Luke Hespanhol
email: luke.hespanhol@sydney.edu.au

Peter Dalsgaard
email: dalsgaard@cavi.au.dk

Social Interaction Design Patterns For Urban Media Architecture

Keywords: Social interaction, media architecture, media façades, urban HCI, responsive environments, proxemics

Media architecture has emerged as a relevant field of study within HCI since its inception at the turn of the century. While media architecture has the potential to radically affect the social space into which it is introduced, much research in the field was initially carried out through experimental installations in public spaces, often with higher emphasis on examining the properties of this novel type of interface, rather than examining the impact it had on the social context. In this paper, we look back at the field and analyze interactive urban media architecture covering a period of fifteen years of practice with a particular emphasis on how installations have influenced modes and patterns of social behaviour. We classify nine representative installations according to their physical layout, interaction strategies and types of interface. We focus on how these installations were perceived and used by their respective audiences and outline six modes of social interaction that unfold with these installations. From this analysis, we derive seven social interaction patterns, which represent different strategies for designing and employing media architecture to influence social interaction.

Introduction

Media architecture is an emergent field in the intersection of HCI, design, architecture, urban planning, art and sociology. While the use of media technologies in architecture has a long historyexemplified by the neon signs at Times Square in New York dating back to the 1920sthe field we know as media architecture emerged when designers and architects began to embed digital technologies, in particular displays, into the built environment. Decreasing prices of important media architecture components, such as display and sensor technologies, has led to uptake in industry, and it is now a prominent feature in many cities in the world. Simultaneously, the interest in media architecture has grown in the HCI community, and in addition to a dedicated conference series [START_REF]Media Architecture Biennale[END_REF], media architecture research is represented at many HCI conferences, including INTERACT. Typical of emergent disciplines related to HCI, the initial focus in many academic contributions was on technical aspects and the potentials of this novel form of interface. To the extent that social aspects of media architecture have been examined, it has primarily been in relation to specific installations. From these contributions, it is clear that social aspects play a very large role in how media architecture is perceived and used, and that interaction designers working in media architecture need an understanding of both technical and social dimensions to develop successful installations.

In this paper, we therefore offer an overview and analysis of social interaction in media architecture through a comparison of nine representative cases from the relatively brief history of the field. We combine the findings on social interaction from each original case study to offer two complementary contributions: firstly, a framework outlining six different modes of social interaction in relation to media architecture: appreciation, self-expression, playfulness, collective narratives, triangulation, and negotiation of space; secondly, a set of seven social interaction design patterns for media architecture, which represent different strategies for designing media architecture to achieve specific types of social interaction: shadow playing, remote control, smooth operator, soapbox, amusement park, swarm, and automatic gate.

The intended audience of the paper is HCI researchers working within the field of media architecture, who may employ the framework and design patterns, examine the relations between the technical and social aspects of media architecture and to categorise and analyse further installations. In addition, the social interaction design patterns may be of value for HCI practitioners, since they indicate specific strategies for developing media architecture installations.

Background: Social Interaction and Media Architecture

Goffman's seminal works in the dynamics of interaction in public spaces and the consequent impact on human social behaviour provide a suitable framework for analysis of urban interventions with media architecture. In Relations in Public [START_REF] Goffman | Relations in Public[END_REF], Goffman classifies agents in a social context as being either 'singles' (individuals by themselves) or in a 'with' (two or more individuals together). Such a configuration has direct implications on the relationship between the various individuals in a public space: for example, people generally feel more comfortable with approaching 'singles' than groups. Likewise, individuals in a 'with' relationship would behave differently than they would if they were singles in the same space. The notion of design patterns, inspired by the work of Alexander [START_REF] Alexander | A Pattern Language: Towns, Buildings, Construction[END_REF], provides a frame for our analysis, which at the same time is directed towards applying insights in practice. Alexander's patterns for architecture were developed from studies of architectural practice and history, describing recurring configurations of the built environment that over the course of time had proven to have specific effects, such as courtyards as places that facilitate gatherings and social exchanges. Likewise, in the mid-90s, Gamma et al. [START_REF] Gamma | Design Patterns: Elements of Reusable Object-Oriented Software[END_REF] the so-called Gang Of Four (GoF) -inspired by Alexander's work, adapted the concept of patterns to software engineering. By presenting a series of design patterns as reusable solutions for recurrent software development problems, their work underpinned the wide adoption of object-oriented programming and paved the way for greater reusability and scalability of computer systems. While the history of media architecture is much shorter, we can nevertheless begin to derive recurring patterns for social interaction. As in the works by Alexander and the GoF, these patterns can serve as strategies for designers of future media architecture installations.

Social aspects of media architecture have been addressed in many existing contributions; however, they have mainly addressed phenomena and design strategies in singular or few cases. For example, Dalsgaard & Halskov [6] propose that designers can work towards 'situational interaction flexibility' as a strategy for allowing people to ease in and out of specific social configurations around installations; however, this has not yet been replicated. Some social aspects discussed in the literature have started out as singular observations, and have since then been found to hold across different cases. The 'honeypot effect' [START_REF] Brignull | Enticing People to Interact with Large Public Displays in Public Spaces[END_REF], which proposes that the presence of people interacting with an installation will entice new users to start interacting to a higher degree than if nobody was using it, is one such aspect that has been observed in multiple cases. This is related to the observation by Mueller et al. [START_REF] Mueller | Looking glass: A field study on noticing interactivity of a shop window[END_REF] that many people notice the interactive potentials of a media architecture installation by observing other people interacting with it. Yet, an analysis of the recurring patterns of social interaction posed by media architecture design solutions is still lacking in the literature. Our approach in this paper has been to derive social interaction modes and patterns across a range of cases, with respect to common design variables such as layout of the public space and strategies for interaction and feedback. To the extent that existing literature corroborates our findings, we will discuss these in the relevant parts of the analysis.

Methodology

Our approach to analysing social interaction design patterns is based on a study of nine prominent examples of media architecture. We based the selection of works on two main design factors common to any responsive media architecture installation in public space: (a) the type of interface; and (b) the public space layout. We describe below each of those factors, the selection process and the steps taken in our analysis.

Types of Interfaces

Hespanhol & Tomitsch [START_REF] Hespanhol | Strategies for Intuitive Interaction in Public Urban Spaces[END_REF] argue that interactive behaviour around urban media architecture may emerge intuitively as a product of its level of accessibility and the type of feedback it offers, therefore leading to particular modes of social interaction. Analysing a variety of responsive public spaces, they propose a classification of interfaces into three types: performative, allotted and responsive ambient. Performative interfaces are defined as those where the interactive zones are well delimited, yet restricted to a small number of participants, resulting in a natural division of the public into 'performers' (active participants) and 'spectators' (passive participants). Allotted interfaces share the same basic characteristics as performative ones, however are large enough to accommodate a population of participants, so that each no longer has full visibility of the interface. Instead, participants operate locally on their own section of the interface, with interaction therefore distributed across the environment. The third category, responsive ambient, refers to urban interfaces which track and react to the presence of people, however offer indirect and generic feedback rather than responding to specific individuals. We employ this classification in our analysis since it offers a framework for understanding how the design of media architecture may influence the social behaviours of the local public. To that end, we include in our analysis a selection of works that represent a balance of the different types of interfaces (see Table 1).

Types of Spatial Layouts

In categorising and analysing the spatial layouts, we employ two categories: (1) plaza, and (2) thoroughfare. These are based on the concepts of "spatial nodes" and "links" (respectively), as described in urban planning works such as Hillier & Hanson's The Social Logic of Space (1984) [START_REF] Hillier | The Social Logic of Space[END_REF]. We define a plaza as a wide, open public space where a large number of citizens potentially congregate, facilitating social encounters as well as passive social practices as people watching or even loitering. A thoroughfare, by contrast, is a transit area connecting plazas, therefore characterized by the continuous flux of passers-by walking from one destination to another. In our selection, the plaza examples outweigh the thoroughfares; this is representative of the fact that media architecture is more often placed in such settings.

Selection of Case Studies

We adopted the large survey on media architecture by Haeusler et al. [START_REF] Haeusler | New Media Facades: A Global Survey[END_REF], as well as more recent literature, as departing points for our selection process. Within that framework, and to the extent possible, we gave preference to works we had the chance to visit in situ, either on their original setup (Aarhus By Light [START_REF] Dalsgaard | Designing Urban Media Façades: Cases and Challenges[END_REF], Solstice LAMP [START_REF] Hespanhol | Strategies for Intuitive Interaction in Public Urban Spaces[END_REF], Interference [START_REF] Kollision | Interference[END_REF], The Climate on The Wall [START_REF] Dalsgaard | Designing Urban Media Façades: Cases and Challenges[END_REF] and Chromapollination [START_REF] Hespanhol | Strategies for Intuitive Interaction in Public Urban Spaces[END_REF]) or as a subsequent iteration (SCSD [START_REF] Behrens | Smart Citizen Sentiment Dashboard: A Case Study Into Media Architectural Interfaces[END_REF]), since those could offer us first hand access to evaluate the social interaction impact on their visitors. The selection of remaining works was largely determined by the availability in the literature of an indepth account of their social interaction dynamics. Due to length constraints, we decided to include only three more representative works (MyPosition [START_REF] Valkanova | MyPosition: Sparking Civic Discourse by a Public Interactive Poll Visualization[END_REF], Body Movies [START_REF] Lozano-Hemmer | Body Movies. Relational Architecture 6[END_REF] and iRiS [START_REF] Boring | Multi-User Interaction on Media Facades through Live Video on Mobile Devices In[END_REF][START_REF] Wiethoff | Designing Interaction with Media Façades: A Case Study[END_REF]), limiting the number of case studies presented in this paper to nine. While it is important to acknowledge the limitations inherent to this selection process, we will argue that our analysis can provided the basis for further identification of social interaction patterns in urban media architecture. The balance between the different types of interfaces and spatial layouts adopted as selection criteria provides a representative sample of the works found both in the comprehensive field survey by Haeusler et al. [START_REF] Haeusler | New Media Facades: A Global Survey[END_REF] as well as in more recent literature.

