Distributed competition to compute localized scheduling

Résumé : L'usage de protocoles MAC à endormissement cyclique permet un allongement de la durée de vie des réseaux de capteurs. Cet accroissement de la durée de vie du réseau est obtenue au dépend de la capacité en traffic des réseaux et entraîne une mauvaise adaptabilité aux charges de traffic variables, en particulier les traffics en rafale. En effet, les mécanismes de contention existants ne permettent pas une allocation dynamique et efficace de plusieurs paires de communications(récepteur/émetteur). Dans cet article, le médium est partagé en plusieurs canaux logiques (par exemple:division en temps/fréquence) et nous proposons d'allouer dynamiquement ces ressources aux noeuds en faisant la demande. A ces fins, nous proposons un algorithme, Extended Slot Selection (ESS), combinant contention et ordonnancement, qui alloue plusieurs paires émetteur/récepteur aux canaux logiques disponibles.

Mots-clés : partage de médium, contention, allocation localisée. , [START_REF] Ye | Medium access control with coordinated adaptive sleeping for wireless sensor networks[END_REF], [START_REF] Rhee | Z-MAC: a hybrid MAC for wireless sensor networks[END_REF], [START_REF] Bachir | Micro-frame preamble mac for multihop wireless sensor networks[END_REF], [5], [6]. However, existing localized contention algorithms for MAC protocols only allow for the selection of a single sender. In LPL protocols, this often implies that a single transceiver/receiver pair is allowed to communicate in a given wake-up period (as multiple contention phases are deemed too expensive). Consequently, the bandwidth utilization is very limited as shown in fig. 1 and the latency is in the order of several sleep periods for congested networks. This cripples WSN performances and advocates for algorithms that permit to simultaneously select multiple sender nodes and allocates them to the available logical channels. This paper proposes a localized medium sharing algorithm using a joint contention/scheduling scheme and is organized as follows: section 2 exposes the addressed problematic and the assumptions made in this work, section 3 reviews existing medium sharing algorithms, section 4 exposes our proposal and its design, section 5 presents simulations that assess the benefits of our proposal, section 6 investigates the energy cost of the proposal and its adaptability to dynamic topologies. Finally, section 7 unveils future work and summarizes this article.

Problem statement

As stated in sec. 1 and illustrated by fig. 1, existing duty-cycled protocols present poor bandwidth utilization and poor latency since they allocate only one logical channel per contention phase and because the wake-up period is much larger than the typical packet duration. We believe that alleviating this limitation is equivalent to answering the following problem: Given N nodes competing to access the medium, how to share the medium such that N' nodes get dedicated logical channels for their transmissions?

Requirements

Considering WSNs specificities, the solution should exhibits the following properties: p 1 it must allow for the scheduling of multiple transmitter/receiver pairs in a given wake-up period.

p 2 dimensioning of the algorithm must only depend on traffic load, not on network properties such as diameter and degree.

p 3 it must only rely on localized information (localized algorithm).

p 4 it must self-adapt to fast-varying, bursty traffic.

p 5 it must cope with network dynamicity.

p 6 it must grant fair access among nodes.

Assumptions

Most WSN applications strongly rely on network synchronization. For example, in metering applications as in urban networks [START_REF] Dohler | Routing requirements for urban low-power and lossy networks[END_REF] or industrial networks [START_REF] Pister | Industrial routing requirements in low-power and lossy networks[END_REF], data relevance is often limited to a given time-frame and outdated packets are dropped for energy and congestion considerations. Such mechanisms compel packet sources and all the forwarding nodes to share a common time reference so as to compare timestamps to their clock, thus requiring network synchronization. Network synchronization also allows for various optimizations such as synchronous sleep schedules that shorten wake-up guard times or time-spread transmission schedules for congestion mitigation. Therefore, we will make the assumption that a network time synchronization mechanism such as [START_REF] Sommer | Gradient clock synchronization in wireless sensor networks[END_REF] or [START_REF] Maróti | The flooding time synchronization protocol[END_REF] is operating on the network.

Related work

Although quite different in nature, access control algorithms can be classified into three categories regarding the way nodes compete to access to the medium: deterministic, random and hybrid access.

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Deterministic access algorithms

Deterministic access relies on pre-established schedules of node emissions such as in [START_REF] Pister | TSMP: Time synchronized mesh protocol[END_REF]: each wake-up period is dedicated to a specific node providing it with a collision free time slot. However, these algorithms poorly adapt to unpredictable, bursty traffics and to network changes or imply frequent re-schedules, at high energy cost. In order to mitigate this poor scalability, one could also think of extending the scheduling algorithm to address multiple logical channels in a single wake-up period. However, this would force nodes to listen to all available logical channels which is either not feasible (eg: multiple frequency channels at once) or extremely energy inefficient. Therefore, these algorithms meet the key properties p 1 and p 6 but fail to achieve p 2 , p 3 p 4 and p 5 .

