Multi-Layered Perceptual Model for Haptic Perception of Compliance

Abstract

Haptic rendering of compliance is widely used in human–computer haptic interaction. Haptic impressions of virtual objects are usually controlled through rendering algorithms and devices. However, subjective feelings of compliance are easily affected by physical properties of objects, interactive modes, and so on. So it is important to ascertain the mapping relations between controlled physical parameters and subjective perceptual feelings. In this paper, a multi-layered perceptual model was built based on psychophysical experiments to discuss these relationships in a simplified scene. Interactive signals of physical stimuli are collected by the physical receptor layer, handled by the subjective classifier layer and finally generate the evaluation results of compliance. The physical perceptual layer is used to extract useful interaction features affecting perceptual results. The subjective classifier layer is used to analyze the perceptual dimensionality of the compliance perception. The final aim of the model is to determine the mapping relationships between interaction features and dimensions of perception space. Interactive features are extracted from the interaction data collected during the exploring process, perceptual dimensionality of the compliance perception was analyzed by the factor analysis method, and relations between hierarchical layers were obtained by multi-linear regression analysis. A verification test was performed to show whether the proposed model can predict the perceptual result of new samples well. The results indicate that the model was reliable to estimate the perceptual results of compliance with an accuracy of approximately 90%. This paper may contribute a lot to the design and improvement of human-computer interaction and intelligent sensing system.

1. Introduction

Haptic feedback rendering system is widely used in the areas of teleoperation, virtual surgery, rehabilitation training, virtual reality, and so on. Haptic rendering technology gives us the ability to control the haptic impression a rendering object creates by controlling parameters of object attributes, interactive modes, and so on. In order to develop haptic rendering systems with good sense of reality, it is significant to ascertain the relationships between the controlled parameters and subjective perceptual feelings.

1.1. Factors and Cues of Compliance Perception

The perception of compliance is a kind of subjective assessment of the amount of deformation on an object in response to an applied force [1]. Deformation can be elastic, viscous, or otherwise [2]. Compliance relates to the ability of the object to deform under pressure and is usually expressed by stiffness and Young’s modulus. “Hardness” and “softness” are words to describe subjective compliance perception that people have when they handle compliant materials hard or soft [3]. In our daily life, most of the compliant objects have more than one feeling beside hardness and softness, for example, viscosity and friction. Lots of researches [4,5,6] have focused on the perceptual mechanism between physical parameters of objects and perceptual results for the haptic perception of texture. Some researches [7] fond that, the perception of texture was a comprehensive haptic perceptual property, roughness/smoothness, hardness/softness, and coldness/warmness are basic perceptual dimensions in the perception space of textures. However, the compliance perception, as another important haptic property, though some researches have studied the perceptual characteristics, few of them paid attention to the psychophysical dimensions of compliance and the relationships between interactive data and perceptual results quantificationally through modeling and analysis.

A series of psychophysical experiments have been conducted to measure the perceptual ability of the compliance and test how different factors affected perceptual ability. Blair and Coppen [8] tested some rubber cylinders with elastic moduli of around 1.5 MPa and reported the Weber fraction was 0.13. Koçak, Palmerius, and Forsell [9] reported that in continuous pressure model (discriminate a stiffness change during exploration without taking the probe away from the object), the participants showed a significantly better discrimination performance. Beside experimental conditions, factors from the participants would also affect the perception results. Kaim and Drewing [10] reported that high peak force resulted in better discrimination performance for less compliant rubber stimuli (0.15 mm/N) while discrimination performance did not vary within the investigated range of peak forces for more compliant stimuli (1.5 mm/N). In conclusion, the compliance perception is a complex and important haptic perceptual property and could be influenced by many experimental and subjective factors, such as types of materials, interaction modes or force values applied by users.

