Mohamed Hilal

Thierry Brossard

Jean Cavailhès

Daniel Joly

François P Tourneux

Pierre Wavresky

Landscape metrics for determining landscape prices

This paper reviews the landscape metrics used in economic valuations of farmland and forest landscapes. The diversity of landscape indicators used in the literature (land use, landscape composition, distance from landscape assets, seen landscapes: viewshed, objects, shapes) sometimes leads to divergent results. After reviewing the literature, we present a numerical model of the visible landscape, combining satellite images and digital elevation models. Associated with a hedonic model of price valuation, this geographical model offers the flexibility required to test several landscape metrics. The results show that although these have substantial effects on the estimation of the price of a particular landscape seen from a given house, the hedonic prices obtained have rather similar mean values.

INTRODUCTION

Urban sprawl extending out into peripheral rural areas is a familiar phenomenon in most developed countries [START_REF] Caruso | La diversité des formes de la périurbanisation en Europe[END_REF]. 'Open space is often cited as a primary attractor of urban and suburban residents to exurban areas located just beyond the metropolitan fringe' [START_REF] Irwin | The problem of identifying land use spillovers: measuring the effects of open space on residential property values[END_REF]. In France, Le [START_REF] Jeannic | Trente ans de périurbanisation : extension et dilution des villes[END_REF] speaks of the taste for 'living in a natural setting far from the bustle of cities, in spacious houses with gardens while holding on to the source of remuneration of a job in town'. The choice between living in the city or in the surrounding countryside involves striking a balance between better access to employment and to cultural and educational infrastructures that weigh on the urban side and to environmental and landscape amenities that weigh on the side of peripheral rural areas.

The management of urban and periurban amenities is a major ongoing concern for local public authorities [START_REF] Clay | Scenic landscape assessment: the effects of land management jurisdiction on public perception of scenic beauty[END_REF][START_REF] Davodeau | Les paysages, une nouvelle préoccupation dans la gestion des espaces périurbains[END_REF]. For geographers, this research area is very well documented because of collaboration with ecologists, urbanists or public operators, giving rise to objectively-based criteria for evaluating landscapes. Economists too endeavour to evaluate landscape prices; they have an interest in integrating the results of the geographic approach, provided that it engenders the quantitative indices they need to construct their econometric models. To this end, geographical information systems (GISs) are the essential technical resource for characterising landscape using numerical data filed in ad hoc bases.

GISs can produce a large quantity of indices on various landscape properties; but their sheer abundance may cause confusion. This is why this paper proposes to review the landscape metrics used in economic evaluations of farmland and forest landscapes. Section 1 presents the main economic results derived from a review of the literature, arranged by the type of metric used. Section 2 looks at how GISs and the experimental capacities they provide can be used to construct a formal approach to landscapes providing insight into the diversity of landscape composition through quantitative indices. Section 3 tests how robust or how volatile the selected landscape measures are by using the hedonic price method and discusses the results.

INDICATORS USED IN ECONOMIC EVALUATIONS OF LANDSCAPE

In the literature on the economic value of farmland and forest landscapes, indicators of landscape attributes long remained rather crude. They were confined to three types of variable: distance from particular objects such as golf courses, parks, open spaces, forests, etc. insofar as the view of a landscape is dependent on such distance; land use (farming, forestry, etc.) analysed in two dimensions from maps, aerial photographs and satellite images; landscape composition indices that add shape variables to the foregoing. More recently, recourse to numerical geographic information means more elaborate landscape indicators can be constructed that measure what an observer at the ground level sees in three dimensions.

Distance from landscape objects

Economists have long included in hedonic price models landscape variables of the presence of an object that may be thought of as an amenity (golf courses, parks, open spaces) or a nuisance (waste disposal sites, electricity pylons, roads) or of distance of housing from such objects as measured on a map. This is a far cry, then, from landscapes, although it is sometimes the view of such objects that is under study.

Distance provides an understanding that most landscape goods have a double status [START_REF] Correll | The effects of greenbelts on residential property values: some findings on the political economy of open space[END_REF][START_REF] Lee | Efficient configuration of a greenbelt: theoretical modelling of greenbelt amenity[END_REF]. A green belt is both a pure public good by the view one has of it and an impure public goods since, because of remoteness, some consumers can be excluded from it. Distance also makes it possible to analyse the effect on house prices of proximity to objects such as open areas, when the scale is very large [START_REF] Hobden | Green space borders: a tangible benefit? Evidence from four neighbourhoods in Surrey, British Columbia, 1980-2001[END_REF][START_REF] Mooney | The influence of riparian protection measures on residential property values: the case of the Oregon plan for salmon and watersheds[END_REF][START_REF] Thorsnes | The value of a suburban forest preserve: Estimates from sales of vacant residential building lots[END_REF]. Distance also lends itself to studies of the decreasing effect on property prices when an object is moved away, which also applies to nuisances [START_REF] Garrod | Using contingent ranking to estimate the loss of amenity value for inland waterways from public utility service structures[END_REF][START_REF] Hamilton | Do high voltage electric transmission lines affect property values?[END_REF] or to amenities such as wetlands [START_REF] Mahan | Valuing urban wetlands: a property price approach[END_REF], green areas [START_REF] Bolitzer | The impact of open spaces on property values in Portland, Oregon[END_REF][START_REF] Morancho | A hedonic valuation of urban green areas[END_REF][START_REF] Smith | Treating open space as an urban amenity[END_REF] or woodlands (Tyrvainen and Miettinen, 2000).

These few examples are just a small fraction of the very many studies of economic evaluation using distance variables. The results are fairly homogeneous from one study to another. These works show in particular that the effects of distance are generally very localised: sometimes one has to be adjacent to the object in question or with a few tens of metres of it at most, to perceive its positive (open spaces) or negative (electricity pylons) effects. Beyond a few tens of metres, or one or two hundred metres at most, consumers seem indifferent to the presence of this sort of object.

Land use without shape variables

Economic evaluations of the price of land use by different types of land cover (forests, farmland, gardens, roads, etc.) are very plentiful. The method consists first in delimiting a perimeter around a house, usually a circle or an administrative unit and then measuring the presence, the proximity or the proportion of land-use categories within that perimeter. The number of categories varies with studies. For example, Kestens et al. (2001) 2002) use 31 categories (trees located in front of or behind property, ranked by size, species, distinctions in floral compositions, hedges, laid-out pavements, the presence of a patio, balcony arrangements, etc.). Classifications are usually made on the basis of maps, aerial photographs or satellite images. The method therefore provides complete cover of a study region so it can be mapped and each house located in its two-dimensional landscape setting so that it can be examined whether or not prices incorporate landscape attributes. The landscape analysed is greatly simplified since no measure of shape is involved, although this is essential for characterising landscape.

Studies based on land use generally conclude that the presence of farmland, woodland or forests command positive hedonic prices. Thus, among others, Des [START_REF] Des Rosiers | Landscaping and house values: an empirical investigation[END_REF] show that, near Quebec (Canada), the presence of landscape attributes (forests, landscaped gardens, hedges) raises house prices. Other workers, however, report contrasting results, as is the case of Garrod and Willis (1994), [START_REF] Irwin | The effects of open space on residential property values[END_REF], [START_REF] Palmer | Using spatial metrics to predict scenic perception in a changing landscape: Dennis, Massachusetts[END_REF], [START_REF] Smith | Treating open space as an urban amenity[END_REF] and [START_REF] Dumas | Les indices de l'écologie du paysage à l'épreuve du marché foncier[END_REF]. The results for farmland are also ambiguous as they differ from one study to another. [START_REF] Roe | The effects of farmland, farmland preservation, and other neighborhood amenities on housing values and residential growth[END_REF] report a positive hedonic price, contrary to [START_REF] Smith | Treating open space as an urban amenity[END_REF], while for [START_REF] Paterson | Out of sight, out of mind? Using GIS to incorporate visibility in hedonic property value models[END_REF] the effect is not significant either way.

Landscape ecology indices

Beyond land use measured by the presence, proximity and proportion of the different types of land cover, shape analysis has come under study as it makes it possible to get closer to what a landscape is through the analysis of the number of patches of each type, their shape, the complexity or the uniformity of the compositions arising from them, etc. [START_REF] Gustafson | Quantifying landscape spatial patterns: what is the state of the art?[END_REF].

Here we look at works using land-use data and landscape-ecology indices and ignore those based on photographs [START_REF] Palmer | Using spatial metrics to predict scenic perception in a changing landscape: Dennis, Massachusetts[END_REF]. [START_REF] Geoghegan | Spatial landscape indices in a hedonic framework: an ecological economics analysis using GIS[END_REF], in a multidisciplinary programme of landscape economics and ecology, use landscape-ecology indices which they integrate in a hedonic model. The study area is a watershed of 1000 square miles in Maryland, near to Washington D.C. [START_REF] Acharya | Valuing open space and land-use patterns in urban watersheds[END_REF] conduct similar work to estimate the value of identical or similar attributes in a watershed in Connecticut. [START_REF] Geoghegan | Spatial landscape indices in a hedonic framework: an ecological economics analysis using GIS[END_REF] define eight types of land use and introduce into the regression a land-use diversity index and a fragmentation index calculated within a 400-m circle around the house (supposedly corresponding to the field of view) and with a 1-km circle (walking range). [START_REF] Acharya | Valuing open space and land-use patterns in urban watersheds[END_REF] also define two circles, of 800 m and 1.6 km that similarly are supposed to correspond to the viewshed and to the walking range; to these indices they add richness, measured by the number of categories of land use found in each circle relative to the maximum possible number of categories and an interaction between richness and population density in order to distinguish between the richness of rural and urban landscapes.

