James T Murphy
email: jmurphy@sb-roscoff.fr

Mark P Johnson

Frédérique Viard

A theoretical examination of environmental effects on the life cycle schedule and range limits of the invasive seaweed Undaria pinnatifida

Keywords: Macroalgae, seaweed, life cycle schedule, individual-based model, agent-based model, ecological niche, species distribution

Invasive macroalgae form a substantial component of marine invaders at a global level. However, it is poorly understood how the complex interactions between local environmental conditions and life cycle dynamics contribute to invasion success from a mechanistic viewpoint. The aim of this study was to use a model (UndariaGEN) that incorporates a detailed representation of the individual heteromorphic life history stages (sporophytes and gametophytes) of the species in order to explore how interactions between these components contribute to the overall population dynamics. The latest version of the model was validated against field data from a real-life population in Brittany, France. This was followed by an assessment of the role of temperature limitations in determining its potential global range and then a more detailed examination of how environmental factors affect the life cycle dynamics of U. pinnatifida across a range of conditions characteristic of European populations. In terms of both relative abundance and recruitment, the model matches closely the patterns observed from field studies in Brittany, France (R 2 =0.98 respectively). Furthermore, the model predicted theoretical temperature limits for growth (9.1 -22.5 o C) match closely the actual current global range limits for the species (9.5 -22.4 o C) reported in the literature. In addition, the size of the species' ecological niche is shown to be directly related to the amplitude in seasonal variation of temperature. This demonstrates that U. pinnatifida has a wider ecological niche in conditions of high seasonality; this finding is consistent with theories that propose the heteromorphic life cycle may have evolved as an optimal growth strategy for highly seasonal environments.

Introduction

Ecological niche models (ENMs) (and related species distribution models, SDMs) are commonly used to predict the potential range limits of invasive species based on their ecological niche characteristics in their native range [START_REF] Peterson | Predicting the Geography of Species' Invasions via Ecological Niche Modeling[END_REF]. These approaches can be used to produce accurate habitat suitability maps assuming that ecological niches represent long-term stable constraints on a species' potential geographic range [START_REF] Marcelino | Ecological niche models of invasive seaweeds[END_REF]. Most of these approaches are correlative in nature, by statistically linking spatial environmental data to species distribution records. However, attempts have also been made to incorporate some mechanistic links between the functional traits of organisms and their environments into SDMs [START_REF] Kearney | Mechanistic niche modelling: combining physiological and spatial data to predict species' ranges[END_REF]. The addition of physiological information can enable more robust predictions of range shifts, particularly in novel or non-equilibrium contexts such as invasions or in response to environmental change, by taking into account niche shifts during biological invasion [START_REF] Broennimann | Evidence of climatic niche shift during biological invasion[END_REF]; [START_REF] Urban | The cane toad's (Chaunus [Bufo] marinus) increasing ability to invade Australia is revealed by a dynamically updated range model[END_REF].

In this study, an individual-based modelling approach was used to represent the population dynamics of the study species from a bottom-up perspective [START_REF] Deangelis | Individual-based models and approaches in ecology: populations, communities and ecosystems[END_REF][START_REF] Grimm | Individual-based Modeling and Ecology[END_REF]. This approach means that the individuals of the population are explicitly modelled with individual growth rates and environmental response parameters as opposed to assuming global population averages. This approach allows one to differentiate between the responses of the various life history stages of the species. This is particularly important among species with heteromorphic life cycles consisting of more than one alternating stage or phase. These are characterised by life history stages that often differ in their growth parameters, ecological niches and physical properties [START_REF] Ebenman | Evolution in organisms that change their niches during the life-cycle[END_REF][START_REF] Eckert | Effects of the planktonic period on marine population fluctuations[END_REF][START_REF] Thornber | Functional properties of the isomorphic biphasic algal life cycle[END_REF]. For example, in the case of many kelp species, the gametophyte and sporophyte stages can differ in their genetic properties (haploid versus diploid), size (microscopic versus macroscopic) and ecophysiology (temperature response profiles, photosynthetic properties) (Morita et al. 2003a, b). For this study, an updated version of an individual-based model framework which we previously developed, called UndariaGEN (Murphy et al. 2016a), was used to investigate the population dynamics of the invasive kelp species Undaria pinnatifida in a coastal environment.

Undaria pinnatifida is native to the coastal regions of eastern Asia, including Japan, Korea and China, where it is commonly cultivated for culinary purposes [START_REF] Ohno | Undaria cultivation 'wakame'. Seaweed cultivation and marine ranching[END_REF][START_REF] Shao-Jun | Study on gametophyte vegetative growth of Undaria pinnatifida and its applications[END_REF]. This edible seaweed was accidentally introduced to Europe in the 1970's in the Étang de Thau, France, followed by deliberate introduction for cultivation purposes in Brittany, France in 1983 [START_REF] Daguin | Microsatellites isolation and polymorphism in introduced populations of the cultivated seaweed Undaria pinnatifida (Phaeophyceae, Laminariales)[END_REF][START_REF] Floc | The Japanese brown alga Undaria pinnatifida on the coast of France and its possible establishment in European waters[END_REF][START_REF] Voisin | Differential shuffling of native genetic diversity across introduced regions in a brown alga: Aquaculture vs. maritime traffic effects[END_REF]. Subsequent to these introductions it has expanded its range in Europe, invading artificial and natural coastal habitats and extending from southern Italy in the south to the United Kingdom and Ireland in the north [START_REF] Cecere | Undaria pinnatifida (Fucophyceae, Laminariales) spread in the central Mediterranean: Its occurrence in the Mar Piccolo of Taranto (Ionian Sea, southern Italy)[END_REF][START_REF] Minchin | The invasive brown alga Undaria pinnatifida (Harvey) Suringar, 1873 (Laminariales: Alariaceae), spreads northwards in Europe[END_REF][START_REF] Peteiro | A new record of the introduced seaweed Undaria pinnatifida (Laminariales, Phaeophyceae) from the Cantabrian Sea (northern Spain) with comments on its establishment[END_REF]). Furthermore, it has also arisen as an invasive threat in other regions of the world such as New Zealand and North America [START_REF] Hay | The Asian kelp Undaria pinnatifida (Phaeophyta: Laminariales) found in a New Zealand harbour[END_REF][START_REF] Silva | First Report of the Asian kelp Undaria pinnatifida in the Northeastern Pacific Ocean[END_REF]. Its ability to colonize artificial and disturbed habitats, its high fertility and its fast-growing strategy are among the characteristics that explain its ranking as the third most invasive seaweed, in a species trait analysis of 113 introduced macroalgae, in Europe [START_REF] Nyberg | Can species traits be used to predict marine macroalgal introductions?[END_REF].

