Tengfei Chang
email: tengfei.chang@inria.com

Mališa Vučinić

| Xavi

Vilajosana Guillén

Diego Dujovne

Thomas Watteyne
email: thomas.watteyne@inria.com

6TiSCH Minimal Scheduling Function: Performance Evaluation

Keywords: 6TiSCH, Scheduling, Latency, Overhead, SF

6TiSCH is a standardization group within the Internet Engineering Task Force (IETF) that works on IPv6-enabled Time-slotted Channel Hopping (TSCH) networks. The 6TiSCH protocol stack, designed by the standardization work at the IETF, has direct applicability to low-power Internet of Things (IoT) use cases, including smart factory, building, infrastructure and home applications. A key component of the 6TiSCH stack is the Minimal Scheduling Function (MSF). MSF implements a traffic adaptation algorithm which allocates link-layer resources, i.e. cells in the TSCH schedule, according to the traffic load. MAX_NUMCELL is an important parameter defined in the MSF draft standard which determines the length of the running window used to measure cell usage. MSF draft standard does not recommend a value of MAX_NUMCELL to use. This paper provides recommendations on how to choose the value of MAX_NUMCELL, validated through simulation. For periodic traffic, setting MAX_NUMCELL to at least 4 times the traffic load is recommended to increase efficiency. For bursty traffic, we show that setting MAX_NUMCELL to a small value achieves a low end-to-end latency but at high communication overhead. In addition, an improved version of MSF is implemented and tested, which shows a 44% reduction in the communication overhead, considering MAX_NUMCELL = 4, while maintaining the same end-to-end latency.

INTRODUCTION

The Internet Engineering Task Force (IETF) and the 6TiSCH working group have produced a set of specifications that constitute the 6TiSCH protocol stack [START_REF] Vilajosana | 6TiSCH: Industrial Performance for IPv6 Internet-of-Things Networks[END_REF] . The stack is designed for critical industrial automation, and is hence also applicable to smart building, infrastructure and home automation applications. The stack foundation is the Time-Slotted Channel Hopping (TSCH) mode of operation at the MAC layer, which enables ultra low-power operation. As a time-frequency division multiple access technique, the quantum of TSCH bandwidth is called a cell. A cell is uniquely defined by a time and channel offset, describing the time instant and the frequency at which the transmission to a neighbor node takes place. The set of all cells that a 6TiSCH network node, the mote, has installed is called a schedule. The allocated cells provide a deterministic communication pattern for the motes to schedule bandwidth according to the traffic being transmitted.

The 6top Protocol 2 (6P) enables the motes to manage the schedule in a distributed manner through a handshake mechanism. The Scheduling Function (SF) defines the rules that trigger 6P transactions. As a successful outcome of these 6P exchanges, the schedule of the neighboring motes is populated with cells. A number Scheduling Functions have already been proposed [START_REF] Chang | 6TiSCH Minimal Scheduling Function (MSF)[END_REF][START_REF] Duquennoy | Orchestra: Robust Mesh Networks Through Autonomously Scheduled TSCH[END_REF][START_REF] Domingo-Prieto | Distributed PID-Based Scheduling for 6TiSCH Networks[END_REF][START_REF] Palattella | Traffic Aware Scheduling Algorithm for reliable low-power multi-hop IEEE 802.15.4e networks[END_REF][START_REF] Accettura | Decentralized Traffic Aware Scheduling for multi-hop Low power Lossy Networks in the Internet of Things[END_REF] .

FIGURE 1

MSF adapts the number of cells a mote allocates to a neighbor according to the number of frames they exchange.

Here, MAX_NUMCELL = 6. A cell is added when the cell usage is over 75%, and deleted when below 25%.

They were designed to address different application types and particular network requirements. The Minimal Scheduling Function (MSF) [START_REF] Chang | 6TiSCH Minimal Scheduling Function (MSF)[END_REF] is the only scheduling function adopted for standardization at the IETF. MSF is designed for best-effort traffic, where most traffic is periodic monitoring, with occasional bursts. MSF uses two types of cells: autonomous and negotiated. Autonomous cells provide the minimal bandwidth required for bidirectional communication between two neighbors. Negotiated cells are installed in response to dynamic bandwidth requirements. Bandwidth requirements are estimated based on the traffic load, by measuring the usage of the installed cells over some sliding window. If the load is above a pre-defined threshold, a new cell is added. If the load is below some other threshold, a cell is removed from the schedule. If the load is between those two thresholds, no action is taken. This is illustrated in Fig. 1.

The MSF draft standard 3 defines a number of parameters that influence network performance. It does not, however, provide any recommendation on the duration of the sliding window, a parameter that is critical for responsiveness to variation in traffic. The MSF draft standard refers to the sliding window length as MAX_NUMCELL, the term we use throughout this paper.

The contribution of this paper is twofold. First, we provide recommendations on how to choose the value of MAX_NUMCELL, based on simulation results. Second, we propose an advanced version of MSF which reduces the number of 6P exchanges to adapt to bursty traffic.

The remainder of this paper is organized as follows. Section 2 gives on overview of the MSF's traffic adaptation algorithm. Section 3 provides simulation results and insights on the performance of MSF using different MAX_NUMCELL values. Finally, Section 4 summarizes and concludes this paper.

TRAFFIC ADAPTATION OF MSF

The MSF draft standard defines the rules for motes in the network to follow, both when first booting and when adapting to traffic. In this section, we only focus on the MSF's adaptation algorithm.

MSF adds or deletes a cell according to the cell usage. Cell usage is calculated as NumCellUsed divided by NumCellElapsed. MSF maintains those two counters. When the current cell is a negotiated transmit cell, NumCellElapsed increments. When a frame is sent on that transmit cell, regardless of whether it is acknowledged, NumCellUsed increments. When NumCellElapsed hits the pre-defined value MAX_NUMCELL, MSF calculates the cell usage. It decides to issue a 6P request to add a cell if it is over 75%, or delete a cell if it is below 25%. After computing cell usage (and possibly adding/deleting cells), NumCellElapsed and NumCellUsed reset to zero and the process repeats.

Fig. 1 shows an example of how MSF adds and deletes negotiated cells to adapt to traffic changes. Assuming the slotframe is 5 slots long, and MAX_NUMCELL =6. The mote already has one transmit cell to its parent. At the beginning, the traffic load of the mote is 1 packet/slotframe. After 6 slotframes, NumCellElapsed reaches 6 and cell usage is 100% (6/6). MSF starts a 6P transaction with its parent and adds one transmit cell. The goal of this paper is to show how the value of MAX_NUMCELL influences the performance of MSF, and to recommend a range of values meeting the expected performance for specific scenarios. Performance results are obtained using the 6TiSCH simulator [START_REF] Municio | Simulating 6TiSCH networks[END_REF] , and are presented in Section 3.

EVALUATION

Simulation Results

The 6TiSCH simulator implements the complete 6TiSCH protocol stack; it is written in Python and includes an implementation of the latest version of the MSF draft. The log files generated by the simulator record each event from each mote. Events include: an app packet generated by a mote, an app packet received by the root, a 6P request frame sent by a mote, a cell added in the schedule and a cell deleted from the schedule.

The simulator supports multi-processing, so we run it on many cores in a cluster system, reducing total simulation time.

We target two traffic patterns: periodic, where each mote generates one packet per slotframe and bursty, where one mote generates 20 packets per minute at the start of each minute.

We consider the two Key Performance Indicators (KPIs) for MSF:

• the number of 6P messages sent by each mote, which we aim to reduce,

• the end-to-end latency of each packet, which we aim to shorten.

