Stephanie Jerônimo De Almeida

Zenilton Kleber Gonçalves
email: zenilton@pucminas.br

Do Patrocínio

Simon Malinowski
email: simon.malinowski@irisa.fr

Silvio Jamil
email: sjamil@pucminas.br

F Guimarães

Guillaume Gravier

Carolina Jerônimo
email: carolinajeronimo@gmail.com

Zenilton K G Patrocínio

A novel method for temporal graph classification based on transitive reduction Carolina

Keywords: Temporal Graph, learning on dynamic graphs, temporal graph classification

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

I. INTRODUCTION

Graphs are used to represent linked data in various domains such as social network [START_REF] Amara | Multilingual topic modeling for tracking covid-19 trends based on facebook data analysis[END_REF], [START_REF] Akhtar | Graph tools for social network analysis[END_REF], disease analysis [START_REF] Karaivanov | A social network model of covid-19[END_REF], [START_REF] Zhu | Interpretable learning based dynamic graph convolutional networks for alzheimer's disease analysis[END_REF], [START_REF] Wang | scgnn is a novel graph neural network framework for single-cell rna-seq analyses[END_REF], bioinformatics [START_REF] Zhang | Graph neural networks and their current applications in bioinformatics[END_REF] and computer vision [START_REF] Pradhyumna | Graph neural network (gnn) in image and video understanding using deep learning for computer vision applications[END_REF]. Modeling problems in terms of graphs allows for a deep understanding of the relationship between all elements. In social networks, for example, we can estimate how much a part of the network influences another [START_REF] Yang | Networks: An introduction by mej newman[END_REF]. In graph theory, a vertex is reachable from another if there exists, at least, a path between these two vertices. The existence of two or more different paths between two vertices, called the transitive relation, may be seen as redundant (or ambiguous) making it more difficult to understand the graph behavior. The result of the Transitive Reduction (TR) of a directed graph is a subgraph that has the same reachability relation as the original graph but with the Fig. 1: Example of a transitive reduction in which the paths between all vertices are preserved. If there exists a path between two vertices in (a), then as can be seen in (b), there is one only path between them in the simplified graph. minimum number of edges, preserving all paths between nodes in the original graph but removing unnecessary edges that can be inferred from the remaining edges [START_REF] Aho | The transitive reduction of a directed graph[END_REF]. An example of TR is shown in Figure 1, in which the edge a-d was removed because there is a path from a to d passing through c or b. TR is useful in many applications, such as database optimization, model checking, and software verification [START_REF] Clough | Transitive reduction of citation networks[END_REF], [START_REF] Dubois | Transitive reduction for social network analysis and visualization[END_REF], [START_REF] Tang | One edge at a time: A novel approach towards efficient transitive reduction computation on dags[END_REF].

Many real-world graphs are not Directed acyclic graphs (DAGs), for which TR is well-defined. Besides, most of those graphs have cycles, such as social networks and transportation networks, and TR may not be applicable or helpful in these cases. This motivates our decision to use TR for temporal graphs. When working with temporal graphs, since one cannot come back in time, it is possible to avoid cycles and be able to use the TR algorithm. And, as it will be shown, avoiding redundancy can improve the accuracy of classification tasks.

The idea of temporal graphs (also known as dynamic, evolving, or time-varying graphs), was first introduced in 1997 [START_REF] Harary | Dynamic graph models[END_REF] discussing applications of temporal graphs and highlighting the great importance of a systematic treatment of the subject. The study in [START_REF] Mesbahi | On a dynamic extension of the theory of graphs[END_REF] provided tools and algorithms needed for controlling a system with dynamic graphs, the authors also studied on invariance and the reachability properties of these structures, proposing formalism leading to appropriate methods for dynamic graph problems. In an effort to integrate different notations found in the fields of delaytolerant networks, the authors in [START_REF] Casteigts | Timevarying graphs and dynamic networks[END_REF] introduced the definition of Time-Varying Graphs. Later, a lot of works studied how algorithms for statics graphs can be applied with dynamics ones, such as shortest path [START_REF] Wu | Path problems in temporal graphs[END_REF], traveling salesman [START_REF] Michail | Traveling salesman problems in temporal graphs[END_REF], and minimum spanning trees [START_REF] Huang | Minimum spanning trees in temporal graphs[END_REF]. It is interesting to note that temporal graphs are present in numerous domains, such as communication networks [START_REF] Tadić | Dynamics of directed graphs: the world-wide web[END_REF], [START_REF] Ozcan | Centrality and scalability analysis on distributed graph of large-scale e-mail dataset for digital forensics[END_REF], [START_REF] Oettershagen | Classifying dissemination processes in temporal graphs[END_REF], biological systems in cell and microbiology, neural networks [START_REF] Meng | Protein interaction networks: centrality, modularity, dynamics, and applications[END_REF], [START_REF] Paulevé | Reconciling qualitative, abstract, and scalable modeling of biological networks[END_REF], and economics [START_REF] Barunik | Dynamic networks in large financial and economic systems[END_REF], [START_REF] Nonejad | An overview of dynamic model averaging techniques in time-series econometrics[END_REF]. In the literature, a lot of works have focused on how to apply learning tasks to such temporal graphs. Classification of the outbreak of fake news disseminated on social networks, the proliferation of disease, prediction of rerouting traffic, and video classification are examples of such learning tasks that can be applied to temporal graphs. Hence, while there are existing techniques for classifying temporal graphs, it is necessary to note that many of these methods are computationally expensive. This work addresses this issue by reducing this complexity while maintaining reasonable accuracy.

Therefore, the major goal of our work is to analyze the impact of the TR algorithm on temporal graphs for a classification task on real-world data sets, providing a performance comparison of different techniques in terms of accuracy and computational efficiency across different classification tasks. Furthermore, we introduce a novel method for mapping temporal graphs to static ones based on TR that outperforms existing techniques in terms of classification tasks. Experimental results demonstrate the effectiveness of the proposed approach in improving the accuracy of supervised classification for temporal graphs while maintaining reasonable computational efficiency. To the best of our knowledge, this work is the pioneer in the merging directed line graph and TR for classifying temporal graphs and our main contributions are the following: (i) a comprehensive analysis of how the removal of transitive edges affects the classification performance of temporal graphs; (ii) introduction of a novel mapping method of temporal graphs to static ones that outperforms existing techniques in terms of classification accuracy and running time; and (iii) a comprehensive evaluation using graph kernels on real-world data sets.

This work is organized as follows. In Section II, some related works are described. Section III presents the main concepts related to graph theory that are necessary for understanding the proposed method for classifying temporal graphs, which is described in Section IV. A quite extensive evaluation is given in Section V. And finally, some conclusions are drawn in Section VI.

II. RELATED WORK

In [START_REF] Tannier | Evaluating temporal graphs built from texts via transitive reduction[END_REF], the authors focus on the problem of evaluating the relation between events in a given text, which is important for information extraction. They argue that finding a common comparison referent at the text level is not straightforward and propose a shift from event-based measures to measures on a minimal underlying temporal graph, which is the TR of the graph of relations between event boundaries. They support their proposal by investigating its properties on synthetic data and a well-known temporal corpus. The authors aim to accomplish two things: to find a graph that is easy to compute and to eliminate the bias introduced by measures that do not consider the combinatorial aspect of agreement on transitive closure graphs. In [START_REF] Clough | Transitive reduction of citation networks[END_REF], the authors proposed a method for analyzing directed acyclic graphs that take into account causality and highlight causal structure. The method is illustrated using citation networks from academic papers, patents, and US Supreme Court verdicts. The proposed approach is based on TR to remove unnecessary edges from a directed acyclic graph, which reveals the fundamental causal structure of the network. The authors have demonstrated how TR can identify differences in citation practices among different areas and can correct for the effect of a document's age on its citation count. Finally, the authors used TR to analyze null models of citation networks to illustrate the lack of causal structure in these models. Moreover, from the proposed method, it was possible to see that TR helped to reveal the causal skeleton of the network, after which standard network analysis tools may be used to analyze the network.

