Fuzzy Logic for Culture-aware Robotics Barbara Bruno1 , Fulvio Mastrogiovanni1 , Federico Pecora2 , Alessandro Saffiotti2 , and Antonio Sgorbissa1? 1 University of Genova, Dept. DIBRIS, Via Opera Pia 13, 16145 Genova, Italy {barbara.bruno,fulvio.mastrogiovanni,antonio.sgorbissa}@unige.it 2 Örebro University, AASS Cognitive Robotic Systems Lab, Fakultetsgatan 1, S-70182 Örebro, Sweden {fpa,asaffio}@aass.oru.se Abstract. In the context of culture-aware robotics, we propose a method for the explicit, on-line mapping between cultural variables and robot behaviour parameters which relies on the linguistic variable formalism, fuzzy clustering and the principles of fuzzy controllers. As a case study, we consider the adaptation of the Human-Robot conversational distance to Hofstede’s cultural dimension of Individualism. Keywords: Human-Robot Interaction, Fuzzy Controllers. 1 Introduction In 2013, in an experiment involving Arab and German participants, people were asked to place a Nao robot at a suitable distance to hold a conversation with them [1]. The participants placed the robot at a distance they deemed appro- priate for a conversation among two persons, unconsciously assuming that a robot shouldn’t be too far from a human. Anthropomorphism, albeit important in the interaction between humans and robots [2], is not the only key factor. In the study about appropriate distances, experimenters found a significant differ- ence in the behaviour of the Arab and the German participants, with the latter placing the robot much farther (approx. 85 cm) than the former (approx. 65 cm), in accordance with the social norms of their respective cultures [3]. Culture comprises both nation-wide aspects and individual traits, measured with quan- titative variables, such as net income, but also nominal variables, which only allow for differentiation, such as gender, and ordinal variables, which only allow for differentiation and ordering of values, such as the OCEAN factors describ- ing personality traits [4] or Hofstede’s dimensions for the cultural categorization of countries [5]. The influence of a person’s culture on his attitude towards a robot is the subject of ongoing research [6, 7]. However, culture-dependent robot behaviours are often implicitly set by designers, which makes it hard to adapt robots to a different culture. ? This work was partially supported by a grant of the Fondazione/Stiftelsen C.M. Lerici awarded to the first author. 2 Bruno et al. We propose a method allowing for the automatic, on-line tuning of culture- dependent robot parameters in accordance with a cultural assessment which is explicitly expressed in terms of standard cultural variables. To this aim, we pro- pose the linguistic variable formalism as a unifying representation of cultural variables, and linguistic fuzzy controllers for the definition of the mapping be- tween cultural aspects and robot behaviours. 2 Method The proposed method requires: i) the definition of the cultural variable domain and the acquisition of a training set of data points over the domain; ii) the def- inition of the robot behaviour parameter with the linguistic variable formalism; and iii) the description of the relation between the cultural variable and the parameter in the form of if-then rules for a fuzzy controller. To illustrate the approach, let us consider a personal mobile robot, engaging an assisted person in a conversation. One of the parameters of such a behaviour is the conversational distance P . Literature specifies that suitable values lie within the range P = [0.45m, 1.2m] [8] and that this parameter is directly correlated with Hofstede’s dimension of Individualism C [1, 9, 10]. The mapping between the two variables is only known in a qualitative form: countries with a high individualism score tend to have larger values for the con- versational distance than countries with a low individualism score. Literature provides the Individualism scores of 110 countries3 . Our goal is to define a compact, complete and explicit mapping between C and P , induced by qualitative knowledge like the one above, which allows the robot to tune the conversational distance in accordance with the user’s nationality. 