Radu Cârpa
email: radu.carpa@ens-lyon.fr

Marcos Dias

D E Assunção

Olivier Glück
email: olivier.gluck@ens-lyon.fr

Laurent Lefèvre
email: laurent.lefevre@inria.fr

Jean-Christophe Mignot
email: jean-christophe.mignot@ens-lyon.fr

Evaluating the Impact of SDN-Induced Frequent Route Changes on TCP Flows

Traffic engineering technologies such as MPLS have been proposed to adjust the paths of data flows according to network availability. Although the time interval between traffic optimisations is often on the scale of hours or minutes, modern SDN techniques enable reconfiguring the network more frequently. It is argued, however, that changing the paths of TCP flows too often could severely impact their performance by incurring packet loss and reordering. This work analyses and evaluates the impact of frequent route changes on the performance of TCP flows. Experiments carried out on a network testbed show that rerouting a flow can affect its throughput when reassigning it a path either longer or shorter than the original path. Packet reordering has a negligible impact when compared to the increase of RTT. Moreover, constant rerouting influences the performance of the congestion control algorithm. Designed to assess the limits on SDN-induced reconfiguration, a scenario where the traffic is rerouted every 0.1s demonstrates that the throughput can be as low as 35% of that achieved without rerouting.

I. INTRODUCTION

Although the "best effort" service that the Internet Protocol (IP) provides has been able to handle a wide range of workloads, current infrastructure requires intelligent traffic management to enable sharing resources between delay critical audio flows and best-effort traffic. Traffic engineering techniques are widely used by network operators to provide better Quality of Service (QoS) via fine-grained traffic management. MultiProtocol Label Switching (MPLS) is the classical protocol used to this end.

MPLS-based traffic engineering allows for dynamically configuring how the traffic flows across the network based on available network resources, and for changing the flow routes dynamically to meet QoS requirements. Unfortunately, route changes induce jitter that can perturb overlaying transport protocols. For example, the TCP congestion control algorithm may improperly assume that some data segments are lost and hence reduce its sending rate. Hence, in industrial deployments, operators usually avoid frequent route changes.

Software Defined Network (SDN), on the other hand, enables applications to reconfigure the network more frequently than previous technologies. Large service providers, including Microsoft [START_REF] Hong | Achieving high utilization with software-driven wan[END_REF] and Google [START_REF] Jain | B4: Experience with a globallydeployed software defined wan[END_REF], have deployed SDN in production environments, thus increasing the efficiency of their network. However, the frequency of reconfigurations that these organisations perform is as low as once every 5 minutes [START_REF] Hong | Achieving high utilization with software-driven wan[END_REF],

and not all data flows are rerouted during these optimisations because doing so could severely hurt the performance.

The question this work aims to address is therefore: what would be the impact of highly frequent rerouting decisions on network traffic? To answer this question, we evaluate the impact of frequent route changes on the performance of Cubic TCP flows. We consider backbone networks with high aggregation level employing SDN to improve their energy efficiency. Previous work introduced algorithms to minimise the number of active system components to reduce its overall energy consumption [START_REF] Carpa | Improving the energy efficiency of software-defined backbone networks[END_REF], [START_REF] Cârpa | Responsive algorithms for handling load surges and switching links on in green networks[END_REF]. Flows are aggregated on a subset of links to put network elements in sleep mode, hence maximising the utilisation of active components. Frequent and fast route changes are needed for both reacting quickly to traffic bursts and waking up sleeping devices to avoid congestion. The changes are made either to provide more network capacity or to detect network over-subscription and thus turn off active components to save energy.

By evaluating whether it is efficient to change the routes of TCP flows frequently, the analysis focuses on the impact of packet reordering due to the difference in delay between two routes. The employed methodology may be relevant and applicable to any network operator willing to perform aggressive traffic engineering with frequent route changes.

The rest of the paper is structured as follows. Section II presents the context and background while Section III describes the theoretical analysis. Experimental setup and validations are presented and analysed in Section IV. Section V discusses related work and Section VI concludes the paper.

II. CONTEXT AND BACKGROUND

A. Context

We consider an operator with an energy-efficient network whose resources are underutilised at times, and the data flows are hence rerouted to release links and/or devices that can henceforth be turned them off. When an increase of network traffic is detected, the network devices are switched back on, and the flows are rerouted accordingly to avoid congestion.

The operator employs SDN, a recent trend in network management that advocates for separating the control plane from the data plane. The goal is to move towards more programmable networks, where very sophisticated optimisation can be done online by a centralised controller, which has global knowledge of the network status and can automatically react to changes by rerouting data flows. For example, after moving to SDN, Google advertises near 100% link utilisation in their network while maintaining a very high quality of service [START_REF] Jain | B4: Experience with a globallydeployed software defined wan[END_REF].

B. Rerouting and congestion control

The TCP's congestion control algorithm, defined in the RFC 5681, is an important, and one of the most complex, features of modern TCP implementations. This algorithm tries to split the network capacity fairly among all the flows traversing it. Under such algorithm, the sender keeps a congestion window that is dynamically modified depending on the network conditions. The source cannot send more data than what fits in this window during a Round-Trip Time (RTT). The maximum instantaneous throughput, B, of the TCP connection parameters, is thus limited by the size of the congestion window: B = W • M SS/RT T , where W is the size of the congestion window in segments, M SS is the maximum segment size in bytes.

TCP was initially designed for standard IP routing assuming that all packets follow the same route towards a destination. Packet reordering and route changes were considered rare. However, in an SDN network, the controller may frequently shift a TCP flow to an alternative path to optimise the overall network throughput. The route change can impact the throughput of a TCP flow in two different ways:

• When the new route has higher RTT, the sender increases the size of the congestion window to maintain the same sending rate. For example, if the RTT doubles, the throughput will be halved and will gradually increase with the growth of the TCP congestion window; a direct application of the equation B = W • M SS/RT T . • The receiver will see packets arriving out of order if the new route has a lower RTT. TCP's congestion control algorithms will assume the worst-case scenario and will see this as an indication of packet loss due to congestion. Figure 1 illustrates the problem considering a sample backbone network. The link ab becomes available for transmission and hence packets 2 and 3 take a route shorter than packet 1. The packets will arrive out of order at the receiver who will use duplicate ACKs to notify the sender about a problem. The sender will reduce the size of the sending window to avoid congestion, hence decreasing the transmission rate. Given that a large part of network traffic is often carried by a small number of large, long-lived flows [START_REF] Qian | A flow-based performance analysis of tcp and tcp applications[END_REF], rerouting these flows may significantly reduce the overall network throughput. We encountered this problem in practice on a testbed set up to evaluate energy-efficient algorithms [3] [4]. These algorithms, implemented in the SDN controller ONOS [START_REF]Introducing ONOS: A SDN network operating system for service providers[END_REF], were reacting too fast to a reduction in the network throughput caused by a recent rerouting. While the algorithms tried to take advantage of this transitory condition to save energy, they created undesired traffic oscillations.

C. Congestion Control Algorithms

The analysis focuses on the impact of route changes on the default TCP congestion-control mechanism in the Linux kernel. As of writing, the default algorithm used in all the tested devices is Cubic, with a fallback to Reno, including devices using kernel versions 3.2, 3.13, 4.4 and 4.7. Figure 2 shows how the congestion window of a single flow evolves when using the Reno and Cubic congestion control algorithms.

1) RENO: The standard TCP congestion control mechanism is Reno, which is still the fallback algorithm in the Linux kernel. This algorithm is straightforward and comprises two phases: 1) the slow start phase, where the window size doubles each RTT; and 2) the congestion control phase where the window size increases by one each RTT or is divided by two under packet loss.

The slow start phase allows for a fast increase in the window size at the beginning of communication. It corresponds to the spike at time 0 in Figure 2. In this work, we ignore the slow start.

Although TCP Reno behaves well under small Bandwidth x Delay Product (BDP), it severely underutilises the channel on long fat networks -i.e. networks with high data rate and high delay -because it linearly grows the windows by one every RTT when recovering from packet loss. In Figure 2, Reno takes more than a minute to grow the congestion window and fill a 100Mbps link with an RTT of 50ms.

2) CUBIC: The Cubic TCP algorithm [START_REF] Ha | Cubic: A new tcp-friendly high-speed tcp variant[END_REF] aims to avoid the shortcomings of Reno and achieve high data rates in networks with large BDP. At the same time, when Cubic detects a network with small BDP, it tries to mimic Reno's behaviour emulated by a mathematical model. This happens, particularly in local high-speed, low-delay, networks.

As its name suggests, the algorithm grows the congestion window by using a cubic function of the elapsed time from the last loss event y = (∆t) 3 . More precisely, using a shifted and scaled version y = 0.4 • (∆t -RT T -K) 3 + W max . In this equation, ∆t is the time passed from the last loss event, W max is the size of the congestion window just before the loss event. K = (W max /2) 1/3 is a parameter that depends on W max .

Cubic is also less aggressive in reducing the congestion window at a loss event. Compared to Reno, which halves the window, Cubic reduces it by 20%.

III. ESTIMATING THE BEHAVIOUR OF TCP CUBIC UNDER ROUTE CHANGES

A. Recovering from a Loss The growth function, y, used to increase the congestion window depends mainly on W max , which makes it easy to estimate the time taken by TCP cubic to restore its congestion window after a loss. To do that, we ignore the insignificant dependency on RTT in the Cubic function and solve

W max = y = 0.4 • (∆t -K) 3 + W max . We obtain ∆t = K = (W max /2) 1/3 .
Moreover, the RTT and the bottleneck bandwidth are used to determine the maximum window size. This estimation is valid only if no further loss is detected. Figure 3 contains a counter-example where TCP would reach W max at approximately t = 36s, but a loss happened at ∼34s and forced the congestion control algorithm to reduce the size of the congestion window.

While particularly interested in the time needed to recover from a loss, we note that route changes may generate false loss events when the TCP flow shifts towards a shorter route -shorter here meaning a route with smaller round trip time. At such time, the size of the congestion window is reduced, and cubic tries to grow it back. Intuitively, the throughput of a TCP flow may drop significantly if the SDN controller attempts a second re-optimisation in the meantime.

