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

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.

Appendix: Notations for wrist orientation

Endpoint space representations

The 3D orientation of a rigid body such as the human wrist can be described by means of a 3 × 3 rotation matrix R (satisfying ortho-normality R ^T R = I and ‘right-handedness’ det R =  +1). A rotation is physically determined once the rotation axis $\user2{n}$ ($|\user2{n}|=1$) and the rotation angle θ are known, thus can be described by a rotation vector $\user2{r} = \theta \user2{n}$. The rotation matrix R corresponding to a rotation vector $\user2{r}$ can be computed via the Rodrigues’ formula (Murray et al. 1994):

$$ R = \exp(\widehat{\user2{r}}) = I +\sin|\user2{r}|\frac{\widehat{\user2{r}}}{|\user2{r}|}+ (1-\cos|\user2{r}|)\frac{\widehat{\user2{r}}^2}{|\user2{r}|^2}, $$

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where the skew-symmetric matrix $\widehat{\user2{r}}$ is defined through:

$$ \widehat{\cdot}:\user2{r}=\left[\begin{array}{l} r_x\\r_y\\r_z\\ \end{array}\right]\longrightarrow\left[\begin{array}{lll}0&-r_z &r_y\\ r_z &0 &-r_x\\ -r_y &r_x &0\\ \end{array}\right]=\widehat{\user2{r}}. $$

Conversely, for a given rotation matrix R, the corresponding rotation vector can be computed as:

$$ \user2{r} = \log_\vee (R) = \frac{\theta}{2\sin \theta} \left[\begin{array}{l} R_{3,2}-R_{2,3}\\R_{1,3}-R_{3,1}\\R_{2,1}-R_{1,2}\\ \end{array}\right] $$

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where θ = arccos((trace(R) − 1)/2), valid for θ < π.

Forward kinematics

Following the axes conventions of Wu (2005), the wrist orientation R can be computed as the ordered product of three rotations:

$$ R (\varvec{\theta}) = \exp(- \widehat{\user2{e}}_{\user2{x}} \theta^{\rm PS}) \exp(\widehat{\user2{e}}_{\user2{z}}\theta^{\rm FE}) \exp(\widehat{\user2{e}}_{\user2{y}}\theta^{\rm RUD}) $$

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and the corresponding rotation vector is:

$$ FK(\varvec{\theta}):= \log_\vee \left(R (\varvec{\theta})\right). $$

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Inverse kinematics

If the wrist orientation is given by the rotation matrix R, the joint angles $\varvec{\theta}=[\theta^{\rm RUD}\,\theta^{\rm FE}\,\theta^{\rm PS}]^T$ can be computed as the Euler angles (Shoemake 1985). Although, in general, there are several solutions, the biomechanical range of motion of the wrist (see Table 1) yields a unique joint angles configuration for any reachable orientation:

$$ IK(\user2{r}) = \varvec{\theta}= \left[\begin{array}{l} \hbox{atan2}\left(R_{1,3}, R_{1,1}\right)\\ \hbox{arcsin}\left( - R_{1,2} \right)\\ \hbox{atan2}\left( -R_{3,2}, R_{2,2} \right)\\ \end{array}\right]. $$

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