3 Cultural Variables as Linguistic Variables In Fuzzy Logic, a linguistic variable [11] can be expressed as the quadruple: hC, C, LC, µLC i (1) where C is the name of the variable (e.g., Age), C is its domain (e.g., [0, 117] years), LC is the set of linguistic values LC that C can take (e.g., {baby, teenager, adult, elderly}) and µLC is the membership function defining the relationship between a linguistic value and the domain values. In the case of nominal vari- ables we can define a one-to-one mapping between LC and C (e.g., LCGender = CGender = {f emale, male}). In the case of ordinal variables, such as Individu- alism (for which C = [0, 100]), the number of linguistic values LC to consider and their relation with the domain values is less obvious. Most studies arbi- trarily impose LC = {low, medium, high} [9, 12], with a crisp mapping to the domain which only depends on its range. However, we argue that the introduced 3 Publicly available at: http://www.geerthofstede.com/dimension-data-matrix Fuzzy Logic for Culture-aware Robotics 3 discontinuities may be unnatural since the range is arbitrary, and propose the extraction of the linguistic values from available data. We denote with C T = {cT1 , . . . , cTi , . . . , cTI } the set of data points to use for estimating LC and all corresponding µLC . The set can contain publicly available data, as well as user-specific information; moreover, it can be updated at any time, thus allowing for continuous learning and adaptation. We use as training dataset C T for the Individualism variable the aforementioned publicly available scores of 110 countries. The x-axis of Figure 1 spans the domain C and the blue dots mark the 110 scores (e.g., cT1 = 6 corresponds to the score of Guatemala and cT2 = 8 corresponds to the score of Ecuador). 1 0.9 0.8 0.7 CT 0.6 µLC1 (points) 0.5 µLC1 (function) 0.4 µLC2 (points) µLC2 (function) 0.3 0.2 0.1 0 0 10 20 30 40 50 60 70 80 90 100 Fig. 1. Clustering & Fuzzification of the training set C T of a cultural variable C. We propose a three-step procedure for the automatic estimation of LC and all corresponding µLC on the basis of C T , based on the intuition that we can define the linguistic values LC as clusters on C T . We use Subtractive Clustering [13] for the estimation of the number of clusters to use and Fuzzy C-means Clus- tering [14] for the association of the points to the clusters. More specifically, the algorithm computes for each point cTi its membership value µi,LC to each cluster LC. In our case: i) the Subtractive Clustering algorithm identifies 2 as the opti- mal number of clusters; ii) Fuzzy C-means Clustering computes the membership values µi,LC1 to cluster LC1 (orange dots) and the membership values µi,LC2 to cluster LC2 (purple dots). Finally, we approximate each membership function µLC with a two-terms Gaussian function defined as: (c−β1 )2 (c−β2 )2 − 2 − 2 γ1 γ2 µLC = α1 e + α2 e (2) where c spans the domain C and the parameters α1 , β1 , γ1 , α2 , β2 , γ2 are esti- mated to best fit the distribution of the values µi,LC . In Figure 1, the yellow line corresponds to the two-terms Gaussian function µLC1 , while the pink line corresponds to the two-terms Gaussian function µLC2 . 4 Bruno et al. 120 1 conversational distance 110 0.8 100 0.6 µnear 90 0.4 µfar 80 70 0.2 60 0 50 50 60 70 80 90 100 110 120 0 20 40 60 80 100 P individualism (a) (b) Fig. 2. (a) Description of the conversational distance as a linguistic variable. (b) Map- ping between the Individualism and conversational distance given by (3). 4 Parameters setting with Fuzzy Controllers Linguistic fuzzy controllers allow for describing relations between linguistic vari- ables in natural language, by means of if-then rules specified over the respective linguistic values [15]. Let us assume that we describe parameter P with the lin- guistic variable hP, P, LP, µLP i, with LP = {near, f ar} as shown in Figure 2(a); then we might define the rules as: ( if C is LC1 then P is near (3) if C is LC2 then P is f ar Fuzzy controllers require the specification of: i) the fuzzification method, which takes a value c∗ ∈ C in input and computes the linguistic value LC ∗ it corresponds to; ii) the inference method, which solves the set of if-then rules, and iii) the defuzzification method, which finally computes the value p∗ ∈ P on the basis of the linguistic values LP ∗ activated by the rules. We use a fuzzy controller relying on Mamdani implication and composition-based inference, and on the Center-of-Area defuzzification method. Then, the set of rules specified in (3) generates the continuous mapping C → P shown in Figure 2(b). As an example, c∗ = 38 (Arab countries) is mapped to p∗ = 69.9cm, while c∗ = 67 (Germany) is mapped to p∗ = 100.7cm. A mock-up system has been implemented in MATLAB (R2014a), making use of the Fuzzy Logic Toolbox (2.2.19) and the Curve Fitting Toolbox (3.4.1). 5 Conclusions Cultural adaptation of robots is an important but under-addressed problem. We have presented an approach to dynamic, on-line cultural adaptation based on the mapping of cultural variables to parameters of robots behaviours. We have illustrated our approach on a simple case involving one variable and one parameter, but work is under way to generalize this approach to consider several cultural variables and several behavioural traits. 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