Some trends and tools for the study of time-delay systems


Moad Kissai
email: moad.kissai@ensta-paristech.fr

Bruno Monsuez
email: bruno.monsuez@ensta-paristech.fr

Adriana Tapus
email: adriana.tapus@ensta-paristech.fr

Current and Future Architectures for Integrated Vehicle Dynamics Control

Keywords: Control Architectures, Over-Actuated Systems, Coordination, Control Allocation, Vehicle Dynamics

Today's vehicles use coordination strategies downstream their subsystems to avoid conflicts. This is possible as far as a very limited number of integrated subsystems is concerned. This paper discusses the potential of this approach and proposes an eventual substitute. A brief review is given to speculate about the architectures' compatibility with respect to control problems. It appears that autonomous vehicles require additional subsystems for safety. Therefore, an upstream approach is no longer appropriate to handle vehicle's subsystems interactions.

I. INTRODUCTION

In the past years, a large interest has been given to autonomous vehicles. Automation promises safer and smarter vehicles. In this context, several researches has been carried out in robotic vision, decision algorithms, big data management, and others. As soon as the vehicle is "aware" of its environment and a trajectory is planned, the chassis system should be actuated in a way to ensure reference 1 tracking with the desired dynamics. To do so, the vehicle has more than one subsystem per control axe [START_REF] Soltani | Low Cost Integration of Electric Power-Assisted Steering (EPAS) with Enhanced Stability Program (ESP)[END_REF]. This redundancy aims to ensure safety and improve other performances as handling and manoeuvrability. Therefore, vehicle's subsystems interact and conflicts could occur. To avoid this, automotive engineers use rule-based strategies to coordinate the different subsystems [START_REF] Selby | Intelligent Vehicle Motion Control[END_REF]. This requires an expert knowledge on vehicle dynamics to predict the influence of each actuators over an another, and then on the overall vehicle. For example, the work in [START_REF] Velardocchia | Control systems integration for enhanced vehicle dynamics[END_REF] used Active Differential (AD), Electronic Stability Control (ESC), and Torque Vectoring (TV) to improve the vehicle lateral performances. A simple method based on prioritizing one system over another has been used. A more complex method based on Artificial Neural Network had been adopted in [START_REF] Nwagboso | Development of neuralnetwork control of steer-by-wire system for intelligent vehicles[END_REF]. This method consists of simple averaging or via a nonlinear interpolation function weights. These functions could be chosen to ensure smooth transitions between coordination modes. A larger review could be found in [START_REF] Kissai | Review of Integrated Vehicle Dynamics Control Architectures[END_REF].

However, as broad as an expert knowledge could be, it cannot foresee all the possible situations. Thus, optimization is not explicitly formalized. We cannot prove that the downstream coordination is the best solution. Consequently, a single 1 Here, it is the trajectory planned.

objective is aimed most of the time [START_REF] Ivanov | Systematization of Integrated Motion Control of Ground Vehicles[END_REF], while secondary objectives could be achieved as far as the vehicle is overactuated. Control objectives should then be formalized to enable optimization. This can only be ensured if we act differently on the subsystems references. The coordination should then be upstream the subsystems. Because of the overactuation, the system of equations to be solved has more unknowns than equations. Moreover, each actuator has its own limits. Command vectors are then constrained. This is called the control allocation problem [START_REF] Bodson | Evaluation of optimization methods for control allocation[END_REF]. This problematic has already been encountered in flight-control systems. While ganging has been used in many of these systems, control allocation methods have become necessary regarding advanced aircraft with more numerous actuators [START_REF] Bodson | Evaluation of optimization methods for control allocation[END_REF].

In the automotive sector, optimization methods are considered too complex and too time-consuming to be implemented in a vehicle [START_REF] Ivanov | Systematization of Integrated Motion Control of Ground Vehicles[END_REF]. However, dramatic increases in computing speed and algorithms efficiency have been elaborated. In this paper, we discuss the capacity of the upstream approach to handle more complex coordination difficulties with respect to the downstream approach. Architecture models based on the upstream approach offer the possibility to incorporate optimization-based control allocation algorithms. Better solutions are provided, secondary objectives are ensured, and performance is then enhanced. Moreover, fault-tolerance is naturally managed and reconfiguration methods are more efficient [START_REF] Tor | Control allocationA survey[END_REF]. These characteristics constitute a major aspect for autonomous vehicles safety.

We start by presenting the downstream approach in Section II. The upstream approach is then presented in Section III. Our work in progress is described in Section IV. Finally, conclusions are drawn and future works are outlined (see Section V).

II. DOWNSTREAM COORDINATION ARCHITECTURE

Here, additional layer is added to manage the subsystems influence on the overall system. Coordination is ensured downstream the subsystems. In [START_REF] Selby | Intelligent Vehicle Motion Control[END_REF], this is referred to a "bottom-up approach". In fact, this is about taking two or more wellunderstood controllers and apply them to the vehicle. As a multi-variable system is concerned, each controller could deteriorate the performance of the other. Automotive engineers use then their expertise to foresee these interactions. This enables the development of rule-based strategies to mitigate the subsystems conflicts. Fig. 1 illustrates this architecture in a more general scenario, when a future vehicle could integrate a 4-Wheel Drive (4WD), Torque Vectoring (TV), Electric Power-Assisted Steering (EPAS), 4-Wheel Steering (4WS), Electronic Stability Program (ESP), and Adaptive Suspension (AS) systems. Four coordinator types have been distinguished in [START_REF] Gordon | Integrated control methodologies for road vehicles[END_REF]: Pure Subsumption (Fig. 2 -(A)), Largest Modulus Activation (Fig. 2 -(B)), Artificial Neural Network (Fig. 2 -(C)), and Fuzzy Logic Control (Fig. 2 -(D)).

Regarding the pure subsumption approach, the highest level non-zero command takes precedence over of the other subcommands [START_REF] Velardocchia | Control systems integration for enhanced vehicle dynamics[END_REF]. In the largest modulus activation, several high level commands are considered, and the one with the highest modulus takes precedence over the rest [START_REF] Gordon | Integrated control methodologies for road vehicles[END_REF]. The Artificial Neural Network consists of simple averaging or via a nonlinear interpolation function weights [START_REF] Nwagboso | Development of neuralnetwork control of steer-by-wire system for intelligent vehicles[END_REF]. The Fuzzy Logic uses "easily understood" rule-based coordination functions, where the highest level predominates and smooth transitions are ensured [START_REF] Wang | Agent-based coordination framework for integrated vehicle chassis control[END_REF].

Each of these coordinators requires several studies in advance to predict the eventual conflicts. Different scenarios should be imagined to find the best solution. Rule-based strategies are then developed to mitigate the conflicts issued from the scenarios imagined. For single-objective control problems, one could develop a global overview of the possible problems that could occur. Regarding multiple-objective control problems, rules are less obvious to develop. An optimization problem with multiple cost functions and constraints should be formalized. Interactions at the vehicle level should be taken account of. A global multi-variable controller is then developed. Command requests are then distributed in an optimum manner to the different standalone controllers. In other words, the coordination should be made upstream the subsystems.

III. UPSTREAM COORDINATION ARCHITECTURE

Here, the coordination layer is located upstream the standalone subsystems. The command requests are distributed in a way to avoid conflicts before their occurrence. Relations between control commands and performance outputs are mathematically described to find an optimal solution or a sub-optimal one. In [START_REF] Selby | Intelligent Vehicle Motion Control[END_REF], this is referred to as a "top-down" approach, as a general supervisor is synthesized to distribute the commands to the different subsystems. Fig. 3 illustrates this concept.

Three structures have been distinguished in [START_REF] Kissai | Review of Integrated Vehicle Dynamics Control Architectures[END_REF]: centralized control, supervisory control, and multi-layer architecture. Next, each structure is described.

A. Centralized Control

A central global controller is synthesized to control the overall vehicle. As multiple objective are concerned, the controller must be multi-variable. Its outputs are then transmitted to the different subsystems as illustrated in Fig. 4.

However, if one want to add an additional subsystem, the whole control synthesis should be done once again. This architecture is not flexible. As long as autonomous vehicle control architecture is not frozen, flexibility is necessary. A distributed method is then more attractive.

B. Supervisory Control

In order to ensure flexibility, controllers are separated and a supervisory layer is added (see Fig. 5).

This has additional advantages:

• Fault-tolerance: it ensures a minimum of operations safety even if the high-level controller fails, • Extensibility: it can be evolved to a multi-layer hierarchical structure to add more functionalities, • Modularity: it allows manufacturers and suppliers develop independently complementary control algorithms. As an example, in [START_REF] Vivas-Lopez | Global Chassis Control System Using Suspension, Steering, and Braking Subsystems[END_REF], the authors used three main layers with two levels of abstraction: Decision Layer, Control Layer, and Physical Layer.

C. The Multi-layer architecture

As a result of extensibility, the supervisory control can be generalized to a multi-layer architecture. Each function of the control process is separated. As multi-variable systems are concerned, functions with the same nature are grouped into different layers:

• Layer 1 : Generation of vehicle motion reference,

• Layer 2 : Decision on the control mode,

• Layer 3 : Calculation of the generalized forces and moments through the high-level controllers, • Layer 4 : Distribution of the commands to the available actuators through control allocation logic, • Layer 5 : Control of stand-alone subsystems, • Layer 6 : Execution of the various operations through smart actuators.

D. Discussion

Upstream architecture could be used for single-objective control problems [START_REF] Falcone | Integrated braking and steering model predictive control approach in autonomous vehicles[END_REF]. But as long as optimization methods are considered too complex, their implementation should be justified, which is not the case for single-objective control problems.

In the past years, control allocation techniques became more preponderant to face over-actuation problems [START_REF] Soltani | Low Cost Integration of Electric Power-Assisted Steering (EPAS) with Enhanced Stability Program (ESP)[END_REF], [START_REF] Heo | Integrated chassis control for enhancement of high speed cornering performance[END_REF]. For example the authors of [START_REF] Heo | Integrated chassis control for enhancement of high speed cornering performance[END_REF] used an Integrated Chassis Control (ICC) strategy to improve cornering performance in high speed by combining the ESC, the 4-Wheel Drive (4WD) and the Active Roll Control (ARC) systems. The control architecture is composed of thee parts: a supervisory controller that determines the target vehicle motions, the upper-level controller that calculates the target forces and moment and the lower-level controller that optimally distributes the actuator inputs. However, as pointed out by [START_REF] Ivanov | Systematization of Integrated Motion Control of Ground Vehicles[END_REF], all the results are made by simulation only. There is a clear lack of experimental results and benchmark requirements that allow comparison between the different methods.