The annual (or bi-annual) life cycle of U. pinnatifida is typical of brown kelp seaweeds with two heteromorphic phases: a microscopic haploid gametophyte stage and a large (1-3 m in length) diploid sporophyte stage [START_REF] Clayton | Evolution and Life Histories of Brown Algae[END_REF][START_REF] Thornber | Population ecology of the invasive kelp Undaria pinnatifida in California: environmental and biological controls on demography[END_REF][START_REF] Wallentinus | Alien Species Alert: Undaria pinnatifida (wakame or Japanese kelp). ICES Cooperative Research Report[END_REF]). These forms differ in their growth requirements and responses to environmental factors such as light and temperature (Morita et al. 2003a, b). The gametophyte stage is likely to facilitate the introduction steps and subsequent spread through ballast water or as epiphytes on ship hulls and oyster shells [START_REF] Mineur | Experimental assessment of oyster transfers as a vector for macroalgal introductions[END_REF][START_REF] Mineur | Hull fouling on commercial ships as a vector of macroalgal introduction[END_REF]. Gametophytes are also capable of delaying their development possibly forming a 'gametophyte bank', analogous to seed banks in terrestrial plants [START_REF] Carney | Cryptic Processes in the Sea: A Review of Delayed Development in the Microscopic Life Stages of Marine Macroalgae[END_REF].

However, despite their importance in influencing the overall population dynamics at both local and regional scales and thus their critical role in invasion success [START_REF] Couceiro | The Ectocarpus life cycle in the field supports the hypothesis of niche separation between gametophytes and sporophytes[END_REF][START_REF] Edwards | The Role of Alternate Life-History Stages of a Marine Macroalga: A Seed Bank Analogue[END_REF], gametophytes are surprisingly hardly considered directly in empirical or theoretical studies.

The aim of this project was to use a model that incorporates a detailed representation of the life history stages of U. pinnatifida with individual gametophytes and sporophytes represented as autonomous agents with independent behaviour. The effects of environmental parameters such as light and temperature on the growth and development of these stages could then be parameterised independently for each stage, using empirical data from the literature, thus allowing population behaviour to be traced back to the individual components. Three seasonally-variable environmental parameters, which have been shown to play a key role in determining the growth potential of U. pinnatifida, were examined: light availability, water temperature, and day length (or the photoperiodic effect) [START_REF] Báez | Macro-environmental modelling of the current distribution of Undaria pinnatifida (Laminariales, Ochrophyta) in northern Iberia[END_REF][START_REF] Campbell | Seasonal Patterns in the Photosynthetic Capacity, Tissue Pigment and Nutrient Content of Different Developmental Stages of Undaria pinnatifida (Phaeophyta: Laminariales) in Port Phillip Bay, South-Eastern Australia[END_REF][START_REF] Choi | Growth and reproductive patterns of Undaria pinnatifida sporophytes in a cultivation farm in Busan, Korea[END_REF][START_REF] James | Using satellite-derived sea surface temperature to predict the potential global range and phenology of the invasive kelp Undaria pinnatifida[END_REF][START_REF] Thornber | Population ecology of the invasive kelp Undaria pinnatifida in California: environmental and biological controls on demography[END_REF]). There are extensive literature studies on the impact of these abiotic factors on individual growth stages of U. pinnatifida [START_REF] Choi | Effects of daylength, irradiance and settlement density on the growth and reproduction of Undaria pinnatifida gametophytes[END_REF]Morita et al. 2003a, b;[START_REF] Pang | Photoperiodic long-day control of sporophyll and hair formation in the brown alga Undaria pinnatifida[END_REF][START_REF] Pang | Responses of vegetative gametophytes of Undaria pinnatifida to high irradiance in the process of gametogenesis[END_REF][START_REF] Shao-Jun | Study on gametophyte vegetative growth of Undaria pinnatifida and its applications[END_REF]. However, our understanding of the complex spatiotemporal interactions between these factors and the microscopic and macroscopic stages of the life cycle is limited.

Methods

A detailed description of the underlying model framework and the technical processes involved in its design and parameterisation have been published previously (and presented with updates in Appendix I) (Murphy et al. 2016a). For this study, we upgraded this model and focus on a case study analysis using the model to examine the role of environmental conditions in determining the current range of U. pinnatifida in Europe. The model has two measures of invasive potential: (a) the traditional measures of population growth (changes in abundance and recruitment over time), and (b) temporal changes to the seasonal schedule of recruitment of the various life history stages.

Therefore, in order to factor in these differences between life history stages and between individual sporophytes and gametophytes the IBM approach was chosen. The environment is modelled as a spatially-explicit, discrete two-dimensional grid in order to take into account heterogeneities in the physical surroundings (at a spatial resolution of 0.25 m 2 ). Meanwhile, time is represented as discrete time steps with environmental parameters such as temperature and light levels updated on a daily basis. This is to enable seasonal changes in the water conditions to be accounted for and the growth responses of the agents to be updated at a relatively fine-grained scale.

A broad technical overview of the model is included in Appendix I for reference. In the following sections, we highlight the key changes made for the latest version of the model, developed for this study. This upgraded version (v0.6.4) is available under a GNU General Public License Version 3 on the GitHub repository hosting service (https://github.com/murphyjtm/undariaGEN) (Murphy 2016).

Sporophyte and gametophyte agents

The IBM approach involves a bottom-up modelling technique whereby the main components simulated are the individual life history stages, in this case the diploid sporophytes and haploid gametophytes of the U. pinnatifida life cycle. Each individual sporophyte or gametophyte is represented as an independent agent in the model with its own set of unique parameters (e.g. growth rate, temperature response curve, etc.). Individual variability is an important component of an IBM model and therefore a Mersenne Twister random number generator is used to vary the growth properties for each individual agent (by sampling from a normal distribution around a population mean) [START_REF] Matsumoto | Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator[END_REF].

Depending on the type of the agent (sporophyte or gametophyte), the behaviour and responses to local environmental parameters are calculated based on a different set of sub-models, unique to each agent type, which are fitted to ecophysiological data gathered from the literature (see Appendix I). At each time step, the growth rate of each agent is calculated separately according to its type (gametophyte or sporophyte), its own internal state (size, maturity, etc.), and the local environmental conditions (light, temperature, day length) at that point in space and time. The test simulations involved up to 3.0x10 5 agents growing concurrently and independently interacting with their environment and each other. Execution time was approximately 15 minutes per simulated year (on a quad-core Intel Xeon E3 1270 @3.4GHz desktop computer) The results were then aggregated and population-averaged results, which are an emergent property of all the agents interacting together, were collected for further analysis and presentation.

Modification 1: Calculation of photosynthetically available radiation

In the current and the previous versions of the model, the growth of agents is determined by the availability of light (since they depend on photosynthesis). In the model, there is a parameter for the light attenuation coefficient for photosynthetically available radiation (KdPAR) which determines the amount of light attenuation, with increasing depth in the water column [START_REF] Saulquin | Estimation of the diffuse attenuation coefficient KdPAR using MERIS and application to seabed habitat mapping[END_REF]. This is used to calculate the residual energy available for photosynthesis by U. pinnatifida agents, depending on their depth, according to the following equation:

𝐸(𝑧) = 𝐸(0)𝑒 -𝑧𝐾 𝑑𝑃𝐴𝑅 (1)
where z is the depth (m) of the seaweed below the water surface, E(0) is the level of irradiance at the water surface, and E(z) is the energy available for photosynthesis at depth z. For the case studies in this paper, the depth (z) was set to 1.0 m and the attenuation coefficient (KdPAR) to 0.4 for conditions representative of the coastline of Brittany, France [START_REF] Saulquin | Estimation of the diffuse attenuation coefficient KdPAR using MERIS and application to seabed habitat mapping[END_REF]).