In [START_REF] Saramäki | Weighted temporal event graphs[END_REF], the authors have proposed a method for representing and analyzing temporal event data using weighted temporal event graphs, which are directed acyclic graphs where nodes represent events, and edges represent temporal orderings between the events. The weights of the edges capture the temporal distances between the events. To construct the graphs, the authors first constructed a directed graph of all possible temporal orderings between events. This graph can contain many transitive edges, which are edges that are implied by other edges in the graph and they used TR to simplify the graph and remove these transitive edges being easier to analyze and interpret. They also demonstrated the effectiveness of the weighted temporal event graphs approach on real-world datasets from social media and finance.

In [START_REF] Guidi | Parallel string graph construction and transitive reduction for de novo genome assembly[END_REF], the authors have addressed the challenge of de novo genome assembly (computational biology), which involves decoding the sequence of an unknown genome from short sequences. They introduced new distributed-memory parallel algorithms for overlap detection and layout simplification steps of genome assembly. The algorithms are based on linearalgebra operations over semirings using 2D distributed sparse matrices, which reduces the need for different data structures in different steps of genome assembly. The proposed approach includes a novel distributed memory algorithm for the TR of the overlap graph, which simplifies the graph and makes it easier to resolve inconsistencies and create contigs.

Most related works have specific applications and datasets in mind, such as information extraction from text, citation networks, and social media/finance data. In contrast, this work aims to provide a more generalized approach applicable to a wider range of classification tasks on real-world datasets. This includes the exploration of mapping temporal graphs to static graphs and assessing the impact of transitive reduction, which surpasses existing techniques in terms of classification accuracy.

III. FUNDAMENTAL CONCEPTS

Several definitions to model formally discrete temporal graphs are proposed in the literature. In [START_REF] Xuan | Computing shortest, fastest, and foremost journeys in dynamic networks[END_REF], timed evolving graphs are defined as a system G ′ composed by a graph G = (V, E), an ordered sequence of its subgraphs

S g = (G 1 , G 2 , • • • , G t ), then G ′ = (G, S g ).
The edge weights represent the traversal time. This notation is useful when one wants to predict the topology dynamics at different time intervals since the paths are restricted to never move into edges that existed only in past subgraphs. Similar to the previous definition, the authors in [START_REF] Tang | Temporal distance metrics for social network analysis[END_REF] defined a temporal graph by a sequence of time windows of snapshots of the network at that time interval. This work uses the definition proposed in [START_REF] Oettershagen | Classifying dissemination processes in temporal graphs[END_REF] considering temporal graphs with edges existing at specific integral points in it and node labels. This definition allows us to model the problem, transforming it into static graphs to apply transitive reduction.

State-of-the-art techniques for temporal graph classification make use of graph kernels to compare different graphs. When graphs are represented as sets of features, such as node or edge attributes, important structural information about the graphs is lost, such as the presence of cycles, paths, or other higherlevel patterns. Graph kernels, on the other hand, capture this structural information by computing a similarity score between pairs of graphs based on the graph structure, without requiring explicit feature representations. However, classical graph kernels can only be applied to static graphs. For temporal graph classification, temporal graphs are first mapped into static graphs before applying the kernels for classification. There are some mapping methods shown in [START_REF] Kostakos | Temporal graphs[END_REF], [START_REF] Michail | An introduction to temporal graphs: An algorithmic perspective[END_REF], and [START_REF] Wu | Path problems in temporal graphs[END_REF]. One of the best methods, in terms of temporal graph classification, to map temporal graphs into static graphs is called Direct Line Graph [START_REF] Oettershagen | Temporal graph algorithms[END_REF].

The fundamental concepts about temporal graphs, graph kernels, transitive reduction, and directed line graph are given in this section.

A. Temporal Graph

Let G = (V, E) be an undirected (static) graph in which V is a finite set of vertices and E a finite set of undirected edges defined by

E ⊆ {{u, v} ⊆ V | u ̸ = v}. A labeled, undirected (static) graph (G, l
) is a pair of an undirected graph and a labeling function l : V ∪ E → Σ that assigns a label to each vertex or edge of G, in which Σ is a finite alphabet. In a directed graph the set of edges E is defined by

E ⊆ {(u, v) ∈ V × V | u ̸ = v}. A (static) walk in a graph G
is an alternating sequence of vertices and edges connecting consecutive vertices, but for simplification of the notation, we omit the edges. The length of a walk

(v 1 , v 2 , • • • , v k+1 ) is k.
Now, let G = (V, E) be a temporal graph in which V is a finite set of vertices and E is a finite set of undirected temporal edges e = ({u, v}, t) with u and v in V , u ̸ = v and the availability time (or time stamp) t ∈ N. A labeled, undirected, temporal graph G = (V, E, l ′ ) consists of a temporal graph G = (V, E) and a labeling function l ′ : V ∪ T → Σ that assigns a label to each vertex at each time step t ∈ T = 1, . . . , t max + 1 with t max being the largest timestamp of any e ∈ E.

For a temporal graph the number of edges is not polynomially bounded by the number of vertices. A temporal walk of length k is an alternating sequence of vertices and temporal edges

(v 1 , e 1 = (v 1 , v 2 , t 1 ), v 2 , • • • , e k = (v k , v k+1 , t k ), v k+1 ) such that t i < t i+1 for 1 ≤ i < k. Moreover, for a temporal walk, the waiting time at vertex v i with 1 < i ≤ k is t i -(t i-1 + 1). The set of temporal walks (of length k) in a temporal graph G is denoted by W tmp (G) (W k tmp (G))
. Finally, we define the function L that maps a temporal walk w to the label sequence

L(w) = (l(v 1 , t 1 ), l(v 2 , t 1 + 1), l(v 2 , t 2 ), l(v 3 , t 2 + 1), • • • , l(v k , t k ), l(v k+1 , t k + 1)).

B. Transitive Reduction

Let G = (V, E) be a directed graph with vertex set V and edge set E. A (directed) path from vertex u to vertex v in a graph G is a walk from u to v without repetition of vertices. The TR of G is another directed graph G ′ = (V, E ′ ) such that:

• G ′ has the same vertex set V as G.

• For every pair of distinct vertices u, v ∈ V , if there is a directed path from u to v in G, then there is also a directed path from u to v in G ′ . • G ′ has the minimum number of edges among all graphs satisfying the first two conditions. To define E ′ , we first define the relation R on V as follows: for u, v ∈ V , uRv if and only if there is a directed path from u to v in G. Then, the transitive closure of R is the smallest transitive relation R ′ on V such that R ⊆ R ′ . We can define the edge set E ′ = {(u, v) ∈ E : u¬R ′ v}. In other words, E ′ contains only the edges in E that are not necessary for maintaining the reachability relation in G. The resulting graph G ′ = (V, E ′ ) is the transitive reduction of G [START_REF] Aho | The transitive reduction of a directed graph[END_REF]. The transitive reduction is well-defined only for DAGs.

The complexity of the transitive reduction algorithm depends on the size and structure of the input graph. Generally, the algorithm has a worst-case time complexity of O(n 3 ), in which n is the number of vertices in the graph. However, several optimizations and heuristics can be applied to improve the performance of the transitive reduction algorithm in practice. For example, the algorithm can be modified to use a breadth-first search instead of a depth-first search, which reduces the worst-case time complexity to O(n 2 log n) for sparse graphs. In addition, the algorithm can be parallelized to take advantage of multi-core processors, or specialized hardware such as GPUs, to further improve performance. A comparison (considering time and space complexity) of several algorithms on TR computation is presented in [START_REF] Tang | One edge at a time: A novel approach towards efficient transitive reduction computation on dags[END_REF].

C. Graph Kernels

A graph kernel is a function k : G × G → R that maps pairs of graphs to a real number representing the dissimilarity between two graphs. Graph kernels need to have some properties:

(i) symmetry -k(G 1 , G 2 ) = k(G 2 , G 1 ) for all graphs G 1 and G 2 ; (ii) positive semi-definiteness -k(G 1 , G 2 ) ≥ 0 for all pairs of graphs G 1 and G 2 , and k(G 1 , G 2 ) = 0 if and only if G 1 = G 2 ; and (iii) compositional -for any function f : V → R, the kernel k f (G 1 , G 2 ) = u,v∈V f (u)f (v)k G (u, v) is also a valid kernel,
where k G (u, v) is the kernel value between nodes u and v in the graphs G 1 and G 2 . We briefly summarize two well-known kernels for static graphs.

