Lourdes Diaz Olvera
email: lourdes.diaz-olvera@entpe.fr

Assogba Guézéré
email: guezereassogba@gmail.com

Didier Plat
email: didier.plat@entpe.fr

Pascal Pochet
email: pascal.pochet@entpe.fr

Improvising intermodality and multimodality. Empirical findings for Lomé, Togo

Keywords: intermodality, multimodality, transport integration, mobility, motorbike taxis, Lomé

While achieving integrated transport is an increasingly important policy goal in the cities of the North, it is of little concern to African decision-makers. Nevertheless, the scarcity of transport has prompted African city-dwellers to create "de facto" integration, which has, until now, received little scholarly attention. This paper describes the intermodal and multimodal practices of the inhabitants of Lomé, the capital of Togo, where the motorbike taxi plays a dominant role. Intermodality, i.e. the use of at least two transport modes in the course of a single trip, allows individuals to travel throughout the city but requires trade-offs between transport cost, duration and physically exhausting conditions. Multimodality, which is defined as the use of a different mode or combination of modes for outbound and return trips, is observed to some extent among the working population but even more amongst schoolchildren, usually in order to reduce the cost of transport. The consequences of individuals' "de facto" intermodal and multimodal practices in terms of mobility and money and time budgets highlight the strategic importance of formal transport integration for African city-dwellers.

Introduction

In the cities of the North, the integration of the different types of passenger transport services to form a coherent system is the subject of much concern and an increasing number of applications. Integration sets out to enhance simultaneously the sustainability of transport systems and accessibility which allows the provision of good transport services at a low cost. At a fundamental level, integration can be seen as a set of technical and organizational tools that seek to make the best use of existing transportation networks, by combining them with the use of personal modes when this is possible. Integration may also take more ambitious forms and attempt to align transport policies with those for other sectors.

However, in the cities of sub-Saharan Africa, where walking is the rule and the majority of motorised trips are made by informal public transport, integration arouses very little interest on the part of either the public authorities or the operators. Nevertheless, when we examine transport demand, daily travel includes practices which can be regarded as "de facto" integration, instituted by the users themselves: the successive use of several transport modes in the course of the same trip (which may include long segments on foot), the use of different modes for the outbound and return journeys. This paper sets out to improve our understanding of the "de facto" integration of daily travel in the cities of sub-Saharan Africa through the study case of Lomé, the capital of Togo. It examines the modal practices of public transport users. How do citizens use the different transport modes on a day-to-day basis? How do they combine them in order to reach their destination? Do they use different modes or sequences of modes for their outbound and return trips? How does their choice of transport mode affect their travel times and their monetary expenditure on daily travel?

The analyses presented in this paper are based on empirical data collected in the course of a survey conducted in Lomé in late 2011. During the survey, 1228 individuals who were travelling in the conurbation were asked to describe their current trip, their customary practices when travelling from home to work or school and back, the availability of personal vehicles, and the accessibility of their home. The conventional data were elicited about their trips: the origin and destination, the modes used, the trip purpose, the price paid and the duration of the trip. It was consequently possible to carry out detailed analysis of modal choice, the price/travel time trade-offs, the difficulties they experienced and the way they adapted to them.

Section 1 deals with the different types of transport integration and defines the concepts of intermodality and multimodality as we have used them in this paper. Section 2 describes the study area, the survey methodology and the data set. Intermodal trips are analysed in Section 3, taking account of the combination of modes and the duration, distance and cost of the trip. Section 4 sheds light on multimodal practices by comparing the combinations of modes used during the outbound and return trips for work-and school-related trips. These analyses reveal the additional money and time costs incurred as a result of such intermodal and multimodal practices which are improvised by users in the absence of any formal support from the transport sector. The conclusion highlights the need for formal transport integration in order to improve the travel conditions in sub-Saharan African cities.

From transport integration to intermodality and multimodality

The public authorities in the developed world advocate transport policies that are based on integration because of two major factors: the urgent need to limit car traffic in order to create more sustainable cities and the limited funding available for public transport as a result of the financial crisis. However, the term transport integration may refer either to highly operational measures focused on transport supply or to strategic matters that involve comprehensive public policies [START_REF] May | The principles of integration in urban transport strategies[END_REF]. It is thus important to distinguish between different forms of transport integration [START_REF] Potter | On transport integration: a contribution to better understanding[END_REF][START_REF] Viegas | From multimodality to seamless intermodality in urban transport: a difficult transition[END_REF]. The first rung of an integrated transport system is internal integration, which sets out to achieve an internally consistent transport system (Givonni and Banister, 2010;[START_REF] Preston | Measuring the costs and benefits of integrated transport policies and schemes[END_REF]. The second rung aims to integrate transport policies and land-use. Finally, the third rung attempts to harmonise multi-sectoral policies (for example transport and local economic development, transport and access to healthcare). In this paper we shall deal only with internal integration.

Internal transport integration may be either modal or functional [START_REF] Potter | On transport integration: a contribution to better understanding[END_REF]. Modal integration aims to facilitate transfers between different transport modes by improving interchange points and synchronizing the timetables of different services. Functional integration involves integrating pricing so that a change in transport mode does not generate potentially dissuasive additional user costs. In both cases the primary aim is to promote the successive use of several transport modes during the same trip [START_REF] Potter | On transport integration: a contribution to better understanding[END_REF][START_REF] Hine | Seamless, accessible travel: users' views of the public transport journey and interchange[END_REF][START_REF] Gorter | Analysis of travellers' satisfaction with transport chains[END_REF][START_REF] Martens | The bicycle as a feedering mode: experiences from three European countries[END_REF][START_REF] Martens | Promoting bike-and-ride: the Dutch experience[END_REF]. Such successive use can be referred to by the terms "'multimodal' and 'intermodal', which are used interchangeably" (Givoni and Banister, 2010: 5). In other cases multimodality refers to the use of different transport modes at different times of the day or the week, depending on the trip purpose or the location of the destination [START_REF] Nobis | Multimodality: facets and causes of sustainable mobility behaviour[END_REF][START_REF] Parkhurst | Intermodal personal mobility: a niche caught between two regimes[END_REF]. The ability to distinguish between intermodality and multimodality would seem to be important when analysing daily mobility, as they result from partially different modal choices and behavioural adaptations in response to constraints. In this paper we shall use the term intermodality to describe the chaining of transport modes within the same trip and multimodality to describe differences in modal use between outbound and return trips.

It is important to make a distinction between intermodality and multimodality as the corresponding data are collected and processed in a different manner. Multimodality is easier to identify by observing the sequence of trips, but to measure it accurately data must be collected over a long enough period of time. The underestimation of intermodality can be significant because conventional surveys pay little attention to short walking trips and underestimate the role of walking segments during mechanized trips [START_REF] Behrens | Meta-analysis of travel of the poor in West and Southern African cities[END_REF][START_REF] Clifton | Capturing and representing multimodal trips in travel surveys[END_REF]. Trips in which mechanized modes and walking are combined "[…] are generally not regarded as intermodal trips in the technological sense, nor within the consciousness of most travellers, transport planners and politicians" (Parkhurst et al. 2012: 309). As a consequence, the role of walking in the daily travel of city-dwellers is seriously underestimated. In a study of the Netherlands, [START_REF] Rietveld | Non-motorised modes in transport systems: a multimodal chain perspective for The Netherlands[END_REF] has shown that walking becomes the most used mode when pedestrian segments are considered, its share rising from 18 to 59%. In the cities of sub-Saharan Africa, walking is not only the sole mode of transport for many trips, it may also be the only possible way of reaching public transport stops, as in Douala or Conakry (Sitrass, 2004a[START_REF] Sitrass | Poverty and Urban Mobility in Douala (Report SSATP n° 09/04/Dla)[END_REF]. In Douala, for example, trips in which a single public transport segment is combined with at least one walking segment lasting more than 5 minutes are three times more numerous than trips that require the successive use of several public transport vehicles. These commonplace practices show that urban residents are genuinely implementing modal integration. Nevertheless, in the cities of developing countries, which are highly fragmented [START_REF] Balbo | Urban fragmentation as a research hypothesis: Rabat-Salé cases study[END_REF] and whose transport systems rely essentially on informal operators [START_REF] Cervero | Informal transport: A global perspective[END_REF], thinking about intermodality in transport policies is at a much less advanced stage [START_REF] Yeh | A study on feasability of passenger intermodal transport in city of the developing worl[END_REF][START_REF] Rivasplata | Intermodal transport centres: towards establishing criteria[END_REF]. In African cities, urban transport suffers from a lack of organization and regulation which pushes informal operators into fierce competition with each other [START_REF] Cervero | Informal Transport in the Developing World[END_REF]. However, this situation does not exclude the possibility of transport modes and operators working together. In order to ensure their business remains profitable, operators must provide a minimum number of interconnections between public transport vehicles serving different routes. The informal sector's ability to organize itself leads to possible connections between different transport modes, as studied by [START_REF] Audard | Système de transport en commun et auto-organisation: Le cas de Brazzaville[END_REF] in Brazzaville. The presence of the different transport modes at major stops and terminals means that these operate, de facto, as multimodal interchanges.

Intermodal trips are still complicated and costly. Public transport users in cities such as Conakry, Douala and Nairobi all suffer from the poor organization of passenger pick-up and drop-off and the shortening and division of routes into fare sections. They have to make unplanned walking segments or change vehicle, which increases the cost of travel (Sitrass, 2004a[START_REF] Sitrass | Poverty and Urban Mobility in Douala (Report SSATP n° 09/04/Dla)[END_REF][START_REF] Mccormick | Paratransit business strategies: a bird'seye view of matatus in Nairobi[END_REF]. Likewise, in Dar es Salaam, passengers frequently need to chain several dala dala journeys in order to travel to work, paying each time they board a vehicle, which makes commuting expensive [START_REF] Nkurunziza | Modelling commuter preferences for the proposed Bus Rapid Transit in Dar-es-Salaam[END_REF]. The marked sprawl that affects African cities and the frequent need to use several vehicles to reach a destination make transport an increasingly expensive item of household expenditure and increase transport time budgets, particularly for workers and students living in the outskirts.