In the latest version of the model, an additional parameter has been incorporated to represent scale effects on photosynthetic performance between individual plants and communities.

It has been demonstrated that the light saturation point (Ik) for macroalgal communities can be several times higher than for individual thallus pieces tested in the laboratory [START_REF] Binzer | From thallus to communities: scale effects and photosynthetic performance in macroalgae communities[END_REF]. Therefore, in order to make a more realistic representation of natural conditions, we have included an additional parameter to vary the light saturation point according to the community structure. For the test cases described in this paper, this value was calculated by fitting to field data collected from Brest harbour, France (see appendix Table S1). This involved varying the light saturation factor from 1.0 (no competitive inhibition of light) to 10.0 (high saturation point due to complex community structure) and comparing the R 2 values. A value of 3.3 resulted in a seasonal pattern of recruitment that most closely matched the natural population in Brest (Fig. 1).

Modification 2: Seasonal effects on juvenile mortality among sporophytes

The death of a sporophyte agent is determined either when it reaches the natural end of its lifespan, assumed to be after it has matured and released all its spores, or through premature death or removal. Field studies have shown that up to 70% of sporophyte recruits die or are dislodged within one month of their appearance [START_REF] Voisin | Les processus d'invasions biologiques en milieu côtier marin: le cas de l'algue brune Undaria pinnatifida, cultivée et introduite à l'échelle mondiale[END_REF]. Therefore, premature death (by various means such as competition or physical dislodgement) represents a significant proportion of the deaths in a population. To account for this, field data (Murphy et al. 2016b) collected on a population of U.

pinnatifida in Brest, France, was used to create an age to mortality curve (Weibull distribution) which determines the probability of premature death as a function of the age of the sporophyte.

In earlier versions of the model, there was a single age to mortality curve used to describe the probability of premature death among recruits. However, in the latest version, it has been updated to take into account seasonal variation in the probability of mortality (due to changes in competition). For example, in July and August there is a peak in the mortality of young recruits (<1 months old), which was not taken into account in the previous version.

Therefore, a series of 12 individual Weibull functions (for each month of the year) were fitted to the mortality data in order to capture the seasonal variation in the probability of premature death. Cosine curves were used to describe the change in the shape (k) and scale () parameters of the Weibull curve as a function of the day of the year:

𝑦 = 𝐴 cos[𝜔(𝑥 -𝛼)] + 𝐶 (4)
where x is the day of the year, A is the amplitude,  is the horizontal phase shift, C is the vertical offset and  is the angular frequency (2/365). This functional relationship could then be used in the model to generate a probability of premature mortality based on the time of the year and the age of the sporophyte, rather than assuming a constant probability.

Validation versus field data from Brest harbour, France

The updated version of the model was first validated against field data collected from a population of U. pinnatifida growing in Brest harbour, France. The field experiment and raw data are summarized in (Murphy et al. 2016b). This consisted of monthly abundance and recruitment data for the species recorded at Brest harbour, France. Temperature data for the site were sourced from a SOMLIT (Service d'Observation en Milieu Littoral) buoy (INSU-CNRS 2015). Global solar irradiance data (in order to calculate the light availability at the sea surface) were obtained using the CALSOL online application (Institut National de L'Energie Solaire, CEA-CNRS) [START_REF] Ines | Logiciel CALSOL[END_REF]. Finally, day length

The first scenario (referred to as "North") involved environmental parameters representative of the northern edge of U. pinnatifida's current reported range in Europe (Northern Ireland). This test case is characterised by a relatively low mean water temperature (11.2 o C) and light availability (9.1 MJ m -2 day -1 ), as well as strong seasonal variation in day length due to the latitudinal location (54.6 o N). An alternative scenario, representative of the southern range limits of U.

pinnatifida, was also analysed (referred to as "South"). This represents conditions of relatively high water temperature (19.8 o C), high light availability (15.1 MJ m -2 day -1 ) and lower seasonal variation in day length (latitude = 42.5 o N). These latter conditions are similar to those present towards the southern limit of U. pinnatifida's current range in Europe (Ionian Sea, southern Italy) [START_REF] Cecere | Undaria pinnatifida (Fucophyceae, Laminariales) spread in the central Mediterranean: Its occurrence in the Mar Piccolo of Taranto (Ionian Sea, southern Italy)[END_REF]. As a control, the results from the simulation of Brest harbour are included, referred to as "Central" since it is approximately in the middle of the species' current range distribution in Europe along the North-South gradient. Brittany is also representative of the biogeographic transition between Lusitanian and Boreal Province in the NE Atlantic [START_REF] Spalding | Marine Ecoregions of the World: A Bioregionalization of Coastal and Shelf Areas[END_REF]). This test case represents an intermediate scenario in terms of water temperature (13.3 o C), light availability (11.4

MJ m -2 day -1 ) and day length (latitude = 48.4 o N).

Results

Validation versus field data from Brest harbour, France

The predicted relative monthly abundance and recruitment data for sporophytes of U.

pinnatifida were compared to field data from Brest harbour, France (Fig. 1). In terms of both relative abundance (total number of sporophytes) and recruitment (number of newly recruited sporophytes, >5 cm in length) the model matches closely the patterns observed in the real life population (R 2 =0.98 respectively). It also shows good quantitative agreement with field data in terms of age to maturity

(1.6±0.1 vs. 1.9±0.5 months, mean±SE) and months mature (1.5±0.1 vs. 1.45±0.4 months, mean±SE) (Fig. 2a).

The results of validating the model versus field data show a substantial improvement in the accuracy of predictions compared to the previously published version of the model: R 2 values increased from 0.84 to 0.98 and 0.85 to 0.98 for abundance and recruitment respectively (Murphy et al. 2016a). This result suggests that the majority (84%) of monthly variation in abundance of U.

pinnatifida sporophytes can be explained by seasonal changes in environmental parameters (light, temperature and photoperiodic effect). However, increased juvenile mortality in the summer could be due to competitive interactions and community effects potentially offsetting the effects of high recruitment.

Predicted temperature range of U. pinnatifida

Globally, U. pinnatifida has been found in areas with mean water temperatures (seasonal range between min and max included in brackets) as low as 9.

Case studies: Effect of environmental conditions on life cycle schedule

There are clear differences in the seasonal patterns of abundance and recruitment between the three case study populations (Fig. 4). For example, the northern population is predicted to peak in abundance later in the year (April) compared to the southern population (March) (Fig 5a), and has lower overall recruitment compared to both the central and southern populations (Fig 5b). There are also distinct differences in the seasonal growth rates of the individual sporophyte and gametophyte agents (Fig. 5). In general, the populations further south are predicted to reach optimal growth rates earlier in the spring, before temperature becomes a limiting factor. Furthermore, individuals in the south have a shorter predicted lifespan since they reach maturity and release their spores earlier in the year (Fig. 2b) due to spore release triggered by warming temperatures [START_REF] Suto | On shedding of zoospores in some algae of Laminariaceae-2[END_REF].

