Design and verification of a simple 3D dynamic model of speed skating which mimics observed forces and motions

Advice about the optimal coordination pattern for an individual speed skater, could be addressed by simulation and optimization of a biomechanical speed skating model. But before getting to this optimization approach one needs a model that can reasonably match observed behaviour. Therefore, the objective of this study is to present a verified three dimensional inverse skater model with minimal complexity, which models the speed skating motion on the straights. The model simulates the upper body transverse translation of the skater together with the forces exerted by the skates on the ice. The input of the model is the changing distance between the upper body and the skate, referred to as the leg extension (Euclidean distance in 3 D space). Verification shows that the model mimics the observed forces and motions well. The model is most accurate for the position and velocity estimation (respectively 1.2% and 2.9% maximum residuals) and least accurate for the force estimations (underestimation of 4.5–10%). The model can be used to further investigate variables in the skating motion. For this, the input of the model, the leg extension, can be optimized to obtain a maximal forward velocity of the upper body.     

Coordination of leg muscles during speed skating

Five speed skaters of elite performance level and six speed skaters of trained level were subjected to an inverse dynamical analysis during speed skating. Push-off forces were registered by means of special skates. Myoelectric activity (EMG) of ten leg muscles and cinematographic data were recorded. Linked segment modelling yielded net joint moments and joint powers. The speed skating technique is characterized by a typical horizontal position of the trunk and a suppression of a plantar flexion during the push-off. This technique, necessary to reduce external friction, constrains the transfer of rotation in joints to translation of the mass center of the body. In spite of constrained push-off, the EMG levels of the leg muscles show a proximo-distal temporal order which to a certain extent is comparable to that previously found in an unconstrained vertical jump. This proximo-distal sequence is also reflected by the time courses of the net moment and net power output in hip, knee and ankle joints. The temporal sequence in activation levels of activated muscles is not different between elite and trained speed skaters. The difference in performance level between these groups obviously has an origin in the ability of the elite speed skaters to realise larger net joint moments. Differences in net joint moments and in kinematics result in a higher power output and a lower air frictional force for the elite than for the trained speed skaters.     

An optimisation-based model for full-body upright reaching movements

An optimal simulation 3D model for full-body upright reaching movements was developed using graphic-based modelling tools (SimMechanics) to generate an inverse dynamics model of the skeleton and using parameterisation methods for a sensory motor controller. The adaptive weight coefficient of the cost function based on the final motor task error (i.e. distance between end-effector and target at the end of movement) was used to correct motor task error and physiological measurements (e.g. joint power, centre of mass displacement, etc.). The output of the simulation models using various cost functions were compared to experimental data from 15 healthy participants performing full-body upright reaching movements. The proposed method can reasonably predict full-body voluntary movements in terms of final posture, joint power, and movement of the centre of mass (COM) using simple algebraic calculations of inverse dynamics and forward kinematics instead of the complicated integrals of the forward dynamics. We found that the combination of several control strategies, i.e. minimising end-effector error, total joint power and body COM produced the best fit of the full-body reaching task.     

Improving joint torque calculations: optimization-based inverse dynamics to reduce the effect of motion errors

The accuracy of joint torques calculated from inverse dynamics methods is strongly dependent upon errors in body segment motion profiles, which arise from two sources of noise: the motion capture system and movement artifacts of skin-mounted markers. The current study presents a method to increase the accuracy of estimated joint torques through the optimization of the angular position data used to describe these segment motions. To compute these angular data, we formulated a constrained nonlinear optimization problem with a cost function that minimizes the difference between the known ground reaction forces (GRFs) and the GRF calculated via a top-down inverse dynamics solution. To evaluate this approach, we constructed idealized error-free reference movements (of squatting and lifting) that produced a set of known “true” motions and associated true joint torques and GRF. To simulate real-world inaccuracies in motion data, these true motions were perturbed by artificial noise. We then applied our approach to these noise-induced data to determine optimized motions and related joint torques. To evaluate the efficacy of the optimization approach compared to traditional (bottom-up or top-down) inverse dynamics approaches, we computed the root mean square error (RMSE) values of joint torques derived from each approach relative to the expected true joint torques. Compared to traditional approaches, the optimization approach reduced the RMSE by 54% to 79%. Average reduction due to our method was 65%; previous methods only achieved an overall reduction of 30%. These results suggest that significant improvement in the accuracy of joint torque calculations can be achieved using this approach.     

