Augmenting Perceived Length of Handheld Controllers: Effects of Object Handle Properties

In the realm of virtual reality (VR), shape-changing controllers have emerged as a means to enhance visuo-haptic congruence during user interactions. The major emphasis has been placed on manipulating the inertia tensor of a shape-changing controller to control the perceived shape. This paper delves deeper by exploring how the material properties of the controller’s handle, distinct from the inertial information, affect the perceived shape, focusing on the perceived length. We conducted three perceptual experiments to examine the effects of the handle’s softness, thermal conductivity, and texture, respectively. Results demonstrated that a softer handle increases the perceived length, whereas a handle with higher thermal conductivity reduces it. Texture, in the form of varying bumps, also alters the length perception. These results provide more comprehensive knowledge of the intricate relationship between perceived length and controller handle properties, expanding the design alternatives for shape-changing controllers for immersive VR experiences.


INTRODUCTION
In virtual reality (VR), users commonly interact with virtual objects using handheld controllers.However, conventional controllers have fxed sizes and shapes, resulting in incongruent visual-haptic user experiences for visually diverse virtual objects.To address this challenge, researchers proposed novel haptic controllers called shape-changing controllers.These controllers attempt to provide congruent visual-haptic experiences by altering their physical attributes, such as length, weight, imbalance, and volume (see Section 2.3).This pursuit of multisensory consistency can improve the realism and immersiveness of VR experiences.
Shape-changing controllers primarily achieve a consistent visualhaptic correspondence by two methods.First, they replicate the mass distribution of a virtual object based on the concept of dynamic touch [75].It requires manipulating the physical shape of the controller [41,56,57,82].Second, they modify the shape of the controller's handle held in the user's hand, typically by adjusting the handle volume with pin or planar arrays [20,67,72,81] or pneumatic actuators [43,80].For the frst method, recent research developed computational models to determine a haptic shape optimal for a visual shape, or vice versa [16,50]; see Section 2.2.However, the second method, which utilizes the controller handle, has received limited attention.To our knowledge, there is no prior research addressing the infuence of handle properties on shape perception during dynamic touch, except for handle diameter [6,49].
In this paper, we explore the efects of the material properties of the handle held in the hand on object shape perception, particularly the perceived length of the object.The moments of mass distribution of objects, which have a critical efect on the perceived length, were maintained constant in our research.We conducted three perceptual experiments, which focused on the efects of handle softness (Exp.1; Section 3), thermal conductivity (Exp.2; Section 4), and texture (Exp.3; Section 5); these are essential psychophysical dimensions for tactile perception [46].We summarize and compare the efects of the three handle properties and then delve into the design implications for handheld controllers (Section 6).Our research outcomes ofer novel scientifc insights into length perception through dynamic touch, with potential applications to shape-changing controllers.

RELATED WORK
This section introduces haptic perception under diferent touch modes, focusing on shape perception through dynamic touch.It is followed by a review of shape-changing controllers and haptic perceptual cues for several properties.

Touch Modes
When humans interact with external objects, they obtain haptic information using one or a combination of three touch modes: passive, active, and dynamic touch [19].In passive touch, the information fows in through skin deformation (cutaneous sense) without any voluntary movement of joints or muscles.Active touch, on the other hand, involves haptic perception derived from both skin deformation and joint movements (kinesthetic sense).These two touch modes allow humans to perceive various haptic properties, such as volume, weight, and texture, through corresponding exploratory procedures (EPs) [39].In contrast, dynamic touch provides haptic information of objects' shape through minimal body movements, such as holding and wielding, without using EPs.
Most previous studies concentrated on object shape perception through dynamic touch when users wield an object.In addition, our work assesses the shape perception in conjunction with the object's handle properties, which are mediated by both passive and active touch.

Shape Perception via Dynamic Touch
2.2.1 Physical Properties.Consider an object held in the hand and rotating around one point, the center of rotation (COR).A person holds and wields the object for its shape perception through dynamic touch.This process is related to the three types of moments of mass distribution: the zeroth (mass; ) [33,34], frst (static moment; M) [5,33,34], and second moment (moment of inertia; MOI) [4,13,33,34,74,75].
In physics, the second moments of mass distribution are generally represented using an inertia tensor: The inertia tensor consists of three diagonal terms called the moments of inertia (MOIs) and six of-diagonal terms called the products of inertia (POIs), which are symmetric.These moments of mass distribution are physical invariants of the object and related to shape perception, such as length, shape, and weight, alone or in combination.A full review of the above concepts is available in [75].
The characteristics of the grip grasped by the hand are also relevant to the perceived shape of an object.For example, the perceived length decreases if the handle diameter increases [6,49].While this topic has pertinent implications for shape-changing controller designs, it has received relatively less attention for research.In this study, we consider three more material properties: the softness, thermal conductivity, and texture of the handle.

