current two-legged walking machines differs from that of humans, although the kinematic structures of these machines' legs frequently imitate human limbs. This paper presents a method of generating the trajectories of hip and knee joint angles resulting in a gait pattern similar to that of a human. For this purpose the solutions of coupled van der Pol oscillator equations are utilised. There is much evidence that these equations can be treated as a good model of the central pattern generator generating functional (also locomotional) rhythms in living creatures. The oscillator equations are solved by numerical integration. The method of changing the type of gait by changing appropriate parameter values in the oscillator equations is presented (change of velocity and trajectory of leg-ends). The results obtained enable enhanced control of twolegged walking systems by including gait pattern generators which will assume a similar role to that of biological generators.     

Using linkage models to explore skull kinematic diversity and functional convergence in arthrodire placoderms

Biomechanical models offer a powerful set of tools for quantifying the diversity of function across fossil taxa. A computer‐based four‐bar linkage model previously developed to describe the potential feeding kinematics of Dunkleosteus terrelli is applied here to several other arthrodire placoderm taxa from different lineages. Arthrodire placoderms are a group of basal gnathostomes showing one of the earliest diversifications of jaw structures. The linkage model allows biomechanical variation to be compared across taxa, identify trends in skull morphology among arthrodires that potentially influence function and explore the role of linkage systems in the early evolution of jaw structures. The linkage model calculates various kinematic metrics including gape angle, effective mechanical advantage, and kinematic transmission coefficients. Results indicate that the arthrodire feeding system may be more diverse and complex than previously thought. A range of potential kinematic profiles among arthrodire taxa illustrate a diversity of feeding function comparable with modern teleost fishes. Previous estimates of bite force in Dunkleosteus are revised based on new morphological data. High levels of kinematic transmission among arthrodires suggest the potential for rapid gape expansion and possible suction feeding. Morphological comparisons indicate that there were several morphological solutions for obtaining these fast kinematics, which allowed different taxa to achieve similar kinematic profiles while varying other aspects of the feeding apparatus. Mapping of key morphological components of the linkage system on a general placoderm phylogeny illustrates the potential importance of four‐bar systems to the early evolution of jaw structures. J. Morphol. 271:990–1005, 2010. © 2010 Wiley‐Liss, Inc.     

Linkage mechanisms in the vertebrate skull: Structure and function of three‐dimensional,parallel transmission systems

Many musculoskeletal systems, including the skulls of birds, fishes, and some lizards consist of interconnected chains of mobile skeletal elements, analogous to linkage mechanisms used in engineering. Biomechanical studies have applied linkage models to a diversity of musculoskeletal systems, with previous applications primarily focusing on two‐dimensional linkage geometries, bilaterally symmetrical pairs of planar linkages, or single four‐bar linkages. Here, we present new, three‐dimensional (3D), parallel linkage models of the skulls of birds and fishes and use these models (available as free kinematic simulation software), to investigate structure–function relationships in these systems. This new computational framework provides an accessible and integrated workflow for exploring the evolution of structure and function in complex musculoskeletal systems. Linkage simulations show that kinematic transmission, although a suitable functional metric for linkages with single rotating input and output links, can give misleading results when applied to linkages with substantial translational components or multiple output links. To take into account both linear and rotational displacement we define force mechanical advantage for a linkage (analogous to lever mechanical advantage) and apply this metric to measure transmission efficiency in the bird cranial mechanism. For linkages with multiple, expanding output points we propose a new functional metric, expansion advantage, to measure expansion amplification and apply this metric to the buccal expansion mechanism in fishes. Using the bird cranial linkage model, we quantify the inaccuracies that result from simplifying a 3D geometry into two dimensions. We also show that by combining single‐chain linkages into parallel linkages, more links can be simulated while decreasing or maintaining the same number of input parameters. This generalized framework for linkage simulation and analysis can accommodate linkages of differing geometries and configurations, enabling novel interpretations of the mechanics of force transmission across a diversity of vertebrate feeding mechanisms and enhancing our understanding of musculoskeletal function and evolution. J. Morphol. 277:1570–1583, 2016. © 2016 Wiley Periodicals, Inc.     