Analytical Approach

We have examined the academic contributions describing each installation, the types of interfaces and spatial layouts that it represents and, to the extent that this is described in the literature, the designers' intentions with regards to affecting and/or shaping social interaction. That analysis led us to define six recurring modes of social interaction. By further observing the ways those have been recurrently combined across the selected works, we then identified seven distinct strategies, here labelled design patterns, commonly implemented in the installations to elicit intended modes of social interaction.

Case Studies

The goal of this section is to present a sample of previous and current works in the field (9 in total), illustrating various approaches to augmenting the urban environment. We describe each in regards to (1) the types of interfaces, interactive strategies and spatial layouts; and (2) how social interactions unfolded around it.

Table 1. Media architecture installations.

Smart Citizen Sentiment Dashboard (SCSD)

The Smart Citizen Sentiment Dashboard (SCSD) [START_REF] Behrens | Smart Citizen Sentiment Dashboard: A Case Study Into Media Architectural Interfaces[END_REF] introduced the notion of media architectural interfaces (MAIs), a design approach where a tangible user interface (TUI), positioned on public space, functions as a mediator for people to interact with content in a media façade (the carrier). The TUI was an analogue console allowing users to select one of five discussion topics (environment, transport, safety, public space and housing), and use radio frequency identification (RFID) cards to select among three different moods: happy, indifferent or sad. RFID cards are used in many cities as digital tickets for the public transport system and building access, and thus constitute a familiar instrument for interaction with the urban realm; the design goal, therefore, was to opportunistically intervene in the social situation of people walking along a busy street for a brief civic poll. SCSD was deployed in 2013 to the very large (3700 sqm.) media façade of a commercial building in the largest avenue of Sao Paulo, Brazil, running for three weeks. Due to the scale of the façade and the narrow available space in the sidewalk in front of it, the TUI was positioned across the avenue in a small square near the entrance to the local metro station. A variety of emerging behaviour by passers-by and participants could be observed. A large proportion (28%) explored the interface playfully, expressing conflicting moods to the same topics. The majority (72%) of participants, however, did express meaningful opinions. Awareness about the visualisations was also verified across the broader surrounding space, with a particular prevalence of people taking photos of the façade. According to the authors, however, given the small scale of the TUI compared to the high visibility of the façade, it was clear that most passers-by were unaware of the mechanisms of interaction or even the meaning of the façade graphics, rather enjoying them as visually appealing ambient art.

MyPosition

MyPosition [START_REF] Valkanova | MyPosition: Sparking Civic Discourse by a Public Interactive Poll Visualization[END_REF] is a system designed for civic participation in public spaces. It consists of a polling interface back projected into a 5x2 meters canvas, integrated to depth-cameras used to track passers-by. The interface displays a polling question at the top and graphics depicting a 4-point scale: 'strongly agree', 'agree', 'disagree' and 'strongly disagree'. Each section consists in a visualisation of the votes it has received, each vote displayed as a colourful tile. When a participant is detected in the area in front of the screen, the section corresponding to the position they occupy is animated to indicate preference. They can then vote by positioning themselves along the display in front of a desired section and raising their arms for 2s, during which a dwelling animation is displayed as feedback. Only one participant can interact with the system at any given time. After a vote is cast, a new tile is added to the chosen option. Three modalities of visual feedback are available: (1) identical tiles for all participants;

(2) each tile with the participant silhouette; (3) each tile with the participant image captured by the cameras.

The system was deployed as a field study for a week in 2013, at the cafeteria foyer in a large university in Berlin, Germany. It revealed that the playfulness of the interface was not a hindrance to participation, with people interviewed stating they actually meant the answers they gave. Notably, when participants were identified (i.e. their votes were displayed as tiles containing their images) interaction events decreased and were more evenly distributed across the four options, which might be due to an increased accountability regarding the votes cast. In a broader sense, however, the au-thors' observed social interaction across the whole precinct, beyond the direct interaction zones: there was a considerable level of discussion, social learning and teaching as well as nudging among the public [START_REF] Valkanova | MyPosition: Sparking Civic Discourse by a Public Interactive Poll Visualization[END_REF]. The display was not only noticed but also successfully promoted active discussion about the proposed topics.

Body Movies

Body Movies [START_REF] Lozano-Hemmer | Body Movies. Relational Architecture 6[END_REF] is an interactive installation by artist Rafael Lozano-Hemmer. Large-scale portraits of people are projected onto an urban wall, blanked by strong white lights. As passers-by walk and position themselves in front of the lights, they cast their shadows onto the wall and reveal the images underneath. Since its first inception in 2001 at the V2 Grounding, Rotterdam, The Netherlands, Body Movies has been exhibited multiple times worldwide. Social interaction generally emerges with ease, prompted by the familiarity of shadow playingthe work has famously attracted large crowds, with people readily congregating in front of the exhibition wall and moving around the space in order to cast shadows of different sizes. Notably, however, interaction through the shadow playing itself often becomes the greatest appeal of the work, as opposed to the unveiling of the underlying images, as per the artist's original concept [START_REF] Bullivant | Responsive environments: architecture, art and design[END_REF]. Rather than detracting from the experience, such a spontaneous manifestation actually enriches it, not only by allowing individuals and small groups of friends to express themselves playfully in public, but also reportedly bridging the social gap between non-acquaintances. As it is typical of performative scenarios, interaction unfolds mainly in the area directly in front of the projections, via triangulation [START_REF] Memarovic | Using Public Displays to Stimulate Passive Engagement, Active Engagement, and Discovery in Public Spaces[END_REF]: people communicating through their representations on the façade. Conversely, the spots along the periphery of the space turn into comfort spaces [START_REF] Fischer | Urban HCI : Spatial Aspects in the Design of Shared Encounters for Media Façades[END_REF] where passive appreciation and social commentary can take place.

Aarhus By Light

Aarhus By Light [START_REF] Dalsgaard | Designing Urban Media Façades: Cases and Challenges[END_REF] was an interactive media façade created in 2008 with 180 square meters of semi-transparent LED screen, distributed in a non-rectangular pattern behind the glass front wall of the Musikhuset in Aarhus, Denmark. The façade faced a public park where interactive zones were clearly identified by colourful mats placed on pathways leading to the venue. Aarhus By Light was one of the first public interfaces designed to support a "walk-up-and-use" experience, enabling full body interaction with a media façade via computer vision. It was designed so that passers-by in the park or going to the Musikhuset could quickly figure out the mechanisms of interaction by providing them with a clear and direct feedback in the shape of their silhouettes magnified and integrated with the visual graphics in the façade. When no person interacted with the installation, the façade displayed small, animated social creatures moving through it. When individuals were detected, the creatures would become aware of their presence, come closer to their silhouettes and start playing with them. Social interaction unfolded seamlessly, with both individuals and groups interacting via their representations on the allotted façade. The work also managed to successfully address the various regions of proxemics around the venue: people walking along the streets around the park would become aware of the Musikhuset by observing both the animated façade and the crowd gathering in front of it; people in the park would engage both in active and passive interaction with the work, and those inside the venue could watch the animations in the façade from behind, highlighting the building itself as destination. The work successfully fulfilled its goal of transforming a whole urban precinct and the social dynamics around it with interactive technology.

Solstice LAMP

Solstice LAMP [START_REF] Hespanhol | Strategies for Intuitive Interaction in Public Urban Spaces[END_REF] was a large-scale interactive installation developed by the Design Lab, University of Sydney, Australia, for the 2013 edition of the local Vivid Sydney festival. The work was designed to reactivate a normally underused square facing a 100 meters tall skyscraper in the centre of the city. It consisted of two sections: (1) two interactive rectangular zones in the building forecourt, delimited by projections on the floor; and (2) laser projections on the entire building façade, derived from the interaction unfolding at the forecourt. Equipped with overhead depth-view cameras and data projectors, the installation tracked people within the interactive zones and projected halos around their bodies. If people got close enough, their halos would merge as a feedback to the fact they were then a single entity. During a period of one minute, each halo would become increasingly brighter and eventually pop, moving away from their hosts towards the building and then up towards the top of the building. At that point, a new interactive cycle would start at the forecourt. People could enter the interactive zones from all directions and the work would respond to as many participants as it could fit simultaneously, characterizing the interface as allotted [START_REF] Hespanhol | Strategies for Intuitive Interaction in Public Urban Spaces[END_REF]. Interaction with the work varied across demographics and different situations. The authors observed three main social interaction patterns: (1) children attracting adults;

(2) tentative single 'exploration' versus expansive group 'performance'; (3) "body gloss". The first refer to the fact children would readily break into interaction upon reaching the environment, with many parents noticeably observing them 'trying' the environment before joining them in. The second points to the remarkable difference between the observed behaviour of individuals and that of groups engaging in interaction: when the space was already full of other people by the time they start interacting, participants tended to demonstrate restrain and self-consciousness. However, when engaging with the work as part of a larger group of friends, they would try expansive movements like dancing, jumping or even running around. Finally, the third pattern was the amplification of what Goffman termed 'body gloss' [START_REF] Goffman | Relations in Public[END_REF], i.e. a visual reinforcement of personal space, here prompted by the projected graphics rather than emphatic body language: people not acquainted to each other would often apologise when their halos merged, even without any physical touch actually happening.

iRiS

iRiS [START_REF] Boring | Multi-User Interaction on Media Facades through Live Video on Mobile Devices In[END_REF][START_REF] Wiethoff | Designing Interaction with Media Façades: A Case Study[END_REF] is a system for remote interaction with a media façade via an application running on a smartphone, which allows participants to visualize the façade on their devices, superimposed with a user interface for drawing over the image. It identifies the colour selected by the user and the region of the image they are drawing over and turns the corresponding pixels in the façade accordingly. Multiple users can interact simultaneously with the façade, which is shared on a first-come, first-served basis.