B. Relevant Linux Implementation Details

The TCP implementation in the Linux kernel incorporates many optimisations [START_REF] Sarolahti | Congestion control in linux tcp[END_REF]. One of the optimisations, which is very important for our work, is the possibility to undo an adjustment to the congestion window. This implementation tries to distinguish between packet reordering and loss by using the "Timestamp" TCP option. When the sender detects that a past loss event was actually a false positive due to packet reordering, the algorithm reverts the window size to the value used before the reduction. As a result, packet reordering may have much less impact compared to standard TCP.

C. Multiple TCP Flows on a Bottleneck Link

As mentioned earlier, TCP cannot send more data than the size of the congestion window per RTT. As a result, when the sender's window W is smaller than W max , TCP spends part of the time waiting for acknowledgements without sending any data. Figure 4 illustrates this case, summarising data collected using the tcpdump application to capture the packets of a TCP flow over a second. The figure shows the times packets were captured. Approximately 2/3 of the time, there was no packet passing through the network because the congestion window is too small (only 150 segments). Under such conditions, a congestion window of 431 segments was needed to fill the bottleneck link.

If a route change happens when a flow is waiting for acknowledgements, there is no re-ordering problem since no packet is sent. Hence, the probability that a TCP destination will receive packets out of order increases as W approaches W max . Respectively, when TCP backs off and lowers its sending rate by reducing W , it also reduces the probability to be impacted by the rerouting.

If a large number of TCP flows share the same bottleneck link, the flows spend a lot of time waiting for the ACKs. Hence, rerouting a large bunch of TCP flows at the same time may have less impact on the overall network throughput than rerouting a single TCP flow.

This TCP behaviour may change in the future. The benefit of smoothing the transmission over time by employing traffic pacing was shown to have a beneficial effect on network performance [START_REF] Gharakheili | Edge versus host pacing of tcp traffic in small buffer networks[END_REF] and state-of-art congestion control algorithms, like BBR [START_REF] Cardwell | Bbr: Congestion-based congestion control[END_REF], use this technique to avoid the bufferbloat problem.

IV. EXPERIMENTAL EVALUATION

A. Experimental Setup

We consider backbone, highly over-provisioned, networks where resource capacity is not a limiting factor. The transmission speed is either bounded by a congested link in the aggregation, or by a very high propagation delay, i.e. by the size of the maximum allowed congestion window. No congestion ever occurs on the backbone links. We construct the topology presented in Figure 5. The examined scenario consists of multiple TCP sources sending data to remote clients. The transfers pass through a backbone network employing traffic engineering: the two parallel 10Gbps links. The traffic is routed through 2 alternative paths within the backbone network, emulated by these 2 links. The paths have different delays. One of the backbone paths has no additional delay (0ms), while the second path induces an additional delay of X ms to simulate a longer route. We use tc-netem for this purpose. Each of the two paths (links) has sufficient capacity to transfer the flows. The bottleneck capacity is outside the backbone network. Congestion never happens on these 2 links.

The sources are aggregated in groups. The flows from 2 different groups never share a bottleneck point. However, two flows of the same group compete for the bandwidth allocated to the group. Each group has a different path RTT, uniformly distributed in the interval [0ms, 120ms], which adds to the delay of the backbone paths. The goal is to recreate the scenario where flows with different end-to-end delay coexist in the backbone network.

Concerning the physical infrastructure: nodes in the topology correspond to Ubuntu 14.04 Linux servers running Open vSwitch v2.4 [START_REF] Pfaff | The design and implementation of open vSwitch[END_REF] (manually compiled for Ubuntu 14.04). The network is controlled by the ONOS SDN controller (v1.7) via an out-of-band control connection. We wrote an ONOS SDN application to reroute the flows between the 2 backbone links.

The backbone links use 10G optical Ethernet interconnects. The connections to the TCP sources use 1G Ethernet ports. In the physical topology, both 10G and 1G network interconnects pass through Dell Ethernet switches. The point-to-point links are emulated using vlans on these hardware switches. This permits to reconfigure the network topology remotely. Unfortunately, the queuing disciplines on these switches are opaque and cannot be fine tuned. To avoid perturbations, we use tctbf (Token Bucket Filter) to limit the speed of each group to 100Mbps. In such a way, the data passing through a hardware switch is always much below the link capacity. Moreover, this allows avoiding perturbations due to the Ethernet flow control mechanisms: PAUSE frame. We use a drop-tail bottleneck queue having the size 0.1 * BDP.

B. Rerouting Independent Flows

This section analyses the impact of rerouting multiple independent TCP flows and how the total backbone throughput changes as a consequence of rerouting.

We generate 20 unsynchronised flows, one flow per group, that start at random times within a 30-second interval. We run the transfer for 400 seconds to allow TCP flows to converge to a steady state. Afterwards, we reroute the traffic every 200 seconds and measure how the aggregated throughput changes at during rerouting.

Figure 6 shows how the aggregated throughput varies when the second backbone path induces an additional delay of 10ms, 30ms, 50ms, 70ms or 90ms.

At t = 600, the flows are rerouted towards the path with lower delay. Respectively, at t = 400 and t = 800, the flows are rerouted towards the longer path. This experiment confirms our expectations, that rerouting the flows impacts their throughput both when moving towards a longer route and when moving towards a shorter one.

Nevertheless, packet reordering, which happens at t = 600, has a less pronounced impact than expected. Theoretically, Cubic would reduce the transmission window to 80% of its size when a loss is detected. Then it would gradually increase the window back to the size before the loss. However, the Linux TCP can detect packet reordering in some cases. In particular, this happens when the difference of delay between two routes is small compared to the delay of the fastest route. Hence, a small variation of delay, like 10ms, has a marginal impact on the total throughput of the flows. The next section gives more insights on this optimisation. Rerouting towards the link with higher RTT substantially reduces the aggregate throughput of the 20 flows (t = 400 and t = 800). The larger the delay difference between the two routes, the bigger the drop. In this case, rerouting is transparent to the congestion control algorithm. The drop comes from the unexpected increase of the delay. As a consequence, the size of the transmission window must be increased to allow transmission at the same speed.

C. Rerouting Flows Sharing a Bottleneck Point

To evaluate the impact of sharing a bottleneck point where flows inside a group compete for bandwidth, we fix the RTT of the second path to 50ms and vary the number of flows per group. With one flow per group, 20 flows traverse the network whereas, under 9 flows per group, there is a total of 180 flows.

Figure 7 shows how competing flows impact the recovering speed after rerouting. The throughput drop caused by rerouting towards a shorter route becomes quickly insignificant (t = 600s) in contrast to rerouting towards a longer path (t = 400s). With the increase in the number of flows sharing a bottleneck link, each of these flows gets a smaller proportion of the total bottleneck bandwidth. The flows hence spend more time waiting for acknowledgements than actually sending data. At rerouting, fewer flows see their packets arrive out-oforder. Moreover, the flows that do not experience out-of-order arrivals can increase their sending rate by using the capacity released by flows that have just reduced the sending window. As a consequence, the total aggregated throughput on the backbone links is almost not impacted.

When rerouting flows towards a longer path, the aggregate throughput recovered slightly quicker when the number of competing flows per group increases. It is due to non-linear growth function used by Cubic. Having more flows increases the probability that some of them will be in the "aggressive growth" phase of the cubic function.

D. One TCP Flow under Frequent Rerouting

The previous sections showed that packet reordering has negligible impact compared to the increase of RTT. In this section, we assess whether very frequent route changes may reorder enough packets to perturb the congestion control algorithm even if the delay difference between the two considered routes is small compared to the RTT.

While focusing on analysing a single TCP flow, we use iperf to transfer 2Gbytes of data and measure its mean throughput (on testbed from Figure 5). A big enough transfer size was chosen to reduce the impact of the slow start phase on the mean throughput. Considering a maximum bottleneck throughput of 100Mbps, each transfer takes approximately 3 minutes if transferred at full speed. The flow is frequently rerouted between the two paths. We tested with periods of rerouting going from once every 15 seconds, to as low as once every 0.1s.

Figure 8 summarises the results. Considering five experiment runs, each data point in the graph corresponds to mean throughput, i.e. the amount of transferred data divided by the total transmission time. The error intervals show the absolute minimum and maximum values recorded. The baseline without rerouting shows the average throughput of a TCP flow when it always passes through the same path. The baseline is the mean of 20 experiments. The inner plot zooms in on the region with frequent route changes. Frequent route changes have a large impact on the performance of the congestion control algorithm. When the traffic is rerouted once every 0.1s, the throughput is as low as 35% of the througtput without rerouting. We can also observe that with a route change every 2 seconds, the throughput is around 60% compared to the throughtput without rerouting. Although, from a practical point of view, we believe that these frequencies of rerouting are extreme (and not very realistic), they were included here to evaluate the limits on traffic rerouting. To re-optimise the network at such frequencies, a lot of control messages must be transmitted between the SDN controller and the switches, creating a flood of control traffic. Moreover, the network optimisation problems are usually computationally intensive and take the time to find a good solution.

It is worth noting that the biggest drop in throughput is observed when a second rerouting happens before Cubic recovers from the first rerouting. Previously, we gave an estimation of the recovery time in Table I.

Figure 9 shows two interesting results for higher RTT: 1) Frequent rerouting has a beneficial effect under an RTT of 170, observed at period = 0.6s (the first ellipse) where the mean throughput of a flow rerouted every 0.6s is higher than the average throughput of a flow that is always transmitted over the same path. 2) Under RTT=480, the throughput of the flows is also less impacted when rerouted frequently. These results are consistent among the multiple transfers. The error intervals are very tight and difficult to see in the figure . To explain this behaviour, we analyse in detail the evolution of the congestion window and packets traversing the network. We choose to concentrate on the points marked with ellipses in Figure 9.

a) Inspecting the beneficial effect at RTT=170: Figure 10 gives more insights on this case. The dashed line shows the evolution of the congestion window without rerouting. At times t = 23, t = 43, etc, the congestion control algorithm does not behave as expected. Instead of reducing the congestion window by 20% as dictated by the Cubic algorithm, the drop is much higher. Hence Cubic claims bandwidth too aggressively and huge number of packets are lost when reaching the bottleneck capacity. Moreover, Linux TCP does not employ a multiplicative decrease technique of reducing the congestion window. Instead, it reduces it additively at every second duplicate ACK received. These two causes together imply a bad performance at RTT=170.