Constructing genetic linkage maps under a tetrasomic model

An international consortium has launched the whole-genome sequencing of potato, the fourth most important food crop in the world. Construction of genetic linkage maps is an inevitable step for taking advantage of the genome projects for the development of novel cultivars in the autotetraploid crop species. However, linkage analysis in autopolyploids, the kernel of linkage map construction, is theoretically challenging and methodologically unavailable in the current literature. We present here a theoretical analysis and a statistical method for tetrasomic linkage analysis with dominant and/or codominant molecular markers. The analysis reveals some essential properties of the tetrasomic model. The method accounts properly for double reduction and incomplete information of marker phenotype in regard to the corresponding phenotype in estimating the coefficients of double reduction and recombination frequency and in testing their significance by using the marker phenotype data. Computer simulation was developed to validate the analysis and the method and a case study with 201 AFLP and SSR markers scored on 228 full-sib individuals of autotetraploid potato is used to illustrate the utility of the method in map construction in autotetraploid species.     

Neck muscle paths and moment arms are significantly affected by wrapping surface parameters

In this paper, we studied the effects of wrapping surfaces on muscle paths and moment arms of the neck muscle, semispinalis capitis. Sensitivities to wrapping surface size and the kinematic linkage to vertebral segments were evaluated. Kinematic linkage, but not radius, significantly affected the accuracy of model muscle paths compared to centroid paths from images. Both radius and linkage affected the moment arm significantly. Wrapping surfaces that provided the best match to centroid paths over a range of postures had consistent moment arms. For some wrapping surfaces with poor matches to the centroid path, a kinematic method (tendon excursion) predicted flexion moment arms in certain postures, whereas geometric method (distance to instant centre) predicted extension. This occurred because the muscle lengthened as it wrapped around the surface. This study highlights the sensitivity of moment arms to wrapping surface parameters and the importance of including multiple postures when evaluating muscle paths and moment arm.     

Neck muscle paths and moment arms are significantly affected by wrapping surface parameters

In this paper, we studied the effects of wrapping surfaces on muscle paths and moment arms of the neck muscle, semispinalis capitis. Sensitivities to wrapping surface size and the kinematic linkage to vertebral segments were evaluated. Kinematic linkage, but not radius, significantly affected the accuracy of model muscle paths compared to centroid paths from images. Both radius and linkage affected the moment arm significantly. Wrapping surfaces that provided the best match to centroid paths over a range of postures had consistent moment arms. For some wrapping surfaces with poor matches to the centroid path, a kinematic method (tendon excursion) predicted flexion moment arms in certain postures, whereas geometric method (distance to instant centre) predicted extension. This occurred because the muscle lengthened as it wrapped around the surface. This study highlights the sensitivity of moment arms to wrapping surface parameters and the importance of including multiple postures when evaluating muscle paths and moment arm.     

Synthesis of human walking: A planar model for single support

A mathematical model for the single support phase of normal, level, human walking is formulated. The motion of the lower extremity is synthesized using a preprogrammed set of inputs, recognized by the model as a simple collection of applied joint moments.

Two mechanisms are forwarded as candidates for producing the observed peaks in the vertical ground reaction. The first, stance knee flexion-extension, generates the necessary level of whole-body vertical acceleration during the initial region of single support (opposite toe-off to heel-off). A model accounting for the determinants of foot and knee interaction then predicts the second peak to be the result of an increasing ankle moment in the region from heel-off to opposite heel-strike.     

