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Curve fitting problem: torque–velocity relationship with polynomials and Boltzmann sigmoid functions

Milanov P., Koroleova G., Mavrevski R., Pencheva N.

Acta of Bioengineering and Biomechanics

2018

Vol. 20, no. 1

169--184

The aim of this study was to investigate the curve fitting and model selection problem of the torque–velocity relationship of elbow flexors and extensors in untrained females. The second goal was to determine the optimal models in different function classes and the best, among the optimal ones. Lastly, test the best models to predict the torque were tested. Methods: Using the polynomials (second – fourth degree) and Boltzmann sigmoid functions, and a different presentation of data points (averages, a point cloud, etc.), we determined the optimal models by both error criteria: minimum residual sum of squares and minimum of the maximal absolute residue. To assess the best models, we applied Akaike and Bayesian information criteria, Hausdorff distance and the minimum of the smallest maximal absolute residue and the predictive torque–velocity relationships of the best models with torque values, calculated beyond the experimental velocity interval. Results: The application of different error and model selection criteria showed that the best models in the majority of cases were polynomials of fourth degree, with some exceptions from second and third degree. The criteria values for the optimal Boltzmann sigmoids were very close to those of the best polynomial models. However, the predicted torque–velocity relationships had physiological behavior only in Boltzmann’s sigmoid functions, and their parameters had a clear interpretation. Conclusion: The results obtained suggest that the Boltzmann sigmoid functions are suitable for modeling and predicting of the torque–velocity relationship of elbow flexors and extensors in untrained females, as compared to polynomials, and their curves are physiologically relevant.

Evaluation of the effectiveness of training on a machine with a variable-cam

Urbanik C., Staniszewski M., Mastalerz A., Karczewska M., Lutosławska G., Iwańska D., Madej A., Ostrowska E., Gwarek L., Tkaczyk J.

Acta of Bioengineering and Biomechanics

2013

Vol. 15, no. 4

93--102

The aim of the study was to assess the effectiveness of the training of elbow flexors through the use of 2 machines, one of which was equipped with a disc plate of constant radius, the other one with a variable-cam having a radius adjustable to muscle strength. The experiment included 45 men divided into 3 equal groups: training group A (variable-cam), training group B (circle), and control group C. The training lasted for 8 weeks, 3 times a week. In order to control the effects, the values of peak torque and power of the flexor muscles of the elbow were isokinetically measured for the angular velocities of 30°/s and 60°/s. Also taken were anthropometric measurements of the arm and the creatine kinase (CK) activity in the blood plasma. As a result of the training, significant increases of biomechanical values were noted only in group A: power increased over 20%, the peak torque over 14%. After the training, significant increases of arm circumference in the relaxed position were noted in group A (17 mm), as well as in group B (11 mm). Also, some changes in CK activity were observed between Monday and Friday in a training week. On the basis of the experimental measurements, it may be ascertained that training elbow flexor muscles on a machine with a variable-cam is more efficient for increases in strength and power, as well as for some anthropometric parameters, than training on a machine with a disc plate.