A geometrical model of speed skating the curves

The centripetal force in speed skating the curves has to be delivered by the push off force which also does the external work to maintain the speed. Based on the geometry of the speed skating oval and the sideward push off characteristics in speed skating, a mathematical model of the power output in skating the curves was deduced. The power required to follow the curve is dependent on the mean speed in the curve, the work per stroke and the radius of the speed skating oval. Measurements (by means of film and video analysis) during the 5000 m races at the European Championships for ladies (n = 16) yielded on the one hand power from the geometrical model and on the other hand power losses due to air- and ice- friction. The difference between power delivered and power lost is used by the skaters to increase their speed. The difference between predicted power and measured power used to increase the kinetic energy of c.g. was only 3% thereby providing strong support for the validity of the model. The analysis suggested that skaters who want to accelerate in the curves should increase their work per stroke. The model can be a useful tool to provide insight into this form of human locomotion and its optimization under competitive conditions.     

A neuromusculoskeletal tracking method for estimating individual muscle forces in human movement

A neuromusculoskeletal tracking (NMT) method was developed to estimate muscle forces from observed motion data. The NMT method combines skeletal motion tracking and optimal neuromuscular tracking to produce forward simulations of human movement quickly and accurately. The skeletal motion tracker calculates the joint torques needed to actuate a skeletal model and track observed segment angles and ground forces in a forward simulation of the motor task. The optimal neuromuscular tracker resolves the muscle redundancy problem dynamically and finds the muscle excitations (and muscle forces) needed to produce the joint torques calculated by the skeletal motion tracker. To evaluate the accuracy of the NMT method, kinematics and ground forces obtained from an optimal control (parameter optimization) solution for maximum-height jumping were contaminated with both random and systematic noise. These data served as input observations to the NMT method as well as an inverse dynamics analysis. The NMT solution was compared to the input observations, the original optimal solution, and a simulation driven by the inverse dynamics torques. The results show that, in contrast to inverse dynamics, the NMT method is able to produce an accurate forward simulation consistent with the optimal control solution. The NMT method also requires 3 orders-of-magnitude less CPU time than parameter optimization. The speed and accuracy of the NMT method make it a promising new tool for estimating muscle forces using experimentally obtained kinematics and ground force data.     

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.     

Knee moment outcomes using inverse dynamics and the cross product function in moderate knee osteoarthritis gait: A comparison study

Inverse dynamics are the cornerstone of biomechanical assessments to calculate knee moments during walking. In knee osteoarthritis, these outcomes have been used to understand knee pathomechanics, but the complexity of an inverse dynamic model may limit the uptake of joint moments in some clinical and research structures. The objective was to determine whether discrete features of the sagittal and frontal plane knee moments calculated using inverse dynamics compare to knee moments calculated using a cross product function. Knee moments from 74 people with moderate knee osteoarthritis were assessed after ambulating at a self-selected speed on an instrumented dual belt treadmill. Standardized procedures were used for surface marker placement, gait speed determination and data processing. Net external frontal and sagittal plane knee moments were calculated using inverse dynamics and the three-dimensional position of the knee joint center with respect to the center of pressure was crossed with the three-dimensional ground reaction forces in the cross product function. Correlations were high between outcomes of the moment calculations (r > 0.9) and for peak knee adduction moment, knee adduction moment impulse and difference between peak flexion and extension moments, the cross product function resulted in absolute values less than 10% of those calculated using inverse dynamics in this treadmill walking environment. This computational solution may allow the integration of knee moment calculations to understand knee osteoarthritis gait without data collection or computational complexity.     

A sensitivity analysis of the calculation of mechanical output through inverse dynamics: a computer simulation study

The purpose of this study was to systematically determine the effect of experimental errors on the work output calculated using two different methods of inverse dynamics during vertical jumping: (a) the conventional (rotational) method and (b) the translational method. A two-dimensional musculoskeletal model was used to generate precisely known kinematics. Next, the location of each joint center (JC) and the location of each segment's center of mass (CM) were manipulated by +/-10% of segment length to simulate errors in the location of joint centers (delta JC) and errors in the location of segment's center of mass (delta CM), respectively. Work output was subsequently calculated by applying the two methods of inverse dynamics to the manipulated kinematic data. The results showed that the translational method of inverse dynamics was less sensitive (up to 13% error in total work output) to delta JC and delta CM than the rotational method (up to 28% error in total work output). The rotational method of inverse dynamics was particularly sensitive to simulated errors in JC.     