A low level of modal integration and an absence of functional integration are thus the general rule in African cities. The blossoming of Bus Rapid Transit projects in many cities has scarcely stimulated any debate on the need for transport sector integration that includes informal transport [START_REF] Behrens | An evaluation of policy approaches to upgrading and integrating paratransit in African urban public transport systems: results of the first round of a Delphi survey[END_REF]. Formal modal integration set up by the authorities seems to be fairly rare, in particular because interchange stations are more a setting for confrontation between stakeholders and for social and political conflict than genuine intermodal interchange [START_REF] Lombard | Introduction. Quand le transport est d'abord un lieu ! Autrepart[END_REF][START_REF] El Hadji | Le transport routier au Sénégal: Problématique de la gestion des gares routières[END_REF]. The introduction of integrated fares seems unattainable in the short or medium term, as a result of the fragmentation of the sector and the limited regulatory capacities of the public authorities. While a traveller may negotiate the fare of each successive leg of a trip, this is not the case at all times and for all destinations and, obviously, the amount of the fare does not take account of the other segments in the trip or the user's ability to pay. [START_REF] Salon | Mobility, poverty, and gender: travel 'choices' of slum residents in Nairobi, Kenya[END_REF] have shown that, in the case of Nairobi, affordability, more than transport provision, hinders access to public transport for poor slum-dwellers. However a close examination of daily mobility in Lomé shows that intermodal and multimodal practices are improvised, but widespread.

Study area and empirical material

The conurbation of Lomé had a population of 1.5 million in 2010, which amounts to slightly less than a quarter of the resident population of Togo (République Togolaise, 2011). Administratively, it is divided into a central municipality, the city of Lomé proper, which is home to 57% of the conurbation's population, and seven peripheral cantons. Urban sprawl has been restricted to the west by the border with Ghana and to the south by the Atlantic Ocean (Figure 1). The most important zones for business activities and jobs are near the ocean: the autonomous port of Lomé which handles four-fifths of Togo's foreign trade, the administrative district, and the central market of Adawlato. The network of main roads consists essentially of a few radial roads and a ring road, which are all asphalt-paved. The rest of the road network near the centre and in the outskirts consists of sand or earth roads some of which are completely flooded in the rainy season. During this period they turn into genuine swamps making motor vehicle travel difficult or even impossible. The general state of the road network therefore limits the possibilities for travel, especially by four-wheeled vehicle. As in other West African cities, household ownership of personal vehicles is low in Lomé [START_REF] Diaz Olvera | The puzzle of mobility and access to the city in Sub-Saharan Africa[END_REF]. In 2011, public transport supply was provided essentially by shared taxis and motorbike taxis, with the minor presence of the few buses on the "experimental route" operated by Sotral, a company set up in 2007. Shared taxis are saloon cars which either cruise for clients or operate on fixed routes which usually start from the central market. In the early 1990s these were the only taxis in Lomé, but they were rapidly replaced by motorbike taxis which have now become the dominant form of public transport [START_REF] Guézéré | Territoires des taxis-motos à Lomé: de la pratique quotidienne à la recomposition des espaces urbains et des liens sociaux[END_REF]. The oleyias or zemidjans, as motorbike taxis are known locally, are thus the main suppliers of public transport, both in the city centre and in the outskirts. We conducted a survey in 2011 to study how the inhabitants of Lomé use the different public transport modes and to identify intermodal and multimodal practices.

The survey covered individuals of 13 years of age and over who were travelling in the conurbation, and focused on public transport users. It was carried out during a week (excluding Sunday) in public space at 10 key points distributed throughout the conurbation. A total of 1,220 individuals, 68% of whom were economically active, were asked to give details about their socio-economic characteristics, the trip they were making and their habits with regard to work-or school-related trips. For the trip the individual was making when he/she was intercepted for the interview, information was collected about all the modes used during the trip, including all walking segments of at least 10 minutes made before or after travel in public transport vehicles. In the case of individuals' customary work-and school-related trips, the survey also collected information on the combinations of transport modes used, distinguishing between outbound and return trips.

The purpose of the survey was to identify the combination of transport modes used in daily travel and did not seek to be statistically representative of the population of Lomé. Individuals who did not travel on the survey day were not included and members of the working population had a higher probability of being interviewed. The underestimation of pedestrian trips, an essential component of daily mobility in African cities, was also a methodological choice. The fact that the interviews were conducted in the vicinity of key transport nodes meant that internal trips within the individuals' residential district, which are almost always on foot, were virtually excluded from the survey. This choice does not interfere with the survey's goal. After consolidation of the collected data we were left with a sample of 3,984 trips to study the intermodal and multimodal practices of urban transport users.

Chaining different modes: a costly necessity

The modal distribution of "current" trips1 reflects the domination of motorbike taxis (Table 1). The oleyias are used in slightly more than eight out of every ten public transport trips2 . However, their share drops when intermodality is taken into account. Less than a quarter of the trips in which oleyias are involved are intermodal while this applies to more than half of the trips involving shared taxis. These figures reveal two important types of intermodality in Lomé, each of which applies to more than 10% of public transport trips:

-intermodality between public transport and walking, i.e. the combination of one public transport leg and one (rarely two) pedestrian legs lasting at least 10 mins; -intermodality between public transport modes, i.e. the combination of two (rarely three) public transport legs (usually by oleyia and shared taxi), sometimes with a pedestrian leg lasting at least 10 mins. Intermodal practices involving a personal motorbike and a public transport mode are fairly rare: individuals who get a ride on a friend's motorbike in order to shorten the distance and reduce the cost of the rest of the trip on public transport, or motorbike drivers who park their vehicle and complete their trip to the city centre by motorbike taxi because they are afraid of the traffic or the theft of their machine. The absence of intermodal practices involving a private car is simply due to the low rate of car ownership.

Intermodality is valuable

The making of intermodal trips is closely linked to the distance covered. They are made on the longest and most complex trips between the central municipality of Lomé and its outskirts -the cantons in the conurbation (Figure 2). Trips which combine oleyia and shared taxi are particularly common. On the other hand, the proportion of intermodal trips becomes much smaller for internal trips within the municipality of Lomé or its peripheral cantons. In the outskirts, the higher frequency with which oleyia use requires walking as an access or egress mode is explained by the low residential densities, the limitations of the road network and the longer distances between the loading point and one's home.

Lomé -Lomé

Cantons -Cantons Lomé -Cantons The boxplot of travel distances3 show that short and medium distance trips are the market segment for which the use of oleyias is the most common (Fig. 3a). Three-quarters of all oleyia trips, with or without the use of walking as a feeder mode, involve distances of under about 4 kilometres. Shared-taxi use, whether or not it is combined with an oleyia involves longer distances. However, oleyias dominate transport supply to such an extent that they are still used for more than half the trips that exceed 5 km. A similar hierarchy is apparent with increasing trip duration (Figure 3b). The oleyia, with or without the use of walking as a feeder mode, gradually gives way to the shared taxi, and then to the combined use of the oleyia and the shared taxi. Logically, adding a walking segment at the beginning or end of an oleyia trip results in a lengthening of door-to-door travel times, but this use of walking for part of the trip does not increase the travel domain of the motorbike taxi in terms of distance. In distance terms, the travel domains of shared taxis and the combination of shared taxi and oleyia are fairly similar: 50% of all the trips are between 4 and 8 km in the case of the first and 50% are between 5 and 9 km in the case of the second. However, in terms of duration, intermodal oleyia-shared taxi trips often take twice as long, often lasting 40 mn.

Previous research on daily mobility conducted in other cities of sub-Saharan Africa show the way an individual's neighbourhood and its immediate vicinity dominate the spatial distribution of daily trips, but also the need to travel throughout the metropolitan area due to the functional specialization of urban areas [START_REF] Diaz Olvera | The puzzle of mobility and access to the city in Sub-Saharan Africa[END_REF]. The results given above for Lomé show that intermodal combinations, especially those that involve more than one public transport mode, make it possible for city dwellers to travel to parts of the city that are far from their home. In spite of the fact that transport services are fragmented and uncoordinated, intermodality is valuable for city-dwellers… but it comes at a price.

Intermodality is costly

The fragmented nature of public transport supply and the fact that the fares charged by informal operators are not controlled mean that fares increase automatically with distance, even if negotiation between the driver and the passenger is often possible leading to variations in the price charged for a given distance [START_REF] Guézéré | Oléyia" (taxi-moto): Acteurs et usagers d'un mode de transport artisanal récent à Lomé[END_REF]. Figure 5a explains why the oleyia, when used on its own is above all used for short and medium distances. On average, it is less expensive than the other modes or modal combinations up to a crow's flight distance of 2.5 km, after which it is increasingly out-priced by the shared taxi. Above a distance of 6.5 km, oleyia fares are barely lower than the cost of combined use of the oleyia and the shared taxi. This notwithstanding, the conditions that apply to the use of different modes allow us to see why motorbike taxis are often perceived as inexpensive. If we consider the median trip distance for each mode, the median oleyia trip costs under 200 CFA Francs (0.30 €4 ), that by shared taxi costs 250 CFA Francs (0.38 €) and that by oleyia and shared taxi combined, 420 CFA Francs (0.64 €; see Figure 5a).