Shape-Changing Controllers
One class of shape-changing controllers vary their moments of mass distribution during dynamic touch and keep their handle properties constant.For instance, diverse changes in perceived shape can be made by weight-shifting mechanisms-1D mass movement [82], 2D movement involving two masses [41,57], and 3D movement with one heavy mass [56].Some controllers transmit dynamic haptic sensations using powerful airfow mechanisms [25,29,70].The other class of controllers adjusts the grip's shape, where a good example is volume changing.They can be implemented using pin arrays [20,71,72,81], planar arrays [67], pneumatic actuators [43,80], and even tape mechanisms for portable design [84].
To our knowledge, no controllers and their computational models for shape perception exist that modulate the handle's material properties.It can cause inconsistency between the visual and haptic perception of a virtual object.

Haptic Perceptual Cues
An object has many physical properties, e.g., viscoelasticity and friction.The object also has many perceptual attributes, e.g., softness and roughness.In principle, each physical property can afect each perceptual property, but the extent of contribution often varies considerably.If the efect is substantial, the physical property is regarded as a cue for that perceptual attribute.This section reviews how softness, temperature, and texture serve as perceptual cues for some perceptual properties 1 .
2.4.1 Sofness.The viscoelasticity of a deformable object is a key physical variable for softness perception.Perceived softness is also determined by other cues, including displacement, force, contact area, and skin deformation, mediated by both kinesthetic and cutaneous senses [3].Due to this complexity, we control the softness of an object by changing its material in this study.
The softness of an object afects the perception of several other haptic attributes.For example, a soft material attenuates the propagation of vibrations, leading to a degradation in vibration perception sensitivity [11,51].The softness of an object also alters the perceived texture [37,78] and friction [27] of a surface.Moreover, softness is signifcantly related to the human sense of pleasantness [53].
2.4.2Temperature.Temperature perception of a physical object is contingent on its thermal conductivity and heat capacity, mediated by the two kinds of thermoreceptors that respond to heat energy gain and loss, respectively [24,32].Previous studies reported that temperature can infuence tactile perception of several attributes.For example, cold sensations make humans perceive the object as harder [68,79] and heavier [10,65].Object temperature also afects pressure perception [64], tactile acuity [65], vibrotactile sensitivity [21], and the perceived roughness of a surface [22].

Surface
Texture.Haptic perception of an object's surface texture relies on both spatial tactile cues for coarse features and 1 Tactile perception mechanisms involve a combination of four psychophysical dimensions: softness (or hardness), warmth (related to temperature), roughness (related to texture), and friction [46].This research focuses on the frst three properties, excluding friction.vibrational cues for fne features [2,26,58].However, the perceptual efects of surface texture on other haptic attributes have received relatively less attention.The only examples to our knowledge are those related to how the rough or smooth surface texture of a handheld object afects the weight perception [14,15,30,40].

EXP. 1: EFFECT OF SOFTNESS
This experiment aimed to measure how the softness of an object handle changes its perceived length during dynamic touch.

Participants.
Eighteen volunteers (nine females and nine males; mean age 24.9 years) participated in this experiment.No participants reported sensorimotor abnormalities.Before the experiment, participants were provided with detailed information about the experiment's goals and procedure through a written document and then signed a consent form.Each participant was compensated with approximately USD 15.All experiments in this paper were approved by the Institutional Review Board at the author's institution (PIRB-2022-E036).