The results show that in Maryland, near Washington, the immediate proximity of open spaces slightly raises land values, but that the effect becomes negative for more distant open spaces.

Fragmentation and landscape diversity around the house have a negative effect on land values, except very close to and very far from Washington. In the Connecticut watershed studied by Acharya and Bennett, the diversity and richness have a negative hedonic price and the interaction between richness and density has a slightly positive price. The negative effect of diversity is consistent with zoning policies which are designed to segregate the different types of land use spatially. The positive effect of the interaction supposedly reflects a positive price of richness in an urban environment (habitat, gardens, shops) and a negative price of low richness in rural settings (uniform landscapes). The proportion of open spaces in the circles has a positive effect on price, with a parabolic shape.

The introduction of landscape composition variables or shape variables marks a significant advance in the economic evaluation of landscapes. Landscape ecology indices are generally used, since this is the discipline that provided the statistical indicators and tools for using them such as Fragstats [START_REF] Mcgarigal | Fragstats: Spatial pattern analysis program for quantifying landscape structure[END_REF]. It may be that the pioneering paper by [START_REF] Geoghegan | Spatial landscape indices in a hedonic framework: an ecological economics analysis using GIS[END_REF] also helped to direct research along these lines.

However, the use of such indices must be discussed for three reasons. First, it assumes that people obey the same determinants as other animal species, as expressed by [START_REF] Geoghegan | Spatial landscape indices in a hedonic framework: an ecological economics analysis using GIS[END_REF]; yet, the aesthetic preferences of human beings are not of the same nature as the search for specific ecosystems and biotopes of other animal species. Secondly, the landscapeecology variables have no connection with those used previously by urban planners, landscape architects, sociologists and economists and that sought to reflect aesthetic, psychological or cultural notions. Thirdly, landscape ecology can be used to produce variables seen from above and not seen from the ground: landscapes seen by an observer from the ground are very different from those seen from above. For example, where there may be a forest seen from above, someone in the forest will only see the few trees immediately around them.

The view of landscape

The view of a landscape is a much more complicated variable to compute than the attributes that have been analysed so far. It was long impossible to reconstruct the view from a large number of observation points. That is why, until recently, photographs of landscapes were used in survey protocols where respondents were asked to attribute scores to them or to evaluate them. In recent years, new approaches have made it possible to reconstruct views of landscapes from any point in space through suitable numerical modelling of landscape.

Preference analysis from photographs

Many studies investigate consumer preferences for landscapes from photographs. The method generally used involves explaining, on the basis of several sets of variables, a score given by respondents to a photograph.

For example, [START_REF] Kaplan | Environmental preferences. A comparison of four domains of predictors[END_REF] compare the role of four sets of variables in explaining a score given by students to landscape photographs in a rural area of Michigan: physical attributes (slope, relief, edge contrast, diversity, etc.); land use (arable land, grassland, brush, wasteland, etc.); landscape understanding and discovery variables (coherence, order, readability, complexity, mystery) and perception variables (field of view, texture). The authors conclude expressing some surprise because of the low predictive effect of the physical or land-use variables that are used in most studies of this type. It may be that an analysis of photographs by landscape architects and rated by psychology students introduces a bias, since these are not ordinary consumers of landscapes.

Recently, [START_REF] Kaplan | Out in the country: sprawl and the quest for nature nearby[END_REF] have shown that, for people from a rural area near Detroit (USA), having a view of a 'natural landscape' from home is more important than having space (plot size, living area). Other works of this type have distinctive methods, such as the work of [START_REF] Johnston | Spatial factors and stated preference values for public goods: considerations for rural land use[END_REF] that analyses the development of a housing project on Rhode Island (USA) by presenting plans and models to future residents. They show that households prefer a continuous, low-density project with fragmented plots separated by hedges.

The price of the landscape seen

New approaches mean that the view of landscape can now be reconstructed from any point in space by suitable numerical modelling of landscape. In these recent approaches, the actual view is introduced in the two-dimensional world described by a land-use layer, as presented earlier, by including the third dimension in the analysis through topography. Here what is investigated is the effect of the viewshed and, within the viewshed and depending on the different studies, the effect of the areas seen and put to various uses, the effect of the view of objects and of their composition, of topographical shapes, depending on the distance between the observer and the components of the field of view.

The study made of the value of view in the Rocky Mountains (Wyoming, USA) by [START_REF] Bastian | Environmental amenities and agricultural land values: a hedonic model using geographic information systems data[END_REF] was the first to combine a method for reconstructing the view of the landscape with a hedonic price estimation. The environment of 158 plots is characterised from satellite images (25-m pixels) allowing 10 classes of land cover to be defined and from a 30-m digital elevation model. The combination of the two layers of geographic information allows the landscape to be reconstructed and landscape variables calculated: viewshed, relief, land use, diversity, break lines. The authors conclude that few of these variables are significant except for landscape diversity, which is greatly appreciated. [START_REF] Paterson | Out of sight, out of mind? Using GIS to incorporate visibility in hedonic property value models[END_REF] also use satellite images and a digital elevation model to analyse landscape seen in a rural region of Connecticut (USA). They obtain results that they differ with the type of variable (satellite view or ground view). [START_REF] Cavailhès | Les paysages périurbains et leur prix[END_REF][START_REF] Cavailhès | Pricing the homebuyer's countryside view[END_REF][START_REF] Cavailhès | GIS-based hedonic pricing of landscape[END_REF] and Brossard et al. (2008aBrossard et al. ( , 2008b) ) study the hedonic price of landscapes seen in periurban areas around Dijon and Besançon (France) using a method for modelling the view from the ground [START_REF] Joly | A quantitative Approach to the visual evaluation of landscape[END_REF]. They show that beyond a few tens or hundreds of metres, the planes of view and the objects in them do not significantly influence house prices. However, trees or fields in view close to houses have positive hedonic prices and roads negative hedonic prices. The view itself matters in that nearby trees that are hidden from view have a lower hedonic price than trees that are in view. Landscape ecology indices indicate positive prices for complex landscape shapes (mosaics, non-connectedness, etc.). In this paper, we extend these works by introducing new landscape measures.

What should be remembered from this overview of the literature?

First, that what is visible must be finely considered in landscape evaluation models.

Admittedly analysis based on an overhead view of land use and topography is useful for defining the actual layout of the landscape in the geographical area, but it leads to approximations when attempting to bring out the criteria that underpin the value of the landscape. The mental processes by which a value is attributed to landscape is driven by many sensory experiences where the view from the ground is decisive. So by formalising the rules of visual composition of the landscape this reality can be approached objectively and the preferred features of the landscape identified. In this exercise, spatial and thematic precision of information is of great importance, because it must distinguish between close-up views and fine landscape objects that greatly structure the view from the ground. It is understandable that this requirement of precision implies designing state-of-the-art methods and using suitable instruments. Accordingly we are now seeing the emergence of principles for a coherent approach combining numerical data, quantitative methods, instrumentation by information systems and proceeding from conceptual thinking on measurement in respect of landscapes.

CALCULATING LANDSCAPE METRICS FROM A NUMERISED LANDSCAPE

Landscape modelling

Our landscape analysis method is based on a geographic information system (GIS) that manages multi-source data, including satellite pictures and a digital elevation model (DEM).

We synthesise several approaches that are required to produce landscape indicators (see [START_REF] Joly | A quantitative Approach to the visual evaluation of landscape[END_REF] for a full description of the method):

(i) We formalise the reversibility relations between an observer and the landscape to be seen.

We thus show the difference between the 'view from above', such as that in an aerial photograph, and the 'view from within', seen from the ground, where it is blocked by relief or tall objects that bring in the horizon. The 'view from within' is analysed by coupling it with the DEM, which describes the relief, and with satellite images, which are used for defining land use and the height of associated objects.

(ii) A landscape is arranged into grounds 1 , from near to far. An object, such as a tree, is not the same for an observer depending on whether it is nearby (it is then a single object) or far away (it becomes a part of a forest). We allow for this property by using multiple-scale methods that vary the nature of objects with distance.

(iii) The view of a landscape may be an amenity, but the possibility of being seen by others (passers-by on a road, the next-door neighbour) may be a nuisance. But the first view, or active view, is not symmetrical with the second, or passive view because of masks (a forest may be seen from far off, but someone walking in it sees only a small patch of undergrowth).

Numerical modelling of landscape allows us to measure the area seen by an observer virtually positioned at each point in the space in question and turning through 360° (active view). The same operation is repeated for each pixel and a symmetrical approach can be taken for the passive view.

In this way, the visual reality of the landscape is decomposed and may be translated into numerical indices. A virtual landscape is thus obtained for each pixel in the study area. The georeferencing of points where houses are located provides knowledge of the landscape around them, such as it is perceived by the householders moving around their plot.

