Abstract
We have developed an inverse dynamics model of unrestrained natural reaching movements. Such movements are usually not planar and often involve complex deformation of the shoulder girdle as well as rotary and linear torso motion. Our model takes as its input kinematic data about the positions of the finger, wrist, elbow, left and right acromion processes, and the sternum and produces the torques and forces developed at the shoulder, elbow, and wrist joints. The model can also be used to simulate the consequences of introducing passive torso rotation or linear acceleration on arm movements and to simulate the consequences of applying mechanical perturbations to the reaching limb. It separately quantifies the contributions of inertial forces resulting from torso rotation and translation. In experimental paradigms involving arm movements, different dynamic components can be present such as active or passive torso rotation and translation, external forces and Coriolis forces. Our model provides a means of evaluating the different sources of force and the total muscle force needed to control the trajectory of the arm in their presence.
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Notes
From the equations in Zatsiorsky and Seluyanov (1985) we calculated that for an average person the two transverse momenta of inertia of the upper arm are approximately 140 kg cm2 and 130 kg cm2, respectively, with 20% variation for different body heights and weights. Consequently we model the upper arm as a cylindrical inertial ellipsoid.
The vertical component of the angular momentum is \({\bf h}_{\bf z} \cong \left({I_{{\rm T}} -I_{{\rm T}}} \right)\sin(\alpha)\cos(\alpha)\sin(\vartheta)\dot {\alpha}+ \left({I_{{\rm L}} \sin(\alpha)^2+}{I_{{\rm T}} \cos(\alpha)^2} \right)\left({\dot {\vartheta}^r+\dot {\vartheta}_{{\rm T}} +\omega} \right).\) From the data in Zatsiorsky and Seluyanov (1985), we calculated that I L is about 40 kg cm2 ± 16% across a variety of body heights and weights. Consequently, \(I_{{\rm L}} \approx {I_{{\rm T}}} \mathord{\left/ {\vphantom {{I_{{\rm T}}} 3}} \right. \kern-\nulldelimiterspace} 3.\) For reaching movements in typical paradigms, kinematics variations values are: \(\alpha \approx 55{\text{--}}35^{\circ}, \dot{\alpha}\approx 0{\text{--}}15^{\circ}/{\rm s}, \vartheta^r=0{\text{--}}90^{\circ},\) and \(\dot {\vartheta}^r=0\text{--}70^{\circ}/{\rm s}.\) With these numerical values the angular momentum can be simplified as follows, \({\bf h}_{\bf z} \cong \frac{I_{{\rm T}}}{3}\sin(2\alpha)\sin(\vartheta)\dot {\alpha}+\frac{I_{{\rm T}} }{3}\left({\sin(\alpha)^2+3\cos(\alpha)^2} \right)\left({\dot {\vartheta }^r+\dot {\vartheta}_{{\rm T}} +\omega} \right).\) In addition, the term in \(\dot {\alpha}\) is ten to 20 times smaller than the term including \(\dot {\vartheta}^r\) when the torso is not rotating \((\dot {\vartheta}_{{\rm T}} =\omega =0).\) Therefore, an excellent approximation for the vertical component of the angular momentum is \({\bf h}_{\bf z} \cong \frac{I_{{\rm T}}}{3}\left({\sin(\alpha )^2+3\cos(\alpha)^2} \right) \ \left({\dot {\vartheta}^r+\dot {\vartheta }_{{\rm T}} +\omega} \right)\).
\(\left. {\frac{{\rm d}{\bf h}}{{\rm d}t}} \right|_z =\mathop {\bf h}^\prime_z.\)
All forces F are (9 × 1) arrays.
References
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Acknowledgments
We thank Dr. Marco Quadrelli, Jet Propulsion Laboratory, California Institute of Technology for useful discussions during the development of our arm model. Support was provided by a grant from the National Institutes of Health, R01AR4854601.
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Bortolami, S.B., Pigeon, P., DiZio, P. et al. Dynamics model for analyzing reaching movements during active and passive torso rotation. Exp Brain Res 187, 525–534 (2008). https://doi.org/10.1007/s00221-008-1323-y
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DOI: https://doi.org/10.1007/s00221-008-1323-y