domain: spi phys math info

text:
L Brevault

M Balesdent

R Wuilbercq

N Subra

S Oriol

C Bonnal

M Dodds

N Taylor

R Varvill

Conceptual design of a Two-Stage-To-Orbit vehicle using SABRE engines

Aérospatiales -ONERA) share a common interest for future space launch architectures and technologies. Preliminary results of a joint study are presented in this paper with the main objective of assessing and quantifying the potential advantages and benefits of air-breathing propulsion technology developed by REL named SABRE (Synergetic Air-Breathing Rocket Engine) for future launch architectures. In that purpose, a conceptual study of two alternative Two-Stage-To-Orbit (TSTO) vehicles is carried out using the SABRE air-breathing rocket engine and a LOx/CH4 rocket engine (for the second stage) for the injection of a 15t payload into low Earth orbit. Different analyses are carried out using a coupled multidisciplinary process involving propulsion, aerodynamics, sizing and trajectory optimizations. Some key influential parameters and intricate behaviors are analyzed through sensitivity to the stage separation conditions. As a result, these analyses provide insight into technological requirements to achieve the target mission using these types of advanced propulsion technologies.

Introduction

The French Space Agency (CNES), the UK Space Agency (UKSA), Reaction Engines Limited (REL) and the French Aerospace Lab (ONERA) have a common interest in researching future space launch architectures and technologies, including the impact of using air-breathing propulsion systems. The primary objective of this joint study is to assess and quantify the potential advantages and benefits of air-breathing propulsion technology developed by REL named SABRE (Synergetic Air-Breathing Rocket Engine) for future European launch architectures. In that purpose, a conceptual study of two alternative Two-Stage-To-Orbit (TSTO) vehicles is carried out using the SABRE engine for the first stage [START_REF] Hempsell | Progress on the skylon and sabre[END_REF] and a LOx/CH4 rocket engine for the second stage. [START_REF] Iannetti | Prometheus, a low cost lox/ch4 engine prototype[END_REF] Launch vehicle reusability has the potential to lower the cost of access to space and significantly to increase the efficiency of space transportation. Reusable first stages combined with expendable upper stages are a promising first step towards fully reusable launch vehicles. Whether an air-breathing Reusable Launch Vehicle (RLV) could come to be accomplished mainly depends on the developments and research progresses of air-breathing combined engines. Various air-breathing combined cycles have been proposed. [START_REF] Daines | Combined rocket and airbreathing propulsion systems for space-launch applications[END_REF][START_REF] Víctor | Simulation, Design and Analysis of Air-Breathing Combined-Cycle Engines for High Speed Propulsion[END_REF][START_REF] Mcclinton | Airbreathing hypersonic technology vision vehicles and development dreams[END_REF][START_REF] David M Van Wie | Hypersonic airbreathing propulsion[END_REF] REL has conducted lots of fundamental and key technology studies around SABRE engine. [START_REF] Hempsell | Progress on the skylon and sabre[END_REF] SABRE engine presents the interest of being able to fly up to around 25km and Mach number 5.0 in air-breathing mode, limiting the need for liquid oxidizer during the atmospheric flight. Moreover, it combines air-breathing and rocket cycles into a single engine, removing the need to have two separated propulsion systems depending on the trajectory phase.

The goal of the present study is to analyze and compare two reusable launch vehicle concepts in terms of both feasibility, reusability and payload performance using SABRE technologies. The study is on-going and the presented results are preliminary. More analyses involving economical aspects, operational conditions and a development roadmap are under assessment.

Different past studies have focused on the use of SABRE engines, [START_REF] Barry M Hellman | Two stage to orbit conceptual vehicle designs using the sabre engine[END_REF][START_REF] Longstaff | The skylon project[END_REF][START_REF] Varvill | A comparison of propulsion concepts for ssto reusable launchers[END_REF] or alternative Rocket-Based Combined Cycle engines [START_REF] Zhou | A preliminary research on a two-stage-to-orbit vehicle with airbreathing pre-cooled hypersonic engines[END_REF][START_REF] Zhou | Preliminary analysis for a two-stage-to-orbit reusable launch vehicle[END_REF] for TSTO vehicles. These studies differ by vehicle layouts, target mission, trajectory envelop and integration of the second stage.

For the two studied concepts, this paper presents the design assumptions, simulation tools, vehicle characteristics and configurations, and reference trajectories. The paper is organized as follows. In Section 2, the mission specifications and design assumptions are introduced. In Section 3, the design method and the different modeling tools are detailed for the propulsion, the aerodynamics, the geometry and sizing, the mass budget and the trajectory simulation. Then, in Section 4, the preliminary design and trajectory results are presented for the two concepts and a first comparison of the two configurations is presented in Section 5. In Section 6, preliminary results on Return To Launch Site (RTLS) are described. Finally, in Section 8, the on-going works around this project are presented and the future works are detailed.

Mission specifications and design assumptions

The present study focuses on TSTO vehicles with the first stage being reusable and the second stage expendable. The considered reference mission for the reusable TSTO vehicle is to inject a payload of 15t into LEO at 400km on a circular equatorial orbit. It corresponds to a reference mission, but flexibility in terms of payload mass and target orbit is sought.

The architecture configurations for reusable TSTO vehicle are numerous, and this paper focuses on winged RVL. In order to identify promising architecture choices, a literature review 3-5, 8, 10, 12, 17, 20, 23, 24, 29, 32, 35-40, 44, 45 of proposed concepts and existing vehicles has been carried out (Figure 1). These concepts have been reviewed with respect to the possibility to perform the target mission and to integrate two key technologies: SABRE (for the first stage) and a low cost LOx/CH4 gas generator cycle rocket engine (for the second stage). Moreover, specific attention has been made on abort scenarios and consequences in terms of architecture choices. Following the literature review, two kinds of concepts may be suggested. As a reusable TSTO vehicle is designed using SABRE engines, high staging Mach number for the separation of the first and second stages is sought in order to minimize costs and efforts on the expandable second stage and to maximize the benefits from the reusable first stage.. The design of these concepts should be driven by a number of considerations such as the possibility (and types) of abort missions, the aerodynamics efficiency, the controllability as well as the thermal handling, due to the high Mach number at staging and the re-entry phase. The second stage is considered as expendable and its design is kept as simple as possible to minimize its cost using a low cost LOx/CH4 gas generator cycle rocket engine. Concerning the operational conditions in terms of abort, different criteria may be considered. The first criterion is relative to the identification of the different possible failures. In the concepts under study, the main identified sources of failure seem to be relative to the propulsion system (in air-breathing or rocket modes). It is necessary to identify the nature of the failure and its time of occurrence during the mission in order to define a set of abort scenarios and their aftermaths on the vehicle architectures. For these different abort scenarios, the second criterion that has to be considered is the part of the launch vehicle that should be recovered. Indeed, the question of recovering only the first stage or the payload and / or the second stage is crucial in terms of the architecture choices for the considered mission. Finally, the location of the landing site (near the launch site or a distant site requiring a large cross-range capability) is a key-driver to the architecture choice. Depending on the failure and its time of occurrence, different options can be considered: either maximizing the use of aerodynamics forces, or using the air-breathing capability (or even the rocket capability if possible and air-breathing is in failure mode), to land either at a distant site or near the launch site. Based on these considerations, two kinds of architectures may be suggested and are briefly described in the following.

Payload-bay configuration

This concept consists of a winged first stage with the second stage (and payload) located in a payload bay inside the first stage. This architecture shares similarities with the SKYLON [START_REF] Hempsell | Skylon users manual[END_REF] but some differences in terms of geometrical shape are introduced (left of Figure 7). This architecture allows to enhance the aerodynamics efficiency (see Section 3.2), to set the location of the second stage such as the discrepancy of the center of gravity (CoG) locations before and after the stage separation is reduced, to ease the search of a trimmed configuration for both ascent and RTLS as the aerodynamics configuration and the CoG location is similar for both phases and to avoid the need of fairing for the second stage. Nevertheless, this concept requires integrating the second stage inside a payload bay, even if it is relatively small due to the high Mach number at staging and the use of Lox/CH4 propellant for the second stage. Adding bay doors to the vehicle increases the structural and the separation maneuver complexities. Moreover the internal volume required to accommodate the second stage leads to a large first stage which may be more expensive to acquire and more complex to design and operate. In terms of abort scenarios, having the second stage inside the first one may make more complex abort mission strategies (difficulty to jettison the elements inside the main stage in some cases of abort mission at high dynamic pressure involving the recovery of only the first stage). A consequence is for the first stage to be designed to land carrying the upper stage and payload filled with propellants, thus generating specific risks to be mitigated during abort situations.