The Graphlet kernel computes the similarity between two graphs counting the occurrences of graphlets of different sizes. A k-graphlet is a connected subgraph H of G with k vertices, denoted by H ∈ G k . More formally, let G be a graph with vertex set V (G) and edge set E(G), and let G k be the set of all k-graphlets of G. The graphlet degree vector of G is a vector of counts of each k-graphlet in G, denoted by

g k (G) = (g 1 (G), g 2 (G), ..., g |G k | (G)).
The graphlet kernel between two graphs G and H is then defined as the inner product of their graphlet degree vectors:

K(G, H) = k Φ(k)g k (G) • g k (H) (1)
in which Φ(k) is a weighting function that assigns different weights to graphlets of different sizes, and • denotes the inner product between two vectors, see [START_REF] Shervashidze | Efficient graphlet kernels for large graph comparison[END_REF] for more. The Weisfeiler-Lehman subtree kernel is a kernel function that computes the similarity between two graphs based on their shared subtrees. The Weisfeiler-Lehman subtree kernel proceeds iteratively, refining a labeling function for the vertices of each graph using information about the local neighborhood of each vertex [START_REF] Schulz | A generalized weisfeiler-lehman graph kernel[END_REF]. At each iteration t, the labeling function h t : V → N k maps each vertex v ∈ V to a vector of k integer labels that encode the frequency of certain local subtrees in the neighborhood of v.

k W L (G 1 , G 2 ) = ∞ t=0 α t v∈V h (1) t (v) • h (2) t (v) (2)
in which h

t (v) and h

t (v) are the label vectors for vertex v in graphs G 1 and G 2 at iteration t, respectively. The parameter α is a damping factor that controls the weight given to iterations at different depths in the subtree hierarchy.

D. Directed Line Graph

A line graph of a static graph G is a graph whose vertices are the edges of G that are connected if they share a vertex in G. The authors in [START_REF] Oettershagen | Classifying dissemination processes in temporal graphs[END_REF] used DL to encode the temporal information in a study of the dissemination process. Each temporal edge is represented by two vertices and a temporal walk can be performed, since a walk in the DL graph is related to the temporal walks in the original temporal graph having the same label sequence, being able to model waiting times and keeping the temporal information. Following [START_REF] Oettershagen | Classifying dissemination processes in temporal graphs[END_REF], Directed line graph expansion (DL) can be defined as follows: given a temporal graph (G, l), the directed line graph expansion

DL(G, l) = (G ′ , l ′ ) in which G ′ = (V ′ , E ′ )
l ′ n t -→ uv = (l(u, t), l(v, t + 1)
) is set. Figure 2 shows an example of the directed line transformation. The edge value is related to waiting time: if the previous node edges are consecutive (e.g., edges 2 and 3 in Figure 2a), then there is no waiting time between edges, represented by value 1, otherwise, 0 is used.

IV. TRANSITIVE REDUCTION ON TEMPORAL GRAPHS

For decreasing the size of graphs and keeping just one path between vertices, if there exists, we can apply transitive reduction. Here, we will study the impact of TR on temporal graphs. For that, temporal graphs are mapped to static graphs so that conventional static kernels, for example, graphlet or Weisfeiler-Lehman subtree kernel can be applied.

In [START_REF] Oettershagen | Temporal graph algorithms[END_REF], the authors have proposed different methods for mapping temporal graphs into static ones. The DL approach outperformed other methods capable of fully encoding the temporal information. As in the DL transformation the edges are vertices, and the number of vertices is |E|. The maximal number of edges is reached when each vertex of the original temporal graph for every incoming edge can be combined with all outgoing edges [START_REF] Oettershagen | Temporal graph algorithms[END_REF], then the number of edges in the DL is O(|E| 2 ) leading to a quadratic blowup with regard to the number of temporal edges. Proposition 1 shows that the conversion of a temporal graph to DL leads to a DAG, then we are able to apply the TR algorithm to deal with the size of edges.

Proposition 1 (Directed line graph): Let (G, l) be a temporal graph. The resulting DL(G, l) is a Directed acyclic graph (DAG). Proof: Suppose there exists a directed cycle in DL(G, l). Let n t -→ uv be a vertex on the cycle with the earliest timestamp t. Let n s -→ xy be the last vertex of the cycle that connects n t -→ uv . Since there is a path from n s -→ xy to n t -→ uv , we know that y = u and s > t. However, n t -→ uv represents a temporal edge (u, v, t), and n s -→ xy represents a temporal edge (x, y, s). Therefore, u = y = x and t > s. But this contradicts the fact that n t -→ uv has the earliest timestamp on the cycle. Therefore, there are no directed cycles in DL(G, l), and DL(G, l) is a DAG. ■ To the best of our knowledge, this work proposes a novel methodology for temporal graph classification that makes use of directed line graph and TR. Figure 3 schematically represents the different pipelines used in this work to perform the classification task, yiedling three possibilities. The first one, called baseline [START_REF] Oettershagen | Temporal graph algorithms[END_REF], is represented by the gray blocks in Figure 3, in which temporal graphs are mapped to static graphs using the DL transformation. Then, the normalized Gram Matrix is computed using two graph kernels (Graphlet and Weisfeiler-Lehman subtree kernel) for the learning task. The second one, called TR-based method, is represented by gray and red blocks in Figure 3, we propose to apply TR after the DL transformation in order to reduce the size of these graphs. The impact of TR on classification accuracy and running time compared to baseline is analyzed in the experimental results. Finally, we propose also a method, called DLTR-based method (illustrated by yellow blocks), which is an extension of the TR-based method that combines the DL and TR methods. We detail below how this extension is realized.

There are different ways to remove transitive edges, but the TR algorithm used in the TR-based method removes all transitive edges, which is more complex, performing a depth-first search or breadth-first search on the graph, and computationally expensive for large graphs. Additionally, the algorithm requires the use of data structures such as stacks or queues to keep track of the visited nodes and the order in which they were visited. The concept in the DLTR-based method is to remove a certain amount of transitive edges using the simplest implementation of the TR algorithm. The basic idea of the TR algorithm is that for each set of three nodes (x, y, z) ∈ V ′ if there is a path xy, yz we can remove xz if it exists. However, as shown in Proposition 2, it is impossible to have transitive edges with only three nodes, it is only possible with at least four nodes, as in Figure 4b. Having more than three loops can be time-consuming and does not guarantee the removal of transitive edges.

Proposition 2: Let DL(G, l) a directed line expansion of a temporal graph (G, l). If there exists a transitive edge in DL(G), this transitive edge involves at least four consecutive nodes in DL(G, l). and DL conversion with temporal graph smoothing (c). The dotted lines represent transitive edges that can be removed.

Proof: Let v 1 = n t -→ uv , v 2 = n s -→ vx and v 3 = n p -→ xv with t < s < p be 3 connected vertices, v 1 , v 2 , v 3 ∈ V ′ and u, v, x ∈ V .
To create a transitive edge we need to connect v 1 with v 3 . However, this leads to a contradiction, as v 1 corresponds to a temporal edge (u, v, t) and v 3 corresponds to a temporal edge (x, v, p) and they are not consecutive in (G, l) since the temporal edge v 3 starts with vertex x not v. To be able to construct a transitive edge in DL(G, l) we need to create a temporal edge starting with v and ending with u with a time higher than p. With this, we create a vertex v 4 = n n -→ vu , n > p connected to v 3 and now we are able to create a transitive edge from v 1 to v 4 . ■ To deal with this, we have used graph simplification (that we denote temporal graph smoothing), adapted from graph smoothing. Graph smoothing, also known as smoothing away or smoothing out, is the process of replacing edges incident at a vertex of degree 2 by a single new edge and removing the vertex. When running the DL algorithm, each temporal edge is transformed into a node. Before creating the node, for each edge n t -→ uv we check if there exists a n p -→ vu with p > t. This means that, instead of creating two consecutive nodes, we create just one that corresponds to the last temporal edge n p -→ vu . The label of a node in the directed line graph is a combination of the labels of its corresponding edges in the temporal graph as shown in section III-D. Maintaining the same idea, the node label that corresponds to the first temporal edge is combined with the one of the last node. For example, in Figure 4c the nodes ab and ac were not created since for ab, ba exists and for ac, ca exists.

