Estimation of errors in mechanical efficiency

Errors in measurements of mechanical work, net energy expenditure and mechanical efficiency (ME) were calculated, when subjects performed isolated eccentric or concentric muscle actions and combinations of these actions [stretch-shortening cycle (SSC) exercises] with a special sledge apparatus. The relative error of mechanical work was 6.1%. When estimating the error of energy metabolism from oxygen consumption the error would be about 4% (McArdle et al. 1981). The maximum error of ME was the sum of these two values (10.1%). Obviously the error of ME was less than 5%, because 30 muscle actions were averaged and, in addition, the errors of mechanical work and energy expenditure were not in the same direction every time. It was concluded that mechanical work can be determined accurately when the force is measured as a function of the moved distance of the sledge. Thus calculation of ME can be performed quite reliably in isolated eccentric and concentric exercises. The greatest problems were, however, in the SSC exercises, where the errors were higher, because of the problems of dividing the net energy expenditure into eccentric and concentric phases. Therefore, further developments must be made to minimize the errors in measurement and calculation during SSC-exercise.     

The simulation of aerial movement--III. The determination of the angular momentum of the human body

A method is presented for determining the angular momentum of the human body about its mass centre for general three-dimensional movements. The body is modelled as an 11 segment link system with 17 rotational degrees of freedom and the angular momentum of the body is derived as a sum of 12 terms, each of which is a vector function of just one angular velocity. This partitioning of the angular momentum vector gives the contribution due to the relative segmental movement at each joint rather than the usual contribution of each segment. A method of normalizing the angular momentum is introduced to enable the comparison of rotational movements which have different flight times and are performed by athletes with differing inertia parameters. Angular momentum estimates were calculated during the flight phases of nine twisting somersaults performed on trampoline. Errors in film digitization made large contributions to the angular momentum error estimates. For individual angular momentum estimates the relative error is estimated to be about 10% whereas for mean angular momentum estimates the relative error is estimated to be about 1%.     

The gait of children with and without cerebral palsy: work, energy, and angular momentum

This paper describes a method to characterize gait pathologies like cerebral palsy using work, energy, and angular momentum. For a group of 24 children, 16 with spastic diplegic cerebral palsy and 8 typically developed, kinematic data were collected at the subjects self selected comfortable walking speed. From the kinematics, the work-internal, external, and whole body; energy-rotational and relative linear; and the angular momentum were calculated. Our findings suggest that internal work represents 53% and 40% respectively of the whole body work in gait for typically developed children and children with cerebral palsy. Analysis of the angular momentum of the whole body, and other subgroupings of body segments, revealed a relationship between increased angular momentum and increased internal work. This relationship allows one to use angular momentum to assist in determining the kinetics and kinematics of gait which contribute to increased internal work. Thus offering insight to interventions which can be applied to increase the efficiency of bipedal locomotion, by reducing internal work which has no direct contribution to center of mass motion, in both normal and pathologic populations.     

A platform for dynamic simulation and control of movement based on OpenSim and MATLAB

Numerical simulations play an important role in solving complex engineering problems and have the potential to revolutionize medical decision making and treatment strategies. In this paper, we combine the rapid model-based design, control systems and powerful numerical method strengths of MATLAB/Simulink with the simulation and human movement dynamics strengths of OpenSim by developing a new interface between the two software tools. OpenSim is integrated with Simulink using the MATLAB S-function mechanism, and the interface is demonstrated using both open-loop and closed-loop control systems. While the open-loop system uses MATLAB/Simulink to separately reproduce the OpenSim Forward Dynamics Tool, the closed-loop system adds the unique feature of feedback control to OpenSim, which is necessary for most human movement simulations. An arm model example was successfully used in both open-loop and closed-loop cases. For the open-loop case, the simulation reproduced results from the OpenSim Forward Dynamics Tool with root mean square (RMS) differences of 0.03° for the shoulder elevation angle and 0.06° for the elbow flexion angle. MATLAB's variable step-size integrator reduced the time required to generate the forward dynamic simulation from 7.1s (OpenSim) to 2.9s (MATLAB). For the closed-loop case, a proportional-integral-derivative controller was used to successfully balance a pole on model's hand despite random force disturbances on the pole. The new interface presented here not only integrates the OpenSim and MATLAB/Simulink software tools, but also will allow neuroscientists, physiologists, biomechanists, and physical therapists to adapt and generate new solutions as treatments for musculoskeletal conditions.     