IV. CURRENT RESEARCHES

Our purpose is to prove that the upstream coordination architecture will become necessary to handle interactions of the vehicle's subsystems. This could motivate the different stakeholders to start thinking on the standardization of an integrated vehicle dynamics control architecture that suits best autonomous vehicles. Therefore, both architectures should be developed to enable their comparison. This requires different modelling approaches and different control synthesis techniques. Nevertheless, we can allow ourselves to provide a qualitative comparison to explain our motivations.

A. Qualitative Comparison

Three characteristics are chosen that interest most a car manufacturer: complexity, cost, and potential.

1) Complexity: This is the main drawback of the upstream architecture. While the downstream approach uses understood rule-based strategies to manage subsystems interactions, the upstream approach rely on first a MIMO 2 controlled based on a non-linear vehicle model, and then on optimization techniques that should find a solution in real-time operations. But again, the more numerous and complex interactions become, the more the upstream approach is pertinent.

2) Cost: This also does not favour the upstream approach. This latter requires additional high-level controller(s) and may even need additional sensors or estimators [START_REF] Heidrich | Hardware-in-the-loop test rig for integrated vehicle control systems[END_REF]. In contrast, the downstream approach requires only a coordination strategy between the standalone subsystems and the vehicle. This is why car manufacturers use this approach until these days.

3) Potential: In the downstream approach, automotive engineers use their expertise to design real-life scenarios. These scenarios help to study the eventual interactions that could occur between the subsystems. Rule-based strategies are then developed to manage these interactions. As engineers cannot foresee all the possible situations, errors could be generated in some cases. The more interactions become numerous and complex, the more errors are probable to occur. In this way, the upstream approach would become necessary. This approach depicts mathematically the different interactions and quantifies the dynamic couplings. Only this way one can hope to find an optimal solution, or at least a sub-optimal one.

B. Motivation

As we have mentioned, the main drawbacks of the upstream approach that have hindered its development are complexity and cost. However, today's Electronic Control Units (ECUs) have became faster, less cumbersome, and cheaper. Moreover, control allocation methods have been reviewed in [START_REF] Bodson | Evaluation of optimization methods for control allocation[END_REF], and tested in a Hardware-In-the-Loop (HIL) procedure in [START_REF] Heidrich | Hardware-in-the-loop test rig for integrated vehicle control systems[END_REF]. These methods appear to be more suitable for over-actuated systems. A simple Linear Programming (LP) or Quadratic 2 Multiple Inputs Multiple Outputs Programming (QP) could be executed in a few milliseconds with a limited number of iterations.

These new opportunities could finally get rid of the main drawbacks of the upstream approach. More objectives could be handled and more performances could be achieved. Autonomous vehicles could then be controlled in a larger dynamic range. In this context, we aim to develop both architectures to enable their effective comparison and hope to convince car manufacturers to switch to the upstream approach.

Next, a starting working example is described. This example represents a starting point of our plan.

C. A Working Example

We choose to start with only two subsystems. These subsystems should respect the following criteria:

• They should interact often to evaluate the potential of coordination approaches, • They should be able to influence more than one control axe to be able to study a multi-objective problem, • They should be mature enough to be able to test them in a real vehicle. For these reasons, we choose the 4-Wheel Steering (4WS) and the brake-based Vehicle Dynamics Control (VDC)3 systems. Several vehicles are equipped by these subsystems in the same time 4 . While the 4WS system uses the rear steering to generate a steering angle, the VDC uses the differential braking 5 to generate a yaw moment. Both systems influence the vehicle's yaw rate. Only VDC influence the longitudinal acceleration. As both systems are interacting, we can expect from the 4WS to influence indirectly the longitudinal acceleration.

As each architecture uses different control methods, the problem remaining is which control method to choose in each architecture to enable reliable comparison of both approaches. For simplicity, we can start with basic methods of each structure. Pure subsumption in a downstream approach can be compared to a centralized control structure in an upstream approach.

In the downstream approach, a reduced vehicle model could be considered to synthesize standalone controllers. A pure subsumption strategy is then added to handle interactions. In contrast, as long as a centralized is concerned, a more complete vehicle model should be considered. Therefore, a MIMO controller have to be synthesized taking account of the eventual interactions. Fig. 6 summarizes our ongoing work.

D. Challenges 1) Reliable Comparison:

As we have mentioned, it is hard to be fair to each of the architectures when choosing a control technique. Several techniques exist in both approaches. While some techniques could be used in both approaches, others can only be applied in a specific architecture. We are currently testing the techniques described in Fig. 6. One can also test Fuzzy logic in both approaches as this technique has been applied with the different architectures ( [START_REF] Wang | Agent-based coordination framework for integrated vehicle chassis control[END_REF] for the downstream architecture and [START_REF] Boada | Integrated control of front-wheel steering and front braking forces on the basis of fuzzy logic[END_REF] for the upstream one). And finally, we can also choose some specific and strong techniques that could only be used in one architecture or another to justify the architecture, e.g. Control Allocation for the upstream architecture.

2) Real-Time Implementation: Several architectures were only validated through simulations [START_REF] Ivanov | Systematization of Integrated Motion Control of Ground Vehicles[END_REF]. As coordination strategies are located in an inner loop, computations must be fast (around 100 Hz according to [START_REF] Bodson | Evaluation of optimization methods for control allocation[END_REF]). This puts a real challenge to automotive engineers and limits their choices. In this context, Linear Programming could be the key. In [START_REF] Bodson | Evaluation of optimization methods for control allocation[END_REF] for example, several control allocation techniques were transformed from a quadratic formulation to a linear one. Optimization is still ensured through few hypothesis, and computation is mush more faster.

3) Multi-Sense: As different subsystems with different dynamics are integrated, different motion feelings could be generated. Command requests should be distributed in a way to favour one behaviour over another. Driving modes can be taken account of in the mean of weighted functions. However, drivers do not have the same definition of comfort or excitement. Weighted functions have to be adaptable. One could think about evolutionary algorithms to enable a vehicle to adapt to its driver.

V. CONCLUSION AND FUTURE WORKS

In this paper, two major architectures for integrated vehicle dynamics control have been distinguished. Control techniques of each of these architectures have been differentiated. To enable their reliable comparison, a working example has been proposed and its related challenges exposed. This is an ongoing work that we hope will help us to get more evidence to convince the different stakeholders to favour one architecture over another.

To do this, we will start by proposing different vehicle models. One to interpret driver commands (reference vehicle model), one for downstream coordination synthesis, and another one for a centralized MIMO controller synthesis. A new tire model will also be proposed. This model should take account of the dynamic couplings and friction limits to develop realistic coordination strategies. Then using the different techniques presented in this paper, we expect to offer a consistent architecture model for autonomous vehicles.

Fig. 1 .

 1 Fig. 1. Structure of the downstream coordination approach [5].

Fig. 2 .Fig. 3 .

 23 Fig. 2. Coordinator types of the downstream coordination approach [5].

Fig. 4 .

 4 Fig. 4. Centralized control structure [5].

Fig. 5 .

 5 Fig. 5. Supervisory control structure [5].

Fig. 6 .

 6 Fig. 6. Ongoing work: comparison of downstream and upstream architectures.

It is just another appellation of the ESP used by Renault.

Renault Talisman for example.

Brake at each wheel independently.


Moad Kissai
email: moad.kissai@ensta-paristech.fr

Bruno Monsuez
email: bruno.monsuez@ensta-paristech.fr

Adriana Tapus
email: adriana.tapus@ensta-paristech.fr

Review of Integrated Vehicle Dynamics Control Architectures

Keywords: Control Architectures, Chassis Systems, Systems Coordination, Vehicle Dynamics Control, Control Allocation

Most of the chassis systems are developed by automotive suppliers to improve a specific vehicle performance. Drivers, and consequently vehicle manufacturers, are more concerned by the overall behaviour of the Vehicle. Many coordination architectures have been proposed in the literature in order to integrate different chassis systems in a single vehicle. In this paper, these architectures are compared and discussed. Two major classes are proposed: Downstream and Upstream Coordination. The purpose of this classification is to help car manufacturers and suppliers standardize an Integrated Vehicle Dynamics Control architecture for faster and more flexible designs.

I. INTRODUCTION

In today's automotive sector, driving experience, vehicle safety, and environmental protection are key competition criteria between automakers. In this context, Original Equipment Manufacturers (OEMs) offer constantly new attractive subsystems. Advances in automotive Electronic Control Units (ECUs), sensors, and actuators make the cost of vehicle embedded systems constantly decreasing. The number of chassis control systems grows rapidly as well as the number of competing systems in a single vehicle [START_REF]Chassis Handbook -Fundamentals, Driving Dynamics, Components, Mechatronics, Perspectives[END_REF].

However, each vehicle subsystem has an independent control logic to accomplish a specific objective. These objectives compete and may implement contradictory logic: if a brakebased yaw control is used to reduce oversteer or understeer, the longitudinal acceleration demand in a cornering operation will deteriorate. Since new hardware is always expensive, improvements must be provided by the synergies that already exist between subsystems [START_REF] Coelingh | Open-interface definitions for automotive systems -Application to a Brake by Wire System[END_REF]. If instead of implementing competing systems, subsystems get coordinated, over-actuation would offer new opportunities to improve the overall system's safety and performance [START_REF] Zhang | Bibliographical review on reconfigurable faulttolerant control systems[END_REF]. This could be done by developing a supervised overall vehicle control system, where information is shared by many subsystems, less resources are required, and computational costs are decreased therefore avoiding unnecessary duplication [START_REF] Coelingh | Open-interface definitions for automotive systems -Application to a Brake by Wire System[END_REF]. This has been given the name of interoperability in [START_REF] Gordon | Integrated control methodologies for road vehicles[END_REF].