Rerouting the flow in the meantime perturbs the Cubic protocol and avoids this unwanted behaviour. It is due to the optimisation in the Linux kernel which detects out-oforder deliveries thus reducing the impact of the rerouting. The evolution of the congestion window in the case when the route changes every 0.6s can be seen on the continuous, blue, line of Figure 10. For example, at t = 8, a duplicate ACK is detected as being due to reordering and not to a loss, the congestion window is restored to the size before the reduction.

We mentioned in an earlier section that the TCP sender might "miss" a rerouting because the sender alternates between sending packets and waiting for ACKs. If the route change happens when the sender waits for ACKs, no packets will arrive out-of-order. The blue line illustrates this case. At small window size, between t = 0 and t = 30, the sender spends a lot of time waiting. As a result, the probability to be impacted by a rerouting is small. The bigger the congestion window, the higher the probability to be impacted by a rerouting. Starting with t = 120, an equilibrium is created. The sender struggles to grow the window any further: any rerouting has a high probability of reordering packets. b) Inspecting the unexpected behaviour at RTT=480: Figure 11, which analyses the evolution of the congestion windows, shows that the window remains constant if the routes do not change. This behaviour is expected as the transmission is limited by the size of the congestion window, which in turn is constrained by the size of the in-kernel buffer reserved for the TCP connection.

If we compare the evolution of congestion window in case of rerouting, we observe that the Linux Kernel's optimisation, which detects out-of-order packets, is more susceptible to work as expected at frequent route changes. This unexpected behaviour is consistent among multiple transfers.

However, rerouting once every 15 second allows the flow to converge towards the maximum window size. Any bigger rerouting period (20s, 30s, etc) will have less impact on the average throughput. The worst case scenario occurs at a period of 14s.

V. RELATED WORK

Although the impact of packet reordering on the performance of TCP flows has been extensively analysed by the research community, to the best of our knowledge, previous work has not specifically evaluated the impact of frequent flow rerouting. Moreover, most of the existing work employed discrete-event simulation for evaluating the impact of link fluctuations on TCP flows [START_REF] Ranadive | Some observations on the effect of route fluctuation and network link failure on tcp[END_REF], [START_REF] Baccelli | Interaction of tcp flows as billiards[END_REF]. The present work considers a real life SDN controller and testbed.

Bennett et al. [START_REF] Bennett | Packet reordering is not pathological network behavior[END_REF] were among the first to highlight the frequency of packet reordering and its impact on the throughput of TCP flows. Their results showed that packet reordering has a very strong effect on a network's performance. Laor and Gendel [START_REF] Laor | The effect of packet reordering in a backbone link on application throughput[END_REF] have experimentally measured this impact on a testbed and also concluded that even a small rate of packet reordering could significantly affect the performance of a high bandwidth network. As a result, a lot of work was performed to increase the tolerance of TCP to packet reordering [16] [17] [18] [START_REF] Ludwig | The eifel algorithm: Making tcp robust against spurious retransmissions[END_REF].

Finally, a solution both to detecting unneeded retransmissions caused by packet re-ordering and to undoing the congestion window was proposed and implemented. The problem of packet reordering was less considered until the introduction of Multi-Path forwarding, where packets of a single flow are split among different paths [START_REF] Karlsson | Impact of multi-path routing on tcp performance[END_REF] [START_REF] Dixit | On the impact of packet spraying in data center networks[END_REF]. It is worth noting that different seetings lead to different conclusions, for example general networks vs fat trees. Karlsson et al. [START_REF] Karlsson | Impact of multi-path routing on tcp performance[END_REF], for example, concluded that multi-path forwarding reduces the throughput of TCP flows and that mitigation techniques implemented at the transport layer in the Linux kernel are not effective in reducing the impact of packet reordering. On the other hand, Dixit et al. [START_REF] Dixit | On the impact of packet spraying in data center networks[END_REF] affirm that multi-path forwarding in fat-tree data-center networks has little impact on the throughput. In any case, modern multi-path forwarding techniques try to act on a flowlet level and avoid reordering [START_REF] Katta | Hula: Scalable load balancing using programmable data planes[END_REF].

What differentiates our case from the multi-path forwarding in data-center networks is the fact that packet reordering does not arrive on a per-packet base. Rerouting a flow induces bursts of packets arriving out of order. As a result, our work shows that transport level mitigation techniques are effective as long as route changes do not happen too often.

VI. CONCLUSION

This work analysed the impact of frequent route changes on Cubic TCP flows. Although, to the best of our knowledge, the literature does not confirm that packet reordering caused by route changes has a noticeable impact on the network performance, the networking community usually considers it as an implicit truth.

By attempting to quantify this impact on a real testbed, experiments showed that packet reordering incurred by rerouting towards a route with lower delay has actually marginal impact on the throughput of rerouted flows. This negligible impact is partially due to optimisations introduced by Linux kernel developers. As a result, the sender can distinguish between a packet loss and reordering and recovers efficiently with negligible drops in transmission speed. Only when rerouting happens unrealistically frequently, we observe a degradation of the throughput of individual flows, but even so, the effect decreases as the flow aggregation level increases.

We also showed that shifting the traffic towards a path with longer RTT has, however, a negative influence on the throughput of TCP flows. SDN-based traffic engineering techniques must be constrained to avoid large variations in the end-to-end delay when rerouting the flows.

In any case, the traffic engineering SDN applications must be made aware of the short drop of network throughput following a rerouting and must limit the frequency of network re-optimisations. Otherwise, this throughput drop caused by the reaction of the TCP congestion control mechanisms may be incorrectly interpreted as a need for further optimisation of network flows.

Figure 1 .

 1 Figure 1. Rerouting towards a lower RTT ⇒ packet reordering.

Figure 2 .

 2 Figure 2. TCP congestion control; RTT = 50ms; 100mbps link.

Figure 3 .

 3 Figure 3. Function used by the Cubic algorithm; RTT = 50ms.

Figure 3

 3 Figure3illustrates the cubic function when applied to part of the trace from Figure2. At time t = 29.43, a loss is detected when reaching the bottleneck capacity of a network path. Cubic hence reduces the congestion window to 80% of W max .The growth function, y, used to increase the congestion window depends mainly on W max , which makes it easy to estimate the time taken by TCP cubic to restore its congestion window after a loss. To do that, we ignore the insignificant dependency on RTT in the Cubic function and solveW max = y = 0.4 • (∆t -K) 3 + W max . We obtain ∆t = K = (W max /2) 1/3 .Moreover, the RTT and the bottleneck bandwidth are used to determine the maximum window size. TableIestimates the size of the congestion window needed to transmit at 100Mbps with 1448 byte segments. It also uses the equation ∆t = K

Figure 4 .

 4 Figure 4. Alternated sending and waiting phases.

Figure 5 .

 5 Figure 5. Testbed overview

Figure 6 .

 6 Figure 6. Throughput drop when rerouting 20 independent cubic flows

Figure 7 .

 7 Figure 7. Impact of the number of flows in each of 20 groups.

Figure 8 .

 8 Figure 8. Throughput of a TCP flow. Periodic rerouting between 2 routes with different RTT; Low RTT.

Figure 9 .

 9 Figure 9. Throughput of a TCP flow. Periodic rerouting between 2 routes with different RTT; High RTT.

Figure 10 .Figure 11 .

 1011 Figure 10. Evolution of the congestion window at RT T = 170ms.

  TableIestimates the size of the congestion window needed to transmit at 100Mbps with 1448 byte segments. It also uses the equation ∆t = K to assess the time taken by the Cubic algorithm to grow the window back to W max after a loss. * the default maximum size of the kernel buffer is too small to allow full speed communications with RT T = 500ms. In this case, the transmission speed is limited by the size of the kernel buffer; approximately 2100 segments.

	RTT (ms)	10	50	180	500
	Wmax (segment)	87 431 1550 4316.3 *
	Time to recover (s) 3.5	6 13 Table I 9.2
	ESTIMATED Wmax AND TIME NEEDED TO RECOVER AFTER A LOSS,
	100MBPS BOTTLENECK LINK.

*

ACKNOWLEDGEMENTS

This work is supported by the ELCI project, a French FSN ("Fonds pour la Société Numérique") project that associates academic and industrial partners to design and provide software environment for very high performance computing. This work is also supported by the CHIST-ERA STAR [23] project. We also thank Daniel VALENTIN for implementing the ONOS application used for traffic rerouting.


