Stéphane Azefack

Kamel Aouiche
email: kamel.aouiche@gmail.com

Jérôme Darmont
email: jerome.darmont@univ-lyon2.fr

Dynamic index selection in data warehouses

Analytical queries defined on data warehouses are complex and use several join operations that are very costly, especially when run on very large data volumes. To improve response times, data warehouse administrators casually use indexing techniques. This task is nevertheless complex and fastidious. In this paper, we present an automatic, dynamic index selection method for data warehouses that is based on incremental frequent itemset mining from a given query workload. The main advantage of this approach is that it helps update the set of selected indexes when workload evolves instead of recreating it from scratch. Preliminary experimental results illustrate the efficiency of this approach, both in terms of performance enhancement and overhead.

Introduction

A data warehouse is generally modeled by a starlike schema that contains a central, typically very large fact table, and several dimension tables that describe the facts [START_REF] Kimball | The Data Warehouse Toolkit[END_REF]. An analytical query over such a model necessitates very costly join operations between the fact table and dimension tables.

Selecting suitable physical structures that improve system performance is the role of data warehouse administrators. However, given the wide development of data warehouses, as well as their structural and operational complexity, minimizing the administration function is a crucial issue.

In this context, we have proposed an automatic bitmap join index selection method based on frequent itemset mining from a given query workload [START_REF] Aouiche | Automatic Selection of Bitmap Join Indexes in Data Warehouses[END_REF]. Attributes that frequently appear together in queries indeed constitute good candidate indexes; and bitmap join indexes are particularly appropriate to data warehouses [START_REF] Sarawagi | Indexing OLAP Data[END_REF]. However, this approach is static: if the input workload significantly evolves with time, we must rerun the whole process to preserve performance.

In this paper, we improve our approach by two aspects. First, we replace the frequent itemset mining technique we used (namely, Close [START_REF] Pasquier | Efficient mining of association rules using closed itemset lattices[END_REF]) with an incremental frequent itemset mining technique, so that the selected index configuration can be updated instead of being recreated from scratch. Second, instead of mining closed frequent itemsets, we mine maximal frequent itemsets that are less numerous and help build better indexes. Finally, to the best of our knowledge, this is the first attempt at dynamically selecting indexes in data warehouses.

The remainder of this paper is organized as follows. We present the state of the art regarding both index selection in data warehouses and incremental frequent itemset mining in Section 2. Then, we detail our approach in Section 3 and discuss related experimental results in Section 4. We finally conclude this paper and provide research perspectives in Section 5.

Related work

Index selection in data warehouses

The index selection problem has been studied for many years in databases, but adaptations to data warehouses are few. In this particular context, research studies may be clustered into two families: algorithms that optimize maintenance cost and algorithms that optimize query response time. In both cases, optimization is realized under storage space constraint. In this paper, we are particularly interested in the second family of approaches, which may be classified with respect to the way a set of candidate indexes and the final configuration of indexes are built.

A set of candidate indexes may be built manually by the administrator, according to his expertise of the workload [START_REF] Frank | Adaptive and automated index selection in RDBMS[END_REF][START_REF] Choenni | On the selection of secondary indices in relational databases[END_REF]. This is both subjective and quite hard to achieve when the number of workload queries is very large. In opposition, candidate indexes may also be extracted automatically by syntactically analyzing the workload [START_REF] Chaudhuri | An efficient costdriven index selection tool for Microsoft SQL server[END_REF][START_REF] Valentin | DB2 advisor: An optimizer smart enough to recommend its own indexes[END_REF][START_REF] Golfarelli | Index selection for data warehousing[END_REF].

There are also several methods for building the final index configuration from candidate indexes. Typically, greedy algorithms increasingly select indexes minimizing workload cost until it does not decrease anymore [START_REF] Frank | Adaptive and automated index selection in RDBMS[END_REF][START_REF] Choenni | On the selection of secondary indices in relational databases[END_REF][START_REF] Chaudhuri | An efficient costdriven index selection tool for Microsoft SQL server[END_REF]. Classical optimization algorithms have also been used to solve this problem, such as knapsack resolution [START_REF] Valentin | DB2 advisor: An optimizer smart enough to recommend its own indexes[END_REF] and genetic algorithms [START_REF] Kratika | A genetic algorithm for the index selection problem[END_REF].

Incremental frequent itemset mining

Many algorithms have been proposed in the literature for incrementally mining frequent itemsets. They reuse the frequent itemsets discovered before transaction database update to compute new frequent itemsets. Updating the set of frequent itemsets is very costly, though.

To reduce the problem's dimensionality, closed or maximal frequent itemsets may be mined instead of all frequent itemsets. A frequent itemset I is closed if it contains all the items that occur in every transaction in which I is present. A maximal frequent itemset is a frequent itemset that has no frequent superset.

Most closed frequent itemset mining algorithms exploit concept lattices. The main incremental approach [START_REF] Valtchev | Generating frequent itemsets incrementally: two novel approaches based on Galois lattice theory[END_REF] manages lattice updates (unchanged, updated and inserted nodes into the lattice). However, its complexity is quadratic with respect to the number of elements in the concept lattice [START_REF] Valtchev | Incremental maintenance of association rule bases[END_REF] and the number of closed frequent itemsets may become very large with respect to database size [START_REF] Zaki | Fast Vertical Mining Using Diffsets[END_REF].

To the best of our knowledge, the only incremental maximal itemset frequent mining approach, GenMax, exploits a backtracking algorithm to prune the search space as soon as possible with respect to previous iterations [START_REF] Gouda | GenMax: An Efficient Algorithm for Mining Maximal Frequent Itemsets[END_REF]. This algorithm also improves support computation and optimizes short term mining.