Experimental Conditions.
The experiment had a within subject factorial design with two independent variables: MOI and SOFT-NESS.Since the primary MOI has a dominant efect on the perceived length (the perceived length increases as MOI increases; Section 2.2), we included MOI as one independent factor.The MOI had three levels: 8, 16, and 32 ×10 4 g•cm 2 , ranging from small handheld controllers [50] to large objects like a tennis racket [69].The physical objects made to have the three MOIs are shown in Figure 2 and  3.Each object consisted of three components: a handle, a hollow aluminum rod, and two brass disks.All aluminum rods were 50 cm long.The MOI was adjusted by moving the positions of the two brass disks.The physical properties of all objects are described in Appendix A.
The default handle was 3D-printed using ABS (thus rigid) with a length 10 cm, weight 73 g, and diameter 4 cm.We made another handle using ABS but with a smaller diameter (3.2 cm).This handle was inserted into a silicon shell (thickness 4 mm) to vary its softness.We fabricated two silicon shells: one made of EcoFlex 0010 (Shore 00-10) and the other of Dragon skin FX-Pro (Shore 2A). 2 Therefore, the SOFTNESS of the handle was varied in three levels: very soft (EcoFlex 0010), soft (Dragon skin FX-Pro), and rigid (ABS-A100).They were softer, as soft as, and stifer than the human fnger pad [12], respectively.Their stifness values were measured to be 1.8 N/mm, 3 N/mm, and 16 N/mm, respectively.
3.1.3Procedure.Participants sat comfortably in a chair with armrests and were blindfolded to eliminate visual cues (Figure 4).The experimental procedure followed magnitude estimation with modifed modulus [18,23].In each trial, participants were presented with a pair of physical objects.One object, designated as the reference (rigid; MOI = 16×10 4 g•cm 2 ), was held in one hand, while the other object was held in the other hand for comparison.Participants wielded the two objects freely as long as they needed.
They were asked to assign a positive number (modulus) to the perceived length of the reference object.Then, they verbally provided a positive number that represented the perceived length of the comparison object relative to the reference object by scaling it proportionally to the established modulus.The experimenter then changed the comparison object to another for the next trial.
The experiment was structured into six sessions, each comprising two blocks.Every block consisted of nine trials, encompassing all conditions.The order of the presented conditions was balanced using a Latin square.In one block, participants held the reference with their left hands, while in the other block, they held it with their right hands.The order of the hands was also balanced across participants.Consequently, this design resulted in 108 trials (9 conditions × 6 sessions × 2 blocks; 12 repetitions per condition).Participants had a break after every session.The entire experiment took 50 min on average.

Data Analysis.
Most participants were unfamiliar with evaluating the length of an object while wielding it.Thus, we discarded 2 Shore hardness is a standard measure of the resistance that a material has to indentation. 2 Participants were instructed to maintain a consistent grip position throughout the object manipulation.We imposed no limitations on the pivot and rotation directions because free manipulation yields superior or comparable perceptual performance in shape perception than constrained interactions [47,74].the data collected in the frst session (two repetitions) regarding it as practice.We used the data collected in the other fve sessions (10 repetitions) for data analysis.
In magnitude estimation experiments, data standardization is necessary to compensate for scale variations across participants.We adopted the mean deviation standardization method for visualizing scale-built data and used the geometric mean standardization method for statistical analysis (detailed in Han et al. [23]).

Results
Experimental results are summarized in Figure 5 in a log-log scale; a logarithmic relationship was observed between perceived length and MOI [34,49,62,75].
Two out of the nine data sets violated the normality assumption according to the Shapiro-Wilk test, but all the data met the assumption by the Kolmogorov-Smirnov test.We checked the sphericity of the data using Mauchly's test and applied the Greenhouse-Geisser correction if the assumption was violated.Then, we conducted a two-way repeated-measures ANOVA on the perceived length.Both MOI and SOFTNESS were statistically signifcant ( (1.31, 22.35) = 222.60,< 0.0001, 2 = 0.929, 2 = 0.909; (1.63, 27.74) = 22.65, < 0.0001, 2 = 0.601, 2 = 0.057).The interaction term was not signifcant ( (3.1, 52.64) = 2.196, = 0.079).We conducted Tukey's post-hoc multiple comparison tests for the signifcant cases, and these results are also represented in Figure 5.
The above results indicate that the length of a handheld object perceived by dynamic touch is infuenced by both the object's MOI and the softness of its handle.Consistent with previous studies on length perception (Section 2.2), participants perceived longer lengths in cases of larger MOI.The object handle's softness also increased its perceived length; however, the efect of the softness was less pronounced than that of MOI.Compared to the rigid objects, SOFTNESS increased the perceived length from 7% (small MOI) to 9% (large MOI).