I. INTRODUCTION

T HE control of mechanical systems subject to unilateral constraints has been the object of many studies in the past fifteen years. Such systems, which consist of three main ingredients (see (1) below) are highly nonlinear nonsmooth dynamical systems. Theoretical aspects of their Lyapunov stability and the related stabilization issues have been studied in [START_REF] Brogliato | Absolute stability and the Lagrange-Dirichlet theorem with monotone multivalued mappings[END_REF], [START_REF] Leine | Stability properties of equilibrium sets of nonlinear mechanical systems with dry friction and impact[END_REF], [START_REF] Leine | Stability and Convergence of Mechanical Systems with Unilateral Constraints[END_REF], [START_REF] Tornambé | Modeling and control of impact in mechanical systems: Theory and experimental results[END_REF]. The specific yet important task of the stabilization of impacting transition phases was analyzed and experimentally tested in [START_REF] Lee | Bang-bang impact control using hybrid impedance/time-delay control[END_REF], [START_REF] Pagilla | An experimental study of planar impact of a robot manipulator[END_REF], [START_REF] Sekhavat | Impact stabilizing controller for hydraulic actuators with friction: Theory and experiments[END_REF], [START_REF] Van Vliet | Experimental validation of contact dynamics simulation of constrained robotic tasks[END_REF]- [START_REF] Xu | Experimental study of contact transition control incorporating joint acceleration feedback[END_REF]. From the point of view of tracking control of complementarity Lagrangian systems along general constrained/unconstrained paths, such studies focus on a module of the overall control problem. The problem of robust impact detection with only position measurement received attention in [START_REF] Bourgeot | Fast algebraic impact times estimation for a linear system subject to unilateral constraint[END_REF]. One of the first works formulating the control of complete robotic tasks via unilateral constraints and complementarity conditions was presented in [START_REF] Huang | Time optimal control for a robotic contour following problem[END_REF]. In that work the impacts were considered inelastic and the control problem was solved using a time optimal problem. The tracking control problem under consideration, involving systems that undergo transitions from free to constrained motions, and vice-versa, along an infinity of cycles, was formulated and studied in [START_REF] Brogliato | On the control of finite-dimensional mechanical systems with unilateral constraints[END_REF] for the 1-dof (degree-of-freedom) case and in [START_REF] Bourgeot | Tracking control of complementarity lagrangian systems[END_REF] for the -dof case. Both of these works consider systems with only one unilateral frictionless constraint. In this paper we not only consider the multiconstraint case but the results in Section VII relax some very hard to verify conditions imposed in [START_REF] Bourgeot | Tracking control of complementarity lagrangian systems[END_REF] to assure the stability. Moreover the accurate design of the control law that guarantees the detachment from the constraints is formulated and incorporated in the stability analysis for the first time. Considering multiple constraints may be quite important in applications like virtual reality and haptic systems, where typical tasks involve manipulating objects modelled as rigid bodies [START_REF] Faulring | Haptic display of constrained dynamic systems via admittance displays[END_REF] in complex environments with many unilateral constraints. We note that in the case of a single nonsmooth impact the exponential stability and bounded-input bounded state (BIBS) stability was studied in [START_REF] Menini | Exponential and BIBS stabilisation of one degree of freedom mechanical system subject to single non-smooth impacts[END_REF] using a state feedback control law. A study for a multiple degree-of-freedom linear systems subject to nonsmooth impacts can be found in [START_REF] Menini | Dynamic position feedback stabilisation of multidegrees-of-freedom linear mechanical system subject to nonsmooth impacts[END_REF]. That approach proposes a proportional-derivative control law in order to study BIBS stability via Lyapunov techniques. Other approaches for the tracking control of nonsmooth mechanical systems can be found in [START_REF] Galeani | Trajectory tracking for a particle in elliptical billiards[END_REF], [START_REF] Leine | Uniform convergence of monotone measure differential inclusions: with application to the control of mechanical systems with unilateral constraints[END_REF], [START_REF] Menini | Asymptotic tracking of periodic trajectories for a simple mechanical system subject to nonsmooth impacts[END_REF], and [START_REF] Pagilla | Control of contact problem in constrained euler-lagrange systems[END_REF]. The analysis and control of systems subject to unilateral constraints also received attention in [START_REF] Bentsman | Dynamical systems with active singularities elastic type: A modeling and controller synthesis framework[END_REF].

This paper focuses on the problem of tracking control of complementarity Lagrangian systems [START_REF] Moreau | Unilateral contact and dry friction in finite freedom dynamics[END_REF] subject to frictionless unilateral constraints whose dynamics may be expressed as [START_REF] Acary | Numerical Methods for Nonsmooth Dynamical Systems[END_REF] where is the vector of generalized coordinates, is the positive definite inertia matrix, represents the distance to the constraints, is the matrix containing Coriolis and centripetal forces, contains conservative forces, is the vector of the Lagrangian multipliers associated to the constraints and is the vector of generalized torque inputs. For the sake of completeness we precise that denotes the Euclidean gradient where represents the vector of partial derivatives of w.r.t. the components of . We assume that the functions are continuously differentiable and that for all with . It is worth to precise here that for a given function its derivative w.r.t. the time will be denoted by . For any function the limit to the right at the instant will be denoted by and the limit to the left will be denoted by . A simple jump of the function at the moment is denoted . The Dirac measure at time is . Definition 1: A Linear Complementarity Problem (LCP) is a system given by [START_REF] Ballard | Formulation and well-posedness of the dynamics of rigid body systems with perfect unilateral constraints[END_REF] which is compactly re-written as [START_REF] Bentsman | Dynamical systems with active singularities elastic type: A modeling and controller synthesis framework[END_REF] Such an LCP has a unique solution for all if and only if is a P-matrix, i.e., all its principal minors are positive [START_REF] Facchinei | Finite-Dimensional Variational Inequalities and Complementarity Problems, ser. Springer Series in Operations Research[END_REF].

The admissible domain associated to the system (1) is the closed set where the system can evolve and it is described as follows:

where considering that a vector is non-negative if and only if all its components are nonnegative. In order to have a well-posed problem with a physical meaning we consider that contains at least a closed ball of positive radius.

Definition 2: A singularity of the boundary of is the intersection of two or more codimension one surfaces . The presence of may induce some impacts that must be included in the dynamics of the system. It is obvious that allows both simple impacts (when one constraint is involved) and multiple impacts (when singularities or surfaces of codimension larger than 1 are involved). Let us introduce the following notion of -impact.

Definition 3: Let be a fixed real number. We say that a -impact occurs at the instant if where , .

If

the surfaces are stroked simultaneously and a impact occurs. When the system collides in a neighborhood of the intersection . Definition 4: [START_REF] Hiriart-Urruty | Fundamentals of Convex Analysis[END_REF], [START_REF] Moreau | Unilateral contact and dry friction in finite freedom dynamics[END_REF] The tangent cone to at is defined as: where is the index set of active constraints. When one has and . The normal cone to at is defined as the polar cone to

The collision (or restitution) rule in [START_REF] Acary | Numerical Methods for Nonsmooth Dynamical Systems[END_REF], is a relation between the post-impact velocity and the pre-impact velocity. Among the various models of collision rules, Moreau's rule is an extension of Newton's law which is energetically consistent [START_REF] Glocker | Concepts for modeling impacts without friction[END_REF], [START_REF] Mabrouk | A unified variational model for the dynamics of perfect unilateral constraints[END_REF] and is numerically tractable [START_REF] Acary | Numerical Methods for Nonsmooth Dynamical Systems[END_REF]. For these reasons throughout this paper the collision rule will be defined by Moreau's relation [START_REF] Moreau | Unilateral contact and dry friction in finite freedom dynamics[END_REF] (4) where is the post-impact velocity, is the pre-impact velocity and is the restitution coefficient. Denoting by the kinetic energy of the system, we can compute the kinetic energy loss at the impact time as [START_REF] Mabrouk | A unified variational model for the dynamics of perfect unilateral constraints[END_REF] (5)

The collision rule can be rewritten considering the vector of generalized velocities as an element of the tangent space to the configuration space of the system, equipped with the kinetic energy metric. Doing so (see [6, section 6.2]), the discontinuous velocity components and the continuous ones are identified. Precisely, , where represents the unitary normal vectors (see Definition 4) and represents mutually independent unitary vectors such that . In this case the collision rule (4) at the impact time becomes the generalized Newton's rule , where is the restitution coefficient w.r.t. the surface . For the sake of simplicity we consider in this paper that all the restitution coefficients are equal, i.e., . Remark 1: 1) If and the angle then in the neighborhood of one has .

2) Let

. The case is called a plastic impact and the case is called an elastic impact. In the first case the normal component of the velocity becomes zero and in the second case the normal component of the velocity changes only its direction and preserves its magnitude. As we can easily see from (5) in the second case there is no loss of kinetic energy at the impact moment.

3) One recalls that we deal with frictionless unilateral constraints. Some frictional contact laws that fit within the nonsmooth mechanic framework (1) can be found in [START_REF] Leine | Stability properties of equilibrium sets of nonlinear mechanical systems with dry friction and impact[END_REF]. The structure of the paper is as follows: in Section II one presents some basic concepts and prerequisites necessary for the further developments. Section III is devoted to the controller design. In Section IV one defines the desired (or "exogenous") trajectories entering the dynamics. The desired contact-force that must occur on the phases where the motion is constrained, is explicitly defined in Section V. Section VI focuses on the strategy for take-off at the end of the constraint phases. The main results related to the closed-loop stability analysis are presented in Section VII. One example and concluding remarks end the paper.

The following standard notations will be adopted: is the Euclidean norm, and are the vectors formed with the first and the last components of , respectively.

is the normal cone to at [START_REF] Hiriart-Urruty | Fundamentals of Convex Analysis[END_REF], [START_REF] Moreau | Unilateral contact and dry friction in finite freedom dynamics[END_REF] when satisfies , and represent the smallest and the largest eigenvalues of a symmetric matrix, respectively.

II. BASIC CONCEPTS

A. Typical Task

Following [START_REF] Brogliato | On the control of finite-dimensional mechanical systems with unilateral constraints[END_REF] the time axis can be split into intervals and corresponding to specific phases of motion. Due to the singularities of that must be taken into account, the constrained-motion phases need to be decomposed in sub-phases where some specific constraints are active. Between two such sub-phases a transition phase occurs only when the number of active constraints increases. This means that a typical task can be represented in the time domain as [START_REF] Brogliato | Nonsmooth Mechanics[END_REF] where the superscript represents the set of active constraints during the corresponding motion phase, and denotes the transient between two phases when the number of active constraints increases. Without loss of generality we suppose that the system is initialized in the interior of at a free-motion phase. The impacts during involve constraints ( -impacts). Furthermore we shall prove that the first impact of is a -impact with bounded by a parameter chosen by the designer. When the number of active constraints decreases there is no impact, thus no other transition phases are needed. We note that corresponds to free-motion . Since the tracking control problem involves no difficulty during the phases, the central issue is the study of the passages between them (the design of transition phases and detachment conditions), and the stability of the trajectories evolving along (6) (i.e., an infinity of cycles). It is noteworthy that the passage consists of detachments from some constraints. In Section VI we consider that constraints are active and we give the conditions to smoothly take-off from of them. It is clear that once we know how to do that, we can manage all the transitions mentioned above. Throughout the paper, the sequence will be referred to as the cycle of the system's evolution. For robustness reasons during transition phases we impose a closed-loop dynamics (containing impacts) that mimics somehow the bouncing-ball dynamics (see e.g., [START_REF] Brogliato | Nonsmooth Mechanics[END_REF]).

B. Exogenous Signals Entering the Dynamics

In this section we introduce the trajectories playing a role in the dynamics and the design of the controller. Some instants that will be used further are also defined. • denotes the desired trajectory of the unconstrained system (i.e., the trajectory that the system should track if there were no constraints). We suppose that for some , otherwise the problem reduces to the tracking control of a system with no constraints.