Generally speaking, force and displacement changing of fingertips, shape and size over the contact area, and the deformation of the object are necessary haptic cues for estimation and discrimination of compliance. These cues can be further subdivided into kinaesthetic cues and cutaneous cues [11]. Kinaesthetic cues relate to the perception of force on, and movements of, limbs or fingers, while cutaneous cues relate to the perception of the perception of the pressure on the skin. Srinivasan and LaMotte [12] proved that cutaneous information was both necessary and sufficient for accurate softness perception for surface deformable objects. However, for deformable objects having rigid surfaces, the addition of kinaesthetic displacement information was necessary for the compliance discrimination [2]. In the experimental conditions of this paper, stimuli are springs cells having rigid surface, so both of the cutaneous and kinaesthetic information were considered in the analysis process.

In order to determine the specific mapping relationships from physical stimuli to subjective perceptual results of compliance, it is not enough to just analyse these basic haptic cues qualitatively. Some of those basic cues were proved to be effective in certain compliance discrimination scenarios but not in others. It is necessary to extract effective features from the interaction data according to specific experimental condition and perception tasks. Studies [13] have shown that time-dependent cues, usually information about force and deformation changing with time, could convey compliance in a manner more efficiently and rapidly. Lawrence [14] introduced the quality metric “rate-hardness” and linked it to the human perception of hardness of the rigid surface via a psychophysical study. Based on Lawrence’s work, Han and Choi [15] proposed the metric named “extended rate-hardness” for perceived hardness. In addition, they also listed and analysed some common interaction features such as peak force, peak force rate, measured stiffness, and contact velocity.

However, most of these researches only reported the correlation between physical interaction features and perceived results, few of them build models to describe the relationship quantificationally, just like researches conducted for haptic texture perception. Therefore for haptic perception of compliance, it is still unclear the exact relationship between physical measures and perceptual results. This is the problem that our works trying to discuss in simplified scenes.

1.2. Computational Models for Compliance

There are different methods to describe the relationships between objective physical measures and subjective perceptual results. The original researchers [16] used the Stevens’ law as a psychophysical function to convert physical stimuli into subjective perceptual intensity. This method mapped each different of physical quantity into a proportion in perceptual responds. But such a simple mechanism of static mapping would lose lots of useful information brought by time-series cues of the interaction process (such as force and movement), which has been confirmed to have significant influence on haptic perception of compliance [13,14,15]. In order to combine movement and force information together instead of static perception models, Rank and Hirche [17] proposed a kind of dynamic state observer model to predict the perceptual thresholds varying with the specific interaction movement with the environment. But their models were only effective for exploring pattens restricted to sinusoids of different amplitudes and frequencies, other movements must be confirmed for a more general applicability to haptic teleoperation systems. Di Luca et al. [18] narrated that, due to the limitations of the experimental environment and conditions, subjects usually cannot obtain all the perceptual information they need for compliance evaluation, so they face the task of inferring the perceptual attribute from the incomplete signals. In order to improve the performance of compliance perception, it is particularly useful for human perception system to combine information derived from multiple sensory channels and previous experience. The human brain would make best guesses and correct and update the estimate based on as much as information available. These indicated that the human haptic evaluation of compliance is more similar to a decision mechanism model based on interaction information during the process. The information can be obtained not only from haptics, but also from other sensory channels such as vision and audition.

Di Luca [2] concluded that human perceptual system handled the information in two terms, the combination and the integration. But it is still not entirely clear whether the information was combined before integrated, or integrated before combined. They proposed a model of human perception of compliance based on Bayesian decision theory. The model indicates how complementary and redundant information should be treated. However, their model was more suitable for interaction of multi-sensory fusion, such as touch and vision. The effect is still need to be verified for the perception of haptic single channel. Some researches also showed that, when receiving external interaction information, the human brain obtains the feeling regularity of stimuli thorough a multi-layer network model instead of processing the data directly [19]. Different kinds of mechanical and thermal receptors beneath the skin are stimulated by compliant objects and make corresponding responses. The patterns of stimulation are transferred through different channels and mapped on to outer areas of the brain and are processed at a first level related to organoleptic judgments [20]. Then different kind of these judgements are passed to other areas of the brain, combined, and later may also be compared to memories to finally create subjective perceptual feelings [21]. This conclusion indicates that a multi-layered model was consistent with the mechanism of haptic perception and suitable to describe the relationships from physical stimuli to subjective perceptual feelings. Multi-layered model and analysis of correlation and regression are popular methods to describe the relationship between physical measures (independent variables) and human subjective response. They are basic methods of quantifying or translating physical measures to subjective response used in many previous works [22,23,24,25] for haptic rendering and evaluation systems.