Dynamic index selection strategy

In this section, we present the extension of our automatic join index selection method based on frequent itemset mining [START_REF] Aouiche | Automatic Selection of Bitmap Join Indexes in Data Warehouses[END_REF]. Its principle is to exploit transaction logs (i.e., the set of all queries processed by the system) to recommend an index configuration improving data access time.

This new approach is subdivided into six steps (Figure 1): (1) a workload Q we suppose representative of system usage is extracted from system logs; (2) so-called indexable attributes are extracted from Q and structured in a suitable data mining context QA; (3) incremental frequent itemset mining is applied on QA, exploiting a knowledge base P that stores information regarding previous executions of this step; (4) emerged (new) frequent itemsets are analyzed to generate new candidate indexes; declined (now infrequent) itemsets correspond to indexes to be dropped; retained (still frequent) itemsets correspond to candidate indexes to retain; the whole set of candidate indexes is labeled I C ;

(5) since disk space is constrained, I C is pruned using cost models; (6) the resulting index configuration I is finally effectively updated. The whole process then reiterates after a period of time set by the administrator. We detail its steps in the following sections.

Workload extraction

System workload is typically accessible from the host database management system's transaction log. A given workload Q is supposed representative if it has been measured during a time period the warehouse administrator judges sufficient to anticipate upcoming transactions.

Since we are more particularly interested in analytical query performance and not warehouse maintenance, we only consider interrogation query workloads in this paper. These queries are typically composed of join operations between the fact table and dimensions, restriction predicates, and aggregation and grouping operations. More formally, an analytic query q may be expressed as follows in relational algebra:

q = π G, M σ R (F >< D 1 >< … >< D n );
where G is the set of attributes from dimensions D 1 , …, D n that are present in q's grouping clause, M is a set of aggregate measures from fact table F and R a conjunction of predicates over dimension attributes.

Workload analysis

Attributes a j that may support indexes belong to the sets G and R defined in Section 3.1 [START_REF] Chaudhuri | An efficient costdriven index selection tool for Microsoft SQL server[END_REF][START_REF] Valentin | DB2 advisor: An optimizer smart enough to recommend its own indexes[END_REF][START_REF] Golfarelli | Index selection for data warehousing[END_REF]. We reference them in a "query-attribute" binary matrix QA whose rows represent workload queries q i ∈ Q and whose columns are indexable attributes a j . The general term QA ij of this matrix is equal to one if attribute a j is present in query q i , and to zero otherwise. This data structure or extraction context directly corresponds to attribute-value tables that are exploited by data mining algorithms.

Maximal frequent itemset mining

In the static version of our approach, we have used the Close closed frequent itemset mining algorithm [START_REF] Pasquier | Efficient mining of association rules using closed itemset lattices[END_REF] to obtain a set of candidate indexes I C . In this dynamic extension, we replace it by the GenMax incremental, maximal frequent itemset mining algorithm [START_REF] Gouda | GenMax: An Efficient Algorithm for Mining Maximal Frequent Itemsets[END_REF].

In our context, workload Q can be very large and evolves with time. We selected GenMax because it can determine, in a short time, all maximal frequent itemsets from large databases, by optimizing support computation and infrequent itemset pruning through a backtracking process. Moreover, queried data from Q are typically correlated, which leads to a dense extraction context QA. Since incremental frequent itemset mining may produce a number of closed frequent itemsets exponentially greater than the number of maximal frequent itemsets [START_REF] Zaki | Fast Vertical Mining Using Diffsets[END_REF], GenMax helps produce a smaller quantity of candidate indexes, which reduces the dimensionality of index selection and improves scalability.

In summary, GenMax, in a first iteration, exploits an input transaction database D to produce a knowledge base P that stores, e.g., the list of maximal frequent itemsets from D, non-maximal, but frequent itemsets, the number of transactions in D, etc. P is then exploited and updated in the next iterations. At each of these iterations, the list of new transactions d + and the list of deleted transactions d -are used to compute the updated transaction database ∆ = (D ∪ d + ) -d -. Frequent itemset computation is then performed on ∆, using P, to minimize accesses to D.

Candidate indexes generation

The application of GenMax onto matrix QA helps obtain: a set I + of emerged frequent itemsets, which were infrequent in P but become frequent in ∆; a set I - of declined frequent itemsets, which were frequent in P but become infrequent in ∆; and a set I 0 of retained frequent itemsets, which are frequent in both P and ∆. Then, the set of candidate indexes is I C = (I ∪ I + ) -I -, where I is the current index configuration. Note that I 0 is not used to compute I C , but is nonetheless recorded in P.

Candidate indexes selection

The number of candidate indexes in I C is generally proportional to the size of workload Q. Thus, it is not feasible to build all the proposed indexes because of system limitations (e.g., a limited number of indexes per table) or storage space constraints. To circumvent these limitations, we exploit cost models that help greedily select the most advantageous indexes. These models estimate storage space occupied by bitmap join indexes, data access cost whether using these indexes or not, and index maintenance cost. Due to space constraints, we cannot elaborate on these cost models in this paper, but the interested reader can refer to [START_REF] Aouiche | Automatic Selection of Bitmap Join Indexes in Data Warehouses[END_REF] for complete details.

Index configuration update

Applying index selection (Section 3.5) on I C outputs a new index configuration I'. To update the current index configuration I, we must eventually: create all indexes i ∈ I' such that i ∉ I, i.e., all emerged indexes i ∈ (I' -I); drop all indexes i ∈ I such that i ∉ I', i.e., all declined indexes i ∈ (I -I'); and reset I to I'.