Without lack of generality, we consider the canonical case of 4-hop linear topology (5 motes). This allows us to observe only the impact of the scheduling approach, and presents a worst case scenario compared to for example load-balancing traffic across multiple routing parents. Mote 0 is the root node, mote 1 is the first hop node, etc. In the periodic traffic case, each of the 4 motes generates data. This results in the first-hop mote having to send 4 frames per slotframe to the root. In the bursty traffic case, only the fourth hop mote generates traffic. The generality is because of the link local nature of MSF protocol. For most of the application of 6TiSCH network, they are upstream traffic. Each mote sends packet periodically to the root. From the view of routing, it is a linear topology for each of the motes. Though two motes may select a common mote as their parent, which turns to a star topology, the common mote does not differentiate the source of the traffic, either from previous hop or generated by itself. In this case, from the point of MSF, it equals to the second hop mote in a linear topology. The MAX_NUMCELL is an input parameter of link local decision for the mote who is trying to send packets. Hence the performance of different values of MAX_NUMCELL does not vary depending on the topology.

We repeat the simulation for MAX_NUMCELL set to 4, 8, 16 and 32. For each value of MAX_NUMCELL, we run a simulation for a period equivalent to 3600 slotframes, with slot duration set to 10 ms and slotframe length 101 slots. To ensure our results are statistically valid, we conduct 1000 runs for each of the example values of MAX_NUMCELL, and present all results with 99.3% confidence intervals. Table 1 summarizes the simulation settings.

The typical KPI of a scheduling function, such as packet delivery rate (PDR) or energy consumption, are not chosen directly to evaluate the result. In fact, the MAX_NUMCELL is not affected by the packet loss because of the nature of MSF scheduling mechanism. The MSF adds/removes cells according to number of packet sent. In case of a < 100%, the number of packet sent will be translated as number of packet expected to send divides PDR. For example, the packet sending rate of 1 packet per slotframe at = 50%, the performance then can be referred to packet sending rate of 2 packets per slotframe at = 100%. The simulation in the paper fixed the PDR to 100% but added different packet sending rates at each hop of the network. Plots only for Mote 4. Comparing to regular MSF (Fig. 4), end-to-end latency is equivalent, but the number of 6P messages is halved.

The energy consumption of MSF mostly comes from two parts: the number of cells scheduled and the overhead. As the scheduling mechanism of MSF indicated it schedules more cells than required to compensate latency performance, at the cost of additional energy. The MAX_NUMCELL has a limited effect on the number of cells scheduled. The energy consumption related to MAX_NUMCELL is mostly introduced by the transaction overhead, which is the number of 6P messages sent. This is discussed following.

Periodic Traffic Pattern

Fig. 2 shows the number of 6P messages sent by each mote. Using MAX_NUMCELL = 4 results in a higher number of 6P messages, compared to larger values of MAX_NUMCELL. For mote 1, the traffic load is 4 packets per slotframe, thus the mote needs to schedule 8 cells to meet the ideal 50% cell usage. By using MAX_NUMCELL = 4, MSF deletes a cell when 4 cells are unused, representing a 0% cell usage. However, the following 4 cells could be used for transmitting purposes, thus generating a 100% cell usage. As a consequence, MSF may trigger a command to add a cell right after. The probability of adding a cell right after increases when the value of MAX_NUMCELL is reduced.

Fig. 3 shows the result of end-to-end latency for motes at each hop. For each mote, changing MAX_NUMCELL has a very limited effect on improving end-to-end latency. This is because MSF reserves additional cells for the case when traffic load increases. Hence, the delay caused by the 6P transaction is very small for the periodic traffic application. Notice that the average latency using MAX_NUMCELL = 32 is a little bit lower than using MAX_NUMCELL = 16, but with a larger variation. This is because the packets are generated at rate of 20 packets per minutes which is less than 60 slotframes. The generating interval has a variation of 20%, ranging of 48 to 72 slotframes interval. When using larger MAX_NUMCELL such 32, the 20 packets generated at first time may be not all sent out yet when the next 20 packets are generated. Hence the previous added cells in schedule are kept, which reduces part of the latency caused by 6P transactions overhead.

Assuming the traffic load is [packets per slotframe], the recommended value of MAX_NUMCELL setting should be at least 2 × ( ÷ 50%), which corresponds to 4 × . Here, 50% is the average of cell usage and the 2 × is double the sample rate to increase the accuracy. For example, for mote 1, its link to the root has a traffic load of 4 packets per slotframe. To reach a stable cell usage of 50%, it needs to schedule 8 cells in the schedule. So the final MAX_NUMCELL value for mote 1 should be set to > 16 to have a higher sampling precision of cell usage among the 8 cells.

Bursty Traffic Pattern

For the bursty traffic pattern, the delay on 6P transactions shows up as indicated in Fig. 4. Fig. 4(b) shows that the end-to-end latency increases when increasing the MAX_NUMCELL value. Fig. 4(a) shows the number of 6P messages sent by mote 4. When MAX_NUMCELL is set to 4, the end-to-end latency is reduced to 0.5 s compared to using MAX_NUMCELL = 8, which doubles the 6P frame overhead.

An Improved MSF

One of the main issues of MSF is the generation of a large amount of 6P messages because MSF only adds or deletes one cell at a time. The mote may need several 6P transactions to reach the stable status (25% to 75% of cell usage).

One possible improvement to MSF is to let the mote add or delete multiple cells at a time to reach the ideal 50% cell usage. Base on this core idea, we design an advanced version of MSF: A-MSF, which adapts to traffic as follows:

• when cell usage > 75%, add transmit cells, with = ℎ ⋅ ( ∕0.5 -1)

• when cell usage < 25%, delete transmit cells, with = ℎ ⋅ (1 -∕0.5)

To compare the performance to MSF, we implement A-MSF and conduct the same experiment for the bursty traffic pattern case. Fig. 5 shows the result of A-MSF using the same setting of Fig. 4. A-MSF yields the same end-to-end latency as MSF, while reducing the number of 6P messages by 44% (resp. 26%) when using MAX_NUMCELL = 4 (resp. MAX_NUMCELL = 8). For other values of MAX_NUMCELL, the number 6P messages sent by A-MSF and MSF is equivalent.

CONCLUSION

This paper provides recommendations on how to set the value of MAX_NUMCELL when using MSF. It considers two common traffic patterns. For periodic traffic, it is recommended to use a larger MAX_NUMCELL so as to reduce 6P traffic. The value should be set at least 4 times the link traffic load measured in packets per slotframe. For bursty traffic, a small MAX_NUMCELL value reduces the end-to-end latency, but exhibits a large numerb of 6P messages. We proposed A-MSF, extension of MSF which yields 44% less 6P messages than MSF, with comparable end-to-end latency. Based on these results, we are working with the 6TiSCH standardization team to introduce the changes of A-MSF into the next version of the 6TiSCH MSF standard.

FIGURE 2

 2 FIGURE 2 Periodic Traffic. Number of 6P messages sent through each simulation. One plot per mote.

4 FIGURE 3 FIGURE 4

 434 FIGURE 3 Periodic Traffic. End-to-end latency. One plot per mote.

FIGURE 5

 5 FIGURE 5 Bursty traffic, using Advanced MSF. Number of 6P messages sent by Mote 4 (left) and end-to-end latency (right).Plots only for Mote 4. Comparing to regular MSF (Fig.4), end-to-end latency is equivalent, but the number of 6P messages is halved.