Presence and abundance of seen and unseen objects on field depth

The model allows us to make an inventory of landscape characteristics that can be modulated depending on the objectives. For any point in the study, its breadth of outlook can be calculated; we can detail the way landscape is arranged in the viewshed, in open views extending to the distant horizon or dominated by nearby masks, etc. Landscape is divided up into six grounds whose content and transparency vary with distance (Skov-Petersen 2007).

The analysis is broken down ground by ground with special attention to their content (agriculture, forests, buildings, roads, lakes, etc.).

Modelling and the formal ordering that it underpins allow us to transcribe these different terms of landscape categorisation into numerical form. In this way, we can chose to find them a visual expression in the form of a map or graph; or they can equally be stored in numerical form, because that is how they are transferred as explanatory variable for econometric modelling.

[Insert Figure 1 here]

Angular area related to vertical masks

The number of cells seen by type and their distribution in the different grounds provides an approximation of the landscape that is to be discussed. For example, agriculture seen from the distance may take up much of the viewshed when measured in square kilometres, but it is speak of 'ground', by analogy with painting, for the seen area within each buffer. perceived at a very low incident angle in the visual volume of the landscape. By contrast, a nearby house or tree may entirely fill the view, despite the small area they occupy. To allow for such effects, we calculate angular areas that are obtained by finding the product of the vertical and horizontal angular components of an object (it is possible to differentiate between what is due to the height of objects and to relief).

[Insert Figure 2 here]

We construct a cone whose summit ô corresponds to the observer's eye and base to the visible section of a cell, a square with sides ca and cb. The angular product aôb * aôc corresponds to its angular surface area. The angular area of a type of landscape object seen from an observation point is the sum of the angular areas of all the cells of this type composing its visual field.

The two foregoing approaches-grounds and angular area-may be combined: for identical angular areas, one type of object may not have the same effect for an observer depending on how close it is. It is therefore important to calculate the angular area of each type of object for the groundto which it belongs (the calculation is made for grounds 1, 2 and 3).

Distance from objects seen

By measuring the distance between an observation point and each cell of a given type, we can calculate a 'weighed aggregate' distance of this type, by dividing each cell by this distance and summing the ratios. For reasons of computation time, and in view of earlier econometric results (Cavailhès and Joly, 2007), the weighed aggregate distance was calculated for cells in view less than 280 m from an observation point.

Likewise for angular areas, the weighted aggregate distance from a type of object can be decomposed by grounds. This is tantamount to specifying the ground: the cells in view are defined not only by their inclusion in a particular ground but also, within this ground, by the distance at which they are located.

COMPARISON OF THE HEDONIC PRICES OBTAINED

The econometric model

We use the usual hedonic price equation:

i i i b X P ε + = ln
, where P i is the price of real-estate i, X i the matrix of explanatory variables (including an intercept), b the vector of parameters to be evaluated, and ε i an error term. Application of this method to property values raises econometric problems that we discuss briefly. Besides the usual issues of heteroscedasticity of residuals and multicolinearity between covariates, the method involves processing spatial correlations between residuals (houses being spatialised goods) and the endogeneity of certain regressors (mainly due to the simultaneous choice of house price and some of its attributes).

Endogeneity 'may be addressed by means of spatial fixed effects. This rests on the assumption that the spatial range of the unobserved heterogeneity / dependence is specific to each spatially delineated unit' [START_REF] Anselin | Errors in variables and spatial effects in hedonic house price models of ambient air quality[END_REF]. Following this method, we introduce into the equation a variable m j characterizing the commune j:

ij j j ij ij m b b X P ε + + = ln
that captures the effects of attributes whose values are shared by observations located in this commune, including badly measured or omitted variables. This makes it possible to not have spatial autocorrelations between the residuals of observations situated in different communes (a nullity test of Moran's index confirms this is so). We also calculate Moran's index for observations of a single commune, with a contiguity matrix where all transactions less than 200 m away are considered to be neighbouring. The significance test of this index makes it possible to determine whether or not there are spatial autocorrelations and if so to make allowance for them.

Endogeneity of some covariates is allowed for by the 'instrumental method' which allows us to check, using the Hausman's test, whether a covariate is endogenous. The estimation is then made in two steps: projection of this endogenous variable on exogenous instruments (characteristics of buyers and sellers), and then use of this projection in the main equation, estimated by the 2SLS. Sargan's test indicates whether the instruments are exogenous or not.

Study region and data

The study region is in Burgundy (France) around the city of Dijon. The outer limit of the area is bounded by an access time to Dijon of less than 33 minutes or a distance by road of less than 42 kilometres. The inner bound is that of the built-up area of Dijon (excluded from the study area because our landscape analysis models do not apply well to spaces where building heights are very variable). The area covers 3408 km² and numbers 140 703 inhabitants.

The property data are from real-estate lawyers (notaires) and pertain to 2665 sales between 1995 and 2002. They have been precisely located by ascribing geographical coordinates to them, so each house could be matched with the landscape on view from it.

Results

The results for housing attributes and other location variables are analogous to those obtained elsewhere [START_REF] Cavailhès | GIS-based hedonic pricing of landscape[END_REF]. The purpose of this paper being to test the results obtained using different landscape metrics, we present only the results relating to these variables.

Quantities of landscape seen according to the metrics

The three types of landscape metrics lead to measurements of seen objects that differ markedly in magnitude. The correlation between the number of tree-covered cells seen at less than 70 m and their angular area is quite low (R² = 0.47). The correlation is better between this number of cells and their weighted aggregate distance (R² = 0.76). These results are fairly logical since, for a tall object, the angular area varies greatly with its distance from the observer. By examining the connection with farmland seen at less than 280 m away, better statistical relations are obtained because, on the one hand, farmland is a flat object and because, for the other hand, we are interested in a more extended horizon. The R² value is 0.62 (angular area) or 0.94 (weighted aggregate distance).

Overall econometric results

Three types of model were estimated: a fixed-effects model estimated by the OLS and with the instrumental method by the 2SLS, and lastly a random-effects model. The instrumental method shows that the floor space is endogenous and Sargan's test indicates that the other covariates are exogenous; the landscape variables are all exogenous, contrary to results reported for the United States [START_REF] Irwin | The effects of open space on residential property values[END_REF][START_REF] Irwin | The problem of identifying land use spillovers: measuring the effects of open space on residential property values[END_REF]. In all three cases the model is homoscedastic and the Moran's index test on neighbouring residuals (within and between communes) shows that these are not spatially autocorrelated. For all three estimations, the orders of magnitude of the coefficients, the signs and the standard variations are quite close to one another.

Since one variable is endogenous and the random-effects model is statistically less sure than the fixed-effects model estimated by the instrumental method (cf. [START_REF] Cavailhès | GIS-based hedonic pricing of landscape[END_REF], we present only the main results of the latter model in Table 1. Moreover, the estimates are made for angular area from zero to infinity and for distances from cells from 0 to 280 m, then by decomposing these variables by the ground to which they belong (cf. supra).

[Insert Table 1 here]

The results in Table 1 concern only landscape variables with significant parameters. Full results using distance buffers are reported in [START_REF] Cavailhès | GIS-based hedonic pricing of landscape[END_REF]. They are similar to those in Table 1 column (1), showing that the omission of non-significant landscape variables did not affect too much the parameters. To save space, the non-landscape variables (characteristics of the property and its location) have not been reported (see full results in [START_REF] Cavailhès | GIS-based hedonic pricing of landscape[END_REF].

Let us note, lastly, when reading this table that only the Student's t values are of interest: the parameters, measured in different units (number of cells, square degrees, number of cells weighted by distance) cannot be compared from one column to the next.

The adjusted R² values of the five regressions shown in Table 1 are quite similar and very close to 0.66. However, insofar as house prices depend mainly on the house's characteristics (size, comfort, etc.) and its location rather than with the landscape seen, it is preferable to examine the adjusted partial R² value of landscape composition variables to compare the effect of the metrics. This varies from 0.02 to 0.03, showing that these variables 'explain'

nearly 2-3% of the variance of the model.

The partial adjusted R² values of the landscape variables are slightly better for the ground metric (0.03), because objects that are present in the grounds but out of sight (tree-covered cells, farmland) are added to the objects seen. If we confine ourselves to seen objects alone, which make up the landscape seen in the strict sense, the ground metric leads to worse partial R² and partial (0.019) adjusted R² values (0.017). It is better to measure seen landscapes by angular areas (for all objects in view from 0 to infinity: partial adjusted R² =0.024) or by weighted aggregate distances (from 0 to 280 m: partial adjusted R² =0.024). Lastly, it is by combining inclusion within grounds and angular areas and above all weighted aggregate distances that the best partial adjusted R² values are obtained. This is not surprising because the latter two metrics combine the advantages of grounds and the position of objects within them. It is therefore a more precise measure. We obtain adjusted partial R² values that rise to 0.024 and 0.026.

[Insert Table 2 here]

Landscape objects influencing price

By examining the number of objects seen for the grounds in which they are included (Table 1, column 1), we observe, as in [START_REF] Cavailhès | GIS-based hedonic pricing of landscape[END_REF], significant parameters for trees located less than 70 m away, for farmland between 70 and 180 m (positive effects) and for the view of roads and railway lines up to 280 m away (negative effect); this effect is not significant when these infrastructures are present but unseen. The absence of any effect beyond a few tens of metres may be explained by the study area having a 'non-descript' farmland and forest landscape with no particularly remarkable features such as a view of the sea or of snowcapped mountains, etc.