Random access algorithms

Random access relies mostly on CSMA algorithms and can also be sub-divided into two sub-types: unsynchronized and synchronized algorithms.

Unsynchronized algorithms

Unsynchronized algorithms such as described in [START_REF] Polastre | Versatile low power media access for wireless sensor networks[END_REF], [START_REF] Bachir | Micro-frame preamble mac for multihop wireless sensor networks[END_REF] and [5] are based on the preamble sampling algorithm. Nodes wake up periodically but at different times, the period being often the same for all nodes. When a node has data to send, it first listens to the channel: if it's idle, the node transmits either a long preamble whose size is larger than the wake-up period as in [START_REF] Polastre | Versatile low power media access for wireless sensor networks[END_REF] or send a sequence of short preambles until the destination wakes up and responds to the preamble [START_REF] Bachir | Micro-frame preamble mac for multihop wireless sensor networks[END_REF], [5]. If the medium is occupied, the node backs off and retries later. Thus, once a node initiates the algorithm by starting to send preambles, it inhibits other senders, that delay their transmissions. Therefore, this contention algorithm allows only for a single sender/receiver pair which violates desired key property p 1 .

Synchronized algorithms

Synchronized algorithms benefit from a shared knowledge of time by setting rendez-vous points in time for congestion resolution algorithms. A widely-used algorithm in Low Power MAC protocols relying on nodes synchronization consists in the usage of fixed contention windows to grant access to the available channel as described in [START_REF] Jamieson | Sift: A MAC protocol for event-driven wireless sensor networks[END_REF], [START_REF] Zakhia | CONTI: Constant-time contention resolution for WLAN access[END_REF] and [START_REF] Galtier | Graphs and Algorithms in Communication Networks: Studies in Broadband, Optical, Wireless, and Ad Hoc Networks[END_REF]. The fixed contention window algorithm involves a fixed-size time frame subdivided into K time slots. Nodes willing to transmit on the medium, called contending nodes, will pick one or more of these slots and mark them by an occupancy signal, possibly random sequences of symbols. When not transmitting, nodes listen to check for occupancy of the other slots. Then, the decision to transmit is computed locally at each node based on its choices and other occupied slots. Therefore, these algorithms do not exhibit the key property p 1 but enforce p 2 , p 3 , p 4 , p 5 and p 6

1 . As described in sec. 4, our proposal relies on this kind of algorithm so we deem necessary to review them here. Single slot choice algorithms In congestion algorithms such as SMAC [START_REF] Ye | Medium access control with coordinated adaptive sleeping for wireless sensor networks[END_REF] and SIFT [START_REF] Jamieson | Sift: A MAC protocol for event-driven wireless sensor networks[END_REF], nodes only pick one slot that they mark with an occupancy signal while they listen during the others. Node(s) occupying the first nonempty contention slot decide(s) to transmit, the others pass. Therefore, in fig. 2, node n1 decides to transmit in the emission slot while n2, n3 and n4 pass, hence the grayed chosen slot.

Longest burst algorithms

The HIPERLAN [START_REF]Standard. HIgh PErformance Radio Local Area Network(HIPERLAN) Type 1; Functional Specification[END_REF] protocol defines a congestion resolution algorithm based on a longest bursts policy. Contending nodes pick an integer number i in the [1,CW]2 interval that defines their respective burst length and occupy the medium in the i first slots of the contention window. At the end of their own burst, nodes listen for other nodes' bursts. If the channel is idle until the end of the window, a node decide to transmit otherwise it passes. This is also equivalent to: the node(s) with the longest burst decide(s) to transmit while the others pass. Fig. 3 illustrates this kind of policies.

Binary countdown algorithms

The CONTI protocol [START_REF] Zakhia | CONTI: Constant-time contention resolution for WLAN access[END_REF] and its improvment [START_REF] Galtier | Graphs and Algorithms in Communication Networks: Studies in Broadband, Optical, Wireless, and Ad Hoc Networks[END_REF] consider the K slots of the contention window as K rounds of selection. At the first slot, each contending node emits an occupancy signal with a probability p. When a node does not transmit, it listens for other signals: if there is one, it withdraws from the contention process otherwise it participates to the Figure 4: binary countdown algorithms next round, and reiterates until the end of the contention window. Fig. 4 gives an example of such a contention algorithm.