  4, 8, 16, 32 duration per run 1 hour (3600 slotframes, each slotframe 101 slots, each slot 10 ms) number of runs 1,000 runs for each value of MAX_NUMCELL

3 slotframes later, after the second transmit cell is scheduled, NumCellElapsed name settings event to log app.tx, app.rx, tsch.add_cell, tsch.delete_cell, sixp.tx traffic pattern periodic, bursty KPIs num. of cells scheduled, num. of 6P messages sent, end-to-end latency MAX_NUMCELL

TABLE 1

 1 Simulation Settings.


Yanyan Yuan
email: yuan.yanyan@just.edu.cn

Bingye Zhang

Philippe Jonnard

Karine Le Guen

Yuchun Tu

Chao Yan
email: chaoyan@just.edu.cn

Rui Lan

Structure and optical properties of CrO x N y films with composition modulation

Keywords: chromium oxynitride, optical properties, composition modulation, XPS

The composition and structure properties of the CrO x N y films deposited by RF reactive magnetron sputtering were investigated and correlated with their optical properties. Two kinds of targets Cr 2 O 3 and CrN compounds with variation of N 2 and O 2 gas flow were used to prepare two series of samples. The phase structure, chemical composition and optical properties of CrO x N y films were characterized by X-ray diffraction, X-ray photoelectron spectroscopy and ultraviolet-visible-near-infrared spectrophotometer, respectively. The films mainly consist of Cr 2 N and CrO 3 phases.

The Cr/ON ratio derived by XPS varies with the variation of nitrogen with Cr 2 O 3 target.

But only lower nitrogen content is detected with variation of oxygen using CrN target.

The CrO x N y films with higher nitrogen content possess high absorbance around at 35% in the visible light range no matter the target.

Introduction

Transition metal nitrides and oxides have received considerable interests due to their versatile properties. They are frequently applied in optical devices, such as solar cell electron transport materials [START_REF] Zheng | Metal Oxide CrO x as a Promising Bilayer Electron Transport Material for Enhancing the Performance Stability of Planar Perovskite Solar Cells[END_REF][START_REF] Martin | The contribution of grain boundary barriers to the electrical conductivity of titanium oxide thin films[END_REF] . On the other hand, transition metal nitrides such as CrN, TiN, ZrN exhibit excellent physical properties like relatively high hardness, high melting point and good chemical stability. Furthermore, for some of them their performance can be improved by ternary, quaternary and multicomponent thin films preparation through addition of other refractory materials [START_REF] Ju | Influence of vanadium incorporation on the microstructure, mechanical and tribological properties of Nb-V-Si-N films deposited by reactive magnetron sputtering[END_REF][START_REF] Ju | Microstructure, oxidation resistance, mechanical and tribological properties of Ti-Y-N films by reactive magnetron sputtering[END_REF][START_REF] Xu | Influence of silicon content on the microstructure, mechanical and tribological properties of magnetron sputtered Ti-Mo-Si-N films[END_REF][START_REF] Yu | Influence of Al content on microstructure, mechanical and tribological properties of Ti-W-Al-N composite films[END_REF] .

In addition, a combination of transitional metal nitrides and oxides as oxynitrides promises the possibility to tune the structural, optical and electrical properties in a wide range as desired for various applications, like hard coatings, decorative coatings as well as solar devices. See for example, some works on these TiO x N y [7,8] , ZrO x N y [9,10,11] , TaO x N y [START_REF] Chung | Effect of reactive gases flow ratios on the microstructure and electrical resistivity of Ta-N-O thin films by reactive co-sputtering[END_REF] films.

However chromium nitrides and oxides with potential properties such as high chemical stability, wear and corrosion resistance as well as optical properties have received great attention.

The chromium oxynitride films, whose properties could be tuned between their corresponding oxides and nitrides, can be formed through the addition of a small amount of oxygen into chromium nitrides or nitrogen into the oxides. They possess many potential interesting applications due to their versatile properties, like electrical, optical, mechanical that the chromium oxides show potential application in planar perovskite solar cell as electron transport material for enhancing the stability of solar cells [START_REF] Zheng | Metal Oxide CrO x as a Promising Bilayer Electron Transport Material for Enhancing the Performance Stability of Planar Perovskite Solar Cells[END_REF] . It has been reported that chromium oxynitride (CrO x N y ) films can often be prepared by using reactive magnetron sputtering [START_REF] Sushant | Structural and optical investigation of sputter deposited hydrophobic chromium oxynitride films[END_REF] and pulsed laser deposition techniques [START_REF] Suzuki | Influence of oxygen content on structure and hardness of Cr-N-O thin films prepared by pulsed laser deposition[END_REF] . Suzuki reported that the oxygen content in the Cr(N,O) film prepared by pulse laser deposition can significantly improve its hardness up to be 30 GPa [START_REF] Suzuki | Controlling oxygen content by varying oxygen partial pressure in chromium oxynitride thin films prepared by pulsed laser deposition[END_REF] . The chromium oxynitride films have different colors with variation of thickness and composition as decorative coatings [START_REF] Mientus | Optical and electronic properties of CrO x N y films, deposited by reactive DC magnetron sputtering in Ar/N 2 /O 2 (N 2 O) atmospheres[END_REF] . Wierzchon et al. reported that the CrO x N y films show higher corrosion resistance as well as better adhesion and uniform structure than that of Cr 2 O 3 films [START_REF] Wierzchon | Corrosion resistance of chromium nitride and oxynitride layers produced under glow discharge conditions[END_REF] . K Sushant et al. reported that the transmittance and band gap values of CrON films increase with the increase of oxygen partial pressure [START_REF] Sushant | Ramesh Chandra,Structural and optical investigation of sputter deposited hydrophobic chromium oxynitride films,[END_REF] .

This paper study the CrO x N y thin films prepared by RF magnetron sputtering with Cr 2 O 3 and CrN targets using oxygen and nitrogen as reaction gases. The structure, surface composition and optical properties of the CrO x N y thin films were investigated by grazing incidence X-ray diffraction (GIXRD), x-ray photoelectron spectroscopy (XPS) and ultraviolet-visible-near-infrared spectrophotometer, respectively. The main goal of the present paper is to develop a method to slightly control the N/O ratio in the films.

Experimental details

The chromium oxynitride thin films were deposited on Si (111) substrates and ITO-glass substrates by using radio frequency (RF) reactive magnetron sputtering technology (JCP-500M3 Beijing Technol. Co. LTD., China) with Cr 2 O 3 (99.5%) and CrN (99.5%) targets.

The base pressure was around at 6.0×10 -4 Pa before deposition. The working gas was argon (99.999%) with reactive gases oxygen and nitrogen keeping at a constant working pressure of 1.0 Pa. The target-substrate distance was about 100 mm. When the Cr 2 O 3 or CrN was used as target, nitrogen or oxygen flow rate varied with the value of 5, 8, 10 sccm and 3, 5, 8, 10sccm

respectively, but kept the total constant flow rate of 25.0 sccm. The power applied on all the targets was set as 100 W. All samples were deposited at room temperature. Five minutes pre-sputtering was performed on the targets before depositing all samples.

The phase structures of the films were measured via grazing incidence X-ray diffraction (GIXRD) using D8 Advance A25 with Cu Kα radiation (λ = 0.1542 nm) at 40 kV and 10 mA.

The grazing incidence angle was set as 4 • .

The surface chemical composition and chemical bonding states of the CrO x N y thin films were investigated through XPS using Escalab 250Xi system (Thermo Scientific) equipped with an Al Kα source (1486.6eV). The surface chemical composition was quantified using Avantage software. The absorbance of the CrO x N y thin films was observed using an ultraviolet-visible-near-infrared spectrophotometer (UV-3600, Shimadzu). The atomic ratios in the samples will be discussed in the following part.