Experiments

Experimental conditions

To illustrate the advantage of our dynamic index selection approach over our static one, we ran tests on a 1 GB data warehouse implemented within Oracle 9i, on a Pentium Dual Core 1.6 GHz PC with 1 GB RAM and a 120 GB IDE disk under Windows XP Pro. Our test data warehouse is derived from Oracle's, whose classical sales star schema is composed of one fact table and five dimensions. We have defined an initial workload Q1 of 30 analytical queries involving aggregation operations and multiple joins between the fact table and dimension tables. Then, we defined four evolutions of Q1 (Q2 to Q5), so that some frequent attribute sets emerge, some decline, and others remain frequent. Due to space constraints, we reproduce here neither the full data warehouse schema nor the detail of each workload, but they are available on demand.

Results

We ran our tests for each workload Q1 to Q5; without indexing (for reference), with static and dynamic indexing; and for an arbitrary minimum support value (when mining frequent itemsets) of 0.05 that is low enough to produce a fair number of candidate indexes. In each of these tests, we measured index selection time (with both our static and dynamic approaches), index creation (static approach) or update (dynamic approach) time under Oracle, and workload execution time. The results we obtained are plotted in Figures 2, 3 and4, respectively. All three figures feature workloads (Q1 to Q5) on the X axis and execution times on the Y axis (in milliseconds, seconds and minutes, respectively).

Figure 2 shows that the overhead of our dynamic approach, in term of index selection alone, is about 5.2 times greater than that of our static approach, on an average. However, this is not due to the algorithms' intrinsic complexity, but to our implementations. The static approach has been implemented and optimized in PHP, while our newer, dynamic approach is implemented in Java (which appears slower than PHP on a standalone workstation) and not optimized yet. Both their execution times remain in the same order of magnitude (hundreds of milliseconds here), though.

On the other hand, Figure 3 shows that index update time is about 9.5 faster with our dynamic approach on an average, and even about 11.5 times faster if Q1's execution (and thus initial index configuration creation) is excluded from computation. Since index update runs in tens of seconds in our examples, the main overhead of our method lies here, and the enhancement brought by dynamicity is obvious.

Figure 4 eventually shows that exploiting maximal frequent itemsets (dynamic approach) instead of closed frequent itemsets (static approach) helps select more pertinent indexes, since response time is slightly better (about 13% on an average) in the dynamic case. This is presumably because fewer candidate indexes are generated and then selected, which simplifies index choice at query optimization time.

Conclusion and perspectives

In this paper, we have presented a dynamic bitmap join index selection method for data warehouses that is based on incremental frequent itemset mining from a given query workload. The main advantage of this approach is that it helps update the set of selected indexes when workload evolves instead of recreating it from scratch.

Our first experiments (which we aim to extend, notably by scaling them up, to complete our approach's validation) indeed show that introducing dynamicity helps reduce index maintenance overhead. Furthermore, exploiting maximal instead of closed frequent itemsets also helps improve the index configuration's quality, and hence query response time.

Furthermore, note that our approach is purposely modular and generic. Each step (frequent itemset mining, candidate indexes selection…) exploits interchangeable tools. The data mining technique and cost models we use are indeed not related to any system in particular and could easily be replaced by other, more efficient methods if necessary.

Eventually, a critical issue when using automatic, dynamic optimization strategies is to master system overhead, and in particular determine when the administrator should run the incremental index update process. Pursuing this lead is our main research perspective. Studies related to session detection that are based on entropy computation [START_REF] Yao | Machine Learning Approach to Identify Database Sessions Using Unlabeled Data[END_REF] could be very useful for this sake.

Figure 1 .

 1 Figure 1. Dynamic index selection strategy

Figure 4 .

 4 Figure 2. Index selection time


Rohit Chandra
email: ol.chandra@gmail.com

Juan Antonio Corrales-Ramon
email: juan-antonio.corrales-ramon@sigma-clermont.fr

Youcef Mezouar
email: youcef.mezouar@sigma-clermont.fr

Resolved-Acceleration Control of Serial Robotic Manipulators Using Unit Dual Quaternions

Keywords: Robots manipulators, motion control systems, torque control "

A new method for resolved-acceleration control of serial chain manipulators has been proposed which uses dual quaternion representation of screw-based motion variables, i.e. pose, velocity and acceleration. The corresponding coupled controller considers both translational and orientational errors simultaneously for trajectory tracking and utilizes spatial acceleration to compute the feedforward compensation term for feedback linearization. The proposed coupled control law was validated on a robotic arm along a pre-defined trajectory. The controller demonstrated an improved trajectory tracking performance as compared to the conventional decoupled resolved-acceleration controller which treats translational and orientational error separately.

INTRODUCTION

Robotic manipulators are often required to follow the desired trajectory composed of position as well as orientation set points, along with the desired profile of linear and angular velocities for tighter control and smooth motion. Due to intuitive ease of trajectory specification and the control of end-effector interaction with the environment, a task space controller is often preferred over a joint space controller. Operational space control [START_REF] Khatib | A unified approach for motion and force control of robot manipulators: The operational space formulation[END_REF], [START_REF] Nakanishi | Operational space control: A theoretical and empirical comparison[END_REF]) and resolved-acceleration control [START_REF] Caccavale | Resolved-acceleration control of robot manipulators: A critical review with experiments[END_REF], [START_REF] Luh | Resolvedacceleration control of mechanical manipulators[END_REF]) have been the two preferred methods for task space control in previous decades. They differ primarily in their treatment of singularity, apart from the dynamic formulation.

Traditionally the trajectory tracking strategies in the case of robotics manipulation involve the decoupling of position and orientation variables. However, it is desired to strive for a coupled controller to take into account the inherent effect of rotation on the translation of a rigid body as mentioned by [START_REF] Han | Kinematic control of free rigid bodies using dual quaternions[END_REF]. The basis of such control was given by [START_REF] Bullo | Proportional derivative (pd) control on the euclidean group[END_REF], along with a strategy to deal with coupled trajectory tracking problem for a rigid body using the homogeneous transformation matrix (HTM) representation of pose. The difference in the convergence behavior of coupled and decoupled control of linear and angular terms were reported in [START_REF] Han | Kinematic control of free rigid bodies using dual quaternions[END_REF] for the motion on SE(2) of a ground omnidirectional robot they concluded that the coupled treatment is a better choice when the synergy of position and orientation is important, i.e. for applications where the position and orientation setpoints are desired to be achieved simultaneously, for instance, robotic welding of a curved surface.