With the other metrics, these results remain stable overall: the significance of the parameters, are very similar in the different equations. The main difference is related to the adoption of a quadratic from for trees. When a wooded landscape is measured by its angular area, a saturation effect occurs, making the quadratic term significant: it is known that an abundance of nearby trees brings in the horizon and it is understandable that this is reflected by depreciation of the property.

We notice also that the use of finer variables, combining inclusion within a ground and a measure of location in the ground (aggregate distance or angular area) leads to a refinement of certain results: it is no longer farmland within 71 and 280 m that contributes significantly to price but only farmland between 71 and 140 m when location within grounds is measured. A few variations are also observed for road networks, but not for tree-covered areas: it is always the first 70 m that count.

Comparison of results for the metrics

Since the estimated parameters are not directly comparable because of the different measurement units, we present in Table 3 the effect on property prices of an increase of one standard variation of landscape variables.

For communication networks the results are pretty much identical, whichever landscape measure is used.

[Insert Table 3 here]

For tree-covered areas, the highest result is obtained with the angular surface metric when the location in the ground 0-70 m is taken in consideration: the adjusted partial R² is equal to 0.053. The adjusted partial R² is equal to 0.042 with the total angular surface. The results with the weighted aggregate distance are lower, the hierarchy being the same: 0.042 when the location in the ground 0-70 m is taken in consideration, 0.024 otherwise. The adjusted partial R² is only 0.02 with the ground metric (ground 0-70 m). This seems to indicate that for a tall object like a tree, a wood or a copse, which may mask the view and occupy a large proportion of the very near landscape, the distance should be measured quite precisely: inclusion within the first 70 m buffer is too coarse a variable. The somewhat mediocre correlation observed between this latter variable and those that take account of position within the buffer (R² = 0.47 or 0.76) is reflected by the marked spread of the results, since they may be more than doubled depending on the metric.

For farmland, the total angular areas or weighted aggregate distance up to 280 m away affect price by 2 to 3%, which is the same value obtained with the metrics for the 71-140 m ground, which in turn is less than the number of cells in the 71-280 m ring (+4.6% for one standard deviation). These differences are less important than for trees, which may be explained by a different perspective effect for a flat object that is incapable of masking the view.

The hedonic price of a reduction of view

The price of a marginal loss of landscape, due, say, to a new building closing out 10% of the view, is calculated for each house of the database from values predicted by the models: the price of this loss of landscape is equal to the sum of quantities of each object weight by its price.

We only use here the three variables for which we use different metrics: tree cover, farmland and networks that are actually in view, excluding objects that are present but unseen.

Figure 3 shows as XY graphs the results for the 2665 observations on which the estimates were made. These graphs also show the R² values for the hedonic prices of the three landscape variables analyses, estimated point by point, and which are 0.48, 0.58 and 0.78. The first two values are quite low.

[Insert Figure 3 here] Despite this rather weak link for individual hedonic results, the aggregate results are quite similar, as shown in Tables 2 and3: around 4% per additional standard variation of treecovered areas, 2-4% for the same variation of farmland in sight, and -1% for one standard deviation of additional transport network. It might be thought that the geographical method of measuring landscapes has substantial effects on price estimations of a particular landscape seen from a particular house but that such individual differences fade when we reason in terms of aggregates: the individual deviations due to the metric used are absorbed by the residual of regression and the statistical estimation of the 'mean' effect depends little on the way landscape is measured.

CONCLUSION

In the economic evaluation of landscapes, it is difficult to express the view of landscape into quantitative variables so as to introduce it into econometric models. Geographers use a battery of approaches ranging from analysis of photographs to quantitative geography models using satellite images and digital elevation models. The explanatory variables of landscape value range from psychology (mystery, order, harmony, etc.) to landscape ecology. However, little economic valuation works have yet been done.

Here we examine the results of econometric estimations obtained by using three metrics

  use four categories of land use (trees, lawns, wasteland and artificialised ground) whereas Des Rosiers et al. (

  relating to landscape attributes: (i) division of the viewshed into grounds according to their remoteness, (ii) visual scope measured by angular areas of vertical masks due to relief and objects, and (iii) the aggregate area of objects seen weighted by their distance from the observer. These three metrics quantify the objects seen differently, as is clearly shown by comparison of individual results. The landscape metric has important effects on price orthophotograph (a) Land cover. (b) Landscape seen. (c) Exposure to view.

Figure 1 .

 1 Figure 1. Landscape models around a transaction up to 280 m away (Buffers 1, 2 and 3).Each point is surrounded by landscape features described in terms of land use. Diversity and abundance can be specified by counting these features. In (a) the landscape, seen from overhead, is completely accessible to view and exhibits the structure of the orthophotograph despite the coarser resolution. In (b), the plan shows the segment of landscape that can be seen in three dimensions by an observer located at the centre of the circles; notice that most of the surrounding landscape is masked. In (c), all the pixels from where an observer can see the central point are shown. Figuresfigures (b) and (c) cannot be superposed: so seeing and being seen are complementary and non-symmetrical terms.

FigureFigures 3 .

 3 Figure 2. Angular area of vertical masks

Table 1 .

 1 Econometric Results

			(1)		(2)		(3)		(4)		(5)
			buffers	angular surface (total)	weighted agregate distance (total)	angular surface distributed by grounds	weighted agregate distance distributed by grounds
	adjusted R²		0,66262	0,65915	0,65737	0,65962	0,66085
			estimate	Student' t	estimate	Student' t	estimate	Student' t	estimate	Student' t	estimate	Student' t
	TREES SEEN	< 70m	0,004075	2,0			0,091744	2,2	0,016021	2,9	0,188559	3,6
	(TREES SEEN)²	< 70m							-0,00069	-2,0
	TREES SEEN* LOT/LSPACE	< 70m	-0,00014	-1,3			-0,00659	-3,3	-4,43E-07	-3,7	-0,00937	-4,1
	TREES UNSEEN	< 70m	0,001784	3,6
	TREES UNSEEN* LOT/LSPACE	< 70m	-0,00007	-4,2
	TREES SEEN	0-40km			0,014791	2,2
	(TREES SEEN)²	0-40km			-0,00086	-1,7
	TREES SEEN* LOT/LSPACE	0-40km			-0,00041	-2,9
	AGRI SEEN	71-280m 0,000139	4,3
	AGRI SEEN * LOT/LSPACE	71-280m -0,00643	-5,8
	AGRI SEEN * POSUD	71-280m -0,00005	-1,7
	AGRI SEEN	71-140m							0,013381	2,4	0,024959	3,1
	AGRI SEEN * LOT/LSPACE	71-140m							-0,00156	-4,3	-0,00144	-5,2
	AGRI SEEN * POSUD	71-140m							-0,00773	-1,6	-0,01317	-1,5
	AGRI UNSEEN	71-280m 0,000037	2,9
	AGRI UNSEEN * LOT/LSPACE	71-280m -0,00214	-2,5
	AGRI SEEN	0-40km			0,072276	2,4	0,005784	2,6
	AGRI SEEN * LOT/LSPACE	0-40km			-0,008	-4,0	-0,0004	-4,5
	NETWORKS SEEN	0-280m	-0,00025	-1,9
	NETWORKS UNSEEN	0-280m	0,00007	1,3
	NETWORKS SEEN	71-140m							-0,06713	-2,4
	NETWORKS SEEN	0-70mm									-0,01875	-2,2
	NETWORKS SEEN	0-40km					-0,01414	-2,1

Table 2 .

 2 Partial R² and adjusted partial R² of the landscape variables

	effect of
	1std change

Table 3 .

 3 Effect on the house price of an increase of one standard deviation of landscape variables depending on the metric used.

APPENDIX : LANDSCAPE VARIABLES USED

Landscape variables: according to rings: < 70 m, 70-140 m, 140-280 m, 280-1200 m, 280-1200 m,1.2-6 km, 6-40 km. Cells SEEN and UNSEEN are distinguished. Rings may be merged ABBREVIATION DEFINITION

  TREE number of cells of tree-covered area (R_TREE: rate of these cells) TREE * LOT/LSPACE number of cells of tree-covered area * LOT/LSPACE AGRI number of cells of agriculture (R_AGRI: rate of these cells) AGRI * LOT/LSPACE number of cells of agriculture * LOT/LSPACE AGRI * POSUD number of cells of agriculture * class UD of the zoning scheme NETWORK TRANSPORT number of cells of road/railroad (R_NETWORKS: rate of these cells) BUILT number of built cells (R_BUILT: rate of these cells) BUSH number of cells of bush (R_BUSH: rate of these cells) WATER number of cells of water DECID_PACHES number of patches of deciduous trees within a 70 m radius DECID_EDGE length of deciduous wood edges within a 70 m radius (m) AGRI_PACHES number of patches of crops betwen 70 -140 m COMPACT compactess index (0=compact forms; 1=elongate forms), < 70 m BUILT ANGULAR SURFACE angular surface made by built cells (square degrees) AGRI ANGULAR SURFACE angular surface made by cells of agriculture (square degrees) TREE ANGULAR SURFACE angular surface made by tree-covered cells (square degrees) BUSH ANGULAR SURFACE angular surface made by cells of bushes (square degrees) NETWORK ANGULAR SURFACE angular surface made by cells of roads and railroads (square degrees) WATER ANGULAR SURFACE angular surface made by cells of water (square degrees)

For land cover, we use the term 'buffer' for a ring around transactions. For viewshed we


Anh Duc

Isabelle Ta

Rogelio Fantoni

Lozano

Modeling and Control of a Tilt tri-rotor Airplane

A helicopter offers the capability of hover, slow forward displacement, vertical take-off and landing while a conventional airplane has the performance of fast forward movement, long reach and superior endurance. The aim of this paper is to present the modelling and control of a tilt trirotor UAV's configuration that combines the advantages of both rotary wing and fixed wing vehicle.