In-front configuration

To avoid the complexity which stems from the presence of the payload bay and the abort capabilities, an alternative configuration could be to have a tube and wing architecture and integrate the second stage in front of the first stage as conventional expandable launchers (right of Figure 7). This configuration presents advantages to separate the second stage in a similar fashion as classical launch vehicles, to avoid the need for a payload bay (benefit in terms of structural complexity, avoid specific mechanisms for the second stage jettisoning), to reduce the volume of the first stage (no need to store the second and payload inside the first stage), to allow the possibility to jettison the upper stage and payload in case of abort mission, in order to safely recover the first stage that is the most expensive part of the launch vehicle.

This concept requires additional fairing for the second stage, resulting in additional mass that is not recovered. Moreover, the centering possibilities and controllability (for both ascent and return phases) are key-aspects due to the fact that the aerodynamic configuration differ significantly between ascent with payload in front and descent without.

For the presented results in the following, the study focuses on a vehicle that takes-off from Kourou and RTLS is the prefered recovery option (that will be studied in the future works as explained in Section 6).

Design method and modeling tools 3.1 Propulsion

The SABRE engine (Figure 2) is able to fly to over Mach 5 and an altitude of around 25 km in air-breathing mode. This allows to reduce the propellant consumption compared to the use of traditional full rocket engine in the atmospheric phases of the trajectory.

A simplified view of the SABRE engine cycle is represented in Figure 3. [START_REF] Hempsell | Progress on the skylon and sabre[END_REF] It uses sub-cooled liquid hydrogen as fuel and sub-cooled liquid oxygen as the oxidizer in rocket mode. In the rocket mode, the engine operates as a closed cycle rocket engine. In the air-breathing mode, the liquid oxygen flow is replaced by the atmospheric air. The airflow goes into the engine via an intake and is cooled by a pre-cooler heat exchanger. This latter is part of a closed cycle helium loop using the hydrogen fuel as the heat sink before it enters the combustion chamber. After cooling the air is compressed and fed to the combustion chamber. [START_REF] Hempsell | Progress on the skylon and sabre[END_REF] For the propulsion performance simulation, the present study is based on a SABRE engine deck, built by REL, that provides engine performances as a function of key parameters. Normalized specific impulse (Isp), gross thrust and fuel mass flow rates as a function of altitude and Mach number are illustrated in Figures 4, 5 and6. The engine deck used for the study represents the best estimate of the SABRE installed nacelle performance available at the start of the current project. This engine deck was used as the basis for the SKYLON System SRR, performed by ESA in 2016. The toolset used was extensively reviewed by ESA experts as part of this process. Although validation data must await results from REL ongoing experimental programmes (HTX, DEMO-A, DEMO-N) [START_REF] Varvill | Sabre technology development: Status and update[END_REF] , [START_REF] Payne | Sabre demo-a -key challenges for a ground demonstrator[END_REF] at present it is believed that the model represents a good level of performance fidelity for the installed nacelle systems, suitable for preliminary systems studies such as that reported here.

The deck is based on Equivalent AirSpeed (EAS) and altitude as independent variables. The nacelle flow rates (air and fuel), gross thrust, and drag terms (momentum, spillage, external and base drags) are provided as dependent variables. System performance is calculated via an REL code which solves generalised, multiple, coupled networks of fluid paths, each containing an arbitrary number of components. All physical components within the engine (heat exchangers, turbomachinery, combustion zones and intakes and nozzles) are modeled based on physically dimensioned models and include reasonable loss/efficiency assumptions, although some simplifications are used for ducting and other connecting components. The deck supplied is generated using perfect gas models for all core flows, and a semiperfect gas assumption within the combustion zones. Enthalpy is conserved at system and subsystem boundaries. More recent models now include complete real gas flow solutions, but these are not available at the start of this study. Currently, the engine deck does not include the effect of incidence or sideslip on intake performance, and hence is not suitable for modelling high incidence flight. External and spillage drag estimates are provided by numerical simulation of the nacelle intake system, validated by ongoing experiments performed by Gas Dynamics and Bayern Chemie. [START_REF] Barth | Progress on the development and testing of the sabre air intake[END_REF] The ascent trajectory typically assumes the engine to be operating at maximum throttle throughout the ascent. For return to base and/or abort analysis, the ability to throttle the engine in some manner is key. The SABRE engine is more complex than a conventional turbojet or turbofan, and hence multiple throttling strategies are possible. That most often used within REL assumes the use of a simple choke on the core flow path, which in effect biases the flow towards the ramjet and away from the core engine, while maintaining full capture. As the specific thrust of the ramjet is lower, this reduces the thrust, and as the fuel flow rate in the nacelle is dominated by the flow through the core, reduces fuel flows and potentially increases Isp. An engine deck with throttle capability was generated for this project, although it uses a lower fidelity tool with much simplified core flow model. This deck has been validated at the 100% throttle point by comparison with the ascent deck defined above, and found to produce errors in fuel burn over the entire ascent of less than 1% (admittedly errors for just the air-breathing segment are higher).

For the second stage, a low cost LOx/CH4 gas generatoir cycle rocket engine is used to scale the engine with the mission requirements. This engine [START_REF] Iannetti | Prometheus, a lox/lch4 reusable rocket engine[END_REF][START_REF] Iannetti | Prometheus, a low cost lox/ch4 engine prototype[END_REF] is a methane-oxygen propelled engine, which is interesting in terms of compactness for the second stage compared to hydrogen-oxygen engines. Its characteristics and performances are provided by CNES. So far, a 150 kN thrust is chosen for this second stage engine. This thrust is assumed to be close enough to an optimum regarding the first stage high velocity staging, and the 15t payload objective on LEO orbit. It uses a cycle comparable to the one that is studied in the Prometheus(c) program. It uses a gas genrator cycle, single shaft single turbine turbopump. It develops a 150 kN thrust with a targeted 365s specific impulse in vacuum conditions, and could benefit from throttling and versatility capacities. However, this engine cannot derive directly from the Prometheus since its thrust will keep one order of magnitude lower than the Prometheus's one. Therefore a specific dimensionning work was conducted by CNES in order to define preliminary propulsive characteristics for this engine as shown in Table 1.

Geometry and sizing

Two configurations are studied for the reusable TSTO vehicle: a payload-bay configuration and an in-front configuration. These concepts differ by their integration of the second stage with respect to the first stage. In the payload-bay configuration, the second stage is integrated inside the first stage in the payload-bay located near the center of gravity (which evolves during the flight) while for the in-front configuration, the second stage is located in front of the first stage as in conventional launcher. Moreover, to perform a sensitivity analysis with respect to the aerodynamical shape, two different shapes are analyzed. On the one side, the in-front configuration is a tube and wing concepts similar in shape to SKYLON. [START_REF] Hempsell | Progress on the skylon and sabre[END_REF][START_REF] Hempsell | Skylon users manual[END_REF][START_REF] Mehta | Skylon aerospace plane and its aerodynamics and plumes[END_REF] For the in-front configuration, the main differences consist of the integration of the second stage that is not located in a payload-bay.

On the other side, the payload-bay concept presents an alternative aerodynamical shape. In order to improve thermal characteristics during reentry a flattened underside and elliptical nose is considered along with forward strakes to improve flight qualities and control as they provide forebody lift and nose-up pitching moment therefore canards are not required. In addition, the rear fuselage is flattened into a pen-nib shape in order to accommodate a flap for aero-control during high angle of attack reentry. Most of these aerodynamical improvements could be transfered to the in-front configuration, but in order to carry out a sensitivity analysis with respect to the aerodynamical shape (during ascent, re-entry, gliding phase, etc.) only the payload-bay configuration is considered with these geometrical characteristics.

In terms of tools to design the in-front and payload bay configurations, the geometry and sizing disciplines is a chain from geometrical modeler, to a parametric definition of the vehicle up to mesh generation for the aerodynamics discipline (Figure 8). Two geometrical modelers (OpenVSP 28 and FreeCAD [START_REF] Falck | Freecad [How-To[END_REF] ) have been used to create a parametric model for the vehicle configurations. This step consists in building a parametric model of the vehicle by defining sizing variables (e.g., diameter, vehicle length, wing surface, wing root chord) that may be affected during the design process. By modifying such variables, it enables to reconstruct a coherent vehicle when some dimensions are changed.