Hybrid algorithms

Hybrid algorithms such as [START_REF] Ye | Medium access control with coordinated adaptive sleeping for wireless sensor networks[END_REF], [START_REF] Rhee | Z-MAC: a hybrid MAC for wireless sensor networks[END_REF], [START_REF] Sitanayah | Emergency response mac protocol (ER-MAC) for wireless sensor networks[END_REF] use a mixed TDMA-CSMA scheme: a slot assignment is first performed to allocate dedicated time slots to nodes as in TDMA algorithms. Then, if a slot owner does not use it, a random access algorithm takes place between nodes that need access to the medium. This way, such algorithms provide a more adaptive solution to the problem than purely deterministic algorithms. However, the dimensioning of the time slots and the time-frame remains problematic. In order to re-allocate slots that are not used by their owners, time-slots must be long enough to embed a contention window. Time-frames must also provide a dedicated slot to each node or face fairness issues. Therefore, hybrid algorithms fail to meet p 2 , p 5 or p 6 .

Existing solutions and adequacy to requirements

As seen previously, existing deterministic contention algorithms do not meet p 2 , p 3 , p 4 and p 5 whereas random access algorithms fail to meet p 1 . Finally hybrid algorithms do not achieve p 2 , p 5 or p 6 . Therefore, none of the reviewed algorithms satisfies our requirements, thus prompting for a new algorithm that satisfies them all.

4 Proposed algorithm

Key Concept of the algorithm

In existing fixed congestion window algorithms, the decision process relies on the information that another node marked a preceding slot in the contention window. This leads to binary decisions, ie. transmit or go back to sleep. We propose to generalize this decision process by using the information on the number of occupied slots preceding current node's slot. This number is then used to build a localized scheduling of the nodes to the available logical channels. More specifically, we propose to use a counter c that is managed locally at each node. This counter is used to keep track of the number of already occupied slots in the contention window, as seen by each node. Nodes emitting before the first increment of the counter decide to use the first logical channel, nodes that incremented the counter only once choose the second logical channel and so on until all the C log logical channels get used (see fig. 5 for more details).

Detailed algorithm

Our proposal relies on a 3-tiers extended selection algorithm called Extended Slot Selection (ESS) described below and illustrated by fig. 5. Contending nodes participate in tier 1 and 2 while non-sending nodes sleep. In tier-3, all nodes are awake.

tier 1

Similarly to single slot choice algorithms, contending nodes pick a slot i in a tier-1 contention window in which they emit an occupancy signal. When a node does not transmit it listens for other nodes' transmissions. For each slot j, j < i in which an occupancy signal has been heard, it increments a counter c of initial value 1.

tier 2

Each contending node whose counter c is less than a given threshold3 C log then competes in the c th tier-2 contention window. When a node loses in one Tier-2 contention round, it may participates to the next available Tier-2 round if any. Hopefully, at the end of tier-2, up to C log logical channels will be occupied by single nodes.

tier 3

The last step of this algorithm consists in the advertisement of packets' destinations by their senders. The node of the first logical channel will announce first, followed by the one of second logical channel and so on. This way, destination nodes will be able to deduce which logical channels they should listen to.

5 Performance evaluation: Simulation

Scenario and model

As depicted in [START_REF] Dohler | Routing requirements for urban low-power and lossy networks[END_REF] and [START_REF] Pister | Industrial routing requirements in low-power and lossy networks[END_REF], wireless sensor networks may present very dense node deployments and neighborhood of several hundreds sensors are to be expected. This performance evaluation will then consider a neighborhood of up to N max = 500 nodes that are all within communication range of one another. All nodes in the neighborhood will want to access the medium. Moreover, we consider that two nodes emitting simultaneously results in a collision at the receiver (no capture effect).

Notations

We consider a N nodes neighborhood. The tier-1 contention window is of size K 1 and the tier-2 contention window of size K 2 . C log logical channels are available for transmission.

Simulation environment

Simulations are performed using the WSNet Simulator [START_REF] Elyes | Impact of the physical layer modeling on the accuracy and scalability of wireless network simulation[END_REF]. Relevant simulator parameters are given in table

ESS parameter dimensioning

The SIFT protocol [START_REF] Jamieson | Sift: A MAC protocol for event-driven wireless sensor networks[END_REF] recommends a contention window of size 32 for neighborhoods up to 512 nodes. So as to exhibit the benefit of the ESS algorithm, we have chosen to restrict the number of slots per logical channel to N Slots log = 16, half of what is recommended in SIFT. For the simulations, we chose K 1 = 128, K 2 = 12 and C log = 32, that configuration having proven to be efficient4 .

Measurements

In order to evaluate the ESS algorithm, we have conducted measurements of the following indicators:

• Throughput : computed at the sink node. The packet size being constant, we consider the packet rate instead of the bit rate. We compute the average number of packets successfully sent per node. Obtained values are then compared to the theoretical maximum throughput for a C log = 32 logical channels and for a single logical channel medium access protocol such as SCP-MAC [6] or SIFT [START_REF] Jamieson | Sift: A MAC protocol for event-driven wireless sensor networks[END_REF].

• Collision rate: we keep track of the number of collisions and also compute the ratio #(packets.successfully.sent)/#(packets.sent).

