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Trajectory of the index finger during grasping

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Abstract

The trajectory of the index finger during grasping movements was compared to the trajectories predicted by three optimization-based models. The three models consisted of minimizing the integral of the weighted squared joint derivatives along the path (inertia-like cost), minimizing torque change, and minimizing angular jerk. Of the three models, it was observed that the path of the fingertip and the joint trajectories, were best described by the minimum angular jerk model. This model, which does not take into account the dynamics of the finger, performed equally well when the inertia of the finger was altered by adding a 20 g weight to the medial phalange. Thus, for the finger, it appears that trajectories are planned based primarily on kinematic considerations at a joint level.

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Acknowledgments

This research was supported in part by the German–Israeli Project Cooperation (DIP) and by the Moross Laboratory at the Weizmann Institute of Science. Tamar Flash is an incumbent of the Dr. Hymie Morros Professorial chair. We thank Armin Biess for his assistance in generating the predicted trajectories.

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Correspondence to Jason Friedman.

Appendix: dynamic equations

Appendix: dynamic equations

Complete details of the derivation can be found in Friedman (2007). The 3 × 3 inertia matrix for a 3 joint finger is given by

$$ \begin{aligned} M_{ 1, 1} & = (2lc_{3} c_{3} l_{2} + l_{2}^{2} + l_{1}^{2} + 2lc_{3} l_{1} c_{23} + lc_{3}^{2} + 2l_{2} l_{1} c_{2} )m_{ 3} \\ & \quad + m_{ 1} lc_{ 1}^{ 2} + I_{z 1} + I_{z 2} + I_{z 3} + 2m_{ 2} c_{ 2} l_{ 1} lc_{ 2} + m_{ 2} lc_{ 2}^{ 2} + l_{ 1}^{ 2} m_{ 2} \\ M_{ 1, 2} & = (l_{2}^{2} + lc_{3}^{2} + 2lc_{3} c_{3} l_{2} + lc_{3} l_{1} c_{23} + l_{2} l_{1} c_{2} )m_{ 3} \\ & \quad + m_{ 2} c_{ 2} l_{ 1} lc_{ 2} + m_{ 2} lc_{ 2}^{ 2} + I_{z 2} + I_{z 3} \\ M_{ 1, 3} & = (lc_{3} + c_{3} l_{2} + l_{1} c_{23} )m_{ 3} lc_{ 3} + I_{z 3} \\ M_{ 2, 2} & = (l_{2}^{2} + lc_{3}^{2} + 2lc_{3} c_{3} l_{2} )m_{ 3} + m_{ 2} lc_{ 2}^{ 2} + I_{z 2} + I_{z 3} \\ M_{ 2, 3} & = (lc_{3} + c_{3} l_{2} )m_{3} lc_{3} + I_{z 3} \\ M_{ 3, 3} & = m_{ 3} lc_{ 3}^{ 2} + I_{z 3} \\ \end{aligned} $$
(15)

where l are the lengths of the phalanges, lc are the lengths to the center of the phalanges, c i is the cosine of joint angle i, c ij is the cosine of angle i + j, s i is the sine of joint angle i, and m are the masses of the phalanges.

The other three terms of the inertia matrix can be found from the symmetry property of the inertia matrix.

The Coriolis and centrifugal forces are given by

$$ \begin{aligned} C_{1,1} & = - \dot{\theta }_{2} l_{1} (m_{2} s_{2} lc_{2} + m_{3} l_{2} s_{2} + m_{3} s_{23} lc_{3} ) - \dot{\theta }_{3} m_{3} (lc_{3} s_{3} l_{2} + l_{1} s_{23} lc_{3} ) \\ C_{1,2} & = - \dot{\theta }_{1} l_{1} (s_{2} m_{2} lc_{2} + m_{3} l_{2} s_{2} + s_{23} m_{3} lc_{3} ) \\ & \quad - \dot{\theta }_{2}l_{1} ( m_{2} s_{2}lc_{2} + m_{3} l_{2} s_{2} + m_{3} s_{23} lc_{3}) - \dot{\theta }_{3} m_{3}( s_{3} l_{2} lc_{3} +l_{1} s_{23} lc_{3} ) \\ C_{1,3} & = - m_{3} lc_{3} (\dot{\theta }_{1} + \dot{\theta }_{2}+ \dot{\theta }_{3}) ( s_{3} l_{2} + l_{1} s_{23} ) \\ C_{2,1} & = \dot{\theta }_{1} (m_{2} s_{2} l_{1} lc_{2} + m_{3} l_{1} l_{2} s_{2} + l_{1} s_{23} m_{3} lc_{3} )-\dot{\theta }_{3} ( m_{3} s_{3} l_{2} lc_{3} ) \\ C_{2,2} & = - \dot{\theta }_{3} m_{3} s_{3} l_{2} lc_{3} \\ C_{2,3} & = - m_{3} s_{3} l_{2} lc_{3} (\dot{\theta }_{1} + \dot{\theta }_{2} + \dot{\theta }_{3}) \\ C_{3,1} & = m_{3} lc_{3}(\dot{\theta }_{1}s_{3} l_{2} + \dot{\theta }_{1}l_{1} s_{23}+\dot{\theta }_{2}s_{3}l_{2} ) \\ C_{3,2} & = m_{3} s_{3} l_{2} lc_{3} (\dot{\theta }_{1} + \dot{\theta }_{2}) \\ C_{3,3} & = 0 \\ \end{aligned} $$

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Friedman, J., Flash, T. Trajectory of the index finger during grasping. Exp Brain Res 196, 497–509 (2009). https://doi.org/10.1007/s00221-009-1878-2

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