Once the sizing variables have been modified and the geometrical model rebuilt, data and geometrical mesh may be generated for the aerodynamics discipline (e.g. CFD meshes). This entire chain of geometry and sizing is then Figure 8: Geometry and sizing discipline integration from OpenVSP/FreeCAD to meshing generation for aerodynamics discipline wrapped into OpenMDAO [START_REF] Gray | OpenM-DAO: An Open-Source Framework for Multidisciplinary Design, Analysis, and Optimization[END_REF] as a component in order to have the possibility to define some of the sizing variables as design variables to enable a control by an optimization process.

Aerodynamics

The estimation of the aerodynamics of such launch vehicles is challenging. Indeed, the vehicles face hot reentry, resulting in a need to estimate its aerodynamics performance for high angles of attack (inducing separation flow, complex leeside flow, etc.), in a broad Mach number range : from a "standstill" to hypersonic speeds and in a transatmospheric flight path, from a continuum to a rarefied regime. In addition, the base drag in the non-propelled flight has be accounted especially at low speed. As the project is focused on the conceptual design phase, a multi-fidelity approach has been used in order to combine data from preliminary tools (SHAMAN and MISSILE [START_REF] Denis | Onera's aerodynamic prediction code-missile[END_REF] ) with some Euler CFD calculations. [START_REF] Palacios | Stanford university unstructured (su2): Analysis and design technology for turbulent flows[END_REF] SHAMAN is an engineering-level code which applies a combination of well-proven Local Surface Inclination (LSI) methods on an unstructured surface grid to compute the aerodynamics coefficients. Whenever the vehicle is not propelled (e.g., part of its reentry or for some abort scenarios), a base drag term needs to be added relying on an empirical correlation of C p,b as a function of M ∞ . This term can take up a significant part of the total drag especially at subsonic and low supersonic speed. Other empirical relations were used to account for the effect of shear stress (parasite drag) on the full drag coefficient estimates: laminar and turbulent flat plate relations applied to all wetted-area of the vehicle.

Mass budget

The mass budget discipline is in charge to estimate the inert mass of the vehicles including the primary structure (e.g., tanks, wings, interstage) and the secondary masses (landing gears, avionics, power, etc.). Estimating the mass of such vehicles is challenging due to the absence of existing similar vehicle and the lack of data from past vehicles. For the conceptual design phase, Mass Estimating Relationships (MERs) have been used from the literature. [START_REF] Castellini | Multidisciplinary design optimization for expendable launch vehicles[END_REF][START_REF] Cerro | Structural weight estimation for launch vehicles[END_REF][START_REF] Gary | Hasa: Hypersonic aerospace sizing analysis for the preliminary design of aerospace vehicles[END_REF][START_REF] Io Macconochie | Techniques for the determination of mass properties of earth-to-orbit transportation systems[END_REF][START_REF] Rohrschneider | Development of a mass estimating relationship database for launch vehicle conceptual design[END_REF] The mass budget estimation is decomposed according to the different components of the launch vehicles. Three main types of inputs are required for MERs: propulsion and geometry disciplinary outputs (diameters, lengths, surfaces and volumes of components, etc.), trajectory loads (e.g., axial and lateral accelerations, dynamic pressure, heat flux) and technological choices (structural material, tanks type and layout, etc.). The implemented MERs are based on a comprehensive set collected from various sources. [START_REF] Castellini | Multidisciplinary design optimization for expendable launch vehicles[END_REF][START_REF] Cerro | Structural weight estimation for launch vehicles[END_REF][START_REF] Gary | Hasa: Hypersonic aerospace sizing analysis for the preliminary design of aerospace vehicles[END_REF][START_REF] Io Macconochie | Techniques for the determination of mass properties of earth-to-orbit transportation systems[END_REF][START_REF] Rohrschneider | Development of a mass estimating relationship database for launch vehicle conceptual design[END_REF] The SABRE engine mass is provided by Reaction Engine Limited and the second stage rocket engine mass by CNES.

Trajectory simulation

The vehicles are modeled as a point mass in an Earth rotating frame with 3 degrees of freedom for all phases of the mission. The state vector contains seven variables for positions, velocities and the vehicle mass. The equations of motion are integrated using a 5 order Runge-Kutta technique. The vehicle control variables are the throttling of the engine, the pitch angle and the bank angle of the vehicle. The propulsion system is non -gimbled, the thrust vector produced is aligned with the longitudinal body axis of the vehicle. In order to define the optimal trajectory with the optimal vehicle configuration, the trajectory control variables are defined as design variables in the multidisciplinary design optimization problem. Moreover, several constraints are added to ensure the satisfaction of the mission (e.g., injection of the payload to the target orbit, return to the launch site) along with the satisfaction of the maximal allowed loads undertaken during the flight (axial and lateral loads, dynamic pressures, etc.).

The trajectory starts with an horizontal lift-off (not simulated, the trajectory is initialized at 50m of altitude and a Mach number of 0.5) and then an air-breathing flight powered by SABRE is considered. This phase lasts until around 25km and Mach number 5.0. At these conditions, the SABRE engine is switched to rocket mode and uses the liquid oxydizer stored in the first stage. Then, a more standard rocket phase is done until the propellant used for the ascent is burnt. Then, a short ballistic flight is carried out to increase the altitude in order to satisfy a jettison condition on the heat flux for the second stage. Then, the second stage is jettisoned and carries on with its rocket flight. Finally, a coasting phase is performed in order to reach the final orbit that is circularized. In the meantime, the first stage re-enters into the atmosphere and returns to launch site using powered SABRE engines in air-breathing mode.

Multidisciplinary Design Analysis and Optimization (MDAO)

In order to identify the optimal architectures and to carry out parametric analyses, the multidisciplinary simulation is done using OpenMDAO [START_REF] Gray | OpenM-DAO: An Open-Source Framework for Multidisciplinary Design, Analysis, and Optimization[END_REF] to integrate the different disciplines and to perform launch vehicle design and trajectory This multidisciplinary approach allows to account for the interdisciplinary couplings in the design process to ensure the interdisciplinary feasibility.

Optimization results and discussions

In order to compare the two concepts, MDAO is carried out to optimize the launch vehicle trajectory and propellant masses needed for the mission. The optimization problem consists in minimizing the Gross Lift-Off Weight (GLOW) while ensuring the payload injection into the target orbit and the satisfaction of the maximal allowed loads. Different staging conditions between the first and the second stages are considered (velocity of 5500m/s, 6000m/s and 6500m/s) in order to evaluate its impact on the launch vehicle design. The results obtained with the payload-bay configuration are presented in Section 4.1 and the in-front configuration in Section 4.2.

The preliminary results presented in the following Sections do not account for the required propellants for the RTLS, only account the propellant masses for the ascent is considered. In future works, the analyses will account for these additional propellant masses (see Section 8 on the perspectives). Then, an increase of the altitude with a high rate corresponding to the switch from air-breathing to rocket modes for the SABRE engines as it can be seen in the mass flow rate. Following the rocket phase of the first stage, the SABRE engines are cut-off and a short ballistic flight is carried out to increase the altitude. Then the two stages are separated and the second stage engine is ignited until the start of the coasting phase (which is not represented on the Figures). Altitude evolution for the ascent trajectory for the payload-bay configuration (without coasting phase) Figure 16:

Velocity evolution for the ascent trajectory for the payload-bay configuration (without coasting phase) Table 2 presents the GLOW and the distribution of the propellant masses between the first and second stages and between air-breathing and rocket mode for the SABRE engines. The GLOW increases with the Mach staging condition from 368t for 5500m/s to 425t for 6500m/s. It can be seen that the second stage uses a small amount of propellants, facilitating its integration in the payload-bay of the first stage especially with the high density of methane.

Gliding re-entry trajectories

In order to compare the aerodynamical shapes between the payload-bay and the in-front configurations, a gliding reentry is performed considering two scenarios: maximization of the down-range and maximization of the cross-range. Moreover, these studies enable to illustrate the reachable footprint in case the SABRE engines do not re-ignite after atmospheric re-entry. The two problems consist in the maximization of the down-range and cross-range using the angle of attack and the bank angle as control variables. Figures 18, 19 First, the different bumps on the altitude plots for the down-range and the cross-range maximizations are characteristics of classical skipping re-entry trajectories that use bounces on the atmosphere to increase the range covered. As expected, the higher the staging condition, the higher the down-range and cross-range reached by the vehicle during the re-entry.