Muscle contributions to whole-body sagittal plane angular momentum during walking

Walking is a complex dynamic task that requires the regulation of whole-body angular momentum to maintain dynamic balance while performing walking subtasks such as propelling the body forward and accelerating the leg into swing. In human walking, the primary mechanism to regulate angular momentum is muscle force generation. Muscles accelerate body segments and generate ground reaction forces that alter angular momentum about the body’s center-of-mass to restore and maintain dynamic stability. In addition, gravity contributes to whole-body angular momentum through its contribution to the ground reaction forces. The purpose of this study was to generate a muscle-actuated forward dynamics simulation of normal walking to quantify how individual muscles and gravity contribute to whole-body angular momentum in the sagittal plane. In early stance, the uniarticular hip and knee extensors (GMAX and VAS), biarticular hamstrings (HAM) and ankle dorsiflexors (TA) generated backward angular momentum while the ankle plantar flexors (SOL and GAS) generated forward momentum. In late stance, SOL and GAS were the primary contributors and generated angular momentum in opposite directions. SOL generated primarily forward angular momentum while GAS generated backward angular momentum. The difference between muscles was due to their relative contributions to the horizontal and vertical ground reaction forces. Gravity contributed to the body’s angular momentum in early stance and to a lesser extent in late stance, which was counteracted primarily by the plantar flexors. These results may provide insight into balance and movement disorders and provide a basis for developing locomotor therapies that target specific muscle groups.     

Modelling the maximum voluntary joint torque/angular velocity relationship in human movement

The force exerted by a muscle is a function of the activation level and the maximum (tetanic) muscle force. In "maximum" voluntary knee extensions muscle activation is lower for eccentric muscle velocities than for concentric velocities. The aim of this study was to model this "differential activation" in order to calculate the maximum voluntary knee extensor torque as a function of knee angular velocity. Torque data were collected on two subjects during maximal eccentric-concentric knee extensions using an isovelocity dynamometer with crank angular velocities ranging from 50 to 450 degrees s(-1). The theoretical tetanic torque/angular velocity relationship was modelled using a four parameter function comprising two rectangular hyperbolas while the activation/angular velocity relationship was modelled using a three parameter function that rose from submaximal activation for eccentric velocities to full activation for high concentric velocities. The product of these two functions gave a seven parameter function which was fitted to the joint torque/angular velocity data, giving unbiased root mean square differences of 1.9% and 3.3% of the maximum torques achieved. Differential activation accounts for the non-hyperbolic behaviour of the torque/angular velocity data for low concentric velocities. The maximum voluntary knee extensor torque that can be exerted may be modelled accurately as the product of functions defining the maximum torque and the maximum voluntary activation level. Failure to include differential activation considerations when modelling maximal movements will lead to errors in the estimation of joint torque in the eccentric phase and low velocity concentric phase.     

A dynamic optimization technique for predicting muscle forces in the swing phase of gait

The muscle force sharing problem was solved for the swing phase of gait using a dynamic optimization algorithm. For comparison purposes the problem was also solved using a typical static optimization algorithm. The objective function for the dynamic optimization algorithm was a combination of the tracking error and the metabolic energy consumption. The latter quantity was taken to be the sum of the total work done by the muscles and the enthalpy change during the contraction. The objective function for the static optimization problem was the sum of the cubes of the muscle stresses. To solve the problem using the static approach, the inverse dynamics problem was first solved in order to determine the resultant joint torques required to generate the given hip, knee and ankle trajectories. To this effect the angular velocities and accelerations were obtained by numerical differentiation using a low-pass digital filter. The dynamic optimization problem was solved using the Fletcher-Reeves conjugate gradient algorithm, and the static optimization problem was solved using the Gradient-restoration algorithm. The results show influence of internal muscle dynamics on muscle control histories vis a vis muscle forces. They also illustrate the strong sensitivity of the results to the differentiation procedure used in the static optimization approach.     