This will influence the vehicle's overall behavior. Thus, the high-level control architecture should be designed by car manufacturers. As different suppliers are concerned, a standardization of actuators, sensors, and software components interfaces is crucial. The development of this architecture requires a close collaboration between vehicle manufacturers and their suppliers. Therefore, any proposed solution should respect the intellectual property rights (IPR) of both sides [START_REF] Navet | Automotive embedded systems handbook[END_REF]. In this context, three essential capabilities have been cited in [START_REF] Coelingh | Open-interface definitions for automotive systems -Application to a Brake by Wire System[END_REF]:

• Adaptability and dynamic reconfiguration to face environment changes, drivers behaviors, and failures [START_REF] Wills | An open platform for reconfigurable control[END_REF],

• Plug-and-play extensibility to rapidly insert additional technologies without redesigning the whole architecture, • Openness to support various systems from different sources.

The main purpose of this work is to propose a new classification of integrated control architectures to help designers choose a combination approach for a specific need or for the overall system. This would not be the first try, as in [START_REF] Gordon | Integrated control methodologies for road vehicles[END_REF] a classification of integrated control methodologies for road vehicles has been made. The architectures have been differentiated as centralized, supervisory, hierarchical, and coordinated control. However, no discussion is provided so as to assign an architecture to a specific goal. Another recent classification was proposed in [START_REF] Ivanov | Systematization of Integrated Motion Control of Ground Vehicles[END_REF], where the classification has been simplified to affect the architectures to the desired goals. Two classes have been proposed: single-criterion and multicriterion integration motion control. In the single-criterion integration motion control approach, the integrated systems are combined to improve one single aspect of the vehicle. In the multi-criterion integration motion control approach, the combined systems aim to manage different concurrent vehicle dynamics. However, for each class, we could find different combination methodologies depending on the problem's complexity.

A main difference if not the main difference that we distinguished from the different architectures reviewed, is the position of the coordination layer in the control flow [START_REF] Selby | Intelligent Vehicle Motion Control[END_REF]. Two main approaches could be adopted to deal with the subsystems combination problem: either we treat the interactions and the eventual conflicts after the subsystems' operations or we control the commands distribution to the different subsystems in order to generate the desired synergies and avoid conflicts. We call the first approach systems downstream coordination, and the second, systems upstream coordination.

We start with a presentation of the downstream coordination approach in Section II and we explain why it was preferred by auto-makers until these days. We then give a presentation of the upstream approach and justify why it should be investigated and replace the downstream coordination approach (see Section III). In Section IV, we propose a general architecture where the different researches seem to converge to and we discuss the open challenges arising from its use. Finally, conclusion is be drawn and some future works are outlined (see Section V).

II. SYSTEMS DOWNSTREAM COORDINATION

We choose the appellation "systems downstream coordination" because this approach consists of studying the standalone chassis systems interactions after the commands generation by these subsystems. The coordination is made downstream. To schematize this approach, in Fig. 1 we reproduce the simple structure given by [START_REF] Selby | Intelligent Vehicle Motion Control[END_REF], where this approach is called the "bottom-up approach". Fig. 1. Structure of the downstream coordination approach (adapted from [START_REF] Selby | Intelligent Vehicle Motion Control[END_REF]).

In [START_REF] Gordon | Integrated control methodologies for road vehicles[END_REF] and [START_REF] Duffie | Fault-tolerant heterarchical control of heterogeneous manufacturing system entities[END_REF], this is rather called a decentralized control. It follows a parallel structure where each subsystem works separately. Controllers have to work simultaneously. They have their own information system and Electronic Control Unit (ECU), which require more costs and space [START_REF] Vivas-Lopez | Global Chassis Control System Using Suspension, Steering, and Braking Subsystems[END_REF]. The only way to add "integration" in the process is by allowing additional communications between the controllers as the Fig. 2 shows. Fig. 2. Decentralized control structure (adapted from [START_REF] Gordon | Integrated control methodologies for road vehicles[END_REF]).

This architecture is more suited for some manufacturing systems [START_REF] Duffie | Fault-tolerant heterarchical control of heterogeneous manufacturing system entities[END_REF]. where it is called heterarchical. The fact that there is no global supervision makes this architecture particularly less suitable for vehicle safety control.

In order to develop a coordination strategy, interactions between the subsystems are studied. Automakers engineers use their "expert knowledge" to develop handling controllers. This is achieved through a rule-based controllers arbitration deduced from the subsystems interactions studies [START_REF] Selby | Intelligent Vehicle Motion Control[END_REF]. In this context, four coordinator types have been distinguished in [START_REF] Gordon | Integrated control methodologies for road vehicles[END_REF]: Pure Subsumption, Largest Modulus Activation, Artificial Neural Network, and Fuzzy Logic Control.

A. Pure Subsumption

Regarding the pure subsumption approach, the highest level non-zero command takes precedence over of the other subcommands (see Fig. 3). In this context, the work in [START_REF] Velardocchia | Control systems integration for enhanced vehicle dynamics[END_REF] used Active Differential (AD), Electronic Stability Control (ESC), and Torque Vectoring (TV) to improve the vehicle lateral performances. A simple method based on prioritizing one system over another has been used. If the yaw torque demand can be satisfied by the AD, then the ESC and TV will not be activated. Otherwise, the rest of the yaw torque demand will be equally shared between the ESC and the TV systems. Both vehicle performance and safety have been improved within this method.

B. Largest Modulus Activation

In the largest modulus activation, several high level commands are considered, and the one with the highest modulus takes precedence over the rest (see Fig. 4). Both pure subsumption and largest modulus activation methods are characterized by modes switching. These switches generate undesirable transients that could destabilize the overall system. In this context, the Artificial Neural Network and the Fuzzy Logic Control were introduced.

C. Artificial Neural Network

The Artificial Neural Network consists of simple averaging or via a non-linear interpolation function weights (see Fig. 5). These functions could be chosen to ensure smooth transitions between coordination modes, actuator saturation avoidance, etc. [START_REF] Nwagboso | Development of neuralnetwork control of steer-by-wire system for intelligent vehicles[END_REF].

D. Fuzzy Logic

The Fuzzy Logic uses "easily understood" rule-based coordination functions (see Fig. 6). Fig. 6. Fuzzy Logic coordination (adapted from [START_REF] Gordon | Integrated control methodologies for road vehicles[END_REF]).

Here again, the highest level predominates but smooth transitions are ensured [START_REF] Gordon | Integrated control methodologies for road vehicles[END_REF]. In [START_REF] Wang | Agent-based coordination framework for integrated vehicle chassis control[END_REF] for example, a fuzzy logic scheme and weighting factors are used to coordinate the different systems. The controller agents computes combined control signals for the steering angle and the wheel torque depending on the targeted performances priority. The main advantage is the conflict mitigation, for example, when the braking controller has to track simultaneously the yaw rate reference and the longitudinal acceleration demand.

E. Architecture's Compatibility

From the examples in the previous subsections, it appears that the downstream coordination approach is more suitable for single objective coordination control. For multi-objective control, Other coordination algorithms could be cited as the Linear Quadratic Regulator (LQR), Lyapunov functions, sliding mode control, adaptive control, etc. These optimization methods have the major advantage of dynamic coordination, which ensure dynamic transitions. They offer possibilities for robust and fault-tolerant vehicle control systems without apriori knowledge of anticipated failure modes [START_REF] Gordon | Integrated control methodologies for road vehicles[END_REF]. Thus, the redundancy (that could be very expensive) only concerns the basic hardware needs. These methods are rather used in an upstream coordination approach.

III. SYSTEMS UPSTREAM COORDINATION

Here, the coordination is carried out upstream the subsystems. The commands are distributed in a way to avoid the conflicts downstream the subsystems. A multivariable controller is placed between the driver/pilot commands and the chassis systems. The controller design is based on a coupled nonlinear vehicle model that give insights about the possible conflicts before reaching them. In [START_REF] Selby | Intelligent Vehicle Motion Control[END_REF], this approach is called "top-down approach". To illustrate this approach, we reproduce the structure given in [START_REF] Selby | Intelligent Vehicle Motion Control[END_REF] (Fig. 7). Fig. 7. Structure of the downstream coordination approach (adapted from [START_REF] Selby | Intelligent Vehicle Motion Control[END_REF]).

In accordance to this approach, three architectures have been distinguished in [START_REF] Gordon | Integrated control methodologies for road vehicles[END_REF] and [START_REF] Duffie | Fault-tolerant heterarchical control of heterogeneous manufacturing system entities[END_REF]: centralized control, supervisory control, and decentralized control. As we have mentioned, the decentralized control rather corresponds to downstream coordination. We present only the centralized control (A) and the supervisory control (B).

A third class could be distinguished, which is more of an extension of the supervisory control, called the multi-layer architecture (C).

A. Centralized Control

A central global controller is responsible of taking all the control decisions. In general, this controller follows the global multivariable control formalism as it has been realized in [START_REF] Brennan | Integrated vehicle control via coordinated steering and wheel torque inputs[END_REF]- [START_REF] Nagai | Integrated robust control of active rear wheel steering and direct yaw moment control[END_REF]. Fig. 8 illustrates this concept.

However, it has been pointed out in [START_REF] Kelling | The BRAKE project centralized versus distributed redundancy for brake-by-wire systems[END_REF] that any desired fail-safe redundancy of micro-controllers or power converters increases rapidly the cost of the control components. For that matter, a distributed control method has been preferred, which can be assimilated to a supervisory control [START_REF] Gordon | Integrated control methodologies for road vehicles[END_REF].

B. Supervisory Control

Supervisory control represents an intermediate between the centralized and decentralized control. Indeed, a supervisory layer is added to a decentralized structure to add more information in the process. Fig. 9 illustrates this approach. Fig. 9. Supervisory control structure (adapted from [START_REF] Gordon | Integrated control methodologies for road vehicles[END_REF]).

Three advantages could be deduced from this architecture:

• Fault-tolerance: it ensures a minimum of operations safety even if the high-level controller fails, • Extensibility: it can be evolved to a multi-layer hierarchical structure to add more functionalities, • Modularity: it allows manufacturers and suppliers develop independently complementary control algorithms. Using this structure, [START_REF] Vivas-Lopez | Global Chassis Control System Using Suspension, Steering, and Braking Subsystems[END_REF] used three main layers with two levels of abstraction:

1) Decision Layer: Identifies the current driving situation first, then decide how to coordinate the subsystems actions, 2) Control Layer: Transforms the control objectives generated by the Decision Layer to references for each of the local controllers, 3) Physical Layer: Contains simply the different actuators and sensors. It should be noted that the decision layer plays a major role to ensure the overall system safety. It is responsible of two main tasks: classifying the current driving situation and deciding how coordination should be made. For example, in [START_REF] Vivas-Lopez | Global Chassis Control System Using Suspension, Steering, and Braking Subsystems[END_REF], a k-means data-based algorithm and a decision logic module based on a set of heuristic rules have been used 1 .