• Fairness: illustrated by the computation of Jain's fairness index [START_REF] Rk Jain | The Art of Computer Systems Performance Analysis: Techniques for Experimental Design, Measurement, Simulation, and Modeling[END_REF].

Simulation results

Throughput

Figure 6: Throughput: simulation results Fig. 6 exposes the average number of packets received at the sink per sending node. Since the simulations last 1000 sleep periods, each node submits 1000 packets to the Medium Access Control protocol and up to 32 000 packets reach the sink node. Values obtained by simulation for the ESS algorithm are depicted by crosses. The upper curve represents the theoretical maximum average number of packets received at the sink per sending nodes for a C log = 32 logical channels medium. The lower curve provides the same value for a single logical channel. Simulations show that our proposal is near-optimal regarding the throughput with chosen C log , K 1 and K 2 and outperforms all protocols with a single logical channel allocation algorithm.

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Collisions

Fig. 7 illustrates the collision rate that was observed during the simulation. Vertical bars represent the number of lost packets due to collisions divided by the number of packets sent over all logical channels. With the chosen protocol parameters, this ratio is below 1.4% for all simulations and below 0.1% for in the 50 to 500 nodes range. Crosses report the raw number of collisions observed during the simulations and the line provides a linear approximation of the number of collisions. The correlation factor of 0,978 indicates that the collision rates follows a near linear progression for increasing neighborhood sizes in the range of interest.

Fairness

Each round of contention/selection is independent of another, therefore the algorithm must be totally fair when all nodes use the same parameters. The simulations confirmed it: the Jain index stayed above 0.99 in all scenarios.

Discussions

Energy

As stated in 1, energy efficiency is a major concern in Wireless Sensor Networks therefore we deemed necessary to evaluate the cost of ESS and compare it to SCP-MAC [6] that is known to perform well under the same assumptions.

no traffic

When there is no traffic, the ESS energy cost is due to the synchronization mechanism and to the channel sampling. Since the synchronization mechanism is out of scope and is used in both ESS and SCP-like protocols, we will just consider the latter. This process requires all nodes to wake up periodically and sample the medium for activity, which cost energy. ESS and SCP-MAC require a different number of samplings per wake-up period. In SCP-MAC, receiver nodes must sample each and every slot of the post wake-up contention window. In ESS, a receiver node goes back to sleep as soon as it has determined the first advertisement slot to be empty.

Light traffic

Non-zero traffic adds cost in the case of both protocols. In ESS, there is a cost implied at the senders that contend in the tier-1 and tier-2 contention and a cost at the receivers that must listen to the advertisement of the destinations. In SCP-MAC, similar costs happen at the transmitters for the pre and post wake-up contention windows and at the receivers to listen to the post wake-up contention window and the packet header. Again, ESS cost is either less or equivalent to that of SCP, depending on contention windows sizes.

Heavy traffic, congestion

In case of heavy traffic, ESS energy cost increases because receivers listen to more advertisements whereas SCP-MAC cost for receivers stays constant. However, this increase in cost is factored over the number of transmissions that will occur in the same period. Indeed, ESS is able to schedule up to C log concurrent communication pairs whereas SCP-MAC can only establish one. This means that SCP-MAC will need up to C log wake-up periods to handle the same amount of traffic, each period causing additional cost for waking up the radio. Again, in case of heavy traffic, ESS is more energy efficient than SCP-MAC.

Network dynamicity

WSNs often present varying topologies during their life-time, such as node additions or removals. As stated in 3.1, this can cripple MAC performances (energy/collision rate), especially for deterministic and hybrid ones. As it does not rely on a reservation scheme, ESS is insensitive to the node IDs present in a given neighborhood, ie: changing a node MAC address or exchanging nodes won't affect its performances. ESS, despite using fixed size congestion windows, performs also very well in a network with varying neighborhood densities. As section 5.4.2 shows, ESS collision rate varies only slightly while the load (number of contending nodes) dramatically increases. Being insensitive to nodes IDs, ESS is therefore able to cope with varying densities, thus making it suitable for realistic conditions.

Conclusion and future work

ESS is an algorithm that allows for a better usage of the medium capacity by allocating multiple logical channels in a single contention/allocation process. Simulations have shown that ESS exhibits very low collision rates and almost optimal throughput. ESS is also fair, energy efficient and scales very well for neighborhoods up to hundreds of nodes. Therefore, ESS fulfills all requirements described in section 2. We have proposed an algorithm and shown its efficiency by simulating a single neighborhood with increasing numbers of competing nodes. We are currently evaluating the same algorithm in multi-hop networks with realistic traffic patterns. We will also derive an optimal distribution for the first contention window and derive optimal K 1 /K 2 and C log values for a given maximum collision rate and population of nodes. This algorithm also offers interesting opportunities for QoS mechanisms that we will evaluate. Lastly, various optimizations can be envisioned on top of ESS.