Under certain conditions (in terms of maximal loads undertaken during the re-entry), in case of impossibility to re-ignite the SABRE engines, the preliminary analyses (Figure 24) show that it is possible to reach Cape Verde island or Ascension island to land the first stage which is an important aspect for abort scenarios. 3 presents the GLOW and the distribution of the propellant masses between the first and second stages and between air-breathing and rocket mode for the SABRE engines. The GLOW increases with the Mach staging condition from 330t for 5500m/s to 391t for 6500m/s. The second stage is similar to the payload-bay configuration as the staging conditions are identical. The gliding performances of the in-front configuration are worst than the payload-bay concept and this is due to the aerodynamical shape improvements of this latter. The re-entry trajectories present the same skipping bumps as for the payload-bay concept. Similarly to the payload-bay configuration, under certain conditions (in terms of maximal loads undertaken during the re-entry), in case of impossibility to re-ignite the SABRE engines, the preliminary analyses (Figure 24) show that it is possible to reach Cape Verde island, Azores or Ascension island to land the first stage depending on the staging conditions.

Comparisons and analyses

From the preliminary analyses, the two configurations are able to inject a 15t payload into a 400km circular orbit taking-off from Kourou, without consideration of the return to launch site. The two configurations present similar results for the ascent trajectories, but some specificities differentiate them especially for the re-entry. Based on the current modeling fidelity, as presented in Figure 35, the payload-bay configuration tends to present a higher GLOW compared to the in-front for the same 2 nd stage separation condition. However, the mass estimation needs to be refined in order to capture the mass differences with a higher fidelity (see future works). For the re-entry, for both the downrange and the cross-range maximizations in gliding mode (Figure 36), the payload-bay concept covers a higher range and this is due to the improvements in terms of aerodynamical shapes including the flattened underside, the forward stakes and the rear fuselage flattened into a pen-nib shape. Some of these improvements may be applied to the in-front configuration to improve its glider capabilities.

This improved gliding capabilities are interesting especially for the abort scenario in case the SABRE engines do not re-ignite during the re-entry. It offers the possibility to reach a distant landing site that is not Kourou (e.g., Azores, Cape Verde, Ascension island) where a runway of at least 3km is available.

Return to launch site

Simulations of powered RTLS using SABRE engine in air-breathing mode have been performed to provide a first estimate of the quantity of propellant needed to return to Kourou after the stage separation. After the 2 nd stage separation, the first stage performs a high angle of attack re-entry into the atmosphere and then starts to carry out a u-turn. When the conditions for re-ignitition of the SABRE engines in air-breathing mode are reached (around 25km, Mach 5.0), a powered RTLS is performed (Figure 37). Further works on that aspect are currently under development.

Figure 37: Illustration of return to launch site with a propelled air-breathing phase using SABRE engines. Orange trajectory corresponds to the ascent phase, then cyan trajectory is the ballistic phase, the re-entry and u-turn of the first stage while the blue trajectory corresponds to the air-breathing propelled phase to return to the launch site.

Study at Reaction Engines Limited

A parallel conceptual study is being conducted at REL using an independent toolset, whilst applying common toplevel assumptions. This approach allows for direct comparison of the results and preliminary data has demonstrated a strong correlation between the two studies. The REL results will be published at a later date once the study has been concluded.

Moreover, a sensitivity to the second stage engine is under analysis, with a low cost LOx/CH4 gas generator cycle rocket engine of 180kN of thrust. This choice enables to compare in further analysis with a full LOX/LH2 TSTO system using the well known 180kN Vinci(c) engine that will propel Ariane 6 Upper Liquid Propulsive Module. [START_REF] Alliot | Vinci®, the european reference for ariane 6 upper stage cryogenic propulsive system[END_REF]

Conclusions and perspectives

This paper presented an on-going study between CNES, REL, ONERA and UKSA on launch vehicle design of reusable TSTO vehicles integrating SABRE technologies. The present study focused on two TSTO vehicles to evaluate the potential interest of the combination of SABRE and a low cost LOx/CH4 gas generator cycle rocket engine on an integrated concept. Two alternative concepts are analyzed: a payload-bay configuration and an in-front configuration.

Under the current assumptions of SABRE and the low cost LOx/CH4 gas generator cycle rocket engine engine performances, at the conceptual design level, a TSTO vehicle seems to be a feasible option for the considered mission. The two studied concepts present important similarities but also specificities especially with respect to the second stage integration and their performances in gliding mode. Among the different analyses, a key aspect that is highlighted is the importance of aerodynamical shape design to gain important gliding capabilities in case of SABRE engine deficiencies during the re-entry. Different ascent trajectories considering various staging conditions have been presented for the two concepts along with some re-entry trajectories in gliding mode. Moreover, preliminary trajectory analyses showed the possibility to reach distant landing site in gliding mode under certain conditions. Some key influential parameters and intricate behaviors have been, analyzed, more particularly through a sensitivity to the stage separation conditions for the two concepts.

This on-going project has more analyses to perform including the full RTLS phase simulation and optimization but also some abort scenario simulations. It will also be necessary to study the controllability of such vehicle especially for the in-front concept due to the different vehicle configurations for the ascent and the re-entry (aerodynamical performances, evolution of the center of gravity, etc.). As a result, these preliminary analyses provide insight into technological requirements to achieve the target mission using advanced propulsion technologies. The objective of this project is to finalize a conceptual study of these vehicles including also recovery and refurbishment assessments, economical aspects, operational conditions as well as a preliminary development roadmap.

Figure 1 :

 1 Figure 1: Literature review of the proposed winged RLV concepts

Figure 2 :

 2 Figure2: SABRE engine architecture[START_REF] Hempsell | Progress on the skylon and sabre[END_REF] Figure3: SABRE engine simplified cycle[START_REF] Hempsell | Progress on the skylon and sabre[END_REF]

Figure 4 :Figure 5 :

 45 Figure 4: Illustration of the normalized Isp provided by the SABRE simulation deck as a function of Mach and altitude in air-breathing mode

Figure 6 :

 6 Figure 6: Illustration of the normalized fuel mass flow rate provided by the SABRE simulation deck as a function of Mach and altitude in air-breathing mode

Figure 7 :

 7 Figure 7: Illustration of the two first alternative configurations (payload-bay on the left and in-front on the right) represented without control surfaces

Figure 9 :

 9 Figure 9: SHAMAN principle

Figure 10 :

 10 Figure 10: Example of CFD Euler calculation

Figure 11 :

 11 Figure 11: Example of normalized lift-to-drag improvements combining SHAMAN and Euler CFD as a function of angle of attack and Mach number.

Figure 12 :

 12 Figure 12: Example of Euler CFD calculations for different angles of attack and Mach numbers.

Figure 13 :

 13 Figure 13: Example of trajectory simulations (ascent phase in left, and downrange re-entry in right).

Figure 14 :

 14 Figure 14: Integrated simulation multidisciplinary process.

  CONCEPTUAL DESIGN OF A TWO-STAGE-TO-ORBIT VEHICLE USING SABRE ENGINES 4.1 Payload-bay configuration 4.1.1 Ascent trajectories The altitude, velocity and mass as functions of time for the three considered staging conditions are represented in Figures 15, 16 and 17. Considering these Figures, the different phases of the trajectory are distinguished: first an ascent phase up to 25km of altitude with a constant altitude phase allowing to increase the velocity up to Mach 5.

Figure

  Figure 15:Altitude evolution for the ascent trajectory for the payload-bay configuration (without coasting phase)

  Figure 15:Altitude evolution for the ascent trajectory for the payload-bay configuration (without coasting phase)

Figure 17 :

 17 Figure 17: Mass evolution for the ascent trajectory for the payloadbay configuration (without coasting phase)

  and 20 present the re-entry with maximization of the down-range while Figures 21, 22 and 23 illustrate the cross-range maximization.

Figure 18 :Figure 19 :

 1819 Figure 18: Altitude for the reentry trajectory for the payload-bay configuration, maximization of the down-range Figure 19: Velocity for the reentry trajectory for the payload-bay configuration, maximization of the down-range

Figure 20 :Figure 21 :Figure 22 :

 202122 Figure 20: Longitude and latitude for the reentry trajectory for the payload-bay configuration, maximization of the down-range

Figure 23 :

 23 Figure 23: Longitude and latitude for the reentry trajectory for the payload-bay configuration, maximization of the cross-range

Figure 24 :

 24 Figure 24: Possible landing sites depending on the staging velocity for the -bay concept for 5500m/s and 6000m/s.

Figure 25 :Figure 26 :Figure 27 :

 252627 Figure 25: Altitude for the ascent trajectory for the in-front configuration Figure 26: Velocity for the ascent trajectory for the in-front configuration Figure 27: Mass for the ascent trajectory for the in-front configuration Table3presents the GLOW and the distribution of the propellant masses between the first and second stages and between air-breathing and rocket mode for the SABRE engines. The GLOW increases with the Mach staging condition from 330t for 5500m/s to 391t for 6500m/s. The second stage is similar to the payload-bay configuration as the staging conditions are identical.