The work was trialled at the Ars Electronica Festival 2010, in Linz, Austria, running at the exterior walls of the Ars Electronica Center. Evaluation conducted by the authors found that participants perceived the façade as (1) easy-to-learn, but also that (2) it might leave users unaware of the actions of others [START_REF] Boring | Multi-User Interaction on Media Facades through Live Video on Mobile Devices In[END_REF]. Concurrent users could see multiple actions unfolding on the screen, but were not always able to see who was responsible for them. Likewise, access to the interface was also sometimes denied to new users. The lack of social protocol for such an unfamiliar public space would often lead to user frustration when participants were not acquainted with each other.

Interference

Interference [START_REF] Kollision | Interference[END_REF] is a permanent interactive installation by Danish company Kollision in a tunnel for pedestrians and bicycles under an avenue in Kolding, Denmark.

Opened in 2014, it consists of colourful light panels installed along the walls of the underpass, equipped with a series of trip-wire infrared sensors. Passers-by are detected as they move, with the light panels in front of them turning on in response. The light effects follow the direction of their movement, progressively illuminating the way ahead. If many people occupy the tunnel at the same time, the same interaction pattern is applied simultaneously to each of them, resulting in a fleeting social interaction that amplifies their awareness of each other: if people are moving in the same direction, whoever is in front is "warned" about the presence of those coming behind; if they move in opposite directions, their light beams collide in anticipation of their physical encounter. The social experience of sharing the public space is thus highlighted by the mutual interference of the light patterns created by each individual.

The Climate on the Wall

The Climate on the Wall [START_REF] Dalsgaard | Designing Urban Media Façades: Cases and Challenges[END_REF] was an installation conceived to be in operation during the climate conference Beyond Kyoto, in Aarhus, Denmark, in 2009. It consisted in a large projection on a wall of the Ridehuset, a prominent historical building in a busy intersection in the centre of the city. The projection would show falling speech bubbles containing terms employed in the ongoing climate debate. Dedicated software used cameras installed along the façade to detect people going through the sidewalk. They could then 'grab' a word, carry it along and reposition it on the wall, thus contributing to the climate debate by forming phrases expressing their views. The interface was perceived as playful and occasionally passers-by would stop by and interact with the projections. More often then not, however, implicit or inadvertent interaction [START_REF] Mueller | Looking glass: A field study on noticing interactivity of a shop window[END_REF] would take place: people would unwillingly engage in interaction just by walking passed the wall, with some noticing they had carried a speech bubble along with them only after it had already happened. The lack of proper grammar seemed to hinder the formation of coherent sentences but, most importantly, the best spot to observe the evolution of graphics on the screen was from the opposite side of the streeti.e., by people watching the façade, not those actually interacting with it. Despite the general awareness about climate debate sparked by the installation, the social conversation via the media façade seemed to have felt short of expected [START_REF] Dalsgaard | Designing Urban Media Façades: Cases and Challenges[END_REF].

Chromapollination

Chromapollination [START_REF] Hespanhol | Strategies for Intuitive Interaction in Public Urban Spaces[END_REF] was a responsive ambient sculpture developed as an urban activation study by the Design Lab, University of Sydney, Australia, for the 2012 edition of the local Vivid Sydney festival. The public space was an underutilised thoroughfare beneath an elevated highway in the centre of the city. The installation consisted of three large-scale sculptures of beds of dandelion flowers, embedded with fibre optics and ultrasound motion detectors. Above, a triangular LED ceiling seamlessly connected the sculptures. As people walk through and around them, their movement was detected, creating 'digital wind'a stream of light flowing overhead from one flower bed to another, carrying the colour of the source dandelion to be mixed with that of the target, which would then be 'pollinated' by changing colour.

Responses to the detected movement were, by design, subdued and delayed, in order to avoid people interacting for too long and cluttering the public thoroughfare. Most commonly, couples or small groups of friends and families would stop for close appreciation and photos of themselves near the work, while 'singles' would usually observe and take photos or videos from a relative distance.

Analysis

From the juxtaposition of the key features of the nine installations summarised in Table 1, certain recurrent design strategies and resulting forms of social interaction can be identified. For example, most urban interventions designed for performance (e.g. Body Movies, Aarhus By Light or Solstice LAMP) are placed in urban spaces that can be characterized as plazas, while responsive ambient media architecture (such as Interference or Chromapollination) has been often used as a way to enhance the experience of thoroughfares. Likewise, full body interactions are clearly more effective to entice explicit and inadvertent interaction as a way to facilitate collective participation, leveraging from established norms of social behaviour in public spaces [START_REF] Mueller | Looking glass: A field study on noticing interactivity of a shop window[END_REF]. Allotted interfaces making use of full body interaction, in particular, reduce the social pressure and risk of embarrassment by allowing a larger number of people to perform simultaneously. In that sense, they can be considered as a less disrupting intervention on the dynamics of the social space, while still supporting both direct interaction among people and indirect interaction via triangulation [START_REF] Memarovic | Using Public Displays to Stimulate Passive Engagement, Active Engagement, and Discovery in Public Spaces[END_REF] through the interfaces.

Consoles (as used in SCSD) and mobile devices (as in iRiS) both constitute TUIs functioning as mediators to the interaction with the media architecture.

Social Interaction Modes

By comparing the installations and analysing how the social interaction unfolded around them, we propose that six modes of social interaction are present in the cases:

(1) appreciation, (2) self-expression; (3) playfulness; (4) collective narratives; (5) triangulation; and (6) negotiation of space. These modes can be further grouped into three broader categories: appreciation and self-expression into spectacle; playfulness and collective narratives into creativity; triangulation and negotiation of space into conversation. Table 2 displays how each mode maps to the media architecture installations considered in our analysis. We discuss each of these categories below, and will later reference them to derive social interaction design patterns.

Table 2. Social interaction modes mapped to the media architecture installations.

Spectacle. Creating spectacle is inherent to media architecture, especially when it is interactive. Just like street performers, media architecture disrupts the normal flow of urban activities, turning passers-by into spectators and, depending on its nature, active performers. Accordingly, we can consider two aspects of the spectacle offered by media architecture: appreciation (i.e. the passive spectatorship of the urban intervention) and self-expression (active interaction via performance or casting of votes, for example). As Table 2 indicates, although all works analysed prompted public appreciation, not all of them led to self-expression. Noticeability of the interaction zones is critical for active participation, particularly in regards to the spatial ratio between those zones and the media architecture. As pointed out by Fischer & Hornecker [START_REF] Fischer | Urban HCI : Spatial Aspects in the Design of Shared Encounters for Media Façades[END_REF], for media façades the display spacei.e. the space around the façade where people can see itis often much larger than the interactive zones, making it difficult for distant observers to perceive the interactive nature of the work. Installations such as Body Movies, Aarhus By Light and MyPosition, for example, successfully addressed this problem by positioning the interactive zones in prominent, wide and therefore very visible areas right in front of their façade. The interactive console used in SCSD, however, was much harder to be spotted amidst the busy urban environment of the large avenue in Sao Paulo. Equally important to making the interactive zone visible to potential participants, however, is to position it so that the feedback provided by the architecture can clearly communicate its intention: direct and uniquely addressed to an individual, if to convey agency and identity; or delayed and defused across the space if to communicate ambient response. While MyPosition, for example, was highly successful in its use of feedback to communicate interactivity to casual passers-by, guiding them through the various steps required for voting via the interface, The Climate on the Wall's feedback was more visible to spectators on the other side of the street than to the actual (and often unaware) participants near the façade. Embodied allotted interfaces, such as Solstice LAMP and Interference, offer a neat solution for both problems: they are wide enough to be noticed from a distance and, for having the interactive zone largely coinciding with the media architecture itself, they make the effects of participation obvious to participants. Moreover, at a closer distance, visibility is equally given to the environment and the people within it, highlighting their roles as agents over the immediate urban space. Such increased tangibility results in an additional benefit: learnability by demonstration. As pointed out by Mueller et al. [START_REF] Mueller | Looking glass: A field study on noticing interactivity of a shop window[END_REF], many people notice interactivity by observing other people interacting. Allotted interfaces, being able to accommodate a large number of simultaneous participants, are therefore highly effective in promoting social interaction, creating a positive feedback loop whereby many people interacting make even more people not only aware of the interaction but also able to readily join in. The shared interface also helps to reduce the fear of public embarrassment [START_REF] Rico | Designing for Performative Interactions in Public Spaces[END_REF] that can potentially hinder participation by people less comfortable to perform in public.

Creativity. While media architecture can be subject to appreciation and lead to individual self-expression, it is not necessarily playful or able to accommodate simultaneous interaction by multiple agents. SCSD and MyPosition are examples of works intentionally designed to be predominantly functional (albeit enjoyable) and restricted to individual interaction. Interactive public art and urban activation projects (such as Aarhus By Light, Solstice LAMP, Interference and Body Movies), on the other hand, often rely on playfulness to subvert ordinary street practices. As Huizinga [START_REF] Huizinga | Homo Ludens: A Study of the Play Element in Culture[END_REF] pointed out, play is a cultural construct perceived as not "real" or "ordinary" life; it refers to activities that are intrinsically motivated, situated outside of everyday life and with no direct benefit or goal. By enabling a playful situation, interactive media architecture can achieve a relaxation of the established social norms, easing the constraints for social encounters to emerge. When the interface also allows for simultaneous interaction by multiple people, collective narratives may emerge out of the constraints and suspension of disbelief posed by the responsive environment, akin to what is observed in alternative or mixed reality games [START_REF] Rettberg | Collective knowledge, collective narratives, and architectures of participation[END_REF].

