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.     

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.     

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.     

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.     

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.     

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.     

Haptic probing: perceiving the length of a probe and the distance of a surface probed.

Knowing about the properties of objects by wielding them and knowing about the distances of surfaces by striking them with objects as probes are examples of dynamic or effortful touch. Six experiments focused on the invariant mechanical parameters that couple the time-varying states (displacements, velocities) of hand-held rods to the time-varying torques and forces imposed upon them by wielding and probing. There were three major conclusions. First, when a probe is wielded without contact, perceived probe length is a function of the probe's rotational inertia; however, with contact, perceived probe length is affected by the rotational inertia and the distance of the point of contact from the probe's center of percussion. Second, when a surface is struck with a probe, perceived surface distance is affected by the probe's rotational inertia and the angle of inclination of the probe at contact. Third, under seemingly identical conditions of probing, either probe length or surface distance can be perceived selectively without confusion. Results were discussed in terms of haptic information, haptic attention, and the dynamics of probing.     

Haptic probing: Perceiving the length of a probe and the distance of a surface probed

Knowing about the properties of objects by wielding them and knowing about the distances of surfaces by striking them with objects as probes are examples of dynamic or effortful touch. Six experiments focused on the invariant mechanical parameters that couple the time-varying states (displacements, velocities) of hand-held rods to the time-varying torques and forces imposed upon them by wielding and probing. There were three major conclusions. First, when a probe is wielded without contact, perceived probe length is a function of the probe’s rotational inertia; however, with contact, perceived probe length is affected by the rotational inertia and the distance of the point of contact from the probe’s center of percussion. Second, when a surface is struck with a probe, perceived surface distance is affected by the probe’s rotational inertia and the angle of inclination of the probe at contact. Third, under seemingly identical conditions of probing, either probe length or surface distance can be perceived selectively without confusion. Results were discussed in terms of haptic information, haptic attention, and the dynamics of probing.     

Berkeleian Principles in Ecological Realism: An Ontological Analysis

Three experiments are reported, which examined the relation between the percep- tion of the distance reachable with a hand-held rod that can be wielded but not seen and the rod's resistance to having its rotational speed changed by application of a torque. In these experiments, subjects wielded any given rod about an axis intermediate between its endpoints. The subject's task was to adjust a visible, movable surface to coincide with where he or she could reach with the given rod if allowed to hold it at its proximal end. Two experiments considered the effects of wielding a rod at different orientations to the pull of gravity. Rods were wielded either within a plane roughly perpendicular to the ground or within a plane roughly parallel to the ground. Plane of wielding did not affect the patterning of perceived reachable distances as a function of the various conditions, which included variations in the positioning of grasp and the positioning of a mass affixed to the rods. The patterning of the moments of inertia associated with the various conditions determined the patterning of perceived reachable distances. The third experiment restricted wielding to a plane roughly parallel to the ground and varied how the rods were grasped, either overhand or underhand. The variation in grasp amounted to a variation in the neuromuscular patterning associated with the wielding of any given rod. Perceived reachable distances proved to be indifferent to the overhand versus underhand contrast.     

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.     

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.     

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.     

Perceiving the lengths of rods wielded in differentmedia

In two experiments, we investigated the ability of participants to report the lengths of rods wielded in air or water. Homogeneous aluminum rods were employed in Experiment 1. The inertia of the rods was manipulated in Experiment 2 through the use of attached masses. Although the torques required in order to wield rods in water are substantially greater than those required to wield rods in air, the perceived lengths of rods wielded in the two media were very similar. Perceived length was found to be a function primarily of inertia in both media. The experiments also revealed a small influence of resistance due to the denser medium of water. The results demonstrate the ability of perceivers to extract a physical invariant from a complex array of forces. The discussion is focused on the role of invariants in dynamic touch.     