So in this paper, a multi-layered model for perceptual compliance based on interaction cues was built to explain the relationship between physical measures and perceptual feelings during haptic rendering perception. A hierarchical layered structure of the compliance perception was proposed to describe the perception mechanism (Section 2.1). Psychophysical experiments were conducted to collect the interactive data and estimated results of compliance perception (Section 2.2). Factors of interactive features and perceptual dimensionality were analysed and determined by factor analysis method (Section 3.1 and Section 3.2). Relationships between hierarchical layers were obtained by multi linear regression analysis (Section 3.3). A verification test was also performed to show whether the proposed model can predict the perceptual result of new samples (Section 4). Though the experiment results and the application of the multi-layered model may be limited by the experimental environment and the performance of the device, it may give some advice to the design and improvement of haptic rendering system or intelligent haptic perception for robots.

2. Method, Experiments, and Data Collection

2.1. Method and Concept

Based on previous researches, a hierarchical layered structure of the compliance perception and physical measures was proposed in this paper as a framework for the multi-layered model, as shown in Figure 1. In the hierarchical layered structure, the physical perceptual layer is corresponding with the process of compliant stimuli being perceived by the mechanical receptors beneath the skin. The subjective classifier layer is corresponding with the process of patterns of stimulation being mapped on to the brain and integrated to generate the final estimated result. By extracting interaction features from the data collected in the exploring precess, redundant information is treated and time-series cues are fully taken into consideration and characterized. The process of perceptual synthesis for compliance is also characterized by the process of perceptual dimension analysis of perception space.

This paper focused on the compliance perception of objects having rigid surface. Interactive information from other perceptual channels such as vision and audition was not taken into consideration. So the haptic information such as position and force cues necessary for the compliance estimate is the input of the proposed model and transmitted according to the term of combination of interaction information. According to previous researches, the source information may have different ways of expression and measurement, which were proposed as effective metrics or features. In addition, compliance is a comprehensive perceptual feeling that consists of different dimensions of feelings. So in order to take all these factors into consideration, the interaction features and perceptual dimensionality are two important intermediate layers for transmitting information. Some concepts and function explanations of the multi-layered model are as following:

Physical stimuli: Physical stimuli are samples provided in this paper for the subjects to perceive. The compliance of these samples are generated through a haptic rendering device by controlling the parameters which reflect the properties of the objects or experimental conditions. For example, for the compliance perception, stiffness and damping coefficient are concerned physical parameters for object properties and exploring speed for experimental conditions.

Interactive features: Interactive features are metrics that dominating for the perceptual results extracted from interactive data, which are source information necessary for an accurate estimation of compliance. As mentioned in the introduction, force–time and position–time cues such as peak force and rate hardness are most common effective features that have high correlation with perception results. In practical perception tasks, property parameters of samples are usually difficult to measure, while interaction data are easier to obtain from data record, so in the multi-layered model, interaction features are the input variables.

Perceptual dimensionality: Perceptual dimensions are different perceptual properties that form the comprehensive perception space for a kind of haptic feeling. For example, elastic and viscous deformation are most common kinds of deformation for the compliance perception, and for haptic texture, the dimensions are roughness/smoothness, hardness/softness, and coldness/warmness. Perception dimensionality is usually determined by psychophysical experiments or factor analysis based on interaction features.