The neighbors' edges from the first temporal edge are propagated to the last one to keep the structure. The result is a compact graph (see Figure 4c), and the TR algorithm can be applied to remove the transitive edges (dotted line in Figure 4c) in a much faster way. In the next section, we will study the impact of these two novel methods (TR-and DLTRbased methods) on both classification accuracy and efficiency.

V. EXPERIMENTS

In this Section, we describe the experiments and the obtained results taking into account real-world datasets.

A. Datasets

In order to provide a comparative analysis between the different strategies described before, we have used six different databases provided by TUDataset [START_REF] Morris | Tudataset: A collection of benchmark datasets for learning with graphs[END_REF]. This benchmark provides temporal graph classification datasets derived from Tumblr, Dblp, Facebook as well as contacts between students at MIT, in a Highschool, and visitors at the Infectious exhibition, for dissemination process study. For each dataset, a dissemination process simulation was done, in which nodes are infected at different time instants, providing two classification tasks for each dataset. The first task involves discriminating between temporal graphs with vertex labels resulting from a dissemination process and those without. To accomplish this, the authors ran a Susceptible-infected (SI) simulation with fixed parameters on half of the dataset and used it as the first class. The second class was made up of the remaining graphs. For each graph in the second class, the authors counted the number of infected vertices, reset the labels, and then randomly infected a number of vertices at a random time. The second task involves discriminating between temporal graphs that differ in the dissemination process itself. For this task, the authors ran the SI simulation with different parameters for each of the two subsets. For both subsets, I = 0.5 (initial infection rate), but for the first subset, the authors set p = 0.2 (infection probability), and for the second subset, p = 0.8. The simulation runs repeatedly until at least |V | × I vertices are infected or no more infections are possible, for example, if a graph has 100 vertices and the initial infection rate is set to 0.5, then initially 50 vertices are infected. In order to stop the SI simulation, at least 50 × 0.5 = 25 vertices (i.e., 50% of the total number of vertices) need to be infected. The simulation continues until either this condition is met or no more infections are possible.

B. Graph Kernels

As a baseline we use the 3-node Graphlet (GL) and the Weisfeiler-Lehman subtree (WL) kernels on static graphs obtained by interpreting the timestamps as discrete edge labels, and assigning to each vertex the concatenated sequence of its labels. The source code is provided by [START_REF] Morris | Tudataset: A collection of benchmark datasets for learning with graphs[END_REF].

C. Experimental Setup

The normalized Gram matrix was calculated for each kernel and then we used the C-SVM implementation of LIB-SVM [START_REF] Chang | Libsvm: a library for support vector machines[END_REF] to determine the classification accuracies. We performed 10-fold cross-validation to select the C parameter from the range of 10 -3 , 10 -2 , ..., 10 2 , 10 3 on the training folds. We repeated the 10-fold cross-validation ten times with different random folds to obtain the average accuracies and standard deviations. The number of iterations for the Weisfeiler-Lehman subtree kernel (from 0 to 5) was selected through 10-fold cross-validation. A NetworkX implementation of a transitive reduction will be used in the TR-based method [START_REF] Hagberg | Networkx[END_REF].

D. Evaluation and Discussion

Table I shows that TR algorithm improved the classification accuracy. This improvement is more significant when the dataset has a bigger quantity of edges (Highschool and MIT). Reducing the graph size can help to eliminate noise and irrelevant information from the data, making it easier for the SVM to identify the patterns and features that are truly relevant to the classification task. This can lead to a more accurate classification because the SVM is working with a more focused and relevant set of features. Additionally, reducing the graph size can also reduce the risk of overfitting, which occurs when a model becomes too complex and starts to fit the noise in the data instead of the underlying patterns [START_REF] Procházka | Scalable graph size reduction for efficient gnn application[END_REF]. Overfitting occurs when a model fits too closely to the training data and fails to generalize to new data. In the case of graph kernels, the size and complexity of the graph can affect the performance of the kernel function, and a smaller graph can improve the kernel's ability to generalize to new data by reducing the number of possible patterns or subgraphs that TABLE I: Classification accuracy in percent and standard deviation for the first classification task. DL, is only the directed line approach, DL + TR, is the transitive reduction algorithm applied after the directed line transformation and DLTR, is the TR algorithm applied at the same time of DL. For the second classification task, Table II shows that reducing the size of the graph also improves accuracy, but when we do the temporal smoothing process, the graphs lose some relevant information about the graph structure for classification. However, the accuracy of the DLTR-based method is very similar to the baseline when the overall quantity of edges is not high, leading to consider a slight loss of accuracy. In general, the second classification task poses a greater challenge for the temporal approaches that reach lower accuracy than the first classification task. Especially, the MIT data set seems to be hard. In Figure 5 we can see that TR improved accuracy compared to the baseline and when compared to the DLTRbased (see the third plot), the accuracy is quite similar, gaining in running time.

As we can see in Table III, the size of the graphs decreases when the transitive reduction algorithm is applied, which was expected. The reduction in size is most significant in the DLTR-based method, in which both the line graph conversion and transitive reduction are applied simultaneously. In the TR-based method, the reduction in edge count varies depending on the dataset, but it is significant in the case of the Tumblr and Highschool dataset, where the number of edges decreased from 412,892 to 205,357 and from 2,079,062 to 360,895. This is a reduction of almost 50% and 83% in edge count, respectively. The reduction in the number of vertices is zero, as transitive reduction does not affect the number of vertices in a graph. Compared to the TR-based method, the DLTR-based method resulted in a further reduction in the number of edges and vertices for all datasets. The reduction in the number of edges is most significant in the case of the Highschool dataset, where the number of edges decreased from 360,895 to 252,081, which is a further reduction of about 30%. The reduction in the number of vertices is also significant for all datasets, with reductions ranging from about 3% to almost 70% for the MIT dataset. Overall, with the DLTR-based method, we can significantly reduce their size while still maintaining their essential properties for classification tasks. The reduction in size can be more or less significant depending on the dataset, but it is generally more significant for larger graphs.

Table IV shows that the time spent to perform transitive reduction separately is significantly longer than the time necessary to convert the temporal graph to a directed line graph in both baseline and DLTR-based. This is particularly evident in the Facebook dataset, in which transitive reduction takes over 9000 milliseconds while baseline and DLTR-based take only 90 and 248 milliseconds, respectively. However, it is important to note that the TR-based method takes longer than the baseline and DLTR-based method in all datasets. This is likely due to the additional step of performing transitive reduction on the already converted directed line graph. Overall, we can see that the DLTR-based method is the most efficient in terms of both accuracy and time, as it combines the conversion and transitive reduction steps, resulting in faster processing time and improved accuracy.

In Table V, the time required for the classification step of the three methods is presented. One can see that the DL-GL kernel is faster than the DL-WL kernel for all datasets and all methods. This happens because the Weisfeiler-Lehman kernel involves the computation of the neighborhood of nodes and updating labels iteratively, while the graphlet kernel only counts the number of isomorphic graphlet subgraphs, which could also contribute to the speed difference. Regarding the running times, the DLTR-based method is generally slower than the other two methods, which is also expected as it involves an additional step of transitive reduction and temporal graph smoothing. However, the difference in running times between the TR-based method and the DLTR-based method is not significant in most cases, except for the MIT dataset in which the DLTR-based method is much faster than TRbased. Overall, we can conclude that the DLTR-based method offers improved accuracy at the cost of longer running times compared to other methods, but the difference in running times is not significant in most cases. improved classification accuracy, especially for larger graphs, since real-world graph datasets can be incredibly large and complex, containing millions or even billions of nodes and edges like transportation networks, financial transaction networks, and communication networks. The proposed method achieved similar or better accuracy compared to the stateof-the-art methods while also being computationally more efficient. Moreover, the proposed method provides a significant reduction in the size of the graph, making it more manageable for further analysis and visualization. Finally, this work contributes to the development of graph-based machine learning methods for temporal graph data and can be applied to a wide range of temporal graph data sets, such as social networks and communication networks, to improve their classification performance. Thus, this work's findings have the potential to benefit a wide range of applications, from predicting disease outbreaks to detecting online fraud. Future research opportunities in this area include practical implementation and application of different transitive reduction techniques to determine their real-world performance. Additionally, exploring the effectiveness of the method on graphs with multiple edge types or different structures and investigating its potential for classification tasks in temporal graph databases, such as video classification, where each video can be modeled as a temporal graph, can be valuable. In conclusion, the method proposed is a promising avenue for future research on temporal graph classification and has the potential to be further optimized and improved in future studies.

