End-state comfort and joint configuration variance during reaching

Abstract

This study joined two approaches to motor control. The first approach comes from cognitive psychology and is based on the idea that goal postures and movements are chosen to satisfy task-specific constraints. The second approach comes from the principle of motor abundance and is based on the idea that control of apparently redundant systems is associated with the creation of multi-element synergies stabilizing important performance variables. The first approach has been tested by relying on psychophysical ratings of comfort. The second approach has been tested by estimating variance along different directions in the space of elemental variables such as joint postures. The two approaches were joined here. Standing subjects performed series of movements in which they brought a hand-held pointer to each of four targets oriented within a frontal plane, close to or far from the body. The subjects were asked to rate the comfort of the final postures, and the variance of their joint configurations during the steady state following pointing was quantified with respect to pointer endpoint position and pointer orientation. The subjects showed consistent patterns of comfort ratings among the targets, and all movements were characterized by multi-joint synergies stabilizing both pointer endpoint position and orientation. Contrary to what was expected, less comfortable postures had higher joint configuration variance than did more comfortable postures without major changes in the synergy indices. Multi-joint synergies stabilized the pointer position and orientation similarly across a range of comfortable/uncomfortable postures. The results are interpreted in terms conducive to the two theoretical frameworks underlying this work, one focusing on comfort ratings reflecting mean postures adopted for different targets and the other focusing on indices of joint configuration variance.

Appendix 1

Uncontrolled manifold analysis (more details in Scholz and Schoner (1999) and Scholz et al. 2000)

The reference joint configuration vector ($\overline{{\underline{\theta } }}$) was calculated at each time step of a pointing posture by averaging joint angles across trials. We used forward kinematic model parameters (Scholz et al. 2000) to create Jacobian matrix ($\underline{\underline{J}} (\underline{{\overline{\theta } }} )$). This matrix determines how infinitesimal deviations of each joint from $\overline{{\underline{\theta } }}$ affect performance variable (pointer position or pointer orientation). Subsequently, the null-space of $\underline{\underline{J}} (\underline{{\overline{\theta } }} )$ was numerically obtained from the singular value decomposition, solving

$$ \underline{\underline{J}} (\underline{{\overline{\theta } }} ) \cdot \underline{\underline{E}} = 0, $$

(5)

where each column of $\underline{\underline{E}}$ is a basis vector. The null-space of $\underline{\underline{J}} (\underline{\theta } )$ represents configurations of joint angles that keep the performance variable unchanged; it was used as a linear approximation of the UCM. The deviation of joint angles from the reference configuration ($\underline{\theta } - \underline{{\overline{\theta } }}$) were projected onto the null-space at each time step, using:

$$ \underline{\theta }_{\text{UCM}} = \underline{\underline{P}} \cdot (\underline{\theta } - \underline{{\overline{\theta } }} ), $$

(6)

where $\underline{\underline{P}}$ is the projection matrix

$$ \underline{\underline{P}} = \underline{\underline{E}} \cdot (\underline{\underline{E}}^{T} \cdot \underline{\underline{E}} )^{ - 1} \cdot \underline{\underline{E}}^{T} . $$

(7)

The projection of the deviations of the joint configuration from its average value on the orthogonal subspace ($\underline{\theta }_{\text{ORT}}$) was computed as:

$$ \underline{\theta }_{\text{ORT}} = (\underline{\theta } - \underline{{\overline{\theta } }} ) - \underline{\theta }_{\text{UCM}} . $$

(8)

The following formulas for calculating per dimension variance along $\underline{\theta }_{\text{UCM}}$ and $\underline{\theta }_{\text{ORT}}$ projections were used

$$ V_{\text{UCM}} = \frac{{\sum {\left| {\underline{\theta }_{\text{UCM}} } \right|}^{2} }}{{(n - d) \cdot N_{\text{trial}} }}, $$

(9)

$$ V_{\text{ORT}} = \frac{{\sum {\left| {\underline{\theta }_{\text{ORT}} } \right|{}^{2}} }}{{d \cdot N_{\text{trial}} }}. $$

(10)

The total variance $V_{\text{TOT}}$ was computed as

$$ V_{\text{TOT}} = \frac{{\sum {\left| {(\underline{\theta } - \underline{{\overline{\theta } }} )} \right|{}^{2}} }}{{n \cdot N_{\text{trial}} }}. $$

(11)