Results and discussion

X-ray photoelectron spectroscopy

Figure 2 shows the XPS spectra of the Cr2p, O1s and N1s core level of the CrO x N y thin films prepared by using CrN target with variation of the oxygen flow rate. Regarding the Cr2p spectra, shown in Figure 2 (a) the peak position shift significantly toward high binding energies when the oxygen flow rate exceeds 8 sccm. According to the literature [START_REF] Milosev | Comparison of TiN, ZrN and CrN hard nitride coatings: Electro chemical and thermaloxidation[END_REF] , the shift According to literature [20], the component at 576.0 eV is attributed to Cr-O bond from Cr 2 O 3 .

In Figure 3 Regarding the samples prepared using CrN target, it can be seen that all the samples have lower N content even without oxygen introduction. The Cr/N ratio in the target is around 2:1 before sputtering measured by EDX. When the samples are prepared using CrN target without oxygen, the N content is quite low and much less than the O content. The Cr/NO ratio in the films is about 1:3 and is very different from the target value. Moreover, the N content in the films decreases a lot with the introduction of oxygen compared with the N content without oxygen. The main reason for the low N content is due to the oxidation as well as the loss of nitrogen during sputtering process.

Grazing incidence X-ray diffraction (GIXRD)

Figure 5 shows the phase structures of the chromium oxynitride thin films. In addition, the XRD pattern of the Si substrate is also shown here for comparison. There is no diffraction peak of the Si substrate in the measurement range. The samples prepared using Cr 2 O 3 target with N 2 reactive gas show an intense peak located at around 2θ=57.5° which corresponds to CrO 3 (003). Otherwise, a shoulder located at 2θ=56° along the strong peak appeared, which probably corresponds to Cr 2 N (112). A weak peak located at 2θ=77.5° probably corresponds to CrO 3 (402). The results are in agreement with the XPS-derived Cr/O ratio. When the sample prepared using CrN target with 10 sccm of oxygen flow rate, the peak position is same as these prepared using Cr 2 O 3 target. However, the sample deposited using Cr 2 O 3 target with 5 sccm of oxygen flow rate shows an intense peak located at 2θ=56°, which corresponds to Cr 2 N (112). It is not in agreement with the XPS-derived atomic ratio. But one thing should be noted that the XPS-derived atomic ratio is from the sample surface. However the XRD pattern shows the whole sample information.

Optical properties

To observe the effect of composition on the optical properties of the samples, the absorbance spectra was measured and shown in Figure 6.

Disclosure statement

No potential conflict of interest was reported by the authors.

Figure 1

 1 Figure1shows the XPS spectra of the Cr2p, O1s and N1s core levels of the CrO x N y thin

  Figures 3 and 4 present the fitted spectra of the samples prepared using Cr 2 O 3 and CrN

  (b), for the samples prepared without nitrogen, the fitting of the O1s peak clearly shows the existence of three peaks components. The component at 532.5eV corresponds to N-O bond and the components located at around 531.6 and 530.7eV correspond to O-H and Cr-O bond, respectively. The existence of O-H bond is due to absorption water on the sample surface. The O1s spectra show a broad feature with increase of nitrogen flow rate indicating the high oxygen content. Note that the samples prepared with nitrogen flow rate of 10 sccm and 5 sccm, the O1s peak is fitted with four peaks components. It indicates the change of O chemical bond state in the sample. The N1s fitting spectra are shown in Figure 3(c). The high nitrogen content in the sample prepared with 10 sccm of nitrogen is confirmed by the broad feature. Meanwhile N1s peak is fitted with four peaks components. The component at 399.9eV corresponds to the N-O bond.The other components at 398.0, 396.8 and 398.8eV probably correspond to the Cr-N from Cr 2 N and CrN x , respectively.

Figure 4

 4 Figure4shows fitted XPS spectra for the CrO x N y films prepared using CrN target. The

Figure 6 (

 6 a) show the absorbance spectrum of the chromium oxynitride thin films prepared with Cr 2 O 3 target with variation of nitrogen. The results present that the absorbance increases slightly with decrease of the nitrogen flow rate in the range of 300-800 nm. The absorption edge of the sample prepared with 5 sccm of nitrogen is located at around 450 nm and the absorption edge shifts toward the short wavelengths direction as the nitrogen flow rate increases. Figure6 (b)shows the absorbance spectra of the chromium oxynitride thin films deposited using CrN target with variation of oxygen. The absorption edge is located at about 400 nm and the absorbance is around 0.2% in the visible range for all samples. There is no obvious shift of the absorption edge with increase of the oxygen flow rate.4.ConclusionsWe study the effects of the reactive atmosphere on the crystal structure, chemical state and composition as well as optical properties of the CrO x N y films. The XRD results demonstrated the coexistence of Cr 2 N and CrO 3 compounds of all samples. The films prepared using Cr 2 O 3 target possesses the preferred CrO 3 (003). However, the preferred orientation of the film prepared using CrN target is converted from Cr 2 N (112) to CrO 3 (003) as the oxygen flow rate increases. XPS results indicate that the composition of the CrO x N y films can be easily modulated by using the Cr 2 O 3 target than CrN target. Absorbance of the CrO x N y films prepared with Cr 2 O 3 target shows no significant change only the sample with high N content possess high absorbance around 35% in the visible light range.

Figure 1 .

 1 Figure 1. XPS spectra recorded from chromium oxynitride thin films prepared by using Cr 2 O 3

Figure 2

 2 Figure 2 XPS spectra recorded from the chromium oxynitride thin films prepared using CrN

Figure 3 .

 3 Figure 3. Fitted XPS spectra for the chromium oxynitride thin films prepared using Cr 2 O 3

Figure 4 .

 4 Figure 4. Fitted XPS spectra for the chromium oxynitride thin films prepared using CrN target

Figure 5 .Figure 6 .

 56 Figure 5. GIXRD patterns of the CrO x N y films prepared using Cr 2 O 3 (a: N 2 : Ar=8:17sccm; b: N 2 : Ar=10:15 sccm) and CrN targets (c: O 2 : Ar=5:20sccm; d: O 2 :Ar=10 : 15 sccm), e: Si substrate.

  and tribological properties achieved by tuning N/O composition.Chromium oxynitride can be considered as a mixture of CrN and Cr 2 O 3 rather than a single phase chromium oxynitride. Therefore, it has the hardness of Cr 2 O 3 and the toughness of CrN materials. Otherwise, CrN x shows a metallic-like (ρ≈6.4×10 -4 Ωcm,ⅹ≈0.93) or semiconducting behavior(ρ>1×10 -2 Ωcm,ⅹ≈1.06) depending on its stoichiometry, while CrO

x is an insulator(Eg≈4eV, ρ Cr2O3 >>1Ωcm). The CrO x N y thin films combining both materials show the possibility to tune the energy band gap. Recently it has been reported by

Zheng et al.

Table 2

 2 listed the atomic ratio of each element in chromium oxynitride thin films derived from XPS spectra using Avantage software. It is shown that the surface chemical composition of the CrO x N y films prepared using Cr 2 O 3 target show a variation of Cr/NO ratio dependence on the reactive gases. The Cr/O ratio is about 1:4 when the sample is prepared using Cr 2 O 3 target with only argon, which is quite different from the Cr/O ratio of the target. It means that the oxidation occurred on the surface of the samples when they are stored in air. However, the N content in the films decreases with increase of nitrogen flow rate. The Cr/NO ratio slightly increase with increase of nitrogen flow rate. There is no doubt that N atoms are introduced into the films and the N content varies depending on the variation of the nitrogen flow rate even the O content is always higher than N content. The main reason for higher O/Cr ratio

than N/Cr is due to the large difference in electronegativity between Cr and O which makes Cr-O bonds easier to be formed.