Modelling of the mesophilic anaerobic co-digestion of olive mill wastewater with olive mill solid waste using anaerobic digestion model No. 1 (ADM1)

The anaerobic digestion model No. 1 (ADM1), conceived by the international water association (IWA) task group for mathematical modelling of anaerobic digestion processes is a structured generic model which includes multiples steps describing biochemical and physicochemical processes encountered in the anaerobic degradation of complex organic substrates and a common platform for further model enhancement and validation of dynamic simulations for a variety of anaerobic processes. In this study the ADM1 model was modified and applied to simulate the mesophilic anaerobic co-digestion of olive mill wastewater (OMW) with olive mill solid waste (OMSW). The ADM1 equations were coded and implemented using the simulation software package MATLAB/Simulink. The most sensitive parameters were calibrated and validated using updated experimental data of our previous work. The results indicated that the ADM1 model could simulate with good accuracy: gas flows, methane and carbon-dioxide contents, pH and total volatile fatty acids (TVFA) concentrations of effluents for various feed concentrations digested at different hydraulic retention times (HRTs) and especially at HRTs of 36 and 24 days. Furthermore, effluent alkalinity and ammonium nitrogen were successfully predicted by the model at HRTs of 12 and 24 days for some feed concentrations.     

Leapfrog Rotational Algorithms

Abstract

A new implicit rotational integrator for the orientation of rigid molecules is introduced, and compared with an existing explicit integrator. Both algorithms are categorised as leapfrogs since the quantities saved between time steps are on the on-step orientation and the mid-step angular momentum. Orientations may be expressed in terms of principal axis vectors or, as in the implementations used here, quaternions. Thermostatted versions of the algorithms as well as conventional energy-conserving versions are described. The algorithms are extensively tested in simulations of liquid water, the aim being to study the effect of increased time steps on a range of measured properties. The implicit algorithm is superior to the explicit algorithm, and can be used with time steps up to 3 fs with energy-conserving dynamics. When thermostatted, it may be used with time steps up to at least 6 fs.     

Dynamic balance during running using running-specific prostheses

Running is beneficial for physical, social, and emotional health, and participating in physical activity, including running, is becoming more popular for people with an amputation. However, this population has a greater risk of falling relative to people without an amputation, which may be a barrier to running. Understanding how dynamic balance is maintained during running is important for removing this barrier. To investigate dynamic balance, we quantified whole-body angular momentum in eight people with a unilateral transtibial amputation (TTA) using running-specific prostheses (RSPs) compared to eight people without TTA during running at 2.5, 3.0, and 3.5 m/s. People with TTA had greater ranges of whole-body angular momentum compared to people without TTA in the frontal and sagittal planes (p < 0.01). These greater ranges resulted from smaller peak medial, lateral, and braking ground reaction forces from the amputated leg compared to the intact leg and people without TTA. Reduced RSP mass relative to the biological leg also influenced whole-body angular momentum as evidenced by smaller ranges of amputated leg angular momentum compared to the intact leg in the frontal and sagittal planes. Smaller amputated leg angular momentum corresponded with smaller contralateral arm angular momentum in the sagittal plane (p < 0.01). People with TTA maintain balance during running with altered muscle coordination and prosthesis characteristics. Restoring mediolateral force generation through prosthetic design advances may help in regulating the frontal plane component of whole-body angular momentum for people with TTA, with potential to improve their ability to maintain balance during running.     

On the estimation of joint kinematics during gait.

In gait analysis, the concepts of Euler and helical (screw) angles are used to define the three-dimensional relative joint angular motion of lower extremities. Reliable estimation of joint angular motion depends on the accurate definition and construction of embedded axes within each body segment. In this paper, using sensitivity analysis, we quantify the effects of uncertainties in the definition and construction of embedded axes on the estimation of joint angular motion during gait. Using representative hip and knee motion data from normal subjects and cerebral palsy patients, the flexion-extension axis is analytically perturbed +/- 15 degrees in 5 degrees steps from a reference position, and the joint angles are recomputed for both Euler and helical angle definitions. For the Euler model, hip and knee flexion angles are relatively unaffected while the ab/adduction and rotation angles are significantly affected throughout the gait cycle. An error of 15 degrees in the definition of flexion-extension axis gives rise to maximum errors of 8 and 12 degrees for the ab/adduction angle, and 10-15 degrees for the rotation angles at the hip and knee, respectively. Furthermore, the magnitude of errors in ab/adduction and rotation angles are a function of the flexion angle. The errors for the ab/adduction angles increase with increasing flexion angle and for the rotation angle, decrease with increasing flexion angle. In cerebral palsy patients with flexed knee pattern of gait, this will result in distorted estimation of ab/adduction and rotation. For the helical model, similar results are obtained for the helical angle and associated direction cosines.(ABSTRACT TRUNCATED AT 250 WORDS)     

Whole-body angular momentum during stair walking using passive and powered lower-limb prostheses*