Finally, the effect of environmental conditions on gametophyte fertility (and thus the formation of a new generation of sporophytes) was compared across the three locations (Fig. 6). For the central and northern populations, there are two clear annual peaks in the fertility of gametophytes, in October and February respectively. However, the southern population lacks the autumn peak due to inhibition of gametogenesis and therefore has a narrower window of opportunity for forming a new generation of sporophytes. This also explains the characteristic pattern of recruitment observed in Figure 4b where the southern population lacked a secondary minor peak in recruitment in late November.

Discussion

The close agreement between the predicted abundance and recruitment values with field data for sporophytes growing in Brittany, France (Fig. 1) is particularly interesting in light of the diverse sources of input data used to parameterise the model. For the most part, the ecophysiological data used to parameterise the model came from individuals sampled in the study species' native range of eastern Asia (due to a lack of equivalent information for European populations). For example, the temperature response curves for the gametophytes and sporophytes came from individuals sampled in Japan (Morita et al. 2003a, b). Similarly, the effects of light and day length on gametophyte growth and reproduction were based on studies in the Korean peninsula [START_REF] Choi | Effects of daylength, irradiance and settlement density on the growth and reproduction of Undaria pinnatifida gametophytes[END_REF]. Finally, photosynthesis-irradiance curves for the sporophytes were sourced from studies in south-eastern Australia [START_REF] Campbell | Seasonal Patterns in the Photosynthetic Capacity, Tissue Pigment and Nutrient Content of Different Developmental Stages of Undaria pinnatifida (Phaeophyta: Laminariales) in Port Phillip Bay, South-Eastern Australia[END_REF].

Therefore, it is noteworthy that the model predicts a pattern of monthly recruitment and abundance that closely matches the patterns observed among a population growing in western France (R 2 =0.98). This suggests potentially strong phenotypic conservation in terms of its response to environmental factors since the species' introduction to Brittany, France over 40 years ago. This may be explained by a large physiological niche for U. pinnatifida. Broad physiological tolerance is an important component of invasiveness in terrestrial plants [START_REF] Higgins | Invasive plants have broader physiological niches[END_REF]. But there are currently no similar studies available for macroalgae, and a lack of experimental data on the basic physiological responses of U. pinnatifida populations in Europe compared to those in its native range.

Similarly, the model predicted temperature limits match closely with the current known global distribution for U. pinnatifida when taking into account both the mean water temperature and the amplitude of seasonal variation (Fig. 3a). In particular, the size of the species' ecological niche is shown to be directly related to the amplitude in seasonal variation of temperature (Fig. 3b).

This demonstrates that U. pinnatifida has a wider ecological niche in conditions of high seasonality, and is consistent with theories that propose the heteromorphic life cycle, characteristic of species such as this, may have evolved as an optimal growth strategy for highly seasonal environments [START_REF] Bessho | Heteromorphic and isomorphic alternations of generations in macroalgae as adaptations to a seasonal environment[END_REF].

However, there are a number of other indirect effects which must be taken into account when determining the theoretical range limits of U. pinnatifida, for example how changes in the seasonal pattern and timing of recruitment might affect its competitive ability versus native species.

These changes can affect its ability to invade natural habitats, even when the temperature conditions are suitable for recruitment. For example, [START_REF] James | Using satellite-derived sea surface temperature to predict the potential global range and phenology of the invasive kelp Undaria pinnatifida[END_REF] discussed how seasonal changes to phenology may affect impacts from invasion. To explore this, a number of test cases scenarios were designed to represent the conditions populations are exposed to at the current northern and southern range limits of U. pinnatifida in Europe.

The differences in growth and recruitment patterns (Fig. 4) can be explained by changes in the fundamental growth rates of the U. pinnatifida agents in response to the environmental parameters. The predicted growth rates of both the gametophyte and sporophyte stages of the life cycle are limited by sub-optimal temperature and irradiance conditions in the north (Fig. 5). There is also a delay in the shedding of spores due to inhibition under cooler water conditions in early summer which may explain the delayed pattern of recruitment in the northern population (<12 o C) [START_REF] Suto | On shedding of zoospores in some algae of Laminariaceae-2[END_REF]. However, there is another notable feature in the predicted recruitment, for both the Central and Northern populations, evident in Figure 4b. That is the presence of a minor secondary recruitment peak in November, in addition to the primary spring peak in April. This secondary annual peak in recruitment is absent from the predicted results for the southern population.

This secondary peak of recruitment has been observed in the field data from the population surveyed in Brest harbour (Fig. 1b) and in populations introduced in other temperate regions where temperatures vary annually between 10 and 21°C (N America; [START_REF] Thornber | Population ecology of the invasive kelp Undaria pinnatifida in California: environmental and biological controls on demography[END_REF]). This pattern of recruitment can be explained by differences in the seasonal patterns of gametogenesis (Fig. 6). The model predicts a delay of several months in the onset of maturity among U. pinnatifida gametophytes on the southern range edge. This results in a substantial change in the seasonal pattern of gametogenesis that, if exhibited in natural populations, could significantly impact their life cycle schedule and affect their potential for establishment in lower latitudes. U. pinnatifida is considered an opportunistic and pioneering species and the ability to recruit earlier in the year is thought to confer on it a competitive advantage compared to native macroalgal species in Brittany, France [START_REF] Grulois | Mosaic genetic structure and sustainable establishment of the invasive kelp Undaria pinnatifida within a bay (Bay of St-Malo, Brittany)[END_REF]. However, in the Mediterranean Sea, a delay in gametogenesis could play a critical role in limiting the ability of U. pinnatifida to supplant native species from their ecological niche.

Indeed, although the first reported occurrence of U. pinnatifida in Europe occurred in Étang de Thau, on the Mediterranean coast of France in the 1970's, it has failed to establish itself in the surrounding region subsequently [START_REF] Floc | The Japanese brown alga Undaria pinnatifida on the coast of France and its possible establishment in European waters[END_REF]. It is still present in Thau lagoon but only in a small number of discrete locations and it has not expanded outside the lagoon (F. Viard, pers. obs.).

Potential explanations for this have included nutrient limitations due to the oligotrophic nature of this region of the Mediterranean Sea [START_REF] Báez | Macro-environmental modelling of the current distribution of Undaria pinnatifida (Laminariales, Ochrophyta) in northern Iberia[END_REF]. Indeed, populations have been found in nutrient-rich polluted waters in the Venice lagoon and the Ionian Sea off southern Italy [START_REF] Cecere | Undaria pinnatifida (Fucophyceae, Laminariales) spread in the central Mediterranean: Its occurrence in the Mar Piccolo of Taranto (Ionian Sea, southern Italy)[END_REF][START_REF] Curiel | Distribution of introduced Japanese macroalgae Undaria pinnatifida, Sargassum muticum (Phaeophyta) and Antithamnion Pectinatum (Rhodophyta) in the Lagoon of Venice[END_REF][START_REF] Curiel | The introduced alga Undaria pinnatifida (Laminariales, Alariaceae) in the lagoon of Venice[END_REF]. Other factors such as chlorophyll concentration and shading have also been shown to be important for the growth of the species [START_REF] Báez | Macro-environmental modelling of the current distribution of Undaria pinnatifida (Laminariales, Ochrophyta) in northern Iberia[END_REF].