I. INTRODUCTION

Nowadays, Unmanned Aerial Vehicle (UAV) possess several applications in the military and civil domains: surveillance, detection, reconnaissance, search and rescue operations... The objective of this work is to design a miniature aircraft which has the performance of an airplane for the forward flight and the capacity of a helicopter for the hover flight. In the past, this configuration has motivated lots of researchers because this type of aircraft does not require a runway and the ability of hover makes it very useful for aerial surveillance missions. In [START_REF] Oner | LQR and SMC Stabilization of a New Unmanned Aerial Vehicle[END_REF], ongoing works on the development of a quadrotor aerial vehicle having a tilt-wing mechanism using an LQR and a sliding mode controller for the stabilization of the attitude and the altitude are presented via vertical take-off and landing simulations. Other simulation results of the transition to level flight of a tiltwing aircraft which uses a gain-scheduled, multi-variable H ∞ control law are obtained in [START_REF] Dickeson | Robust LPV H∞ Gain-Scheduled Hover-to-Cruise Conversion for a Tilt-Wing Rotorcraft in the Presence of CG Variations[END_REF]. More longitudinal dynamic simulation results are found in [START_REF] Xili | Transition Flight Control of Two Vertical/Short Takeoff and Landing Aircraft[END_REF], in which nonlinear approaches are used for the autonomous transition of two vertical/short take-off and landing aircraft (a fixed-wing fighter equipped with a vectored thrust and lift fan and a tilt rotor). In [START_REF] Ta | Modeling and Control of a Convertible Mini-UAV[END_REF], we proposed a convertible tail-sitter UAV which has the capability of transition between the vertical and horizontal flight. However, in order to make a transition, the whole fuselage of the convertible airplane must rotate to the horizontal. In this paper, we propose a configuration of the tilt-rotor mini aircraft capable of performing the hovering, transition, vertical take off and landing. In the phases of transition, the fuselage still remains in the horizontal, then the transition is more stable than the convertible airplane. The detailed mathematical model is obtained by the equations of Newton-Euler and the aerodynamic formulas. In terms of control, we propose a simple nonlinear PID control using Neural Network to stabilize the attitude and the altitude and a motion profile generator to realize the phases of take-off, landing and transition.

The paper is organized in five sections. The second section is devoted to the presentation of the platform. The mechanical structure and the aerodynamic model of the aircraft will be presented in the third section. The fourth section contains our contributions, namely the presentation Simulation results are presented in the fifth section. Finally, the paper ends with some conclusions.

II. TILT TRI-ROTOR AIRPLANE

1) Platform: The tilt tri-rotor airplane has been built and completely developed at Heudiasyc laboratory (see Figure 1). It has a total weight of about 1250g and uses one 2200mAh LiPo battery. In the vertical flight (Figure 2), the altitude is controlled by collective thrust of the three brushless electric DC Motors. The roll motion is achieved with the difference of thrust between the two front motors: the left rotates in the clockwise direction while the right in counterclockwise direction. The rear motor compensates the force generated by two front motors to stabilize the pitch angle of the airplane.

After take-off, the two front motors tilt forward thanks to an electromechanical system. The rear motor reduces its rotational speed according to the tilt angle δ of the two font motors. When δ reaches zero degree, i.e. the axis of both front motors coincides with the axis of the fuselage, the rear motor will stop completely. The airplane then flies horizontally like a conventional aircraft (Figure 3) with a cruise speed to the target. In horizontal flight, the roll motion is controlled by moving the ailerons differentially, the pitch motion is controlled by moving the elevator and the yaw motion is controlled by moving the rudder. When the vehicle reaches the target, the two front motors tilt to the vertical and the vehicle returns to the helicopter mode and lands to the ground vertically. Note that the rotational speeds of both front motors are equal in the phases of transition and horizontal flight.

2) Embedded Control System: An Inertial Measurement Unit (Microstrain, 3DM-GX3-25) is used to obtain the attitude. It weighs just 11.5g and is composed of three triaxial accelerometers and angular rate gyros as well as three orthogonal magnetometers. A Digital Signal Controller (Microchip, dsPIC33FJ256GP710) was selected to implement the Embedded Control System. The measurement used for the altitude is obtained from the low cost ultrasonic sensor (SRF10) which has a range of 10cm to 3m.

III. MATHEMATICAL MODEL

From the design model of the tilt rotor airplane described above, the Newton-Euler equations of motion describing the six degrees of freedom of the system can be separated in translational motion (Σ P ) and rotational motion (Σ A ):

Σ P : ˙ p f = R v b ˙ v b = R T g f + 1 m F b A,T -ω × v b (1)
Σ A : Θ = J (Θ) ω I ˙ ω = -ω × I ω + Γ A,T (2)
where: R is the rotation matrix from E b to E f ; p f represents the position of the airplane in E f ; v b is the linear velocity vector presented in

E b ; m = diag(m) ∈ R 3×3 where m (≃ 1250g
) is the weight of the vehicle; g f is the vector of the gravity acceleration in E f ; ω is the angular velocity vector; Θ = φ θ ψ is the vector of the Euler angles; I ∈ R 3×3 represents the inertial matrix of the airplane;

F b A,T = F b A + F b
T is the vector of the external aerodynamic forces and the thrusts expressed in the coordinate E b ; Γ A,T = Γ A + Γ T is the external torque vector; J (Θ) is the Jacobian matrix defined as:

J (Θ) =   1 tan θ sin φ tan θ cos φ 0 cos φ -sin φ 0 sin φ cos θ cos φ cos θ   (3)
Note that the singularity of the attitude representation by Euler angles can be eliminated if the airplane's orientation in space is limited. In our case, since the fuselage is always in horizontal (-45 • θ 45 • ), the Euler angles are used to represent the attitude due to their simplicity. At first, we analyse the forces F b T and the torques Γ T generated by the actuators. Each motor produces a force T i parallel to its axis of rotation, and a reactive torque Q i opposite to the direction of rotation. Since the rotor speed reaches very high values (more than 200 rad/sec), we can approximate the forces T i and the reactive torques Q i generated by the motors:

T i = b i s 2 i ; Q i = k i s 2 i ( 4
)
where s i is the rotational speed of rotor; b i and k i are two positive parameters depending on the density of air, the radius, the shape, the pitch angle of the blade and other factors [START_REF] Fay | Derivation of the aerodynamic forces for the mesicopter simulation[END_REF]. The combination of the forces T i and the reactive torques Q i is given by:

       F b T = (T 1 + T 2 ) cos δ 0 (T 1 + T 2 ) sin δ + T 3 T Γ T =   (T 2 -T 1 ) l m sin δ + (Q 2 -Q 1 ) cos δ T 3 l 2 -(T 1 + T 2 ) l 1 sin δ (T 1 -T 2 ) l m cos δ + (Q 2 -Q 1 ) sin δ + Q 3   (5)
In the vertical mode (tilt angle δ = 90 • , Figure 2), the equation ( 5) becomes:

       F b T = 0 0 T 1 + T 2 + T 3 T Γ T =   (T 2 -T 1 ) l m T 3 l 2 -(T 1 + T 2 ) l 1 (Q 2 -Q 1 ) + Q 3   (6)
In the horizontal mode (δ = 0 • , T 3 = Q 3 = 0, Figure 3), the equation ( 5) becomes:

F b T = (T 1 + T 2 ) 0 0 T Γ T = (Q 2 -Q 1 ) 0 (T 1 -T 2 ) l m T (7)
The aerodynamic forces F b A and torques Γ A depend on the working mode of the airplane. In all phases of flight, we assume there is no wind in the environment. So when the airplane flies vertically, there are no aerodynamic forces and torques because the speed of the vehicle is minor. However in transition and horizontal phases, the air velocity will be equal to the relative velocity of the UAV but in the opposite direction : V air = -V . The vector of the air velocity creates with the airplane an angle of attack α (Figure 3). In these modes, the aerodynamic forces and torques are given by:

       F b A =   -T b w + T s ar + T s al + T b e +
P s i = 1 2 ρv 2 in S s i C P δ i and T s i = 1 2 ρv 2 in S s i C T δ i
are respectively the lift force and the drag force created by slipstream with i = al, ar, e, r (left aileron, right aileron, elevator and rudder); ρ is the density of air;

C P δ i = C P i δ δ i ; C T δ i = C T i δ δ i ; δ i
and S i are the deflection angle and the area of control surface; v in = 2Ti ρAM + V 2 is the air velocity created by the propeller (inflow or induced velocity).