Conversation. As Goffman [START_REF] Goffman | The Presentation of Self In Everyday Life[END_REF] pointed out, when faced with an unfamiliar social setting, people scan each other for clues about what the appropriate behaviour is. In the case of interactive public environments, unless the rules are explicitly stated via call to actions [START_REF] Valkanova | MyPosition: Sparking Civic Discourse by a Public Interactive Poll Visualization[END_REF] they often end up emerging on the spot and collectively as people try to make sense of an otherwise unusual situation. This process of sensemaking often emerges among strangers as a consequence of triangulation [START_REF] Memarovic | Using Public Displays to Stimulate Passive Engagement, Active Engagement, and Discovery in Public Spaces[END_REF], a social situation where conversation is started out of mutual interest in the content itself displayed by the interface. In that sense, allotted interfaces may have different results depending on how much visible participants are to each other: iRiS, although playful and enabling self-expression, was occasionally perceived as frustrating since participants could not always see physically who they were interacting with on the screen. In Solstice LAMP and Interference, on the other hand, participants were always on sight of each other and therefore negotiation of space unfolded according to established social norms. Technology, in those cases, supported and promoted social interaction through the mutual awareness of other people in the space. Table 3. Patterns derived from layout factors (spatial layout, interaction and feedback), on the left, and social interaction modes (spectacle, creativity and conversation), on the right.

Social Interaction Design Patterns

When compared with the interaction strategies and spatial layouts of the selected media architecture installations, the social interaction modes discussed above indicate a series of recurring strategies employed by the designers. We propose that those strategies can be characterised as social interaction design patterns for eliciting specific forms of crowd behaviour and, as a consequence, the social identity of the public space. In that sense, our proposed patterns are conceptually closely related to Alexander's pattern language for architecture [START_REF] Alexander | A Pattern Language: Towns, Buildings, Construction[END_REF] than to the software design patterns proposed by the GoF [START_REF] Gamma | Design Patterns: Elements of Reusable Object-Oriented Software[END_REF], consisting in reusable strategies to use digital media for designing the interaction among people and between them and their surrounding built envi-ronment. From Table 2, we derived the patterns by grouping either recurring layout factors (i.e. spatial layout, interaction strategy and type of feedback) or social interaction modes. We then named each pattern by using metaphors to familiar concepts that operate in a similar fashion, with corresponding social reactions.

Grouping recurring layout factors (Table 3, Shadow Playing. This pattern captures the recurrent combination of full-body interaction and direct feedback (Table 3, left). It prescribes the use of full body interaction and immediate visual feedback to create a 1-to-1 relationship between individuals and their graphical representation in the media architecture (as in Body Movies, Aarhus By Light and Solstice LAMP). Previous studies have found that shadow playing is an effective way to communicate interactivity to passers-by via inadvertent interaction, with mirror images, in particular, being more effective than silhouettes and avatars [START_REF] Mueller | Looking glass: A field study on noticing interactivity of a shop window[END_REF]. By minimizing noise and latency on the communication, the interface enables a high sense of agency and identity among participants, who feel like they are controlling the media content directly. Social interaction therefore unfolds both in the physical and digital environments simultaneously, via triangulation (and possibly negotiation of space) between individuals and their "projected selves". Apart from spectacle, this pattern often entails playfulness and enables the emergence of collective narra-tives, with multiple rows of participants tending to form around the interactive zones [START_REF] Mueller | Looking glass: A field study on noticing interactivity of a shop window[END_REF]. For that reason, this pattern typically requires a plaza spatial layout, so that multiple participants can be accommodated, allowed to move freely and have good visibility of their digital representations on the media architecture.

Remote Control. In this pattern, a tangible user interface (TUI) is employed as a mediator between the public and the media architecture [START_REF] Behrens | Smart Citizen Sentiment Dashboard: A Case Study Into Media Architectural Interfaces[END_REF], usually in a plaza spatial layout (for the same reasons as Shadow Playing). Two opposite design strategies have traditionally been adopted: centralised remote controls (such as the console used in SCSD) or distributed portable devices (often mobile devices, as in iRiS). In both cases, discoverability about the interactive aspects of the architecture becomes a recurrent challenge: how to easily communicate to the public they can use smartphones to interact with the content on the façade, or indicate the existence and location of a mediating console? Both scenarios also pose trade-offs for the design. Interaction via mobile devices allows simultaneous participation, but negotiating time on the interface may be problematic; moreover, it is usually pre-empted by the system itself rather than a result of a tacit social agreement between the participantswho, by the way, often cannot see each other. Interaction via a centralised TUI, on the other hand, solves such an accessibility problem, however eliminates the possibility of more elaborate social interaction to unfold via triangulation with the media architecture.

Smooth Operator. This pattern describes the delayed feedback to full-body tracking unfolding in a thoroughfare, responding to movements of passers-by in a covert way (Table 3, left). While affected by the presence of passers-by, a smooth operator does not respond direct to them, and therefore is perceived as an independent entity, not controlled by the public (or at least not entirely). Typically, its responses to inputs from the public is delayed and, often, not in their field of view (e.g. the landing effect [START_REF] Mueller | Looking glass: A field study on noticing interactivity of a shop window[END_REF] verified in Chromapollination and The Climate on the Wall). Smooth operators are normally designed for thoroughfares and regions of intense pedestrian traffic, raising awareness about the public urban space yet without compromising the flow of people through it. 3, right, this pattern represents installations enticing predominantly individual expression and its appreciation, but with little playfulness, collective narratives or direct social conversation. Its goal is to offer an opportunistic encounter whereby citizens can informally express their opinions about a local topic without too much disruption to their daily activities. Public consultation can therefore happen in a lightweight yet regular basis, providing local government with invaluable feedback without excessive interference on the social dynamics of the community. As such, soapboxes tend to be designed for quick but reliable individual interaction, and seem to be more effective when ludic elements are downplayedin other words, they benefit from a lack of playfulness, increasing the focus on their actual functionality. Yet, as in any interface, user experience should be taken into account. For example, according to the designers of MyPosition the interface was deliberately designed to create a good balance between playful interaction and meaningful participation. The goal was to enable incidental interaction leading to an enjoyable voting experience, yet avoiding the inadvertent interaction observed in The Climate on the Wall [START_REF] Valkanova | MyPosition: Sparking Civic Discourse by a Public Interactive Poll Visualization[END_REF]. Amusement Park. The amusement park is an urban media architecture installation that promotes a wide array of social interaction, reaching out to all three proposed domains: spectacle, creativity and conversation (Table 3, right). On those settings, there is not only individual self-expression and its appreciation by bystanders, but also collective play leading to co-created narratives, mediated via triangulation with the media architecture and, occasionally (as it is the case of embodied allotted interfaces, such as Solstice LAMP), negotiation of the physical space during the experience. It is characterized for being a disruptive urban intervention that captures the attention of passers-by and turns some of them into performers, while others participate passively as spectators. As a playful environment, it entices suspension of disbelief and, consequently, subversion of pre-established social norms, with new ones defined on the spot as a result of the social interaction itself. Performative and allotted interfaces are the most effective choices for designing amusement parks.

Soapbox. As shown in Table

Swarm. This pattern refers to environments where people interact via triangulation with the architecture, negotiating the physical space in the process (Table 3, right). A swarm is an allotted interface built upon full body interaction. Generally deployed to plazas, it consists in an interface large enough to accommodate many simultaneous participants, each involved in embodied interaction with their own section of the interface. Feedback is such that participants are continuously aware of both their own representation on the interface and those of the peers in their immediate vicinity. Consequently, local actions of one individual may imply in reactions by their immediate neighbours, which will then somehow impact their own neighbours, in a process resembling a swarm of insects or school of fishes. Works like Solstice LAMP and Interference are good examples of swarms, successfully assisting the autonomous negotiation of space by participants by making use of technology to create 'body gloss' [START_REF] Goffman | Relations in Public[END_REF] (i.e. the emphasis and exaggeration of bodily presence). In addition, the social relationships among participants may also be highlighted by the feedback they receive. For instance, Solstice LAMP's feedback mechanism of merging halos when people got close enough to each other indicated the transition between a state where people were perceived by the environment as separate individuals (or 'singles', as defined by Goffman [START_REF] Goffman | Relations in Public[END_REF]) to a scenario where they could be socially characterized as a group (in a 'with'). As observed, people would often adjust their behaviour as a result of the new social configuration. Automatic Gate. As indicated in Table 3, right, this pattern describes environments with varying levels of playful feedback and appreciation yet no room for selfexpression or collective narratives. An automatic gate is a media architecture installation implemented in thoroughfares and areas of high pedestrian activity, which people attend to with peripheral attention. It is characterised by being strongly blended into the built environment, to the point of being almost integrated to it. It responds to pedestrians in a subtle, seamless and non-disruptive way, changing the environment appearance either in front or behind them. Interference and Chromapollinationenhancing pedestrianised areas without structurally altering themconstitute good examples. In other cases, however, such a pattern emerges unintentionally: for example, when interaction fails to be understood or noticed, the work may be inadvertently interpreted as ambient, as was the case with The Climate on the Wall.

