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## Forward Dynamics of Two-Dimensional Skeletal Models. A Newton-Euler Approach

Mathematical modeling and computer simulation play an increasingly important role in the search for answers to questions that cannot be addressed experimentally. One of the biggest challenges in forward simulation of the movements of the musculoskeletal system is finding an optimal control strategy. It is not uncommon for this type of optimization problem that the segment dynamics need to be calculated millions of times. In addition, these calculations typically consume a large part of the CPU time during forward movement simulations. As numerous human movements are two-dimensional (2-D) to a reasonable approximation, it is extremely convenient to have a dedicated, computational efficient method for 2-D movements. In this paper we shall present such a method. The main goal is to show that a systematic approach can be adopted which allows for both automatic formulation and solution of the equations of kinematics and dynamics, and to provide some fundamental insight in the mechanical theory behind forward dynamics problems in general. To illustrate matters, we provide for download an example implementation of the main segment dynamics algorithm, as well as a complete implementation of a model of human sprint cycling.

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## Three-Dimensional Dynamic Analysis of Wheelchair Propulsion

A three-dimensional (3-D) inverse dynamic model of wheelchair propulsion was developed using the Newton-Euler method based on body coordinate systems. With this model, the arm was assumed to be three rigid segments (hand, forearm, and upper arm) connected by the wrist, elbow, and shoulder joints. A symbolic method was adopted to generate the equations of motion. The model was used to compute the joint forces and moments based on the inputs obtained from a 3-D motion analysis system, which included an instrumented wheelchair, video cameras, and a data acquisition system. The linear displacements of markers placed on the joints were measured and differentiated to obtain their velocities and accelerations. Three-dimensional contact forces and moments from hand to handrim were measured and used to calculate joint forces and moments of the segments.

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## Neuromusculoskeletal Modeling: Estimation of Muscle Forces and Joint Moments and Movements from Measurements of Neural Command

This paper provides an overview of forward dynamic neuromusculoskeletal modeling. The aim of such models is to estimate or predict muscle forces, joint moments, and/or joint kinematics from neural signals. This is a four-step process. In the first step, muscle activation dynamics govern the transformation from the neural signal to a measure of muscle activation—a time varying parameter between 0 and 1. In the second step, muscle contraction dynamics characterize how muscle activations are transformed into muscle forces. The third step requires a model of the musculoskeletal geometry to transform muscle forces to joint moments. Finally, the equations of motion allow joint moments to be transformed into joint movements. Each step involves complex nonlinear relationships. The focus of this paper is on the details involved in the first two steps, since these are the most challenging to the biomechanician. The global process is then explained through applications to the study of predicting isometric elbow moments and dynamic knee kinetics.

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## A Multisegment Dynamic Model of Ski Jumping

This paper presents a planar, four-segment, dynamic model for the flight mechanics of a ski jumper. The model consists of skis, legs, torso and head, and arms. Inputs include net joint torques that are used to vary the relative body configurations of the jumper during flight. The model also relies on aerodynamic data from previous wind tunnel tests that incorporate the effects of varying body configuration and orientation on lift, drag, and pitching moment. A symbolic manipulation program, “Macsyma,” is used to derive the equations of motion automatically. Experimental body segment orientation data during the flight phase are presented for three ski jumpers which show how jumpers of varying ability differ in flight and demonstrate the need for a more complex analytical model than that previously presented in the literature. Simulations are presented that qualitatively match the measured trajectory for a good jumper. The model can be used as a basis for the study of optimal jumper behavior in flight which maximizes jump distance.

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## Shorter Oars Are More Effective

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## External Effects in the 400-m Hurdles Race

A mathematical model based on a differential equation of motion is used to simulate the 400-m hurdles race for men and women. The model takes into account the hurdler’s stride pattern, the hurdle clearance, and aerobic and anaerobic components of the propulsive force of the athlete, as well as the effects of wind resistance, altitude of the venue, and curvature of the track. The model is used to predict the effect on race times of different wind conditions and altitudes. The effect on race performance of the lane allocation and the efficiency of the hurdle clearance is also predicted. The most favorable wind conditions are shown to be a wind speed no greater than 2 m/s assisting the athlete in the back straight and around the second bend. The outside lane (lane 8) is shown to be considerably faster than the favored center lanes. In windless conditions, the advantage can be as much as 0.15 s for men and 0.12 s for women. It is shown that these values are greatly affected by the wind conditions.

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## Three-Dimensional Bobsled Turning Dynamics

We have developed a mathematical model and computer simulation of three-dimensional bobsled turning. It is based on accurate descriptions of existing or hypothetical tracks and on dynamic equations of motion including gravitational, normal, lift, drag, ice friction, and steering forces. The three-dimensional track surface model uses cubic spline geometric modeling and approximation techniques. The position of the sled on the track is specified by the two variables α and β in the along-track and cross-track directions, differential equations for which govern the possible motions of the sled. The model can be used for studies involving (a) track design, (b) calculation of optimal driver control strategies, and (c) as the basis for a real-time bobsled simulator. It can provide detailed quantitative information (e.g., splits for individual turns) that is not available in runs at actual tracks. The model also allows for comparison of driver performance with the numerically computed optimum performance, and for safe experimentation with risky driving strategies.

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## Inverse Dynamics Modeling of Paralympic Wheelchair Curling

m and ψ m a x m . Inverse Dynamics Inverse dynamics analysis is a mathematical technique through which resultant forces and moments about individual joints are calculated by solving the Newton-Euler equations of motion given the kinematics and inertial parameters of adjacent body segments. 3

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## Mechanisms of Performance Improvements Due to a Leading Teammate During Uphill Cycling

and resistive force when a leader follows a teammate (drafting effect) would represent the main biomechanical benefit of cycling performance. 1 However, the biomechanical equations of motion demonstrate that, when cycling uphill, most of the resistive force is due to gravitational resistance. 2 , 3

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## The Effects of Filter Cutoff Frequency on Musculoskeletal Simulations of High-Impact Movements

If marker and force platform data are low-pass filtered at different cutoff frequencies, inconsistencies can arise in the equations of motion. This occurs because the peak segmental accelerations are attenuated more at impact than the peak ground reaction forces (GRFs). These inconsistencies create