Based on the hierarchical layered structure, the final estimated results of the compliance perception can be expressed as:

$$\left(\begin{array}{c}{I}_{d1}\\ I_{d2}\\ \cdots \\ I_{dm}\end{array}\right)=T_{mn}·\underset{Factors}{\underbrace{\left(T_{nk}·\stackrel{features}{\overbrace{\left(\begin{array}{c}{f}_1\\ f_2\\ \cdots \\ f_k\end{array}\right)}}\right)}}+C$$

(1)

and the factors as:

$$Factors=\left(\begin{array}{c}{F}_1\\ F_2\\ \cdots \\ F_n\end{array}\right)=\left(\begin{array}{cccc}{t}_{11}& t_{12}& \cdots & t_{1k}\\ t_{21}& t_{22}& \cdots & t_{2k}\\ \cdots & \cdots & \cdots & \cdots \\ t_{n1}& t_{n2}& \cdots & t_{nk}\end{array}\right)·\left(\begin{array}{c}{f}_1\\ f_2\\ \cdots \\ f_k\end{array}\right)$$

(2)

In Formulas (1) and (2), the $I_{d1}$ to $I_{dm}$ are perceptual intensity for corresponding dimensions in the perceptual space of compliance. Factors $F_1$ to $F_n$ are orthogonal factors used in the analysis of perceptual dimensions, they are generated by linear combination of interaction features having high correlation with the corresponding factor. $f_1$ to $f_k$ are effective interaction features extracted from interaction data. m, n, and k are the number of perceptual dimensions of compliance, the number of factors used the intermediate transition layer, and the number of interaction features extracted respectively. $T_{nk}$ is the transition matrix that recombines each feature component and generates orthogonal transition factors related to perceptual dimensions, it represents the transform in the physical perception layer of the hierarchical structure. $T_{mn}$ is the transition matrix used to transform these factors into perceptual dimensions, which corresponds with the transform in the subjective classification layer. C is the constant matrix. In this way, interactive features (layer of physical perception) are transformed and mapped into subjective perceptual feelings (layer of subjective classification) through the factors and dimensions (intermediate layers).

Several steps were finished in this paper to analyse the relationships from objective interaction data to subjective estimated results by building the multi-layered model based on psychophysical experiments, where proper physical parameters were controlled to generate test samples. Firstly, interaction cues such as force, deformation, and velocity information were collected by sensors in the experiments. Then interaction features dominating for perceptual results were extracted based on correlation analyses from the interaction data. Thirdly, perceptual dimensionality was analysed through the psychophysical experiments, where adjective labels and corresponding rating scores were used by subjects to describe perceptual feelings for compliance and explain meanings of possible perceptual dimensions of the perception space. Finally, the methods of factor analysis and the regression analysis were conducted to determine the relationships between each layers. Based on the steps above, a multi-layered model was built to describe the relationships between interactive data and subjective perceptual feelings for compliance.

2.2. Experiments and Data Collection

Participants

14 participants (nine males and five females) from Southeast University aged on average 25.6 (24–28 years old) took part in the experiments. All of them were right-handed, reported having no cutaneous or kinesthetic problems, were ignorant of the purpose of the experiments, and signed informed-consent forms before the experiments, and they were compensated for the work in the experiments.

Stimuli

In haptic rendering of compliance, a model of stiffness, damping and inertia can cover most compliant objects in our daily life. When interacting with compliant objects with a rigid surface such as keyboard keys, the inertia is too small to influence the perceptual results. In contrast, the palpation speed is a factor that can’t be ignored [26]. Taking the mechanical keyboard as an example, researches [27] measured that the average stiffness, damping and pressing speed were about 0.1 N/mm, 0.002 Ns/mm, and 80 mm/s, and they would change with the kinds of keys and users typing habit. For our experiments, it is hard to control these parameters of compliant samples easily as we want by finding proper objects in the real world. So psychophysical experiments of this paper were conducted through a haptic rendering device (Geomagic Touch) to control physical parameters expediently. Three levels of each parameter were selected according to perceptual range and limitations of device performance, so there were 27 samples in total for participants to perceive and rate in the experiments. The stiffness levels were 0.05 N/mm, 0.10 N/mm, and 0.15 N/mm. The damping levels were 0.001 Ns/mm, 0.003 Ns/mm, and 0.005 Ns/mm. The exploring speed levels were set as 40 mm/s, 80 mm/s, and 120 mm/s. There was a speed guiding ball in the rendering environment to guide the users exploring the compliance samples as evenly as possible at the controlled speeds (seen in Figure 2). The 27 test samples combined by the three kinds of parameters were presented to the participants in a disordered sequence so as to eliminate the influence of regular pattern of sample combination, the 27 test samples and corresponding parameters are shown in

Table 1. Compliant samples used in the experiment.