The coupled treatment of position and orientation variables using spatial dynamics concepts given by Feath-erstone (2010a,b) and compact representation of screw displacements using unit dual quaternions (UDQs) are the motivations for the development of the proposed new approach of UDQ based resolved-acceleration control. The interpretation of screw-based dynamics using spatial dynamics Featherstone (2010aFeatherstone ( ,b, 2001) ) led to compact and more tractable formulations of kinematic and dynamic equations. The simplicity is evident from the formulations of the Jacobian and the derivative of the Jacobian for a kinematic chain as compared to the approach followed by [START_REF] Bruyninckx | Symbolic differentiation of the velocity mapping for a serial kinematic chain[END_REF].

The UDQ based approach for kinematic modeling of serial manipulator provides storage and computational advantage, as demonstrated by [START_REF] Özgür | Kinematic modeling and control of a robot arm using unit dual quaternions[END_REF]. We also combine the approach given by Featherstone (2010a,b) for the computation of Jacobian derivative with the UDQ based kinematic formulation developed by [START_REF] Özgür | Kinematic modeling and control of a robot arm using unit dual quaternions[END_REF]. The control law for task-space trajectory tracking by a robotic serial manipulator is based on the work of [START_REF] Wang | Unit dual quaternion-based feedback linearization tracking problem for attitude and position dynamics[END_REF], where UDQ based error dynamics as derived from the UDQ based dynamics for a single rigid body for the attitude and position tracking. In our implementation, we derive the error dynamics of an end-effector in a desired frame attached to the base of a manipulator using a slightly different definition of the UDQ based pose error. Hence it allows for an intuitive representation of trajectory which is more prevalent in robotics literature, compared to the error represented in the body frame in [START_REF] Wang | Unit dual quaternion-based feedback linearization tracking problem for attitude and position dynamics[END_REF].

The contributions of this paper can be summarized as:

• Extension of the kinematics developed by [START_REF] Özgür | Kinematic modeling and control of a robot arm using unit dual quaternions[END_REF] for the computation of the derivative of Jacobian for a serial manipulator using the theory of spatial dynamics given by [START_REF] Featherstone | A beginner's guide to 6-d vectors (part 2)[tutorial[END_REF].

The proposed approach for the computation of the Jacobian derivative is computationally advantageous. • A screw-based resolved-acceleration controller using UDQ representation to compute the joint torque commands for the end-effector trajectory tracking. The coupled controller designed for a serial manipulator to follow the end-effector trajectory treats both orientation and position set-points jointly, thus addressing the inherent effect of rotational motion on translation motion.

In the following section (section 2) the theoretical background and the design of resolved-acceleration controller in explained. The derivation of UDQ error dynamics based control Law is given in section 3. The experimental validation and related discussions are given in section 4. Section 5 presents the conclusion and discussion about the current work.

THEORY AND PROBLEM FORMULATION

Resolved-Acceleration Control

The dynamic model of a serial manipulator consisting on n joints in joint space is given as: H(q) q + C(q, q) q + G(q) = Γ (1) where H(q) is a symmetric and positive definite inertia matrix. C(q, q) consists of Coriolis and centrifugal terms, whereas G(q) represents gravity vector. Γ is a vector of joint driving torques. The joint positions, velocities and acceleration are represented above with q, q, and q, respectively.

Assuming perfect dynamic compensation, the driving torques can be computed with a new control input as: Γ = H(q) Ĵ-1 (q)(â cmd -J(q, q) q) + C(q, q) q + G(q)

(2)

The (â cmd -J(q, q) q) terms actually replaces q in (1) using following equation relating the joint velocities to the end-effector velocity, and its derivative:

ω = Ĵ q (3) â = Ĵ • q + J • q (4)
The goal of resolved acceleration control is to find the acceleration command (â cmd ) for (2) from the desired trajectory consisting of desired acceleration, velocity, and position of end-effector. Note that the dual velocity and acceleration terms, i.e. ω and â, has different meaning in the context of this paper, and is explained in the next section.

Mathematical and Theoretical Preliminaries

In this section, the concepts required for subsequent formulation are given. It includes: the meaning of mathematical notations used; relevant background related to the dual-quaternions; the formulation of screw-based forward and inverse kinematics concepts proposed in [START_REF] Özgür | Kinematic modeling and control of a robot arm using unit dual quaternions[END_REF]; and, concepts and formulations related to the spatial acceleration.

Notations: Dual quaternions and dual vectors have been notated with bold characters with hat, e.g. x and ŝ, respectively. Dual numbers have been named with variables with hats (e.g. θ), whereas dual number arrays have been notated with underlined bold characters with hats (e.g. θ). Arrays of real number variables are named using underlined bold characters (e.g. θ) and vectors and quaternions are notated with bold characters (e.g. l and q respectively).

