# Pointing with the wrist: a postural model for Donders’ law

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## Abstract

The central nervous system uses stereotypical combinations of the three wrist/forearm joint angles to point in a given (2D) direction in space. In this paper, we first confirm and analyze this Donders’ law for the wrist as well as the distributions of the joint angles. We find that the quadratic surfaces fitting the experimental wrist configurations during pointing tasks are characterized by a subject-specific Koenderink shape index and by a bias due to the prono-supination angle distribution. We then introduce a simple postural model using only four parameters to explain these characteristics in a pointing task. The model specifies the redundancy of the pointing task by determining the one-dimensional task-equivalent manifold (TEM), parameterized via wrist torsion. For every pointing direction, the torsion is obtained by the concurrent minimization of an extrinsic cost, which guarantees minimal angle rotations (similar to Listing’s law for eye movements) and of an intrinsic cost, which penalizes wrist configurations away from comfortable postures. This allows simulating the sequence of wrist orientations to point at eight peripheral targets, from a central one, passing through intermediate points. The simulation first shows that in contrast to eye movements, which can be predicted by only considering the extrinsic cost (i.e., Listing’s law), both costs are necessary to account for the wrist/forearm experimental data. Second, fitting the synthetic Donders’ law from the simulated task with a quadratic surface yields similar fitting errors compared to experimental data.

## Keywords

Human wrist Kinematic redundancy Donders’ law Hopf fibration Multi-objective optimization## Notes

### Acknowledgments

This study was partially funded by the Academic Research Fund (AcRF) Tier1 (RG 40/09), Ministry of Education, Singapore, and by the EU FP7 VIACTORS project. The authors are grateful to Ms Kelly Savin for proofreading the manuscript.

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