Intermodal trips that involve the successive use of an oleyia and a shared taxi are systematically more expensive because of the summing of the costs for each mode. The resulting expenditure is a considerable drain on household budgets as we can gauge from two straightforward comparisons. First, the median cost of a trip, 420 CFA Francs (0.64 €) is almost equivalent to the price of a bowl of maize that provides a meal for 8 to 10 people5 . Second, a worker on the minimum wage (28,000 CFA Francs, 42.7 €) who travels from home to work and back by oleyia and shared taxi twenty times a month would only have 11,000 CFA Francs (16.8 €) per month left to spend on other items, including food and housing. These estimations, even though they only refer to hypothetical situations, show the level of constraints such modal practices can generate when they are repeated frequently. This helps us understand why, in order to reduce the cost of their daily travel, citizens are attracted to the other main intermodal strategy based on combining the oleyia and walking (Figures 5a and5b). The cost differential between the two types of intermodal trips (oleyia combined with shared taxis or with walking) increases very rapidly once the trip duration exceeds 20 mins. Walking slightly flattens the price-distance curve. However, this statement needs some qualification. On the one hand, once the distance exceeds 3.5 km, the fare for an oleyia + walking trip increases much more slowly with distance than the fare for a trip by oleyia on its own which becomes more expensive once the distance exceeds 4 km (Figure 5a). On the other hand, the price-trip duration curve confirms that the way to reduce the cost of oleyia trips is to combine them with walking (Figure 5b). Walking has two different functions in intermodal trips in Lomé:

-as a feeder mode, in order to get to the nearest oleyia loading point. This practice has little impact on the distances covered or the cost of transport. It has more severe consequences for persons who live in the outskirts. In the city centre there is always an oleyia station very close to an individual's home. However, in the outskirts one respondent in ten had at least a ten minute walk to reach one, and this proportion may reach one in five in the most peripheral cantons.

-as a partial substitute for public transport use, to shorten the motorized segment of the trip so as to limit the cost or because no motorbike taxi is available, for example in distant or reputedly dangerous areas, particularly at night. Walking does not alter the total trip distance but it has a real impact on the relationship between the price and the trip duration because as the total duration of the trip increases the expenditure tends to stabilize. Walking implies however a greater physical effort and discomfort.

This dual function of walking in intermodal trips obviously goes hand in hand with a considerable reduction in speed, as can be seen for oleyia + walk intermodal trips (Table 2).

The outcome is a marked deterioration in the overall performance of the intermodal combination when this is measured by the price/speed ratio. A similar, but even more marked, deterioration is observed in the case of the intermodal combination oleyia + shared taxi. While this combination is as fast as using a motorbike taxi on its own, it cannot reach the speed of shared taxis used on their own. The price/speed ratio is roughly twice as high for an oleyia + shared taxi intermodal trip than when the two modes are used separately. City-dwellers can also adapt to the major constraints that restrict their travel on public transport by using a different mode or combination of modes for their outbound and return journey. A description of multimodal practices provides a complementary insight into the use of walking to partially replace public transport modes.

Exploring different possibilities to be able to travel

The Lomé survey provides information about the outbound and return trips of respondents' customary home-work and home-school mobility (locations of home and work or school, transport mode or sequence of modes, duration and cost). The survey methodology was not intended to capture occasional multimodal practices but customary ones. It is therefore likely that it underestimates the variability between outbound and return trip.

According to the usual practices reported by the respondents, the return trip is frequently different from the outbound trip. Less than three-quarters of home-work trips and only half of home-school trips have outbound and return trips that are strictly identical in terms of mode, duration and price (Table 3). When the outbound and return trips differ we can identify two main types: those for which the modal choice is the same but the price and/or duration differs and those in which modal use differs. The size of these two categories is similar for the working population, but the second category is much more frequent in the case of schoolchildren and students.

When individuals use the same modes for both directions of their trip, changes in travel conditions are more the outcome of the lower availability or lower modal efficiency than a price-time trade-off. Amongst the working population, in almost all cases the situation is worse on the return trip either because of longer durations or higher costs. The same applies to schoolchildren and students, mainly due to longer durations.

The use of different modes on the return trip is either due to price-time trade-offs because time constraints are less pressing after work or school or because the operating conditions of public transport change according to the time of day. When a different mode is used for the return trip, the duration and cost of the trip change. In the case of half of the workers and three-quarters of the schoolchildren and students, the changes result in less expensive travel on the return trip and longer durations because individuals walk part of the way. More expensive return journeys are considerably more frequent among workers than young people in education. The difference between the two groups is due to the low personal incomes of the latter which puts them under even greater pressure to limit their expenditure. These multimodal strategies implemented by the inhabitants of Lomé are consistent with what has been observed in other African cities during exceptional events that interfere with customary practices and reveal their underlying rationale. For example, in Douala, between 2003 and 2005, the repair of the only bridge over the Wouri, which connects the right bank to the left bank where the principal urban facilities are located, disrupted the travel of residents crossing the river. The banning of motorbike taxis and shared taxis from the area where the works were taking place and the introduction of a pay bus to cross the river led passengers to adopt a variety of tactics to minimize the rise in transport costs. The most frequent of these was to transfer to the least expensive modes at the cost of increased travel times [START_REF] Diaz Olvera | Quand tout ne tient qu'à un pont ! Réfection d'ouvrage et dysfonctionnements urbains à Douala[END_REF].

Trade-offs between the different modes, whether in the context of intermodal or multimodal practices, take very different forms to those observed in the countries of the North where time is one of the principal parameters of urban life [START_REF] May | La ville éclatée[END_REF] and where time savings are of central importance. The operation of the transport systems in sub-Saharan African cities does not allow time-space compression, which is an extremely important trend in contemporary cities in the developed world [START_REF] Janelle | Space-time constructs for linking information and communication technologies with issues in sustainable transportation[END_REF]. In African cities, price-time trade-offs appear to be very much determined by the extremely low incomes of most public transport users. The result is that city-dwellers have very limited margins of manoeuvre with respect to the many constraints that relate to time, finance, or public transport supply.

Conclusion

Data on daily travel in Lomé allowed us to study practical examples of transport integration. The intermodal and multimodal practices of the city's residents demonstrate that they make a compromise between the external constraints imposed by the way the transport system operates, the location of activities and their personal resources. When they engage in such practices, users incur additional money or time costs, as well as additional physical difficulties, particularly when the modal combination includes long walking segments. The flexibility of motorbike taxis tends to limit the need for intermodal practices in the case of short or medium distance trips. However, for long distance intra-urban trips, such as the trips between the central municipality and the peripheral cantons, the chaining of several modes of transport, motorized or not, becomes inevitable. Intermodal and multimodal practices are equal expressions of how difficult it is for users to pay for their public transport trips and the limited availability of transport in some locations, such as peripheral zones, or at some times of the day or the night.

These findings on intermodality and multimodality in Lomé show that the lack of a public policy to foster integration in the urban transport sector has a major impact on the daily behaviours of the city's residents, in particular those who are poor or who reside in the outskirts. Although individuals may attempt to improvise modal integration using the forms of organization that have been put in place by the operators, functional integration is obviously completely beyond them. Intermodality and multimodality thus involve constant trade-offs between the availability of transport, and the monetary cost and the duration of the trip.

Ambitious forms of integration, for example between transport policies and land-use, or transport policies and policies for other sectors, currently appear to be out of reach for African national and sub-national governments which are frequently politically and economically weak. In a context in which poverty is prevalent, improving accessibility to the resources and opportunities afforded by the city is nevertheless a vital aim for development strategies. Public transport policies that set out to bring about better integration between the various components of transport have an important role to play in achieving this goal [START_REF] Vasconcellos | Urban Transport, Environment and Equity. The Case for Developing Countries[END_REF][START_REF] Dimitriou | Conclusions: emergent crucial themes[END_REF]). An integrated transport policy requires that the public authorities play an active role in the organisation and regulation of public transport. Such a policy must be based on a hierarchical transport network which substantiates the complementary nature of the different transport modes (motorized and non-motorized, private and public). Moreover, it requires a "paradigm shift" (Pendakur, 2011: 225) in order to assign a major role to walking that is in line with its actual importance in urban travel in Africa. This change must be accompanied by the willingness to make public transport affordable for lowincome groups. Whatever the local situation regarding the organization of transport in the cities of sub-Saharan Africa, moving towards both modal integration that includes nonmotorized modes and functional integration appear to be essential in order for transport policies to play a role in poverty reduction.

Figure 1 .

 1 Figure 1. The conurbation of Lomé: administrative areas, main road network and business zones, and survey locations

Figure 2 .

 2 Figure 2. Distribution of public transport trips in Lomé according to origin and destination Note: S. Taxi=Shared taxi Source: 2011 Lomé Survey

Figures

  Figures 3a and 3b. Boxplots of trip distance and duration by public transport mode in Lomé Note: S. Taxi=Shared taxi Source: 2011 Lomé SurveyFigure4presents the travel domains of the different public transport modes in Lomé, in terms of travel distance and travel time. Logically, adding a walking segment at the beginning or end of an oleyia trip results in a lengthening of door-to-door travel times, but this use of walking for part of the trip does not increase the travel domain of the motorbike taxi in terms of distance. In distance terms, the travel domains of shared taxis and the combination of shared taxi and oleyia are fairly similar: 50% of all the trips are between 4 and 8 km in the case of the first and 50% are between 5 and 9 km in the case of the second. However, in terms of duration, intermodal oleyia-shared taxi trips often take twice as long, often lasting 40 mn.

Figure 4 .

 4 Figure 4. The travel domains of public transport modes in Lomé Interpretation example: 50% of the oleyia trips involve a crow's flight distance of between 1.7 and 4.2 km; 50% have a duration of between 9 and 20 min. S. Taxi=Shared taxi Source: 2011 Lomé Survey

  Figures 5a and 5b. Price according to trip distance and duration on public transport in Lomé Note: XOF=CFA Francs. 100 CFA Francs=0.152 Euros. S. Taxi=Shared taxi Source: 2011 Lomé Survey

Table 1 .