We have used the same analysis as in [START_REF] Ta | Modeling and Control of a Convertible Mini-UAV[END_REF] and aerodynamics formulas in [START_REF] Mc Cormick | Aerodynamics Aeronautics and Flight Mechanics[END_REF].

Then, in the vertical mode (δ = 90 • ) the external torque vector Γ A,T is given by:

Γ A,TV = Γ T =   b s 2 2 -s 2 1 l m bs 2 3 l 2 -b s 2 1 + s 2 2 l 1 k s 2 2 -s 2 1 + ks 2 3   (11)
In the transition mode, the external torque and external force depend on several aerodynamic parameters. Due to space limitations, the expressions are given in [START_REF] Ta | Avion convertible à décollage et atterrissage vertical[END_REF].

IV. CONTROL STRATEGY

In the longitudinal dynamics, the tilt-rotor airplane performs the following phases of flight: vertical take-off, hover mode, transition to the horizontal, horizontal flight, transition to the vertical and vertical landing.

A. Control Law 1) Altitude Control:

The control of the altitude is used when the airplane flies vertically but it is not used in the transition and horizontal flight. Extracting the translational motion on the z axis of E f from equation (1) yields:

mz = T cos θ cos φ -mg ( 12
)
where T is the total thrust force of the three motors. A simple PD control can guarantee the convergence of the altitude to the desired altitude z d :

T = T 1 + T 2 + T 3 = k z p (z d -z) + k z d ( żd -ż) + mg cos θ cos φ (13)
but for security reasons and in order to improve the performance, a PID control with a saturation is used:

T = sat T k z p (z d -z) + k z d ( żd -ż) + k z i (z d -z) + mg cos θ cos φ (14) T = sat T k z p e z p + k z i e z i + k z d e z d + mg cos θ cos φ ( 15
)
where sat M (x) is a nonlinear saturated sigmoid function:

sat M (x) = M 1 -e -2x/M 1 + e -2x/M (16) Note that lim x→±∞ (sat M (x)) = M sign(x)
, where sign(x) is the signum function defined as sign(x) = {1 if x 0; -1 if x < 0}, then M is the boundary saturation. The sigmoid function sat M (x) becomes linear when M goes to infinity.

2) Attitude Control: The attitude control is used to stabilize the UAV when it takes off and lands vertically. However in the transition phase, two front motors must tilt forward with the same rotational speed. So only the pitching movement is stabilized in the transition phases, the rolling and yawing motions are controlled manually. The bounded attitude control PID is used:

Γ j = sat Γj k j p e j p + k j i e j i + k j d e j d ( 17
)
with j = φ, θ, ψ and Γ j are the boundary of the control torques around the axes x, y, z.

3) Adaptative Control Using Neural Network: A modification of control parameters for PID control must be adjusted to realize more accurate control. It is obvious that the PID control method is limited because the parameters k p , k d , k i are constant. This means that the control output is not adaptable and optimal in any case. An intelligent adaptative control, which has the adaptability of control parameters to minimize the position error, is used. With the capacity of learning and adaptability of neural network, the controller can solve these problems. The parameters PID will be tuned adaptively and optimally in order to minimize the position error with respect to external perturbations.

Figure 4 shows the structure of the PID control using neural network for one degree of freedom. Here we take the pitch angle for example, the control method will be applied similarly for other degrees of freedom (altitude, roll and yaw). The block diagram of neural network is shown in Figure 5. Here, k p , k d , k i , e p , e i and e d are proportional, integral and derivative gains, the system error between desired and actual output, the integral of the system error and the difference of the system error, respectively. Neural networks are trained by the conventional back propagation algorithm to minimize the system error between the desired and actual output. In Figure 5, the input signal of the sigmoid function in the output layer becomes:

x(k) = k θ p (k)e θ p (k) + k θ i (k)e θ i (k) + k θ d (k)e θ d (k) (18)
where,

               e θ p (k) = θ d (k) -θ(k) e θ i (k) = k n=1 e θ p (n)∆T e θ d (k) = e θ p (k) -e θ p (k -1) ∆T (19)
∆T is the sampling time, k is the discrete sequence; θ d (k) and θ(k) are the desired and actual pitch angle. To tune the gains of the PID controller k p , k i , k d , the Gradient Descent method using the following equation was applied:

k θ j (k + 1) = k θ j (k) -η θ j ∂E(k) ∂k θ j ( 20
)
for j = p, i, d where η θ p , η θ i , η θ d are learning rates determining the convergence speed, and E(k) is a cost function. The PID parameters are updated at the step k in order to decrease the cost function. Since the control purpose is to decrease the

(k) = 1 2 (θ d (k) -θ(k)) 2 .
According to [START_REF] Thanh | Techical note: Nonlinear PID control to improve the control performance of 2 axes pneumatic artificial muscle manipulator using neural network[END_REF], the parameters are updated for j = p, i, d:

k θ j (k + 1) = k θ j (k) + η θ j e θ j (k)e θ j (k) ∂Γ θ (k) ∂x (21)
where :

∂Γ θ (k) ∂x = ∂(sat Γ θ x(k)) ∂x = ∂ ∂x Γ θ 1-e -2x(k)/Γ θ 1+e -2x(k)/Γ θ = 4e -2x(k)/Γ θ 1+e -2x(k)/Γ θ 2 (22)
Then the PID parameters are tuned at every loop.

We can notice that when the learning rates η j (with j = p, i, d) are minor, the convergence speed will be slow, which will decrease the adaptation of system. But if a high η j is used, the Neural Network adapts more quickly but the cost function will overpass more easily the minimum value and the gains cannot converge to the needed values. In order to improve the Gradient Descent algorithm, the learning rates η j are also updated adaptively as follow:

η j (k + 1) = (1 + ς j sign (E(k + 1) -E(k))) η j (k) (23)
where ς j are positive parameters.

B. Trapezoidal Trajectory of Velocity -Servo Control

In the case when the vehicle autonomously takes off, lands and makes a transition, the controller requires the next desired altitude and the next desired tilt angle. Then a trajectory generation algorithm must be used for optimum motion control. We have used the same motion profile algorithm that controls the speed and acceleration as in [START_REF] Ta | Modeling and Control of a Convertible Mini-UAV[END_REF].

The trajectory generator will produce trapezoidal shaped velocity curves for a long movement and triangular curves for a short movement where maximum velocity was not reached.

The control law of the altitude will force the vehicle to track the new desired altitude created by the trajectory generator in order to realize the process of take-off and landing. Figure 6 shows the rotor's tilt angle generated by the motion profile algorithm and its adaptation when the airplane makes transition to the horizontal flight. During the transition to the level flight when both front rotors start tilting forward, the force of motors on the x axis appears and then the aircraft speed increases. The aerodynamic lift force appears, however we can not measure or estimate this force. So the control law of the pitching movement must be used. From ( 5) the pitching moment equation is:

Γ C θ = T 3 l 2 -(T 1 + T 2 ) l 1 sin δ (24)
where Γ C θ is the pitching torque deduced from the attitude control law to stabilize the aircraft, T 1 = T 2 are the thrusts of two front motors which are given by the operator via R/C receiver. By using this method, the operator can adjust the UAV's speed during the transition. Knowing that the tilt angle δ decreases from 90 • to 0 • , so the third motor's speed must vary according to the equation:

T 3 = Γ C θ + (T 1 + T 2 ) l 1 sin δ l 2 (25)
While the force generated by the motors in the z axis decreases (T 3 and (T 1 + T 2 ) sin δ), the force in the x axis will increase. Therefore the aerodynamic force F z A at the moment will increase in order to compensate the reduction of F z T thanks to the high velocity of the vehicle.

V. SIMULATION RESULTS

A. Longitudinal Flight Simulation

In this section, simulation results from Matlab-Simulink and MSC. Visual Nastran 4D simulator are presented to demonstrate the performance of the control strategy. In this simulation, we assume that the airplane is stable in lateral dynamics and the reaction of the tilt angle's servo motor is much faster than the vehicle's dynamic. It means that we exam here only the longitudinal dynamics and the tilt angle's servo adapts immediately with respect to the desired tilt angle. According to the logics of transition (1 = transition to level flight, 0 = transition to hover) and of vertical flight (1 = take off, 0 = landing)(Figure 8), we can divide the phases of flight as follows:

(1). Take-off (from the beginning to second 3): Initially, the vehicle takes off automatically up to 2m, the altitude velocity has a trapezoidal form, it accelerates and reduces the speed when the vehicle reaches 2m.

( order to have a high angle of attack. This pitch angle can be increased more but 2 • is sufficient to prove the performance of the control strategy. The vehicle loses the altitude at the beginning of the transition because its low speed cannot permit to create the sufficient lift force. In fact, we can increase the rotational speed of both front rotors so that the airplane can avoid losing the altitude during the transition but here they are still kept at the value (349 rad/s) at which they contribute to the total thrust force equal to the airplane's weight in the vertical flight (Figure 10). During the transition, the rotation speed of the rear motor reduces to 0 rad/s according to the equation (25) and the airplane accelerates thanks to the increase of thrust force F x T (see Figure 9). The transition terminates at second 8.3 when the tilt angle reaches 0 • .