C. The Multi-layer architecture

For more flexibility and more comprehensive control, the functional requirements should rather be separated while ensuring a supervised control. Fig. 10 illustrates this method. 1 Comparison of decision techniques is beyond the scope of this paper.

Each layer has a specific function [START_REF] Soltani | Low Cost Integration of Electric Power-Assisted Steering (EPAS) with Enhanced Stability Program (ESP)[END_REF]: 1) Layer 1: Generation of vehicle motion reference, 2) Layer 2: Decision on the control mode based on the vehicle state recognition, 3) Layer 3: Calculation of the generalized forces and moments at the vehicle's center of gravity through the high-level controllers, 4) Layer 4: Distribution of the commands to the available actuators in an optimal or sub-optimal way through control allocation logic, 5) Layer 5: Control of stand-alone subsystems to follow the commands that comes from the layer 4, 6) Layer 6: Execution of the various operations through smart actuators composed of low-level effectors (e.g. electric motor, hydraulic valve ...etc.) and their own controllers.

D. Architecture's Compatibility

This approach can be used for single objective coordination control. For example, lateral performances have been improved by integrating Active Front Steering (AFS) and the brakebased Dynamic Yaw Control (DYC). In this context, model predictive control [START_REF] Falcone | Integrated braking and steering model predictive control approach in autonomous vehicles[END_REF], model-following control [START_REF] Abe | An integration of vehicle motion controls for full drive-by-wire vehicle[END_REF], and even fuzzy logic control in an upstream coordination structure [START_REF] Boada | Integrated control of front-wheel steering and front braking forces on the basis of fuzzy logic[END_REF], [START_REF] Hou | Integrated chassis control using ANFIS[END_REF] were used. These studies showed not only improvement of lateral stability, but also manoeuvrability and agility were enhanced. However, the current cost of this architecture do not justify the achieved gains.

Most of interesting studies carried out using this approach concern the multiple objective coordination control. In this context, various integration methods have been adopted. For example, to deal with lateral and vertical integration, active suspension has been combined with brake-based control in [START_REF] Gspr | Active suspension in integrated vehicle control[END_REF]. A Linear Parameter Varying (LPV) design with faulttolerant control has been used. The commands distribution is made by using three weighting functions for lateral acceleration, heave acceleration, and suspension deflection. The results showed attractive improvements in rough surface conditions.

In the past years, control allocation techniques became more preponderant, e.g. [START_REF] Soltani | Low Cost Integration of Electric Power-Assisted Steering (EPAS) with Enhanced Stability Program (ESP)[END_REF], [START_REF] Knobel | Optimized force allocation -A general approach to control and to investigate the motion of over-actuated vehicles[END_REF]- [START_REF] Shyrokau | Integrated vehicle dynamics control based on control allocation with subsystem coordination[END_REF]. For example the authors of [START_REF] Heo | Integrated chassis control for enhancement of high speed cornering performance[END_REF] used an Integrated Chassis Control (ICC) strategy to improve cornering performance in high speed by combining the ESC, the 4-Wheel Drive (4WD) and the Active Roll Control (ARC) systems. The control architecture is composed of thee parts: a supervisory controller that determines the target vehicle motions, the upper-level controller that calculates the target forces and moment and the lower-level controller that optimally distributes the actuator inputs. The same architecture has been adopted in [START_REF] Shyrokau | Integrated vehicle dynamics control based on control allocation with subsystem coordination[END_REF]. The subsystems coordination algorithm uses restriction weights that changes according to the performances targeted. The investigations were carried out by a hardware-in-the-loop test rig, which demonstrated potential enhancement for different performances. Regarding the optimization methods, we can cite 3 approaches [START_REF] Van Den | Global chassis control and braking control using tyre forces measurement[END_REF]: Update laws [START_REF] Gerard | Optimal Control Allocation on Over-Actuated Vehicles -Maximizing vehicle performance and safety through advanced vehicle control strategies[END_REF], Repeated optimization [START_REF] Bodson | Evaluation of optimization methods for control allocation[END_REF], [START_REF] Hrkegrd | Dynamic control allocation using constrained quadratic programming[END_REF] and Precomputed laws [START_REF] Borrelli | An mpc/hybrid system approach to traction control[END_REF]. However, as pointed out by [START_REF] Ivanov | Systematization of Integrated Motion Control of Ground Vehicles[END_REF], all the results are made by simulation only. There is a clear lack of experimental results and benchmark requirements that allow comparison between the different methods.

IV. COMPARISON AND CHALLENGES A. Relevance of the new classification

We recall that two enriching reviews of integrated architectures have already been proposed in [START_REF] Gordon | Integrated control methodologies for road vehicles[END_REF] and [START_REF] Ivanov | Systematization of Integrated Motion Control of Ground Vehicles[END_REF]. In [START_REF] Gordon | Integrated control methodologies for road vehicles[END_REF], emphasis has been put on the architectures' topology. The authors pointed out two major extremes, the fully decentralized control and the fully centralized control, considering the supervisory control as an intermediate between them. This allowed the authors to highlight the benefits of a multi-layer architecture and to draw specific requirements related to the architecture topology: modularity, simplicity, fault-tolerance, and openness. However, no relations were proposed between the different structures and control objectives. As "simplicity" should be one of the architecture's requirements, a multi-layer architecture may seem exaggerated for single-objective control problems.

The authors of [START_REF] Ivanov | Systematization of Integrated Motion Control of Ground Vehicles[END_REF] focused rather on control objectives. Two classes have been proposed: single-criterion and multicriterion. They studied a large amount of examples for both classes, but conclusions were limited to the control methods and challenges. They particularly highlighted that most singlecriterion solutions are using rule-based methods (e.g. fuzzy logic control), while recent researches around multi-criterion control are using optimization methods, especially control allocation. Unlike [START_REF] Gordon | Integrated control methodologies for road vehicles[END_REF], no discussion about the architectures topology is provided. We recall that fuzzy logic could be used downstream the stand-alone subsystems for single-objective control [START_REF] Selby | Intelligent Vehicle Motion Control[END_REF], or upstream the stand-alone subsystems for multiobjective control [START_REF] Boada | Integrated control of front-wheel steering and front braking forces on the basis of fuzzy logic[END_REF].

In order to gather the different stakeholders to start thinking on the standardization of integrated vehicle dynamics control architectures, we have to make a bridge between the two types of classifications. For this reason, we believe that the architecture topology should be related to the control objective. As we are interested in subsystems coordination, the topologies classification should be linked to the coordination layer position with respect to stand-alone subsystems. In this context, two approaches have been distinguished in [START_REF] Selby | Intelligent Vehicle Motion Control[END_REF]: the "bottom-up" approach (which is called in this paper downstream coordination) approach, and the "top-down" approach (referred to the upstream coordination). However, in [START_REF] Selby | Intelligent Vehicle Motion Control[END_REF], only the centralized control structure was given to the top-down approach. The concluding remarks were that the top-down approach is unfavourable from an industrial perspective for its difficulty and cost2 . The bottom-up approach was then chosen and Fuzzy Logic with β-phase plane control was selected. The main differences that we provide with respect to [START_REF] Selby | Intelligent Vehicle Motion Control[END_REF] are a broader distinction of each approach and their suitability for control objectives. Therefore, the control designer would be able to automatically choose a standardized architecture depending on its objectives. The new classification proposed in this paper seems to be relevant within this framework. Upstream or downstream coordination not only designate where, but also how the coordination is managed. This has allowed us to link each approach to control objectives. When the coordination is made downstream, rule-based control design is used. This technique can handle well single-objective control problems. As more interactions are added, it is hard to foresee the different couplings induced, so it is more complicated to formalize additional rules to achieve safe coordination. When the coordination is made upstream, the control design is based on a coupled vehicle model. Different interactions could be predicted. With an adapted optimization technique, multi-objective control problems could be handled.

B. Comparison of the two approaches 1) Complexity:

The main advantage of the downstream approach is the low complexity as long as the number of interactions between competing systems are low. The design methodology consists on first studying the interactions between two or more systems and then establishing adequate rules to benefit from their potential synergies. This could be done for example by using a fuzzy logic approach or the βphase plane control [START_REF] Selby | Intelligent Vehicle Motion Control[END_REF]. On the other hand, complexity is the main drawback of the upstream approach. Firstly, the coordination lies on a MIMO (Multiple Inputs Multiple Outputs) controller based on a coupled non-linear vehicle model. Then, optimization techniques are used to solve the problem, in realtime. However, the more complex the interaction between the competing systems get nonetheless with regards to emerging behaviour prediction, the more this upstream approach is pertinent.

2) Cost: Here, also the downstream approach is more attractive. In fact, without modifying the structure of the subsystem control logic, automobile manufacturers can proceed to bulk purchasing from OEMs taking advantage from the economy of scale. Moreover, the architecture does not require additional controllers, but simply a coordination strategy made downstream. Unfortunately, this is not the case for the upstream approach. Not only additional high-level controller(s) is needed, but it may also require additional sensors or estimators [START_REF] Heidrich | Hardware-in-the-loop test rig for integrated vehicle control systems[END_REF]. For these reasons, auto-makers are reluctant to implement this approach in real vehicles.

3) Potential: The downstream approach is based on "expert knowledge" methods [START_REF] Selby | Intelligent Vehicle Motion Control[END_REF]. This consists in using some preliminary use-case studies and control designers' expertise to develop arbitration strategies. This approach is not well formalized because, theoretically, we cannot cover all the possible use-cases. It is hard to measure this approach potential, so we are not able to know if optimal results are achieved, or at least if better results could be obtained. Another issue is the fact that the subsystems control laws may then be based on different reduced vehicle models 3 . For instance, the development of an Active Rear Steering (ARS) is based on just the simple bicycle model [START_REF] Brennan | Integrated vehicle control via coordinated steering and wheel torque inputs[END_REF]. As a result, putting together different systems based on different behaviour models does not ensure proper operation of the overall vehicle system. In contrast, the upstream approach depicts mathematically the dynamic interactions, which is more suitable for numerical processes. Couplings can be quantified, so we can have some insights about the possibility of finding an optimum solution, a sub-optimal solution or no solution at all. The conflicts are rather prevented than mitigated. For this reason, upstream coordination techniques have more potential in handling multiobjective control problems.