 1 Distribution of public transport trips in Lomé according to type of mode

		% of total public	% of total	% of total
		transport trips	oleyia trips	shared taxi trips
	Oleyia	64.6	77.8
	Oleyia + Walking >=10 mins	7.6	9.1
	Oleyia + Other Mode	1.6	2.0
	Oleyia + Shared taxi	9.2	11.1	36.3
	Shared taxi	12.1		47.8
	Shared taxi + Walking >=10 mins	3.9		15.4
	Shared taxi + Other mode	0.1		0.5
	Other public transport modes*	0.9
	Total	100.0	100.0	100.0
	* Mainly Sotral buses and buses for university students.
	Source: 2011 Lomé Survey

Table 2 .

 2 Median speed and price estimates for public transport modes, in Lomé

		Oleyia	Oleyia +	Oleyia +	Shared Taxi
			Walk	Shared Taxi
	Median Speed (km/h)	14.2	10.5	14.9	17.7
	Median Price FCFA	200	200	450	300
	(Euro)	(0.30)	(0.30)	(0.69)	(0.46)
	Median Price / Median Speed
	FCFA / (km/h)	14.1	19.0	30.2	16.9
	(Euro / (km/h))	0.0215	0.0290	0.0460	0.0258
	Source: 2011 Lomé Survey

Table 3 .

 3 Comparisons between customary outbound and return trips for work and school in Lomé according to mode, trip duration and cost (%)

	Mode(s), Duration, Cost	Home -Work	Home -School
	Identical outbound and return trips	71	52
	(same mode(s), duration and cost)
	Same mode(s), different duration and/or cost for return	14	17
	trip
	Longer duration, same cost	56	75
	Same duration, more expensive	30	13
	Other cases	14	12
	Different mode(s) for return trip	15	31
	Longer duration, less expensive	44	64
	Shorter duration, more expensive	24	5
	Longer duration, more expensive	11	5
	Other cases	21	26
	Source: 2011 Lomé Survey

The "current" trip is the trip the respondent was making when he or she was intercepted for the survey.

If we consider legs rather than trips, oleyias accounted for 76% of the legs in "current" public transport trips and 83% of the public transport legs made in the last seven days.

The distance used here is the crow's flight distance between the centroids of the origin and destination zones.

100 CFA Francs=0.152 Euros.

Based on our field observations at the time of the survey.


L Marioni
email: luca.marioni@mines-paristech.fr

F Bay

E Hachem

Numerical stability analysis and flow simulation of lid-driven cavity subjected to high magnetic field

Keywords: Magnetohydrodynamics, lid-driven cavity, braking, variational multiscale approach

In this work we study the flow of a conducting fluid inside a two-dimensional square domain. The problem is solved by using a variational multiscale finite element approach. The study focuses on a high magnetic interaction parameter range and high Reynolds number. Under the imposition of a high magnetic field, the flow gets regularized, but fast transient phenomena take place, which could lead to numerical errors. An expression to compute the maximum time-step which guarantees convergence in explicit schemes is proposed and validated through numerical tests.

INTRODUCTION

The flow in a cavity induced by the motion of one of the walls is a classical benchmark for fluid mechanics analyses and it has been widely studied in literature [START_REF] Ghia | High-resolution for incompressible flow using the navier-stokes equations and a multigrid method[END_REF][START_REF] Sahin | A novel fully-implicit finite volume method applied to the lid-driven cavity problem. part i: high reynolds number flow calculations[END_REF][START_REF] Zhang | Numerical simulation of 2d square driven cavity using fourth-order compact finite difference schemes[END_REF]. The most interesting part of the flow is the area close to the corners, where high turbulence and discontinuous velocity occur [START_REF] Botella | Benchmark spectral results on the lid-driven cavity flow[END_REF]. The flow has been studied at different range of Reynolds number (Re) up to 100000 in [START_REF] Bruneau | The behaviour of high reynolds flows in a driven cavity[END_REF]. Approaches similar to the one used in the present work have been used to study Reynolds from 10000 in [START_REF] Gravemeier | A three-level finite element method for the instationary incompressible navier-stokes equations[END_REF][START_REF] Masud | A multiscale finite element method for the incompressible navier-stokes equations[END_REF] and up to 50000 in [START_REF] Hachem | Stabilized finite element method for incompressible flows with high reynolds number[END_REF][START_REF] Coupez | Solution of high-reynolds incompressible flow with stabilized finite element and adaptive anisotropic meshing[END_REF]. The number of induced eddies increases with the increasing of the Reynolds number, their size decreases and the flow turns to be chaotic [START_REF] Peng | Transition in a 2-d lid-driven cavity flow[END_REF]. This flow can be braked by the superimposition of a constant magnetic field [START_REF] Yu | The effect of a uniform magnetic field on vortex breakdown in a cylinder with rotating upper lid[END_REF]. This case has been used as benchmark for different applications, from biomagnetic fluids [START_REF] Tzirtzilakis | Biomagnetic fluid flow in a driven cavity[END_REF], to nanofluids [START_REF] Kefayati | Lattice boltzmann simulation of natural convection in nanofluid-filled 2d long enclosures at presence of magnetic field[END_REF]. The stabilisation due to the external magnetic field in the cavity has also been studied [START_REF] Shatrov | Three-dimensional linear stability analysis of lid-driven magnetohydrodynamic cavity flow[END_REF][START_REF] Sivasankaran | Hydro-magnetic combined convection in a lid-driven cavity with sinusoidal boundary conditions on both sidewalls[END_REF][START_REF] Yu | The effect of an external magnetic field on oscillatory instability of convective flows in a rectangular cavity[END_REF]. Deeper investigations have been done on the natural convected cavity under an external magnetic field both numerically [START_REF] Hakan | Mhd mixed convection in a lid-driven cavity with corner heater[END_REF][START_REF] Garandet | Buoyancy driven convection in a rectangular enclosure with a transverse magnetic field[END_REF][START_REF] Ozoe | The effect of the direction of the external magnetic field on the three-dimensional natural convection in a cubical enclosure[END_REF][START_REF] Yu | Numerical investigation of natural convection in a rectangular cavity under different directions of uniform magnetic field[END_REF] and experimentally [START_REF] Davoust | Buoyancy-driven convection with a uniform magnetic field. part 2. experimental investiga-tion[END_REF]. At the best of our knowledge, the flow behaviour at high Stuart number has not been object of studies. The idea is to use this benchmark to test the numerical coupling between the fluid flow and the magnetic field at different configurations. The framework of this work is braking application in material forming, in order to decrease the turbulence. In these kind of applications, the magnetic Reynolds number is usually Rem << 1, so the magnetic field convection has a small role in the development of the magnetic field. A given turbulence level can be obtained by imposing a certain magnetic field. The problem is that this level does not depend only on the Stuart number, N = Ha 2 /Re. By rewriting N in terms of material parameters we have N = (B 2 Lσ)/(ρu), so it does not depend on the fluid viscosity. Thus a high Stuart number allows us to brake the flow, but the expected braking will occur at different level of N , depending on the material viscosity. For low viscosity fluids, Re will increase, while N will not, so high B will be needed to completely brake the flow. Therefore, we want to study the numerical behaviour of the solution when the imposed magnetic field is high, which induces a fast transient state in the flow. All the simulations in this work have been performed by using CimLib R which is a set of libraries developed in the Lab and used as base for the commercial software TherCast R .

MATHEMATICAL MODEL

The considered test case has been sketched in figure 1. The 2D square cavity has length L = 1; no-slip boundary conditions have been imposed at three borders while a constant tangent unity velocity has been imposed at the last border. The zero level pressure has been imposed to the vertex opposite to the applied shear velocity. The gravity is set to zero and the magnetic field is imposed in different configurations.

A newtonian, incompressible and conductive fluid has been considered. The density has been set to ρ = 10 3 kg/m 3 and the dynamic viscosity µ has been set in accordance to the target Reynolds number Re = ρuL µ . The electric conductivity is set to σ = 7.14 × 10 5 Ωm. The external magnetic field (B 0 ) is constant; the Reynolds magnetic number is assumed to be less than one, hereby the induced magnetic field produced by the conductor's motion is neglected. Thus, the interaction between the electromagnetic fields (EMF) and the flow consists in the Lorentz force:

F L = j × B 0 (1)
with j to be the electric current density computed according to the Ohm's law:

j = σ(-∇φ + u × B 0 ) (2)
where φ is the electric scalar potential. The potential has to respect the current density conservation condition

∇ • j = 0 ∆φ = ∇ • (u × B 0 ) (3)
which is naturally satisfied for any in-plane magnetic field vector superimposed to a 2D flow, since

∆φ = ∂ z (u x B y -u y B x ) ≡ 0 (4)
The Lorentz force is added in Navier-Stokes equations as a volumetric force term:

ρ(∂u + u • ∇u) -∇ • (2µε(u) -pI) = F L ∇ • u = 0 (5)
The equations are solved by a variational multiscale approach in the framework of stabilized finiteelements. For the detailed analysis of the method, we recall to [START_REF] Hachem | Stabilized finite element method for incompressible flows with high reynolds number[END_REF]. In this work we limit to notice that the electromagnetic force is tracked explicitly in time, so it affects the Galerkin resolution and the stabilization terms as a known source term.