Figure 1 :

 1 Figure 1: bandwidth utilization limitation

Figure 2 :Figure 3 :

 23 Figure 2: single slot choice algorithms

Figure 5 :

 5 Figure 5: Detailed algorithm

Figure 7 :

 7 Figure 7: Collisions: simulation results

Table 1 :

 1 simulation parameters

	1.
	propagation model	infinite range
	collision model	full
	simulation time	120000s
	wake-up period	120s
	guard time (between slots)	100µs
	slot time	450µs
	data	1packet/120s, constant size
	C log	32

Table 2 :

 2 1, as illustrated by table 2. Fulfillment of the requirements

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RR n°7522

under the assumption that all nodes share the same algorithm and parameters and that they do not depend on previous contention resultsRR n°7522

contention window slots number INRIA

a discussion will be provided in 5.2INRIA

the optimal ratio K 1 /K 2 and C log = 32 for a given collision rate and population N will be addressed in a future work.RR n°7522


E Piron

; D Miclet

; L Leveillé

; D Clochard

; S Villette

Mineral spreader eco-design: method and real application examples

Keywords: Centrifugal spreading, eco-design, spread pattern

This paper shows how information enclosed in the 3D spread pattern measurements obtained with the CEMIB test bench can be used to support scientifically spreader improvements. The main aspects to consider during this first step of improvement are described and explained using real examples. The importance of an adapted choice of mean projection distance is shown, as well as the necessary choice of working width ranges for each disc… Then specifications are expressed, as robustness evaluation of the transverse curves, since the beginning of the design. Relation with fertiliser characteristics is introduced, as well as the development of a spreading simulator. Results, mainly articulate around the SULKY X Range spreader design, highlight the technical choices made by SULKY, and the performances reached, as well for main field or border applications.

Introduction

Centrifugal spreaders used for mineral applications are nowadays relatively complex, equipped with electronic devices and computers. They generally use 2 different spinning discs turning in opposite directions, positioned under a hopper, to spread fertiliser granules on the ground after a ballistic flight. Because of their simplicity in the concept, they are robust regarding different fertiliser types, easy to use, very efficient as well considering spreading performances or working velocity, and finally cheap regarding all these advantages. Since the beginning of the use of this spreading process, many improvements have been proposed and developed: from the single spinning disc to the double discs, from simple mechanical and hand made settings to several electronic ones, with computer assistance directly useful from the cab. Many reasons guided these successive innovations, either to improve the spreading performances, i.e. the different distribution homogeneity, or to make the different functions easier to use. Advances have very often been made step by step, and directly linked to the sprayer working width improvements: spreaders tramlines are often the same than sprayers ones. In addition, regarding the distribution improvements, the only available development tool was the transverse bench, described in the EN13739-2 standard. Even if it represents a good way to evaluate real spreader performances, this method doesn't give real knowledges about the 3D spread pattern and its determinants. As the standard method integrates the spreader displacement parameter, it's almost impossible to go back to the 3D spread pattern. Nevertheless, this one is of major interest for the understanding of physical phenomena, for conception assistance, or to develop improvements. Improving spreaders is also a complex but very important procedure for manufacturers. Even if they built up experience and if the Computer-aided Design (CAD) helps them to simulate a global mechanical work, no method is really available to predict the most important aspect, i.e. the spreading distribution quality.

Nevertheless, alternative ways have been explored to reduce the number of needed real tests, and to better take into account the physical phenomena and their eventual interactions, i.e. mechanical disc and vane characteristics, fertiliser physical properties … A particularly interesting way is the use of Discrete Element Method (DEM) simulations, in particular presented by [START_REF] Van Liedekerke | A discrete element model for simulation of a spinning disc fertilizer spreader I. Single particle simulations[END_REF], 2009). A virtual spreader could be obtained, allowing the research of the best global organisation of all the spreading devices (vanes, drop zone shape and position, angular velocity …). At the present time, this method can be used to help the design of a particular piece. Nevertheless, firstly, calculation time necessary to simulate an entire spreader is too important (linked to the number of particles) regarding the number of different simulation cases which should be necessary to obtain a "real" virtual spreader. Secondly, the model needs to be adapted to real tests to give similar answers. When a manufacturer wants to improve a part of his spreader, or a new function, he writes specifications to achieve, and then would like to reach them with the maximum of reactivity and control. The aim of this paper is to show how the information enclosed in the 3D spread pattern can offer a more professional way to the manufacturers to improve centrifugal spreaders as well as fertiliser properties: from the first step of increased knowledge to the development step, and finally to the final checkings. Improvements have to be understood under the environmental aspects, i.e. in global goals of application defaults minimisation in the field. Due to economical reasons, for example to minimize global development costs or building ones, spreaders are designed around an identical base, as well for the smallest or the biggest ones. The only different pieces are the discs, the arrangement around the hopper … The paper is organised around the real development and improvement of the Sulky X Range spreader.