C. Summary

It appears that a compromise should be made between complexity, cost, and potential. To avoid complexity, the control designer could prefer a downstream approach as long as the problem consists on a single-objective control. As more interactions are added, upstream approach becomes necessary for safety matters.

Although the upstream approach may seem expensive, the ECUs have became faster, less cumbersome, and cheaper in the past years. Moreover, various algorithms have been developed and tested. According to [START_REF] Soltani | Low Cost Integration of Electric Power-Assisted Steering (EPAS) with Enhanced Stability Program (ESP)[END_REF], [START_REF] Chen | Integrated control of vehicle system dynamics: Theory and experiment[END_REF], [START_REF] Tjnns | Nonlinear and Adaptive Dynamic Control Allocation[END_REF]- [START_REF] Shuai | Lateral motion control for four-wheel-independent-drive electric vehicles using optimal torque allocation and dynamic message priority scheduling[END_REF], control allocation methods are more suitable for this problem, especially for over-actuated vehicles. These methods have been reviewed and compared in [START_REF] Bodson | Evaluation of optimization methods for control allocation[END_REF], [START_REF] Johansen | Control Allocation -A survey[END_REF]. Linear Programming (LP) and Quadratic Programming (QP), could be executed in a few milliseconds with a limited number of iterations, which real-time computations require. These advances may finally convince auto-mobile manufacturers to adopt this approach.

As the main aim of this paper is to invite auto-makers and suppliers to standardize one overall architecture, one approach should be put on the spotlight. With the arrival of autonomous vehicles, the virtual pilot should take into account multiple objectives at the same time. As we have mentioned, downstream coordination is more suitable for single objective control where the human pilot deals with the other performances. For example, in a high speed cornering manoeuvre, the driver controls the longitudinal acceleration while the active steering could be combined with the brake-based yaw control to control the lateral dynamics and stabilize the vehicle. In autonomous driving, these two objectives should be fulfilled by the virtual pilot. Interactions should be predicted, and conflicts should be avoided rather than mitigated. Consequently, as we are moving towards the autonomous driving, the upstream coordination approach will become necessary.

For this reason, vehicle manufacturers and suppliers should prepare a common overall architecture. This architecture should have the following criteria:

• Adaptability to face environment changes and drivers behaviors [START_REF] Coelingh | Open-interface definitions for automotive systems -Application to a Brake by Wire System[END_REF],

• Fault-tolerance to propose some degraded modes and ensure a minimum of safety, • Dynamic reconfiguration to ensure soft switching and prevent loss of stability [START_REF] Wills | An open platform for reconfigurable control[END_REF],

• Extensibility to rapidly insert additional technologies without redesigning the whole architecture [START_REF] Coelingh | Open-interface definitions for automotive systems -Application to a Brake by Wire System[END_REF],

• Modularity to ensure flexibility, • Openness to support various systems from different sources without jeopardizing the intellectual property rights of the different stakeholders [START_REF] Coelingh | Open-interface definitions for automotive systems -Application to a Brake by Wire System[END_REF].

As a consequence, the multi-layer architecture with the control allocation method seems to be a better choice to fulfill these criteria [START_REF] Knobel | Optimized force allocation -A general approach to control and to investigate the motion of over-actuated vehicles[END_REF]- [START_REF] Van Den | Global chassis control and braking control using tyre forces measurement[END_REF], [START_REF] Reinold | Closed-loop control with optimal tireforce distribution for the horizontal dynamics of an electric vehicle with single wheel chassis actuators[END_REF]- [START_REF] Shyrokau | Vehicle motion control with subsystem prioritization[END_REF]. For example, the advanced approach presented in [START_REF] Shyrokau | Control allocation with dynamic weight scheduling for two-task integrated vehicle control[END_REF]- [START_REF] Shyrokau | Vehicle motion control with subsystem prioritization[END_REF] uses vehicle state estimator based on extended Kalman filter, high-level controller of vehicle general motion, middle-level control allocation and lower-level controllers for each subsystem as the Fig. 11 shows.

It should be noted that this architecture has been validated in a Hardware-In-the-Loop (HIL) procedure [START_REF] Heidrich | Hardware-in-the-loop test rig for integrated vehicle control systems[END_REF]. For all these reasons, we believe that this architecture is worth investigating and should be implemented in a real vehicle. D. Open challenges 1) Real vehicle implementation: As we mentioned, the first challenge is to be able to implement this architecture in a real vehicle. Real-time computation of an optimization method is usually a challenging task. In Integrated Vehicle Dynamics Dynamics Control, the challenge gets bigger as the coordination technique is located in an inner loop. Consequently, a higher rate is required (around 100 Hz according to [START_REF] Bodson | Evaluation of optimization methods for control allocation[END_REF]). Control allocation with linear or quadratic programming could be the solution [START_REF] Bodson | Evaluation of optimization methods for control allocation[END_REF], [START_REF] Johansen | Control Allocation -A survey[END_REF]. Another issue is to bring together manufacturers and suppliers to collaborate. That goes beyond the scope of our work. Nevertheless, attractive results for both sides could be the first step.

2) "Adaptability": As long as over-actuated systems are concerned, multiple solutions could be found for an optimization problem. Secondary objectives could therefore be achieved. Allocation can be used to favour one solution over another according to the desired behaviour of the vehicle. Consequently, different "feelings" can be generated to realize the same manoeuvre. While controlling the yaw rate using the 4-Wheel Drive system could give the vehicle a sporty behaviour, using the 4-Wheel Steering system for the same manoeuvre could rather give a comfortable behaviour. This is of major importance for autonomous vehicles where the challenge is not only the trajectory following but also how the vehicle follows this trajectory. Control allocation introduces new opportunities to make drivers accept autonomous vehicles. Using specific weighting functions to favour different subsystems combinations could be used to generate feelings of comfort and security, and therefore make drivers trust more their vehicles. However, the intra and inter-individual variability between the drivers is also a challenge. Motions that generate excitement for some people could generate fear among others. So allocation should be adaptable and change over time and maybe even learn from its driver's preferences. Evolutionary algorithms and artificial intelligence could be an interesting approach to investigate in this field too.

V. CONCLUSION AND FUTURE WORKS

In this paper, a new classification of integrated vehicle dynamics control architectures has been proposed. Two major classes related to the coordination logic have been outlined: the downstream coordination architecture and the upstream coordination architecture. These two classes have been compared. The downstream coordination requires less cost and less complexity but it is limited to single-objective control coordination. In contrast, the upstream coordination can handle multi-objective control coordination but it is more complex and requires more costs. These last drawbacks could be overcame by the advances made in electronic and software engineering.

The literature shows that the integrated vehicle dynamics control architectures are still an object of research. There is a clear lack of benchmark standards and common test procedures to validate the integrated subsystems coordination methods. This paper aims to invite auto-mobile manufacturers and suppliers to adopt an upstream approach and to standardize its structure.

We recognize that more evidence are needed to convince the different stakeholders to favour one architecture over another. That is why our future works will concern the development of the multi-layer architecture with control allocation methods and their comparison with the downstream approach.

The first step of control synthesis is modelling. Regarding ground vehicles, tires are the sole effectors. Distribution of control commands is mainly constrained by tires' potential. As far as combined slip is concerned, this potential varies. Tires' stiffness and maximum efforts should be updated on-line. In this way, we can favour the tires with greater potential. For these reasons, a new tire model is under development. This model is linear in order to facilitate control synthesis, and parameter varying to depict combined slip.

Fig. 3 .

 3 Fig.3. Pure Subsumption coordination (adapted from[START_REF] Gordon | Integrated control methodologies for road vehicles[END_REF]).

Fig. 4 .

 4 Fig.4. Largest Modulus Activation coordination (adapted from[START_REF] Gordon | Integrated control methodologies for road vehicles[END_REF]).

Fig. 5 .

 5 Fig. 5. Artificial Neural Network coordination (adapted from [4]).

Fig. 8 .

 8 Fig.8. Centralized control structure (adapted from[START_REF] Gordon | Integrated control methodologies for road vehicles[END_REF]).

Fig. 10 .

 10 Fig. 10. Multi-layered control architecture [19].

Fig. 11 .

 11 Fig. 11. General architecture of Integrated Motion Control [42].

This has changed today.

This is certainly the case when produced by different suppliers but may also be the case for a single supplier.

ACKNOWLEDGEMENT

The authors would like to thank Dr. Xavier Mouton and Dr. Didier Martinez from the Group Renault for their enriching discussions about today's automotive industry challenges.


Moad Kissai
email: moad.kissai@ensta-paristech.fr

Bruno Monsuez
email: bruno.monsuez@ensta-paristech.fr

Adriana Tapus
email: adriana.tapus@ensta-paristech.fr

Didier Martinez
email: didier.d.martinez@renault.com

A New Linear Tire Model With Varying Parameters

Keywords: Tire Modeling, combined slip, linearization, global chassis control, AMESim c simulation

A large interest has recently been given to global chassis control. This requires a tire model adapted to control synthesis that can represent combined slip. In this paper, a new suitable tire model for vehicle coupled operations control is proposed. The existing models are either too simple to represent coupled behaviors or too complex to allow control synthesis. This new tire model takes benefits from both approaches, as it is linear with varying parameters. Simulations of coupled operations proved the model accuracy in the tire stable range and its promising potential for global chassis control synthesis.

I. INTRODUCTION

The development of chassis systems has been ongoing since the late 1970's [START_REF] Selby | Intelligent Vehicle Motion Control[END_REF]. Most of these systems are designed for single-objective performances. This is about to change with the arrival of autonomous vehicles. In fact, the virtual pilot has to handle combined operations. For example, let us consider the high speed cornering maneuver. In the present situation, the human pilot controls the longitudinal speed while active steering systems (e.g., the 4-Wheel Steering (4WS) system [START_REF] Brennan | Integrated vehicle control via coordinated steering and wheel torque inputs[END_REF]) or, in hazardous situations, brake-based yaw control systems (e.g., The Electronic Stability Control (ESC) system [START_REF] Velardocchia | Control systems integration for enhanced vehicle dynamics[END_REF]) control the lateral dynamics to ensure the vehicle stability. In contrast, the virtual pilot has to handle multi-objective control problems. Therefore, couplings should be taken into account.