The results in the current study represent a potential alternative explanation, for the lack of expansion in Mediterranean populations, through disturbance to the life cycle schedule of the species. A delay of a few months in the predicted onset of gametogenesis under the warmer southern conditions could be a critical factor in reducing the population's ability to settle and establish itself opportunistically in the natural habitat. This also agrees with theoretical studies that have indicated macroalgal species with heteromorphic life cycles, such as U. pinnatifida, may be better adapted to the more strongly seasonal environments of northern latitudes [START_REF] Bessho | Heteromorphic and isomorphic alternations of generations in macroalgae as adaptations to a seasonal environment[END_REF]. This could limit the species competitive ability to establish itself in lower latitudes even in cases where nutrient limitation is not a factor.

With regard to the northern range limits of U. pinnatifida in Europe, currently the most northerly reported population is in Northern Ireland where mean annual water temperatures are approximately 11-12 o C [START_REF] Minchin | The invasive brown alga Undaria pinnatifida (Harvey) Suringar, 1873 (Laminariales: Alariaceae), spreads northwards in Europe[END_REF]. This is within the tolerance limit for the sustainability of a population according to the model predictions (Fig. 3). This also agrees with other studies, using habitat suitability models, which have indicated that temperature may not be a limiting factor for its distribution in the British Isles (Yesson et al. 2015). In terms of gametogenesis, populations are also predicted to maintain the characteristic bi-annual peak in gametophyte fertility (Fig. 5). This means that it also has the potential to be an opportunistic invader through early recruitment.

Therefore, under this model, the environmental conditions are predicted to be suitable for the continued spread of the species as an invasive threat in the British Isles, particularly under scenarios of increasing sea water temperatures [START_REF] Mackenzie | Daily ocean monitoring since the 1860s shows record warming of northern European seas[END_REF]. However, expansion to waters with a mean annual temperature <10 o C will depend on the seasonal variation in the temperature. When the seasonal amplitude in water temperature is <3.5 o C, then the minimum temperature a population is predicted to survive in is >9.1 o C (Fig. 3a). However, in the case of a high variation between summer and winter temperatures (amplitude 5. These model results indicate that both direct and indirect effects on the life cycle schedule may determine the future expansion potential of U. pinnatifida in Europe and elsewhere. These types of subtle interactions are not as readily apparent when considering the invasion from a population perspective only. This modelling approach represents a promising framework that could be applied to other species of macroalgae with heteromorphic life histories in order to understand better their growth and life cycle dynamics and how these contribute to their potential distribution range under variable climatic conditions.

Future work will require a detailed analysis of the role of other factors such as nutrient limitations and biotic interactions versus direct and indirect effects of environmental factors on the life cycle schedule of U. pinnatifida in order to build a complete picture of the invasion dynamics.

This model is somewhat limited in that it focuses on seasonal changes in light, temperature and day length only. Nevertheless, the results raise interesting points about the role of changes in the life cycle schedule and seasonal cycle on the invasive potential of species such as U. pinnatifida.

Potential temporal changes in the life cycle schedule of introduced species outside their native range can have important impacts when it comes to accurately predicting their range limits and should be considered in future modelling studies. pinnatifida population in a simulated harbour environment versus field data from Brest harbour in Brittany, France [START_REF] Voisin | Les processus d'invasions biologiques en milieu côtier marin: le cas de l'algue brune Undaria pinnatifida, cultivée et introduite à l'échelle mondiale[END_REF]. Predicted values represent mean (±S.D.) from four simulated years.

Recruitment

Field Data Predicted R 2 = 0.98

  5 o C (6.0 -13.0 o C, in Southern Patagonia, Argentina, Lat. -47.75 Long. -65.92) and as high as 22.4 o C (16.8 -28.0 o C, in its native range of Japan, Lat. 31.85 Long. 130.15) (Dellatorre et al. 2014; Martin, Cuevas 2006; Saito 1975). When simulations were carried out with seasonal temperature conditions representative of these regions (amplitude = 3.5°C in Southern Patagonia, and 5.5°C in southern Japan), the model predicted temperature limits for growth were 9.1 o C (5.6 -12.6 o C) and 22.5 o C (17 -28 o C) respectively, which closely matches the actual range limits (9.5 & 22.4 o C) above (Fig 4a). In addition to the mean annual water temperature, the effect of the amplitude of the seasonal change in water temperature (between the warmest and coolest months of the year) was explored in more depth. The results indicate that the theoretical thermal niche of U. pinnatifida increases as a function of the amplitude of the seasonal change in temperature (Fig. 3b).

  5 o C), it is possible that populations of U. pinnatifida could survive in conditions where the mean annual temperature is as low as 8 o C. In this case, other factors such as light or nutrient availability might become more important limiting factors.

Fig. 1 :

 1 Fig. 1: Model predicted values for (a) relative abundance and (b) relative recruitment of an U.

Fig. 2 :

 2 Fig. 2: (a) Predicted age to maturity and months mature for U. pinnatifida sporophytes compared to

Fig. 3 :Fig. 4 :Fig. 5 :Fig. 6 :

 3456 Fig. 3: (a) Predicted rate of population expansion (log increase in annual recruitment) for U.

Table 1 :

 1 Environmental input parameters for three locations representing the current known geographic range of Undaria pinnatifida in European coastal waters. North = Northern Ireland; Central = Brittany (Brest Harbour), France; South = Mediterranean Sea (Taranto, Italy). Global solar irradiance data from CALSOL online application (Institut National de L'Energie Solaire, CEA-CNRS) (INES 2015). Day length calculated based on sunrise/sunset records from the U.S. Naval Observatory's Astronomical Applications Department (USNO 2015). Temperature data: Met Eireann (Irish National Meteorological Service), SOMLIT (Service d'Observation en Milieu Littoral, INSU-CNRS), Dellatorre et al. (2014).

	Location	Latitude	Temperature ( o C)	Irradiance	Day Length (h)
		(Degrees N)			(MJ m -2 day -1 )
			Mean	Amplitude Mean Amplitude Mean Amplitude
	North	59.6	11.17	3.26	9.1	8.65	12.3	4.82
	Central	48.4	13.33	4.14	11.36	9.05	12.3	3.77
	South	40.4	19.8	6.3	15.1	8.8	12.2	3

Acknowledgements

This research is supported by an Irish Research Council ELEVATE international career development fellowship, co-funded by Marie Curie Actions under the European Union's Seventh Framework Programme. The field data presented here, which were collected in Brest, were obtained as part of the PhD thesis of Marie Voisin who benefitted from a Ph.D. fellowship ("Renouvellement des Compétences" Program) from the Region Bretagne. MV and FV are thankful to the many people from the Department AD2M of the Station Biologique de Roscoff who provided help for the surveys carried out in the field.

was calculated based on sunrise/sunset records from the U. S. Naval Observatory's Astronomical Applications Department (USNO 2015).

