Assume a Spherical Chicken: Analytic Constraints,Inertia Tensor Information,and Wielded Rod Length Perception

Information associated with the inertia tensor is the preeminent explanation for haptic perception of object properties, notably wielded rod length. Critics counter that tensorial-based information requires non-tensorial supplementation (mass, torque). However, those critiques omit important constraints. With relevant constraints included, the inertia tensor alone completely specifies rod length. I list constraints inherent (but tacit) in haptic rod length perception, and show that object properties associated with the inertia tensor are invariant, even with constraints removed, by involving (a) longitudinal moment equivalents for rod mass and (b) derivatives of moments with respect to varying rotation axes. Analytic outcomes show tensorial-based information is a robust basis for wielded rod length perception, and suggest open questions for empirical exploration.     

The effect of density and diameter on haptic perception of rod length

Three experiments on the effect of density and diameter on haptic perception of rod length are reported. In Experiment 1, the subjects wielded visually occluded rods of different densities. Perceived length was found to be affected by the density of the rod regardless of the actual length. In Experiment 2, three aluminum rods of different lengths with handles of four different diameters were wielded. Perceived length of the rod was found to be shorter as the diameter of the handle with which it was wielded increased. A diameter—length illusion was thereby produced. In Experiment 3, visually occluded rods of different diameters but of the same moment of inertia about thex-axis were wielded with the right hand, and tubes of different diameters were felt with the left hand. The subjects were instructed that their right hand was grasping a handle, and that the actual diameter of the rod could be felt with the left hand. Rods were perceived to be shorter if a larger diameter was felt with the left hand. The results showed that perceived length is not just a function of actual rod length, and that it is not accounted for by inertia only. The results are further discussed in terms of the nature of invariants and the effect of knowledge on perception.     

Which mechanical invariants are associated with the perception of length and heaviness of a nonvisible handheld rod? Testing the inertia tensor hypothesis

It has been suggested that the inertia tensor governs many instances of haptic perception. However, the evidence is inconclusive because other candidate mechanical parameters (i.e., invariants) were not or were insufficiently controlled for in pertinent experiments. By independently varying all candidate mechanical parameters, the authors were able to test the role of the inertia tensor relative to that of other mechanical parameters. The results showed that length perception during rod wielding is not governed by the inertia tensor alone but also by the static moment. In contrast to previous reports, length perception during rod holding and heaviness perception during rod wielding were found to be unrelated to the inertia tensor and strongly related to the static moment.     

Role of the inertia tensor in perceiving object orientation by dynamic touch.

Ss wielded an occluded L-shaped rod and attempted to perceive the direction in which the rod was pointing with respect to the hand. The pattern of the rod's different resistances to rotation in different directions, quantified by the inertia tensor, changes systematically with the rod's orientation. Perception of orientation by wielding is possible if the tissue deformation consequences of the rod's inertia tensor are detectable. It was shown that perceived orientation was a linear function of actual orientation for both free and restricted wielding and for rods of different-size branches. The eigenvectors of the inertia tensor were implicated as the basis for this haptic perceptual capability. Results were discussed in reference to information-perception specificity and its implications for effortful or dynamic touch.     

Perceptual behavior: recurrence analysis of a haptic exploratory procedure

Various object properties are perceptible by wielding. We asked whether the dynamics of wielding differed as a function of the to-be-perceived property. Wielding motions were analyzed to determine if they differed under the intention to perceive or not perceive rod length (experiment 1), to perceive object height versus object width (experiment 2), and to perceive the length forward of where the rod was grasped versus the position of the grasp (experiment 3). Perceiving these different properties is known to depend on different components of the object's inertia tensor. Analyses of the subtle recurrent patterns in the phase space of the hand motions revealed differences in wielding across the different perceptual intentions. Haptic exploratory procedures may exhibit distinct exploratory dynamics.     