Table 1

 1 Binding energy and chemical bond of the fitted peak components of the Cr2p and O1s spectra of the samples. BE: binding energy (eV); CB: chemical bond (corresponded compounds are in the parentheses). The samples prepared using Cr 2 O 3 target with nitrogen flow rate of 0, 5, 8 and 10 sccm are labeled as S10, S11, S12 and S13, respectively. The samples prepared using CrN target with oxygen flow rate of 0, 5, 8 and 10 are labeled as S20, S21, S22 and S23 respectively.

	Samples			Cr 2p			O 1s
		BE(eV)	577.1	579.0		530.7	531.6	532.5
	S10	CB	Cr-O-N	Cr-O(CrO 3 )	Cr-O (CrO 3 )	O-H	O-N
		BE(eV)	576.0	576.9	579.2	530.2	531.2 531.9 532.7
	S11	CB	Cr-O (Cr 2 O 3 )	Cr-N (Cr 2 N)	Cr-O (CrO 3 )	Cr-O (CrO 3 )	O-H	O-N
		BE(eV)	576.0	576.8	579.0	530.3	531.2	532
	S12	CB	Cr-O (Cr 2 O 3 )	Cr-N (Cr 2 N)	Cr-O (CrO 3 )	Cr-O (CrO 3 )	O-H	O-N
		BE(eV)	575.9	577.0	578.9	529.9	530.5 531.2 532.1
	S13	CB	Cr-O (Cr 2 O 3 )	Cr-O-N	Cr-O (CrO 3 )	Cr-O (Cr 2 O 3 )	Cr-O (CrO 3 )	O-H	O-N
		BE(eV)	574.4	576.7	579.6	530.0 530.7	531.3 532.0
	S20	CB	Cr-Cr	Cr-N (Cr 2 N)	Cr-O (CrO 3 )	Cr-O (Cr 2 O 3 )	O-H	O-N
		BE(eV)	576.1	577.2	579.4	529.8 530.4	531.1 532.5
	S21	CB	Cr-O (Cr 2 O 3 )	Cr-N (Cr 2 N)	Cr-O (CrO 3 )	Cr-O (Cr 2 O 3 )	O-H	O-N
		BE(eV)	576.4	577.5	578.2 579.6	530.2 530.7	531.4 532.5
	S22	CB	Cr-O (Cr 2 O 3 )	Cr-O-N	Cr-O (CrO 3 )	Cr-O (Cr 2 O 3 )	O-H	O-N
		BE(eV)	576.7	577.6	578.6 579.7	530.2 530.7	531.5 532.7
	S23	CB	Cr-O (Cr 2 O 3 )	Cr-O-N	Cr-O (CrO 3 )	Cr-O (Cr 2 O 3 )	O-H	O-N

Table . 2

 . Atomic ratio of the chromium oxynitride thin films derived from XPS spectra.

				Atomic ratio (at.%)
	Target	Atmosphere
			Cr	O	N
		Ar=20sccm	19.5	80.5	--
		N 2 :Ar=5:20sccm	14.1	59.7	26.2
	Cr 2 O 3
		N 2 :Ar=8:17sccm	20.4	58.9	20.7
		N 2 :Ar=10:15sccm	23.5	66.1	10.4
		Ar=20sccm	29.0	64.0	7.0
		O 2 :Ar=3:22sccm	24.9	72.7	2.4
	CrN	O 2 :Ar=5:20sccm	25.7	71.8	2.5
		O 2 :Ar=8:17sccm	21.8	75.1	3.1
		O 2 :Ar=10:15sccm	22.3	74.2	3.5

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No.

51702133) and by the Priority Academic Program Development of Jiangsu Higher Education

Institutions (PAPD).


niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Mazen Zein
email: mazen.zein@irccyn.ec-nantes.fr

Philippe Wenger

Damien Chablat

Non-Singular Assembly-mode Changing Motions for 3-RPR Parallel Manipulators

When moving from one arbitrary location at another, a parallel manipulator may change its assembly-mode without crossing a singularity. Because the non-singular change of assembly-mode cannot be simply detected, the actual assembly-mode during motion is difficult to track. This paper proposes a global explanatory approach to help better understand non-singular assembly-mode changing motions for 3-RPR planar parallel manipulators. The approach consists in fixing one of the actuated joints and analyzing the configuration-space as a surface in a 3-dimensional space. Such a global description makes it possible to display all possible non-singular assembly-mode changing trajectories.

Introduction

Most parallel manipulators have singularities that limit the motion of the moving platform. The most dangerous ones are the singularities associated with the direct kinematics, where two assembly-modes coalesce. Indeed, approaching such a singularity results in large actuator torques or forces, and in a loss of stiffness. Hence, these singularities are undesirable.

There exists three main ways of coping with singularities, which have their own merits. A first approach consists in eliminating the singularities at the design stage by properly determining the kinematic architecture, the geometric parameters and the joint limits [START_REF] Chablat | Architecture Optimization of a 3-DOF Parallel Mechanism for Machining Applications, the Orthoglide[END_REF][START_REF] Arsenault | The synthesis of three-degree-of-freedom planar parallel mechanisms with revolute joints (3-RRR) for an optimal singularity free workspace[END_REF]. This approach is safe but difficult to apply in general and restricts the design possibilities. A second approach is the determination of the singularity-free regions in the workspace [START_REF] Li | Determination of maximal singularity-free zones in the workspace of planar three-degreeof-freedom parallel mechanisms[END_REF][START_REF] Kong | Determination of the uniqueness domains of 3-RPR planar parallel manipulators with similar platforms[END_REF][START_REF] Merlet | Determination of the presence of singularities in 6D workspace of a Gough parallel manipulator[END_REF][START_REF] Li | Determination of maximal singularity-free zones in the six-dimensional workspace of the general Gough-Stewart platform[END_REF][START_REF] Wenger | Definition sets for the direct kinematics of parallel manipulators[END_REF]. This solution does not involve a priori design restrictions but, because of the complexity of the singularity surfaces, it may be difficult to determine definitely safe regions. Finally, a third way consists in planning singularity-free trajectories in the manipulator workspace [START_REF] Merlet | Trajectory verification in the workspace for parallel manipulators[END_REF][START_REF] Bhattacharya | Comparison of an exact and an approximate method of singularity avoidance in platform type parallel manipulators[END_REF][START_REF] Dash | Singularity-free path planning of parallel manipulators using clustering algorithm and line geometry[END_REF][START_REF] Sen | Variational approach for singularity-free-path-planning of parallel manipulators[END_REF]. With this solution one is also faced with the complexity of the singularity equations but larger zones of the workspace may be exploited. This paper addresses a feature that has drawn the interest of quite few researchers but yet may concern many parallel manipulators, even planar ones [START_REF] Innocenti | Singularity-free evolution from one configuration to another in serial and fully-parallel manipulators[END_REF][START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF]: the fact that the manipulator can change its assembly-mode without passing through a singularity. Planar parallel manipulators with three extensible leg rods, referred to 3-RPR 1 , have received a lot of attention because they have interesting potential applications in planar motion systems [START_REF] Innocenti | Singularity-free evolution from one configuration to another in serial and fully-parallel manipulators[END_REF][START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF][START_REF] Hunt | Assembly configurations of some In-parallel-actuated manipulators[END_REF][START_REF] Merlet | Parallel Robots[END_REF][START_REF] Gosselin | Singularity analysis of closed loop kinematic chains[END_REF][START_REF] Gosselin | Solution polynomiale au problème de la cinématique directe des manipulateurs parallèles plans à 3 degrés de liberté[END_REF][START_REF] Sefrioui | On the quadratic nature of the singularity curves of planar three-degree-of-freedom parallel manipulators[END_REF]. As shown in [START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF], moreover, the study of the 3-RPR planar manipulator may help better understand the kinematic behavior of its more complex spatial counterpart, the 6-dof octahedral manipulator. Planar parallel manipulators may have up to six assembly-modes [START_REF] Hunt | Assembly configurations of some In-parallel-actuated manipulators[END_REF]. It was first pointed out that to move from one assemblymode to another, a 3-RPR planar parallel manipulator should cross a singularity [START_REF] Hunt | Assembly configurations of some In-parallel-actuated manipulators[END_REF]. But [START_REF] Innocenti | Singularity-free evolution from one configuration to another in serial and fully-parallel manipulators[END_REF] showed, using numerical experiments, that this statement is not true in general. In fact, an analogous phenomenon exists in serial manipulators, which can move from one inverse kinematic solution to another without meeting a singularity [START_REF] Innocenti | Singularity-free evolution from one configuration to another in serial and fully-parallel manipulators[END_REF][START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF][START_REF] Hunt | Assembly configurations of some In-parallel-actuated manipulators[END_REF]. The non-singular change of posture in serial manipulators was shown to be associated with the existence of points in the workspace where three inverse kinematic solutions meet, called cusp points [START_REF] Omri | How to recognize simply a non-singular posture changing 3-DOF manipulator[END_REF]. On the other hand, McAree and Daniel [START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF] pointed out that a 3-RPR planar parallel manipulator can execute a non-singular change of assembly-mode if a point with triple direct kinematic solutions exists in the joint space. The authors established a condition for three direct kinematic solutions to coincide and showed that a non-singular assembly-mode changing trajectory in the joint space should encircle a cusp point. Wenger and Chablat [START_REF] Wenger | Workspace and assembly-modes in fully parallel manipulators: A descriptive study[END_REF] investigated the question of whether a change of assembly-mode must occur or not when moving between two prescribed poses in the workspace. They defined the uniqueness domains in the workspace as the maximal regions associated with a unique assembly-mode and proposed a calculation scheme for 3-RPR planar parallel manipulators using octrees. They showed that up to three uniqueness domains exist in each singularity-free region. When the starting and goal poses are in the same singularity-free region but in two distinct uniqueness domains, a non-singular change of assembly-mode is necessary. However they did not investigate the kind of motion that arises when executing a non-singular change of assembly-mode. For the particular case of a planar 3-RPR parallel manipulator with similar base and platform, Kong and Gosselin [START_REF] Kong | Determination of the uniqueness domains of 3-RPR planar parallel manipulators with similar platforms[END_REF] showed that there is no need to investigate non-singular assembly changing motion because each singularity-free region corresponds to one uniqueness domain.