Methods and points to consider

This first part describes different sides of a scientific assisted development, allowed by the 3D spread pattern knowledge and the CEMIB (short for CEmagref MIneral Bench): what are the useful informations, which are the technical limitations to consider, and how to establish the link with a simulator.

The 3D spread pattern and the information enclosed

The mathematical analysis of a spread pattern obtained using a centrifugal disc has been performed in years 1990 to 2000, in particular by [START_REF] Colin | Etude du procédé d'épandage centrifuge d'engrais minéraux[END_REF] or [START_REF] Olieslagers | Fertilizer distribution modelling for centrifugal spreader design[END_REF]. They both looked how the fertiliser granules densities were distributed on the ground after being spread by a centrifugal disc. One significant issue has been to express this fertiliser density using two Gaussian equations. The first for the radial distribution (driven by the mean projection distance -P m -and its standard deviation -σ p ) and the second for the angular distribution (driven by θ m and σ θ ). As long as the spreader tests were carried out using the traditional standard EN13739-2 (EN 13739-2, 2003), it was almost impossible to extract this significant information, and so to fit the correct values in order to develop some spreading models.

The development of the Cemib, [START_REF] Piron | Procédé de mesure de la répartition d'épandage pour épandeur de particules[END_REF] deeply changed the way to analyse centrifugal spreading performances. In addition to the practical interest of this device (it allows testing all centrifugal spreaders whatever their performances in a very short test hall measuring 10m * 40m), the Cemib, patented in 2005 by the Cemagref [START_REF] Piron | Procédé de mesure de la répartition d'épandage pour épandeur de particules[END_REF], performs the measurement of the real 3D spread pattern, using a radial method for a radial phenomena. The spreader, coupled on a rotary carrier, spreads its fertiliser constantly while a radial positioned row of instrumented boxes allows a continuous measurement, angle by angle, of the distributed density [START_REF] Piron | Procédé de mesure de la répartition d'épandage pour épandeur de particules[END_REF].

In a very short time (a few seconds or minutes), and without great fertiliser amounts (between 20kg and 80kg regarding the flow rate), the fertiliser density matrix is obtained (Fig. 1), allowing traditional calculations (transverse and overlapped curve, coefficient of variation curve).

Performed with only one fed disc, the measured "map" represents the result of a limited number of variability factors, and it becomes possible to extract models directly from the spread pattern. Fig. 2 illustrates two of these new calculations. Based on initial works of Colin or Olieslagers, we assume that each vane of the disc distributes fertiliser on the ground according to a Gaussian distribution. The program expresses the measured spread pattern in the disc repair, so that it's possible to obtain the fertiliser distributed densities according to their polar coordinates (R, θ). Then it becomes possible to compute consecutively the angular distribution and the radial distribution (Fig. 2), from an experimental point of view. Then, best Gaussian fitting can automatically be computed to obtain in each case the mean value and its standard deviation [(P m , σ p ), (θ m , σ θ )]. In a very short time, these four determinants are obtained, regarding the mechanical spreading characteristics.

Using simple spreading configurations where the mechanical studied parameters are each time single, it's possible to obtain experimental laws according to an experimental reality, perfectly adapted to the spreader to model. Fig. 3 illustrates the technique: based on four different measurements obtained with four different lengths of vane, the variation laws for the parameters P m and θ m are deduced. Here, a first order law can fit correctly data, but it can be necessary to use more complex laws regarding the problem, the fertiliser … The experimental tests number to perform depends on the problem complexity. Projection distance and angle are here expressed regarding the length of vane, and fitted using a first order law.

Robustness of the C.V. curves:

Problem statement

Even if an important issue is to obtain the best C.V. (lowest distribution heterogeneity) as possible, a particular point was to keep the spreading results as robust as possible in different spreading configurations. It means the spreading defaults induced by tramlines variability must be as low as possible.

The Fig. 4 illustrates the problem. Both cases (right and left) of applications are obtained with a spreader correctly set for a 24m working width (C.V. = 4%). The spreading is performed along two trajectories in the field, and with a regularly increasing space between them, from 20m to 30m. The global fertiliser amount is the same in both cases. Nevertheless, under and over applications are significantly different in both cases: on the left, defaults are in the range [80% -120%] while they are in the range [20% -150% or more] on the right. As it's commonly known, the explanation comes from the geometric shape of the transverse distribution curve, which is quite triangular on the left while it is trapezoidal on the right. Over or under applications are diluted on all the area in the first case, but concentrated on limited area in the second case.

