Friedman, J., Latash, M. L., & Zatsiorsky, V. M. (2011). Directional variability of the isometric force vector produced by the hand in multi-joint planar tasks. Journal of Motor Behavior, 43(6), 451–463.
Abstract: Numerous studies have examined control of force magnitude, but relatively little research has considered force direction control. In this study, subjects applied isometric forces to a handle and we compared within-trial variability when producing force in different directions. The standard deviation (SD) of the force parallel to the prescribed direction of force production increased linearly with the targeted force level, as did the SD of the force perpendicular to the instructed direction. In contrast, the SD of the angle of force production decreased with increased force level. In the four (of eight) instructed force directions where the endpoint force was generated due to a joint torque in only one joint (either the shoulder or elbow) the principal component axes in force space were well aligned with the prescribed direction of force production. In the other directions, the variance was approximately equal along the two force axes. The variance explained by the first principal component was significantly larger in torque space compared to the force space, and mostly corresponded to positive correlation between the joint torques. Such coordinated changes suggest that the torque variability was mainly due to the variability of the common drive to the muscles serving two joints, although this statement needs to be supported by direct studies of muscle activation in the future.
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Biess, A., Flash, T., & Liebermann, D. G. (2011). Riemannian geometric approach to human arm dynamics, movement optimization, and invariance. Phys Rev E Stat Nonlin Soft Matter Phys, 83(3 Pt 1), 031927.
Abstract: We present a generally covariant formulation of human arm dynamics and optimization principles in Riemannian configuration space. We extend the one-parameter family of mean-squared-derivative (MSD) cost functionals from Euclidean to Riemannian space, and we show that they are mathematically identical to the corresponding dynamic costs when formulated in a Riemannian space equipped with the kinetic energy metric. In particular, we derive the equivalence of the minimum-jerk and minimum-torque change models in this metric space. Solutions of the one-parameter family of MSD variational problems in Riemannian space are given by (reparameterized) geodesic paths, which correspond to movements with least muscular effort. Finally, movement invariants are derived from symmetries of the Riemannian manifold. We argue that the geometrical structure imposed on the arm's configuration space may provide insights into the emerging properties of the movements generated by the motor system.
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