The ordinary differential equations for musculoskeletal dynamics are often numerically stiff and highly nonlinear. Consequently, simulations require small time steps, and optimal control problems are slow to solve and have poor convergence. In this paper, we present an implicit formulation of musculoskeletal dynamics, which leads to new numerical methods for simulation and optimal control, with the expectation that we can mitigate some of these problems. A first order Rosenbrock method was developed for solving forward dynamic problems using the implicit formulation. It was used to perform real-time dynamic simulation of a complex shoulder arm system with extreme dynamic stiffness. Simulations had an RMS error of only 0.11 degrees in joint angles when running at real-time speed. For optimal control of musculoskeletal systems, a direct collocation method was developed for implicitly formulated models. The method was applied to predict gait with a prosthetic foot and ankle. Solutions were obtained in well under one hour of computation time and demonstrated how patients may adapt their gait to compensate for limitations of a specific prosthetic limb design. The optimal control method was also applied to a state estimation problem in sports biomechanics, where forces during skiing were estimated from noisy and incomplete kinematic data. Using a full musculoskeletal dynamics model for state estimation had the additional advantage that forward dynamic simulations, could be done with the same implicitly formulated model to simulate injuries and perturbation responses. While these methods are powerful and allow solution of previously intractable problems, there are still considerable numerical challenges, especially related to the convergence of gradient-based solvers.

Recent data highlight that competitive skiers face a high risk of injuries especially during off-balance jump landing maneuvers in downhill skiing. The purpose of the present study was to develop a musculo-skeletal modeling and simulation approach to investigate the cause-and-effect relationship between a perturbed landing position, i.e., joint angles and trunk orientation, and the peak force in the anterior cruciate ligament (ACL) during jump landing. A two-dimensional musculo-skeletal model was developed and a baseline simulation was obtained reproducing measurement data of a reference landing movement. Based on the baseline simulation, a series of perturbed landing simulations (n = 1000) was generated. Multiple linear regression was performed to determine a relationship between peak ACL force and the perturbed landing posture. Increased backward lean, hip flexion, knee extension, and ankle dorsiflexion as well as an asymmetric position were related to higher peak ACL forces during jump landing. The orientation of the trunk of the skier was identified as the most important predictor accounting for 60% of the variance of the peak ACL force in the simulations. Teaching of tactical decisions and the inclusion of exercise regimens in ACL injury prevention programs to improve trunk control during landing motions in downhill skiing was concluded.

This paper compares predictions of metabolic energy expenditure in gait using seven metabolic energy expenditure models to assess their correlation with experimental data. Ground reaction forces, marker data, and pulmonary gas exchange data were recorded for six walking trials at combinations of two speeds, 0.8 m/s and 1.3 m/s, and three inclines, -8% (downhill), level, and 8% (uphill). The metabolic cost, calculated with the metabolic energy models was compared to the metabolic cost from the pulmonary gas exchange rates. A repeated measures correlation showed that all models correlated well with experimental data, with correlations of at least 0.9. The model by Bhargava et al. (J Biomech, 2004: 81-88) and the model by Lichtwark and Wilson (J Exp Biol, 2005: 2831-3843) had the highest correlation, 0.95. The model by Margaria (Int Z Angew Physiol Einschl Arbeitsphysiol, 1968: 339-351) predicted the increase in metabolic cost following a change in dynamics best in absolute terms.

Trajectory optimization with musculoskeletal models can be used to reconstruct measured movements and to predict changes in movements in response to environmental changes. It enables an exhaustive analysis of joint angles, joint moments, ground reaction forces, and muscle forces, among others. However, its application is still limited to simplified problems in two dimensional space or straight motions. The simulation of movements with directional changes, e.g. curved running, requires detailed three dimensional models which lead to a high-dimensional solution space. We extended a full-body three dimensional musculoskeletal model to be specialized for running with directional changes. Model dynamics were implemented implicitly and trajectory optimization problems were solved with direct collocation to enable efficient computation. Standing, straight running, and curved running were simulated starting from a random initial guess to confirm the capabilities of our model and approach: efficacy, tracking and predictive power. Altogether the simulations required 1 h 17 min and corresponded well to the reference data. The prediction of curved running using straight running as tracking data revealed the necessity of avoiding interpenetration of body segments. In summary, the proposed formulation is able to efficiently predict a new motion task while preserving dynamic consistency. Hence, labor-intensive and thus costly experimental studies could be replaced by simulations for movement analysis and virtual product design.