Fig. 2 :

 2 Fig. 2: Example of directed line transformation. The walk (n 2 ca , n 3 ab , n 7 bc ) of length 2 in (b) corresponds to the temporal walk (c,(c, a, 2), a,(a, b, 3), b,(b, c, 7), c) of length 2 in the temporal graph (a) [21].

Fig. 3 :

 3 Fig. 3: Overview of the different temporal graph classification pipelines followed in this work.

Fig. 4 :

 4 Fig. 4: Example of temporal graph (a), DL conversion in (b) and DL conversion with temporal graph smoothing (c). The dotted lines represent transitive edges that can be removed.

Fig. 5 :

 5 Fig. 5: Scatter plots to show the accuracy relationship between two different methods. The dots are accuracy in different datasets and using different graph kernels for classification.

Fig. 6 :

 6 Fig. 6: Temporal Graph Classification time of Highschool data set (ms).

  is the directed graph, where every temporal edge ({u, v}, t) is represented by two vertices n

t -→ uv and n t -→ vu and there is an edge from n t -→ uv to n s -→ xy if v = x and t < s. For each vertex n t -→ uv , the label

TABLE II :

 II Classification accuracy in percent and standard deviation for the second classification task. can be learned from the training data. MIT is a large dataset with more than 62 million edges. When we tried to run the Baseline and TR-based method we got OOM (Out of memory), being able to run just the DLTR-based method.

				Datasets
	Kernel	Highschool Infectious	Tumblr	Dblp	Facebook	MIT
	Baseline
	DL-GL	91.05±1.9	88.10±1.9 78.56±1.5 79.68±0.7 75.86±0.4	OOM
	DL-WL	87.16±1.0	80.25±1.6 75.20±2.1 77.18±0.9 80.35±0.7	OOM
	DL + TR-based
	DL-GL	94.61±1.0	83.85±1.8 80.15±1.1 80.09±0.7 74.35±0.3	OOM
	DL-WL	90.61±1.5	81.00±1.2 77.85±0.8 79.77±0.8 82.16±0.7	OOM
	DLTR-based
	DL-GL	88.22±0.8	82.45±1.6 79.91±1.4 78.33±0.6 72.81±0.5 60.61±2.7
	DL-WL	89.11±1.1	79.90±2.4 79.04±0.9 77.27±0.4 77.20±0.6 59.91±4.2

TABLE III :

 III Graph size in terms of the overall quantity of Vertices and Edges.

				Datasets
	Size	Highschool Infectious Tumblr	Dblp	Facebook	MIT
	Baseline
	Sum |E|	2,079,062	918,513	412,892 1,097,872	851,165	62,923,589
	Sum |V |	98,066	91,944	74,520	241,674	267,673	142,508
	DL + TR-based
	Sum |E|	360,895	380,920	205,357	777,212	516,631	-
	Sum |V |	98,066	91,944	74,520	241,674	267,673	142,508
	DLTR-based
	Sum |E|	252,081	436,015	165,119	839,755	531,431	170,080
	Sum |V |	69,197	71,157	63,971	196,154	234,983	74,801

TABLE IV :

 IV Mapping time in milliseconds for the baseline, the TR-based method plus TR and DLTR-based method running time.

				Datasets
		Highschool	Infectious	Tumblr	Dblp	Facebook	MIT
	DL + (TR) 126 + (31510) 68 + (15398) 28 + (2380) 115 + (4737) 90 + (9134) 4939 + (OOM)
	DLTR	147	183	62	298	248	522

TABLE V :

 V Time for classification performance of the three methods, in seconds.

				Datasets
	Kernel	Highschool Infectious Tumblr Dblp Facebook MIT
	DL
	DL-GL	33.07	9.08	4.67	12.44	16.36	-
	DL-WL	13.57	9.22	11.41	32.84	56.06	-
	DL + TR
	DL-GL	2.59	3.07	2.74	9.89	13.57	-
	DL-WL	7.68	7.39	10.48	32.53	54.25	-
	DLTR
	DL-GL	2.29	4.16	2.80	10.56	12.77	1.60
	DL-WL	6.39	7.01	10.04	30.98	55.38	5.38

-CNPq -(PQ 306573/2022-9 and Universal 407242/2021-0), Coordenac ¸ão de Aperfeic ¸oamento de Pessoal de Nível Superior -CAPES -(Grant COFECUB 88887.191730/2018-00) and Fundac ¸ão de Amparo à Pesquisa do Estado de Minas Gerais -FAPEMIG -(Grants PPM-00006-18). a b c d e (a) a b c d e (b)


Moncef Ghiss

Othman Ahmed

Mohamed Amine Chetoui

Da Anabela

Olivier Silva

Boiron

The impact of flexion mechanical loading on the intervertebral disc metabolism


Fadel Tarhini

Reine Talj

Moustapha Doumiati

Multi-Objective Control Architecture for an Autonomous In-wheel Driven Electric Vehicle ⋆

Keywords: Global Chassis Control, Torque Allocation, In-wheel motors, Super-Twisting Sliding Mode, Autonomous Vehicle

This paper investigates the high-level control of an in-wheel-motor-drive autonomous electric vehicle. Four distinct objectives are achieved, including lateral and longitudinal control, as well as stability and maneuverability control. Actuators designated at the low level are active front steering specified for the lateral control and the 4 independent in-wheel motors for the remnant objectives. Stability and maneuverability are realized using the direct yaw control by distributing driving and braking torques among the motors, along with the longitudinal control within a torque allocation unit. In contrast to critical situations, maneuverability is promoted while relaxing the stability objective during normal driving situations. Hence, a decision layer is developed to coordinate the stabilizing and maneuvering objectives on the same actuator, thus a multi-layer Global Chassis Control (GCC) architecture is established. The control architecture is tested and validated within a MatLab/Simulink environment. Simulation results carried out on a full nonlinear vehicle model emphasize the objectives' achievement and demonstrate the superior performance of such system.