Predicted temperature range of U. pinnatifida

In order to explore the role of temperature limitations on the potential expansion of U. pinnatifida in Europe, a series of simulations to predict the relationship between mean water temperature and annual population growth were carried out. The annual population growth rate is defined as the change in the (log) total annual recruitment per year. In this case, factors such as competition and space or nutrient limitation were ignored in order to focus on the impact of temperature. A series of simulations (>150) were carried out to predict how the population growth rate changes over a range of temperatures (7.5-23 o C) and different amplitudes in the seasonal variation of the temperature (1-8 o C). This data was used to calculate the temperature ranges for survival of the species, i.e. where the population growth rate is predicted to be greater than zero.

Case studies: Effect of environmental conditions on life cycle schedule

Following the high level analysis, a number of case study simulations were carried out to explore the effects of environmental conditions on the life cycle schedule of the species. Three test case scenarios ("North", "Central", and "South") were proposed that represent a range of environmental conditions to which populations may be exposed along a latitudinal gradient in Europe. These test cases are characterised by differences in the temperature, irradiance and day length regimes. All other parameters are maintained constant (e.g. biotic parameters of the algae, and other factors such as depth, light attenuation factor etc.). See Table 1 for a comparison of the environmental input parameters that were varied between the three test case scenarios.


Damien Chablat
email: damien.chablat@irccyn.ec-nantes.fr

Philippe Wenger

The Kinematic Analysis of a Symmetrical Three-Degree-of-Freedom Planar Parallel Manipulator

Keywords: Parallel manipulator, Singularity, Aspect, Assembly modes, Working modes

Presented in this paper is the kinematic analysis of a symmetrical three-degree-of-freedom planar parallel manipulator. In opposite to serial manipulators, parallel manipulators can admit not only multiple inverse kinematic solutions, but also multiple direct kinematic solutions. This property produces more complicated kinematic models but allows more flexibility in trajectory planning. To take into account this property, the notion of aspects, i.e. the maximal singularity-free domains, was introduced, based on the notion of working modes, which makes it possible to separate the inverse kinematic solutions. The aim of this paper is to show that a non-singular assembly-mode changing trajectory exist for a symmetrical planar parallel manipulator, with equilateral base and platform triangle.

I. INTRODUCTION

OR two decades, parallel manipulators have attracted the attention of more and more researchers who consider them as valuable alternative for robotic mechanisms [START_REF] Asada | Robot Analysis and Control[END_REF][START_REF] Fu | Robotics: Control, Sensing, Vision, and Intelligence[END_REF][START_REF] Craig | Introduction to Robotics: Mechanics and Control[END_REF]. As stated by a number of authors [START_REF] Tsai | Robot Analysis, The Mechanics of Serial and Parallel Manipulators[END_REF], conventional serial kinematic machines have already reached their dynamic performance limits, which are bounded by high stiffness of the machine components required to support sequential joints, links and actuators. Thus, while having good operating characteristics (large workspace, high flexibility and manoeuvrability), serial manipulators have disadvantages of low precision, low stiffness and low power. Also, they are generally operated at low speed to avoid excessive vibration and deflection. Conversely, parallel kinematic machines offer essential advantages over their serial counterparts (lower moving masses, higher rigidity and payload-to-weight ratio, higher natural frequencies, better accuracy, simpler modular mechanical construction, possibility to locate actuators on the fixed base) that should lead to higher dynamic capabilities. However, most existing parallel manipulators have limited and complicated workspace with singularities, and highly nonisotropic input/output relations [START_REF] Stewart | A Platform with Six Degrees of freedom[END_REF]. Hence, the performances may significantly vary over the workspace and depend on the direction of the motion.

A well-known feature of parallel manipulators is the existence of multiple direct kinematic solutions (or assembly modes). That is, the mobile platform can admit several positions and orientations (or configurations) in the workspace for one given set of input joint values [START_REF] Merlet | Parallel robots[END_REF]. The dual problem arises in serial manipulators, where several input joint values correspond to one given configuration of the end-effector. To cope with the existence of multiple inverse kinematic solutions in serial manipulators, the notion of aspects was introduced [START_REF] Borrel | A study of manipulator inverse kinematic solutions with application to trajectory planning and workspace determination[END_REF]. The aspects were defined as the maximal singularity-free domains in the joint space. For usual industrial serial manipulators, the aspects were found to be the maximal sets in the joint space where there is only one inverse kinematic solution. Many other serial manipulators, referred to as cuspidal manipulators, were shown to be able to change solution without passing through a singularity, thus meaning that there is more than one inverse kinematic solution in one aspect. New uniqueness domains have been characterized for cuspidal manipulators [START_REF] Ph | A new general formalism for the kinematic analysis of all nonredundant manipulators[END_REF], [START_REF] Omri | Analyse Géométrique et Cinématique des Mécanismes de Type Manipulateur[END_REF]. A definition of the notion of aspect was given by [START_REF] Ph | Uniqueness Domains in the Workspace of Parallel Manipulators[END_REF] for parallel manipulators with only one inverse kinematic solution. These aspects were defined as the maximal singularity-free domains in the workspace. A second definition was given by [START_REF] Chablat | Working Modes and Aspects in Fully-Parallel Manipulator[END_REF] for parallel manipulators with several inverse kinematic solutions. These aspects were defined as the maximal singularity-free domains in the Cartesian product of the workspace with the joint space. However, it was shown in [START_REF] Innocenti | Singularity-free evolution from one configuration to another in serial and fullyparallel manipulators[END_REF] that it is possible to link several direct kinematic solutions without meeting a singularity, thus meaning that there exists cuspidal parallel manipulators. This property was found for particular links lengths. However, [START_REF] Mcaree | An explanation of Never-Special Assembly Changing Motion for 3-3 Parallel Manipulators[END_REF] conjectured that such properties cannot exist for symmetrical parallel manipulator. The aim of this paper is to show that a symmetrical 3-DOF planar parallel manipulator can change assembly mode without meeting a singularity. We mean by symmetrical, a manipulator with equilateral base and platform triangles. This paper is organized as follows. Section II describes the planar 3-RRR parallel manipulator studied, which is used all along this paper. Section III recalls the notion of aspect for parallel manipulators. A non-singular assembly-mode changing trajectory is shown for the symmetrical planar parallel manipulator. The workspace and the generalized F aspects are calculated using octree models.