	Stiffness	Damping	Exploring Speed
Samples	k (N/mm)	b (N$\mathit{·}$s/mm)	v (mm/s)
1	0.1	0.005	120
2	0.1	0.001	80
3	0.1	0.001	40
4	0.1	0.003	120
5	0.15	0.001	80
6	0.15	0.003	120
7	0.15	0.005	80
8	0.05	0.003	80
9	0.15	0.003	40
10	0.15	0.001	40
11	0.05	0.005	80
12	0.1	0.005	80
13	0.15	0.003	80
14	0.1	0.001	120
15	0.05	0.001	120
16	0.05	0.005	120
17	0.1	0.005	40
18	0.05	0.001	80
19	0.15	0.005	40
20	0.15	0.005	120
21	0.05	0.005	40
22	0.05	0.005	120
23	0.1	0.003	40
24	0.15	0.001	120
25	0.05	0.001	40
26	0.05	0.003	40
27	0.1	0.003	80

.

Apparatus

As shown in Figure 2, in the virtual rendering environment, each sample was generated in the middle block by the Geomagic Touch combined with a stiffness level and damping level, with the speed guiding ball (on the left) moving at the corresponding speed level. While in the real interaction environment shown in Figure 3, subjects could explore the compliance samples by moving the arms of the Geomagic Touch through the finger sleeve mounted on the arm. When the users pressed the finger sleeve by using their index fingers vertical downward and the proxy ball of the Geomagic Touch in the virtual environment moved into the area of the blocks along the y-axis, the force feedback consisting of the corresponding stiffness force and damping force could be felt by the participants along the y-axis. In addition, the grey blocks would not have visual deformation change because researches indicated that cues changing in visual channel have significant effects on haptic perceptual results [28].

Data collection

During the interaction process, data of force feedback on the fingers, exploring position, and the exploring speed of the finger were collected by corresponding sensors. Specifically, force change data were recorded by the flexible pressure sensor made by Tekscan, whose force range is 4.4 N at the linearity error $\pm 3\%$ of full scale. The thickness of the sensor was 0.208 mm and the diameter of the sensing size was 9.7 mm, so it is feasible to install the sensor inside the finger sleeve through tight constraint to measure the pressure as shown in Figure 3. The moving tracks of the proxy ball were recorded by the Geomagic Touch, so the position and speed of fingertips could be obtained by coordinate transformation from the virtual coordinate to the world coordinate. All the interaction data were recorded at the frequency of 1000 Hz synchronously and converted to the international unit.

Procedure

The method of semantic differential method was used in this part according to Okamoto’s [7] summarization, where they concluded that it’s a widely used method for specifying the dimensions of tactile perception. In the method, subjects rate samples one by one using scales whose range represents the perceptual magnitude of adjectives labels, such as “hard” and “soft”. These adjective labels represent different perceptual feelings and are usually collected in the preliminary experiment from the subjects. Then a factor or principal component analysis would be conducted to determine the proper perceptual dimensions.

There are two experiments in this paper to collect subjective perception results and interaction data. As the participants were not so familiar with the operation of the Geomagic Touch, there was a short training (about 3 min) before the experiments.

In the first experiment, participants could go through all the 27 samples (presented unordered) at the habitual speed. Meanwhile, they were asked to describe the feelings of compliance using adjectives as much as possible without any prior hints. The first experiment took about 25 min per participant. Assumptions were made that human haptic perception of compliance was a combination of different perceptual properties or dimensions, just like haptic perception of texture, so the adjectives were collected and grouped in order to explain the physical meaning of the potential perceptual dimensions. The selected adjectives were also provided to the subjects as the reference perceptual property labels, which indicated that subjects should rate the samples according to the given property labels and focus on the perceptual feelings belong to the corresponding property label.