Introduction

Road safety is a major issue in our community. According to the "National Highway Traffic Safety Administration", human errors contribute to more than 90% of road accidents [START_REF] Rajamani | Vehicle Dynamics and Control[END_REF]]. After the development of increasingly sophisticated driver assistance systems (ADAS) to support the driver, studies are moving towards the autonomous vehicle (AV) to improve road safety and free the driver. Vehicle autonomy is accomplished in consecutive steps including perception, planning, and control. Taking perceptual data as inputs, and considering a preplanned trajectory, the objective of attaining a level of vehicle autonomy lies in controlling the lateral dynamics that ensures a trajectory following. [START_REF] Tagne | Design and comparison of robust nonlinear controllers for the lateral dynamics of intelligent vehicles[END_REF]] proposed to control the lateral acceleration to track a reference one which depends on vehicle longitudinal velocity and road curvature, while the yaw rate is controlled in [START_REF] Atoui | Design and experimental validation of a lateral lpv control of autonomous vehicles[END_REF]] to follow a constructed reference. As an approach to couple the lateral and longitudinal dynamics, [START_REF] Chebly | Coupled longitudinal/lateral controllers for autonomous vehicles navigation, with experimental validation[END_REF]] proposed to follow a trajectory while controlling the vehicle's longitudinal velocity. Following the global energy crisis, electric vehicles (EVs) have attracted wide range of studies in both academia and industry, owing to their ability to run entirely on renewable energy. A heavily researched category of EVs named In-Wheel-Motor-Drive EVs (IWM), have the capabilities to dominate the industry. IWM EV is an overactuated system allowing an independent control of each motor, for both traction and braking purposes. Compared with the ⋆ This work was supported by the Agence Nationale de la Recherche (ANR) traditional center-traction vehicle, IWM EV has shown a better performance [START_REF] Zou | An integrated path-following and yaw motion control strategy for autonomous distributed drive electric vehicles with differential steering[END_REF]] and high potential for reducing the energy consumption [START_REF] Wang | Optimal component sizing of a four-wheel independently-actuated electric vehicle with a real-time torque distribution strategy[END_REF]]. Recently, seeking for better performance and higher efficiency, research is underway to integrate in-wheel motors into the autonomous vehicles. For the IWM AVs, [START_REF] Peng | Path tracking and direct yaw moment coordinated control based on robust mpc with the finite time horizon for autonomous independent-drive vehicles[END_REF]] proposed a robust MPC with finite time horizon for attaining path tracking while coordinating Direct Yaw Control (DYC) to minimize tire usage. [START_REF] Jeong | Integrated path tracking and lateral stability control with four-wheel independent steering for autonomous electric vehicles on low friction roads[END_REF]] present an integrated path tracking using pure pursuit while achieving lateral stability using DYC. An extension of the potential field method from mobile robotics to the on-road autonomous vehicle is done in [START_REF] Li | A potential field approach-based trajectory control for autonomous electric vehicles with in-wheel motors[END_REF]] to minimize the lateral, longitudinal velocity and yaw angle errors for the purpose of trajectory control. The electric vehicle's control can be divided into two levels based on the structure of its actuators. Several control objectives can be ensured at the high level by intelligent coordination between the actuators at the low level, depending on diverse control strategies. A Lyapunov-based control technique is used to ensure stability and maneuverability control objectives at the high level, and realized at the low level based on a dynamic load distribution strategy in [START_REF] Laghmara | Yaw moment lyapunov based control for in-wheel-motor-drive electric vehicle[END_REF]]. [START_REF] Chokor | A comparison between a centralized multilayer l p v / h and a decentralized multilayer sliding mode control architectures for vehicle's global chassis control[END_REF]] present a comparison between a centralized approach based on LP V /H ∞ and a decentralized one using Sliding Mode technique, in order to control the side-slip angle, the yaw rate and the roll angle, using DYC and Active Front Steering (AFS). Authors in [Wang et al. (2018a)] present a coordination control strategy between the differential drive assist steering (DDAS) and vehicle stability control (VSC). Control systems presented in the literature oftenly aim at achieving one or two of the following objectives: lateral, longitudinal, stability, and maneuverability. In the framework of the autonomous in-wheel motor drive electric vehicle, a novel multi-layer GCC architecture is developed in this paper, considering four separate high-level objectives stated as: lateral, longitudinal, stability, and maneuverability control. A decentralized approach is adopted using the Super-Twisting Sliding Mode Control (STSMC) technique. The developed bottom-up architecture is aided with a decision layer to coordinate between stability and maneuverability. In order to distinguish the superior performance of the in-wheel vehicle, two strategies are compared for the torque distribution among the four motors. The contributions of this research work is combining the four objectives in a global integrated control architecture. Using map-matching, trajectory following is achieved by AFS. Longitudinal control assists with a comfortable drive and enhance the path tracking. A supervision layer is developed to switch between stability and maneuverability using DYC. Longitudinal, stability and maneuverability control objectives are achieved by a torque allocation strategy through the four in-wheel motors. The rest of the paper is structured as follows. Section 2 introduce the developed GCC architecture focusing on the high-level control and presenting the contrasted strategies at the low-level. Results and their discussion are presented in Section 3, while Section 4 concludes the paper work.

2 Multi-Layer GCC Architecture

Vehicle Model

A full non-linear model is adopted from a previous work realized in [START_REF] Chokor | Active suspension control to improve passengers comfort and vehicle's stability[END_REF]]. It consists of combining sub-models representing the vertical, lateral/longitudinal, and the wheel/road contact dynamics. The control of such system is accomplished by forcing the vehicle parameters to converge to a linear characteristics. The well-known bicycle model is used for this purpose. It is a 2 DOF dynamic model for the lateral motion of the vehicle, represented by the vehicle side-slip angle β and the vehicle yaw rate ψ. Its state-space representation is given in (1), where δ c is the controlled steering angle subjected to the vehicle autonomous driving.

ψbic βbic =   -µ l 2 f Cf +l 2 r Cr IzVx µ lrCr-lf Cf Iz -1 + µ lrCr-lf Cf M V 2 x -µ Cr+Cf M Vx   ψbic β bic + µ lf Cf Iz 1 Iz µ Cf M Vx 0 δ c M z
(1) where ψbic and β bic are the yaw rate and the side-slip angle of the bicycle model respectively and V x is the vehicle longitudinal velocity. Parameters description and values are given in Table 1.

a y ≈ V x ( ψ + β) ≤ µg (2a) ψref,max = 0.85 µg V x (2b)
Lateral stability can be related to the lateral acceleration a y , which is limited by the road adherence. Hence, authors in [START_REF] Rajamani | Vehicle Dynamics and Control[END_REF] propose to maintain a y below a threshold depending on the maximal possible adherence (2a) by saturating the yaw rate ψ (2b).

Decentralized GCC Architecture

The vehicle is a complex multi-variable nonlinear system, forcing its control to be extremely difficult. Centralized

Decision Layer

Performance Monitor SI (β, ሶ β)

Control Layer

Reference Linear Bicycle Model

δ c V x ሶ ψ 𝑏𝑖𝑐 STSM ሶ ψ SISO STSM λ ሶ ψ λ β (β, ሶ β) (β, ሶ β) bic

Actuator Layer

In-wheel Motors Actuators AFS and decentralized control came to serve the purpose of controlling the chassis variables by one or several controllers [START_REF] Chen | Integrated vehicle dynamics and control[END_REF]]. In the decentralized control, each variable reflecting a desired objective, is controlled independently using a single control input in a sub-controller. Whereas in the centralized control a unique controller generates the control inputs all together in order to control all the variables of interest. The control variables are all coordinated through the decision layer which anticipates the vehicle situations, thus a multilayer architecture is developed. The main idea behind developing a multilayer control architecture is to decompose the global architecture into several layers (Fig 1). At the middle layer, control commands are generated to ensure the desired performance on the vehicle. The objectives in this work are: lateral control for trajectory following, longitudinal control by tracking a real-time defined velocity profile, and to control the stability and maneuverability of the vehicle.

The developed architecture is aided with a decision layer that monitors the driving situation based on the lateral stability index (SI), to generate time-varying scheduling parameters λ ψ and λ β . Based on these values, the control objectives of stability and maneuverability are promoted/attenuated. In the control layer, three decentralized SISO controllers are presented to independently control the output variables V x , ψ, β and the lateral error e y . A full non-linear vehicle model is used to generate the vehicle's parameters including V x , ψ, β, β, while a reference linear bicycle model is developed to assist the parameters to converge to a familiar linear characteristics ( ψbic , β bic , βbic ). A map-matching module is used to localize the vehicle on a map and determine its lateral error (e y ) with respect to a trajectory formed of recorded data points, and generate a speed profile V x des based on the road curvature and a lateral acceleration criterion. At the actuator layer, the motor actuator of AFS generates a control steering δ a c to track δ c (given by the control layer). The control inputs T m and M z of the longitudinal and stability/maneuverability controllers, are received by a torque allocation unit at the low level, where it generates the driving/braking torques of the four in-wheel motors.

High Level Control (Control Layer)

The high level is composed of three decentralized controllers based on the Super-Twisting Sliding Mode Control (STSMC), that generates control inputs in order to independently control the lateral and longitudinal dynamics, and the stability/maneuverability of the vehicle.

Super-Twisting Sliding Mode Control

The Super-Twisting algorithm is a second order sliding mode control. In spite of perturbations, it generates the continuous control function that drives the sliding variable and its derivative to reach a sliding surface during a finite time. Consider the second order system:

ẍ = f (X, t) + g(X, t)u(t) (3)
where X = [x, ẋ] T ∈ R 2 is the state vector, u is the control input, and f, g are continuous functions. X des is the desired state of X, with

X des = [x des , ẋdes ] T ∈ R 2 .
The error vector E is given by E = X -X des = [e, ė] T ∈ R 2 where e = x -x des and ė = ẋ -ẋdes . Hence, a sliding variable with a relative degree equal to one with respect to the control input is defined :

s = ė + ke (4)
The second order derivative is given by: s(s, t) = Φ(s, t) + ξ(s, t) u(t), (5) where Φ(s, t) and ξ(s, t) are bounded functions. The goal of the Super-Twisting algorithm is to enforce the sliding variable s to converge to zero in finite time. Assume that there exist S 0 , b min , b max , C 0 , U max verifying that for all x ∈ R n and |s(x, t)| < S 0 :

     |u(t)| ≤ U max |Φ(s, t)| < C 0 0 < b min ≤ |ξ(s, t)| ≤ b max (6)
The control input based on the Super-Twisting Sliding Mode algorithm, is given as:

u(t) = u 1 + u 2 u 1 = -α 1 |s| τ sign(s), τ =]0, 0.5] u2 = -α 2 sign(s) (7)
where α 1 and α 2 are positive gains. The following conditions guarantee the finite time convergence:

   α 1 ≥ 4C0(bmaxα2+C0) b 2 min (bminα2-C0) α 2 > C0 bmin (8)
Refer to [START_REF] Utkin | On convergence time and disturbance rejection of super-twisting control[END_REF]] for the convergence analysis. An approximation function s s+ϵ is used to smooth the sign(s) function, where ϵ > 0.