Estimation of external contact loads using an inverse dynamics and optimization approach: General method and application to sit-to-stand maneuvers

This paper presents a general method to estimate unmeasured external contact loads (ECLs) acting on a system whose kinematics and inertial properties are known. This method is dedicated to underdetermined problems, e.g. when the system has two or more unmeasured external contact wrenches. It is based on inverse dynamics and a quadratic optimization, and is therefore relatively simple, computationally cost effective and robust. Net joint loads (NJLs) are included as variables of the problem, and thus could be estimated in the same procedure as the ECL and be used within the cost function.     

Are natural coordinates a useful tool in modeling planar biomechanical linkages?

The paper analyzes the use of natural coordinates in modeling and simulation of musculoskeletal models of the human body. A biomechanical model of the lower extremity of the human body was constructed for the analysis. It consists of three anatomical segments described by eight natural coordinates located at the joints. The developed model was applied to solve three classic dynamics problems of human motion: inverse dynamics, direct dynamics, and static optimization. The analysis covers the raising of a leg together with; the time characteristics of the resultant net torques at the basic joints of the leg (inverse dynamics), the time histories of natural coordinates and their velocities (direct dynamics) as well as the time-varying muscle force patterns (static optimization). In order to check the numerical efficiency of modeling in the natural coordinates' environment the results were compared with the ones received through generalized coordinates. Some conclusions drawn from this comparison and final remarks referring to the biomechanical identification of the analyzed motor task were included in the paper.     

Comparison of methods for the calculation of energy storage and return in a dynamic elastic response prosthesis

The standard method used to calculate the ankle joint power contains deficiencies when applied to dynamic elastic response prosthetic feet. The standard model, using rotational power and inverse dynamics, assumes a fixed joint center and cannot account for energy storage, dissipation, and return. This study compared the standard method with new analysis models. First, assumptions of inverse dynamics were avoided by directly measuring ankle forces and moments. Second, the ankle center of rotation was corrected by including translational power terms. Analysis with below-knee amputees revealed that the conventional method overestimates ankle forces and moments as well as prosthesis energy storage and return. Results for efficiency of energy return were varied. Large differences between models indicate the standard method may have serious inadequacies in the analysis of certain prosthetic feet. This research is the first application of the new models to prosthetic feet, and suggests the need for additional research in gait analysis with energy-storing prostheses.     

Lower-limb joint work and power are modulated during load carriage based on load configuration and walking speed

Soldiers regularly transport loads weighing >20 kg at slow speeds for long durations. These tasks elicit high energetic costs through increased positive work generated by knee and ankle muscles, which may increase risk of muscular fatigue and decrease combat readiness. This study aimed to determine how modifying where load is borne changes lower-limb joint mechanical work production, and if load magnitude and/or walking speed also affect work production. Twenty Australian soldiers participated, donning a total of 12 body armor variations: six different body armor systems (one standard-issue, two commercially available [cARM1-2], and three prototypes [pARM1-3]), each worn with two different load magnitudes (15 and 30 kg). For each armor variation, participants completed treadmill walking at two speeds (1.51 and 1.83 m/s). Three-dimensional motion capture and force plate data were acquired and used to estimate joint angles and moments from inverse kinematics and dynamics, respectively. Subsequently, hip, knee, and ankle joint work and power were computed and compared between armor types and walking speeds. Positive joint work over the stance phase significantly increased with walking speed and carried load, accompanied by 2.3–2.6% shifts in total positive work production from the ankle to the hip (p < 0.05). Compared to using cARM1 with 15 kg carried load, carrying 30 kg resulted in significantly greater hip contribution to total lower-limb positive work, while knee and ankle work decreased. Substantial increases in hip joint contributions to total lower-limb positive work that occur with increases in walking speed and load magnitude highlight the importance of hip musculature to load carriage walking.     