In the second experiment, in each trail, one of the selected adjective labels would be provided to the subject. Participants were firstly asked to perceive all 27 samples and select out the stimuli having the highest and lowest perceptual intensity for the given adjective label as references. According to the results of experiment one, there were five adjective labels in total for the participants to rate. As shown in Figure 2, for a given adjective label needing to be focused, reference samples would be present in the left and right blocks respectively. The score of the references having the lowest perceptual intensity was 0 while the score for the highest intensity was 100. Participants were asked to give a score between 0–100 for the test samples in the middle block comparing with the reference samples. There was no limit on the numbers of visits to the test or reference samples, but the subjects were asked to make a response in 60 s. If the participant finished rating all 27 samples for the current adjective label, data would be saved and the next adjective label would be given to repeat the process until the samples were rated under all the adjective labels. Rating for each adjective labels took about 30 min and the whole procedure took nearly 3 h. Subjects were allowed to take breaks between decisions if they wanted.

Results

In the first experiment, subjects used the adjectives of hard, soft, stick, smooth, rough, elastic, crisp, granular, and unstable to describe their perceptual feelings about the compliance rendering. In relevant research, Rosenberg [29] analysed the perceptual decomposition of virtual haptic surface and concluded that the perceptual compliance feelings could be decomposed into five perceptual stages and properties: the crispness of the initial contact, the hardness of the quasi-static pressing, and the cleanness of the final release. Based on previous researches as well as daily experience, and also in order to cover all possible perceptual dimensions as much as possible, adjectives collected from all the participants were classified into five adjective labels: hardness, viscosity, roughness, crispness, and cleanness. In the second experiment, participants rated the perceptual intensity of these given adjective labels. The results are shown in Figure 4.

3. Analysis and Discussion

According to the process of semantic differential method, factor analysis method was used in this paper to determine the perceptual dimensionality of the compliance perception. Principal component analysis with varimax rotation was performed to classify physical features having strong correlation to ensure no multicollinearity between independent variables. Then multiple linear regression analysis was conducted to show the relations between layers of the multi-layered perceptual model of this paper. The process of the analysis was completed by the software IBM SPSS Statistics 25.

3.1. Perceptual Dimensionality of Compliance

Researches [7,20,30] have shown that the adjective labels can be classified into groups and make it easier to interpret the data by reducing the dimensions through the factor analysis. Figure 5 and

Table 2. Result of principal component analysis for adjective labels.

	Principal Components
Adjective Labels	1	2
Hardness	0.170	0.984
Viscosity	0.851	0.122
Roughness	0.955	0.201
Crispness	0.892	0.409
Cleanness	0.962	0.112

show the results of analysis for perceptual adjective labels. Components have loadings higher than 0.5 are marked in bold. As seen in Table 2, the rotational component matrix indicates that the selected five adjective labels can be expressed by two orthogonal components. After presenting these adjective labels in the rotated spatial component diagram shown in Figure 5, the conclusion can be obtained that there are two basic perceptual dimensions forming the final feeling of compliance. Dimension one is hardness and dimension two is related to the other four adjective labels. Actually, in the experiments, some participants respond that these four adjective labels are reflecting similar perceptual feelings. So the word “viscosity” was selected to represent the feeling property of dimension two in this paper. Dimension one and dimension two are orthogonal, that means that dimension “hardness” and dimension “viscosity” are equally important in the evaluation of compliance in this paper, and they would not have interaction effects on each other.

3.2. Factors of Interaction Features

It’s hard to find regular patterns when using interaction data as independent variable directly. So researches usually extract some effective feature metrics that have high correlation with perceptual results as inputs. What’s more, these features can also take the perceptual characteristics in to consideration and compress the data in the models. Researches [31,32,33] showed that humans usually perceive different haptic properties by using different movement schemes or patterns of contact, named the exploratory procedures. For example, the lateral motion for the texture perception, the static contact for the temperature perception. Klatzky and Lederman [33] concluded that the effective exploratory procedures during compliance judgments was the pressure associated with encoding of compliance, characterized by application of forces to object (usually, normal to surface), while counterforce was exerted (by person or external support) to maintain its position. According to relevant studies mentioned in the introduction (Section 1.1), the conclusion can be obtained that features related to force and position cues varying with time are key features that contribute to haptic perceptual results of compliance. Part of a typical finger force and position changing with time curve in the exploring process and some key stages and features are shown in Figure 6. The method of factor analysis was conducted to analyse the relationships between these features and perceptual results. The results are shown in

Table 3. Results of principal component analysis for interaction features.