Lateral Control

Vehicle autonomy is executed in multiple steps, starting from route planning, followed by a behavioral decision making, a motion planning and finally control. The precontrol steps fall under the decision making of the autonomous vehicle. In this work, the trajectory to be followed is modeled by a parametric curve, connecting a sequence of way-points taken from a predefined map data [START_REF] Said | Local trajectory planning for autonomous vehicle with static and dynamic obstacles avoidance[END_REF]]. The lateral displacement error is computed at a lookahead distance L s from the center of gravity of the vehicle (Fig 2). The purpose is for real-time implementation, in order to take into account the delay of the sensors and controllers/actuators. L s should be tuned as a function of the variation of longitudinal velocity and road curvature. As V x increases, this point should be further from the vehicle CG to respect the response time needed, hence L s should be increased. On the contrary, as road curvature increases, the point to be followed should be closer to the vehicle to avoid following points outside the road, hence CG Vehicle Longitudinal Axis L s Fig. 2. Lateral error computed at a look-ahead distance L s L s should be decreased. In this work and for simplicity, L s is adapted with the longitudinal velocity variation only. The second derivative of the lateral error can be related to the controlled steering angle as defined in [START_REF] Tagne | Design and comparison of robust nonlinear controllers for the lateral dynamics of intelligent vehicles[END_REF]] according to (9).

ëy = - C f + C r M β - l f C f -l r C r M V x ψ -ρV 2 x + C f M δ c (9)
For the lateral control, the objective is to drive e y and its rate to zero. Let's define the sliding variable s y as (10): s y = ėy + k y e y , with k y > 0 (10) where s y has a relative degree equal to 1 w.r.t the control input δ c (9). Hence, sy (s y , t) = Φ y (s y , t) + ξ y (s y , t) δc (t) (11) Φ y (s y , t) and ξ y (s y , t) are bounded functions satisfying conditions of (6). The controlled steering angle δ c is given by

δ c = -α δ,1 |s y | τ δ sign(s y ) -α δ,2 t 0 sign(s y ) dτ, (12)
where α δ,i with i = [1,2] are positive constants satisfying conditions (8). τ δ is a constant in the interval ]0,0.5]. The STSM control input δ c (12) guarantees the convergence of s y to zero in a finite time, hence ėy + k y e y → 0. Once reaching the sliding surface, e y converges exponentially to zero with a rate k y > 0.

Longitudinal Control

Longitudinal control is executed by controlling the longitudinal velocity to track a generated velocity profile. This profile adheres the speed limit of the road (V x lim ) on one hand, and a comfort criterion is used, on the other hand, to keep the lateral acceleration a y under a threshold of 4 m/s 2 to improve vehicle stability and passenger comfort as in [START_REF] Rajamani | Vehicle Dynamics and Control[END_REF]].

V x des = min a ymax ρ , V x lim (13)
The total driving torque can be related to the longitudinal acceleration by ( 14) as it is defined in [START_REF] Chebly | Coupled longitudinal/lateral controllers for autonomous vehicles navigation, with experimental validation[END_REF]]

M ẍ -M ψ ẏ + L * ψ2 - T m r + 4I w ẍ r 2 + δ(F y f l + F y f l ) = 0 L * = l r (m rr + m rl ) -l f (m f l + m f r ) (14
) where I w is the wheel moment of inertia. The objective is to generate a total driving (traction) torque, in order to track the desired longitudinal velocity. For this purpose, let's define an error e x on the longitudinal velocity e x = V x -V x des (15) Hence, when e x is driven to zero, V x will track V x des . So, define the sliding variable

s x = e x + k x e x dt (16)
where s x has a relative degree equal to 1 w.r.t the control input T m (14), hence sx (s x , t) = Φ x (s x , t) + ξ x (s x , t) Ṫm (t) (17) Φ x (s x , t) and ξ x (s x , t) are bounded functions satisfying conditions of (6). The total driving torque T m is given by

T m = -α Tm,1 |s x | τ Tm sign(s x ) -α Tm,2 t 0 sign(s x ) dτ (18)
where α Tm,i with i = [1,2] are positive constants satisfying conditions of (8). τ Tm is a constant in the interval ]0,0.5]. The STSM control input T m (18) guarantee the convergence of s x to zero in a finite time, forcing e x to converge to zero, leading to the tracking of V x to V x,des .

Stability and Maneuverability Control

The DYC controller has to achieve multi objectives on the same actuator, hence a decision layer has been developed to coordinate between the stability and maneuverability control objectives. The main idea is to promote one objective and attenuate the other by multiplying each sliding variable by a scheduled gain named decision variable λ i where i = [ ψ, β]. The decision layer monitors the lateral stability index of the vehicle (19) as defined in [START_REF] Chen | Integrated vehicle dynamics and control[END_REF]], and consequently generates the weighting parameters λ i to promote/attenuate the desired objectives based on a set of coordination rules.

SI = |2.49 β + 9.55β| (19)
s β = ėβ + k β e β = ( β -βref ) + k β (β -β ref ) ( 20b
) Since DYC is responsible for the control of the two state variables ψ and β, let's define a new sliding variable

s ψ,β = c 1 s ψ + c 2 s β , (21)
where c 1 and c 2 are positive constant weights, relatively scaling the sliding variables s ψ and s β . s ψ,β has a relative degree of 1 w.r.t the control input M z , hence

s ψ,β (s ψ,β , t) = Φ ψ,β (s ψ,β , t) + ξ ψ,β (s ψ,β , t) Ṁz (t) (22)
where Φ ψ,β (s ψ,β , t) and ξ ψ,β (s ψ,β , t) are bounded functions satisfying conditions of (6). The corrective yaw moment M z is given by

M z = -α Mz,1 |s ψ,β | τ Mz sign(s ψ,β )-α Mz,2 t 0 sign(s ψ,β ) dτ (

Decision Layer

Developing a decision layer assists in switching between stabilizing and maneuvering objectives. λ ψ and λ β vary between 0 and 1. As λ i approaches 1, the control objective is promoted, whereas it is attenuated at 0. Let's define the reference control inputs as ψref = λ ψ ψbic + (1 -λ ψ ) ψ (24a) 24), when λ i approaches 1, the reference variable turns into the bicycle model variable, and hence the corresponding control objective is promoted. By contrast, when λ i approaches 0, the reference trajectory will be the same as the actual vehicle variable, and the control objective will be relaxed accordingly.

β ref = λ β β bic + (1 -λ β )β (24b) As seen in (
s ψ = ψ -ψref = λ ψ ( ψ -ψbic ) (25a) s β = β -β ref = λ β (β -β bic ) (25b)
The equivalent sliding variables in terms of the bicycle model reference are given in (25). Therefore, the state variables converge to the reference bicycle model if the corresponding decision variable is 1. Otherwise, the state variables remain without control.

Coordination Rules

Stability and Maneuverability are coordinated according to a lateral stability index criterion. For SI < SI, the vehicle is in normal driving situations and no risk for instability, else, the vehicle reaches critical driving conditions and enters the instability zone, so it has to be controlled to come back to the normal driving situation before losing stability completely. If SI ≤ SI, the vehicle is inside the stable zone, hence the DYC controller promotes the maneuverability control (λ ψ = 1) and turn off stability control (λ β = 0). However, if the vehicle is under critical situation SI ≥ SI, then the DYC controller should promote the lateral stability while relaxing maneuverability. A sigmoid function is employed for switching, to enable seamless promotion/attenuation between the low threshold SI = 0.6 and the high threshold SI = 0.8 (Fig 3).