Shoulder joint kinetics and dynamics during underwater forward arm elevation

Aquatic exercises are widely implemented into rehabilitation programs. However, both evaluating their mechanical demands on the musculoskeletal system and designing protocols to provide progressive loading are difficult tasks. This study reports for the first time shoulder joint kinetics and dynamics during underwater forward arm elevation performed at speeds ranging from 22.5 to 90°/s. Net joint moments projected onto anatomical axes of rotation, joint power, and joint work were calculated in 18 participants through a novel approach coupling numerical fluid flow simulations and inverse dynamics. Joint dynamics was revealed from the 3D angle between the joint moment and angular velocity vectors, identifying three main functions—propulsion, stabilization, and resistance. Speeds <30°/s necessitated little to no power at all, whereas peaks about 0.20 W⋅kg⁻¹ were seen at 90°/s. As speed increased, peak moments were up to 61 × higher at 90 than at 22.5°/s, (1.82 ± 0.12%BW⋅AL vs 0.03 ± 0.01%BW⋅AL, P < 0.038). This was done at the expense of a substantial decrease in the joint moment contribution to joint stability though, which goes against the intuition that greater stabilization is required to protect the shoulder from increasing loads. Slow arm elevations (<30°/s) are advantageous for joint mobility gain at low mechanical solicitation, whereas the intensity at 90°/s is high enough to stimulate muscular endurance improvements. Simple predictive equations of shoulder mechanical loading are provided. They allow for easy design of progressive protocols, either for the postoperative shoulder or the conditioning of athlete targeting very specific intensity regions.     

Prediction of antagonistic muscle forces using inverse dynamic optimization during flexion/extension of the knee.

This paper examined the feasibility of using different optimization criteria in inverse dynamic optimization to predict antagonistic muscle forces and joint reaction forces during isokinetic flexion/extension and isometric extension exercises of the knee. Both quadriceps and hamstrings muscle groups were included in this study. The knee joint motion included flexion/extension, varus/valgus, and internal/external rotations. Four linear, nonlinear, and physiological optimization criteria were utilized in the optimization procedure. All optimization criteria adopted in this paper were shown to be able to predict antagonistic muscle contraction during flexion and extension of the knee. The predicted muscle forces were compared in temporal patterns with EMG activities (averaged data measured from five subjects). Joint reaction forces were predicted to be similar using all optimization criteria. In comparison with previous studies, these results suggested that the kinematic information involved in the inverse dynamic optimization plays an important role in prediction of the recruitment of antagonistic muscles rather than the selection of a particular optimization criterion. Therefore, it might be concluded that a properly formulated inverse dynamic optimization procedure should describe the knee joint rotation in three orthogonal planes.     

Estimation of muscle forces in gait using a simulation of the electromyographic activity and numerical optimization

Clinical gait analysis provides great contributions to the understanding of gait patterns. However, a complete distribution of muscle forces throughout the gait cycle is a current challenge for many researchers. Two techniques are often used to estimate muscle forces: inverse dynamics with static optimization and computer muscle control that uses forward dynamics to minimize tracking. The first method often involves limitations due to changing muscle dynamics and possible signal artefacts that depend on day-to-day variation in the position of electromyographic (EMG) electrodes. Nevertheless, in clinical gait analysis, the method of inverse dynamics is a fundamental and commonly used computational procedure to calculate the force and torque reactions at various body joints. Our aim was to develop a generic musculoskeletal model that could be able to be applied in the clinical setting. The musculoskeletal model of the lower limb presents a simulation for the EMG data to address the common limitations of these techniques. This model presents a new point of view from the inverse dynamics used on clinical gait analysis, including the EMG information, and shows a similar performance to another model available in the OpenSim software. The main problem of these methods to achieve a correct muscle coordination is the lack of complete EMG data for all muscles modelled. We present a technique that simulates the EMG activity and presents a good correlation with the muscle forces throughout the gait cycle. Also, this method showed great similarities whit the real EMG data recorded from the subjects doing the same movement.     