Validation of net joint loads calculated by inverse dynamics in case of complex movements: application to balance recovery movements

The joint forces and moments driving the motion of a human subject are classically computed by an inverse dynamic calculation. However, even if this process is theoretically simple, many sources of errors may lead to huge inaccuracies in the results. Moreover, a direct comparison with in vivo measured loads or with "gold standard" values from literature is only possible for very specific studies. Therefore, assessing the inaccuracy of inverse dynamic results is not a trivial problem and a simple method is still required. This paper presents a simple method to evaluate both: (1) the consistency of the results obtained by inverse dynamics; (2) the influence of possible modifications in the inverse dynamic hypotheses. This technique concerns recursive calculation performed on full kinematic chains, and consists in evaluating the loads obtained by two different recursive strategies. It has been applied to complex 3D whole body movements of balance recovery. A recursive Newton-Euler procedure was used to compute the net joint loads. Two models were used to represent the subject bodies, considering or not the upper body as a unique rigid segment. The inertial parameters of the body segments were estimated from two different sets of scaling equations [De Leva, P., 1996. Adjustments to Zatsiorsky-Suleyanov's segment inertia parameters. Journal of Biomechanics 29, 1223-1230; Dumas, R., Chèze, L., Verriest, J.-P., 2006b. Adjustments to McConville et al. and Young et al. Body Segment Inertial Parameters. Journal of Biomechanics, in press]. Using this comparison technique, it has been shown that, for the balance recovery motions investigated: (1) the use of the scaling equations proposed by Dumas et al., instead of those proposed by De Leva, improves the consistency of the results (average relative influence up to 30% for the transversal moment); (2) the arm motions dynamically influence the recovery motion in a non negligible way (average relative influence up to 15% and 30% for the longitudinal force and the transversal moment, respectively).     

Dynamic heat and moisture transfer through multiple clothing layers

1. 1. The purposes of this study are to find out the arrangement effects on the vapor pressure gradient across the cotton–nylon double layer and to elucidate changes in the vapor pressure gradient when an additional third layer covers the double layer.

2. 2. Model tests for single, double and triple layer system and wear test for triple layer clothing were conducted.

3. 3. It was found that up to the second layer, dryness of innermost microclimate could be maintained when cotton faced the skin (C/N).

4. 4. However, when more permeable and hydrophobic third layer (UWF) covers the double layer, the microclimate of C/N is no longer drier than N/C.

5. 5. When nylon is exposed to the skin, a larger drop in vapor pressure across the first two layers occurred for both model and wear test.

6. 6. The innermost microclimate was not necessarily kept dry when the outermost layer dissipated more moisture due to the inefficient distribution of moisture.

Mechanisms of chromosomal aberration production I. Aberration induction by ultraviolet light

We have studied chromosomal aberration production in V-79 Chinese hamster tissue culture cells by UV light administered during the post-DNA-synthetic G₂ phase of the cell cycle. The treatment produced achromatic lesions and some chromatid deletions in the first post-irradiation mitosis, but no isochromatid deletions or chromatid exchange aberrations. In contrast, when G₂ UV-irradiated cells were examined in their second post-irradiation mitosis, there were significant yields of chromatid-type aberrations of all types, including isochromatid deletions and chromatid exchanges.

We have earlier reported²¹ that UV-irradiation during the pre-DNA-synthetic G₁ phase of the cell cycle induces only chromatid aberrations and also that most chromosomal aberration production by UV in G₁ can be photoreactivated in cells possessing the photoreactivating enzyme. We present here a model for chromosomal aberration production by UV. In the model all aberration production is enzymatically mediated, a consequence of the functioning of known molecular repair mechanisms. The important elements in the model are the following:

1. (1) The vertebrate chromosome is mononeme; i.e., contains but a single DNA double helix during the prereplication G₁ phase of the cell cycle.

2. (2) The UV-induced DNA lesion leading to the production of most aberrations is the cyclobutane dimer between adjacent pyrimidines in one polynucleotide strand.

3. (3) Single chain breaks appear at metaphase as achromatic lesions.