Figure 28 :Figure 29 :

 2829 Figure 28: Altitude for the reentry trajectory for the in-front configuration, maximization of the downrange (without coasting phase) Figure 29: Velocity for the reentry trajectory for the in-front configuration, maximization of the downrange (without coasting phase)

Figure 30 :

 30 Figure 30: Longitude and latitude for the reentry trajectory for the infront configuration, maximization of the down-range (without coasting phase)

Figure 31 :Figure 32 :Figure 33 :Figure 34 :

 31323334 Figure 31: Altitude for the reentry trajectory for the in-front configuration, maximization of the crossrange Figure 32: Velocity for the reentry trajectory for the in-front configuration, maximization of the crossrange Figure 33: Longitude and latitude for the reentry trajectory for the infront configuration, maximization of the cross-range

Figure 35 :

 35 Figure 35: Comparison of the GLOW for the two concepts and the different staging conditions.

Figure 36 :

 36 Figure 36: Comparison of the distance covered during gliding re-entry for the two concepts and the different staging conditions.

Table 1 :

 1 Second stage main propulsive characteristics using CNES in house tools for preliminary dimensioning of

	cryogenic engine
	Thrust (vaccum)	150kN
	Engine cycle	Gas generator
	Specific impulse	365s (upper stage)
	Mass target	< 350kg
	Fuel	Ch4 (methane)
	Oxidizer	LOx (oxygen)
	Engine overall length	< 3m

Table 3 :

 3 GLOW and propellant masses for the in-front configuration depending on the staging conditions, AB: airbreathing mode and R: rocket mode Staging condition (velocity) GLOW Propellant mass 1 st stage Propellant mass 2 nd stage

	5500m/s	330t	216t (AB: 26t / R: 190t)	16t
	6000m/s	389t	243t (AB: 30t / R: 213t)	11t
	6500m/s	391t	283t (AB: 36t / R: 247t)	7t
	4.2.2 Gliding re-entry trajectories

DOI: 10.13009/EUCASS2019-454

Acknowledgments

This project is part of an agreement between Centre National d'Etudes Spatiales (CNES), the UK Space Agency (UKSA) and Reaction Engines Limited (REL), this work is funded by CNES for the ONERA works.

The French Space Agency (Centre National d'


Mohamed Maghenem
email: mmaghene@ucsc.edu.

Antonio Loría

Emmanuel Nuño
email: emmanuel.nuno@cucei.udg.mx.

Elena Panteley
email: panteley@centralesupelec.fr.

Distributed full-consensus control of nonholonomic vehicles under non-differentiable measurement delays

Keywords: Formation control, persistency of excitation, Lyapunov design, nonholonomic systems

We address the problem of consensus control of second-order nonholonomic systems via distributed control under the assumption that each vehicle receives measured states from a set of neighbors, with a bounded, time-varying, but nondifferentiable delay. The controller that we propose guarantees full consensus, in the sense that a common consensus point may be reached both in the Cartesian positions on the plane and the orientations of all robots referred to a fixed frame. Our controller is smooth, hence time-varying. Notably, it relies on a property of persistency of excitation. Our main statement guarantees uniform global asymptotic stability. Also, we provide some simulation results to illustrate our theoretical findings.

I. INTRODUCTION

C ONSENSUS control of networked dynamical systems consists in steering certain variables of interest to a nonspecified common value. This problem is well studied and solved for a variety of dynamical systems, most often with linear dynamics, or nonlinear but with reliable interconnections; see for instance [START_REF] Ren | Distributed consensus in multivehicle cooperative control[END_REF], [START_REF] Cao | An overview of recent progress in the study of distributed multi-agent coordination[END_REF] and references therein.

As pointed out in [START_REF] Cao | An overview of recent progress in the study of distributed multi-agent coordination[END_REF], consensus control of nonholonomic vehicles is particularly challenging since these systems are underactuated. But, also, because their nonholonomy prevents stabilization of a set-point via smooth time-invariant feedback [START_REF] Brockett | Asymptotic stability and feedback stabilization[END_REF], [START_REF] Lin | Necessary and sufficient graphical conditions for formation control of unicycles[END_REF]. Indeed, consensus control is inherently a setpoint stabilization problem and it must be distinguished from formation-tracking control [START_REF] Oh | A survey of multi-agent formation control[END_REF].

There are many decentralized controllers (either timevarying or non-smooth) for consensus of nonholonomic mobile robots proposed in the literature; see, for instance, [START_REF] Dimarogonas | On the rendezvous problem for multiple nonholonomic agents[END_REF]- [START_REF] Kranakis | Mobile agent rendezvous: A survey[END_REF] and the references therein, to mention a few. To distinguish these works, it is adequate to regard the type of consensus problem that is addressed. For instance, it may be partial consensus or full consensus. In the first case consensus in the Cartesian coordinates only is of interest while controlling the orientation may be neglected [START_REF] Lin | Necessary and sufficient graphical conditions for formation control of unicycles[END_REF] or it may consist in achieving a desired orientation [START_REF] Maghenem | Consensus of multi-agent systems with nonholonomic restrictions via Lyapunov's direct method[END_REF]. In the case of full consensus the objective is for all robots to acquire common, non pre-specified position and orientation [START_REF] Dimarogonas | On the rendezvous problem for multiple nonholonomic agents[END_REF], [START_REF] Yang | Smooth time-varying formation control of multiple nonholonomic agents[END_REF].

Other distinctions pertain to the hypotheses made regarding the model, the formation, or the scenario. For instance, in [START_REF] He | Consensus-based formation control for nonholonomic vehicles with parallel desired formations[END_REF] a consensus-based formation control problem is solved, albeit for specific parallel formations. In [START_REF] Yang | Smooth time-varying formation control of multiple nonholonomic agents[END_REF] and [START_REF] Peng | Distributed consensusbased formation control for multiple nonholonomic mobile robots with a specified reference trajectory[END_REF], the consensus problem is solved using smooth time-varying control laws that lead the agents to a given formation using only their orientation. In [START_REF] Dong | Consensus of multiple nonholonomic systems[END_REF] consensus is solved for nonhoholonomic systems in chained form, under the assumption that the interconnections are affected by time delays.

In this paper we address the full-consensus problem. In contrast to [START_REF] Dimarogonas | On the rendezvous problem for multiple nonholonomic agents[END_REF] our controller is smooth time-varying and, different from [START_REF] Yang | Smooth time-varying formation control of multiple nonholonomic agents[END_REF], we establish uniform convergence of the error trajectories, which is not guaranteed by the analysis tools used in that reference. We also stress that our controllers rely on persistency of excitation, which has been widely used in the literature since its first use, for tracking control, in [START_REF] Panteley | Exponential tracking of a mobile car using a cascaded approach[END_REF] and for set-point stabilization, in [START_REF] Loría | A new persistency-of-excitation condition for UGAS of NLTV systems: Application to stabilization of nonholonomic systems[END_REF].

As in [START_REF] Dong | Consensus of multiple nonholonomic systems[END_REF], we assume that the communications are affected by delays -see also [START_REF] Wang | Consensus of networked mechanical systems with communication delays: A unified framework[END_REF] and [START_REF] Nuño | Leaderless consensus-based formation control of multiple nonholonomic mobile robots with interconnecting delays[END_REF]. However, in clear contrast with these references, we assume that delays are timevarying and non-differentiable. This is not an assumption of pure theoretical interest. It is motivated by, and apply to, scenarii in which a group of robots that are equipped with necessary sensors to measure their positions and orientations, transmit these data to their reachable neighbors over a wireless network. For instance, this may occur at the beginning of a rendez-vous-and-scout mission in which the robots are too far apart from each other, for relative-distance and line-ofsight sensors to be useful [START_REF] Roza | A smooth distributed feedback for global rendezvous of unicycles[END_REF]. Naturally, the communication channels between any two pair of vehicles is bidirectional, so the graph topology is undirected, but the network introduces time-varying non-necessarily differentiable delays [START_REF] Abdessameud | Synchronization of Lagrangian systems with irregular communication delays[END_REF]. What enables to lift the unrealistic assumption on differentiability of the delays is that, in contrast to the literature (see e.g. [START_REF] Nuño | Leaderless consensus-based formation control of multiple nonholonomic mobile robots with interconnecting delays[END_REF]), our analysis does not involve Barbȃlat's lemma. Indeed, the utilization of this popular tool of analysis entails the recursive differentiation of the system's dynamics and, hence, of the time-delay function. Instead, we employ Lyapunov-Krasovskiȋ's method and output-injection arguments. To the best of our knowledge this has no precedent in the literature (other than [START_REF] Maghenem | Stability and stabilization of networked varying systems[END_REF]).

