The purposes of this investigation were (a) to predict individual muscle forces for maximum effort knee extension exercises using nonlinear optimization mathematics and different objective functions and (b) to estimate the accuracy of these predicted force magnitudes.

One male subject executed maximum effort knee extension exercises on a CYBEX II machine. The knee joint torques (T_k) were calculated from the torques measured by the CYBEX machine (T_c) and film records of the knee extension experiment.

The subject's right leg was modeled as a system consisting of rigid bodies connected by smooth joints. With the help of this model the knee joint torques (T_km) were calculated again. To reduce the differences between T_k and T_km, an optimization algorithm with the objective to minimize the function

[equation]

l was used to obtain the final model.

The forces for the quadriceps muscles were then predicted using a nonlinear optimization approach and three different objective functions:​

[3 equations]

To evaluate the results, the forces predicted were compared to the forces obtained using the final model -- the model forces -- and assuming that each muscle was maximally active throughout the experimental movements.

The force predictions using the first objective function yielded reasonable results for slow movements with a small range of motion. The force predictions using the second and third objective functions yielded reasonable results for slow and fast movements and for the whole range of motion of the knee extensions as long as the model forces for the individual muscles were of similar magnitude.

The results of this study were obtained for a specific type of muscle contraction -- a maximum effort concentric contraction. This was necessary to permit comparison of the forces obtained using the nonlinear optimization algorithm and the model forces and so get an estimate of the accuracy of the predicted forces. Future work should concentrate on testing whether the findings of this study may be generalized to submaximal concentric and eccentric contractions of human muscles.