Quaternions: A Quaternion is a 4-tuple consisting of a scalar and a vector part and is defined as:

q = s + v = s + (v 1 î + v 2 ĵ + v 3 k)
(5) A pure quaternion q has zero as the scalar component and is defined as q = q 1 i + q 2 j + q 3 k. Quaternion product is given as:

q 1 • q 2 = (s 1 s 2 -v T 1 v 2 ) + (s 1 v 2 + s 2 v 1 + v 1 × v 2 ) (6)
Dual Quaternions: A dual quaternions is an 8-tuple and consists of quaternions as their real and dual parts:

x = q r + εq d = (s r + v r ) + ε(s d + v d ) (7)
where ε is the dual number element, such that ε 2 = 0, and ε = 0. The classical conjugate of dual quaternion, used for transforming poses, is given as:

x * = q * r + εq * d = s r -v r + ε(s d -v d ) (8) Dual quaternion product is given as: xa • xb = (q r1 + εq d1 ) • (q r2 + εq d2 ) (9) = q r1 • q r2 + ε(q r1 • q d2 + q d1 • q r2 ) (10)
If the product of a dual quaternion with its classical conjugate has unity norm represented with a dual number, x • x * = x 2 = 1 + ε0, it is defined as a unit dual quaternion.

An adjoint operation for dual quaternions [START_REF] Wang | Unit dual quaternion-based feedback linearization tracking problem for attitude and position dynamics[END_REF]) is defined as:

Ad ( n xm) xa = n xm • xa • n x * m ( 11
)
where n x * m is the classic dual quaternion conjugate of n xm . If xa was a directed line represented using a unit dual vector or a motion screw ∈ M 6 represented with dual vectors, and n xm represented the transformation from frame m to frame n, the adjoint operation given in (11), will transform the corresponding element ∈ M 6 from frame m to n.

A vec x operation is defined for pure dual quaternions (dual quaternions with zero scalar parts in both real and dual components) which extracts the vectors from the dual quaternion product if applied to another pure dual quaternion. This operation is similar to the cross product operation for spatial vectors with motion basis (i.e. vectors ∈ M 6 ). For instance, if x = q r + εq d was a spatial vector representing a dual vector ∈ M 6 (note that we have reduced the dimension from 8 to 6 for x× operation by eliminating the scalar part, for the sake of explanation), x× (cross product operation) is given by Featherstone (2010a) as:

x× = q r × 0 q d × q r × (12)
Pose of a frame is represented using UDQ as:

= exp θ 2 • ŝ = cos θ 2 + ŝ sin θ 2 (13) = cos θ 2 + l sin θ 2 (14) + ε - d 2 sin θ 2 + l d 2 cos θ 2 + m sin θ 2 (15)
where θ ∈ D is a dual angle and ŝ ∈ D 3×1 is a directed line represented using unit dual vector.

θ = θ + ε d, ŝ = l + ε m (16)
where θ is a rotation around the screw axis, d is translation along the same axis, l is the unit direction vector of the screw axis, and m (= p × l) is the moment of this screw axis with respect to the origin of the reference frame, and p is any point lying on the screw axis.

Serial Manipulator Kinematics: For a serial kinematic chain, let xc0 be the initial end-effector configuration, and θ0 be the initial joint configuration such that:

θ0 = θ10 θ20 • • • θn0 T ∈ D n×1 (17)
The end-effector pose xc corresponding to the new joint configuration θ is obtained by multiplying the successive screw displacements caused by the intermediate joints between the base frame and the end-effector link, starting from the link to the base and ending at the joint closest to the end-effector.

xc = δ1 • δ2 • • • • • δn • xc0 (18) δi = exp θi 2 • ŝi0
where θi = ∆θ i for revolute joints, θi = ε ∆d i for prismatic joints. In (18), ŝi0 refers to the unit dual vector representing the initial configuration of the joint screw, and δi represents the screw displacement of link i relative to the link (i -1) resulting from the motion of the previous joint.

The end-effector screw-velocity or spatial velocity (Featherstone (2010a)) represented as a pure dual quaternion, is obtained by adding the twist cause by the intermediate joints. The screw-based Jacobian is given as:

ω = ω v c /b0 = Ĵ θ (19) = [ŝ 1 ŝ2 • • • ŝn ] θ where θ = θ1 θ2 • • • θn T ∈ D n×1
In ( 19), v c /b0 represents the linear velocity of a point attached to the end-effector, which is instantaneously coincident with the origin of the base frame. It will be hereafter notated as v 0 .

The current unit dual vector of joint screw (ŝ i ) for the i th joint, can be obtained from its initial value ŝi0 , by transforming it using the total displacement caused by the previous (i -1) joints, using the adjoint operation.

ŝi = Ad ( b δT (i-1) ) ŝi0 (20)
b δT (i-1) can be obtained as:

b δT (i-1) = δ1 • δ2 • • • • • δi-1 (21)
Note that in the case of the first joint, i.e. ŝ1 , the dual vector of the corresponding joint screw is constant with respect to the reference frame. However, the first link closest to the base frame will move due to the motion of the first joint.

Spatial Acceleration:

A brief discussion about spatial acceleration is provided here to relate our control approach to traditional controllers. All the terms are given in the base frame of the robot. Spatial acceleration is the derivative of screw velocity and defines a helicoidal vector field [START_REF] Featherstone | The acceleration vector of a rigid body[END_REF].

â = ω = ω v0 (22)
The angular acceleration ω of a body in conventional sense and spatial dynamics is the same. However, the linear acceleration of a point attached to the body is quite different in both paradigm.

Let âc /b0 = ω + ε a c /b0 represents the conventional acceleration of a frame c /b0 rigidly attached to the endeffector and instantaneously coincident with the origin of the base frame. The linear acceleration a c /b0 is related to the linear part of spatial acceleration ( v0 ) as :

a c /b0 = v0 + ω × v 0 (23)
Note that the dual part of âc /b0 , i.e. a c /b0 is not a derivative of v c /b0 (i.e. v c0 ). In fact, a c /b0 "refers to the acceleration of an individual body-fixed point at the moment when it happens to be passing through the origin" [START_REF] Featherstone | The acceleration vector of a rigid body[END_REF].