Time-advancing scheme

In this work we use the classical θ-scheme for the variational multi-scale (VMS) algorithm used in [START_REF] Hachem | Stabilized finite element method for incompressible flows with high reynolds number[END_REF]. The main difference is the presence of an additional term, i.e. the Lorentz force; this term is considered as an explicit source term [START_REF] Krasnov | Magnetohyfrodynamic turbulence in a channel with spanwise magnetic field[END_REF], thus it is evaluated at the (n -1) time step and considered constant through the step. This means that, under the hypothesis of a globally braking effect of the magnetic field, we overestimate the force due to the linearization inside the time step. The aforementioned overestimation is not a problem for low level of the magnetic interaction parameter: in this case the transient flow induced by the magnetic field has a characteristic time scale much higher than the time step, classically computed according to the CFL condition:

∆t < C∆x u ( 6
)
where u is the fluid velocity, C is a constant and ∆x is the characteristic dimension of the element.

For the current work we have ∆x ≃ 0.01, u = 1 and C > 1, since C = 1 is the usual limit for explicit time schemes. In [START_REF] Hachem | Stabilized finite element method for incompressible flows with high reynolds number[END_REF] we showed that for the current problem ∆t = 0.1 can be used. Different problems arise when the magnetic interaction parameter is increased. For high N , the transient flow's time scale could be of the same magnitude as the ∆t CF L : in this configuration a different ∆t has to be calculated to both obtain accuracy and convergence. To do this, we consider a purely inertial flow, which is reasonable at high Re. This is also justified in respect to continuous casting, where the metal density is high as well as the turbulence . We want to find the time-step by which the piece-wise constant Lorentz force completely brakes the flow in one time-step. So we compare the work of the external force to the internal energy, approximated to the kinematic energy:

dE = δW. (7)
We consider the force to be constant in time and opposite to the velocity, thus:

1 2 ρu 2 0 = f L s(t). ( 8
)
Physics of fluids (2016) Prepared using fldauth.cls DOI: http://dx.doi.org/10.1063/1.4948433

where s(t) is the displacement of a material particle during the time step. By considering a material particle far enough from the boundary, and by assuming the magnetic field normal to the velocity (most restrictive case), we can reduce the motion of the particle to a constant decelerating motion

s = ut + 1 2 at 2 = ut - 1 2 f L ρ t 2 . ( 9
)
By substituting equations ( 1) and ( 9), we can rewrite equation (8) as:

- 1 2ρ (σB 2 ) 2 ∆t 2 + σB 2 ∆t - 1 2 ρ = 0 ∆t * = ρ 1 σB 2 (10)
where ∆t * is the threshold where the work of the external force is high enough to dissipate all the kinetic energy in one time-step. Similarly, ∆t = 2∆t * will be the time step which will allow the force to induce a flow opposite to the flow at the previous time-step in the areas far enough from the boundary.

Anisotropic mesh adaptation

In [START_REF] Coupez | Metric construction by length distribution tensor and edge based error for anisotropic adaptive meshing[END_REF] a posteriori error estimate based on the length distribution tensor approach and the associated edge based error analysis is proposed. This error estimation algorithm is the base for the anisotropic meshing scheme adopted in the current work. In this section we propose a synthesis of the method based on [START_REF] Coupez | Metric construction by length distribution tensor and edge based error for anisotropic adaptive meshing[END_REF] and further developed in [START_REF] Coupez | Solution of high-reynolds incompressible flow with stabilized finite element and adaptive anisotropic meshing[END_REF] in the case of Navier-Stokes, multi-field re-meshing. Let us introduce the set of fields we want to remesh according to:

u = {u 1 , u 2 , ..., u N }. (11)
Let's than denote the error of each field along the axis as:

e ij = {e 1 ij , e 2 ij , ..., e N ij } (12)
which can be expressed as:

e n ij = g ij • X ij (13)
where X ij is the length of the edge linking the ith node to the jth node, and

g ij = H(u)X ij .
Thus, we can define the error along the edges as

e n ij = g ij • X ij . ( 14
)
Now we want to consider the error in relation to a change in the edges' length, express in term of stretching factors s ∈ R, so that:

Xij = s ij X ij ẽn ij = s 2 ij G • x ij (15)
with Xij and ẽn ij being the target edge length and the target error respectively and G a recoverygradient operator defined in [START_REF] Coupez | Metric construction by length distribution tensor and edge based error for anisotropic adaptive meshing[END_REF].

Let us define the set of stretching coefficients as

S = {s ij ∈ R + i = 1, ..., Λ, j = 1, ..., Λ, Γ(i) ∩ Γ(j) = ∅} (16)
with Λ being the number of nodes of the mesh; we can therefore derive the associate metric M as: For the complete derivation of the metric, please refer to [START_REF] Coupez | Metric construction by length distribution tensor and edge based error for anisotropic adaptive meshing[END_REF].

M i =   1 d j∈Γ(i) s 2 ij X ij X ij   -1 . ( 17
)
Set the metric framework, we need now to choose the input field to remesh according to. For high-Reynolds incompressible flows, we propose the following set of fields:

u(X i ) = N v v i |v i | , N s |v i | max j |v j | (18)
where N s and N v are weighting parameters and v is the velocity field 3. RESULTS

Horizontal magnetic field and validation

The code in absence of magnetic field has been largely validated in literature [START_REF] Hachem | Stabilized finite element method for incompressible flows with high reynolds number[END_REF][START_REF] Coupez | Solution of high-reynolds incompressible flow with stabilized finite element and adaptive anisotropic meshing[END_REF] for the lid-driven case, hence in the current work we limit to validate the case with B 0 = 0. For the first test case, we set B 0 in the same direction as the x axis. In figure 2,u x profile along the vertical middle-line is plotted. The benchmark is taken from [START_REF] Shatrov | Three-dimensional linear stability analysis of lid-driven magnetohydrodynamic cavity flow[END_REF] and Re = 5000 and N = 5, 30 are reported in figure 2(a) and 2(b) respectively. In figure 3 the velocity magnitude in the upper part of the cavity is plotted. We can notice that the braking effect increases as N increases and the vertical velocity (normal to B 0 ) is dumped. This leads to a more horizontal flow and thinner eddies, as shown in figure 4.

Even with higher Re the stabilizing effect of the magnetic field occurs: in figure 5 the evolution of the flow in respect to the magnetic interaction parameter is shown. The number of eddies is directly related to the Stuart number, as reported in figure 6. Note that in figure 6 variable, the number of eddies, is plotted so the linear function represents the tendency of a piecewise constant function. As we see from figure 5, the different N adopted do not represent the last (and most unstable) eddy at the same level of development. The increasing of N is a constrain over the vertical flow which leads to almost horizontal re-circulation patterns. In figure 7 the average slope of the interface between the two upper eddies is plotted; by increasing N an alternating horizontal flow is induced and the vertical re-circulation is confined at the external boundaries. In figure 8 the horizontal velocity along the vertical middle-line is plotted. By increasing the magnetic field, the main horizontal flow moves upwards and the thickness is almost constant. At high N , the eddy is pushed against the lid, its thickness decreases and its velocity increases. Note also that the horizontal velocity of the secondary eddies is much lower than the velocity of the first upper eddy.

As the eddies' thickness decreases with the increasing of B, remesh is fundamental to catch the small eddies without increasing the number of elements. In figure 9 we see how the mesh anisotropically adapts to the flow. From figure 9(c) it is possible to notice that the algorithms does not catch the low velocity eddies by itself. For this reason, the weighting parameter N s and In this way we decrease the importance of the secondary vertical flow, so we create different eddies almost independent; the upper b.c. induces a first eddy, whose bottom part induces the second eddy below and so on. In order to fully interact with the mechanical behaviour we decided to consider a magnetic field facing the primary flow. In the second test case the orientation of the magnetic field was changed, so that B y = const. and B x = B z = 0. In this configuration the primary flow, driven by b.c., must cross the magnetic field, so a higher level of braking is expected. In figure 10 the flow at different Stuart numbers is described. The first effect of the magnetic field is to sharpen the flow at the upper-right corner and to suppress the smallest eddies (figure 10(b)). By increasing the Stuart number the main eddy squeezes along the right boundary, where the kinetic energy input by the b.c. is highest. Finally, in figure 10(f), the flow is controlled by the magnetic field. The fluid coming from the upper boundary deviates in the y direction since the magnetic field obstacles the horizontal flow. Despite this, the flow has to move to the left side because of the boundary conditions on velocity and so it can not form an independent eddy. For this reason a slow, laminar flow is formed in the middle of the cavity. Over this value of N , the vertical eddy's size would decrease as well as the magnitude of velocity in the center band. The height of the right eddy depends on the Reynolds number: as reported in table I a higher Reynolds number leads to a higher vertical excursion of the eddy. It is also shown a very known result, by which a high Stuart number can regularize the turbulent flow and make it steady. In figure 11 the horizontal velocities along the horizontal middle-line are reported. We notice that the constant velocity band in the middle is wider when Re is higher (and µ lower); the low-viscosity flow is therefore constrained into a thin and high eddy close to the right boundary. The same process is clear from the same plot for a higher Stuart number (figure 12). The curve for the same Re (Re = 10000 and Re = 20000) are more braked, thus the velocity picks decrease and the plateau is wider (until almost x = 0.8 against about x = 0.55).

In figure 13 we plot the number of nodes where the velocity normal to the magnetic fields reverses over one time step, i.e. u n • f n > 0; in this case the Lorentz force brakes the flow and then accelerates it inside the same time increment. Since the Lorentz force does not follow the velocity inside the time step, it happens that for fast transient phenomena the ∆t CF L is of the same magnitude than the transient flow itself. From figure 13 we see that ∆t = 3 • 10 the flow in the 55% of the domain. It is important to underline that in a Eulerian framework, this is a natural phenomenon: the velocity variation is referred to the spatial deformation of the flow and not on the material particle itself. In this case the pathological behaviour is evident from the fast increase of the number of these points, as clear from figure 13. To better analyse this problem, we will consider the most restrictive case in section 3.3.