But 3-RPR manipulators with similar base and platform have a major flaw: the manipulator is singular at all positions when the moving platform assumes a zero orientation . On the other hand, planar 3-RRR manipulators and spatial octahedral manipulators with similar base and platform may change their assembly-mode without encountering a singularity [START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF][START_REF] Wenger | The Kinematic Analysis of a Symmetrical Three-Degree-of-Freedom Planar Parallel Manipulator[END_REF]. If, when moving from one arbitrary pose to another, the manipulator changes its assembly-mode without crossing a singularity, the actual assembly-mode during motion is difficult to track even if the initial assemblymode is known, as there is no ways to detect the change of assembly-mode. Therefore, there is a need to understand the non-singular change of assembly-mode. The main goal of this paper is to investigate the non-singular change of assembly-mode in planar 3-RPR parallel manipulators, and to propose an explanatory approach to plan non-singular assembly-mode changing trajectories. The approach consists in fixing one of the actuated joints and analyzing the configuration-space as a surface in a 3-dimensional space. Such a global description makes it possible to display all possible non-singular assembly-mode changing trajectories.

Preliminaries

Manipulators Under Study

Figure1 shows a planar parallel manipulator with three extensible leg rods. The geometric parameters are the three sides of the moving platform d 1 , d 2 , d 3 and the position of the base revolute joint centers defined by A 1 , A 2 and A 3 . The reference frame is centered at A 1 and the x-axis passes through A 2 . Thus, A 1 = (0, 0), A 2 = (A 2x , 0) and A 3 = (A 3x , A 3y ). The joint space Q is defined by the joint vectors q defined by the lengths of the three actuated extensible links:

[ ] T 1 2 3 = ρ ρ ρ q .
The task space is usually defined by the set of vectors

[ ] T x y α = x
where ( , )

x y are the Cartesian coordinates of one point of the platform in the plane, chosen as B 1 in this paper, and α is the orientation of the platform in the plane with respect to the x-axis. In this paper, the task space will be more conveniently defined by

[ ] T 1 1 ρ θ α = x
where (ρ 1 , θ 1 ) are the cylindrical coordinates of B 1 . With these parameters, indeed, it is possible to consider 2-dimensional slices of the joint space and of the workspace by fixing the joint parameter ρ 1 .

x

A 1 A 2 B 2 B 3 B 1 θ 2 d 1 d 3 d 2 1 y A 3 3 2 3 1

Figure 1: The 3-RPR parallel manipulator under study

To illustrate our work, we refer to the same 3-RPR manipulator as the one used in [START_REF] Innocenti | Singularity-free evolution from one configuration to another in serial and fully-parallel manipulators[END_REF][START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF][START_REF] Merlet | Parallel Robots[END_REF], which has the following geometric parameters:

A 1 = (0, 0), A 2 = (

Kinematic relations

The relation between the joint space Q and the output space can be expressed as a system of non-linear algebraic equations, which can be written as: [  ]

( , ) 0 F = x q (1)
T 1 2 3 ρ ρ ρ = q & & & &
is the joint velocity vector.

Singular configurations

The singularities of the 3-RPR planar parallel manipulators have been extensively studied [START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF][START_REF] Gosselin | Singularity analysis of closed loop kinematic chains[END_REF][START_REF] Bonev | Singularity Analysis of 3-DOF Planar Parallel Mechanisms via Screw Theory[END_REF][START_REF] Hunt | Geometry of Mechanisms[END_REF][START_REF] Mohammadi Daniali | Singularity Analysis of Planar Parallel Manipulators[END_REF]. On a singular configuration of the manipulator, matrix A or matrix B or both of them are singular. In this study, only the singularities of A are of interest.

To derive the singularity equations, it is usual to expand the determinant of A. We use rather a geometric approach that does not involve complicated algebraic calculus. The 3-RPR parallel manipulator is on a singular configuration whenever the axes of its three legs are concurrent or parallel [START_REF] Hunt | Geometry of Mechanisms[END_REF] (Fig. 2).

x

A 1 A 2 B 1 2 1 y A 3 3 B 2 B 3 d 1 d 3 d 2 Axis 1 Axis 3 Axis 2 2 1 3

Figure 2: A 3-RPR parallel manipulator in a singular configuration

The equations of the three leg axes can be written as:

1 1 2 2 2 3 3 3 3 3
(Axis 1) : cos( ) sin( )

(Axis 2) : cos( ) ( ) tan( ) (Axis 3) : cos( ) ( ) sin( ) cos( ) x x y y x y x A θ θ θ ⎧ = ⎪ = - ⎨ ⎪ = - + ⎩ (2)
The condition of these three axes to intersect (possibly at infinity) is:

( ) 2 2 3 1 3 3 3 3 1 2 sin( ) sin( ) sin( ) cos( ) sin( ) 0 x x y A A A θ θ θ θ θ θ θ - + - - = (3)
which is the singularity equation of the manipulator. This expression along with the constraint equations of the manipulator (i.e. writing the fixed distances between the three vertices of the mobile platform B 1 , B 2 , B 3 ) allow us to plot the singular curves in 2-dimensional slices of the joint space (ρ 2 , ρ 3 ) and of the workspace (α, θ 1 ) for a fixed value of ρ 1 [START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF][START_REF] Zein | Singular Curves and Cusp Points in The Joint Space of 3-RPR Parallel Manipulators[END_REF].