II. PRELIMINARIES

A. Parallel manipulator studied

The manipulator under study is a planar three-dof manipulator comprising three parallel RRR chains shown in Fig. 1. This manipulator is used to illustrate the example in this paper. This manipulator has frequently studied, in particular in [START_REF] Merlet | Parallel robots[END_REF][START_REF] Borrel | A study of manipulator inverse kinematic solutions with application to trajectory planning and workspace determination[END_REF][START_REF] Ph | A new general formalism for the kinematic analysis of all nonredundant manipulators[END_REF][START_REF] Omri | Analyse Géométrique et Cinématique des Mécanismes de Type Manipulateur[END_REF][START_REF] Ph | Uniqueness Domains in the Workspace of Parallel Manipulators[END_REF][START_REF] Chablat | Working Modes and Aspects in Fully-Parallel Manipulator[END_REF][START_REF] Innocenti | Singularity-free evolution from one configuration to another in serial and fullyparallel manipulators[END_REF][START_REF] Mcaree | An explanation of Never-Special Assembly Changing Motion for 3-3 Parallel Manipulators[END_REF][START_REF] Angeles | Fundamentals of Robotic Mechanical Systems[END_REF][START_REF] Gosselin | The Optimum Kinematic Design of a Planar Three-Degree-of-Freedom Parallel Manipulator[END_REF]. C 3 ) lie at the corners of an equilateral triangle, whose geometric center is 0 (respectively P). Moreover,

A 1 A 2 C 2 C 3 C 1 A 3 B 2 B 3 B 1 O l 1 l 2 l 3 m 2 m 3 P m 1  2  2 x y
1 2 3 6 l l l l     , with l i denoting the length of A i B i , 1 2 3 6 m m m m     , with m i denoting the length of B i C i , 1 2 3 10 r r r r     , with r i denoting the length of A i O i and 1 2 3

s s s s

    , with s i denoting the length of C i P, in units of length that need not be specified in the paper.

B. Kinematic Relations

The velocity p  of point P can be obtained in three different forms, depending on which leg is traversed, namely,

= ( ) + ( ) ( ), [1,3] i i i i i i i i         p E b a E c b E p c    (1)
with matrix E, 01 10

    

E

We would like to eliminate the three idle joint rates 1   , 2   and 3

  from eq. ( 1), which we do by dot-multiplying eq. ( 1) by ()

T ii  cb, thus obtaining ( ) =( ) ( )+( ) ( ), [1,3] T T T i i i i i i i i i i i        c b p c b E b a c b E p c    (2)
Equation ( 2) can now be cast in vector form, namely,  

1 2 3
with and

T T          At Bρ t p ρ        (3)
with ρ  thus being the vector of actuated joint rates. Moreover, A and B are, respectively, the direct-kinematics and the inverse-kinematics matrices of the manipulator, defined as

1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) TT TT TT                   c b c b E p c A c b c b E p c c b c b E p c (4a) 1 1 1 1 2 2 2 2 3 3 3 3 ( ) ( ) 0 0 0 ( ) ( ) 0 0 0 ( ) ( ) T T T           c b E b a B c b E b a c b E b a (4b)

C. Singularities

For the planar manipulator studied, such configurations are reached whenever the axes B 1 C 1 , B 2 C 2 and B 3 C 3 intersect (possibly at infinity), as depicted in Fig. 2. In the presence of such configurations, the manipulator cannot resist any torque applied at the intersection point I.

A 1 A 2 C 2 C 3 C 1 A 3 B 2 B 3 B 1 P I Figure 2: Example of parallel singularity
For the manipulator under study, the serial singularities occur whenever ( ) ( )

T i i i i i i lm    b a c b
for at least one value of i, as depicted in Fig. 3 for i = 1, i.e. whenever the points A i , B i , and C i are aligned.

D. Working modes

The notion of working modes was introduced in [START_REF] Chablat | Working Modes and Aspects in Fully-Parallel Manipulator[END_REF] for parallel manipulators with several solutions to the inverse kinematic problem and whose matrix B is diagonal. A working mode, denoted i Mf , is the set of mechanism configurations for which the sign of jj B ( 1, , jn   for a parallel manipulator with n degrees of freedom) does not change and jj B does not vanish. A mechanism configuration is represented by the vector (X, q), which permits us to locate the mobile platform as well as all the links..

( , ) such that sign( )=cst for 1, , and det( )

0 jj i X q W Q B j n Mf B        



Therefore, the set of working modes ( i Mf , jI  ) is obtained while using all permutations of sign of each term jj B . The manipulator under study has eight working modes, as depicted in Fig. 4, that we call now (a), (b), ..., (h). Each working mode is defined according to the sign of jj B as is given is in table 1. According to each working mode, the parallel singularity locus changes in the workspace, as shown in Fig. 5. In generally, the ability of a parallel manipulator to change its inverse kinematic solution depends on the bounds in the passive and actuated joints. This problem is not taken into account in our study since unlimited joints are assumed.

A 1 A 2 C 2 C 3 C 1 A 3 B 2 B 3 B 1 P Figure 3: Example of serial singularity when 1 A , 1 B and 1 C are aligned (g) (h) (e) (f) (c) (d) (a) (b)
A 1 A 2 C 2 C 3 C 1 A 3 B 2 B 3 B 1 P A 1 A 2 C 2 C 3 C 1 A 3 B 2 B 3 B 1 P Figure 5:
The same platform configuration with two joint configurations (singular on the left and none singular on the right)

E. Octree Models Octree models are hierarchical data structures based on recursive subdivision of the space, respectively [START_REF] Meagher | Geometric Modelling using Octree Encoding[END_REF]. They are useful for representing complex 3 dimensional shapes like workspaces [START_REF] Ph | Uniqueness Domains in the Workspace of Parallel Manipulators[END_REF]. A close method is used in [START_REF] Chablat | Workspace Analysis of the Orthoglide using Interval Analysis[END_REF] that divides the workspace into boxes. This method does not use recursive subdivision but interval analysis methods [START_REF] Merlet | ALIAS: an interval analysis based library for solving and analyzing system of equations[END_REF] to build the dextrous workspace. However, it does not make it possible to perform Boolean operations or to make path-connectivity analysis easily. The first method permits us to calculate easily all kind of space and the computing time is limited as a function of the accuracy. The second one is more exact but requires more ability to be implemented. In both cases, we can characterize spaces whose dimensions are either lengths or angles. Since the structure of the octree model has an implicit adjacency graph, path-connectivity analyses and trajectory planning can be carried out naturally. The optimal construction method of a 2 k -tree is derived from the shape, which recalls the tree. The most interesting approach consists in testing successively all the nodes present in the maximal depth, following an order of numbering which quickly allows nodes to be grouped and thus, simplifies the 2 k -tree. The order of numbering for this algorithm is based on Morton's sweeping [START_REF] Morton | A Computer Oriented Geodetic Data Base and a new Technique in File Sequencing[END_REF]. The inverse or direct kinematic model is used to calculate 2 k -tree. The figure 6 represents the octree model of its Cartesian workspace where the first and the second axis represent the position and the third axis the orientation of the mobile platform.