Conclusion

Successful media architecture combines insights into technical, architectural and social issues. While the field is still relatively new, patterns in how these three aspects can be combined to elicit specific types of behaviour and use are emerging. Our focus has been to understand how different technical and architectural configurationshere labelled types of interfaces and spatial layoutscan lead to specific modes of social interaction, and whether there are recurring strategies for bringing about these social modalities. We find that the field, albeit still in the making, is mature enough to identify recurring design strategies for designing media architecture and analyse the effect they have had on the social aspects in the locations in which they were introduced. Through an analysis of nine media architecture installations that are representative of the field, we have thus identified six distinct modes of interaction, ranging from appreciation to creative expression and dialogue. In continuation, we have identified seven recurring design patterns, which have been employed as strategies to bring about specific modes of social interaction. While these contributions build on analyses of works selected to represent the scope and diversity of media architecture thus far, we do not consider the list of modes and design patterns of social interaction to be exhaustive. A clear limitation lies in our selection of works, which we have limited in order to provide enough details to make the function and purpose of the installations clear; it is plausible that a different or more expansive selection of works could lead to further examples of modes and patterns. Indeed, we would welcome additions to this work, and we expect new modes and patterns to emerge as media architecture continues to evolve. Another limitation in our methodology is that since we have strived to include installations that show the diversity of media architecture, the sample size is not large enough to indicate if specific design patterns are statistically more prevalent or successful. However, this initial identification of social modes and design patterns may lay the ground for more detailed analyses of strategies for influencing and developing modes of social interaction. In addition, we hope that the identification of design patterns can be valuable for interaction designers of future media architecture installations, who strive to bring about specific modes of social interaction.

  left) produced three patterns: (a) Shadow Playing; (b) Remote Control; and (c) Smooth Operator. Grouping the social interaction modes (Table 3, right) produced further four patterns: (a) Soapbox; (b) Amusement Park; (c) Swarm; and (d) Automatic Gate.Figure1summarises the proposed patterns, which can be construed as design strategies for prompting specific forms of social interaction. Below, we discuss each pattern and its effects on social interaction.

Fig. 1 .

 1 Fig. 1. Social interaction design patterns.


Orlando Erazo
email: oerazo@dcc.uchile.cl

José A Pino
email: jpino@dcc.uchile.cl

Pedro Antunes
email: pedro.antunes@vuw.ac.nz

Estimating Production Time of Touchless Hand Drawing Gestures

Keywords: Touchless hand gestures, natural user interfaces, predictive evaluation, user models

Quantitative user models such as CLC, Isokoski's and KLM have been used to estimate the production time of mouse and pen interactions (pointing, clicking, selecting, drawing, writing). In this paper, we assess if these models can be adapted to estimate the production time of touchless hand gestures (air figures of letters and numbers). New parameters were added to the existing models with empirical values drawn from experiments with users. Two metrics were used to evaluate model quality: strength of the relationship between estimated and observed times, and percentage root mean square error. The obtained results support the hypothesis that CLC, Isokoski's and KLM can be adapted to touchless hand gestures. The paper contributes with model modifications and parameters required to estimate the production times of touchless hand gestures.

Introduction

Hardware devices such as Kinect, Leap Motion and Myo have contributed significantly to the emergence of a new class of graphical user interfaces labeled as Natural User Interfaces (NUIs), which most distinctive characteristic is using touchless hand gestures as a new interaction modality. Besides the obvious advantages in entertainment, NUIs are also becoming increasingly popular in fields such as education [START_REF] Blum | mirracle: An Augmented Reality Magic Mirror System for Anatomy Education[END_REF] and healthcare [START_REF] Erazo | Magic Mirror for Neurorehabilitation of People with Upper Limb Dysfunction Using Kinect[END_REF]. However, depending on each particular application, hand gestures may be very taxing on production time, i.e. the time spent by users engendering interactions with an application's user interface. This situation leaves user-interface designers with the difficult task of evaluating which touchless hand drawing gestures are most adequate and how they can be tailored to a user-interface under construction to optimize production time.

Considering the logistic difficulties of doing tests with real users, regarding planning, timing, laboratory setup, recruiting, and conducting experiments, a reasonable assumption is that user-interface designers may find value in instead adopting predictive evaluation. Predictive evaluation provides quantitative indications on how users may perform based on user models instead of real users, thus easing the abovementioned logistic problems.

Of course a consequence of adopting predictive evaluation is that we need quantitative user models. Although some quantitative models have already been proposed that include gestures [START_REF] Barclay | A Quantitative Quality Model for Gesture Based User Interfaces[END_REF][START_REF] Erazo | Estimating the Difficulty of Touchless Hand Gestures[END_REF], they do not support the estimation of production time. However, an extensive body of research in quantitative user models for pointer-and pen-based gestures has already been developed. One of the most influential instances is the KLM (Keystroke Level Model) model originally proposed by Card, Moran and Newell and extended by others [START_REF] Card | The Keystroke-Level Model for User Performance Time with Interactive Systems[END_REF][START_REF] Holleis | Keystroke-Level Model for Advanced Mobile Phone Interaction[END_REF][START_REF] Erazo | Predicting Task Execution Time on Natural User Interfaces based on Touchless Hand Gestures[END_REF]. Another notable example is Fitts' Law [START_REF] Fitts | The information capacity of the human motor system in controlling the amplitude of movement[END_REF], which has been widely used to estimate physical and virtual pointing [START_REF] Pino | Using Kinect for 2D and 3D Pointing Tasks: Performance Evaluation[END_REF][START_REF] Sambrooks | Comparison of gestural, touch, and mouse interaction with Fitts' law[END_REF][START_REF] Schwaller | Pointing in the Air: Measuring the Effect of Hand Selection Strategies on Performance and Effort[END_REF][START_REF] Zeng | Fitts' Law in 3D Space with Coordinated Hand Movements[END_REF]. Some interactions like pointing, clicking and selecting can be done both with a mouse and in the air using hand gestures, and therefore can be modeled using KLM and Fitts' Law, as demonstrated by [START_REF] Pino | Using Kinect for 2D and 3D Pointing Tasks: Performance Evaluation[END_REF][START_REF] Sambrooks | Comparison of gestural, touch, and mouse interaction with Fitts' law[END_REF][START_REF] Schwaller | Pointing in the Air: Measuring the Effect of Hand Selection Strategies on Performance and Effort[END_REF][START_REF] Zeng | Fitts' Law in 3D Space with Coordinated Hand Movements[END_REF]. However, more complex interactions such as drawing characters, numbers and figures require either extending the existing models with new parameters or developing whole new user models. Considering this landscape, the generic goal of the research reported in this paper is to extend the existing models to encompass hand gestures. More specifically, our goal is to extend the CLC (Curves, Line segments and Corners) [START_REF] Cao | Modeling Human Performance of Pen Stroke Gestures[END_REF], Isokoski's [START_REF] Isokoski | Model for Unistroke Writing Time[END_REF] and KLM [START_REF] Card | The Keystroke-Level Model for User Performance Time with Interactive Systems[END_REF] models to estimate the production times of touchless hand gestures. Our research is restricted to gestures performed by young adults in normal health conditions, with basic or no experience with touchless interactions, and using the dominant hand (fingers are not considered). The gestures of interest consist of drawing in the air figures or shapes such as letters or numbers.

Related Work

A commonly accepted strategy to evaluate the usability of a user interface is to use modeling techniques. This strategy presents the advantages of neither depending on real users nor requiring usability experts to participate in the evaluation process. Models can vary in detail and complexity, ranging from descriptive models, which provide a framework for designers to delineate and reflect on usability problems, to predictive models, which use mathematical expressions for estimating user performance [START_REF] Mackenzie | Motor Behavior Models for Human-Computer Interaction[END_REF]. Unlike the descriptive category, predictive models can be used to objectively estimate the required time for performing a set of user interactions. Likewise the descriptive category, predictive models can be applied at early design stages, before starting to develop the real user interface. Some notable user models have been proposed in the research literature. One of the most known and cited models is Fitts' Law [START_REF] Fitts | The information capacity of the human motor system in controlling the amplitude of movement[END_REF], which allows estimating the time to point at a target based on the object size and distance. Taking into account that Fitts' Law is not adequate for certain types of tasks, and based on it, Accot and Zhai [START_REF] Accot | Beyond Fitts' Law: Models for Trajectory-Based HCI Tasks[END_REF] proposed the "steering law" for trajectory-based tasks. This model allows predicting the time to navigate through a two-dimensional tunnel, but it may not be adequate to analyze the trajectory of touchless hand gestures when there are no visual guides.

An alternative was proposed by Isokoski [START_REF] Isokoski | Model for Unistroke Writing Time[END_REF], who introduced a conceptually simple model that predicts production time for unistroke interactions done by expert users. According to this model, a gesture is first decomposed into a number of "needed straight-line segments" which are then counted to estimate the overall timecomplexity of the gesture. The number of considered segments is the minimum necessary to make the gesture recognizable. Additionally, it is assumed that drawing a straight-line segment takes a constant time.

By comparing the model with real-user interactions, Isokoski [START_REF] Isokoski | Model for Unistroke Writing Time[END_REF] measured the strength of the relationship between estimated and observed times (usually noted as R 2 ) to be <0.85. The percentage root mean square error (usually noted as %RMSE) of these measures was 30% [START_REF] Isokoski | Model for Unistroke Writing Time[END_REF].

Although the definition of "needed straight-line segments" is ambiguous [START_REF] Cao | Modeling Human Performance of Pen Stroke Gestures[END_REF][START_REF] Vatavu | Estimating the Perceived Difficulty of Pen Gestures[END_REF] (a procedure describing the reduction of gestures with curves into straight lines is missing), Isokoski's model seems conceptually easy to extend to hand gestures because it requires estimating the constant time to produce a straight line gesture segment from experimental data.

