An Information Space for Partial Length Perception in Dynamic Touch

Abstract

In 2 experiments, participants held a rod, which was occluded from view, at some position along its length and reported the 2 partial lengths—the lengths to the left and to the right of the hand. In Experiment 1, the possible importance of gravitational torque was investigated by comparing judgments made during freely wielding a rod with judgments made during wielding it about a fixed horizontal axis. Performance was better in the free condition, but participants' moderate success in the fixed condition implicated the importance of gravitational torque. In Experiment 2, in which the fixed axis was vertical so no gravitational torque was available, participants could not discriminate longer and shorter sides, though the sum (total rod length) correlated highly with the sum of the 2 partial-length judgments, implicating the importance of inertial moment alone. The results imply that a variable space that captures partial lengths can be derived from moment of inertia and gravitational torque, at least when rods are rotated on a fixed access, and thus when no off-diagonal terms from the inertia tensor are in play.

Notes

¹There has been debate as to what should be considered the center of rotation in computations of inertial variables in dynamic touch experiments. The Connecticut group has favored the wrist (except in experiments that manipulated the joint around which the object was rotated). In contrast, the Amsterdam group has argued on mechanical grounds for taking the end of the rod as the center of rotation and provided a rationale for this choice in the Appendix of van der Langenberg et al. (2006). We prefer to see the center of rotation as an empirical issue, so we attempted to determine its actual location using our experimental setup. We measured the three-dimensionsal positions of Polhemus markers affixed to the wrist and to the rod, 10 cm to the left of the center of the hand. This was done with a 45-cm rod held first at its center and then at its end. These two trials were recorded for each position for each of 3 participants while the participant made a partial length judgment and then while the participant made an array of extreme exploratory movements. We illustrate with the results of 1 representative participant.

For the extreme movements, the rod marker traced out a spherical shape. The center of the sphere—the rotation point—was found using a Matlab minimization program that found the three-dimensional coordinates of the point with the minimum variance in the computed distance to the loci of the markers. For the rod wielded at the middle, the center of sphere was located 3.0 cm in front of, 2.0 cm to the left of, and .5 cm above the center of the wrist joint. For the rod wielded from its end, the center of sphere was located 2.0 cm in front of, 3.0 cm to the left of, and .6 cm below the wrist. The best-fitting centers of the spheres when the participants wielded as they would do when making judgments were comparable but sometimes did not converge on a solution. Inspection of the three-dimensional plots revealed that the natural exploration was sometimes constrained to the x-axis of Figure 1. In any case, there is no evidence to suggest that the wrist is an unreasonable location to use for the center of rotation in the computation of inertial properties in this experimental setup.

²If the system is not at rest, one must include the gyroscopic term, α′ × ([ I ]α′), in which α′ is the angular velocity vector, to complete the Euler dynamical equations (Zatsiorsky, 2002).

³An additional experiment, dropped at the suggestion of reviewers, used a stimulus set in which there was a large difference in ranges between the longer and shorter sides of the rods. With that collection of rods, perception of the length of the shorter side of the rods was poor. For the free-wielding condition, the mean perceived-actual correlations were .848, .794, and .367 for the longer, equal, and shorter sides, respectively. Corresponding means for the fixed-wielding condition were .804, .728, and .316. A possible explanation of the poor performance in perceiving the length of the shorter side was a range effect; the length range of the longer sides was 25 cm whereas the range of the shorter sides was only 7 cm. The current collection of stimuli was selected to make the range of lengths of the longer and shorter sides more nearly equal.

⁴The reader is reminded that the covariates enumerated earlier account for the variance in judgments just as well as the factors listed here. We thank an anonymous reviewer for noting that the superiority of the free-wielding condition could have been due to information about mass, which was unavailable when the rod was supported by the axle.