Perfect braking case

In this section we will present a numerical test in order to better catch the numerical problem shown in figure 13. We propose to consider a Lorentz force opposite to velocity on the whole domain (perfectly braking) and equal to:

f L = σuB 2 . ( 19
)
By this test we will be able to: • Minimize the flow deformation. As stated in section 3.2, "reverse velocity" point could be naturally produced by the flow deformation in Eulerian simulations. By imposing a perfectly braking force, the flow will be slowed but not heavily deformed. The pathological velocity inversion will occur on the whole domain for the same time step, so it will be easily detected.

Physics
• Extend the numerical problem on the whole domain. In the cases presented in sections 3.1 and 3.2, the numerical problems occurred in the regions where magnetic reaction to the flow was maximum, so where the velocity was normal to the magnetic field. By imposing a perfectly braking force we will be able to extend the same conditions to the whole domain.

This numerical test case represents all the physical cases in which the Lorentz force is opposite to the velocity globally (e.g. vertical flow with an horizontal magnetic field) or locally (e.g. cases in sections 3.1 and 3.2).

Several tests similar to the one reported in figure 13 have been conducted and the results are reported in figure 14. In all these configurations we notice that the turn of velocity is much faster than in the case of vertical magnetic field. In figure 13 the velocity turning is smoother because part of the Lorentz force's work is spent to deform the flow; in this case the extra-constraint on deformation leads to an almost instantaneous velocity turning, where the energy dissipated inside the time step is high enough to balance the inertia. We can also notice that the expression [START_REF] Peng | Transition in a 2-d lid-driven cavity flow[END_REF] well predicts the time-step which leads to this phenomenon, represented by the dotted lines (notice that few different time steps were used for each simulation, so the whole band is included between the smaller and larger time steps used). We do not report the complete curves for all the configurations analyzed, but in figure 15 we show the variation of the critical time step over the variation of the main problem variables: from this figure we see the the dependencies are conform to the one adopted in [START_REF] Peng | Transition in a 2-d lid-driven cavity flow[END_REF].

The choice of ∆t affects also the convergence. In figure 16 the convergence of the L 2 error is plotted; the error is here computed in respect to an overkilled solution with ∆t ≪ min(∆t). From this plot we see that for ∆t ≥ 2∆t * the simulation diverges, even if we respect the CFL condition. In this case the Lorentz force (constant inside the time-step) is high enough to reverse the flow and increase the velocity magnitude in the opposite direction, which leads to a diverging acceleration of the flow, thus we have

u n • u n-1 < 0 ∨ ||u n || ≥ ||u n-1 || f n L • u n > 0 ∨ ||f n L || ≥ ||f n-1 L ||. (20)
For ∆t ∈ (∆t * ; 2∆t * ) the solution converges, but the convergence rate is small. In this interval, the transient flow is not well predicted since the punctual velocity reverses its direction at each time-step. The flow will oscillate across an equilibrium point but the reverse velocity's magnitude is still lower than the one at the previous time step, so the oscillations decrease amplitude. This configuration is defined as:

u n • u n-1 ≤ 0 ∨ ||u n || < ||u n-1 || f n L • u n ≥ 0 ∨ ||f n L || < ||f n-1 L ||. (21)
For ∆t < ∆t * the converge rate is higher and no verse-changing takes place and the following conditions are satisfied: In this case the Lorentz force monotonically brakes down the flow. The velocity's evolution is qualitatively convex and monotonically decreasing, so the piece-wise constant approximation of the Lorentz force leads to an over-estimation of the braking effect, which still does not affect the convergence. These considerations are directly connected to the prediction of the transient state; in figure 17, the time required to reach the steady state is plotted. For ∆t = 2∆t * the required time tends to infinity because the system oscillates between two opposite velocity fields. For ∆t < ∆t * the solution tends to a constant value, T s = 0.52 in the current simulation.

u n • u n-1 > 0 ∨ ||u n || < ||u n-1 || f n L • u n < 0 ∨ ||f n L || < ||f n-1 L ||. (22
In figure 18 the horizontal velocity at P (0.5; 0.75) is plotted over the time-steps. As expected, we see that the simulation with ∆t = 0.0072 > 2∆t * diverges and the one with ∆t = 0.0001 < ∆t * converges providing a good description of the transient state. Between these cases, we see that for ∆t ∈ (∆t * ; 2∆t * ) the simulations converge to the correct steady state, but the transient state is oscillatory, due to the numerical problems connected to the time-step. It is also important to note that the velocity, in those cases, does not reverse at each time step forever (like for ∆t = 0.0072 > 2∆t * ): when the velocity is low enough that the viscous energy is dominant in respect to the kinetic energy, the external work is no more sufficient to dissipate the whole energy and equation [START_REF] Peng | Transition in a 2-d lid-driven cavity flow[END_REF] is no longer valid. In figure 18, we see that the simulation with ∆t = 0.006 stops oscillating at the 16th step. For ∆t = 0.0068, this point is reached at the 68th step.

CONCLUSIONS

In this work high Reynolds driven cavity has been studied. We showed how the flow can be braked and the turbulence decreased by different magnetic field configurations. Indeed, high Reynolds number flows need high magnetic field to be braked, but ρ and µ play different roles; while ρ affects both N and Re, µ affects only Re, so the N needed to reduce the turbulence to a certain value depends on the viscosity. For this reason, we studied high Stuart number cases. In these configurations, problems with the prediction of the transient flow arise and the CFL condition is not enough to guarantee the convergence. Therefore an expression to estimate a value of time-step is proposed and it precisely predict and solve the aforementioned problems for high inertia problems.

Figure 1 .

 1 Figure 1. Schematic representation of the lid-driven cavity test: geometrical features and boundary conditions

Physics of fluids ( 2016 ) 30 Figure 2 .

 2016302 Figure 2. Velocity profile along the middle-line: validation

Figure 3 .

 3 Figure 3. Velocity magniture in the upper part of the cavity

Figure 4 .

 4 Figure 4. Isocontour of velocity field magnitude. The isovalues are in logarithmic scale: 20 in the range ||u|| ∈ (0.01; 0.5) and 100 in the range ||u|| ∈ (0; 0.01)

  N n are computed according to the magnetic interaction parameter. The anisotropic non-structured mesh allows the user to well model the boundary layers and the area close to the corners, where instabilities due to the boundary conditions non continuity may occur. Physics of fluids (2016) Prepared using fldauth.cls DOI: http://dx.doi.org/10.1063/1.4948433 A c c e p t e d (a) Velocity streamlines: Re = 10000, N = 0.5 (b) Velocity streamlines: Re = 10000, N = 1 (c) Velocity streamlines: Re = 10000, N = 2 (d) Velocity streamlines: Re = 10000, N = 5 (e) Velocity streamlines: Re = 10000, N = 15 (f) Velocity streamlines: Re = 10000, N = 20

Figure 5 .A c c e p t e d 8 MARIONIFigure 6 .Figure 7 .

 5867 Figure 5. Velocity streamlines of the flow subjected at different external magnetic field's magnitudes

9 -

 9 Physics of fluids (2016) Prepared using fldauth.cls DOI: http://dx.doi.org/10.1063/1.4948433 A c c e p t e d LID-DRIVEN CAVITY FLOW SUBJECTED TO HIGH MAGNETIC FIELD

Figure 8 . 2 Figure 9 .

 829 Figure 8. Horizontal velocity profile along the vertical middle-line

  -3 sec leads to invert Physics of fluids (2016) Prepared using fldauth.cls DOI: http://dx.doi.org/10.1063/1

Figure 10 .

 10 Figure 10. Velocity streamlines of the lid-driven cavity flow under a vertical magnetic field.

Figure 11 .Figure 12 .

 1112 Figure 11. ux plot over the center-line y = 0.5 at different Re and N = 1.785

Figure 13 .

 13 Figure 13. Number of reverse-velocity nodes normalized to the total number of free nodes, N = 7.14

)Figure 14 . 5 • 10 -Figure 15 .A c c e p t e d 10 2 Figure 16 .

 145101510216 Figure 14. Normalized reverse-velocity points in different problem configurations

A c c e p t e d 16 MARIONIFigure 17 .- 1 )Figure 18 .

 1617118 Figure 17. Physical time to reach the steady state in respect to the time-step

Table I .

 I Minimum height of ||u|| = 0.05 isovalue

	Re	y N =7.14	y N =1.785
	200000	0.11	non-steady
	100000	0.165	non-steady
	50000	0.22	0.09
	20000	0.33	0.148
	10000	0.44	0.175
	5000	0.5	0.21
	1000	0.52	0.258

ACKNOWLEDGMENTS

The authors give thanks to Transvalor S.A. for funding this research and for the continuous support which allowed the complete exploitation of its software.


Pierre Renaud
email: renaud@lirmm.fr

Enric Cervera
email: ecervera@icc.uji.es

Philippe Martinet
email: martinet@lasmea.univ-bclermont.fr

Towards a reliable vision-based mobile robot formation control

In this article, a robot formation control strategy based on a vision-based follow-the-leader scenario is proposed, with emphasize on its reliability. On the one hand, perception is enhanced by the control of a motorized zoom. On the other hand, bidirectional and non-oblivious [1] control is implemented, with an odometry-based fault detection of vision-based information and a leader path-planning strategy to improve its visibility. After introduction of the control strategy, extensive experimental results are presented.