II. PROBLEM FORMULATION AND ITS SOLUTION

Let us consider a swarm of N autonomous, force-controlled, vehicles with dynamic model

ẋi = v i cos(θ i ) (1a) ẏi = v i sin(θ i ) (1b) θi = ω i , (1c) vi = u vi (1d) ωi = u ωi , i ≤ N, (1e)
where the variables x i and y i denote the Cartesian coordinates of the middle point on the vehicle's wheels axis relative to a fixed frame, θ i denotes the vehicle's orientation with respect to the abcissae, and v i and ω i denote the forward and angular velocities respectively. The model [START_REF] Ren | Distributed consensus in multivehicle cooperative control[END_REF] corresponds to that of differential-wheel robots, but more general models, which include Lagrangian dynamics in place of Eqs. (1d)-(1e), are available, e.g., in [START_REF] Do | A global output-feedback controller for simultaneous tracking and stabilization of unicycle-type mobile robots[END_REF]. Our main results apply to these models too since they are fully feedback-linearizable. Due to evident physical constraints, consensus for noholonomic vehicles is formulated as the property that all the robots' Cartesian coordinates be equal modulo a certain offset with respect to an unknown barycenter with coordinates z c := [x c y c ] , thereby constituting a formation. Thus, defining such an offset for each robot via the quantities (δ xi , δ yi ), we introduce the displaced-position variables

z i := x i -δ xi y i -δ yi , z = col[z i ]∈ R 2N . (2)
With this notation, the full consensus-formation may be characterized as the property that

lim t→∞ z i (t) = z c , lim t→∞ θ i (t) = θ c (3a) lim t→∞ v i (t) = 0, lim t→∞ ω i (t) = 0 (3b)
where both, the consensus position, z c , and the orientation, θ c , are not imposed a priori, but depend on the network's topology and the systems' initial conditions. We use the qualifier "full" as opposed to "partial" in which case, orientation consensus is not pursued [START_REF] Maghenem | Consensus of multi-agent systems with nonholonomic restrictions via Lyapunov's direct method[END_REF], or simply, orientation control is completely neglected [START_REF] Lin | Necessary and sufficient graphical conditions for formation control of unicycles[END_REF], [START_REF] Roza | A smooth distributed feedback for global rendezvous of unicycles[END_REF]. It might be argued that this problem may be solved by decoupling the consensus and the stabilization problems, i.e., by using an individual controller for each robot to steer it to a consensual reference trajectory ξ i (t) computed using a classical distributed consensus algorithm, such as

ξi = j∈Ni a ij (ξ j -ξ i ), ξ i (0) := [z i (0) θ i (0)] (4)
where a ij ≥ 0; a ij > 0 if the ith and jth vehicles communicate with each other and a ij = 0 otherwise. While such approach could be successfully used for Lagrangian systems (with holonomic constraints), it would appear ineffective for nonholonomic systems since such ξ i (t), in general, may be unfeasible -it may very well not satisfy equations (1a)-(1c). Thus, we define the errors

e i = φ(θ i ) j∈Ni a ij (z j -z i ), (5)
s i = φ(θ i ) ⊥ j∈Ni a ij (z j -z i ) (6)
where

φ(θ i ) = [cos(θ i ) sin(θ i )] and φ(θ i ) ⊥ = [sin(θ i ) - cos(θ i )]. Note that rank[φ(θ i ) φ(θ i ) ⊥ ] = 2 for all θ i ∈ R therefore, z i = z c for all i ≤ N if and only if (s, e) = (0, 0) [12]. That is, consensus is reached if and only if (s, e, θ) = (0, 0, θ c ) and (v i , ω i ) → (0, 0).
Next, in coherence with the setting described in the Introduction, we introduce the following hypothesis on the communications network. Assumption 1 The interconnection graph is undirected, static and connected. Also, the communication during the transfer of information from the jth to the ith agent is affected by a time delay, determined by a function t → T ij (t), not necessarily continuous, but satisfying

ess sup t≥0 |T ij (t)| ≤ T * (7)
where T * < ∞ is known. Now, beyond the mere convergence property expressed by the limits in (3), we solve the following open problem.

Full-Consensus with delays. Consider a network of N nonholonomic robots, under Assumption 1. Then, design a dynamic controller such that all robots positions and orientation converge, globally, uniformly, and asymptotically, to a given formation pattern. More precisely, such that the origin for the closed-loop system,

e i s i (θ i -θ c ) v i ω i = 0, i ≤ N ∈ R 5N ,
is uniformly globally asymptotically stable.

The controller that solves this problem is defined as follows. For each robot (i ≤ N ), the first control input is

u vi = -k di v i + k pi e di , k di , k pi > 0, (8)
where e di corresponds to the error (5) under the effect of measurement delay that is,

e di = φ(θ i ) j∈Ni a ij z j (t -T ij ) -z i . (9)
Correspondingly, based on (6) we introduce

s di = φ(θ i ) ⊥ j∈Ni a ij z j (t -T ij ) -z i . ( 10
)
and the second control input, which is computed dynamically, i.e.,

u ωi = -k ωi ω i -k ωi qi (t)κ i (s di , e di ) - j∈Ni a ij θ i -θ j (t -T ij (t)) + k αi α i - j∈Ni a ij q i (t)κ i (s di , e di ) -q j (t -T ij (t))κ j (s ddj , e ddj ) (11a) αi = -k αi α i -k Ii ω i + ṗ(t)κ i (s di , e di ) (11b)
q (3) i + k αi qi + k Ii qi = ṗ, (11c)
where α i and q i are controller states; k αi , k αi , k ωi , and k Ii > 0 are control gains; q j , θ j and κ j are received directly from the jth neighbor -this is explained farther below, and κ i (s di , e di ) is the ith element of the vector

κ(s d , e d ) := 1 2 (s 2 d1 + e 2 d1 ) • • • (s 2 dN + e 2 dN ) , (12)
where e d := col[e di ] and s d := col[s di ]. The function p : R ≥0 → R, whose derivative appears in (11b) and (11c), is smooth, bounded and has bounded derivatives, while κ j (s ddj , e ddj ) is a quantity not measured, but received from the jth agent, affected by a double delay. This is explained next.

The distributed dynamic controller ( 11) is implemented as follows. On one hand, each robot measures its own states, (z i , θ i , v i , ω i ), and it receives the corresponding positions and orientations from its N i neighbors, albeit with a time-varying delay T ij (t), that is z j (t -T ij (t)) and θ j (t -T ij (t)). Then, the ith robot computes the errors using the fomulae ( 5) and ( 6), which, in view of the delays in the measurements, actually become e di and s di , as in ( 9) and [START_REF] Maghenem | Stability and stabilization of networked varying systems[END_REF]. With the latter, also the coefficients

κ i (s di , e di ) := 1 2 s 2 di + e 2 di , i ≤ N,
are computed. Furthermore, the coefficients q i (t) and qi (t) are generated using (11c). On the other hand, according to the previous rationale, each neighbor robot computes and transmits its corresponding quantities q j (t) and κ j (s dj , e dj ), computed by the jth robot using the information from each of its own N j neighbors. Thus, the data received by the ith agent, with a delay T ij (t), correspond to q j (t -T ij (t)) and

κ j (s ddj , e ddj ) := 1 2 s 2 ddj + e 2 ddj , j ≤ N,
where s ddj and e ddj stand for s dj (t-T ij (t)) and e dj (t-T ij (t)) respectively. It is emphasized that these measurements are affected by a double delay and, mathematically, correspond to

e ddj = -φ(θ j (t -T ij (t))) k∈Nj a jk z j (t -T ij (t)) -z k (t -T ij (t) -T jk (t -T ij (t))) (13)
s ddj = -φ(θ j (t -T ij (t))) ⊥ k∈Nj a jk z j (t -T ij (t)) -z k (t -T ij (t) -T jk (t -T ij (t))) . ( 14
)
The presence of this double delay considerably increases the difficulty to analyze the stability of the closed-loop system, but we stress that the expressions on the right-hand sides of ( 13) and ( 14) are not used in the implementation of the control law (11a); they are only provided for completeness.