• denotes the signal entering the control input and playing the role of the desired trajectory during some parts of the motion.

• represents the signal entering the Lyapunov function. This signal is set on the boundary after the first impact of each cycle. The signals and coincide on the phases while is used to define everywhere and . These three functions coincide only on the phases. Throughout the paper we consider , where is chosen by the designer as the start of the transition phase and is the end of . We note that all superscripts will refer to the cycle of the system motion. We also use the following notations:

• is the first impact during the cycle ; • is the accumulation point of the sequence of the impact instants during the cycle ; • will be explicitly defined later and represents the instant when the signal reaches a given value chosen by the designer in order to impose a closed-loop dynamics with impacts during the transition phases; • is the desired detachment instant at the end of the phase . It is noteworthy that , are state-dependent whereas , and are exogenous and imposed by the designer. To better understand the definition of these specific instants, in Fig. 1 we represent the exogenous signals , , during a sequence when the motion is simplified as follows:

• during the transition phase we take into account only the constraints that must be activated . • at the end of the phase we take into account only the constraints that must be deactivated . The points , , and in Fig. 1 correspond to the moments , , and respectively. We have seen that the choice of plays an important role in the stability criterion given by Proposition 1. On the other hand in Fig. 1 we see that starting from the desired trajectory is deformed compared to . In order to reduce this deformation, the time and implicitly the point must be close to (see also Fig. 4). Further details on the choice of will be given later. Taking into account just the constraints we can identify with the moment when and rejoin at . See also Fig. 4 for an illustration on an example.

C. Stability Analysis Criteria

The system (1) is a complex nonsmooth and nonlinear dynamical system which involves continuous and discrete time phases. A stability framework for this type of systems has been proposed in [START_REF] Brogliato | On the control of finite-dimensional mechanical systems with unilateral constraints[END_REF] and extended in [START_REF] Bourgeot | Tracking control of complementarity lagrangian systems[END_REF]. This is an extension of the Lyapunov second method adapted to closed-loop mechanical systems with unilateral constraints. Since we use this criterion in the following tracking control strategy it is worth to clarify the framework and to introduce some definitions.

Let us define as the complement in of and assume that the Lebesgue measure of , denoted , equals infinity. Consider the state of the closed-loop system in (1) with some feedback controller . Definition 5 (Weakly Stable System [START_REF] Bourgeot | Tracking control of complementarity lagrangian systems[END_REF]): The closed loop system is called weakly stable if for each there exists such that for all , . The system is asymptotically weakly stable if it is weakly stable and . Finally, the practical weak stability holds if there exists and such that for all , . Weak stability is therefore Lyapunov stability without looking at the transition phases. Consider such that there exists strictly increasing functions and satisfying the conditions: and . Definition 6: A transition phase is called finite if it involves a sequence of impact times with the accumulation point (for the sake of simplicity we shall denote the accumulation point by even if ). In the sequel all the transition phases are supposed finite, which implies that (in [START_REF] Ballard | Formulation and well-posedness of the dynamics of rigid body systems with perfect unilateral constraints[END_REF] it is shown that implies that

). The following criterion is inspired from [START_REF] Bourgeot | Tracking control of complementarity lagrangian systems[END_REF], and will be used to study the stability of the system (1).

Proposition 1 (Weak Stability): Assume that the task admits the representation (6) and that a) ; b) outside the impact accumulation phases one has for some constant ; c) for some ; d) the system is initialized on such that ; e) for some and . If then , where is a function that can be made arbitrarily small by increasing the value of . The system is practically weakly stable with . Precisely, the weak stability is characterized by an "almost decreasing" Lyapunov function as illustrated in Fig. 2. Remark 3: It is worth to point out the local character of the stability criterion proposed by Proposition 1. This character is firstly given by condition (d) of the statement and secondly by the synchronization constraints of the control law and the motion phase of the system (see ( 6) and (9) below).

The practical stability is very useful because attaining asymptotic stability is not an easy task for the unilaterally constrained systems described by (1) especially when and is not a diagonal matrix (i.e., there are inertial couplings, which is the general case).

D. Dissipativity and Tracking Versus Stabilization

Let us make a parenthesis to highlight the major discrepancy between the trajectory tracking problem and the stabilization problem. To this aim let us first recall that the dynamics in ( 1) and ( 4) can be equivalently rewritten as the measure differential inclusion [START_REF] Acary | Numerical Methods for Nonsmooth Dynamical Systems[END_REF], [START_REF] Brogliato | Nonsmooth Mechanics[END_REF], [START_REF] Mabrouk | A unified variational model for the dynamics of perfect unilateral constraints[END_REF], [START_REF] Moreau | Unilateral contact and dry friction in finite freedom dynamics[END_REF] (7) where is the differential measure associated with the velocity that is a right-continuous function of local bounded variation, is equal almost everywhere to , is absolutely continuous and . The right-hand-side is the normal cone to the tangent cone, where the cones are as in Definition 4. As shown in [START_REF] Brogliato | Absolute stability and the Lagrange-Dirichlet theorem with monotone multivalued mappings[END_REF] and [7, section 3.9.4, 6.8.2, 7.2.4], a crucial property for stabilization is that the cone complementarity problem [START_REF] Brogliato | On the control of finite-dimensional mechanical systems with unilateral constraints[END_REF] defines a maximal monotone mapping , because the two cones and are polar cones [START_REF] Hiriart-Urruty | Fundamentals of Convex Analysis[END_REF], and . This maximal monotonicity property allows one to use dissipativity arguments in an absolute stability framework to derive a Lyapunov function. Let us consider now the tracking control problem. The new (closed-loop) state vector is . Therefore the right-hand-side of the closed-loop measure differential inclusion becomes the normal cone , with . The sets and are now time-varying, and the monotonicity property is generally lost. This explains why the trajectory tracking problem is much more intricate than its stabilization counterpart.

III. CONTROLLER DESIGN

In order to overcome some difficulties that can appear in the controller definition, the dynamical (1) will be expressed in the generalized coordinates introduced by McClamroch & Wang [START_REF] Mcclamroch | Feedback stabilization and tracking of constrained robots[END_REF], which allow one to split the generalized coordinates into a "normal" and a "tangential" parts, with a suitable diffeomorphic transformation . We suppose that the generalized coordinates transformation holds globally in , which may obviously not be the case in general. However, the study of the singularities that might be generated by the coordinates transformation is out of the scope of this paper. Let us consider . . . , the identity matrix. The new coordinates will be , with

. . . such that 1 .

The tangent cone is the space of admissible velocities on the boundary of .

The controller used here consists of different low-level control laws for each phase of the system. More precisely, the switching controller can be expressed as for for for [START_REF] Brogliato | Absolute stability and the Lagrange-Dirichlet theorem with monotone multivalued mappings[END_REF] where is full-rank under some basic assumptions (see [START_REF] Mcclamroch | Feedback stabilization and tracking of constrained robots[END_REF]). The dynamics becomes [START_REF] Facchinei | Finite-Dimensional Variational Inequalities and Complementarity Problems, ser. Springer Series in Operations Research[END_REF] where the set of complementary relations can be written more compactly as . In the sequel coincides with the fixed-parameter controller proposed in [START_REF] Slotine | Adaptive manipulator control: A case study[END_REF] and the closed-loop stability analysis of the system is based on Proposition 1. First, let us introduce some notations: , , , , where is a scalar gain and , represent the desired trajectories defined in the previous section. Using the above notations the controller is given by [START_REF] Faulring | Haptic display of constrained dynamic systems via admittance displays[END_REF] where is a scalar gain, , and is the desired contact force during persistently constrained motion. It is clear that during not all the constraints are active and, therefore, some components of and are zero.

In order to prove the stability of the closed-loop system ( 9) - [START_REF] Faulring | Haptic display of constrained dynamic systems via admittance displays[END_REF] we will use the following positive definite function: [START_REF] Galeani | Trajectory tracking for a particle in elliptical billiards[END_REF] IV. TRACKING CONTROL FRAMEWORK

A. Design of the Desired Trajectories

In this paper we treat the tracking control problem for the closed-loop dynamical system (9)-( 11) with the complete desired path a priori taking into account the complementarity conditions and the impacts. In order to define the desired trajectory let us consider the motion of a virtual and unconstrained particle perfectly following a trajectory (represented by on Fig. 1) with an orbit that leaves the admissible domain for a given period. Therefore, the orbit of the virtual particle can be split into two parts, one of them belonging to the admissible domain (inner part) and the other one outside the admissible domain (outer part). In the sequel we deal with the tracking control strategy when the desired trajectory is constructed such that:

(i) when no activated constraints, it coincides with the trajectory of the virtual particle (the desired path and velocity are defined by the path and velocity of the virtual particle, respectively); (ii) when constraints are active, its orbit coincides with the projection of the outer part of the virtual particle's orbit on the surface of codimension defined by the activated constraints ( between and in Fig. 1); (iii) the desired detachment moment and the moment when the virtual particle re-enters the admissible domain (with respect to constraints) are synchronized. Therefore we have not only to track a desired path but also to impose a desired velocity allowing the motion synchronization on the admissible domain. The main difficulties here consist of:

• stabilizing the system on during the transition phases and incorporating the velocity jumps in the overall stability analysis;

• deactivating some constraints at the moment when the unconstrained trajectory re-enters the admissible domain with respect to them; • maintaining a persistently constrained motion between the moment when the system was stabilized on and the detachment moment. Remark 4: The problem can be relaxed considering that we want to track only a desired path like (without imposing a desired velocity on the inner part of the desired trajectory and/or a given period to complete a cycle). In this way the synchronization problem (iii) disappears and we can assume there exists a twice differentiable desired trajectory outside that assures the detachment when the force control is dropped. In other words, in this case we have to design the desired trajectory only during phases.

B. Design of and on the Phases

During the transition phases the system must be stabilized on . Obviously, this does not mean that all the constraints have to be activated (i.e.,

). Let us consider that only the first constraints (eventually reordering the coordinates) define the border of where the system must be stabilized. The following methodology will be used to define : 1) During a small period chosen by the designer the desired velocity becomes zero preserving the twice differentiability of . For instance we can use the following definition: which means , and .