I. INTRODUCTION

Control of robot formations has received much attention these past years. Indeed, the use of several coordinated mobile robots enables one to achieve complex tasks [START_REF] Jennings | Cooperative search and rescue with a team of mobile robots[END_REF], [START_REF] Nguyen | Autonomous communication relays for tactical robots[END_REF], [START_REF] Hogg | Algorithms and sensors for small robot path following[END_REF], with a low robot complexity, since each one can be specialized in one task such as navigation or manipulation. The previously proposed control strategies offer different degrees of robustness of the formation, i.e. reliability with respect to real world conditions. Perception, formation and control characteristics, the three main criteria cited in [START_REF] Michaud | Dynamic robot formations using directional visual perception[END_REF] to classify the robot formation control strategy, influence this robustness, as shown in the following.

Using the robot proprioceptive sensors and inter-robot communication, self-localization can be achieved [START_REF] Roumeliotis | Distributed multi-robot localization[END_REF]. Odometry accuracy remains however delicate [START_REF] Goel | Robust localization using relative and absolute position estimates[END_REF]. Dead reckoning is then limited, and will still fail in case of wheel slippage or contact with an obstacle. Vision has been therefore proposed to get an exteroceptive information. With omnidirectional cameras [START_REF] Das | A vision-based formation control framework[END_REF] a 360 • detection is possible, within however a limited distance (in the order of 1.5m in [START_REF] Pereira | Localization and tracking in robot networks[END_REF]). Due to the large field of view, the accuracy may also be limited if a standard camera is used. Directional cameras have complementary performances, with visibility constraints [START_REF] Michaud | Dynamic robot formations using directional visual perception[END_REF], at however a much lower cost [START_REF] Sarcinetti-Filho | Mobile robot navigation via reference recognition based on ultrasonic sensing and monocular vision[END_REF]. Concerning the image processing, color has been widely used to achieve the robots detection [START_REF] Fredslund | A general algorithm for robot formations using local sensing and minimal communication[END_REF], [START_REF] Michaud | Dynamic robot formations using directional visual perception[END_REF], [START_REF] Das | Real-time vision-based control of a nonholonomic mobile robot[END_REF]. The robustness of color detection with respect to light conditions can however be a major source of detection failure [START_REF] Cubber | A bayesian approach for color consistency based visual servoing[END_REF]. Eventually, a global exteroceptive measurement [START_REF] Balch | Behavior-based formation control for multirobot teams[END_REF] is difficult to achieve in indoor applications.

Formation characteristics can improve its reliability. In a unit-referenced formation [START_REF] Balch | Behavior-based formation control for multirobot teams[END_REF], or point-referenced with the use of a virtual structure [START_REF] Lewis An | High precision formation control of mobile robots using virtual coordinates[END_REF], the perception by each robot of the other members enables one to get redundancy which increases the formation robustness. The need for a complete formation perception is however very restricting.

With a neighbor-referenced approach, if the link between the robots is unidirectional [START_REF] Tabuada | Feasible formations of multiagent systems[END_REF], the follower navigation is on the contrary completely dependent on the perception of its neighbors. A neighbor failure cannot be detected and therefore compensated. At the opposite, in the bidirectional formation control proposed in [START_REF] Fredslund | A general algorithm for robot formations using local sensing and minimal communication[END_REF], the failure of one robot is detected by the interruption of the emission of its identification number. Like in [START_REF] Michaud | Dynamic robot formations using directional visual perception[END_REF], the proposed robot attitude is the re-formation of the robots, which is however not adequate with a local loss of information of one robot, because of a vision failure or a temporary occlusion.

Eventually, the ability in the control to manage obstacle avoidance, change of formation geometry can improve the reliability of the formation in unknown environments. The use of a non-oblivious [START_REF] Suzuki | Agreement on a common X-Y coordinate system by a group of mobile robots[END_REF] control scheme, i.e. with a behavior dependent on the past information, is also likely to improve the robustness of the control scheme.

We have proposed in [START_REF] Chiem | Vision-based robot formations with Bezier trajectories[END_REF] a simple and efficient way to control robot formations. Leader-follower control is used, with unidirectional control, using vision-based leader pose estimation. In this article, we develop the reliability of this formation control. On the one hand, the perception is enhanced by the control of a motorized zoom. Leader pose estimation is hence possible in a larger field-of-view, with higher accuracy. On the other hand a bidirectional and non-oblivious control strategy is implemented to improve the reliability of the vision-based scheme: an odometrybased fault-detection is used to detect vision-based pose estimation failure and a novel path-planning scheme is proposed to control the leader and avoid loss of visibility by the follower.

In the second section, the follow-the-leader control scheme is recalled with the description of the development platform. In the third section, the implementation of the zoom control and its advantages are then outlined, before in the fourth section introducing the control strategy proposed to improve the reliability of the vision-based follow-theleader. Extensive experimental results are then given in the fifth section, before concluding on the performance of the robots control and further developments.

II. THE ROBOT FORMATION CONTROL SCENARIO

A. Robot formation control

The control of the formation is using a decentralized algorithm. The leader of the formation is considered spe-cialized in the navigation task, using for instance a laser to build environment maps and achieve the localization task. Each follower robot measures the pose (i.e. position and orientation) of its leader using vision. The follower trajectory is computed by considering a Bezier curve [START_REF] Chiem | Vision-based robot formations with Bezier trajectories[END_REF] between the follower robot current position and the estimated leader relative position.

For the considered unicycle robot model, the Bezier curve allows us to compute trajectories wich are tangent to current leader and follower paths. Assuming the needed angular velocity can be obtained with the follower robot, the computed path is compatible with the nonholonomic constraint. This assumption is generally verified experimentally. Furthermore, the follower is then able to avoid collisions with obstacles bypassed by the leader by properly designing the Bezier curve (figure 2). Finally, formations can be controlled by simply defining virtual leader positions from the current one (figure 2).

B. Hardware platform

Commercial mobile robots Pioneer 2 [18] (figure 3) are used for our application. All the robots are equipped with sonars for obstacle detection and wireless communication (IEEE 802.11b). Control of each robot is performed using an embedded PC.

The leader is equipped with a laser to build environment maps and achieve the localization task. A calibration board is mounted on it, and the follower is equipped with a pan-tilt-zoom camera (Canon VCC4, 4 -64mm zoom) to observe its leader. The calibration board is composed of three different colored areas, which are detected using a blob detector. The pose of the leader robot (d, α) (figure 1) can hence be determined [START_REF] Chiem | Vision-based robot formations with Bezier trajectories[END_REF].

C. Software platform

The formation control is developed in Acromovi [START_REF] Nebot | Agents for cooperative heterogeneous mobile robotics[END_REF], a framework specially designed for the development of Obstacle bypass simulation (90 • turn, radius 2m, speed: 0.15m/s, distance leader/follower: 2m, time increment: 1s). On the top with one follower, on the bottom with three followers. distributed applications for a team of heterogeneous mobile robots. The software architecture gives us the ease of development of cooperative tasks among robots, using an agent-based platform. In particular, communication between robots can be easily integrated to the control scheme.

y f i x f i x f i x l i x f j d y l i x l j y l j α O li O lj O f i y f j O f j

III. PERCEPTION ENHANCEMENT

A. Pan-Tilt-Zoom control

In this section, we propose to use a motorized zoom to lower the detection failure of the calibration board and improve the pose estimation accuracy. The use of zoom has been proposed in the context of active vision [START_REF] Atienza | A practical zoom camera calibration technique: an application of active vision for human-robot interaction[END_REF] or visual servoing, with explicit knowledge of intrinsic parameters from calibration [START_REF] Hosoda | Visual servoing utilizing zoom mechanism[END_REF], [START_REF] Clady | Objet tracking with a pan tilt zoom camera, application to car driving assistance[END_REF], or not [START_REF] Malis | Visual servoing invariant to changes in camera intrinsic parameters[END_REF]. In our context, the communication protocol between the robot and the camera limits the camera control frequency (∼ 1Hz), hence restricting the vision-based camera control to a static look and move strategy [START_REF] Sanderson | Robot vision[END_REF]. Camera intrinsic parameters are

0.01M, 0.01 • X X X X X X X X d (m) α ( • ) 45 • 60 • 1m 0.01m / 0.01m 0.6 • / 0.6 • 0.01m / 0.01m 0.3 • / 0.3 • 2m 0.05m / 0.02m 25 • / 1 • 0.02m / 0.01m 0.3 • / 0.3 • 3m × / 0.03m × / 0.6 • × / 0.04m × / 0.6 •
identified during a prior calibration.

The calibration board center and the height of the center color card in the image are used to compute the pan and zoom corrections to apply to the camera. Since the calibration board center is in our context at the same height that the camera, no tilt correction is needed. Let (f, ∆o l , h) be respectively the current focal length, gap in the image between the camera image center and the calibration board center in the horizontal direction and the size in the image of the central element of the calibration board. The pan and zoom corrections (∆P, ∆f ) to apply at each time step are then computed by first order approximation as:

∆P = ∆o l f ∆f = h des h -1 • f (1)
with h des the desired size in the image of the calibration board central element.

B. Simulation results

The improvement of the pose estimation accuracy has been evaluated experimentally through repeatability measurements in different configurations. In table I the standard deviations of the calibration board angle and distance are indicated for three distinct distances. The fronto-parallel position (α = 0) is not considered because for small values the angle is set to zero.

The use of the zoom improves the ability to detect the calibration board and the accuracy. A significant accuracy improvement of the follower trajectory can also be observed by simulating the trajectory of figure 2 with addition of the experimentally evaluated measurement noise (figure 4). When used, the zoom magnification is approximately equal to 2. Without zoom, the gaps between leader and follower noise are up to 0.3m, which can cause interferences with another robot or an obstacle.

IV. CONTROL ENHANCEMENT

The follower robot is controlled by computing at each time step the vision-based pose estimation as mentioned in the second section, and then the corresponding Bezier path. In this section, we propose two modifications of the control scheme to improve the reliability of the follow-the-leader application. The first one is based on the redundancy of the odometry with respect to the vision-based perception. The second one is an original path-planning strategy to avoid the loss of the leader image by the follower.