Workspace singularities

Figure 3a shows the singular curves in the workspace slice (α, θ 1 ) defined by ρ 1 =17 for the manipulator introduced in Section 2.1. Note that because the space (α, θ 1 ) is a torus (the revolute joints are assumed unlimited) the opposite sides of the square representation in Fig. 3a are actually coincident. Thus, the singularity curves divide the workspace into two connected components called aspects [START_REF] Wenger | Definition sets for the direct kinematics of parallel manipulators[END_REF]. The notion of aspects was first introduced for serial manipulators 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. The aspects were defined as the maximal singularityfree domains in the joint space. The aspects were extended in [START_REF] Wenger | Definition sets for the direct kinematics of parallel manipulators[END_REF] to parallel manipulators with only one inverse kinematic solution such as 3-RPR manipulators. For such manipulators, the aspects are the maximal singularity-free connected regions in the workspace. An equivalent definition was used in [START_REF] Khalil | Kinematic analysis and singular configurations of a class of parallel robots[END_REF] for a special case of parallel manipulators.

For the 3-RPR parallel manipulator at hand, the first (resp. second) aspect is defined by det(A)>0 (resp. det(A)<0),

where A is the Jacobian matrix introduced in Section 2.2.

-3 Figure 3: Singular curves in (α, θ 1 ) for ρ 1 =17 (Fig. 3a). Singular curves in (ρ 2 , ρ 3 ) for ρ 1 =17 (Fig. 3b).

-2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 α 1 Aspect 1 Aspect 2 A s p e c t 1

Joint space singularities

Figure 3b shows the singular curves in the joint space slice (ρ 2 , ρ 3 ) defined by ρ 1 =17. These curves split the joint space slice into several regions with 2, 4 or 6 direct kinematic solutions [START_REF] Zein | Singular Curves and Cusp Points in The Joint Space of 3-RPR Parallel Manipulators[END_REF][START_REF] Zein | An Algorithm for Computing Cusp Points in the Joint Space of 3-RPR Parallel Manipulators[END_REF]. The slice of the joint space is the image by the inverse kinematics of each of aspect 1 and also of aspect 2. That is, plotting either aspect 1 or aspect 2 onto (ρ 2 , ρ 3 )

will define the pattern depicted in Fig. 3b. In each region, the number of direct solutions is equally distributed in the two aspects. In the central 6-solution region, for example, there are 3 solutions in aspect 1 and 3 solutions in aspect 2.

The six points pinpointed with circles are cusp points, where three direct kinematic solutions coincide. As shown in [START_REF] Zein | Singular Curves and Cusp Points in The Joint Space of 3-RPR Parallel Manipulators[END_REF], there may be 0, 2, 4, 6 or 8 cusp points depending on the slice chosen.

Examination of a loop trajectory encircling a cusp point in the joint space

We want to understand how a non-singular change of assembly-mode arises. We consider that no external influence enabling the determination of the assembly mode is possible. We only know that to execute a non-singular assemblymode changing motion, a cusp point must be encircled in the joint space of the manipulator [START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF]. We show in this section that this information is insufficient to determine the actual motion of the manipulator when a cusp point is encircled in its joint space. Let us define a triangular loop trajectory T in a slice of the joint space for 1 17 ρ = , which encircles a cusp point (Fig. 4a). The starting (and final) joint vector [ ]

T 1 2 3 = ρ ρ ρ q is chosen as [ ] T 17 19 17 = q ,
where the direct kinematics at q admits 6 real solutions P i , i=1, 2, …6. Keeping in mind that the opposite sides of the square representation in Fig. 4b are actually coincident, 1 P , 2 P and 3 P are in aspect 1 and the remaining three solutions are in aspect 2. The loop trajectory T crosses the singular curves at four distinct joint configurations, referred to as q a , q b , q c and q d . Along T, the direct kinematics is solved and the solutions are plotted in (α, θ 1 ) (Fig. 4b). According to whether T is executed clockwise or counter-clockwise and according to the initial assembly-mode, a total of 12 motions will result in the workspace. These 12 motions can be classified into the following three types:

• Motions that make the manipulator stop at one of the singular points a S , b S , c S or d S , which are associated with q a , q b , q c and q d , respectively. There are 8 such motions, 4 for each direction of execution of T. When T is executed clockwise the 4 motions are from 2 P to a S , from 1 P to d S , from 5 P to a S and from 6 P to d S . When T is executed counter-clockwise the 4 motions are from 3 P to b S , from 1 P to c S , from 5 P to b S and from 6 P to c S . In each case, the direct kinematic solution associated with the starting assembly-mode is lost at the singular point and this is the reason why the motion stops and T cannot be fully executed. Thus, no assembly-mode changing is feasible with these motions.

• Two loop motions in the workspace starting and ending at 4 P (Fig. 4b). These two loops differ in their direction of execution (clockwise or counter-clockwise), which depends on the direction of execution of T. These motions do not enable the manipulator to change its assembly-mode because the moving platform goes back to its starting pose in the workspace. Unlike the 8 motions described above, for these two loop motions, T is fully executed. These loop motions feature three segments that are associated with the three linear segments of the triangular trajectory T.

• Two non-singular motions that differ only in their direction of motion (from 2 P to 3 P or from 3 P to 2 P , depending on the starting assembly-mode). The path associated with these motions is drawn in dark grey in Fig. 4b. Again, the path is composed of three segments, each associated with a segment of the triangular trajectory T. But in this case, the arrival assembly-mode is different from the starting one. Thus, these two motions are non-singular assemblymode changing motions.

q 2 3 q c q b q a q d -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 α 1 P 2 P 3 P 1 P 4 P 6 P 5 S a S b S d S c ( a ) ( b )
Figure 4: A loop trajectory encircling a cusp point in the joint space section ρ 1 =17 (Fig. 4a) and the associated motions in the workspace (Fig. 4b).

Finally, this analysis raises the following questions or comments:

• Only two assembly-modes, namely 2 P and 3 P , were found to be linkable by a non-singular assembly-mode changing motion. Yet, other non-singular motions should be found in aspect 1 (e.g. between 2 P and 1 P or between 3 P and 1 P ) and even in aspect 2. • What would have been the resulting motions in the workspace if T had encircled another cusp point?

• If one wants to connect two assembly-modes in the workspace associated with the same joint vector q, without crossing a singularity, which cusp point should be encircled?

Clearly, these questions cannot be answered with the sole information provided by the singularity locus in the joint space and in the workspace.

A model for the configuration-space structure 4.1 The configuration-space as a surface in a 3-dimensional space

In the singularity loci shown in Fig. 4, we lose information as these loci result, in fact, from the projection into either (ρ 1 , ρ 2 , ρ 3 ) or (ρ 1 , α, θ 1 ) of the configuration-space CS of the manipulator, which is a 3-dimensional space embedded in the product space of (ρ 1 , ρ 2 , ρ 3 ) and (ρ 1 , α, θ 1 ). When the first leg rod length is fixed, we consider a 2-dimensional slice of the configuration-space, which is thus a surface in a 4-dimensional space (the product space of (ρ 2 , ρ 3 ) and (α, θ 1 )).

Figure 4 represents its projection onto the planes (ρ 2 , ρ 3 ) and (α, θ 1 ). But we cannot depict a surface in a 4-dimensional space.