Discussion
Exp. 1 reconfrmed the critical efect of MOI on the perceived length of a wielded object; its perceived length increases as MOI increases.Additionally, Exp. 1 presented a new fnding about perceived length vs. handle softness.This intriguing discovery may be related to delayed torque.
Time-delayed haptic feedback can alter haptic perception.For example, introducing a temporal delay in haptic response can lead to an altered perception of surface stifness, a phenomenon referred to as delayed stifness [54].This concept extends to visuo-haptic crossmodal perception, where a delay in either visual or haptic feedback can induce a stifness illusion, as applied to augmented reality (AR) applications [9,17,36].In dynamic touch, humans assess the object length by evaluating the torque in conjunction with the angular velocity derived from the object's moments of mass distribution.Unlike a rigid object, the surface deformation of a soft object delays the torque transmission during wielding.This delayed torque may result in the perception of an extended manipulation period, which subsequently contributes to the perception of a greater length.

EXP. 2: EFFECT OF THERMAL CONDUCTIVITY
The hypothesis tested in this experiment was that two identical objects, except for the handle's thermal conductivity, have diferent lengths perceived through dynamic touch.

Methods
The methods common to Exp. 1 are not repeated for brevity.
4.1.1Participants.Eighteen participants (eight males and ten females; mean age 24.9 years) took part in this experiment.None of these participants had taken part in Exp. 1.

Experimental Conditions.
The experiment had a two-factor within-subjects design with two independent variables: MOI and TC (Thermal Conductivity).MOI had three levels as in Exp. 1. TC featured two levels, representing low and high thermal conductivity.As a result, the experiment had six conditions (3 MOI × 2 TC).
The physical objects used are shown in Figure 6.The low conductivity conditions used a 3D-printed handle with standard plastic material (ABS; 0.2 W/mK [28]).For high conductivity, the handle had an aluminum cylindrical cover (thickness 2 mm; 226 W/mK [28]) around a 3D-printed core of diameter 3.6 cm.Both handles had identical specifcations in length (10 cm), weight (73 g), and diameter (4 cm).Consequently, all physical objects exhibited the same mass and static moment, with diferent MOIs and thermal conductivities.

Procedure and Data
Analysis.This experiment had essentially the same procedure as Exp. 1.The reference stimulus had MOI = 1.6×10 4 g•cm 2 and TC = Low.We included 12 repetitions for each experimental condition as in Exp. 1.Since the number of experimental conditions was 6, the total number of trials was 72, and the experiment time averaged around 40 min.Also, the room temperature was controlled to 25 ℃. 3 For data analysis, we used the response data from the latter ten repetitions (the frst two repetitions discarded).
Therefore, we conclude that the perceived length of a wielded object increases with a larger MOI, whereas it decreases with the higher thermal conductivity of its handle.Compared to the objects with low thermal conductivity, an increase in the thermal conductivity of the handle resulted in a decrease in perceived length by 4% at low/high MOI and by 10% at medium MOI.

Discussion
The results of Exp. 2 reconfrmed the infuence of MOI on the perceived length.Furthermore, the results presented the frst evidence that the handle's thermal conductivity, related to temperature perception, can afect the length perception in dynamic touch.Objects with higher thermal conductivity were perceived as shorter.This efect was evident at the low and middle MOIs but less clear at the large MOI.This tendency is similar to the shape perception sensitivity that degrades as MOI increases [50].
We surmise that the above fnding originates from two reasons.First, it can be due to prior knowledge, which signifcantly impacts haptic perception.For example, a person's prior information about the material, such as visually provided diferences [85] or expected compliance diference [31], can modify the perceived softness.In dynamic touch, the object's diameter alters the perceived length [6,49], and prior knowledge can also be responsible for that.Chan reported that descriptive knowledge about the diameter can infuence the perceived length of a wielded rod [6].A metal object (with high thermal conductivity) generally has greater density than a plastic object.This prior knowledge may lead people to infer that the metal object should be shorter to maintain the same moments of mass distribution.After the experiment, we interviewed the participants about the perceived length diferences.Five participants explicitly mentioned that the metal object felt heavier and shorter than the plastic one.Second, the thermal sensations that metal objects feel cold can be relevant.Cold sensations increase the perceived hardness of an object [68,79].It can subsequently reduce the perceived length, as observed in Exp. 1. Cold sensations also make the object feel heavier in weight [10,65].An object with a larger mass is perceived as slightly shorter compared to an object with a smaller mass and the same moments of mass distribution [33,34].Therefore, cold sensations can be responsible for the reduced length perception.

EXP. 3: EFFECT OF SURFACE TEXTURE
The last experiment investigated how the handle's surface texture infuences the perceived length of a handheld object.We hypothesized that textural features would alter the pressure distribution on the hand, which could work as a cutaneous cue for length perception.

Methods
The methods common to Exp. 1 and 2 are omitted for conciseness.