2) The last components of are frozen [START_REF] Glocker | Set valued force laws: Dynamics of non-smooth systems[END_REF] 3) For a fixed the moment is chosen by the designer as the instant when the limit conditions , hold. On we define as a Fig. 3. Design of q on the transition phases I .

twice differentiable decreasing signal. Precisely, denoting , the components of are defined as [START_REF] Glocker | Concepts for modeling impacts without friction[END_REF] where is defined in ( 12) and the coefficients are [START_REF] Huang | Time optimal control for a robotic contour following problem[END_REF] The rationale behind the choice of is on one hand to assure a robust stabilization on , mimicking the bouncing-ball dynamics; on the other hand to enable one to compute suitable upper-bounds that will help using Proposition 1 (hence terms in [START_REF] Glocker | Concepts for modeling impacts without friction[END_REF] with in ( 12)). Remark 5: Two different situations are possible. The first one is given by (see Fig. 3) and we shall prove that in this situation all the jumps of the Lyapunov function in [START_REF] Galeani | Trajectory tracking for a particle in elliptical billiards[END_REF] are negative. The second situation was pointed out in [START_REF] Bourgeot | Tracking control of complementarity lagrangian systems[END_REF] and is given by . In this situation the first jump at in the Lyapunov function may be positive. It is noteworthy that will then have a jump at the time since (see ( 14)). In order to limit the deformation of the desired trajectory w.r.t. the unconstrained trajectory during the phases (see Figs. 1 and3), we impose in the sequel [START_REF] Lee | Bang-bang impact control using hybrid impedance/time-delay control[END_REF] where is chosen by the designer. It is obvious that a smaller leads to smaller deformation of the desired trajectory and to smaller deformation of the real trajectory as we shall see in Section VIII. Nevertheless, due to the tracking error, cannot be chosen zero. We also note that is a practical way to choose .

During the transition phases we define . Assuming a finite accumulation period, the impact process can be considered in some way equivalent to a plastic impact. Therefore, and are set to zero on the right of .

V. DESIGN OF THE DESIRED CONTACT FORCE DURING CONSTRAINT PHASES

For the sake of simplicity we consider the case of the constraint phase with . Obviously a sufficiently large desired contact force assures a constrained movement on . Nevertheless at the end of the phases a detachment from some surfaces has to take place. It is clear that a take-off implies not only a well-defined desired trajectory but also some small values of the corresponding contact force components. On the other hand, if the components of the desired contact force decrease too much a detachment can take place before the end of the phases which can generate other impacts. Therefore we need a lower bound of the desired force which assures the contact during the phases. Dropping the time argument, the dynamics of the system on can be written as [START_REF] Leine | Stability properties of equilibrium sets of nonlinear mechanical systems with dry friction and impact[END_REF] where and . . . . On the system is permanently constrained which implies and . In order to assure these conditions it is sufficient to have . In the following let us denote and where the meaning of each component is obvious.

Proposition 2: On the constraint motion of the closedloop system (17), ( 9), ( 11) is assured if the desired contact force is defined by [START_REF] Leine | Uniform convergence of monotone measure differential inclusions: with application to the control of mechanical systems with unilateral constraints[END_REF] where is the inverse of the Delassus' matrix (see [START_REF] Acary | Numerical Methods for Nonsmooth Dynamical Systems[END_REF], [START_REF] Brogliato | Nonsmooth Mechanics[END_REF] for the definition) and . Proof: First, we notice that the second relation in [START_REF] Leine | Stability properties of equilibrium sets of nonlinear mechanical systems with dry friction and impact[END_REF] implies on (see [START_REF] Glocker | Set valued force laws: Dynamics of non-smooth systems[END_REF]) [START_REF] Leine | Stability and Convergence of Mechanical Systems with Unilateral Constraints[END_REF] From ( 17) and ( 11) one easily gets

Combining the last two equations we obtain the following LCP with unknown :

(20)
Since and hence is a P-matrix, the LCP (20) has a unique solution and one deduces that if and only if [START_REF] Mabrouk | A unified variational model for the dynamics of perfect unilateral constraints[END_REF] with . Since and , (21) rewrites as ( 18) and the proof is finished. It is noteworthy that Remark 6: The control law used in this paper with the design of described above leads to the following closed-loop dynamics on It is noteworthy that the closed-loop dynamics is nonlinear and therefore, we do not use the feedback stabilization proposed in [START_REF] Mcclamroch | Feedback stabilization and tracking of constrained robots[END_REF].

VI. STRATEGY FOR TAKE-OFF AT THE END OF CONSTRAINT PHASES

We have discussed in the previous sections the necessity of a trajectory with impacts in order to assure the robust stabilization on in finite time and, the design of the desired trajectory to stabilize the system on . Now, we are interested in finding the conditions on the control signal that assure the take-off at the end of the constrained phases . We consider the phase expressed as the time interval . The dynamics on is given by ( 17) and the system is permanently constrained, which implies and . Let us also consider that the first constraints have to be deactivated. Thus, the detachment takes place at if which requires . The last constraints remain active which means .

To simplify the notation we drop the arguments and in many equations of this section. We decompose the LCP matrix (which is the Delassus' matrix multiplied by ) as with , and . Proposition 3: For the closed-loop system (17), ( 9), [START_REF] Faulring | Haptic display of constrained dynamic systems via admittance displays[END_REF] the decrease of the active constraints number from to (with ), is possible if [START_REF] Mcclamroch | Feedback stabilization and tracking of constrained robots[END_REF] where and such that . Proof: From ( 11) and ( 17) one gets Therefore the LCP (19) rewrites as [START_REF] Menini | Asymptotic tracking of periodic trajectories for a simple mechanical system subject to nonsmooth impacts[END_REF] Under the conditions and one has with the solution

Thus is equivalent to which leads to the second part of definition [START_REF] Mcclamroch | Feedback stabilization and tracking of constrained robots[END_REF]. Furthermore, replacing in [START_REF] Menini | Exponential and BIBS stabilisation of one degree of freedom mechanical system subject to single non-smooth impacts[END_REF] we get and yields the first part of definition [START_REF] Mcclamroch | Feedback stabilization and tracking of constrained robots[END_REF].

To conclude, the solution of the LCP ( 23) is and is defined by [START_REF] Mcclamroch | Feedback stabilization and tracking of constrained robots[END_REF]. Proposition 4: The closed-loop system (17), ( 9), ( 11) is permanently constrained on and a smooth detachment is guaranteed on ( is a small positive real number chosen by the designer) if (i) is defined on by [START_REF] Mcclamroch | Feedback stabilization and tracking of constrained robots[END_REF] where is replaced by . (ii) On where is a twice differentiable function such that [START_REF] Menini | Dynamic position feedback stabilisation of multidegrees-of-freedom linear mechanical system subject to nonsmooth impacts[END_REF] and . Proof: (i) The uniqueness of solution of the LCP [START_REF] Leine | Stability and Convergence of Mechanical Systems with Unilateral Constraints[END_REF] guarantees that ( 18) and ( 22) agree if . In other words, replacing by in [START_REF] Mcclamroch | Feedback stabilization and tracking of constrained robots[END_REF] we assure a constrained motion on and the necessary conditions for detachment on . (ii) Obviously ( 25) is imposed in order to assure the twice differentiability of the desired trajectory. Finally, straightforward computations show that which means that the detachment is guaranteed and no other impacts occur when the desired acceleration satisfies .

VII. CLOSED-LOOP STABILITY ANALYSIS

In the case , the function in ( 12) can be used in order to prove the closed-loop stability of the system (10), [START_REF] Faulring | Haptic display of constrained dynamic systems via admittance displays[END_REF] (see for instance [START_REF] Brogliato | Dissipative Systems Analysis and Control. Theory and Applications[END_REF]). In the case studied here the analysis becomes more complex.

To simplify the notation is denoted as . In order to introduce the main result of this paper we make the next assumption, which is verified in practice for dissipative systems.

Assumption 1: The controller in [START_REF] Faulring | Haptic display of constrained dynamic systems via admittance displays[END_REF] assures that all the transition phases are finite (see Definition 6) and the accumulation point is smaller than for all . Since outside we will show that the Lyapunov function exponentially decreases, we may presume that all the impacts take place during .

Lemma 1: Consider the closed-loop system ( 9)-( 11) with defined on the interval as in ( 14)-( 13). Let us also suppose that condition (b) of Proposition 1 is satisfied. The following inequalities hold: [START_REF] Moreau | Unilateral contact and dry friction in finite freedom dynamics[END_REF] Furthermore, if and is a -impact one has [START_REF] Pagilla | Control of contact problem in constrained euler-lagrange systems[END_REF] where , and are some constant real numbers that will be defined in the proof Proof: See Appendix A. The main result of this paper can be stated as follows. Theorem 1: Let Assumption 1 hold, and defined as in ( 13)-( 14). The closed-loop system ( 9)-( 11) initialized on such that , satisfies the requirements of Proposition 1 and is therefore practically weakly stable with the closed-loop state and where and are defined in the proof.

Proof: See Appendix B. Remark 7: Since the closed-loop system ( 9)-( 11) satisfies the requirements of Proposition 1 one also deduces , so . In other words the sequence is uniformly upperbounded and the upperbound can be decreased by adjusting the parameters and .

