Abstract: Few computational models have addressed the spatiotemporal features of unconstrained three-dimensional (3D) arm motion. Empirical observations made on hand paths, speed profiles, and arm postures during point-to-point movements led to the assumption that hand path and arm posture are independent of movement speed, suggesting that the geometric and temporal properties of movements are decoupled. In this study, we present a computational model of 3D movements for an arm with four degrees of freedom based on the assumption that optimization principles are separately applied at the geometric and temporal levels of control. Geometric properties (path and posture) are defined in terms of geodesic paths with respect to the kinetic energy metric in the Riemannian configuration space. Accordingly, a geodesic path can be generated with less muscular effort than on any other, nongeodesic path, because the sum of all configuration-speed-dependent torques vanishes. The temporal properties of the movement (speed) are determined in task space by minimizing the squared jerk along the selected end-effector path. The integration of both planning levels into a single spatiotemporal representation simplifies the control of arm dynamics along geodesic paths and results in movements with near minimal torque change and minimal peak value of kinetic energy. Thus, the application of Riemannian geometry allows for a reconciliation of computational models previously proposed for the description of arm movements. We suggest that geodesics are an emergent property of the motor system through the exploration of dynamical space. Our data validated the predictions for joint trajectories, hand paths, final postures, speed profiles, and driving torques.

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.

Abstract: BACKGROUND: Therapists specializing in handwriting difficulties in children often address motor problems including both proximal and distal movements in the upper extremity. Kinematic measures can be used to investigate various aspects of handwriting. This study examined differences in movement patterns in proximal and distal joints of the upper extremity during graphomotor tasks between typically developing children with and without handwriting problems. Additionally, it explored relationships between movement patterns, speed, and legibility of writing. METHODS: Forty-one children, aged 7-11 years, were assessed with the Aleph Aleph Ktav Yad Hebrew Handwriting assessment and the Beery Test of Visual Motor Integration and, based on their scores, were divided into a research group (with handwriting difficulties) and a control group (without handwriting difficulties). Upper extremity joint movement patterns were analyzed with a motion capture system. Differences in the quality of shapes traced and copied on a graphics tablet positioned horizontally and vertically were compared. Between-group differences and relationships with speed and legibility were analyzed. RESULTS: In both groups, there was greater movement in the distal compared to the proximal joints, greater movement when performing the task in a horizontal compared to a vertical plane, and greater movement when tracing than copying. Joint movements in the arm executed scaled-down versions of the shapes being drawn. While the amount of joint displacement was similar between groups, children in the research group showed greater dissimilarity between the drawn shape and the shape produced by the proximal joints. Finally, the drawing measure on the tablet was a significant predictor of legibility, speed of writing, visual motor integration and motor coordination, whereas the dissimilarity measure of joint movement was a significant predictor of speed of writing and motor coordination. CONCLUSIONS: This study provides support for the role of the distal upper extremity joints in the writing process and some guidance to assist clinicians in devising treatment strategies for movement-related handwriting problems. While we observed differences in proximal joint movements between the children with and without handwriting difficulties, the extent to which they are responsible for the differences in drawing quality remains to be determined. Further studies should use a similar methodology to examine additional tasks such as drawing shapes of varying sizes.