Inertial eigenvalues, rod density, and rod diameter in length perception by dynamic touch

Four experiments addressed the relevance of the eigenvaluesI _k of the inertia tensor for perceiving length by dynamic touch. Experiments 1–2 focused on the consequences of limiting variation in the minimum eigenvalueI ₃. Both revealed that perceived length is a function ofI _k . Whether the contribution ofI ₃ is detected, however, depends on the range of values that characterize a particular object set. Experiments 3–4 considered the relationship between an independent index of a rod’s diameter, which does not affectI _k , and actual manipulation of a rod’s diameter, which does affectI _k . Whereas the former appeared as satisfaction of implicit instructions to alter reports of perceived length, the latter entailed actual differences in perceived length in accordance withI _k . Results are discussed with respect to the links among actual length, perceived length, andI _k , as well as, in particular, how these links guarantee that perceived length is in the range of actual lengths.     

Perceiving extents of rods by wielding: haptic diagonalization and decomposition of the inertia tensor

We report two experiments on the length-perception capabilities of the hand-related haptic subsystem. On each trial, a visually occluded rod was wielded by the subject at a position intermediate between its two ends. The position was either 1/2 or 3/4 of the rod's length. On two-thirds of the trials, a weight was attached to the rod at a point either above or below its center of gravity and not coincident with the hand's position. In Experiment 1, the subject's task was to perceive the distance reachable with the portion of the rod extending beyond the position of the grasp. In the second experiment, the subject's task was to perceive the distance reachable with the entire rod if it were held at its proximal end. In Experiment 1, perceived reaching distance was a function of the moment of inertia of the amount of rod forward of the grasp about an axis through the proximal end of the rod segment. In Experiment 2, perceived reaching distance was a function of the moment of inertia of the entire rod about the given axis of rotation intermediate between the rod's ends. The results are discussed in terms of (a) the notion of smart perceptual instruments capitalizing on invariant properties of the inertia tensor and (b) how the haptic decomposition of moments of inertia follows the principle of equivalence of forces.     

Exploration styles modulate length perception through dynamic touch

One of the most surprising capacities of the haptic system is the ability to estimate different properties of objects, like weight or length, through invariants of rotational mechanics that are accessible via the proprioceptive system. This field of research is called Dynamic Touch. In its classical experimental paradigm, the participant firmly grasps a rod that can be wielded but not seen, and he or she tries to match the hand-held rod's length using another rod that can be seen but not wielded. In the experiment reported here, we focus on the role of the exploratory behavior, restricting the wielding in six conditions that vary both the amplitude and the frequency of movements. Increments in the speed of the movement are shown to increase the accuracy in the haptic estimation. It is argued that these results support the moment of inertia as the best informational candidate, given that it is an invariant property that only emerges when rotational torques are applied. Alternative candidates such as static moment or mass are discarded because they do not depend on differential movements.     

Exteroception and exproprioception by dynamic touch are different functions of the inertia tensor

When an object is held and wielded, a time-invariant quantity of the wielding dynamics is the inertia tensorIij. The 3 × 3 quantityIij is composed of moments of inertia (on the diagonal) and products of inertia (off the diagonal). Examination ofIij as a function of different locations at which a cylindrical object is grasped revealed that the products related systematically to grip position (a direction), and both the products and moments taken together related systematically to the extent of the rod to one side of the hand (a magnitude in a direction). In two experiments, observers wielded an occluded rod that was held at an intermediate point along its length and reproduced both the felt grip position and partial rod length. In both experiments, perceived grip position was a function of the rod’s products of inertia and perceived partial rod length was a function of the moments and products. Discussion focuses on the specificity of exteroception and exproprioception toIij.     

Principles of part-whole selective perception by dynamic touch extend to the torso

The haptic subsystem of dynamic touch expresses a novel form of part-whole selective perception. When wielding a nonvisible rod grasped at some intermediate point along its length, an individual can attend to and report the length of a part of the rod (e.g., the segment forward of the hand) or the length of the whole rod. Both perceptions relate to the rod's mass moments about the point of grasp but in systematically different ways. Previous demonstrations of this part-whole selectivity have been in respect to rods grasped by hand or attached to a foot. The authors demonstrated the part-whole selectivity for nonvisible rods attached to the shoulder girdle and wielded primarily by movements of the trunk with benchmark performance provided by the same rods grasped and wielded by hand. Their results suggest that part-whole selectivity is a haptic capability general to the body.     