A. Leader state estimation

Vision is an adequate sensor for the robot control, with accuracy in the order of (1cm, 1 • ) for indoor applications with off-the-shelf cameras. Variable light conditions, calibration board occlusions may however significantly alter the calibration board color cards detection, leading to outliers in the vision-based pose estimation. To improve the follower control reliability, communication between follower and leader and their odometry are hence used to get a redundant leader state estimation.

For sake of simplicity, the calibration board and the camera are supposed located on the rotation axes of the robots. The position of the leader with respect to the follower at time instant j from situation at instant i can be predicted with the relation (figure 1):

O fj O lj | R f j = O fj O fi | R f j +O fi O li | R f j +O li O lj | R f j (2)
with at first order:

O fj O fi | R f j = -v fi ∆t • x f j | R f j O li O lj | fj = v li ∆t • x l i | R f j (3)
where v represent the current translational speed, estimated from odometry.

From eq. ( 2), the pan angle of the camera can also be predicted with:

Θ j = Atan( O i l O j l • y fj O i l O j l • x fj ) (4)
The leader pose estimation with respect to the follower is achieved at each iteration of the control. This enables us to detect occlusions or outliers of the vision-based pose estimation by comparison between the odometrybased estimation and the vision-based one, according to the following pseudo code: The value of the confidence interval is easily set since partial occlusion of the calibration board or color detection failure induces pose error greater than 30cm, to compare with the 1cm vision-based pose accuracy.

Odometry accuracy bias is well known, especially for rotation measurements. It is therefore used as a redundant sensor for fault detection, but no data fusion is achieved between vision and odometry, since no such bias occurs with vision.

B. Path-planning for leader visibility 1) Leader path-planning strategy:

The formation leader has the perception of its environment and its localization, considering that a previous mapping of the environment has been achieved. Several methods have been proposed to ensure the displacement of the robot to a defined goal. The gradient method proposed by in [START_REF] Konolige | A gradient method for realtime robot control[END_REF] present the advantages of a real-time planning taking into account the evolution of the environment, with avoidance of local minimas. In this section we introduce a modification of this planning method to ensure both the planning of the trajectory towards the goal and the vision-based control of a follower robot. The used gradient method is first briefly recalled. The reader is referred to [START_REF] Konolige | A gradient method for realtime robot control[END_REF] for further details.

2) The gradient method principle: The gradient method is based on the evaluation of a navigation function for each path, the optimum being the one minimizing this function. Two costs are computed at each point p i for its evaluation: the intrinsic I(p i ) and adjacent A(p i , p i+1 ) costs. The former represents the cost of being at the point, and the latter the cost of moving from p i to another point p i+1 . It is assumed that these costs are separable, so that along a path constituted by the set of points P = {p 1 , p 2 , ...} the cost is equal to:

F (P ) = p I(p i ) + A(p i , p i+1 ) (5)
The navigation function at a point p is defined as the minimal cost of a path between p and the goal point. The robot path is hence determined by computing at each time step the costs to get the optimal path.

∆ p ′ f p f p ′ -p p ′ f p f p p ′ p ′ f -p f = p ′ -p
3) The visibility constraint: At each position p of the leader (figure 5), the current position of the follower p f is known, either using vision or the estimation described in the previous paragraph. For any point p ′ around the current position of the leader, one can therefore determine the corresponding position p ′ f of the follower and the line ∆ p ′ f p ′ between the robots. At each step, the obstacles are supposed known using the mapping and the perception with the laser and sonars. The associated intrinsic costs can then be computed, so that we can evaluate the loss of the leader image by the follower by the condition:

i I(p i ) > 0, ∀p i ∈ ∆ p ′ f p ′ (6)
If the condition is fulfilled, the intrinsic cost at point p i is set to the cost of an obstacle. Since at each step of the leader trajectory determination the intrinsic costs are computed to determine the path, the visibility of the leader should be ensured.

The existence of a path for the leader with respect of the visibility constraint is not a priori ensured. The gradient method may therefore not find any solution to the planning of the robot leader. In such a case, the planning will be switched to the previous scheme, without considering the additional constraint.

4) Simulation results: In figure 6 two paths are generated to bring the leader robot on the other side of a wall. Without the visibility constraint, the follower loses the leader image during a significant part of the trajectory. Odometry-based trajectory estimation can then be used, but it must be assumed the absence of significant bias of the sensors. The proposed planning strategy avoids such hypothesis, with constant visibility of the leader robot.

The visibility constraint imposes the absence of obstacles between the two robots. One may notice that the angle between the two robots becomes close to 90 • during the leader turn on the figure 6. The calibration board image loss can be avoided using additional calibration boards on the leader sides, or introducing an intrinsic cost linked to the value of the angle between the robots. This method will now be evaluated on our development platform. After validation, planning of a leader with several robots in the formation will be implemented, as well as the angle constraint between the robots.

V. EXPERIMENTAL RESULTS

A. Zoom contribution

To evaluate the contribution of the zoom, in a first experiment the leader and follower are velocity controlled, with a follower having 80% of the leader speed. A 45 • turn is achieved by the leader at 0.2m.s -1 (figure 7). The initial distance is approximately equal to 1m, and increases up to 5m before the leader stops (figure 8).

The control of the focal length allows to use up to a 3× zoom. Without zoom the detection of the leader could not be achieved. The use of the zoom could hence enable us to design formation control with larger flexibility of the distance between the robots, and therefore on formation geometry. In particular, high-speed displacements may be used with large distances between the robots for safety.

The camera is programmed in that experiment to zoom out in case of detection failure. Such a failure occurs several times (figure 8), and outliers in the distance evaluation also appear. The use of odometry-based redundancy enables us to compensate these failures, as demonstrated in the next paragraph.

B. Odometry-based redundancy 1) Long distance following:

With the previous experimental conditions, and the use of the odometry-based fault detection, the distance is now estimated without any outlier (figure 8).

The camera is still programmed to zoom out in case of detection failure, so that the time steps where distance estimation is achieved using odometry can be easily detected. The follower is programmed to stop simultaneously with the leader, so that this experiment is shorter than the previous one (figure 8).

2) Trajectory in the U JI building: Trajectories have been generated in the UJI Computer Science Building, with two consecutive 90 • from the door of the lab. This reproduces the trajectories that would be obtained by planning, if visibility constraints are not taken into account. The follow-the-leader control has been experimented using successively vision, odometry, and their combination with the previously described control scheme. The proposed strategy is the only one which enables the follower to achieve the complete trajectory (figure 9). In particular, the odometry bias (in the order of 35 • for a 360 • rotation) prevented the follower to achieve the leader path.

VI. CONCLUSION

In this article, a robot formation control strategy has been described, with particular developments for its reliability. On the one hand, perception is enhanced by the control a pan-tilt-zoom camera, which gives the follower robot a large field of view (180 • , several meters) and improvement of the leader detection. On the other hand, bidirectional and non-oblivious robot control has been proposed, with the use of odometry to detect outliers in the follower vision-based pose estimation, and the leader path planning taking into account visibility constraint. The presented experimental results have demonstrated the improvement due to the use of zoom and odometry-based fault detection. The enhanced field of perception of the follower robot will enable us to consider new formation geometries, with higher distances between the robots, which now will be implemented. In particular, distances can be set as a function of the leader speed, to get "safe distances" between the robots. Using modified communication between camera and robot, a modified control of the camera will also be used, with a velocity-based control scheme.

Fig. 1 .

 1 Fig. 1. The leader and follower at two time instants i and j, with the Bezier trajectory determined from their positions at time instant i.

  Fig. 2.Obstacle bypass simulation (90 • turn, radius 2m, speed: 0.15m/s, distance leader/follower: 2m, time increment: 1s). On the top with one follower, on the bottom with three followers.

  Fig. 3. Leader and follower robots.

Fig. 4 .

 4 Fig.[START_REF] Hogg | Algorithms and sensors for small robot path following[END_REF]. Simulation of obstacle bypass using the pose measurement noise experimentally evaluated: trace of 50 computed trajectories (simulation conditions of figure2).

for each step, do { 1 .

 1 get previous leader pose value 2. compute odometry-based new leader pose 3. measure the new leader pose from vision 4. compare the two estimations 5. use odometry if vision-based estimation is outside the confidence interval 6. update the leader pose value }

Fig. 5 .

 5 Fig. 5. Computation of the intrinsic cost bound to the visibility constraint.

Fig. 6 .Fig. 7 .

 67 Fig. 6. Simulation of obstacle bypass planning for the leader (Final position of the leader in (0, 1.4), leader path without constraint in continuous line, with constraint in dotted line. Follower positions represented by crosses and circles, Distance leader/follower: 1m)

Fig. 8 .

 8 Fig. 8. Evolution of the vision-based distance estimation and zoom magnification.

Fig. 9 .

 9 Fig. 9. Trajectory in the U JI building.

TABLE I ESTIMATED

 I STANDARD DEVIATIONS OF THE LEADER POSE MEASUREMENTS. CROSSES (×) FOR DETECTION FAILURE OF THE CALIBRATION BOARD. IN ITALIC: WITHOUT ZOOM, IN BOLD: WITH ZOOM. VALUES ROUNDED TO

ACKNOWLEDGMENTS

This research has been carried out at the Robotic Intelligence Laboratory of Universitat Jaume-I. Support for this research is provided in part by the Ministerio de Ciencia y Tecnologia under projects DPI2001-3801, HF2001-0112, and FIT-020100-2003-592 (PROFIT), by the Generalitat Valenciana under projects inf01-27, GV01-244, CTIDIA/2002/195, and by the Fundacio Caixa-Castello under project P1-1B2001-28. The authors gratefully acknowledge this support.