Individuals with a unilateral transtibial amputation have a greater risk of falling compared to able-bodied individuals, and falling on stairs can lead to serious injuries. Individuals with transtibial amputations have lost ankle plantarflexor muscle function, which is critical for regulating whole-body angular momentum to maintain dynamic balance. Recently, powered prostheses have been designed to provide active ankle power generation with the goal of restoring biological ankle function. However, the effects of using a powered prosthesis on the regulation of whole-body angular momentum are unknown. The purpose of this study was to use angular momentum to evaluate dynamic balance in individuals with a transtibial amputation using powered and passive prostheses relative to able-bodied individuals during stair ascent and descent. Ground reaction forces, external moment arms, and joint powers were also investigated to interpret the angular momentum results. A key result was that individuals with an amputation had a larger range of sagittal-plane angular momentum during prosthetic limb stance compared to able-bodied individuals during stair ascent. There were no significant differences in the frontal, transverse, or sagittal-plane ranges of angular momentum or maximum magnitude of the angular momentum vector between the passive and powered prostheses during stair ascent or descent. These results indicate that individuals with an amputation have altered angular momentum trajectories during stair walking compared to able-bodied individuals, which may contribute to an increased fall risk. The results also suggest that a powered prosthesis provides no distinct advantage over a passive prosthesis in maintaining dynamic balance during stair walking.     

A comparison of muscle energy models for simulating human walking in three dimensions

The popular Hill model for muscle activation and contractile dynamics has been extended with several different formulations for predicting the metabolic energy expenditure of human muscle actions. These extended models differ considerably in their approach to computing energy expenditure, particularly in their treatment of active lengthening and eccentric work, but their predictive abilities have never been compared. In this study, we compared the predictions of five different Hill-based muscle energy models in 3D forward dynamics simulations of normal human walking. In a data-tracking simulation that minimized muscle fatigue, the energy models predicted metabolic costs that varied over a three-fold range (2.45–7.15 J/m/kg), with the distinction arising from whether or not eccentric work was subtracted from the net heat rate in the calculation of the muscle metabolic rate. In predictive simulations that optimized neuromuscular control to minimize the metabolic cost, all five models predicted similar speeds, step lengths, and stance phase durations. However, some of the models predicted a hip circumduction strategy to minimize metabolic cost, while others did not, and the accuracy of the predicted knee and ankle angles and ground reaction forces also depended on the energy model used. The results highlights the need to clarify how eccentric work should be treated when calculating muscle energy expenditure, the difficulty in predicting realistic metabolic costs in simulated walking even with a detailed 3D musculoskeletal model, the potential for using such models to predict energetically-optimal gait modifications, and the room for improvement in existing muscle energy models and locomotion simulation frameworks.     

Estimation of Free Energy Systematic Errors in Molecular Simulations of Globular Proteins Surrounded by Finite Water Clusters. One Center Multipole Expansion of Reaction Field Differences

Free energy calculated in simulations on the atomic level (Monte Carlo or Molecular Dynamics) has a systematic error, if the water shell surrounding a globular protein is finite. The error (“cluster error”) is equal to a difference of free energies obtained in simulations with an infinite and finite water shell. In this work a continuum dielectric model was used to estimate the “cluster error”. A multipole expansion of the estimate was performed for a water shell with a spherical outer boundary. The expansion has very simple form. Each term is a product of two functions, one of them depending only on the charge's conformation, and the other one only on dielectric properties of the system. There are two practical uses of the expansion. First, it may be used to estimate the “cluster error” in a simulation already made; second, it may be used to plan a simulation in such a way that the “cluster error” is minimal. Numerical values of the largest terms in the multipole expansion corresponding to a typical system in simulations of globular proteins are given.     

Offset of rotation centers creates a bias in isokinetics: a virtual model including stiffness or friction

The present paper deals with a virtual model devoted to isokinetics and isometrics assessment of a human muscular group in the common joints, knee, ankle, hip, shoulder, cervical spine, etc. This virtual model with an analytical analysis followed by a numerical simulation is able to predict measurement errors of the joint torque due to offset of rotation centers between the body segment and the ergometer arm. As soon as offset is present, errors increase due to the influence of inertial effects, gravity effects, stiffness due to the limb strapping on the ergometer arm or Coulomb friction between limb and ergometer. The analytical model is written in terms of Lagrange formalism and the numerical model uses ADAMS software adapted to multi-body dynamics simulations. Results of models show a maximal relative error of 11%, for a 10% relative offset between the rotation centers. Inertial contributions are found to be negligible but gravity effects must be discussed in regard to the measured torque. Stiffness or friction effects may also increase the torque error; in particular when offset occurs, it is shown that errors due to friction have to be considered for all torque level while only stiffness effects have to be considered for torque less than 25Nm. This study also emphasizes the influence of the angular range of motion at a given angular position.     