One of the most important couplings is the combined slip as the tire is the only effector between the vehicle and the road [START_REF] Soltani | Low Cost Integration of Electric Power-Assisted Steering (EPAS) with Enhanced Stability Program (ESP)[END_REF]. In this context, two major approaches are adopted: empirical methods and theoretical methods. Figure 1 depicts the various differences (taken from [START_REF] Pacejka | Tyre and Vehicle Dynamics[END_REF]).

Six parameters are considered in [START_REF] Pacejka | Tyre and Vehicle Dynamics[END_REF]:

• Degree of fit: model's accuracy with respect to the design objective, required to develop the model. As expected, empirical approaches require an important number of full scale tests with respect to theoretical approaches. These latter are more based on the tire's physical structure theory. Four categories are distinguished in [START_REF] Pacejka | Tyre and Vehicle Dynamics[END_REF]: "from experimental data only", "using similarity method", "through simple physical model", "through complex physical model".

The first category at the extreme left uses regression procedures to develop mathematical formulations whose parameters fit best the measured data. A well-known empirical model is the Magic Formula [START_REF] Pacejka | Tyre and Vehicle Dynamics[END_REF]. This model provides an excellent fit for tire's efforts curves, which makes it more suitable for vehicle motion simulations. The similarity approach uses rather simple distortion and re-scaling methods to develop simpler empirical models. This method is particularly useful when fast computations are needed [START_REF] Pacejka | Tyre and Vehicle Dynamics[END_REF]. A good example would be Dugoff's model, which uses a simpler representation of tire deformation while keeping a good representation of combined slip [START_REF] Dugoff | Tire Performance Characteristics Affecting Vehicle Response to Steering and Braking Control Inputs[END_REF]. But these two categories provide less insight in tire behavior. The relatively simple physical models of the third category are more useful to get better understandings about the tire behavior. In this context, the "brush model" represents a good illustration [START_REF] Pacejka | Tyre and Vehicle Dynamics[END_REF], [START_REF] Svendenius | Tire Models for Use in Braking Applications[END_REF], [START_REF] Svendenius | Brush Tire Model with Increased Flexibility[END_REF]. Regarding the fourth category, interest is given to tire performances related to its construction. More detailed analysis is required and complex finite element based models are usually adopted [START_REF] Pacejka | Tyre and Vehicle Dynamics[END_REF].

On one hand, empirical models rely on experimental measures to make simulation more accurate, and on the other hand, theoretical models rely on physical models to give more insight about the tire behavior and improve its construction. No model however is designed for control synthesis. Physical models are too complex to be implemented in real-time operations, and empirical models use numerous parameters with poor physical meaning, which make them hard to measure in real time.

This paper outlines the design of a new tire model more adapted to control synthesis, and especially, global chassis control. For this purpose, the model should accurately represent the combined slip behavior in a simpler way using real-time measurable parameters. We propose consequently a new model based on Dugoff's model. This model is simple enough to be suitable for control synthesis as it is linear, and depends on measurable or estimable physical varying-parameters, which favor real-time implementation. The method adopted to develop the model is described in Section II. Results and interpretations are given in Section III. In Section IV, the model's relevance is discussed with respect to existing models. Conclusions and future works are consequently drawn at the end of this paper.

II. DEVELOPMENT OF THE NEW TIRE MODEL

First, we propose a set of requirements for the new tire model development. As far as global chassis control is concerned, the new model should:

• Respect the tire physical fundamentals to give enough insight into tire behavior, • Represent as precise as possible combined slip behavior, • Be simple enough for controllability issues, • Depends on a minimum set of parameters that can be measured and estimated so as to favor real-time operations.

A. Tire physical fundamentals

The tire consists of a set of rubber blocks. Two friction mechanisms could be distinguished: shearing and sliding [START_REF] Michelin | Le pneu: L'adhrence[END_REF]. In shearing, the rubber block deforms without sliding with respect to the ground. A resistance force proportional to deformation is generated. When this force reaches a maximum value, called the Coulomb force, the rubber block rubs on the ground and start sliding. If slip value continue growing, the tire suffers from a loss of potential, which causes the instability of the vehicle.

The friction force maximum value is then very important to predict. It depends on two varying-parameters: the vertical load applied by the vehicle and the coefficient of friction that characterizes the condition of both the road and tire rubber [START_REF] Michelin | Le pneu: L'adhrence[END_REF]. Furthermore, longitudinal adhesion and lateral adhesion are competing: they must share the tire friction potential [START_REF] Michelin | Le pneu: L'adhrence[END_REF]. In fact, the overall adhesion is delimited by a "friction ellipse" [START_REF] Pacejka | Tyre and Vehicle Dynamics[END_REF], [START_REF] Svendenius | Tire Models for Use in Braking Applications[END_REF], [START_REF] Wong | Theory of Ground Vehicles, 3rd Edition[END_REF]. A longitudinal force request would consequently penalize a lateral force request if the limits of adhesion are reached. Figure 2 summarizes the previous characteristics. Fig. 2. Combined side force and brake force characteristics (adapted from [START_REF] Pacejka | Tyre and Vehicle Dynamics[END_REF]).

Three regions are distinguished:

• Region 1 : the curve is linear and increasing, the rubber blocks are deforming without sliding (shearing). • Region 2 : the curve is nonlinear and increasing, and ends up reaching a maximum. A portion of the contact area begins to slide. • Region 3 : the curve is decreasing, there is the sliding of the tire. The tire behavior is unstable. The overall behavior of the tire is highly nonlinear.

B. Existing tire models

Three models have received large interest: The Magic Formula of Pacejka [START_REF] Pacejka | Tyre and Vehicle Dynamics[END_REF], the brush model [START_REF] Svendenius | Tire Models for Use in Braking Applications[END_REF], [START_REF] Svendenius | Brush Tire Model with Increased Flexibility[END_REF], and the physical model of Dugoff [START_REF] Dugoff | Tire Performance Characteristics Affecting Vehicle Response to Steering and Braking Control Inputs[END_REF]. Next, SAE ISO notation [START_REF] Svendenius | Brush Tire Model with Increased Flexibility[END_REF] will be used in order to compare these models.

1) The Magic Formula of Pacejka: This model belongs to the empirical approaches category. The tire forces are defined as follows [START_REF] Pacejka | Tyre and Vehicle Dynamics[END_REF]:

Y (X) = y (x) + S v (1)
With:

x = X + S h y = D sin [C arctan {Bx -E (Bx -arctan (Bx))}] (2) (3)
Where:

• Y : F x (longitudinal force), F y (lateral force) or M z (aligning torque), • X : κ (longitudinal slip) or tan α (where α is the sideslip angle), And:

• B : stiffness factor, • C : shape factor, • D : peak value, • E : curvature factor, • S H : horizontal shift, • S V : vertical shift.
Weighting functions are used to take into account the combined slip:

G = D cos [C arctan (Bx)] (4)
2) The Brush model: As its name may reveal, the model is assimilated to a set of brush bristles (Figure 3). Several mathematical formulations were adopted regarding this physical representation [START_REF] Soltani | Low Cost Integration of Electric Power-Assisted Steering (EPAS) with Enhanced Stability Program (ESP)[END_REF], [START_REF] Pacejka | Tyre and Vehicle Dynamics[END_REF], [START_REF] Svendenius | Tire Models for Use in Braking Applications[END_REF], [START_REF] Svendenius | Brush Tire Model with Increased Flexibility[END_REF]. These formulas differ according to the assumptions considered regarding the tire physical characteristics, namely, the friction coefficient and the vertical load distribution. In this paper, we suppose that the friction is isotropic and the tire/ground contact surface is rectangular. Longitudinal stiffness is then different from the cornering one. Pursuing the same development logic in [START_REF] Pacejka | Tyre and Vehicle Dynamics[END_REF] and [START_REF] Svendenius | Tire Models for Use in Braking Applications[END_REF], and considering the engine operation instead of braking operation, we find:

• If ψ (σ x , σ y ) < 1                                  F x (σ x , σ y ) = C s σ x (1 -ψ (σ x , σ y )) 2 + σ x σ 2 x + σ 2 y µF z ψ 2 (σ x , σ y ) (3 -2ψ (σ x , σ y )) F y (σ x , σ y ) = C α σ y (1 -ψ (σ x , σ y )) 2 + σ y σ 2 x + σ 2 y µF z ψ 2 (σ x , σ y ) (3 -2ψ (σ x , σ y )) (5) (6)
• If ψ (σ x , σ y ) ≥ 1            F x (σ x , σ y ) = σ x σ 2 x + σ 2 y µF z F y (σ x , σ y ) = σ y σ 2 x + σ 2 y µF z (7) (8)
Where

ψ (σ x , σ y ) = C s σ x 3µF z 2 + C α σ y 3µF z 2 represents
the adhesion limits when the vertical load distribution is parabolic, and:

• C s : longitudinal stiffness of the tire, • C α : cornering stiffness of the tire, • µ : coefficient of friction, • F z : vertical load, • σ x = κ 1 + κ : theoretical longitudinal slip, • σ y = tan (α) 1 + κ : theoretical sideslip angle.
3) The physical model of Dugoff: Dugoff developed an analytic model based on the classical analysis of Fiala [START_REF] Dugoff | Tire Performance Characteristics Affecting Vehicle Response to Steering and Braking Control Inputs[END_REF]. He assumed a constant friction coefficient and a constant vertical load distribution. These assumptions give:

       F x = C s κ 1 -κ τ F y = C α tan (α) 1 -κ τ (9) (10)
τ is introduced to take into account the combined slip:

τ = (2 -σ) σ if σ < 1 1 otherwise ( 11
)
Where:

σ = (1 -κ) µF z 2 C 2 s κ 2 + C 2 α tan 2 (α) (12)
4) Comparison: The Magic Formula is widely used by car manufacturers to simulate and validate dynamic performances due to its precision. Its parameters are changed offline to take into account the different road conditions. Our goal is to develop a model for control synthesis. The different parameters should be changed online. To do this, we should use physical parameters that could be measured or at least estimated. Consequently, we also use the Magic Formula only for validation.