Selective perception by dynamic touch

Perceiving the length of a rod by dynamic touch is tied to the inertia tensorIij, a quantification of its resistance to rotational acceleration. Perception of the portion extending in front of the grasp has previously been ascribed to decomposing one component ofIij by attention. The tensorial nature of dynamic touch suggests that this ability must be anchored wholly in the tensor. Three experiments show that perceived partial length is a function of two components of the tensor, one tied primarily to magnitude and the other tied primarily to direction, whereas perceived whole length is a function of a magnitude component alone. Dynamic touch is characterized in terms of a haptic perceptual instrument that softly assembles to exploitIij differently depending on the intention, producing 1:1 maps that are appropriately scaled for each intention.     

The situational effects on haptic perception of rod length

Three experiments were conducted to investigate situational effects (manipulation, range, and prior experience) on the haptic perception of rod length. Each rod was held between its two ends with one hand. In Experiment 1, 32 participants judged length of rods using different manipulations. Perceived lengths were found to be dependent on manner of manipulation and not necessarily equal to actual lengths. Different parameters were detected in different manipulations. In Experiment 2, 8 participants judged rod lengths by wielding rods of two ranges: long and short. Perceived length was found to be affected by the range of rods evaluated successively in a single set. In Experiment 3, 9 participants judged rod lengths after an experience of handling dense or light rods. Perceived length was found to be affected by prior experience. Results are discussed in terms of how rod lengths can be perceived accurately by haptic modality without involving direct perception.     

Haptically perceiving the distances reachable with hand-held objects

Nine experiments are reported on the ability of people to perceive the distances reachable with hand-held rods that they could wield by movements about the wrist but not see. An observed linear relation between perceived and actual reaching distances with the rods held at one end was found to be unaffected by the density of the rods, the direction relative to the body in which they were wielded, and the frequency at which they were wielded. Manipulating (a) the position of an attached weight on an otherwise uniformly dense rod and (b) where a rod was grasped revealed that perceived reaching distance was governed by the principal moment(s) of inertia (I) of the hand-rod system about the axis of rotation. This dependency on moment of inertia (I) was found to hold even when the reaching distance was limited to the length of rod extending beyond an intermediate grasp. An account is given of the haptic subsystem (hand-muscles-joints-nerves) as a smart perceptual instrument in the Runeson (1977) sense, characterizable by an operator equation in which one operator functionally diagonalizes the inertia and strain tensors. Attunement to the invariants of the inertia tensor over major physical transformations may be the defining property of the haptic subsystem. This property is discussed from the Gibsonian (ecological) perspectives of information as invariants over transformations and of intentions as extraordinary constraints on natural law.     

Detection of invariants by haptic touch across age groups: rod length perception

This study examined the development in the detection of maximum eigenvalues and static moment as invariants, through a task of perceiving rod length without visual information by haptic touch. 34 participants ages 6 to 83 years participated in the experiment. Their exploratory behavior and perceptions of rod length were analyzed by age group (Children: 6 to 12 years old; Young Adult: 21 to 25 years old; Middle Age: 31 to 56 years old; and Older: 65 to 83 years old). A behavior analysis indicated that use of vertical swinging increased for the Young Adult group and decreased with age for the Older group, whereas Children frequently held the rod without wielding. Analysis showed that, by age, differences in coefficients on the maximum eigenvalue and static moment were parallel with an exploratory behavioral change. Finally, the effect of different exploratory behaviors on length perception was discussed.     

Haptic perception of partial-rod lengths with the rod held stationary or wielded

Three experiments on the haptic perception of partial-rod lengths are reported. The rods were gripped between the two ends and held horizontal. The subjects held the rods stationary; the distribution of mass of the segment in front of the hand was fixed, while the distribution of mass of the segment behind the hand was varied. Perceived forward length was found to be significantly affected by the distribution of mass of the backward segment. Similar results were obtained when the rods were wielded. The results indicated that partial-rod lengths are specified by functions of mechanical perturbations acting on the hand, and not ay the breaking up of the first moment of mass or the moment of inertia of the rod by attention as suggested previously by others. The results are also discussed with respect to invariant detection and attention.     