Experimental Conditions.
One independent variable was MOI with the same three levels as Exp. 1 and 2. The other independent variable was TEXTURE, which had four levels with diferent bump widths and heights: none (N), medium bump (M), small-width bump (SW), and large-height bump (LH).The texture designs are illustrated in Figure 8(A).These textural variations produce diferent pressure distributions on the hand.
The physical objects used in this experiment are shown in Figure 8(B).Objects with the same MOI had diferent handle textures while keeping their actual lengths, static moments, and weights identical.These objects had a 4 cm diameter, but they difered in bump size, increasing the overall handle diameter.Consequently, the diameter of the thickest part of each handle was 4 cm for the fat handle (N), 4.4 cm for the handle with medium (M) and small-width bumps (SW), and 4.8 cm for the handle with large-height bumps (LH).
The experiment had 12 conditions combining 3 MOI levels and 4 TEXTURE levels in a factorial design.

5.1.3
Procedure and Data Analysis.The procedure was essentially the same as Exp. 1 and 2. The reference stimulus had MOI = 16×10 4 g•cm 2 and TEXTURE = N.Since we had the largest experimental conditions in Exp. 3, we decreased the number of repetitions per condition to 10 and the total number of trials to 120 (12 conditions × 10 repetitions).It was to control the experiment time to less than 1 hour.We conducted data analysis using the data obtained from the latter eight repetitions.
The above analysis revealed the following efects of TEXTURE depending on the level of MOI: 1) At the small MOI, there were no signifcant diferences between the texture levels; 2) At the medium MOI, the texture of large-height bumps (LH) decreased the rod's perceived length compared to the textures of medium (M) and small-width bumps (SW); and 3) At the large MOI, the texture of small-width bumps (SW) increased the perceived length of the rod compared to the fat texture (N).

Discussion
In Exp. 3, the MOI of a handheld object changed its perceived length substantially, as expected.Furthermore, the results demonstrated that the efect of surface texture, added as bumps, is signifcant for length perception.The addition of small-width bumps to the handle increased the perceived length of the rod (5-6%).However, the perceived length decreased when large-height bumps were added,  ranging from 3% to 11%, depending on the MOI range.This fnding is a new piece of knowledge for the haptic perception literature.The exact reasons for the texture efects are elusive yet.However, it is noteworthy that the pressure distribution on the fngers and palm occurring by an object with a bumpy surface texture is more complex and irregular than that by an object with a fat surface.Such skin deformation afects weight perception [40,52,66,77], and it can subsequently infuence length perception [33,34] in a causal chain.
Previous studies found that increasing the handle diameter shortens the perceived length of a handheld object [6,49].This pattern contradicts the increased perceived length observed when the smallwidth bumps were added to the handle, as it increased the handle diameter by 4 mm.Therefore, the efects of surface texture observed in this experiment are unique and cannot be attributed to the increased handled diameter.

CONCLUSIONS AND GENERAL DISCUSSION 6.1 Summary of Results
Virtual objects, with diverse shapes and materials, held in the avatar's hand are represented by a physical controller wielded in the user's hand in the real world.In this scenario, shape-changing controllers can enhance the realism of user experiences by rendering similar virtual and real shapes.Furthermore, the introduction of a "material-changing" functionality can elevate the current shape-changing controllers to the next level; see Section 6.3 for relevant haptics technologies.However, shape-changing and material-changing may interfere with each other, and it is essential to understand the efects of these haptic properties on the controller's perceived shape.
Previous research has highlighted the infuence of a handheld object's moments of mass distribution on its perceived length.However, this study took a step forward by investigating whether the material properties of the object can afect its length perception.Major fndings are summarized below: (1) The softness of an object handle has a signifcant efect on the perceived length.Increasing the softness increases the perceived length in the tested MOI range.(2) An object handle with high thermal conductivity, such as metal, reduces the object's perceived length compared to one with low thermal conductivity.(3) Handle texture, particularly bumpiness, is an infuential factor in altering the perceived length.A certain combination of bump width and height extends the perceived length, while excessive bump height reduces the perceived length.
The above results correspond to novel scientifc facts, which ofer new research problems in haptic perception by dynamic touch.We also expect these fndings will provide useful guidelines for designing handheld VR controllers.