Objectives of the manufacturer are also to achieve the more as possible triangular curves in a great range of application cases. When this ratio is equal to 1, the distribution area is in keeping with the wanted working width. When the value is largely superior to 1, it guarantees some significant risks of under and over applications as soon as tramline variations occur.

-The second one is adapted from a criteria introduced by [START_REF] Kweon | Feed Gate Adaptation of a Spinner Spreader for Uniformity[END_REF]. It calculates the ratio between the C.V. integral curve of the measured test around the desired Ww and the C.V. integral curve for a perfect triangular distribution:

∫ ∫ Triangle V C Tested V C . . . .
(expressed in %) (Fig. 5). This criteria is very sensible, with a range of practical results around [100% -2500%]. The greater is this ratio, the more the transverse curve will be sensible to the tramline space variations.

Different discs: an obligation

The projection distance obtained with a spreading configuration (disc, fertiliser …) allows the fertiliser distribution on a determined area and throwing width. According to the above explanations, the projection distance must increase in the same time the wanted working width increases.

Fig. 6: Working widths, spread pattern area, projection distance induced and number of discs: technical limitations.

We consider an angular drop zone variation of the fertiliser on the disc which is a known setting to adjust the working width. With a mean value of 12m for example (the exact value isn't an absolute correct value because it depends on numerous other parameters), it'll be possible to spread correctly between 9 and 24m, using different angular positions for the 3D spread pattern. An angular position near 0° (regard ing the opposite of the travel axis displacement) allows optimised spreading around 9m while an angle of 35 or 40° allows 24m. The corresponding curves will pass from the triangular shape (Ww=9m) to the trapezoidal one (Ww=24m), and the robustness will be good at 9m, and bad at 24m. More extreme angular positions for the 3D spread pattern aren't possible for this type of use, because of very bad shape curves obtained (M shape …). A given mean projection distance also induces a given range of possible working widths, whatever the way setting chosen. Wider performances can't also be obtained without greater mean projection distances. It can be reached, for example, using longer vanes. Each disc of a same spreader will also be designed to work a range of working width, as shown on Fig. 6.

Regarding the ratio

Tw Ww . 2
, and the explanations given in the first part of this paragraph, an almost direct relation can be established with the transversal curve shape (Fig. 6-a). Progressively increasing the spread pattern angle position regarding the axis displacement, a same initial spread pattern will allow working at different working widths. The transverse curve will pass from the "Gaussian" shape (ratio largely inferior to 1, spread pattern too large to work correctly at the given Ww) to the "triangular" shape (ratio around 1, good correspondence between the spread pattern and the wanted Ww), and finally, to the "rectangular" shape -extremely the "M shape" -(ratio largely superior to 1, Ww too important for the given spread pattern). Finally, some working widths (Ww) can correctly be spread with only one disc, while other can be spread using 2 different discs (Fig. 6-b). In this case, C.V. can be the same whatever the used disc, while the robustness cannot. For example, a working width of 36m Ww can be obtained using the blue configuration (P m = Y) or the black one (P m = Z). Using the second configuration (black one) induces a triangular shape for 36m where the blue one induces a rectangular or M shape.

Fig. 7: Product ballistic performances (a), relation with granule physical properties (b) and consequences on their spreadability (c).

In addition, each fertiliser has its own physical characteristics, which explain finally its "spreadability" (spreading ability). Fig. 7-a shows significant ballistic performances differences for various fertilisers. For each of the initial fertiliser velocities at the output of the disc, some fertilisers present greater capabilities to reach important P m values. The mean diameter and its standard deviation, the density, the shape and roughness… are determinant parameters to explain these differences. Using slug pellets, which present relatively small standard deviation around the density 0,75, a direct relation can be established between the obtained P m value and the size of the particles: the more the size increases, the more P m increases (Fig. 7-b). Computing all the setting possibilities for a double disc spreader, the best performances for each Ww between 2 and 50m are extracted. These best performances are reported on Fig. 7-c, after a classification on the particle sizes. Crossing the different informations, it can be concluded to a clear relationship between P m and the spreadability: when P m increases, the range of correct possible Ww (allowing C.V. < 15%) also increases.

The design of a new disc will also need to take into account all these various parameters, from the mechanical device to the physical fertiliser characteristics, in order to find the best set of vane, allowing a correct range of spreading performances with all the fertilisers, with the maximum of robustness…

Virtual development before the real practical development and the validation

Based on all these experimental knowledges obtained using 3D spread pattern, a software can be developed (Fig. 8). It allows performing, without any new tests, some simulations to find, for example, the best P m value to reach by each vane, the best associated spread pattern angular positions …, regarding the range of desired Ww and the variability of spreading fertilisers. It represents, for manufacturers, real time and money saving, without any fertiliser use and recycling… Fig. 8: Virtual assistance to the development, using experimental laws obtained from the 3D spread pattern tests, and using computer simulations.