Cao and Zhai [START_REF] Cao | Modeling Human Performance of Pen Stroke Gestures[END_REF] suggested a model to estimate the production time of single penstroke gestures. The model considers three features found in pen-stroke gestures: Curves, Line segments, and Corners (the reason why the model is referred to CLC).

For any gesture, the production time is calculated by summing up the estimated time durations of all gesture segments (see formula 1 below). The estimated production times of Curve and Line are defined in formulas 2 and 3. (Formula 4 can be used instead of formula 3.) The Corner, which is an abrupt change in stroke direction, was discarded by Cao and Zhai after empirical studies showed its insignificant impact on production times [START_REF] Cao | Modeling Human Performance of Pen Stroke Gestures[END_REF].

The CLC model reveals a strong relationship between estimated and observed times (R 2 > 0.90). Even though the model has been used in several research studies (e.g. [START_REF] Vatavu | Estimating the Perceived Difficulty of Pen Gestures[END_REF][START_REF] Tu | A comparative evaluation of finger and pen stroke gestures[END_REF]), we do not have evidence that it has been used beyond pen-stroke gestures.

𝑇 = ∑ 𝑇(𝑙𝑖𝑛𝑒) + ∑ 𝑇(𝑐𝑜𝑟𝑛𝑒𝑟) + ∑ 𝑇(𝑐𝑢𝑟𝑣𝑒)

(1)

𝑇(𝑐𝑢𝑟𝑣𝑒) = 𝛼 𝐾 𝑟 1-𝛽 (2)
𝑇(𝑙𝑖𝑛𝑒) = 𝑚𝐿 𝑛 (3)
𝑇(𝑙𝑖𝑛𝑒) = 𝑎𝐿 + 𝑏 (4)
Where: α is the sweep angle; r is the radius of the arc; β and K are empirical constants. L is the length of the line; a, b, m and n are empirical constants.

Another well-known user model is the Keystroke Level Model (KLM) [START_REF] Card | The Keystroke-Level Model for User Performance Time with Interactive Systems[END_REF]. KLM defines a set of primitive operations: key press (K); point (P); button press (B); hand movement between keyboard and mouse (H); Drawing (D); and mental preparation (M). For each primitive operation, empirical studies with various types of users allowed to determine the respective average production time constant. The D primitive is the only one relevant to this study, even though it has several constraints worth noticing: drawing is done with the mouse, it only concerns straight-line segments, and it is assumed to be done on a square grid with 0.56cm. According to KLM, the production time of a drawing interaction is defined as a linear function of the number of segments (n D ) and the total length (l D ) of all segments (see formula 5) [START_REF] Card | The Keystroke-Level Model for User Performance Time with Interactive Systems[END_REF].

𝐷(𝑛 𝐷 , 𝑙 𝐷 ) = 𝑎𝑛 𝐷 + 𝑏𝑙 𝐷 (5)
Where: a and b are constants (a = 0.9 and b = 0.16 in the original KLM version [START_REF] Card | The Keystroke-Level Model for User Performance Time with Interactive Systems[END_REF]).

All in all, the Isokoski's, CLC and KLM models advocate: 1) decompose a gesture either into a series of straight-line segments (Isokoski's and KLM) or into a series of curved and straight-line segments (CLC); 2) use a set of formulas and parameters to calculate the production time of each segment, adding them to obtain the overall production time of a gesture; and 3) derive the parameters from empirical studies with various users, so that the formulas adjust to reality. Nevertheless specific parameters for touchless hand gestures do not currently exist and will have to be further researched. We detail that process in the following section.

Hypotheses and Research Design

As mentioned above, there is no evidence that the three described models may be used for touchless hand gestures. Therefore, the following hypothesis should be tested: H1: CLC, Isokoski's and KLM models can be adapted to predict the production time of hand gestures.

The starting point to test this hypothesis is the definition of formulas. Either the original formulas can be applied to hand gestures or they have to be extended to encompass the new conditions imposed by hand gestures. Hence, we introduce a first step in our study where the formulas are evaluated and adapted if necessary. This step is shown in Table 1 (step 1) and denoted as "An" (A -Adaptation).

After adapting the formulas, it is necessary to define new parameters for hand gestures. The second step requires carrying out several experiments with real users (En in table 1, step 2; E -Experiment) and then tuning the parameters so the models may reflect the users' performance.

At this point, the models should be ready to use. However, we still have to consider the quality of the estimations. Thus, we have to evaluate the models, that is, to verify the second hypothesis:

H2: The adapted models can predict the production time of hand gestures with acceptable quality.

We will test this hypothesis using the two metrics also adopted by Cao and Zhai [START_REF] Cao | Modeling Human Performance of Pen Stroke Gestures[END_REF], considering both the strength of the relationship between estimated and observed times (R 2 ), and percentage root mean square error (%RMSE). Furthermore, we will consider that a model has acceptable quality if R 2 and %RMSE are proximate to the values obtained by Cao and Zhai [START_REF] Cao | Modeling Human Performance of Pen Stroke Gestures[END_REF]: R 2 > 0.90 and %RMSE < 30%.

H2 must be tested with a set of experiments with real users (En in Table 1, line 3). With a careful experimental setup, some of the experiments required to test H1 can also be used to test H2. That is the reason why we see in Table 1 that E1 and E2 are shared by steps 2 and 3. This is possible because: 1) the parameters required by CLC and Isokoski's models can be tuned using the same experiments; and 2) we use two different sets of users in E1 and E2, so that the users being used to tune a model can be reused to validate another model, but one set of users is not simultaneously used to tune and validate the same model.

In the remainder of this section we give more details about the experimental setup. The validation of H1 and H2 is discussed in detail in the following sections using the stepwise structure described in Table 1.

Table 1. Research design: adaptations of models (A) and experiments (E).

Models

Steps

CLC Isokoski's KLM 1. Definition of formulas A1 A2 A3
2. Estimation of parameters E1 E1 E2

3. Evaluation of models E2 E2 E3

Apparatus and Method

The hardware setup for the experiments consisted of notebook, Kinect sensor and TV screen mounted in a controlled laboratory setting. The notebook was equipped with an i7 processor and 8GB of RAM. The Kinect sensor was used with a refresh rate of 30 fps, connected to track users' hand position and recognizing gestures, placed at a height of 0.9m and below the TV screen. The TV screen had 42in, 1360×768px resolution. The participants stood in an uncluttered space, 2.5m away from the Kinect sensor.

A custom software tool (Fig. 1) was developed for precisely controlling the experiments. The tool was developed using MS Visual C# and Kinect for Windows SDK V1.8 on Windows 7. The tool logged time marks and hand coordinates while a participant performs a gesture. The Dynamic Time Warping algorithm [START_REF] Senin | Dynamic time warping algorithm review[END_REF] was used for gesture recognition. The tool's interface consisted of an augmented video blending user interface controls and the real environment. Augmented video was adopted in order to avoid participants' distractions while performing the tasks. For instance, a person may judge his/her movements based on a hand cursor and try to make adjustments [START_REF] Sutter | The impact of gain change on perceiving one's own actions[END_REF], especially because of the sensor noise [START_REF] Livingston | Performance Measurements for the Microsoft Kinect Skeleton[END_REF] [10] [START_REF] Zeng | Fitts' Law in 3D Space with Coordinated Hand Movements[END_REF] which should be avoided. We fine-tuned these experimental conditions through a set of trial experiments.

The tool had two additional software modules focused on gesture analysis. The first module allowed recording gestures (Fig. 1-a), while the second one was able to reproduce every user-generated gesture using segmentation and logged hand coordinates (Fig. 1-b). Each gesture instance was segmented in the phases proposed by [START_REF] Mcneill | Guide to Gesture Classification, Transcription, and Distribution[END_REF] to measure the production times of gestures. More specifically, the measured stroke-phase time was defined as corresponding to production time, which does not account for the time spent by the participants in other gesture phases.

The participants in the experiments were University students (33 in total, aged between 17 and 28) invited by email, social networks, etc. The participants were not paid for their participation. Written informed consents were obtained before starting each experiment. A student was allowed to participate in a single experiment.

Before the experiment, each participant received written instructions and an explanation about the research goals. Then, the participant performed some training gestures guided by the software. When the participant indicated s/he was ready, more specific instructions appeared on the TV screen using a PowerPoint slideshow. Enough time was allowed to read the instructions. The instructions required that every gesture should be done inside a red box (gesture input area or gesture space, see Fig. 1-d), having approximately the same size, and balancing speed and accuracy. (The tool adjusted the size of the input area according to the size of the required gesture.) The instructions also noted the participants should use the dominant hand, and should start (preparation phase [START_REF] Mcneill | Guide to Gesture Classification, Transcription, and Distribution[END_REF]) and finish (retraction phase [START_REF] Mcneill | Guide to Gesture Classification, Transcription, and Distribution[END_REF]) a gesture with both hands in a relaxed position below hips.

The tool immediately started the data acquisition phase after displaying the instructions. The tool was programmed to randomly pick a gesture within a gesture set and display it for two seconds (Fig. 1-c). The gesture image was displayed along with a name and a very short description. After the two seconds period, the description disappeared, the red box was displayed, and the participant's gesture was collected (Fig. 1-d). When the gesture was correct, the tool displayed a green check and moved on to the next gesture. When a gesture was wrong, a red-cross mark was displayed, the input was discarded, and the participant had to re-enter the gesture.

Besides a practice session, every experiment included three blocks with gestures to be performed by the participants (e.g. a block only including straight lines). The specific characteristics of these blocks are defined in Sections 5 and 6. The tool included a resting period between blocks of gestures.