4. (4) Dimer removal sometimes leaves unrepaired single chain gaps, possibly as a result of incomplete excision repair.

5. (5) The single-stranded DNA opposite a single chain gap can be cleaved by a single-strand DNAase.

6. (6) Gaps are left in newly synthesized DNA polynucleotide chains opposite defective template chains (i.e., opposite dimers and chain breaks).

7. (7) Double-strand breaks present following local DNA replication may “spread” to the other chromatid by a recombinational process between template and new polynucleotide chains, one from each of the homologous double helices.

The model predicts the occurrence of isoachromatic lesions and of chromatid deletions paired (isolocus) with achromatic lesions. Though often not reported, both do, in fact, occur. In addition, the model accounts for the phenomenon of sister-chromatid exchange as a manifestation of a recombinational, or post-replication, repair mechanism. Finally, the model offers a simple interpretation of chromosomal aberration production by a variety of chemical agents.     

Polymer conformation: 3. Rapid methods for seeking contact-free double helices

A methodology for mapping helix constraint surfaces and contact-free molecular structures, as introduced in parts 1 and 2 of the series^1,2, has been developed and extended to include investigation of stability and stereochemical feasibility of double helices. The motivation for this development was prompted by practical considerations concerning the interpretation of new X-ray fibre diffraction patterns obtained from the gelling polysaccharide agarose³, in the light of previous interpretations⁴ favouring a double helix model for this biopolymer. The method may be of use for objectively scrutinizing double helix structures the interpretation of a number of which, e.g. carrageenan^5,6, poly(methyl methacrylate)^7,10, isotactic polystyrene^11,12, xanthan gum^13,16, appear to be controversial issues.     

Construction of a dense SNP map for bovine chromosome 6 to assist the assembly of the bovine genome sequence

A linkage map was constructed for bovine chromosome 6 (BTA6), using 399 single nucleotide polymorphisms (SNPs) detected primarily from PCR-resequencing. The efficiency of SNP detection was highly dependent on the source of sequence information chosen for primer design (BAC-end sequences, introns or promoters). The SNPs were used to build a linkage map comprising 104 cM on BTA6. The SNP order in the linkage map corresponded very well with radiation hybrid (RH) maps available for BTA6 as well as with expected positions in the human comparative map, but diverged significantly from the current assembly of the bovine genome (Btau_3.1). When performing linkage analysis with the marker order suggested from the Btau_3.1 we observed an expansion of the genetic map from 104 cM to 137 cM, strongly suggesting a reordering of scaffolds in the current version of the bovine genome assembly. The extent of LD on BTA6 was evaluated by calculating the average r ² for SNP pairs separated by given distances. The decline of LD was rapid with distance, such that r ² was 0.1 at 100 kb. Our results indicate that linkage mapping will be a valuable source of information for correcting errors in the current bovine assembly. These errors were sufficiently frequent to be of concern for the accuracy of mapping QTL with panels of SNPs whose positions are based on the current assembly.     

Validation of a calibration technique for 6-DOF instrumented spatial linkages

This paper presents a practical and effective approach to the calibration of instrumented spatial linkages for biomechanical applications. A 6-DOF mechanical linkage with rotational transducers is designed and in-house manufactured for this purpose. In order to assess the validity of the proposed calibration technique and to distinguish between geometrical and electrical parameters uncertainties, high-precision optical encoders are used and calibration is addressed from a kinematic point of view only. The proposed technique is based on a closed-loop method, in which the end segments of the linkage are connected to each other by revolute joints. A parametrical model of the system is formulated using a standard link-to-link transformation matrices approach. Continuous data collection is carried out and a recursive identification of kinematic parameters is implemented by the use of an extended Kalman filter algorithm. Results shows that the proposed technique, despites its simplicity, is effective in improving the accuracy of the system up to its theoretically computed resolution, which limits the performance of the real system.