VIII. ILLUSTRATIVE EXAMPLE

A. Planar Two-Link Rigid-Joint With One Constraint

The main issues of the control scheme proposed in this paper are first emphasized simulating the behavior of a planar two-link rigid-joint manipulator in presence of one unilateral constraint. The lengths of the manipulator's links are set to , and their masses are set to , is the gravity acceleration. Denoting by the joint angle of the link and the moment of inertia of link about the axis that passes through the center of mass and is parallel to the axis, the dynamics of the two-link manipulator is given by (1) with , ,

The dynamics can be rewritten in the cartesian coordinates using the change of variables (28) The admissible domain is the upper half plane (here and

) and the unconstrained desired trajectory is given by a circle that violates the constraint. Precisely, the end effector must follow a half-circle, stabilize on the Fig. 4. Up: The influence of on the real trajectory's deformation for controller's gains set to = 8, = 7. Down: y(t) = q (t) (dashed) and y (t) = (q ) (t) (solid) during the first cycle ( = 0:01). . Using the coordinates is frozen during the transition phases while is defined by ( 14)- [START_REF] Huang | Time optimal control for a robotic contour following problem[END_REF]. Furthermore, the controller is computed by [START_REF] Faulring | Haptic display of constrained dynamic systems via admittance displays[END_REF] where is set to 0.5 and (i.e., the desired contact force corresponding to the constraint ) is given by [START_REF] Leine | Uniform convergence of monotone measure differential inclusions: with application to the control of mechanical systems with unilateral constraints[END_REF] where has a decreasing profile like in item (i) of Proposition 4. The impacts are imposed using the parameter in ( 14)- [START_REF] Huang | Time optimal control for a robotic contour following problem[END_REF]. The numerical simulations are done with the Moreau's time-stepping algorithm of the SICONOS software platform (http://siconos.gforge.inria.fr). The choice of a time-stepping algorithm was mainly dictated by the presence of accumulations of impacts which render the use of event-driven methods difficult [START_REF] Acary | Numerical Methods for Nonsmooth Dynamical Systems[END_REF]. A further reason to choose the SICONOS software platform for the simulation of the complementarity systems is its capability to solve LCPs 2 . Let us set , 10 s the period of each cycle and 30 s the final simulation time. First, let us point out [Fig. 4 (left)] the influence of (i.e., the choice of ) on the deformation of the real trajectory w.r.t. the desired unconstrained one. As we have pointed out in Section IV the deformation gets smaller when decreases. It is noteworthy that the tangential approach 2 The control scheme proposed in this paper may require to solve an LCP of dimension p 10 (reasonable in some control applications). But this requires a specific solver since the usual "hybrid" methods must treat 2 cases and quickly become inefficient [START_REF] Acary | Numerical Methods for Nonsmooth Dynamical Systems[END_REF]. corresponding to lacks of robustness and is unreliable due to the nonzero initial tracking errors.

On Fig. 4 one sees that since before the first impact of each cycle and , there exists a jump at the moment s in , respectively, and both signals are set to zero at s. The jump of induces a positive jump in the variation of (details are in Appendix B) The switches of the controller during the first 10 s are depicted in Fig. 5. Clearly since the velocity jumps, the controller jumps as well.

The Fig. 5 presents the variation of the contact force . One sees that remains 0 during the free motion phases. The contact force is designed as a decreasing linear function during constrained motion phases in order to allow a smooth detachment at the end of these phases. It is worth to mention that the magnitude of depends indirectly on . Precisely, when approaches zero the system tends to a tangential stabilization on , which implies larger values of and consequently smaller length of and smaller magnitude of the contact force measured by (see Proposition 4).

Fig. 6 shows that the tracking error described by the Lyapunov function rapidly decreases and remains close to 0. In other words the practical weak stability is guaranteed. On the zoom made in Fig. 6 one can also observe the behavior of during the stabilization on , that is an almost decreasing function.

B. Planar Two-Link Rigid-Joint Manipulator With Two Constraints

In the sequel we introduce another constraint into the previous dynamics. Precisely we impose an admissible domain . Let us also consider an unconstrained desired trajectory given by the circle that violates both constraints. In other words, the two-link planar manipulator must track a quarter-circle; stabilize on and then follow the line ; stabilize on the intersection of and ; detach from and follow until the unconstrained circle re-enters and finally take-off from in order to repeat the previous steps. Therefore, we have: with , , , , etc. We note that during the system is stabilized on (1-impacts) while during the system is stabilized on ( -impacts). The numerical values used for the dynamical model are again , , and the restitution coefficient . The impacts are imposed by in ( 14) and ( 15) the beginning of transition phases are defined using in [START_REF] Lee | Bang-bang impact control using hybrid impedance/time-delay control[END_REF]. We impose a period of 10 s for two consecutive cycles and we simulate the dynamics during 60 s. Setting the controller gains we see in Fig. 7 (left) that the desired trajectory is accurately followed. The jumps in the variation of the Lyapunov function are pointed out in Fig. 7.

In this case we have imposed a constant contact-force during the motion on the surface (see Fig. 8 (left)) and a decreasing contact-force, that allows a smooth detachment, during the motion on (see Fig. 8 (right)). On Fig. 9 the values of the multipliers and during the transition phase (stabilization in the corner) are depicted.

IX. CONCLUSION

In this paper, we have proposed a methodology to study the tracking control of fully actuated Lagrangian systems subject to multiple frictionless unilateral constraints and multiple impacts. The main contribution of the work is twofold: first, it formulates a general control framework and second, it provides a stability analysis for the class of systems under consideration. It is noteworthy that even in the simplest case of only one frictionless unilateral constraint the paper already presents some notable improvements with respect to the existing works. Precisely, the stability analysis result is significantly more general than those presented in [START_REF] Bourgeot | Tracking control of complementarity lagrangian systems[END_REF] and [START_REF] Brogliato | On the control of finite-dimensional mechanical systems with unilateral constraints[END_REF] and, each element entering the dynamics (desired trajectory, contact force) is explicitly defined. Numerical simulations are done with the SICONOS software platform [START_REF] Acary | Numerical Methods for Nonsmooth Dynamical Systems[END_REF] in order to illustrate the results.

APPENDIX

Proof of Lemma 1: From ( 12) we can deduce on one hand that and on the other hand Since condition (b) of Proposition 1 is satisfied one has and the first two inequalities in (26) become trivial. Let us recall that which implies . Combining this with the first two inequalities in [START_REF] Moreau | Unilateral contact and dry friction in finite freedom dynamics[END_REF] we derive the third inequality in [START_REF] Moreau | Unilateral contact and dry friction in finite freedom dynamics[END_REF].

For the rest of the proof we assume that . Therefore . Since is a continuous function with all the components defined as decreasing functions on , it is obvious that . Furthermore Thus is a -impact with . From Definition 3 one has and using one obtains the first inequality [START_REF] Pagilla | Control of contact problem in constrained euler-lagrange systems[END_REF].

Let us denote . We recall here that was chosen such that . From ( 14), [START_REF] Huang | Time optimal control for a robotic contour following problem[END_REF] and the first inequality in [START_REF] Pagilla | Control of contact problem in constrained euler-lagrange systems[END_REF] On the other hand, straightforward computations show that [START_REF] Slotine | Adaptive manipulator control: A case study[END_REF] Furthermore [START_REF] Tornambé | Modeling and control of impact in mechanical systems: Theory and experimental results[END_REF] Therefore, inserting successively [START_REF] Tornambé | Modeling and control of impact in mechanical systems: Theory and experimental results[END_REF] in [START_REF] Slotine | Adaptive manipulator control: A case study[END_REF] and [START_REF] Slotine | Adaptive manipulator control: A case study[END_REF] in [START_REF] Sekhavat | Impact stabilizing controller for hydraulic actuators with friction: Theory and experiments[END_REF] one arrives at [START_REF] Van Vliet | Experimental validation of contact dynamics simulation of constrained robotic tasks[END_REF] In the sequel let us denote by the sum of all the components of a vector . Taking into account the definition [START_REF] Glocker | Concepts for modeling impacts without friction[END_REF] ) one gets [START_REF] Volpe | A theoretical and experimental investigation of impact control for manipulators[END_REF] where denotes the inertia matrix and is the kinetic energy loss at the impact time .

From (5) one has and (33) becomes . Let us recall that is the percussion vector (see [START_REF] Brogliato | Nonsmooth Mechanics[END_REF]Chapter 1]). In the generalized coordinates introduced in Section III one obtains with . In other words the generalized coordinates introduced in Section III coincide with the so called quasi-coordinates [START_REF] Brogliato | Nonsmooth Mechanics[END_REF] and the vector is equal to (i.e., where denotes the vector with all its components equal zero). Therefore [START_REF] Xu | Experimental study of contact transition control incorporating joint acceleration feedback[END_REF] where we have used and the last equality is stated using the complementarity relation entering the dynamics, which impose that is orthogonal to . The Lyapunov function's jump corresponding to the first impact of each cycle can be computed as (35)

• It is clear that implies and . Thus, the computations for hold also for .

Trajectory

  Tracking Control of Multiconstraint Complementarity Lagrangian Systems Irinel Constantin Morȃrescu and Bernard Brogliato Abstract-In this study, one considers the tracking control problem of a class of nonsmooth fully actuated Lagrangian systems subject to frictionless unilateral constraints. The task under consideration contains both free-motion and constraint-motion phases. A switching controller that guarantees an approximate tracking is designed. Particular attention is paid to transition (impacting and detachment) between different phases of motion. The exogenous signals that assure the stabilization on (take-off from) some constaints are explicitly defined. This paper extends previous works on the topic as it considers more than one constraint for -degree-of-freedom systems. Numerical examples illustrate the main results. Index Terms-Complementarity problem, impacts, Lagrangian systems, nonsmooth systems, stability, tracking control.

Fig. 1 .

 1 Fig. 1. Closed-loop desired trajectory and control signals.

Fig. 2 .

 2 Fig. 2. Typical evolution of the Lyapunov function of weakly stable systems.

  constraintand move on the constraint until the point where the circle re-enters the admissible domain. Thus (6) writes as , with for all , ,

Fig. 5 .

 5 Fig. 5. Up: The switching controller during the first 10 s; Down: Variation of the contact force .

Fig. 6 .

 6 Fig. 6. Variation of the Lyapunov function for = 8; = 7; Zoom: Variation of the Lyapunov function during the phase I .

Fig. 7 .

 7 Fig. 7. Up: The trajectory of the system during six cycles; Down: Zoom on the variation of the Lyapunov function on the first two transition phases.

Fig. 8 .

 8 Fig.8. Up: Variation of the contact force during the motion on 6 ; Down: Variation of the contact force during the motion on 6 .

Fig. 9 .