III. WORKSPACE ANALYSIS

A. Aspect definitions

The notion of aspect was introduced by [START_REF] Borrel | A study of manipulator inverse kinematic solutions with application to trajectory planning and workspace determination[END_REF] to cope with the existence of multiple inverse kinematic solutions in serial manipulators. Recently, the notion of aspect was defined for parallel manipulators with only one inverse kinematic solution to cope with the existence of multiple direct kinematic solutions [START_REF] Ph | Uniqueness Domains in the Workspace of Parallel Manipulators[END_REF] and for parallel manipulators with multiple inverse and direct kinematic solutions (the generalized aspects [START_REF] Chablat | Working Modes and Aspects in Fully-Parallel Manipulator[END_REF]). For the manipulator studied, we use the second definition. The generalized aspects ij A are defined as the maximal sets in WQ  so that

 ij A W Q ;  ij A is connected;    ( , ) such that det( ) 0 ij i A X q Mf    A
In other words, the generalized aspects ij A are the maximal singularity-free domains of the Cartesian product of the reachable workspace (called W) with the reachable joint space (called Q). The projection of the generalized aspects onto the workspace yields the parallel aspects ij WA so that,

 ij WA W  ;  ij WA is connected.
The parallel aspects are the maximal singularity-free domains in the workspace for one given working mode. The projection of the generalized aspects onto the joint space yields the serial aspects ij QA so that,

 ij QA Q  ;  ij QA is connected.
The serial aspects are the maximal singularity-free domains in the joint space for one given working mode. For each working mode, there exists, at least, one aspect where det( ) A is positive and another one where det( ) A is negative. However, such regions can be disjoint. In table 2, we associated the aspects with a working mode for which det( ) A is positive. For the working mode (a), there exist four aspects and for the other ones, there is only one aspect. Due to the symmetrical properties of the mechanism, there exist also 11 aspects where det( ) The calculation of the generalized aspects can be performed by using a 6 2 octree model or two octree models. We use the second method. The first one is the projection of the generalized aspect onto the reachable workspace and the second one the projection onto the reachable joint space for a given working mode and a given sign of det( ) A constant. To obtain these results with an accuracy of 0.093 for the position and 1,4 degrees for the orientation, the computing times is 90 seconds with an AMD Athlon XP processor 2500 + and the maximum memory used is 180 Mb. The connectivity analysis of each domain requires 20 seconds.

A

B. Non-singular posture changing trajectories

In [START_REF] Innocenti | Singularity-free evolution from one configuration to another in serial and fullyparallel manipulators[END_REF], a non-singular posture changing trajectory was found for a 3-RPR planar manipulator. However, it appears that this trajectory passes close to a singular configuration. This property was confirmed in [START_REF] Ph | Uniqueness Domains in the Workspace of Parallel Manipulators[END_REF] for the same manipulator and for a 3-RRR planar manipulator with non-symmetrical geometry [START_REF] Chablat | Les Domaines d'Unicité des Manipulateurs Pleinement Parallèles[END_REF]. According to the assumption in [START_REF] Mcaree | An explanation of Never-Special Assembly Changing Motion for 3-3 Parallel Manipulators[END_REF], we can think that such properties may not exist for the mechanism studied. However, we shown that a non-singular configuration changing trajectories exists. For the following input joint values:

  

Four direct kinematic solutions are found (Figure 15 and Table 3). We notice that solutions 1 and 4 are in the same generalized aspect (The parallel aspect associated is depicted in the figure 8).

Posture (1) Posture ( 2)

Posture ( 3) Posture (4) Figure 15: The four direct kinematic solutions for 1 We find a non-singular continuous trajectory between postures (1) and ( 4) by passing through an intermediate posture (5) (Figure 16) whose position is (-0.987; 1.930) and orientation is 12.35 degrees. Between these three postures, a linear interpolation is defined to stay in the same generalized aspect. The values of det(A), [START_REF] Chablat | Working Modes and Aspects in Fully-Parallel Manipulator[END_REF] B , 22 B and 33 B are evaluated and each value of these indices is normalized by its maximum value as it is shown in figure 17.

C. Characteristic surfaces

To separate the direct and inverse kinematic solutions, the uniqueness domains are determined for the parallel manipulator with one inverse kinematic solution in [START_REF] Ph | Uniqueness Domains in the Workspace of Parallel Manipulators[END_REF] and for parallel manipulator with several inverse kinematic solutions in [START_REF] Chablat | Les Domaines d'Unicité des Manipulateurs Pleinement Parallèles[END_REF]. The boundaries of the uniqueness domains are defined by the characteristic surfaces [START_REF] Chablat | Les Domaines d'Unicité des Manipulateurs Pleinement Parallèles[END_REF]. For the generalized aspect (b), we can compute the characteristic surface that permits us to isolate the assembly modes where it is possible to realize non-singular assembly mode changing trajectories (Figure 18).

IV. SUMMARY AND CONCLUSIONS

A kinematic analysis of a planar 3-RRR parallel manipulator with symmetrical properties was presented in this paper. The eight working modes have been characterized and 22 generalized aspects have been found out. Inside such domains, any continuous trajectories are possible. In such domains, there are non-singular changing trajectories but not any kinematic index can recognize such property. An example of non-singular changing trajectory is given and the characteristic surface are computed which permit, in a future works, to define closely the uniqueness domains of the manipulator studied.

Figure 1 :

 1 Figure 1: The 3-RRR parallel manipulator studied The actuated joint variables are the rotation of the three revolute joints located on the base ( 1 2 3 ,,    ). The Cartesian variables are the coordinate ( , ) xy of the operation point P and the orientation  of the platform. The passive and actuated joints will always be assumed unlimited in this study. Points A 1 , A 2 and A 3 , (respectively C 1 , C 2 and C 3 ) lie at the corners of an equilateral triangle, whose geometric center is 0 (respectively P). Moreover,

Figure 4 :Figure 4

 44 Figure 4: The eight working modes Figure 4 (a) (b) (c) (d) (e) (f) (g) (h) 11 B P P P P N N N N 22 B P N P N N P N P 33 B P P N N P P N N

Figure 6 :

 6 Figure 6: The Cartesian workspace

  is negative. Working modes (a) (b) (c) (d) (e) (f) (g)

Figure 7 :Figure 8 :Figure 9 :Figure 10 :Figure 11 :Figure 12 :Figure 13 :Figure 14 :

 7891011121314 Figure 7: The four parallel aspects for the working mode (a) and det( ) 0  A

Figure 16 :Figure 17 : 33 B

 161733 Figure 16: The intermediate posture (5) for the non-singular changing trajectoryWith this result, we have proofed that a non-singular assembly mode trajectory is possible for a symmetrical planar 3-RRR parallel manipulator. Inside such trajectory, not any kinematic index, derived from the Jacobian matrices, permits us to recognize such property.

Figure 18 :

 18 Figure 18: The parallel singularities and the characteristic surfaces associated with the generalized aspect (b)

Table 1 :

 1 The eight working modes of the manipulators studied with N (resp. P) denoted negative values of

ii B

(resp. positive values)

Table 2 :

 2 The projection of the generalized aspects on the workspace when

	det( ) 0  A	for each working mode

  =5.862610

					,
	23 =1.277470, = 5.213885 
	Posture N°	x	y	 in degrees
	(1)	1.102	1.956	57.50
	(2)	0.705	2.751	46.85
	(3)	4.638	-5.413	32.35
	(4)	-0.357	2.720	26.51

Table 3 :

 3 Four direct kinematic solutions for the same joint values

	A first method to confirm this property is to evaluate the
	determinant of A and ii B (Table 4) and to find out a trajectory
	between these two postures.
	det(A)	11 B	22 B	33 B
	Posture (1) 307.990	-34.132	-34.008	31.827
	Posture (4) 522.868	-33.023	-35.997	21.7203

Table 4 :

 4 Evaluation ofdet( )

	A and ii B for the two pose in the same
	generalized aspect


Toufik Guettari

Dan Istrate

Nesma Houmani

Jérôme Boudy

Suitable segmentation method of a thermal signal representing sleep movement in the bed

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