The conventional dual acceleration with the linear acceleration of a general point i.e. origin of a frame on the endeffector, say c, can be obtained from spatial acceleration (a c /b0 ) as follows:

âc = ωc a c = ω a c /b0 + ω × -→ OC + ω × (ω × -→ OC) (24)
3. DERIVATION OF CONTROL LAW

UDQ based Pose Error

Given two poses, i.e. current (x c ) and desired (x c ), both expressed in a static base frame, there are four possible permutations for error computation. The error UDQ used in our formulation is:

xe = xd • x * c = b xc→d (25)
The screw axis related this error choice represents a screw displacement vector directed from the current frame c to the desired frame d (expressed in the base frame b). It means if we transform the error screw axis derived from xe , from the base frame to the current frame using Ad ( b x * c ) :

c ŝe = Ad ( b x * c ) b ŝe = Ad ( c xb ) b ŝe (26)
and then postmultiply the corresponding UDQ (obtained using ( 13)) to the current frame pose xc , we will get the desired pose:

xc • exp c θe 2 • c ŝe = xd (27)
The same interpretation can also applied for obtaining orientation error in a desired frame when using unit quaternion representation.

Control Law Design:

In this section we derive the error dynamics similar to [START_REF] Wang | Unit dual quaternion-based feedback linearization tracking problem for attitude and position dynamics[END_REF] for the attitude and position tracking problem of a rigid body using feedback linearization.

Taking the derivative of the error UDQ given in ( 25) we obtain:

ẋe = ẋd • x * c + xd • ẋ * c (28)
Now, it has been proved in [START_REF] Han | Kinematic control of free rigid bodies using dual quaternions[END_REF] that:

ẋ = 1 2 ω • x ( 29
)
where ω is the screw velocity of the body, whose pose is represented using x. The conjugate is given as:

ẋ * = - 1 2 x * • ω (30)
Hence ( 28) can be rewritten as:

ẋe = 1 2 xe • ωe (31)
where ωe = Ad (x * e ) ωdωc (32)

Taking the derivative of (32), we obtain: ωe = Ȧd (x * e ) ωd + Ad (x * e ) ωdωc (33) After expanding the involved terms and using the property of UDQs, we obtain:

ωe = Ad (x * e ) ωd -ωc + vec ωc Ad (x * e ) ωd (34)
Now ωc can be substituted with the following acceleration command (â cmd ) for feedback linearisation:

âcmd = 2 Kp • ln xe + Kv • ωe + vec ωc Ad (x * e ) ωd + Ad (x * e ) ωd (
35) Substituting âcmd in (34), we obtain the following error dynamics:

ωe + Kv ωe + 2 Kp ln xe = 0 (36) where the expression (2 ln xe ) refers to the product of dual angle and unit dual vector pertaining to the axis of the screw (see ( 13)),

ln xe = 1 2 θe • ŝe (37)
The details of obtaining the screw parameters from a dual quaternion can be found in [START_REF] Özgür | Kinematic modeling and control of a robot arm using unit dual quaternions[END_REF]. The asymptotic stability of equilibrium point (ln(x e ), ωe ) = ( 0, 0) for the above system has been proven in [START_REF] Wang | Unit dual quaternion-based feedback linearization tracking problem for attitude and position dynamics[END_REF] for an appropriate choice of the gains Kp and Kv . Therefore, the end-effector motion will eventually converge to the desired trajectory for the control command obtained in ( 35). The two equilibria problem for dual quaternion has been discussed in [START_REF] Wang | Unit dual quaternion-based feedback linearization tracking problem for attitude and position dynamics[END_REF], where the system (36) has two identical equilibria at xe = ( Î. 0)and(-Î. 0), which was resolved by multiplying the error UDQ with -1 to make the scalar part of the real quaternion positive.

The âcmd obtained in ( 35) can be used in (2) to compute the joint torque commands for the manipulator. Note that the dual angle θi used in ( 19) is converted to q i in (4), depending on the type of joint (see ( 17)).

Formulation of Jacobian Derivative:

In order to compute the driving torque command in (2), from the control law âcmd obtained in ( 35), the computation of Jacobian derivative ( J) is required. Derivative of the Jacobian of a serial robotic manipulator can be obtained as in [START_REF] Featherstone | A beginner's guide to 6-d vectors (part 2)[tutorial[END_REF], assuming a joint screw axis ŝi given as a dual vector is fixed on the child link i, and noting that ŝi ∈ M 6 :

J = ṡ1 ṡ2 • • • ṡn = [ vec ω1 ŝ1 vec ω2 ŝ2 • • • vec ωn ŝn ] ( 38
)
where ωi represents the screw velocity of an i th link, and can be computed from the twists effected by the joints between the base frame and the i th link (see ( 19)).

EXPERIMENTAL VALIDATION

The controller obtained in section 3.2 was validated in on one of the redundant arms of Baxter dual arm collabo- rative robot [START_REF] Robotics | Baxter collaborative robots for industrial automation[END_REF]). The joint inertia matrix, Coriolis and centrifugal torques, and the gravity torques were obtained using KDL library [START_REF] Smits | Kdl: Kinematics and dynamics library[END_REF], and q was provided by the Baxter ROS interface. The presented implementation does not address redundancy resolution for the extra joint in the used robotic arm, and the starting configuration of the manipulator was carefully chosen to avoid the effects of null space motion on the tracking task. The complete implementation along with the instructions to repeat the experiments and reproduce the results discussed in this section can be obtained from the source 1 . The controller performance was compared with the control law used in [START_REF] Caccavale | Resolved-acceleration control of robot manipulators: A critical review with experiments[END_REF], where the translational and orientational components of the trajectory were treated separately. The Jacobian of the manipulator and the derivative of this Jacobian for the quaternion based controller were obtained from the KDL library since the corresponding terms obtained in ( 19) and ( 38) are screwbased.