 9 Fig. 9. Zoom on the transition phase I with 2 -impacts dots are the impulsive force magnitude at impacts).

  Furthermore andTherefore, the second inequality in[START_REF] Pagilla | Control of contact problem in constrained euler-lagrange systems[END_REF] holds with Proof of Theorem 1: First we observe that conditions (a) and (d) of Proposition 1 hold when the hypothesis of the Theorem are verified. Thus in order to prove Theorem 1 it is sufficient to verify the conditions (b), (c) and (e) of Proposition 1. To this aim we shall also use the function . (b) Using that is a skew-symmetric matrix, straightforward computations show that on the time derivative of the Lyapunov function is given by On the other hand where 01 = max max(M (q)) 1 + 2 2 2 1 ; max (M(q))( 2 + 2)

  First, let us compute the Lyapunov function's jumps at the instants . Using the continuity of the position and the definition of the desired trajectory on the phases (i.e.,

In particular it is implicitly assumed that the functions F (1) in (1) are linearly independent.


Konstantinos Chatzikokolakis

Alexandros Kaloxylos
email: alexandros.kaloxylos@huawei.com

Panagiotis Spapis
email: panagiotis.spapis@huawei.com

Chan Zhou
email: chan.zhou@huawei.com

Ömer Bulakci
email: oemer.bulakci@huawei.com

Nancy Alonistioti

Alexandros Kaloxylos^

Panagiotis Spapis^

Chan Zhou^

Ömer Bulakci^

Context-Aware Location Management of Groups of Devices in 5G Networks

Keywords: location management, location update, paging, 5G cellular networks, cluster-based communication

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

Introduction

According to forecasts [START_REF]White paper[END_REF], by the year 2019, operators worldwide will have to support more than 8 billion of smart phones. This vast increase of wirelessly connected devices increases the control information that has to be exchanged thus stretching the networks' performance to the limits. Additionally, the deployment of small cells will also create additional overhead to the network components [START_REF] Chatzikokolakis | A Survey of Location Management Mechanisms and an Evaluation of their Applicability for 5G Cellular Networks[END_REF]. To efficiently support such communication needs, it is required to optimize control functions for avoiding bottlenecks of control channels. One specific area that calls for improvement is Location Management (LM) [3], which comprises two processes, one related to Location/Tracking Area Update (TAU) and one to location search/paging. The former is an occasional process during which UE regardless if it is in idle or connected state, sends information related to its current location to the network -the user location is linked to Tracking Areas (TA) which are sets of cells, and Tracking Area List (TAL) is a list of TAs assigned to a is a user. When a users moves out of the boundaries of his TAL he performs a TAU and the network assigns to him a new TAL. On the other hand, paging is a process initiated from the network so as to discover the UE when it is in idle state. Between TAU and Paging processes there is a signaling trade-off. In particular, large TAs and/or long TALs reduce the number of updates but increase drastically the number of paged cells and vice versa. Hence, it is important to use LM schemes that keep a concise set of location information for all UEs, while optimizing the signaling overhead for finding them.

Existing solutions for TAU can be roughly classified into two main categories; user-centric approaches that monitor the activity of each user [START_REF] Pollini | A profile-based location strategy and its performance[END_REF] [START_REF] Fu | Group Mobility Management for Largescale Machine-to-Machine Mobile Networking[END_REF], and solutions dictating periodic network reconfiguration based on network traffic to alleviate the overall LM signaling [6] [START_REF] Modarres | Performance improvement of LTE tracking area design: a re-optimization approach[END_REF]. Paging schemes aim to improve the efficiency of the 3GPP standards (i.e. blanket-paging scheme [START_REF]Evolved Universal Terrestrial Radio Access (E-UTRA); User Equipment (UE) procedures in idle mode[END_REF]) that is used in legacy systems by minimizing the paging signaling. In [START_REF] Xiao | Performance Evaluation of Pipeline Paging under Paging Delay Constraint for Wireless Systems[END_REF] and [START_REF] Yi-Bing | A dynamic paging scheme for longterm evolution mobility management[END_REF] sequential paging of subsets of TAL is proposed. These are further extended with Paging Area shuffling and concurrent users' serving [START_REF] Xiao | A Parallel Shuffled Paging Strategy under Delay Bounds in Wireless Systems[END_REF], or hierarchical paging schemes [START_REF] Xiao | Optimal Pipeline Paging Load Balancing for Hierarchical Cellular Networks[END_REF] [START_REF] Xiao | On Hierarchical Pipeline Paging in Multi-Tier Overlaid Hierarchical Cellular Networks[END_REF].

One promising solution for handling the location update and paging challenges in 5G networks relates to group communication. Group based TAU will enable performing updates more often thus significantly reducing the load in the paging channels. Towards this direction, in this paper we propose a Context Aware Location Management (CALM) mechanism in which only one device performs TAU on behalf of a group which is being formed for application specific purposes (e.g., car to car communication) based on context information. The intra cluster communication (for cluster formation and maintenance) takes place using secondary interfaces such as 802.11p.

The rest of the paper is organized as follows. In Section 2 we introduce our novel mechanism for efficient LM of UEs in wireless networks. The method can be applied to vehicular ad-hoc networks, and machine-type communication devices, but can also be applied to human-centric devices. Then, in Section 3 we provide the experimental results that compare our solution against the state of the art solutions and finally, section 4 concludes the paper summarizing our work.

CALM -Context-Aware Location Management

The proposed solution solves the mentioned problems and achieves efficient LM for UEs in wireless networks. The mechanism takes advantage of existing clustering protocols (e.g., the one in [START_REF] Vodopivec | A survey on clustering algoritms for vehicular ad-hoc networks[END_REF]), where grouping is done based on time-stamp, speed, direction, etc. UEs form groups dynamically without network intervention and Cluster Members (CMs) have the possibility to communicate directly without the intervention of the cellular network.

The main goals of the mechanism are to allow a single device, namely Cluster Head (CH), to update the location of all the CMs, thus reducing the TAU overhead for all the members of the cluster, increase the granularity of TAUs and thus increase considerably the paging accuracy. During the initialization phase the CMs of a group will turn off periodic TAU timers and stop sending TAUs when crossing the borders of a TA. From that point on CMs will communicate only with the CH via a secondary interface (e.g. 802.11p). In the proposed scheme when the CH sends group's location performs an RRC connection request but instead of requesting the establishment of a signaling connection so as to perform a higher layer TAU, sends a modified RRC connection establishment request to signify that instead of establishing a signaling connection, the BS should reject this request and notify instead the Location Server about the current location of the group. Whenever the Location Server receives the CH's location from the BS it may proactively produce a new TAL based on this information and the Group ID. Then, since the network is aware of the location and the cluster information, it may accurately determine the Paging Area and there is no need to communicate the updated TAL to the CH or any CM, leading thus to further reduction of signaling. The proposed scheme, called RRC+ since it is based on enhancements of the RRC protocol, can be used on per cell basis or on any other granularity defined by the network, after CH performs cell selection/reselection procedure. Additionally, in order to avoid missing the track of any device when a UE becomes CM it may keep the standard LM functions active until the network has received the information regarding it being member of a group. During this time the UE is still reachable through paging, since the network is aware of the TAL of the device. When a CM leaves the group, based on its location, it determines whether location update is needed or not.

Performance Evaluation

In this section we present the evaluation outcomes of the assessment of the CALM solution that we propose against the Group Mobility Management (GMM) which has proven to be more efficient than 3GPP standards [START_REF] Fu | Group Mobility Management for Largescale Machine-to-Machine Mobile Networking[END_REF]. As described afore, CALM is triggered when the CH has selected to camp to a new eNB. Cell Reselection rate, cluster size and the Location Updates affect the performance of LM techniques and thus, in this section we measure the Location Update and Paging signaling overhead of the CALM solution for various scenarios with different cluster sizes, TAL sizes and Cell Reselection rates. Table 1 summarizes the assumptions of our analysis. 56 minutes In Fig. 1 (a) we present the way the cluster size affects the number of TAUs executed in the GMM mechanism and the messages sent in CALM. Since the traditional TAU does not have the same signaling messages as the Location update scheme we follow, the comparison was made based on signaling events triggered in each case, so as to have a common metric between our solution and the GMM mechanism. As shown in the figure our solution has fewer messages exchange for cluster size greater than 10 devices which is a rather small number taking into account the device density in 5G scenarios [START_REF]Future radio access scenarios, requirements and KPIs[END_REF]. Such gains are achieved because in our solution when a device enters the group it deactivates all its Location Update procedures, while in the GMM mechanism the devices will still have their periodic TAU active. Thus, even for groups with rather small number of group members, our solution outperforms the GMM solution whereas for large numbers of group members, our algorithm achieves very significant gains Although, there are cases that our mechanism increases the signaling overhead for Location Update compared to the GMM solution (e.g., when having small clusters), in paging, the accuracy of our mechanism is always better, as the paging area is too restricted, while in GMM mechanism there is the need to page the whole TAL. Fig. 1 (b) below demonstrates the percentage of signaling reduction for paging procedure for various TAL sizes. The gains range from 60% for small TAL size up to 95% for very large TAL size. It is worth mentioning that our solution eliminates completely the possibility of page misses, due to the mobility context that is always up-to-date in the network. On the contrary, the GMM solution (as well as the other SOTA solutions) fails to avoid page misses, which may occur when a UE leaves a group, because the group maintenance is based on the periodic TAU of each device.

Conclusions

In this paper we have proposed a novel context-aware mechanism for efficient location management of groups of users/devices in wireless networks so as to reduce the overhead of the signaling channels. The method targets vehicular ad-hoc networks, machines, etc. but can also be applied to human-centric devices that move in groups (e.g., crowd movement). Our mechanism exploits already formed groups, for performing group based TAU. Thus, only one device (the cluster head) performs frequent TAUs on behalf of the overall group. This enables reduction on the TAU overhead and significantly precise paging compared to the state of the art solutions.

Fig. 1 .

 1 (a) Location Updates for various cluster sizes (b) Paging signaling

Table 1 .

 1 Parameter Values for CALM Assessment

	Parameter	Assessment Value
	Cluster Size	{1, 5, 10, 20, 30, 40, 50} nodes
	TAL size	{15, 30, 45, 60, 75, 90, 105, 120} cells
	Cell reselection rate	30 per UE/hour [15]
	Location Update rate (for GMM [4])	1.2 per UE/hour [16][17]
	Time window duration	3 hours
	Periodic Location update timer T PLU