Comparative analysis of methods for estimating arm segment parameters and joint torques from inverse dynamics

A common problem in the analyses of upper limb unfettered reaching movements is the estimation of joint torques using inverse dynamics. The inaccuracy in the estimation of joint torques can be caused by the inaccuracy in the acquisition of kinematic variables, body segment parameters (BSPs), and approximation in the biomechanical models. The effect of uncertainty in the estimation of body segment parameters can be especially important in the analysis of movements with high acceleration. A sensitivity analysis was performed to assess the relevance of different sources of inaccuracy in inverse dynamics analysis of a planar arm movement. Eight regression models and one water immersion method for the estimation of BSPs were used to quantify the influence of inertial models on the calculation of joint torques during numerical analysis of unfettered forward arm reaching movements. Thirteen subjects performed 72 forward planar reaches between two targets located on the horizontal plane and aligned with the median plane. Using a planar, double link model for the arm with a floating shoulder, we calculated the normalized joint torque peak and a normalized root mean square (rms) of torque at the shoulder and elbow joints. Statistical analyses quantified the influence of different BSP models on the kinetic variable variance for given uncertainty on the estimation of joint kinematics and biomechanical modeling errors. Our analysis revealed that the choice of BSP estimation method had a particular influence on the normalized rms of joint torques. Moreover, the normalization of kinetic variables to BSPs for a comparison among subjects showed that the interaction between the BSP estimation method and the subject specific somatotype and movement kinematics was a significant source of variance in the kinetic variables. The normalized joint torque peak and the normalized root mean square of joint torque represented valuable parameters to compare the effect of BSP estimation methods on the variance in the population of kinetic variables calculated across a group of subjects with different body types. We found that the variance of the arm segment parameter estimation had more influence on the calculated joint torques than the variance of the kinematics variables. This is due to the low moments of inertia of the upper limb, especially when compared with the leg. Therefore, the results of the inverse dynamics of arm movements are influenced by the choice of BSP estimation method to a greater extent than the results of gait analysis.     

A 'cheap' optimal control approach to estimate muscle forces in musculoskeletal systems

This paper shows a new method to estimate the muscle forces in musculoskeletal systems based on the inverse dynamics of a multi-body system associated optimal control. The redundant actuator problem is solved by minimizing a time-integral cost function, augmented with a torque-tracking error function, and muscle dynamics is considered through differential constraints. The method is compared to a previously implemented human posture control problem, solved using a Forward Dynamics Optimal Control approach and to classical static optimization, with two different objective functions. The new method provides very similar muscle force patterns when compared to the forward dynamics solution, but the computational cost is much smaller and the numerical robustness is increased. The results achieved suggest that this method is more accurate for the muscle force predictions when compared to static optimization, and can be used as a numerically 'cheap' alternative to the forward dynamics and optimal control in some applications.     

Tibio-femoral joint contact forces in sheep

Although the sheep has become a standard model for understanding the mechanical conditions that occur after injury and investigating surgical treatments such as osteochondral defect healing and ligament reconstruction, no study has yet evaluated the contact forces that occur in the sheep tibio-femoral joint in vivo. In this study, bone pins, together with reflective markers, were used to measure the 3D kinematics of three sheep hind limbs, simultaneously with the ground reaction forces during repetitions of gait trials. Joint contact forces were then calculated using inverse dynamics and optimisation techniques. Whilst average peak axial tibio-femoral contact forces of 2.1 body weight (BW) were calculated across the 3 sheep, only small medio-lateral and antero-posterior shear forces, averaging 0.7 BW, were determined. Average knee flexion angles ranging from 49 degrees to 70 degrees were observed. From the forces determined in this study, we have provided a better understanding of the mechanical loading environment that occurs in sheep. This has important implications for the interpretation of knee studies in quadrupeds and their relevance to the clinical situation.     

Ice friction during speed skating.

During speed skating, the external power output delivered by the athlete is predominantly used to overcome the air and ice frictional forces. Special skates were developed and used to measure the ice frictional forces during actual speed skating. The mean coefficients of friction for the straights and curves were, respectively, 0.0046 and 0.0059. The minimum value of the coefficient of ice friction was measured at an ice surface temperature of about -7 degrees C. It was found that the coefficient of friction increases with increasing speed. In the literature, it is suggested that the relatively low friction in skating results from a thin film of liquid water on the ice surface. Theories about the presence of water between the rubbing surfaces are focused on the formation of water by pressure-melting, melting due to frictional heating and on the 'liquid-like' properties of the ice surface. From our measurements and calculations, it is concluded that the liquid-like surface properties of ice seem to be a reasonable explanation for the low friction during speed skating.