Definition of Formulas

This section discusses the definition and adaptation of each model to hand gestures. The tuning and validation of formulas are explained later.

CLC Model

Although Cao and Zhai [START_REF] Cao | Modeling Human Performance of Pen Stroke Gestures[END_REF] adopted several formulas (1, 2, 3 and 4) for the CLC model, there are other options that might be used to improve predictions. Based on regression analysis (discussed later), we suggest that formula 6 can be used instead of formula 2 to estimate the production time of curves. Another simpler formula that may be applied to curves, which Cao and Zhai did not test, is a linear function of the curve's radius and angle (formula 7). These two formulas may contribute to reduce the %RMSE of CLC, but they should be tested against the original one.

𝑇(𝑐𝑢𝑟𝑣𝑒) = 𝛼 𝑎 𝐾 𝑟 1-𝛽 (6)
𝑇(𝑐𝑢𝑟𝑣𝑒) = 𝑚 𝑟 + 𝑛 𝛼

Where: α is the sweep angle; r is the radius of the curve; a, β, K, m and n are empirical constants.

Isokoski's Model

Given the conceptual simplicity of Isokoski's model [START_REF] Isokoski | Model for Unistroke Writing Time[END_REF], we consider it may be straightforwardly adapted to hand gestures. Formula 8 may be applied bearing in mind that hand gestures will be reduced into a sequence of straight lines and an empirical constant is necessary to estimate the time taken to generate every straight line. This reasoning has two implications. First, Isokoski did not provide a constant time for performing a straight-line segment, so we have to estimate that constant. Second, Isokoski' does not provide a clear procedure to reduce curves into straight lines, which may range between 1 (too much error) and an arbitrary large number (less error but more difficult to calculate). We adopt the procedure suggested by Vatavu et al.:

"if the angle α inscribed by an arc was greater than 270° use 3 segments; if α<120° use 1 segment; otherwise use 2 segments" [START_REF] Vatavu | Estimating the Perceived Difficulty of Pen Gestures[END_REF] (p. 97).

𝑇 = #segments * constant_time (8)

KLM

As already mentioned, we are only using the D operator of KLM, and hence using formula 5 to estimate production time of hand gestures. Analyzing the original definition in detail [START_REF] Card | The Keystroke-Level Model for User Performance Time with Interactive Systems[END_REF], we note that formula 5 may not be applicable to gestures with curves and corners. We suggest that curves be approximated to straight-line segments by applying the procedure adopted by Vatavu et al. [START_REF] Vatavu | Estimating the Perceived Difficulty of Pen Gestures[END_REF] (p. 97). Regarding corners, we suggest counting the number of corners (n C ) multiplied by an empirical constant, as shown in formula 9.

𝐷 𝑐 (𝑛 𝐷 , 𝑙 𝐷 , 𝑛 𝐶 ) = 𝑎 𝑛 𝐷 + 𝑏 𝑙 𝐷 + 𝑐 𝑛 𝐶

Where a, b and c are empirical constants.

Estimation of Parameters

In this section we give further insights about formulas 6 to 9 delineated above. The discussion is organized in two steps. In the first step, we describe the experiments (E1 and E2, see Table 1) conducted to obtain empirical data about user's hand gestures. In the second step we present the final formulas with the estimated empirical constants.

CLC Model

Experiment. We repeated most of the experimental process described by Cao and Zhai [START_REF] Cao | Modeling Human Performance of Pen Stroke Gestures[END_REF] using hand gestures to obtain empirical constants for formulas 2, 3, 4, 6 and 7. Experiment E1 involved gathering data for three gesture components: straight lines, curves and corners. This experiment then implied configuring the tool discussed in Section 3 to request participants to produce variations of these individual gesture components (see below). Each participant would produce the same gesture three times in order to increase precision. Trying to avoid learning and/or sequence effects, the order of components was counterbalanced.

Various lengths (L = {0.4, 0.6, 0.8} meters in motor space) and orientations (0, 45, 90 and 135 degrees counterclockwise) were tested when producing straight lines. For curves, various radiuses (r = {0.2, 0.3, 0.4} meters in motor space) and sweep angles (α = {90, 180, 360} degrees) were tested. Start angle (90 degrees) and direction (clockwise) were treated as control variables for curve gestures. Various corner angles (θ = {45, 90, 135} degrees) and directions (CW and CCW) were tested to produce corners. Length was kept constant (0.6cm in motor space). Twelve persons participated in E1 (mean age 21y, σ = 2).

Results. In general, the results obtained from E1 for the CLC model have similar significance to those obtained by Cao and Zhai [START_REF] Cao | Modeling Human Performance of Pen Stroke Gestures[END_REF], which provides a first indication that CLC can be used for estimating production time of touchless hand gestures. In detail:

Straight Lines. We observed statistically significant differences when measuring production time and varying length (F 2,22 = 10.47, p < 0.05) and orientation (F 3,33 = 2.92, p < 0.05). No significant length × orientation interaction effects were found (F 6,66 = 0.32, ns). Fig. 2 shows the relation between length and production time for each orientation. We note that Cao and Zhai [START_REF] Cao | Modeling Human Performance of Pen Stroke Gestures[END_REF] did not take into account orientation in their estimations because its effect was considered smaller than length. We made the same decision due to the similarity of our results, but we also computed the correlation coefficients to confirm it. We found no correlation between orientation and time production (r = -0.022). Finally, performing regression analysis of our experimental data, we obtained the empirical constants shown in formulas 10 and 11.

Where L and T are given in meters and seconds respectively.

Curves. As we expected, measured differences in production time were statistically significant for both radius (F 2,22 = 12.33, p < 0.05) and angle (F 2,22 = 110, p < 0.05). We found significant radius × angle interaction effects (F 4,44 = 3.72, p < 0.05). Fig. 3 shows the relation between sweep angles and production time for each orientation radius. After performing regression analysis with our experimental data, we obtained formulas 12, 13 and 14 below.

Where α, r and T are given in radians, meters and seconds respectively.

Corners. Following Cao and Zhai's method, we computed the "net contribution time" of corners [START_REF] Cao | Modeling Human Performance of Pen Stroke Gestures[END_REF]. Thus, for our experimental data and using the average time to perform a line 0.6cm long (see Table 2): T(corner) = sample production time -2 * 0.627 (seconds). The measured differences in production time were statistically significant for corner angle (F 2,22 = 6.49, p < 0.05), but not for direction (F 1,11 = 2.05, p > 0.05). We found no significant angle × direction interaction effects (F 2,22 = 0.24, ns). Taking into account that the average T(corner) seems to fluctuate around zero (Fig. 4), we made a deliberate simplification (formula 15): to omit corners in the model (Cao and Zhai [START_REF] Cao | Modeling Human Performance of Pen Stroke Gestures[END_REF] made the same decision). Although these results confirm previous preliminary findings, which postulate that corners have influence on production time of hand gestures [START_REF] Erazo | Estimating the Difficulty of Touchless Hand Gestures[END_REF], we think further research is necessary to adequately model the impact of corners in hand gestures.

Isokoski's Model

Experiment. As mentioned above, data obtained from experiment E1 was reused to build an estimation model based on Isokoski's proposal [START_REF] Isokoski | Model for Unistroke Writing Time[END_REF]. Although Cao and Zhai [START_REF] Cao | Modeling Human Performance of Pen Stroke Gestures[END_REF] state that a constant time model should be invalidated, we nevertheless decided to build this model because of its conceptual simplicity. An average time was calculated for each straight line produced by the participants in the experiment (Table 2). Moreover, we estimated a fourth value to evaluate the model with a smaller straightline segment (0.2m). These times must be verified for selecting the best one by using different gestures (see next section).

Results.

KLM

Experiment. Taking into account that experiment E1 is focused on curves, straight lines and corners, we had to perform another experiment (E2) to estimate the empirical constant for the D operator of KLM, given that it is based on the number of segments and the total length of all segments. The experiment consisted of drawing 14 gestures 1 (Fig. 5) in random order. Gestures were performed inside the gesture space, which was a 0.6m square. Twelve persons took part in E2 (mean age 23y, σ = 2). 1 The green point indicates the start of the gesture. Strike-through was selected from [START_REF] Vatavu | Estimating the Perceived Difficulty of Pen Gestures[END_REF].

Results. The procedure adopted by Card et al. [START_REF] Card | The Keystroke-Level Model for User Performance Time with Interactive Systems[END_REF] for drawing straight-line segments (formula 5) was tested with hand gestures. This means that gestures with curves ("question", "three", "eight" and "circle", in Fig. 5) were not used to build the model. The number of segments (n D ) of each gesture produce by participants in the experiment was counted and the total length (l D ) of each gesture was computed (geometrically). Formula 16 was obtained by performing regression analysis. The resulting R 2 value was high (0.988), but we thought the model could still be improved. We obtained a higher R 2 value (0.99, see formula 17) considering the number of corners and using formula 9. (Corners were counted depending on the gesture start point.)

𝐷(𝑛 𝐷 , 𝑙 𝐷 ) = 0.386 𝑛 𝐷 + 0.349 𝑙 𝐷 (𝑅 2 = 0.988)

𝐷 𝑐 (𝑛 𝐷 , 𝑙 𝐷 , 𝑛 𝐶 ) = 0.223𝑛 𝐷 + 0.297𝑙 𝐷 + 0.173𝑛 𝐶 (𝑅 2 = 0.99)

Formula 16 was then tested to estimate the production times of the four gestures with curves shown in Fig 5 , which had not been previously used to estimate the empirical constants (condition D in Fig. 6). Additionally, formula 17 was tested with and without corners (conditions D c and D c *). The obtained results for these three conditions are shown in Fig. 6. These results indicate that these models can also be applied to gestures with curves.