The Cartesian position and quaternion based control law was used for comparative validation of the proposed controller was given as follows in [START_REF] Caccavale | Resolved-acceleration control of robot manipulators: A critical review with experiments[END_REF]:

a c = pd + K Vp ṗe + K Pp p e ωc = ω d + K Vo ω e + K Po v e ( 39
)
where p e refers to the position error of the end-effector frame for a given desired position p d . v e refers to the vector part of the error quaternion computed as q e = q d • q * c , from which the orientation error in the base frame can be derived. a c and ωc are used to obtain the combined acceleration command similar to âcmd , to be used in (2).

Both the controllers were tuned for a stable profile in velocity and to obtain the best tracking performance. The proportional and derivative gain, 190 and 6 respectively, were given in a diagonal matrix form in (35) for the coupled controller using UDQ. The gains for the decoupled controller for the position and orientation error were K Pp = K Po = 270 and for the linear and angular velocity error, K Vp = K Vo = 7.5 in (39), and the control loop frequency for both the controllers was set 200 Hz. Additionally, the conventional decoupled controller was also tested with the same gain as the one chosen for the proposed coupled controller for fair assessment of their performance.

The description of trajectory generation for validation of the controller has been given in Fig. 1. The end-effector was desired to rotate around a pre-selected line, defined the base frame of the robot, while keeping one of its frame axes always pointing towards the line, thus requiring both translational and orientation control. The screw axis corresponding to the line, and taking the rotation (θ traj (t)) as a cubic function of time were used to obtain ) and decoupled (

) controllers. std dq and std kdl refers to the standard deviation of the joint efforts for coupled approach and decoupled approach, respectively. the desired trajectory of the end-effector. The coefficients of the polynomial were computed with four boundary condition: θ trajInit = 0, θ trajF inal = π/2, θtrajInit = 0, θtrajF inal = π/2. The θtraj (t), θtraj (t) and θ traj (t) thus computed, gave spatial acceleration, spatial velocity and pose, when multiplied by the given screw axis. The total duration of the experiment was 18 seconds.

The robot performing the trajectory tracking task can be seen in Fig. 1, and performance of both the controller for pose and velocity tracking have been given in Fig. 2 and 3, respectively. The evolution of position and orientation error norm is given in Fig. 4. The error norm for the position and orientation tracking for both the controllers have been summarized in Table 1. The joint efforts given as an input the joints of the manipulator is given in Fig. 5.

While the coupled controller performance for position tracking is slightly worse than the traditional decoupled controller for both the same and optimal gains, the coupled controller performed better in terms of orientation tracking. The performance is identical in terms of velocity error as is shown in Fig. 3. The commanded joint torques were also close in terms of magnitude, however higher oscillations in the commanded joint torques be observed in Fig. 5 for decoupled controller for joints s0, e0 and w0.

CONCLUSION

A new controller for resolved acceleration control of robotic manipulators was proposed for the trajectory tracking of the serial robotic manipulator using screw theory and concepts from spatial dynamics. Representation of the motion variables with dual quaternions allowed coupled treatment of translational and orientational components of trajectory tracking error. A comparison with a decoupled controller reveals better orientation tracking while achieving identical performance for the translational components. However, a more thorough analysis is re-quired to formulate a better control law that is appropriate to keep the joint torque bounded for this kind of dynamics.

In addition to that, redundancy resolution is needed to utilize the additional degrees of freedom for redundant manipulators, as during the current implementation special attention was taken during the definition of trajectory.

Fig. 1 .

 1 Fig. 1. (a) Desired trajectory descrition; (b),(c) Baxter robot performing trajectory tracking task.

Fig. 2 .

 2 Fig. 2. Pose error (Desired -Current) plot for the coupled () and decoupled ( ) controller with optimally tuned gains.

Fig. 3 .

 3 Fig. 3. Velocity error (Desired -Current) plot for the coupled () and decoupled ( ) controllers with optimally tuned gains.

  Fig. 4. Position and orientation norm error of for the coupled () and decoupled ( ) controllers with optimally tuned gains.

Fig. 5 .

 5 Fig. 5. Joint Effort plot for the coupled () and decoupled () controllers. std dq and std kdl refers to the standard deviation of the joint efforts for coupled approach and decoupled approach, respectively.

Table 1

 1

	. Root mean square error (RMSE) and
	standard deviation (StD) of position (P os)
	and orientation (Orient) errors for the coupled
	controller with optimal gain, and decoupled
	controller with the same and optimal gain.
	Controller	Coupled	Decoupsame	Decoupopt
	RMSE Orient (rad.)	0.1852	0.3436	0.2748
	StD Orient (rad.)	0.0718	0.1553	0.1119
	RMSE P os (m)	0.0089	0.0066	0.0049
	StD P os (m)	0.0026	0.0021	0.0013
	1 https://github.com/rohitChan/ifacWC2020

The authors are supported by the H2020 FoF project SoftMan-Bot. This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement no 869855.


Laurent Morival

Catholics and dissident French royalists from the beginning of the nineteenth century around the journalists F. de La Mennais and P.-S. Laurentie wanted to reconstitute a new Catholic monarchy against the compromise made by Louis XVIII. It was necessary to renew the links with the ancien régime against the revolutionary legacy, and to compensate for the monarchy's more distant flaws, caused according to them by the distancing of the Church's power.

After the July Revolution in 1830, these royalists reconfigured a modern monarchy behind 'Henry V' to make a Third Restoration possible. But there were many disagreements among the Legitimists: between the Parisian "Henriquists" and the absolutist émigrés who favoured Charles X, war raged after 1830, not to mention the opposition they waged against the Orleanists. These neo-legitimists then decided to open up more to modernity and demanded freedoms of association, religion and the press, inspired by the followers of La Mennais. They also sought unity with Catholics and wanted to participate in elections. But this political line was opposed to that of Genoude and the absolutists. Later, Montalembert separated Catholics and royalists and shattered any hope of unity and the creation of a large party under the July Monarchy.