(3). Horizontal flight (from second 8.5 to 12.5): After the transition, the velocity still increases to a certain speed although the rotational speed of the both front motors has been reduced to 270 rad/s (see Figure 10). The velocity then converges to a constant value due to the friction drag. It means that at the moment the longitudinal acceleration of the airplane is zero. In Figure 11, we can notice that the angle of attack also converges to a constant value. Despite this small value (approximately 1.5 • ), it is enough to create the lift force thanks to the high speed of the aircraft (approximately 15m/s)(note that the nominal airplane's speed is about 13m/s). ( 4). Transition to the hover flight (from second 12.5 to 14.5): During the transition, the rear rotor increases the speed from zero to a necessary value in order to maintain the pitch angle at -2 • while the speed of both front rotors still remains at 270 rad/s. The velocity of the airplane decreases because the force F x T reduces rapidly (Figure 9). Therefore the aerodynamic lift force F z A also decreases while the total thrust force of motors F z T increases. ( 5) and [START_REF] Xili | Transition Flight Control of Two Vertical/Short Takeoff and Landing Aircraft[END_REF]. Vertical flight and landing (from second 14.5 to 43): Without the F x T , the aircraft still flies with cruise speed because it still has the momentum in horizontal flight. However its velocity reduces because of the drag force and the pitch angle -2 • . The airplane still has the lift force F z A (Figure 9) thanks to its velocity but this lift force also reduces to zero according to the velocity. When the velocity reaches zero, the desired pitch angle is zero to stabilize the airplane and after that, the airplane lands to the ground at the second 34.

Note that during the period of flight, the sum of the aerodynamic lift force F z A and the total thrust force F z T is always approximate to the weight of the vehicle (mg ≈ 12.26N )(Figure 9). It guarantees that the airplane cannot fall down. The control law of the altitude is inactivated during the horizontal flight and the transition, the profile of the altitude velocity is trapezoidal in the periods of take-off and landing and there is no overshooting.

B. Control performance improved by Neural Network

The Figure 12 shows that the performance of the control was improved by Neural Network when the vehicle realizes 3 times of take-off and landing from 0m to 2m. The curve (1) is the desired position generated by the generator of trajectory and the second one (2) is the adaptation of altitude control when the gains were chosen correctly. The curve (3) is the adaptation when the gains were chosen incorrectly. The adaptation is almost acceptable thanks to the trajectory generator but there are static errors at the beginning. However these static errors are eliminated quickly by the integral control. In the curve (4), the offset of the altitude control mg in the equation (15) was changed to 0.6mg (wrong estimation of system parameters) and the gains still remain incorrect like the case (3). The parameters of control in the test (5) are similar as (4), but here sensor noises and external disturbances (zero mean, bounded noise processes with standard deviations of σ z = 0.03m and σ F z = 4N ) were added to altitude's measure and F z respectively. The conditions of test [START_REF] Xili | Transition Flight Control of Two Vertical/Short Takeoff and Landing Aircraft[END_REF] are the same as (5) (wrong gains and parameters of control, input and output perturbations) but the gains were now corrected adaptively by Neural Network.

CONCLUSIONS AND FUTURE WORKS

In this work, we have presented the dynamic model of a tilt tri-rotor airplane and a simple control strategy to achieve a longitudinal flight path. The obtained simulation results demonstrate that the proposed control strategy could be implemented in reality. The control law is simple and suitable for embedded applications, it does not require a high computational cost for control loop. The proposed approach is currently implemented on the platform and the tests will be presented in a longer version of the paper.

Fig. 1 .

 1 Fig. 1. Prototype of the Tilt tri-rotor Airplane

Fig. 2 .

 2 Fig. 2. Vertical flight (helicopter mode)

Fig. 3 .

 3 Fig. 3. Horizontal flight (airplane mode)

Γ A = P s ar l m + P s al l m P b e + P s e l e -P b w l w P s r l r T ( 8 )

 8 Note that in order to balance in the longitudinal dynamics, the distances l e and l w must satisfy this condition[START_REF] Michael | Flight Dynamics Principles: A Linear Systems Approach to Aircraft Stability and Control[END_REF]: P b e l e = P b w l w → → F b A ≈ -T b w + T b e 0 P b w + P b e T Γ A ≈ P s ar l m + P s al l m P s e l e P s r l r T (10) where

Fig. 4 .Fig. 5 .

 45 Fig. 4. Structure of the nonlinear PID controller using neural network.

Fig. 6 .

 6 Fig. 6. Desired titlt angle generated by the motion profile algorithm and its adaptation

Fig. 8 .

 8 Fig. 7. Longitudinal flight path

Fig. 9 . 3 Fig. 10 .

 9310 Fig. 9. Forces generated in E b axis

Fig. 12 .

 12 Fig. 11. Angle of attack

VII. ACKNOWLEDGMENTS

This work has been supported by the French Armament Procurement Agency (DGA) and the French National Center for Scientific Research (CNRS).


A A Mekonnen

F Lerasle

A Herbulot
email: ariane.herbulot@laas.fr

External Cameras and a Mobile Robot for Enhanced Multi-person Tracking

Keywords: Multi-target Tracking, Multi-sensor Fusion, Automated Human Detection, Cooperative Perception Systems

In this paper, we present a cooperative multi-person tracking system between external fixed-view wall mounted cameras and a mobile robot. The proposed system fuses visual detections from the external cameras and laser based detections from a mobile robot, in a centralized manner, employing a "tracking-by-detection" approach within a Particle Filtering scheme. The enhanced multi-person tracker's ability to track targets in the surveilled area distinctively is demonstrated through quantitative experiments.

INTRODUCTION

Automated multi-person detection and tracking are indispensable in video-surveillance, robotic and similar systems. Unfortunately, automated multi-person perception is very challenging due to variations in human appearance. These challenges are further amplified in robotic platforms due to mobility, limited Field-Of-View (FOV) of on-board sensors, and limited on-board computational resources. Relatively successful multi-person perception systems have been reported in classical video-surveillance frameworks that rely on visual sensors fixed in the environment [START_REF] Hu | A survey on visual surveillance of object motion and behaviors[END_REF]. Even though these systems benefit from global perception from wall-mounted cameras, they are still susceptible to occlusions and dead-spots. To circumvent these shortcomings, we propose a cooperative multi-person perception system consisting of a mobile robot and two wall-mounted fixed-view cameras. This system benefits from the global perception of the wall-mounted cameras and additionally, from the mobile platform which provides local perception, a means for action, and as it can move around, the ability to cover dead spots and possibly alleviate occlusions resulting in enhanced perception capabilities. Similar systems have been proposed in [START_REF] Chia | Cooperative surveillance system with fixed camera object localization and mobile robot target tracking[END_REF] and [START_REF] Chakravarty | External cameras and a mobile robot: A collaborative surveillance system[END_REF]. Contrary to both works, our proposal fuses cooperative information in a centralized manner. The proposed system has the ability to complement local perception with global perception and vice-versa, enhancing each individual approach through cooperation. To the best of our knowledge this cooperative framework has not been addressed in the literature. This paper is structured as follows: architecture of the cooperative system is presented in section 2. Section 3 describes the different detection modalities that drive the multi-person tracker (presented in section 4). Evaluations and results are presented in section 5 followed by concluding remarks in section 6.

ARCHITECTURE

Our cooperative framework is made up of a mobile robot and two fixed view wall-mounted RGB flea2 cameras (figure 1). The cameras have a maximum resolution of 640x480 pixels and are connected to a dualcore Intel Centrino Laptop via a fire-wire cable. The robot, called Rackham, is an iRobot B21r mobile platform. It has various sensors, of which its SICK Laser Range Finder (LRF) is utilized in this work. Communication between the mobile robot and the computer hosting the cameras is accomplished through a wi-fi connection. Each of these parts are discussed in detail in subsequent sections. It is worth mentioning here that the entire system is calibrated with respect to a global reference frame. Both the intrinsic and extrinsic parameters of the fixed cameras are known and in addition the mobile robot has localization module that localizes its pose with respect to the reference frame using laser scan segments.

MULTI-PERSON DETECTION

The perceptual functionalities of the entire system are based on various detections. The detection modules are responsible for automatically detecting persons in the area. Different person detection modalities are utilized depending on the data provided by each sensor.

Leg Detection with LRF: the LRF provides horizontal depth scans with a 180 o FOV and 0.5 o resolution at a height of 38cm above the ground. Person detection, hence, follows by segmenting leg patterns within the scan. In our implementation a set of geometric properties characteristic to human legs and outlined in [START_REF] Xavier | Fast line, arc/circle and leg detection from laser scan data in a player driver[END_REF] are used.

Person Detection from wall mounted cameras: to detect persons using the wall mounted cameras, two different modes are used. First, a foreground segmentation using a simple Σ-∆ background subtraction technique [START_REF] Manzanera | Sigma -delta background subtraction and the zipf law[END_REF] is used. The mobile robot is masked out of the foreground images using its position from its localization module. Second, Histogram of Oriented Gradients (HOG) based person detection [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF] is used. This method makes no assumption of any sort about the scene or the state of the camera (mobile or static). It detects persons in each frame using HOG features. Both detections are projected to yield ground positions, (x, y) G with associated color appearance information in the form of HSV histograms [START_REF] Pérez | Color-based probabilistic tracking[END_REF], of individuals in the area.