The design can also be performed directly by the manufacturer, according to the chosen specifications to achieve. Global performances can be predict and finely adjusted by this way. Using the obtained requirements, the manufacturer can draw the new mechanical layout using CAD (Computed Aided Design) (Fig. 9).

Results

Performing different experimental tests on the CEMIB and extracting the valuable information as described in paragraph 1, then taking into account all the technical aspects and limitations, and finally using the conception method (simulator), three different discs were developed and positioned on the X Range SULKY spreader: the 12-28, the 24-36 and the 32-44 (each time, the number represent the range of Ww of the considered disc). An angular drop zone setting device was placed and optimized from the initial SULKY system (delivery chute), allowing Ww setting from the tractor cab. ), in the same time each of the discs spread predominantly each side of the run.

Main field spreading results

Border spreading results

Researches to obtain main field triangular robust curves lead to a significant problem for border spreading. As it can be seen on the transverse curve of Fig. 10, non negligible fertiliser quantities should be spread out of the field if nothing was changed on the distribution device (the global amount at the right of the dotted red line). In the same time, the quantities distributed in the border distance shouldn't be 100%. Specific devices must also be developed.

In border spreading configuration, at the opposite of main field one, both discs must work differently. One must spread the needed complement to the previous run, while the second must be designed in order to avoid fertiliser distribution out of the field, and keep this fertiliser regularly distributed into the border distance. SULKY decided to develop an innovative solution, patented [START_REF] Leveillé | Machine et procédé d'épandage centrifuge d'engrais ou de produits similaires en grains[END_REF]. The objective is to decrease the standard deviation of the projected distribution, and to correctly place the spread pattern relatively to the travel axis displacement.

As it's illustrated on the Fig. 11, the distribution device which guides the fertiliser from the hopper to the disc can be set differently regarding the wanted configuration. For main field spreading, the fertiliser is guided to the short and the long vane: the obtained spread pattern is large, with important standard deviation and with a spread pattern angle globally placed between 15 and 30° (Fig. 11-a). For border applications, the fertiliser is guided to the short vane and the curved short one: P m and σ p are also less important, adapted to the desired border width. The spread pattern angle position is also advanced in the travel direction (Fig. 11-b). Both requirement of the EN13739 standard can be reached (YOS or EOS), using more or less important proportions of the fertiliser quantities spread by each vanes. The fig. 12 illustrates the results which are obtained with the Tribord system, satisfying the different requirements of the standard on each point. Mechanically easy to drive (small electric motor), it's a compact solution, for which developments has been accelerated by the 3D spread pattern knowledges.

Conclusion

Using real and practical examples, main advantages of the use of the information enclosed in a 3D spread pattern correctly measured has been highlighted. In particular, the information relative to the spatial distribution has been explained, using mathematical and objective values [(Pm, σ p ), (θ m , σ θ )]. In addition to other classical curves (transverse, C.V. …) and using specified quality distribution criteria, these synthetic informations are used in order to predict distribution comportments and to design spreading devices more scientifically, in a global goal to be more environmentally friendly. All the information are also used to take into account physical and technical limits, particularly considering from the design beginning all the linked problems, like robustness evaluation of the device... The X Range spreader from the French manufacturer SULKY has been the support to illustrate the results which can be obtained, in relation with the distribution quality reached in both case of main field and border applications. It's also an example of collaboration between the research world (new method to measure, to simulate, to understand …) and the necessary transfer to the application world, where all concepts have to be adapted.

Fig. 1 :

 1 Fig. 1: Fertiliser density matrix obtained in a few minutes using the CEMIB device.

Fig. 2 :

 2 Fig. 2: Specific analysis realised on the 3D spread pattern measured: Angular and Radial distribution around the disc axis. Experimental values are extracted, as well as the best Gaussian fit, with there driven factors (mean values and standard deviations).

Fig. 3 :

 3 Fig.3: Experimental laws obtained using four different measurements. Projection distance and angle are here expressed regarding the length of vane, and fitted using a first order law.

Fig. 4 : 2 (

 42 Fig. 4: Example of default applications in the field when tramlines aren't regularly spaced, and regarding two different application parameters.

Fig. 5 :

 5 Fig. 5: Calculation illustration of the second robustness criteria.

Fig. 9 :

 9 Fig. 9: Computed Aided Design (CAD) of the new spreader device, performed directly by the manufacturer.Using the obtained requirements, the manufacturer can draw the new mechanical layout using CAD (Computed Aided Design) (Fig.9).

Fig. 10 :

 10 Fig. 10: Example of spread pattern obtained in main field configuration.

Fig. 11 :

 11 Fig. 11: Spreading device developed by SULKY for main field and border spreading.

Fig. 12 :

 12 Fig. 12: Results obtained in border spreading configuration.