Comparison between Physical Properties
Comparing the efects of multiple variables evaluated in separate perceptual experiments is not always straightforward.In our case, the three experiments included the identical conditions of the common variable, MOI.We can leverage it to assess the extent of the three material properties' efects on the perceived length change.  1 summarizes the statistical efect sizes, including partial and generalized 2 , of the four physical properties obtained in our three perceptual experiments.According to 2 , MOI is the most infuential factor in describing the perceived length, followed by SOFTNESS, TC, and TEXTURE.We can also compare the relative efects of each property on perceived length using the ratios of 2 s between MOI and the three material properties.These ratios show that SOFTNESS contributed 6% (relative to MOI) to the perceived length distribution, with TC of 4% and TEXTURE of 5%.These fndings suggest that the three material properties have similar relative efects to the third moment of inertia ( ; 6%) and mass (11%) on length perception [33].
In addition, Figure 10 presents a log-log plot combining the mean perceived lengths measured in all three experiments 4 .It also provides length perception models as the linear functions of log( ) for each level of the material variables obtained by function ftting; their equations are available in Appendix C. Using this plot, we can compare the ratios of increase or decrease in perceived length with respect to the reference object (MOI=16×10 4 g•cm 2 , rigid, low thermal conductivity, and fat), which was used in all experiments.In Exp. 1, the perceived lengths increased from the reference object (log( ℎ) = 1.6504) to the very soft object (1.6851).To achieve the same perceived length change, the MOI of the reference object would need to be increased to 17.72×10 4 g•cm 2 .Thus, the perceived length change resulting from varying the handle softness corresponds to a 1.72×10 4 g•cm 2 increase (11%) in MOI from the reference object with the rigid handle.In Exp. 2, the perceived length of the object with high thermal conductivity (1.1839) was reduced from the reference object with low thermal conductivity (1.2094).The same change in perceived length can be obtained with the object of MOI = 14.96×10 4 g•cm 2 with the low thermal conductivity handle, so a 1.04×10 4 g•cm 2 decrease (7%) in MOI.Finally, in Exp. 3, the largest change in perceived length from the reference object was observed with one that had a handle with small-width bumps.The MOI would need to increase to 17.28×10 4 g•cm 2 to have the same perceived length without adding textures to the handle.This value corresponds to a 1.28×10 4 g•cm 2 (8%) MOI increase.In summary, varying the softness, thermal conductivity, and texture of an object handle brought 11%, 7%, and 8%, respectively, of the corresponding changes in MOI.
All three ratios of perceived length change are similar to or larger than the just noticeable diference (JND) of length perception [50].Therefore, material property changes in softness, thermal conductivity, or texture bring perceptually distinct diferences in shape perception through dynamic touch.Simultaneous modulations in multiple material properties may grant a greater efect, and their extent deserves further quantifcation.

Enabling Technologies for Material-Changing Controllers
The softness, temperature, and texture of a controller handle can be modulated using state-of-the-art technologies in haptics.First, a handle with variable viscoelasticity may be implemented using pneumatic actuators [43,80], particle jamming [63], and magnetorheological elastomers [8].Second, adding a function for changing the handle's temperature and thermal conductivity to handheld controllers is possible using standard Peltier elements [42,44,45].Even fexible ones varying both softness and temperature are available [83].Third, making bumps appear and disappear on the handle surface is possible using a grid-style mechanical structure [20] and a fexible array of liquid pumps [60].Improving and combining these technologies to elicit material-changing sensations from a handheld controller will be a challenging but fascinating research direction.

Future Work
Our research reported in this paper poses more questions than answers.For example, a shape-changing controller can have variable stifness along its entire length except for a rigid handle [55].Shape perception, including length perception, of such fexible handheld objects is an important scientifc theme.Additionally, we can explore the efects of temperature (hot or cold objects), rather than thermal conductivity, on length perception.Lastly, the surface textures tested in this study were relatively large, coarse, and regular.The efects of fne and irregular textures on perceived length are also pertinent to material-changing controllers.

Figure 2 :
Figure 2: Structures of physical objects used in all experiments.The objects consisted of three parts: a handle, an aluminum rod, and two brass disks.MOI was modulated by adjusting the positions of the brass disks.The handle properties were altered by replacing the handle's material.

Figure 4 :
Figure 4: A participant wielding physical objects with two hands without visual cues.

5. 1 . 1
Participants.Eighteen participants (nine females and nine males; mean age 23.3 years) participated in this experiment.None of the participants had participated in Exp. 1 or Exp. 2.

Figure 8 :
Figure 8: Physical objects used in Exp. 3. (A) Structures of the handle with diferent bump shapes.Such bumps correspond to macro-textures [38].(B) Experimental conditions.

Table 1 :
Efect sizes measured in the three perceptual experiments.