To compare the physical models with Pacejka's, the Magic Formula parameters should be interpreted to meet a physical meaning. In [START_REF] Pacejka | Tyre and Vehicle Dynamics[END_REF], Pacejka pointed out that the product BCD is equivalent to the stiffness. Indeed, considering the equation (3), the product BCD represents the tangent at the origin in the same way as C s for the longitudinal force in equation ( 5) or (9), and C α for the lateral force in equation ( 6) or [START_REF] Wong | Theory of Ground Vehicles, 3rd Edition[END_REF]. We use the shape coefficients given in [START_REF] Wong | Theory of Ground Vehicles, 3rd Edition[END_REF] for a vertical load of 4000 N, the stiffness coefficients are then given the value of the product BCD. Although the brush model uses physical parameters, it does not take into account the loss of potential phenomenon when maximum friction force is reached. In [START_REF] Svendenius | Tire Models for Use in Braking Applications[END_REF], this model has been improved by considering a speed dependent friction, and in [START_REF] Svendenius | Brush Tire Model with Increased Flexibility[END_REF], an asymmetric vertical load is considered to improve the peak value location accuracy. However, the complexity also increases.

Dugoff's model suffers from the same drawback. The force's shape is totally inaccurate in the nonlinear region given that the coefficient of friction and the vertical load distribution are considered constant. In [START_REF] Bian | A Dynamic Model for Tire/Road Friction Estimation under Combined Longitudinal/Lateral Slip Situation[END_REF], the authors developed weighting functions to improve the model in the nonlinear region. Accuracy increases, but the model loses its analytic nature.

In control synthesis, these criteria are not crucial. As we mentioned, the third region of Figure 2 is unstable. There is no point to control the vehicle in this region. This behavior should be predicted and avoided. Thereby, the Brush model and Dugoff's model represent good candidates.

These two models respect three out of the four requirements that we proposed at the beginning of this section. They both respect the tire physical fundamentals, they represent precisely enough the combined slip in the stable region and they are dependent of the physical parameters. However, their mathematical formulas are still too complex to manipulate. They should be linearized.

C. Tire physical models linearization

Dugoff's model differs from the brush model by the vertical load distribution assumptions. Dugoff supposes a constant vertical load distribution [START_REF] Dugoff | Tire Performance Characteristics Affecting Vehicle Response to Steering and Braking Control Inputs[END_REF]. In the brush model, a parabolic distribution is considered [START_REF] Svendenius | Tire Models for Use in Braking Applications[END_REF]. Figure 4 shows however that little improvement is brought by this assumption. On the other hand, the formulas (5) and (6) are more complex to manipulate than [START_REF] Michelin | Le pneu: L'adhrence[END_REF] and [START_REF] Wong | Theory of Ground Vehicles, 3rd Edition[END_REF]. The brush model equations' derivatives are too heavy to be exposed. Hence, only Dugoff's model linearization is presented, but both results are illustrated in section III.

According to (11), Dugoff's model has two expressions depending on whether the tire effort has reached its saturation or not. As our concern is the stable part of the tire behavior, it is the expression related to the case where σ < 1 that should be linearized.

1) Longitudinal force linearization: Let κ * be the stable operating point located in the linear region of the tire force curve (Figure 2). After replacing the variable τ in equation ( 9) by its expression in equation [START_REF] Bian | A Dynamic Model for Tire/Road Friction Estimation under Combined Longitudinal/Lateral Slip Situation[END_REF], and σ by its expression in [START_REF] Svendenius | Tire Modeling and Friction Estimation[END_REF], we perform a first-order Taylor series approximation of F x about κ * :

F x (κ) = F x (κ * ) - dF x (κ * ) dκ κ=κ * κ * + dF x (κ * ) dκ κ=κ * κ ( 13
)
As the linear region is concerned, κ * is very small. We verify analytically that:

(κ * ) ⇒ F x (κ * ) ∼ = dF x (κ * ) dκ κ=κ * κ * (14)
Therefore:

(κ * ) ⇒ F x (κ) ∼ = F x (κ * ) κ * κ (15)
Equation [START_REF] Bchle | Slip-Constrained Model Predictive Control Allocation for an All-Wheel Driven Electric Vehicle[END_REF] shows that the term F x (κ * ) κ * corresponds to a longitudinal stiffness. Let us symbolize it C * s . Using Dugoff's model equations [START_REF] Michelin | Le pneu: L'adhrence[END_REF], [START_REF] Bian | A Dynamic Model for Tire/Road Friction Estimation under Combined Longitudinal/Lateral Slip Situation[END_REF], and (12), we find:

C * s (α, µ, F z ) = 4 C 2 s κ * 2 + C 2 α tan 2 (α) -(1 -κ * ) µF z 4 C 2 s κ * 2 + C 2 α tan 2 (α) µF z C s (16)
This longitudinal stiffness depends on varying-parameters α (sideslip), µ (coefficient of friction), and F z (Vertical load). These parameters could be estimated online [START_REF] Soltani | Low Cost Integration of Electric Power-Assisted Steering (EPAS) with Enhanced Stability Program (ESP)[END_REF], [START_REF] Svendenius | Tire Modeling and Friction Estimation[END_REF], [START_REF] Singh | Estimation of tireroad friction coefficient and its application in chassis control systems[END_REF]. It depends also on constant parameters C s (longitudinal stiffness in a non-combined slip), C α (lateral stiffness in a noncombined slip), and κ * . In section IV, an explicit expression will be given to κ * . By setting:

ρ = 4 C 2 s κ * 2 + C 2 α tan 2 (α) -(1 -κ * ) µF z 4 C 2 s κ * 2 + C 2 α tan 2 (α) µF z (17)
it appears that C * s is the product of C s and a coupling term ρ. As a result, the linearized longitudinal force:

F x (κ) = C * s κ (18)
respect the proposed requirements:

• Respect of the tire physical nature as it depends on α, µ, and F z , • Taking into account the combined slip by means of ρ, • Linearity of the formula and therefore simplicity regarding control issues, • Dependency on online estimable parameters [START_REF] Soltani | Low Cost Integration of Electric Power-Assisted Steering (EPAS) with Enhanced Stability Program (ESP)[END_REF], [START_REF] Svendenius | Tire Modeling and Friction Estimation[END_REF], [START_REF] Singh | Estimation of tireroad friction coefficient and its application in chassis control systems[END_REF].

2) Lateral force linearization: The same reasoning is pursued. Here we suppose an α * to be a stable operating point in the tire linear region. Again, we have α * . We proceed then to the same simplifications as ( 14) and [START_REF] Bchle | Slip-Constrained Model Predictive Control Allocation for an All-Wheel Driven Electric Vehicle[END_REF]. Moreover, in the vicinity of α * :

(α ) ⇒ (tan (α) ∼ = α) (19)
Again, a lateral stiffness C * α can be defined using Dugoff's model equations [START_REF] Wong | Theory of Ground Vehicles, 3rd Edition[END_REF], [START_REF] Bian | A Dynamic Model for Tire/Road Friction Estimation under Combined Longitudinal/Lateral Slip Situation[END_REF], and (12):

C * α (κ, µ, F z ) = 4 C 2 s κ 2 + C 2 α α * 2 -(1 -κ) µF z 4 C 2 s κ 2 + C 2 α α * 2 µF z C α ( 20
)
The linearized lateral force is expressed as:

F y (α) = C * α α (21)
The proposed requirements are again fulfilled. The only difference is that C * α depends on κ instead of α. 3) Dynamic saturation: The linearized forces developed are indefinitely increasing. As we have mentionned, the overall adhesion is delimited by a "friction ellipse" [START_REF] Pacejka | Tyre and Vehicle Dynamics[END_REF], [START_REF] Svendenius | Tire Models for Use in Braking Applications[END_REF], [START_REF] Wong | Theory of Ground Vehicles, 3rd Edition[END_REF]. This concept is illustrated in Figure 5. The tire potential is delimited by the product µF z . In general, the friction can be anisotropic. The maximum longitudinal force could be therefore superior to the lateral force maximum. To simplify the problem, we have considered an isotropic friction. Thus, we are facing a "friction circle concept". Despite this simplification, the same logic is applied. When the tire is only solicited longitudinally, the maximum longitudinal force is equal to µF z . If the tire is solicited laterally while accelerating or braking, the overall force moves along the friction circle. The maximum overall tire force is still µF z , but the the maximum longitudinal force is penalized. The different forces saturations should be dynamic:

     F x ≤ (µF z ) 2 -F 2 y F y ≤ (µF z ) 2 -F 2 x ( 22
) (23)
III. SIMULATION OF THE LINEARIZED MODEL By reassembling the equations ( 16), ( 18), ( 20)-( 23), we simulate the new linearized tire model and compare it with Pacejka model using the equations (1)-( 4) in Matlab/Simulink c environment. The linearized Brush-based model is also plotted.

A. Brush-based and Dugoff-based linearized models comparison

Here, we only present the longitudinal force in combined slip with a sideslip angle of 1 • (see Figure 6). The lateral force follows the same behavior with different parameters. Although the brush model considers a parabolic vertical load distribution, once linearized it does not offer much more precision with respect to Dugoff's model. The linearized model based on Dugoff's model provides better simplicity/precision compromise. This latter will be then retained.

B. Linearized Dugoff-based model validation

Note in Figure 6 that the longitudinal force does not reach the maximum of µF z = 4000N because a part of the tire potential is reserved for the lateral force. The curve's slope of the longitudinal force also varies with the sideslip angle because the longitudinal stiffness varies (see Figure 7). Same remark holds for the lateral force with respect to the longitudinal slip (Fig. 8).

That is to say that the linearized Dugoff-based model represent well the combined slip behavior in the stable region of the tire. More realistic simulations are provided in the next section.