	Principal Components
Interaction Features	1	2	3	4
peak force	0.939	0.098	0.029	−0.158
peak force rate (press)	0.914	0.189	0.081	0.234
peak force rate (release)	0.854	0.282	−0.047	0.201
initial speed	0.151	−0.052	0.882	0.305
release speed	−0.001	0.304	0.139	0.845
extended rate-hardness (press)	0.138	0.571	−0.688	0.367
extended rate-hardness (release)	0.441	0.420	−0.090	−0.648
initial force change rate	0.182	0.723	0.171	−0.202
release force change rate	0.555	0.548	−0.117	−0.316
rate-hardness	0.874	−0.098	0.090	−0.214
measured stiffness	0.918	0.165	0.078	−0.237
measured damping	0.058	0.867	−0.076	−0.007
damping force	0.066	0.575	0.767	0.050

. There were four principal components for interaction features with 84.8% of the total variance in this paper. Components that have loadings higher than 0.5 (arbitrary limit) are in bold. Factor one had high correlation with the features peak force, peak force rate of contact, peak force rate of release, rate hardness, and release. Factor two was the dimension having high loadings on initial force force change rate and measured damping. Factor three had high correlation with features of initial speed and extended hardness of contact. Factor four had high loadings with release speed and extended rate-hardness of release.

3.3. Multi-Layered Perceptual Model Based on Regression Analysis

Based on the hierarchical layered structure of the compliance perception and physical measures, multiple linear regression analysis was implemented to analyse the relations between layers of physical parameters, interaction features and perceptual dimensions. In the factor analysis based on principal component analysis with varimax rotation, adjective, and interaction features were grouped according to correlation into different dimensions and feature factors with no multicollinearity. Principal components score of these dimensions and factors can also be used in the regression analysis with a stepwise method. The linear regression analysis between layers was conducted two times in this paper: (1) physical parameters as independent variables and interaction features factors as dependent variables; (2) interaction features factors as independent variables and perceptual dimensions as dependent variables. In this way, the relations between different layers from physical parameters to subjective perceptual results were analysed quantitatively and could be described through a multi-layered perceptual model, shown in Figure 7. Solid and dash lines correspond to the positive and negative coefficients between layers and thickness of the lines represents the values of the standardized coefficients. High values of $R^2$ (0–1) indicate that the model can fit the relationship between independent variables and dependent variables very well. So according to the multi-layered perceptual model, perceptual compliance can be expressed as Formulas (3)–(5).

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{I}_{d1}=Hardness=23.99F_1+8.17F_2+3.29F_3-7.34F_4+44.80\hfill \\ I_{d2}=Viscosity=-2.59F_1+18.88F_2+2.99F_3-1.85F_4+41.98\hfill \end{array}\right.\end{array}$$

(3)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{F}_1=20.54k-45.76b+0.008v-2.36\hfill \\ F_2=1.31k+474.46b+0.003v-1.74\hfill \\ F_3=2.24k+109.33b+0.09v-1.21\hfill \\ F_4=-6.95k-20.93b+0.022v-1.09\hfill \end{array}\right.\end{array}$$

(4)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{F}_1=0.18f_1+40.99f_2+24.15f_3+4.02f_{10}+3.93f_{11}-2.07\hfill \\ F_2=210.91f_8+210.80f_{12}-1.29\hfill \\ F_3=12.31f_4+-5.37f_6+219.9f_{13}-1.43\hfill \\ F_4=16.34f_5-3.30f_6-13.94f_7-1.82\hfill \end{array}\right.\end{array}$$

(5)

Here $I_{d1}$ and $I_{d2}$ are the perceptual intensity of dimensions in perceptual space of compliance. $I_{d1}$ is “hardness” and $I_{d2}$ is “viscosity”. The factors $F_1$ to $F_4$ are four principal components consisting of interaction features according to the results of the factor analysis in Table 2. k, b and v represent the stiffness, damping, and exploring speed respectively. $f_1$ to $f_{13}$ are interaction features used in this paper corresponding with the results in Table 3. If the parameters of the compliant object (k, b) and exploring speed (v) are known, Formulas (3) and (4) can be used together to estimate the perceptual result. In fact, it is usually hard to obtain or measure these basic parameters for new samples in the real word, while it is very convenient to collect interaction data such as force and position by corresponding sensors and then extract interaction features. So in actual application, the inputs of the model are usually useful interaction features and the outputs can be obtained by Formulas (3) and (5).