III. MAIN RESULT

As is well-known, nonholonomic systems are not stabilizable to an equilibrium via smooth feedback, unless it is time-varying. We use a controller of the so-called δ-persistently-exciting type [START_REF] Loría | A new persistency-of-excitation condition for UGAS of NLTV systems: Application to stabilization of nonholonomic systems[END_REF], [START_REF] Maghenem | Consensus of multi-agent systems with nonholonomic restrictions via Lyapunov's direct method[END_REF]. The stabilization mechanism of the control law (11a) relies on the excitation introduced by the function α i , which, roughly speaking, prevails as long as the consensus errors remain away from a δ-neighborhood of the origin. Such excitation is provided by ṗi κ i (s di , e di ) and, more particularly, by ṗi . Indeed, the term ṗi κ i (s di , e di ) is δ-persistently exciting and it may be regarded as an input in equation (11b), which is reminiscent of a low-pass filter with output α i . Therefore, our main statement relies on the following hypothesis.

Assumption 2

The functions p, ṗ, p, and p (3) are bounded and, moreover, ṗ(t) is persistently exciting, i.e., there exist T p and µ p > 0 such that

t+Tp t ṗ(s) 2 ds ≥ µ p ∀ t ≥ 0.
Now, akin to adaptive control theory, a rule of thumb to define ṗi is to make it periodic (although this is by no means necessary) and for it to have at least as many frequencies as robots in the swarm. Other related design parameters are the gains k αi in (11b) and k αi and k Ii in (11c) .

Theorem 1 Consider the systems (1) in closed loop with the distributed controllers ( 8)- [START_REF] Kranakis | Mobile agent rendezvous: A survey[END_REF]. Let Assumption 1 hold and

k ωm ≥ c 1 /c 2 where c 1 = 2T * 2 N 2 ā2 1 6 + 2c 2 , (15a)
c 2 = 3k ωM -2λ N (L) + 3 2 k IM k αm , (15b)
and k m and k M denote, respectively, the smallest and largest values of k i , for all i ≤ N and ā ≥ a ij for all i, j ≤ N . Then, consensus is reached, both in Cartesian positions and orientation angles and the origin for the closed-loop system is uniformly globally asymptotically stable.

The proof of this statement relies on classical stability theory; specifically, on Lyapunov-Krasovskii theory for functional differential equations. First, we derive the closed-loop equations in a suitable form and then, we give the main guidelines to construct a Lyapunov-Krasovskii functional. The latter is done using the elegant Mazenc construction [START_REF] Mazenc | Strict Lyapunov functions for time-varying systems[END_REF], [START_REF] Malisoff | Constructions of Strict Lyapunov functions[END_REF].

A. The closed-loop equations

We start by introducing a multivariable model for the networked vehicles and a compact vector form for the controller (8)- [START_REF] Kranakis | Mobile agent rendezvous: A survey[END_REF].

Consider the equations (1) and let us define

θ := col[θ i ], v := col[v i ], ω = col[ω i ], Φ(θ) = blockdiag[φ(θ i )] ∈ R 2N ×N ,
and the control inputs u v :=col[u vi ] and u ω :=col[u ωi ]. The resulting multiagent system model is

ż = Φ(θ)v (16a) v = u v (16b) θ = ω (16c) ω = u ω . ( 16d
)
We use a Laplacian-based representation of the graph, defined by the matrix

L := [ ij ] ∈ R N ×N , where ij = j∈Ni a ij i = j -a ij i = j . ( 17
)
Under Assumption 1, L is symmetric, it has a single zeroeigenvalue and the rest of the spectrum is positive. Thus, rank(L) = N -1.

For simple integrators, zi , the first limit in (3a) is equivalent to having lim t→∞ [L ⊗ I 2 ]z(t) = 0. In view of the nonholonomy, however, for the system (16a) the counterpart of the relation [L ⊗ I 2 ]z = 0 is that e = 0 and s = 0 simultaneously. To better see this, note that

e = Φ(θ) Lz, s = Φ(θ) ⊥ Lz, (18)
where L := L ⊗ I 2 and the matrix Φ(θ) Φ(θ) ⊥ has full rank.

Next, with the purpose of obtaining a more compact layout of the controller equations, and only for the purpose of analysis, we introduce the change of coordinates,

e θi := θ i + q i (t)κ i (s i , e i ) ( 19
)
so that, substituting θ i = e θi -q i (t)κ i (s i , e i ) and

θ j (t -T ij (t)) = e θj (t -T ij (t)) -q j (t -T ij (t))κ j (s dj , e dj ) (20)
in the third term on the right-hand side of (11a), we obtain

u ωi = -k ωi ω i -k ωi qi (t)κ i (s di , e di ) - j∈Ni a ij e θi -e θj (t -T ij (t)) + k αi α i + j∈Ni a ij q i (t) κ i (s i , e i ) -κ i (s di , e di ) + j∈Ni a ij q j (t -T ij (t)) κ j (s ddj , e ddj ) -κ j (s dj , e dj ) .
Then, let

κ di := κ i (s di , e di ) -κ i (s i , e i ), (21)
κ ddj := κ j (s ddj , e ddj ) -κ j (s dj , e dj ), (22)
a i ( żt ) := j∈Ni a ij t t-Tij (t) żj (σ)dσ (23)
and

á i := j∈Ni a ij q j (t -T ij (t))κ ddj
and let us gather all terms in N -dimensional vectors and appropriate matrices of dimension N × N . That is, we define the control gains Thus, the distributed controller for the multiagent system ( 16) reads

K ω := diag[k ωi ], K I := diag[k I ], K α := diag[k α ], K α := diag[k α ], the matrices Q := diag[q i ], κ d := diag[κ di ], D := diag j∈Ni a ij ,
u ω = -K ω ω + Q(t)κ(s d , e d ) -Le θ + A( ėθt ) +K α α -DQ(t)κ d (s, e, θ, żt ) + B(t) (24a) α = -K α α -K I ω + ṗ(t)κ(s d , e d ).
(24b)

We stress, once more, that these expressions are used only for the purpose of analysis, but the controller is implemented using ( 8), [START_REF] Kranakis | Mobile agent rendezvous: A survey[END_REF] as explained in the previous section.

On the other hand, the distributed controller for the forward motion, u v , is given by

u v = -K p v -K d e d ( 25
)
where

K p := diag[k pi ], K d := diag[k di ] -cf.
Eq. ( 8). Now, let us define

e ω = ω + Q(t)κ(s, e), e α = α + Q(t)κ(s, e) (26)
and the closed-loop states

X t = [v e s ] , X r = [e θ e ω e α ] . (27)
We also introduce the matrices

ē := diag[e i ], s := col[s i ], ēω := diag[e ωi ], κ := diag[κ i ].
With these notations, the closed-loop equations are obtained by replacing (24a) and ( 25) in (16d) and (16b), respectively, and using the expressions in (26). That is,

Ẋt =   -K d -K p 0 0 0 Qκ -ēω 0 -Qκ + ēω 0   X t +   K p Φ A( żt ) Φ LΦv Φ ⊥ LΦv   (28a) Ẋr = F r (X r , e θt ) + Ψ(t, X t , θ, żt ), (28b)
where

F r :=   0 I N 0 -L -K ω I N 0 -K I -K α   X r -   0 A( ėθt ) 0   Ψ :=    Q ēΦ + sΦ ⊥ LΦv Q ēΦ + sΦ ⊥ LΦv -K ω Q + Q κ d Q ēΦ + sΦ ⊥ LΦv -ṗκ d    -   0 B(t) 0  
The first closed-loop equation, (28a), is the same as in [START_REF] Maghenem | Decentralized partial-consensus control of nonholonomic vehicles over networks with interconnection delays[END_REF], where only partial-consensus with measurement delays, but only in the Cartesian coordinates is addressed. Furthermore, in the absence of measurement delays, we recover the closedloop system [12, Eq. ( 21)]. Therefore, we can rely on the analyses in the previous references to make the following reasoning. Firstly, after [START_REF] Maghenem | Decentralized partial-consensus control of nonholonomic vehicles over networks with interconnection delays[END_REF], it may be established that X t → 0 provided that K p and K d > 0 and Q is persistently exciting, that is, if there exist µ and T > 0 such that

t+T t Q(s)ds ≥ µ I ∀ t ≥ 0.
This is the case under Assumption 2.

Furthermore, this holds uniformly in the error trajectories e ω (t) hence, the "translational motion" error dynamics (28a) may be regarded as a system that is decoupled from the "angular motion" error dynamics, (28b). In turn, in the latter equation, Ψ is reminiscent of a (vanishing) perturbation. It may be reasonably assumed that a cascades argument may be invoked to conclude the convergence of all state trajectories to the origin.