Because non-singular assembly-mode motions occur only inside an aspect, a configuration-space should be built for each aspect: it is necessary to build CS 1 and CS 2 , the configuration-space in aspects 1 and 2, respectively. In order to show CS 1 and CS 2 , they should be displayed in (ρ 2 , ρ 3 , α) or in (ρ 2 , ρ 3 , θ 1 ) rather than in the product space of (ρ 2 , ρ 3 ) and (α, θ 1 ). But by doing so one should verify that the third parameter is sufficient to describe fully the configuration of the manipulator in an aspect when arbitrary values of the three actuated joints are given. It has been shown recently that for some 3-RPR manipulators, called degenerate manipulators, two distinct assembly-modes are always associated with the same α and these two assembly-modes may lie in the same aspect [START_REF] Wenger | Degeneracy study of the forward kinematics of planar 3-RPR parallel manipulators[END_REF]. On the other hand, no manipulators exist that have always two distinct assembly-modes per aspect for the same value of θ 1 [3232]. Thus, it is possible to build CS 1 and CS 2 in (ρ 2 , ρ 3 , θ 1 ) but not in (ρ 2 , ρ 3 , α).

Construction of the configuration-space in each aspect

Let us fix ρ 1 to the value corresponding to the current configuration of the manipulator. To build CS 1 and CS 2 , the slice (ρ 2 , ρ 3 ) of the joint space is scanned and the parameter θ 1 is determined by solving the characteristic polynomial in θ 1 [START_REF] Wenger | Degeneracy study of the forward kinematics of planar 3-RPR parallel manipulators[END_REF]. The sign of the determinant of the Jacobian matrix is determined. If this sign is positive, the resulting point (ρ 2 , ρ 3 , θ 1 ) is plotted in CS 1 otherwise it is plotted in CS 2 . Figure 5 shows CS 1 and CS 2 when ρ 1 =17. We have used a CAD interface to build a 3D mesh surface-plot and to enable rotating the viewing perspective. The singular curves, which define the boundaries of CS 1 and CS 2 , have been displayed in bold lines. Their projections onto (ρ 2 , ρ 3 ) are shown in the figure. Assume the manipulator is in the configuration q shown in Fig. 4. The points associated with the six assemblymodes P 1 , P 2 ,…, P 6 are shown (these points have been labeled as P 1 , P 2 ,…, P 6 in Fig. 5 like in Fig. 4b for more simplicity). The first three are in aspect 1 while the last three are in aspect 2. This representation shows all feasible singularity-free motions between P 1 , P 2 and P 3 as well as those between P 4 , P 5 and P 6 .

Examples of non-singular assembly-mode changing trajectories

The Figure 6 shows a non singular assembly-mode changing trajectory connecting the two assembly-modes P 1 and P 2 . We notice that the encircled cusp point is different from the one encircled by the trajectory T shown in Fig. 4a. (Fig. 6a), in (ρ 2 , ρ 3 ) (Fig. 7b).

Finally, Fig. 8 shows a non-singular assembly-mode changing trajectory connecting the two assembly-modes P 1 and P 3 . Figure 8: Non-singular trajectory connecting the two assembly-modes P 1 and P 3 in the first aspect in (ρ 2 , ρ 3 , θ 1 ) (Fig. 8a), in (ρ 2 , ρ 3 ) (Fig. 8b).

We notice that to connect the assembly-modes P 1 and P 3 , the trajectory in the joint space encircles two cusp points as shown in Fig. 8b. In fact, we have verified that any path from P 1 and P 3 must encircle these two cusp points. This fact can be explained by the "layered" structure of the configuration-space surfaces (see comments below). This is a new result, since it was thought that only one cusp point should be encircled when executing a non-singular assembly-mode changing maneuver [START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF]. Clearly, this result could not have been discovered with the sole joint space and workspace representations shown in Fig. 4.

Comments

Using the configuration-space surface model, the problem of which cusp point should be encircled to plan a nonsingular assembly-mode changing motion needs not be solved. However, it is possible to know the response to this question by projecting onto (ρ 2 , ρ 3 ) the path built on CS 1 or CS 2 .

The two configuration-space surfaces CS 1 and CS 2 may be regarded as being composed of three adjacent "layers" L 1 , L 2 and L 3 with respect to the θ 1 -coordinate. These three layers are associated with the three assembly-modes P 1 , P 2 and P 3 .

The first two layers and the last two layers are adjacent and it is possible to move between them with a smooth path. The presence of a cusp point in (ρ 2 , ρ 3 ) accounts for the existence of a continuous link between two layers of CS 1 or of CS 2 .

This is why moving from one layer to another (that is, moving from one assembly-mode to another without meeting a singularity) is equivalent to encircling a cusp point. This is because to move from L 1 to L 3 , one has to go on the intermediate layer L 2 that two cusp points must be encircled in (ρ 2 , ρ 3 ) when connecting the two assembly-modes P 1 and P 3 .

Conclusions

A global explanatory approach was proposed in this paper to help better understand non-singular assembly-mode changing motions for 3-RPR planar parallel manipulators. It has been shown that the joint space and the workspace are not sufficient to describe the non-singular motions between assembly-modes. The proposed approach consists in fixing one of the actuated joint. Then the configuration-space is reduced to two surfaces in a 3-dimensional space, one for each aspect. Such a global description provides sufficient information as to the configuration space topology and makes it possible to display all possible non-singular assembly-mode changing trajectories that operate with one locked actuator. This approach will be extended to the analysis of non-singular assembly-mode changing trajectories in 6-DOF octahedral parallel manipulators. These manipulators are particular Gough-Stewart platforms that feature a triangular base and a triangular moving platform, connected by six double-spherical-joint-ended rods. This manipulator has eight triangular faces (the base-triangle, the platform-triangle and 6 leg-triangles). The idea is to fix all but two joint coordinates and to build the configuration-space surfaces in (ρ 5 , ρ 6 , θ 1 ), where ρ 5 and ρ 6 are the two free joint coordinates and θ 1 is the dihedral angle between the base and one of the leg-triangles.

We think that the work provided in this paper constitutes a first step to the difficult problem of how to identify the different assembly-modes of a parallel manipulator and how to track the assembly-mode during motion.

  15.91,0), A 3 = (0, 10), d 1 = 17.04, d 2 = 16.54 and d 3 = 20.84 in an arbitrary length unit.

Differentiating equation ( 1 )

 1 with respect to time leads to the velocity model: ω is the scalar angular velocity and c & is the two-dimensional velocity vector of the operational point B 1 of the platform. A and B are 3×3 Jacobian matrices which are configuration dependent, and

Figure 5 : 1 ,

 51 Figure 5: The configuration-space CS 1 (Fig. 5a) and CS 2 (Fig. 5b) associated with aspects 1 and 2, respectively, along with the six assembly-modes calculated at

  following three figures display non-singular assembly-mode changing trajectories in aspect 1. The paths are constructed in (ρ 2 , ρ 3 , θ 1 ). We have used a CAD-interface to define the paths from a set of intermediate points and linear segments. In the three examples shown in Figs 6, 7 and 8, we have defined the path with three intermediate points. The projection of the paths onto (ρ 2 , ρ 3 ) are displayed in each figure to show which cusp is encircled in the joint space. The viewpoints have been chosen to show as clearly as possible the paths in CS 1 .

Figure 6 :

 6 Figure 6: Non-singular assembly-mode changing trajectory connecting the two assembly-modes P 1 and P 2 in the first aspect in (ρ 2 , ρ 3 , θ 1 ) (Fig. 6a), in (ρ 2 , ρ 3 ) (Fig. 6b).

Figure 7

 7 Figure7shows a non-singular assembly-mode changing trajectory connecting the two assembly-modes P 2 and P 3 . The

Figure 7 :

 7 Figure 7: Non-singular trajectory connecting the two assembly-modes P 2 and P 3 in the first aspect in (ρ 2 , ρ 3 , θ 1 )(Fig.6a), in (ρ 2 , ρ 3 ) (Fig.7b).

R and P stand for Revolute and Prismatic, respectively. The underlined letter refers to the actuated joint