4. Verification Experiment

In this part, the multi-layered perceptual model was verified. In the verification experiment, nine new samples generated by new parameters (stiffness: 0.04 N/mm, 0.06 N/mm, and 0.09 N/mm; damping: 0.004 Ns/mm, 0.006 Ns/mm, and 0.009 Ns/mm; exploring speed: 80 mm/s) were tested. These samples were perceived and evaluated by the participants in the same experimental conditions. During the progress, interaction data were also collected to extract interaction features. Predicted scores of these new samples could be calculated by using the multi-layered perceptual model. So the reliability of the model could be verified by comparing the estimated results with the real subjects’ response from the experiment. The actual and estimated results were shown in Figure 8, it’s obvious that for the nine new samples in the verification experiments, most of the samples have approximate estimated values of both dimensions compared with real subjects’ response. In order to calculate the estimating errors of the multi-layered model, values of the compliance magnitude from both subjects’ response and the multi-layered model were plotted as x-axis and y-axis with an equivalent reference line, as shown in Figure 9. The average relative errors for the nine new tested samples were 13.3% (std = 0.104) and 6.8% (std = 0.038) for dimensions of “hardness” and “viscosity” respectively. That means that the accuracy rate of the multi-layered model for dimensions of “hardness” and “viscosity” were 86.7% and 93.2% respectively. We perform a single sample t-test (confidence level = 0.95) for the relative errors between the “subjective perceived” and “model predicted” responses: for “hardness”, t = 0.011, p = 0.992 (>0.05) and for “viscosity”, t = −0.003, p = 0.0.997 (>0.05). This indicates that the model of this paper has a acceptable estimated performance for the compliance perception, compared with similar models proposed in previous researches [25,34], whose average accuracy rate were about 80%.

There are several factors that may affect the accuracy of the model in this paper: firstly, the extraction and selection of interactive features. More interactive features extracted from the data do not mean better results. Features having high correlation with subjective perception would contribute a lot to the estimated results. In addition, with the further study of compliance perception mechanism, some new effective features or metrics might be proposed, which would improve the model of this paper. Secondly, the analysis method of factors and dimensions. Factor analysis and linear regressions were used in this paper. These are linear analysis methods. Some nonlinear analysis method or methods based on machine learning may have better fitting effects for the perception of compliance, which is worthy of further study in the future. Furthermore, due to the limitations of experimental conditions, only parameters of stiffness, viscosity, and exploring speed of the compliant objects were controlled in the simple perceptual tasks in this paper. Much more cues should be taken into consideration to build models for more complex perceptual tasks in the future.

5. Conclusions

In this paper, a multi-layered model for perceptual compliance based on interaction cues was proposed to analyse the relationships between physical measures and perceptual feelings. Features extracted from interaction data were the input and subjective responses were the output in this model. A hierarchical layered structure of the compliance perception was used to simulate the perceptual mechanism. Based on analysis results from the experiments, conclusion can be obtained that in the subjective classification layer, subjective feelings of compliance are composed of two basic perceptual dimensions, the ”hardness” and “viscosity”. In the physical perception layer, interaction features related to perceptual results can be grouped and explained by four orthogonal factors. By multi linear regression analysis between layers, dimensions and factors of each layer were connected and the relationships between inputs and outputs were finally determined. Results of the verification experiment showed that the model proposed in this paper can predict the perceptual results very well based on interaction data. This research may contribute to the quantitative estimation of compliance, and provide guidance for designing and improvement of haptic displays and teleoperation systems.