λ β = 1 1 + e -8 SI-SI (SI- SI+SI 2 ) λ ψ = 1 -λ β (26)

Low Level Control (Actuator Layer)

The low-level control represents the actuator layer where the desired objectives are realized by physical systems. Trajectory following is achieved by AFS, while the longitudinal control, along with the stability/maneuverability objectives are executed through a torque allocation among the motors. The performance of the in-wheel driven vehicle and the conventional center-traction vehicle with control systems are contrasted using two alternative approaches.

In-Wheel-Motor-Drive Vehicle

For the in-wheel EV, T m is distributed using driving torques T dij . The strategy is to distribute T m equally on the four motors as in ( 27).

T ij = T m 4 , (27) where i
,j = [r,f],[r,l] = [rear,front],[right,left].
M z is generated using half acceleration and half deceleration on the vehicle's opposite sides, using the four motors. Considering a counter-clock wise (ccw) direction and assuming small δ, M z is generated using

M z t = -F x rl + F xrr -F x f l + F x f l , ( 28
)
where t is the half track of the vehicle. The ratio r/t is multiplied by M z to transform the moment into wheel torques, where r is the wheel's radius. Each wheel on the same side receive the same torque (magnitude and direction), either braking or acceleration. Therefore, M z is generated by

T b il = -r 4t M z (29a) T d ir = r 4t M z (29b)
where T b and T d are the braking and the driving torques.

Traditional Center-Traction Vehicle

In the traditional vehicles, a transmission system is used to distribute torques on the rear wheels via a gearbox. The traditional vehicle approach on control is widely used in literature as in [START_REF] Doumiati | Integrated vehicle dynamics control via coordination of active front steering and rear braking[END_REF], [START_REF] Chokor | A comparison between a centralized multilayer l p v / h and a decentralized multilayer sliding mode control architectures for vehicle's global chassis control[END_REF]].

T ri = T m 2 i = [r,l] (30)
The total driving torque is distributed equally on both of the rear wheels as in (30) while the generation of the yaw moment is achieved through Active Differential Braking (ADB) as in (31) considering a ccw direction. The decentralized architecture is implemented and validated within Matlab/Simulink on a full non-linear model. The autonomous vehicle controlled by the AFS angle δ (Fig. 6), has shown an almost perfect tracking of the desired trajectory (Fig. 4) with a small displacement error (Fig. 7). Lateral control is tested by inducing disturbances and it results in a robust behaviour. The first scenario (Sc1) corresponds to the vehicle tracking the whole trajectory. Longitudinal control is achieved through the distribution of the driving torques for the traditional vehicle (Fig. 8) (half the magnitude for the in-wheel vehicle). Longitudinal velocity of the vehicle is observed to track the reference constructed velocity profile (Fig. 5), with smooth variations to avoid excessive longitudinal acceler- ation. Lateral control assisted by comfortability through longitudinal control, is accompanied by the stability and maneuverability control objectives. Under the same test conditions, due to the relatively low reference velocity, the open-loop system -represented by the controlled vehicle without activating DYC-is coherent with the reference bicycle model. Hence, although there is no significant errors on the yaw rate (Fig. 10) and the side-slip angle (Fig. 11), activating DYC has enhanced the maneuverability and the stability by reducing their correspondent errors with the bicycle model. In this case, no risk on stability has been detected, so DYC has promoted only maneuverability objective.

T b rl = -r t M z T b rr = 0 (31)
To distinguish the effect of DYC in critical situations, a second scenario is introduced (Sc2), where a test is performed with augmenting the desired velocity (to 70 km/h) on the portion indicated on the map, representing a high curvature corner. Maneuverability and stability are switched according to the decision variables in (Fig. 9). We observe a major attenuation of the errors on the yaw rate and the side-slip angle (Figures 12,13). The performance of the in-wheel driven vehicle in achieving the demanding objectives is observed to be supreme to that of the traditional vehicle. The mentioned objectives are achieved through the braking torques for the traditional vehicle (Fig. 14-left)), with a vital attenuation for that of the in-wheel driven vehicle (Fig. 14-right). The last scenario (Sc3) represents a double lane change (DLC) test performed at 22 m/s, to distinguish the performance of the two vehicles. Maneuverability and stability reflected by the yaw rate and the side-slip angle can be observed in (Fig. 15). ψ of the in-wheel driven vehicle has shown a better performance than the traditional one, in tracking the reference bicycle model; Similarly for β.

The lower stability index for the in-wheel vehicle (Fig. 16) resulted in promoting often the maneuverability objective through the decision variables. Consequently, the side-slip angle is attenuated. On the contrary to the traditional vehicle, where the decision variables indicate that it needs more time than the in-wheel vehicle to restore stability, leading to an inversely proportional behaviour (Fig. 18). Furthermore, as a result of ADB, traditional vehicles are penalized in longitudinal dynamics. Fig. 17 explicitly shows the longitudinal dynamics conservence for the inwheel vehicle, counter to the traditional vehicle. Avoiding longitudinal velocity loss involves averting adding traction torques to compensate for the error in longitudinal velocity. As a direct consequence, stability loss is avoided, and the vehicle is more accurately represented by the bicycle model, thus leading to lower stability index. As a result, lower SI frequently promotes maneuverability. This demonstrates the premier advantage of conserving the longitudinal velocity in the in-wheel vehicles. In light of this, the in-wheel driven vehicle has shown a superior performance in achieving the desired objectives.

CONCLUSION

In this paper, a mulit-layer decentralized GCC architecture is developed for lateral, longitudinal, stability, and maneuverability control. The STSMC technique is used to develop a high-level control that comprises the desired objectives. The architecture is aided with a decision layer to switch between the stability and maneuverability objectives according to driving situations. Two strategies are approached at the low level to distinguish the primary advantage of the in-wheel-motor driven vehicles. Following the tendency of the autonomous in-wheel EV for energy economy, an energy efficient strategy for torque allocation is among the work perspectives of this study.

  Fig. 1. Proposed GCC Architecture

  Let's define the state variable X = [ ψ, β] T and the error vector E = [e ψ e β ] T = [ ψψref , β -β ref ] T where ψref and β ref are given in subsection (2.3.4.1). It is desired to control the yaw rate ψ, the side-slip angle and its rate (β, β), hence define the sliding variables s ψ = e ψ = ψψref (20a)

  23)where α Mz,i with i =[1,2] are positive constants satisfying conditions of (8). τ Mz is a constant in the interval ]0,0.5]. The STSM control input M z (23) guarantees the convergence of s ψ,β to zero in a finite time, hence e ψ resp. ėβ + k β e β converge to zero, meaning ψ converges to ψref resp. β exponentially converges to β ref with a rate k β > 0.

Fig. 3 .

 3 Fig. 3. Maneuverability and Stability Switching

  Fig. 4. Trajectory Following (Lateral Control)

  Fig. 10. Vehicle Yaw Rate -Sc1

Fig. 15 .

 15 Fig. 14. Braking Torques Traditional(a), In-wheel(b) -Sc2

Table 1 .

 1 Vehicle Parameters

	Symbol	Description	Value
	M	Vehicle's Total Mass	1286.4 [kg]
	K t	Tire stiffness coefficient	467000 [N/m]
	C t	Tire damping coefficient	500 [N.s/m]
	t f	Half front axle	0.773 [m]
	t r	Half rear axle	0.773 [m]
	l f	Distance between CG and the front axle	1.0385 [m]
	l r I z g	Distance between CG and the rear axle Yaw moment of inertia Gravitational constant	1.6015 [m] 1970 [kg.m 2 ] 9.81 [m/s 2 ]
	µ	Road adherence coefficient	dry surface = 1
	r	Wheel effective radius	0.30759 [m]
	C f	Front tire cornering stiffness	76776 [N/rad]
	C r	Rear tire cornering stiffness	76776 [N/rad]

Acknowledgements

This work is carried out within the framework of the V3EA project "Electric, Energy Efficient and Autonomous Vehicle" (2021-2025), funded by the French National Agency (ANR).