A new approach to the human muscle model

Hill's (1938) two component muscle model is used as basis for digital computer simulation of human muscular contraction by means of an iterative process. The contractile (CC) and series elastic (SEC) components are lumped components of structures which produce and transmit torque to the external environment. The CC is described in angular terms along four dimensions as a series of non-planar torque-angle-angular velocity surfaces stacked on top of each other, each surface being appropriate to a given level of muscular activation. The SEC is described similarly along dimensions of torque, angular stretch, overall muscle angular displacement and activation. The iterative process introduces negligible error and allows the mechanical outcome of a variety of normal muscular contractions to be evaluated parsimoniously. The model allows analysis of many aspects of muscle behaviour as well as optimization studies. Definition of relevant relations should also allow reproduction and prediction of the outcome of contractions in individuals.     

Maximal dismounts from high bar

In men's artistic gymnastics the triple straight somersault dismount from the high bar has yet to be performed in competition. The present study used a simulation model of a gymnast and the high bar apparatus (J. Appl. Biomech. 19(2003a) 119) to determine whether a gymnast could produce the required angular momentum and flight to complete a triple straight somersault dismount. Optimisations were carried out to maximise the margin for error in timing the bar release for a given number of straight somersaults in flight. The amount of rotation potential (number of straight somersaults) the model could produce whilst maintaining a realistic margin for error was determined. A simulation model of aerial movement (J. Biomech.23 (1990) 85) was used to find what would be possible with this amount of rotation potential. The model was able to produce sufficient angular momentum and time in the air to complete a triple straight somersault dismount. The margin for error when releasing the bar using the optimum technique was 28 ms, which is small when compared with the mean margin for error determined for high bar finalists at the 2000 Sydney Olympic Games (55 ms). Although the triple straight somersault dismount is theoretically possible, it would require close to maximum effort and precise timing of the release from the bar. However, when the model was required to have a realistic margin for error, it was able to produce sufficient angular momentum for a double twisting triple somersault dismount.     

A critical examination of the maximum velocity of shortening used in simulation models of human movement

The maximum velocity of shortening of a muscle is an important parameter in musculoskeletal models. The most commonly used values are derived from animal studies; however, these values are well above the values that have been reported for human muscle. The purpose of this study was to examine the sensitivity of simulations of maximum vertical jumping performance to the parameters describing the force–velocity properties of muscle. Simulations performed with parameters derived from animal studies were similar to measured jump heights from previous experimental studies. While simulations performed with parameters derived from human muscle were much lower than previously measured jump heights. If current measurements of maximum shortening velocity in human muscle are correct, a compensating error must exist. Of the possible compensating errors that could produce this discrepancy, it was concluded that reduced muscle fibre excursion is the most likely candidate.     

How early reactions in the support limb contribute to balance recovery after tripping

Tripping causes a forward angular momentum that has to be arrested to prevent a fall. The support limb, contralateral to the obstructed swing limb, can contribute to an adequate recovery by providing time and clearance for proper positioning of the recovery limb, and by restraining the angular momentum of the body during push-off. The present study investigated how such a contribution is achieved by the support limb in terms of response times and muscle moment generation, in order to provide more insight in the requirements for successful recovery after tripping. Twelve young adults repeatedly walked over a platform in which 21 obstacles were hidden. Each subject was tripped over one of these obstacles during mid-swing in at least five trials. Kinematics, dynamics and muscle activity were measured. Very rapid responses were seen in the muscles of the support limb (approximately 65 ms), causing fast increases in muscle moments in the joints during the primary phase of recovery. Especially a large ankle plantar flexion moment (204 Nm), a knee flexion moment (-54 Nm) and a hip extension moment (52 Nm), generated by triceps surae and hamstring muscle activity, brought about the necessary push-off reaction and simultaneously caused a restraining of the forward angular momentum of the body. These required joint moments could be a problem for the elderly, who might not be able to generate such powerful moments. Strength training in these muscle groups may be indicated in elderly subjects to reduce the risk of falling after a trip.