B. Stability analysis

The previous rationale is at the basis of the following statement, which covers the main results in [START_REF] Maghenem | Consensus of multi-agent systems with nonholonomic restrictions via Lyapunov's direct method[END_REF] and [START_REF] Maghenem | Decentralized partial-consensus control of nonholonomic vehicles over networks with interconnection delays[END_REF] and establishes uniform global asymptotic stability of the origin for (28) provided that B ≡ 0. Farther below, Lemma 2 establishes that the overall system is robust with respect to the perturbation B(t) and uniform asymptotic stability is preserved. Lemma 1 Consider the system (28) with B ≡ 0, under Assumptions 1 and 2 and the condition [START_REF] Loría | A new persistency-of-excitation condition for UGAS of NLTV systems: Application to stabilization of nonholonomic systems[END_REF]. Then, the set

D := (X t , X r ) ∈ R 6N : (X t , Le θ , e ω , e α ) = 0 (29)
is uniformly globally asymptotically stable. Sketch of Proof. First, from [START_REF] Maghenem | Decentralized partial-consensus control of nonholonomic vehicles over networks with interconnection delays[END_REF], we recall the functional

V 1 = v K -1 p v + z Lz + 0 -T * t t+θ | ż(τ )| 2 dτ dθ, (30)
which is positive definite and radially unbounded with respect to X t -because λ 2 (L)z Lz ≤ |e| 2 + |s| 2 ≤ λ N (L)z Lz where λ i is the ith eigenvalue-and its total along the trajectories of (28a) satisfies

V1 ≤ -v K -1 p K d v -Y ( żt ) (31)
Y ( żt ) := 1 2ā 2 N T * N j=1 N i=1 a 2 ij t t-Tij żi (τ )dτ 2 .
Hence, V1 is negative semidefinite and it may be established, using standard arguments based on Barbalȃt's lemma, that Y ( żt ) and v → 0.

Next, for the angular-motion dynamics, (28b), we introduce the functional

V 2 := c 2 V 21 + e θ Le ω + c 1 0 -T * t t+h | ėθ (s)| 2 ds dh V 21 := e θ Le θ + |e ω | 2 + e α K -1 I e α (32)
where c 2 and c 1 are positive constants defined in [START_REF] Loría | A new persistency-of-excitation condition for UGAS of NLTV systems: Application to stabilization of nonholonomic systems[END_REF]. V 2 is positive definite, radially unbounded and decrescent; it satisfies

c 2 2 V 21 ≤ V 2 ≤ 2c 2 V 21 + 2c 1 T * t t-T * | ėθ (s)| 2 ds
so V 2 qualifies as a Lyapunov-Krasovskii functional candidate. Then, we use the fact that K ω > I N (see Theorem 1), and ( 15) to find, after some long, but straightforward computations, that

V2 = - 1 2 c 2 e T α K -1 I K α e α + e T ω K ω e ω - 1 2 e T θ L 2 e θ . ( 33
)
Finally, we introduce the following Lyapunov-Krasovskii functional,

V(t, X t , X r , żt , ėθt ) = W (t, X t , V 1 , żt ) + ρ 1 (V 1 )V 2 (X r , ėθt ) + ρ 2 (V 1 )V 1 .
The functional W is given by the following expression, [START_REF] Maghenem | Consensus of multi-agent systems with nonholonomic restrictions via Lyapunov's direct method[END_REF]. The functions ρ 1 , ρ 2 , α, and γ, as well are polynomials of V 1 with positive coefficients. These and the constant ε > 0 are chosen so that V be positive definite and radially unbounded, and its derivative be negative definite. This is possible provided that k ωm ≥ c 1 /c 2 , as required in Theorem 1, and (15) hold -see [START_REF] Maghenem | Stability and stabilization of networked varying systems[END_REF] for detailed lengthy computations. As a matter of fact, these can be chosen so that

W = γ(V 1 )V 1 + V 1 κ(s, e) t+T t Q(s)ds κ(s, e) +α(V 1 )e v -εV 1 e Q(t)s + εb q λ N (L)V 2 1 + λ N (L) + |K p | α(V 1 )V 1 -cf.
V ≤ - µ 4T V 3 - ρ 1 (V ) 8 c 2 e ω K dθ e ω + e θ K pθ e θ - 1 4 γ(V )v K dt K -1 pt v - 1 8 α(V )e K pt e, (34)
so the statement follows.

Lemma 2 Under the conditions of Theorem 1 the set (29) for the closed-loop system (28) is uniformly globally asymptotically stable.

Sketch of Proof. The proof relies on a standard outputinjection argument, akin to [START_REF] Panteley | Relaxed persistency of excitation for uniform asymptotic stability[END_REF]Proposition ] and the fact that B(t) is uniformly bounded and square integrable. Firstly, we observe that the solutions X t (t) of (28a) are uniformly globally bounded; this follows by integrating on both sides of (31). In addition, the nominal system Ẋr = F r (X r , e θt ) in (28b) is input-to-state stable and the "input" Ψ is bounded, so the solutions of (28b), X r (t), are also uniformly bounded. Secondly, some computations establish that there exist a constant c, independent of the initial times, such that

|B(t)| 2 ≤ c k∈Nj a 2 kj t-T * t-T kj -T * żj (δ) 2 ,
so, in view of (31), it follows that B(t) is square integrable. The result follows from this, uniform global boundedness of the solutions, and uniform global asymptotic stability of D for the system with B ≡ 0 (see Lemma 1).

IV. A NUMERICAL EXAMPLE

In this section we illustrate our main result via a numerical example consisting in performing full-consensus-based formation control of four identical unicycle-type robots. They are required to meet at an unknown rendez-vous point, forming a rhomboid. To that end we set the offsets (δ xi , δ yi ), for each i ≤ 4, to (0, 2), (-2, 0), (0, -2), and (2, 0). The initial positions, in the format (x i (0), y i (0)), are set to (5, -4), (7, 3), (-2, -5), and (-4, 5) while all the robots initially point East, i.e., θ i (0) = 0 for all i ≤ 4.

The control gains are set to K p = 60, K d = 40, K α = K α = 40, K I = 8, and K w = 40. The persistently exciting function, ṗ(t), is defined a periodic function of five frequencies, ṗ(t) = π[2 sin(0.5t) + sin(2t) + 3 sin(3t) + sin(t) -sin(5t)].

For simplicity, the time-varying delay is taken equal for all the interconnections. It is generated by a normal Gaussian distribution with a mean of 0.3 and a variance of 0.0003, which generates a function bounded in norm by T * = 0.37. The robots are interconnected in a closed undirected cycle as illustrated in Fig. 1.

Fig. 2 illustrates the paths described by the robots on the plane. The initial positions are marked by '*' and the final orientations are represented by arrows. It can be observed that the robots reach the desired formation with an a priori unknown center at z c ≈ (1.27, -1.61) and a common orientation angle of -177.66 deg.

V. CONCLUSIONS

We presented a decentralized controller for the the fullconsensus problem of multiple second-order nonholonomic vehicles with time-varying delayed communications. That is, our controller guarantees that all vehicles' orientations and Cartesian positions converge to a common equilibrium point. For the Cartesian positions an offset for each vehicle is specified hence, the vehicles may acquire any formation. Only few other results exist in the literature guaranteeing full consensus and with time-varying delays, but none in which delays may be non-differentiable, let alone for second-order systems.

The controller relies on persistency of excitation to overcome the difficulties imposed by the nonholonomy. Although such controllers guarantee strong properties such as uniform convergence of the error trajectories, they are difficult to tune and may entail oscillatory motions. Moreover, our controller relies on state feedback. Research on how to improve performance as well as on output-feedback control designs is to be carried out.

  the vectors A( żt ) := col[a i ( żt )] and B := col[á i ], the error vectors with delays e d :=col[e di ] and s d :=col[s di ], and e θ :=col[e θi ].

Fig. 1 .

 1 Fig. 1. Interconnection Graph and Laplacian matrix

Fig. 2 .

 2 Fig. 2. Trajectories and formation of the network of mobile robots.

This article has been supported by: the Government of Russian Federation (grant 08-08), by the French National Research Agency via the project HANDY (grant number ANR-18-CE40-0010), by ECOS-Nord contract no. M14M02, by the Mexican CONACyT Basic Scientific Research grant CB-282807, and by CEFIPRA under the grant number 6001-A.