Perception of the length of an object through dynamic touch is invariant across changes in the medium

Rotational inertia—a mechanical quantity that describes the differential resistance of an object to angular acceleration in different directions—has been shown to support perception of the properties of that object through dynamic touch (wielding). The goal of the present study was to examine if perception of the length of an object through dynamic touch depends on its rotational inertia, independent of the medium in which it is wielded. The participants (n = 14) wielded 12 different objects held in air or completely immersed in water and reported perceived lengths of those objects. Each object consisted of a rod of a particular density with a particular number of stacked steel rings attached at a particular location along its length. Perceived length was invariant across medium. In addition, a single-valued function of the major eigenvalue, I ₁, and the minor eigenvalue, I ₃, of the rotational inertia, I, of the 12 objects predicted the perceived lengths of those objects in both air and water, and the perceived lengths were invariant across the two media. These results support the hypothesis that the informational support for perception of the length of an object through dynamic touch is invariant across changes in the medium.     

Failure of additivity in bisection of length

Subjects were intructed to select one rod to lie halfway in length between two given rods. These bisection instructions imply an additive model in the subjective metric. However, the data were inherently nonadditive; the length of the bisector could be an increasing or a decreasing function of the length of one given rod, depending on the length of the other given rod. A convexity analysis and a nonmetric analysis both showed that no monotone transformation could make the data additive. The bisection problem is used to contrast the axiomatic and functional approaches to measurement theory.     

Haptically Perceiving the Length of One Rod by Means of Another

Two experiments examined perception of the extent of a target rod that is contacted and wielded by a second probe rod. The equations that define the dynamics of the probe-target system suggest a higher order moment of inertia as the relevant perceptual variable. The particular inertial term implicates parameters of both the target and probe rod. Experiment 1 manipulated the inertia of the target rod and Experiment 2 manipulated the inertia of the probe rod. In both experiments, perceived length was a function of the complex inertial term. Results were discussed in terms of haptic perception at a distance, the equivalence of inert and neural appendages, and the scaling of perceived to actual variables.     

Perceiving the vertical distances of surfaces by means of a hand-held probe.

Nine experiments were conducted on the haptic capacity of people to perceive the distances of horizontal surfaces solely on the basis of mechanical stimulation resulting from contacting the surfaces with a vertically held rod. Participants touched target surfaces with rods inside a wooden cabinet and reported the perceived surface location with an indicator outside the cabinet. The target surface, rod, and the participant's hand were occluded, and the sound produced in exploration was muffled. Properties of the probe (length, mass, moment of inertia, center of mass, and shape) were manipulated, along with surface distance and the method and angle of probing. Results suggest that for the most common method of probing, namely, tapping, perceived vertical distance is specific to a particular relation among the rotational inertia of the probe, the distance of the point of contact with the surface from the probe's center of percussion, and the inclination at contact of the probe to the surface. They also suggest that the probe length and the distance probed are independently perceivable. The results were discussed in terms of information specificity versus percept-percept coupling and parallels between selective attention in haptic and visual perception.     

Transfer of calibration in length perception by dynamic touch

Earlier studies suggested that the calibration of actions is functionally, rather than anatomically, specific; thus, calibration of an action ought to transfer to actions that serve the same goal (Rieser, Pick, Ashmead, & Garing, 1995). In the present study, we investigated whether the calibration of perception also follows a functional organization: If one means of detecting an information variable is recalibrated, are other means of detection recalibrated as well? In two experiments, visual feedback was used to recalibrate perceived length of a rod wielded by the right hand; the recalibration was found to transfer to length perception with the left hand. This implies that calibration in perception is organized functionally rather than anatomically, and supports the general view that calibration applies to functional systems.