MULTI-PERSON TRACKING

Multi-person tracking in our context, is concerned with the problem of tracking a variable number of persons, possibly interacting, in the ground plane. The literature in multi-target tracking contains different approaches but when it comes to tracking multiple interacting targets of varying number [START_REF] Khan | Mcmc-based particle filtering for tracking a variable number of interacting targets[END_REF] has clearly shown that Reversible Jump Markov Chain Monte Carlo -Particle Filters (RJMCMC-PFs) are more appealing taking performance and computational requirements into consideration. Inspired by this, we have used RJMCMC-PF, adapted to our cooperative perceptual strategy, for multi-person tracking driven by the various heterogeneous detectors. The actual detectors are: the LRF based person detector, the foreground segmentation (detection) and HOG based detections from each wall mounted camera. Implementations choices crucial to any RJMCMC-PF are briefly discussed below.

State space: the state vector of a person i in hypothesis n at time t is a vector encapsulating the id and (x, y) position of an individual on the ground plane with respect to a defined coordinate base, x n t,i = {Id i , x n t,i , y n t,i }.

Proposal moves: RJMCMC-PF accounts for the variability of the tracked targets by defining a variable dimension state space. Proposal moves propose a specific move on each iteration to guide this variable state space exploration. In our implementation, four sets of proposal moves, m = {Add, Update, Remove, Swap}, are used. The choice of the proposals privileged in each iteration is determined by q m , the jump move distribution. These values are determined empirically and are set to {0.15, 0.8, 0.02, 0.03} respectively. Equation 1 shows computation of the acceptance ratio, β, of a proposal X * at the n th iteration. It makes use of the jump move distribution, q m ; proposal move distribution, Q m (), associated with each move; the observation likelihood, π(X n t ); and the interaction model, Ψ(X n t ).

β = min   1, π (X * ) Q m * X n-1 t |X * q m * Ψ (X * ) π X n-1 t Q m X * |X n-1 t q m Ψ X n-1 t   (1)
where m ∈ {Add, Update, Remove, Swap} and m * denotes the reverse operation. Update and Swap moves are self reversible.

Add: the add move, randomly selects a detected person, x p , from the pool of provided detections and appends its state vector on X n-1 t resulting in a proposal state X * . The proposal density driving the Add proposal, Q Add X * |X n-1 t , is then computed according to equation 2.

Q Add X * |X n-1 t = ∑ d k d . N d ∑ j=1 N x p ; z d t, j , Σ . 1 -k m N t ∑ j=1 N x p ; Xt-1, j , Σ (2)
where d represents the set of detectors, namely: from laser (l), fixed camera 1 (c 1 ), and fixed camera 2 (c 2 ); d ∈ {l, c 1 , c 2 }, N d the total number of detections in each detector, k d is a weighting term for each detector such that ∑ d k d = 1, N t is the number of targets in the MAP, and k m is a normalization constant. When a new person is added, its appearance is cross-checked with the appearance of persons that have been tracked for re-identification.

Remove: this move randomly selects a tracked person x p from the particle being considered, X n-1 t , and removes it, proposing a news state X * . Contrary to the add move, the proposal density used when computing the acceptance ratio, Q Remove (X * |X n-1 t ) (equation 3), is given by the distribution map from the tracked persons masked by a map derived from the detected passers-by.

Q Remove X * |X n-1 t = 1 -∑ d k d . N d ∑ j=1 N x p ; z d t, j , Σ . k m N t ∑ j=1 N x p ; Xt-1, j , Σ (3)
Update: here, the state vector of a randomly chosen passer-by is perturbed by a zero mean normal distribution.

The update proposal density,

Q update (X * |X n-1 t
), is a normal distribution with the position of the newly updated target as mean.

Swap: the swap move handles the possibility of id switches amongst near or interacting targets. When this move is selected, the ids of the two nearest tracked persons are swapped and a new hypothesis X * is proposed. The acceptance ratio is computed similar to the Update move.

Interaction model (Ψ(.))

: is used to maintain tracked person identity and penalize fitting of two trackers on the same object during interaction. A Markov Random Field (MRF), similar to [START_REF] Khan | Mcmc-based particle filtering for tracking a variable number of interacting targets[END_REF], is adopted to address this.

Observation Likelihood (π(.)): the observation likelihood, in equation 1, is derived from all detector outputs except the laser for which blobs formed from the raw laser range data, denoted as l b , are considered. If the specific proposal move is an Update or Swap move, a Bhattacharyya likelihood measure is also incorporated. Each detection is represented as a Gaussian, N (.), centered on the detection. Representing the measurement information at time t as z t , the observation likelihood of the n th particle X n t at time t is computed as shown in equation 4.

π (X n t ) = π B (X n t ) .π D (X n t ) (4) π B (X n t ) = ∏ M i=1 ∏ 2 c=1 e -λB 2 i,c , if m = Update or Swap 1 , otherwise π D (X n t ) = 1 M M ∑ i=1 ∑ d k d .π x i |z d t , ∑ d k d = 1 π x i |z d t = 1 N d N d ∑ j=1 N x i ; z d t, j , Σ
Above, B i represents the Bhattacharyya distance computed between the appearance histogram of a proposed target i in particle X n t and the target model in each camera c. M represents the number of targets in the particle, and N d the total number of detections in each detection modality d, d = {l b , c 1 , c 2 }, in this case including the measures from the laser blobs. k d is a weight assigned to each detection modality taking their respective accuracy into consideration and x i represents the position of target i in the ground plane.

EVALUATIONS AND RESULTS

To evaluate the performance of our RJMCMC-PF multi-person tracker, three sequences acquired using Rackham and the wall mounted cameras are used. Each sequence contains a laser scan and video stream from both cameras. Sequence I and II contain 200 frames each and consist of two and three targets consecutively. Sequence III is 186 frames long containing four targets moving in the vicinity of the robot. The evaluation is carried out using the CLEAR MOT metrics [START_REF] Bernardin | Evaluating multiple object tracking performance: the CLEAR MOT metrics[END_REF], Multiple Object Tracking Accuracy (MOTA) and Precision (MOTP). To clearly observe the advantages of each sensor modality, the evaluation is carried out by doing the tracking using (1) laser-only information, (2) vision-only data from the two wall mounted cameras, and finally (3) laser and the two cameras cooperatively. A hand labeled ground truth with (x, y) ground positions and unique id for each person is used in the evaluation. Each sequence is run eight times to account for the stochastic nature of the filter. Results are reported as mean value and associated standard deviation in table 1. The results presented in table 1 clearly attest the improvements in perception brought by the cooperative fusion of laser and wall mounted camera percept. The cooperative system consisting of laser and two wall mounted cameras exhibit an MOTA of 0.841 when tracking two targets, 0.793 for three targets. These results clearly indicate the enhanced performance of this system. Sample tracking sequences from sequence II are shown in figure 3 1 . Evidently, the LRF-only has low accuracy owing to the mistakes made with leg like structures in the environment, sensitivity to occlusion, and lack of discriminating information amongst tracked passers-by. The results obtained using the wall mounted cameras show major improvements though their position tracking precision is relatively lower compared to those which include laser measurement. The final tracker runs at 1 f ps. Most of the computation time, ≈ 700ms, is spent on HOG based person detection.

CONCLUSIONS

The work presented herewith makes its main contribution in the vein of multi-person tracking by proposing a cooperative scheme between overhead cameras and sensors embedded on a mobile robot in order to track people in crowds. Our Bayesian data

Figure 1 :

 1 Figure 1: Perceptual platform; static cameras (with rough positions and fields of view) and the mobile robot Rackham.

Figure 2 :

 2 Figure 2: Multi-person detection and tracking system block diagram.

Figure 2

 2 Figure 2 shows block diagram of the envisaged multi-person perceptual system. It has two main parts. The first part deals with automated multiperson detection. The second part is dedicated for multi-person tracking. It takes all detections as input and fuses them in a Particle Filtering framework.Each of these parts are discussed in detail in subsequent sections. It is worth mentioning here that the entire system is calibrated with respect to a global reference frame. Both the intrinsic and extrinsic parameters of the fixed cameras are known and in addition the mobile robot has localization module that localizes its pose with respect to the reference frame using laser scan segments.

Figure 3 :

 3 Figure 3: Multi-person tracking illustrations taken from sequence II at a) frame 60, and b) frame 94.The top row images show camera streams and the bottom shows the ground floor with tracked persons' trajectories superimposed 1 .

Table 1 :

 1 Multi-person tracking evaluation results.

		Laser-only		Fixed Cameras only	Cooperative
	Sequence	MOTP		MOTA	MOTP		MOTA	MOTP	MOTA
		µ	σ	µ	σ	µ	σ	µ	σ
	I	15.62 2.34 0.41 0.05 19.80 0.14 0.79 0.03 17.01 1.87 0.84 0.03
	II	19.90 1.66 0.27 0.07 22.79 1.35 0.70 0.05 17.73 0.50 0.79 0.03
	III	21.94 1.75 0.20 0.07 28.44 1.60 0.46 0.07 21.30 1.34 0.54 0.04

For complete run, visit the URL homepages.laas.fr/aamekonn/videos/ fusion framework with the given sensor configuration enhances typical surveillance systems with only fixed cameras and complete embedded systems without wide FOV and straightforward (re)-initialization ability. The presented results are a clear indication of the framework's notable tracking performance.