Perceiving the lengths of rods wielded in different media

In two experiments, we investigated the ability of participants to report the lengths of rods wielded in air or water. Homogeneous aluminum rods were employed in Experiment 1. The inertia of the rods was manipulated in Experiment 2 through the use of attached masses. Although the torques required in order to wield rods in water are substantially greater than those required to wield rods in air, the perceived lengths of rods wielded in the two media were very similar. Perceived length was found to be a function primarily of inertia in both media. The experiments also revealed a small influence of resistance due to the denser medium of water. The results demonstrate the ability of perceivers to extract a physical invariant from a complex array of forces. The discussion is focused on the role of invariants in dynamic touch.     

Does perceived size depend on perceived distance? An argument from extended haptic perception

Two experiments were directed at the comparison between two perspectives on the perception of size achieved by probing the gap between two occluded distal surfaces by means of a hand-held rod. One perspective was the classical size—distance invariance hypothesis developed for the problem of visual size perception with a central role for perceived distance; the other was the hypothesis that the extended haptic perception of gap size is specific to a physical invariant λ of the dynamics of probing. Experiment 1 examined the relation between haptically perceived gap size and haptically perceived gap distance. No causal connection between the two was found, and all the variance in perceived size was accounted for by?. Experiment 2 manipulated the rotational inertia of the probe. Its effect was different for the two perceptions of size and distance, underscoring their independence. The indifference of perceived size to perceived distance was discussed in reference to identifying invariants for both the haptic and the visual perception of size at a distance.     

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.     

Tensorial basis to the constancy of perceived object extent over variations of dynamic touch

Subjects wielded occluded rods, with or without attached masses, and reported the distances reachable with their distal tips. Experiments 1–3 compared wielding about the wrist, the elbow, and the shoulder. Experiments 4 and 5 compared free wielding, using the whole arm, with wielding only about the wrist. The two comparisons, respectively, were of spatial and temporal variations in the rod’s rotational inertia. Perceived extent was found to be constant in both comparisons. This constancy was tied to the inertia tensorI _ij defined about a point that remains a fixed distance from the object during wielding—an invariant of the spatially and temporally dependent patterning of mechanical energy impressed upon the tissues of the body. Discussion focused on the reciprocal action and perception capabilities of multisegmented limbs, the tensorial relations in the neurobiology of dynamic touch, and the strategy of understanding perceptual constancy through invariants.     

Perceiving the Lengths of Rods That are Held But Not Wielded

The ambiguity inherent in the act of experimental abstraction is discussed particularly with respect to experiments that seem to prove the superfluity of active exploration in perception. For example, in the case of haptic perception of the extent of hand-held rods, the variable of the second moment of mass distribution-the moment of inertia-has been shown to predict perceived length; this variable is inherently active, identifying a system's resistance to rotational acceleration. Other sources have reported that the length of an unseen rod could be perceived even when the rods were not rotated (rendering second moment theoretically inaccessible). The first experiment of this article confirms this ability in the extreme case in which observers are instructed not to move the rod at all. Four more experiments are reported in which the relative roles of the second moment and of the first moment-the other plausible mechanical candidate-are evaluated. The first moment was a better predictor of perceived length in cases in which exploration was restricted, and the second moment was a better predictor in conditions in which exploration was not restricted, although each played some role in all conditions. These results are discussed in terms of the possibility of more than one kind of information specifying the same property.     

Rotational inertia and multimodal heaviness perception

Perceived heaviness of wielded objects has been shown to be a function of the objects’ rotational inertia—the objects’ resistance to rotational acceleration. Studies have also demonstrated that if virtual objects rotate faster than the actual wielded object (i.e., a rotational gain is applied to virtual object motion), the wielded object is perceived as systematically lighter. The present research determined whether combining those inertial and visual manipulations would influence heaviness perception in a manner consistent with an inertial model of multimodal heaviness perception. Rotational inertia and optical rotational gain of wielded objects were manipulated to specify inertia multimodally. Both visual and haptic manipulations significantly influenced perceived heaviness. The results suggest that rotational inertia is detected multimodally and that multimodal heaviness perception conforms to an inertial model.