IV. RELEVANCE OF THE NEW TIRE MODEL

In control synthesis, a linear model of the tire has been used in several works [START_REF] Selby | Intelligent Vehicle Motion Control[END_REF], [START_REF] Soltani | Low Cost Integration of Electric Power-Assisted Steering (EPAS) with Enhanced Stability Program (ESP)[END_REF], [START_REF] Xiong | Research on Robust Control for longitudinal Impact of 4 Wheel-Drive Hybrid Electric Vehicle[END_REF], [START_REF] Bchle | Slip-Constrained Model Predictive Control Allocation for an All-Wheel Driven Electric Vehicle[END_REF]. This model has a simpler representation:

F x = C s κ F y = C α α (24) (25)
The tire behavior is well represented in the linear region, but the model does not take into account the combined slip. This model is more suitable for vehicle stand-alone subsystems. For example, in [START_REF] Xiong | Research on Robust Control for longitudinal Impact of 4 Wheel-Drive Hybrid Electric Vehicle[END_REF], this simple model is used to develop vehicle longitudinal control logic through a 4 Wheel-Drive Hybrid Electric system. In [START_REF] Harada | Analysis of lateral stability with integrated control of suspension and steering systems[END_REF] and [START_REF] Shuai | Lateral motion control for four-wheel-independent-drive electric vehicles using optimal torque allocation and dynamic message priority scheduling[END_REF], a bicycle model, which is based on the simple linear tire model, is used for lateral control performance improvement. With the arrival of autonomous vehicles, the virtual pilot must handle multiobjective control problems. Couplings should be therefore considered, especially at the tire level. For Global Chassis Control (GCC), the linearized Dugoff-based model proposed in this paper is necessary. To illustrate this, a vehicle global dynamic analysis is described next using AMESim c platform.

A. Comparison of linear models

To compare between the two linear models, another requirement should be added. The new linear tire model should be equivalent to the simpler one when there is no combined slip. This requirement enables giving explicit expressions to the different stable operating points κ * and α * .

1) κ * Calculation: Respecting the previous requirement means:

C * s (0, µ, F z ) = C s (26)
According to the expression (16), we find:

κ * = µF z 8C 2 s µF z + 4C s ± (µF z ) 2 + 8µF z C s (27)
Both results are acceptable because µF z + 4C s

(µF z ) 2 + 8µF z C s .
Here, we consider the highest one.

2) α * Calculation: Again, the previous requirement leads to:

C * α (0, µ, F z ) = C α (28)
According to the expression (20), we find:

α * = µF z 2C α (29)
3) AMESim c simulation: The AMESim c demo used describes complete chassis model and provide a global dynamics analysis. Inputs are steering angle and engine torque. A combined slip could then be observed. The vehicle is a Bsegment class car. The model is composed of a chassis model with 15 degrees of freedom with assosicated subsystems: elastokinematics module, suspension (spring, damper, lower and higher end stop, antiroll bar), aerodynamic module, tire, road, sensors, powertrain unit, braking system and powersteering system. It is a highly nonlinear global model whose purpose is to represent as precise as possible the vehicle behavior. Most importantly, it uses Pacejka's model to calculate tires' efforts. We add supplementary modules that include the new linear Dugoff-based tire model and the classic linear tire model and compare it with The Magic Formula of Pacejka. Figure 9 illustrates the longitudinal force and Figure 10 illustrates the lateral force. When only longitudinal force is requested (acceleration or braking), both linear models are close enough to Pacejka's model. The lateral force of the classic linear model starts to deviate as soon as the tire is solicited longitudinally and laterally at the same time. Both longitudinal and lateral forces of the classic linear model diverge at severe conditions, for example in a hard brake maneuver while steering or progressive steering at high velocity. In contrast, the new linearized model follows close enough the Magic Formula. When the tire is solicited longitudinally and laterally at the same time, both longitudinal and lateral stiffnesses are updated and dynamic saturations are recalculated. The values of the forces are then lower.

In control problems, this is crucial. In fact, ignoring combined slip may lead to high tire force demand. This could exceed the saturated value and destabilize the vehicle. The new linearized model proposed in this paper aims preventing this situation.

B. Further improvements

So far, all mathematical formulas developed are expressed in the steady state. In control synthesis, time response is an important criterion. Hence, the transient behavior should be included.

1) Relaxation length: Tire force delay is caused by carcass deflections [START_REF] Pacejka | Tyre and Vehicle Dynamics[END_REF], [START_REF] Svendenius | Tire Models for Use in Braking Applications[END_REF]. This is usually represented by a first order lag [START_REF] Di | Modelling and simulation of the dynamic behaviour of the automobilel[END_REF], [START_REF] Klomp | Safety Margin Estimation in Steady State Maneuvers[END_REF]:

F i = 1 1 + τ i s F ss i (30)
i is either x or y, τ i is the time-constant, s is the Laplace operator, and F ss i is the steady-state force that have been developed in the previous sections.

The carcass deflections change with the vehicle speed [START_REF] Pacejka | Tyre and Vehicle Dynamics[END_REF], [START_REF] Svendenius | Tire Models for Use in Braking Applications[END_REF], [START_REF] Di | Modelling and simulation of the dynamic behaviour of the automobilel[END_REF], [START_REF] Klomp | Safety Margin Estimation in Steady State Maneuvers[END_REF]. Time-constant in equation (30) depends then on vehicle speed:

τ i = 3L i V x ( 31
)
L i is the relaxation length and V x is the vehicle longitudinal speed. The relaxation length represent the distance that tire rolls before the lateral force reaches 63% of its final value [START_REF] Pacejka | Tyre and Vehicle Dynamics[END_REF]. This does not influence much the shape of the tire response as the Fig. 11 shows. This is mainly due to the low value of the relaxation length (around 0.02 m). In control synthesis, the transient behavior could be ignored to prevent high order controllers design.

C. Controllability

The linearized model is developed for control synthesis suitability. A state-space representation can be proposed:

ẋ (t) = A (t) x (t) + B (t) u (t) y (t) = C (t) x (t) (32) (33)
Where:

• x (t) = F x F y : is the state vector,

• y (t) = F x F y : is the output vector,

• u (t) = κ α : is the input vector,

• A (t) =    - 1 τ x 0 0 - 1 τ y    : is the state matrix, • B (t) =    C * s τ x 0 0 C * α τ y   
: is the input matrix,

• C (t) = 1 0 0 1 : is the output matrix.

Without forgetting the constraints (22) and (23).

Note that both state and input matrices contain varying parameters. Although stiffness coefficients depend on input variables, they cannot be splitted as we need the whole expression to be multiplied by its corresponding input variable.

According to the structure of the matrices A, B and C, the system is clearly stable, controllable and observable as long as the varying parameters are positive, which is the case.

Finally, this representation is suitable for modular and flexible control architectures. A simple tire module can be added as the Fig. 12 shows where the tire block contains equations (32) and (33) and constraints [START_REF] Oppenheimer | Control Allocation for Over-actuated Systems[END_REF] and (23).

D. Discussion and application

We recall that this model was developed to face Global Chassis Control issues. As far as the overall chassis is concerned, coordination is a key aspect. Several techniques exist in this context. For example, rule-based strategies using fuzzy logic control downstream the standalone subsystems have been used in [START_REF] Wang | Agent-based coordination framework for integrated vehicle chassis control[END_REF]. Autonomous vehicles require additional subsystems. As the number of subsystems increases, development of such strategies becomes less obvious. Optimization techniques performed upstream the standalone subsystems present a promising alternative in this situation. Moreover, today's vehicles are over-actuated [START_REF] Selby | Intelligent Vehicle Motion Control[END_REF]. That is to say that subsystems control outputs exceed the number of control axes of the vehicle. The solution is therefore not unique. This should not be seen as a disadvantage, but rather as an opportunity to introduce secondary objectives. In this framework, control allocation techniques present more suitable solutions [START_REF] Johansen | Control Allocation -A survey[END_REF].

Control allocation problems are defined as follow [START_REF] Oppenheimer | Control Allocation for Over-actuated Systems[END_REF]: find the control vector, δ ∈ R n such that

B δ = d des (34)
subject to

δ min ≤ δ ≤ δ max ˙ δ ≤ ˙ δ max (35) (36)
where B ∈ R n×m is a control effectiveness matrix, δ min and δ max are the lower and upper position limits respectively, ˙ δ is the control rate, ˙ δ max is the maximum control rate, d des are the desired accelerations, n is the number of control effectors, and m is the number of axes to control (n > m).

For ground vehicles, tires are the sole effectors. Distribution of control commands is mainly constrained by tires' potential. As far as combined slip is concerned, this potential varies. Tires' stiffness and maximum efforts should be updated. For example, in equation [START_REF] Oppenheimer | Control Allocation for Over-actuated Systems[END_REF], F x is constrained by the overall potential (µF z ) and also F y . The lateral stiffness should be updated by taking into account the friction ellipse concept. In this way, we can favour the tires with greater potential.

V. CONCLUSION

A new linear tire model with varying parameters has been developed in this paper. This model is particularly suitable for control problems and especially for global chassis control. Unlike other linear models, our model takes into account the combined slip in the tire's stable region where control is justified. It depends also on physical parameters that can be estimated, which is necessary for online operations.

More evidence is required to validate this model. Unfortunately, experiments were not carried out because they are too tedious for a simple model validation. Our future works will rather focus on the model ability to generate safer control strategies regarding coordinated control for global chassis control.

Fig. 1 .

 1 Fig. 1. Possible Categories of Tire Modeling Approaches [5].

Fig. 3 .

 3 Fig.3. The Brush model[START_REF] Pacejka | Tyre and Vehicle Dynamics[END_REF].

  Figure 4 depicts the differences for the longitudinal force in a combined slip with a sideslip angle equal to 3 • .

Fig. 4 .

 4 Fig. 4. Comparison of models' longitudinal force shapes in combined slip.

Fig. 5 .

 5 Fig. 5. The friction ellipse concept [10].

Fig. 6 .

 6 Fig. 6. Comparison of Pacejka model with both linearized models in combined slip.

Fig. 7 .

 7 Fig. 7. Variation of the longitudinal force slope with respect to sideslip angle (Linearized Dugoff-based model).

Fig. 8 .

 8 Fig. 8. Variation of the lateral force slope with respect to longitudinal slip (Linearized Dugoff-based model).

Fig. 9 .Fig. 10 .

 910 Fig. 9. AMESim c comparison of Pacejka model with linearized models -Front left tire longitudinal force.

Fig. 11 .

 11 Fig. 11. AMESim c comparison of Pacejka model and linearized models with and without transient behavior -Front left tire longitudinal force.

Fig. 12 .

 12 Fig. 12. Linearized tire module with varying-parameters.

•

  Number of full scale tests: amount of tests required to validate the model, • Complexity of formulations, • Effort required to design the model,

• Insight in tire behavior: model's ability of predicting the tire behavior, • Number of special experiments: amount of experiments

ACKNOWLEDGMENT

The authors would like to thank Dr. Xavier Mouton from the Group Renault for his supervision of the Vehicle Motion Control (VMC) project.