Evaluation of Models

The production times of real hand gestures must be compared against predicted values in order to evaluate the adapted models. We tried to reduce the number of experiments to a minimum and thus, we decided to reuse data from experiment E2 to evaluate the adapted CLC and Isokoski's models, whose parameters were developed from E1 using a different cohort and different gestures. Regarding the evaluation of the adapted KLM model, a new experiment had to be setup (E3), since E2 was used to estimate the parameters for this model.

CLC Model

Formula 15 was suggested to estimate production time using the CLC model, with the provision that formulas [START_REF] Sambrooks | Comparison of gestural, touch, and mouse interaction with Fitts' law[END_REF][START_REF] Schwaller | Pointing in the Air: Measuring the Effect of Hand Selection Strategies on Performance and Effort[END_REF] and [START_REF] Zeng | Fitts' Law in 3D Space with Coordinated Hand Movements[END_REF][START_REF] Cao | Modeling Human Performance of Pen Stroke Gestures[END_REF][START_REF] Isokoski | Model for Unistroke Writing Time[END_REF] can be considered options for measuring straight-lines and curves, respectively. Six possibilities can be analyzed to identify the best estimation approach by combining these formulas.

The results obtained from E2 are shown in Table 3 for the six formula combinations. We note that some R 2 values are lower than the baseline (Cao and Zhai's results [START_REF] Cao | Modeling Human Performance of Pen Stroke Gestures[END_REF]), but the obtained %RMSE values are better. Furthermore, the differences between estimates are relatively small. The best results are obtained using the linear model for straight lines (formula 10) and the modified model for curves (formula 13). Fig. 7 displays the predicted versus observed data using this formula combination.

, ( 11) and ( 13) 0.810 15.7 ( 15), ( 10) and ( 14) 0.849 16.9 [START_REF] Mackenzie | Motor Behavior Models for Human-Computer Interaction[END_REF], [START_REF] Schwaller | Pointing in the Air: Measuring the Effect of Hand Selection Strategies on Performance and Effort[END_REF] and ( 14) 0.798 16.9 Fig. 7. CLC model prediction.

Isokoski's Model

E2 also allowed validating the Isokoski's model expressed in formula 8 with the empirical constants defined in section 5.2. The obtained results, shown in Fig. 8, suggest that selecting a constant straight-line length of 0.4m gives the least estimation error. Fig. 9 shows the relationship between predicted and measured production times for the suggested straight-line length (R 2 = 0.935, L = 0.4m, t = 0.544s). Finally, we compared the measured production time with the best results estimated by the CLC and Isokoski's models (Fig. 10). Isokoski's model is slightly better than CLC, but the difference is quite small to choose the best one. Also, we note the worst predictions were made for gestures "three" and "eight", which are outside ±1 SD.

KLM

Experiment E3 was set up in a similar way to E2. Nine participants (mean age 21y, σ = 3), performed the 6 gestures 2 shown in Fig. 11.

Gestures

Observed time

CLC Isokoski's

Stroke times were compared against predicted values using formulas 16 and 17. Before applying the formulas, gestures with curves ("5", "E" and "steep-hill") were reduced into straight lines. Additionally, formula 17 was calculated with and without corners (D c and D c * conditions). For instance, gesture "E" was evaluated using 1 and 0 corners. The obtained results are shown in Fig. 12. The highest R 2 (0.995) value was observed for the D condition, while the lowest (0.947) was observed for D c , even though they were quite approximate. Since the %RMSE was favorable to the D c condition (10.4), we suggest that D c could be considered the best one overall.

General Comparison

In this section we finally compare the three models, using again the data collected in experiment E3 and focused on the formulas and empirical constants that produced the best estimates. Table 4 shows the selected formulas and quality of estimates using the two quality criteria adopted by this study. Fig. 13 provides a more detailed comparison using the observed and predicted production times for the six gestures used to evaluate the estimation models. The highest R 2 value was obtained for the CLC model, but the differences to KLM (D c condition) are quite small. On the other hand, the lowest %RMSE was obtained with KLM (D c ). Consequently, we suggest using KLM (D c ) to predict the production time of hand gestures.

Discussion and Conclusions

In this paper we analyze three estimation models for predicting the production time of users' interactions with other types of user interfaces. We extend these models to hand gestures. Empirical experiments were accomplished to tune and validate the models. The quality of the estimates was evaluated using two criteria: strength of the relationship between estimated and observed times, and percentage root mean square error. In a broad perspective, we can conclude that the three models can be used with hand gestures, which confirms hypothesis H1. Furthermore, we provide new or updated formulas and empirical constants required to use the models with hand gestures. The constant-time estimation model, which was proposed by Isokoski for unistroke writing [START_REF] Isokoski | Model for Unistroke Writing Time[END_REF], is the simplest model. This model is very easy to use because it reduces gestures to straight-line segments, counts them, and uses a constant multiplier that reflects the average time necessary to produce a straight-line segment. The constant multiplier depends on the constant length assigned to a segment.

According to our results, if gestures are drawn inside a square gesture space with 0.6m sides, acceptable results can be achieved with a segment that is 0.4m long. Conversely, the simplifications required by this model lead to erroneous estimations when using variable gesture spaces (i.e. making gestures of different sizes).

An alternative approach, which we also analyzed, consists in using the CLC model, which breaks down gestures into curves, lines and corners [START_REF] Cao | Modeling Human Performance of Pen Stroke Gestures[END_REF]. This model avoids reducing curves and corners to straight-line segments.

We provide new or updated formulas and empirical constants required to use the CLC model with hand gestures. Moreover, our experiments indicate that corners influence production time and therefore should not be neglected. Additionally, slightly different formulas were evaluated, leading us to suggest a new formula for estimating the production time of hand gestures using CLC.

The KLM model also reveals easy to adapt to hand gestures, because it is only based on the number of straight-line segments and the total length of a gesture. Conversely, this strategy also reveals a limitation because KLM's D operator does not Gestures with curves were analyzed as if they were straight lines and with the options of counting or not the number of corners. The obtained results show that counting corners improves the quality of the estimation. Consequently, the adapted KLM formula we suggest counts the number of segments, the total length and the number of corners of a gesture.

Regarding the experiments, we should note the following. First, we could observe a relatively high variation of gesture production times among participants. Although we did not compute a global or final value, the coefficient of variation is, on the average, about 30%. Second, the models were adapted and evaluated only using the gestures' stroke phase, even though they could also be analyzed taking into account a more comprehensive view (e.g. [START_REF] Mcneill | Guide to Gesture Classification, Transcription, and Distribution[END_REF]). Third, the gestures used in our experiments were performed using only the dominant hand, although users may perform gestures with the other hand [START_REF] Annett | Your Left Hand Can Do It Too! Investigating Intermanual, Symmetric Gesture Transfer on Touchscreens[END_REF]. This constraint may have an effect on the estimates (e.g. [START_REF] Zeng | Fitts' Law in 3D Space with Coordinated Hand Movements[END_REF]).

The model we suggest as the best to estimate production time of hand gestures obtained R 2 ≥ 0.947 and %RMSE = 10.4, which are better than the ones obtained by Cao and Zhai [START_REF] Cao | Modeling Human Performance of Pen Stroke Gestures[END_REF] for single pen-stroke gestures. Regarding hypothesis H2, we observe it is validated for the CLC and KLM (D c condition).

We expect to conduct more evaluations in the future with more users and more gestures. We also consider studying other hand gesture types like hover, tap and swipe.

Fig. 1 .

 1 Fig. 1. Interface of the experimental software.

Fig. 2 .

 2 Fig. 2. Straight line production time.

Fig. 3 .

 3 Fig. 3. Curve production time.

Fig. 4 .

 4 Fig. 4. Net corner time contribution. Error bars indicate 1 SD.

Fig. 5 .

 5 Fig. 5. Gestures used in experiment 2.

Fig. 6 .

 6 Fig. 6. Comparison of observed and predicted times of 4 gestures with curves using the D, D c and D c * conditions. Error bars indicate 1 SD.

Fig. 8 .

 8 Fig. 8. Comparison of Isokoski's model prediction errors.

Fig. 9 .

 9 Fig. 9. Isokoski's model prediction.

Fig. 10 .

 10 Fig. 10. Comparison of observed and predicted times using both CLC and Isokoski's models. Error bars indicate 1 SD.

Fig. 11 .

 11 Fig. 11. Gestures used in E3.

Fig. 12 .

 12 Fig. 12. Comparison of observed and predicted times for experiment E3 using D, D c and D c * conditions. Error bars indicate 1 SD.

  Dctake into account other components like corners and curves. Trying to overcome these limitations, we included corners as a third parameter in KLM's estimation formula. The experimental results indicate these modifications provide good results.

Table 2 .

 2 Constant times for Isokoski's model.

	Straight line lengths, L(m)	Observed time, t(s)	SD(s)
	0.2	0.442	NA
	0.4	0.544	0.111
	0.6	0.627	0.180
	0.8	0.738	0.219

Table 3 .

 3 Comparison of CLC model predictions.

	Formulas	R 2	%RMSE
	(15), (10) and (12)	0.834	18.8
	(15), (11) and (12)	0.779	18.8
	(15), (10) and (13)	0.859	15.7

Table 4 .

 4 Comparison of the three models. General comparison of observed and predicted times using CLC, Isokoski's and KLM models. Error bars indicate 1 SD.

	Model name	Formulas	R 2	%RMSE
	CLC	(15), (10) and (13)	0.996	25.7
	Isokoski's	#segments * 0.544	0.881	25.7
	KLM (D c )	(17)	0.947	10.4

Acknowledgments

Orlando Erazo appreciates the financial support provided for this research by SENESCYT, Ecuador.