The Impossible Modernization of Legitimate Monarchy after 1830:

The Journalists Pierre-Sébastien Laurentie and Eugène de Genoude Estelle Berthereau Centre d'histoire du XIXe siècle Sorbonne Catholics and royalists were divided right from the early years of the Bourbon Restoration in France, after the Napoleonic wars. The ultra-royalists defended the ideal of the phantasized Catholic monarchy and Church of the Middle Ages, behind the public figure of Félicité de La Mennais and more modestly next to the young journalist Pierre-Sébastien Laurentie, while royalists supported the pragmatic and modern monarchy behind Louis XVIII. Félicité de La Mennais (1782-1854), who signed 'Lamennais' after 1834, was inventor of social and liberal Catholicism, great polemicist, journalist, theoretician and philosopher. Famous under the Restoration for these volumes of the Essai sur l'indifférence, and favourable to a true Catholic restoration, he decided, seeing the successive kings unable to carry it out, to initiate a 'Catholicization' of liberalism after 1830 behind the Pope. Much less famous, Pierre-Sébastien Laurentie (1793-1876) came from the Gers (South-West of France). Trained by the Church and supported by the Congregation, an influential Catholic association, he began a dual career in Paris in 1817 as a teacher and journalist, and ended up as national editor-in-chief of La Quotidienne.

Around these two people are other members of the Congregation, association coupled with the Knights of Faith (Chevaliers de la Foi) with a desire to promote a traditional Catholic monarchy. These ultras 1 certainly wanted to reconnect with the former monarchy of the ancien régime, but above all they denied the lessons of the French Revolution and compensated for the more distant flaws of the monarchy, which were caused, according to them, by the distance between the Catholic Church's power and royal power in the early modern period. 2 For this reason, they fiercely attacked Louis XVIII's favourite Minister, Decazes, in 1817, who betrayed their conception of the union between the throne and the Catholic Church.

From a modest background, Pierre-Sébastien Laurentie began his career, like his eternal and clever rival Antoine-Eugène Genoud 3 in the royalist and Catholic newspapers and magazines of the Bourbon Restoration. Arrived from the Dauphiné, Antoine-Eugène de Genoude (1792-1849) managed to enter the salons of the suburb of Saint-Germain where he was introduced to Mathieu de Montmorency, the head of the Congregation, by Polignac. 'Doubly ennobled' as Louis XVIII liked to say, Eugène de Genoude abandoned a military career that did not satisfy him 4 to become journalist. At the head of the Knights of Faith, Montmorency introduced him to Chateaubriand who engaged him in the administration of Le Conservateur. He also participated in the newspaper Le Défenseur with La Mennais in 1820. Editor-in-chief of L'Étoile thanks to the ministry Villèle, the polemicist was also the author of a translation of Les Évangiles. As ultra-ministerial, and fervent defender of Villèle, he directed La Gazette de France. He was ordained a priest in 1834 but continued his career as a publicist. He was still the author of many books on the history of Catholicism.

Laurentie and Genoude both met at Le Spectateur a magazine which was published in 1819. This magazine directed, especially for the second version, by Laurentie, the Religious and Political Spectator was probably financed by the Earl of Artois, like Le Conservateur.

Upset by the July Revolution in 1830, these journalists reconfigured a modern monarchy under the Duke of Bordeaux, Henri V, to make a Third Bourbon Restoration possible. René Rémond,8 in his definition of right-wing families, 9 which was further substantiated, presented the different conceptions of the monarchy that then clashed between the Legitimists and the Orleanists with Louis-Philippe at the head of the cadet Bourbon branch. But even within the Legitimists, there were many oppositions: between Parisian 'henriquinquistes', Legitimists in favour of Henri V, and absolutist 'émigrés' who, from abroad, were supporters of Charles X, the war raged after 1830, not to mention the opposition they led against Louis-Philippe. It is now necessary to specify the conceptual changes within this idealized monarchy that these former ultra-dissidents, now neo-legitimists, were able to make after 1830. The opening to the freedoms of association, religion, of the press and more generally to modernity was inspired by La Mennais, while these royalists sought unity with Catholics and wanted to participate in the elections despite the fact that Louis-Philippe was in power.

However, why did these attempts to modernize the legitimate monarchy fail to predominate within the French legitimist movement?

The July Revolution in 1830 put these royalists in competition with each other. They were divided between moderate legitimism and 'Genoudisme', which first withdrew and then became fully part of modernity. These legitimist sensibilities were put to the test in 1835 against absolutist 'émigrés', then especially in 1848 with the Revolution and the establishment of the Second Republic and in 1851 with the Coup d'État of Napoleon III.

Defend the neo-Legitimists after the July Revolution

The July Revolution in 1830, with the fall of Charles X, placed Laurentie against history: journalist by profession, he found himself supporting the legitimate king while the printers and other journalists of Paris were making their revolution 10 against him. He watched helplessly as the king, the Minister Polignac and their immediate entourage failed to act. 11 He saw in it the urgency of a necessary modern redefinition of the legitimate monarchy. He then accepted the question of the oath to Louis-Philippe and, with the famous lawyer Berryer-son, developed a legalistic or parliamentary Legitimism that invited the Legitimists to form themselves into movements in the Chamber of Deputies (I quote Laurentie addressing Berryer): 'I remind you of 1830 and 1831; it is always the same position. Alone, we fought against proud fools, and we made way for a Right wing in a revolutionary chamber'. 12 It meant inciting royalists to run for election and even swear an oath of allegiance to a fought regime, to represent the legitimate monarchy. However, there was no question of adhering to Orleanism. Son of royalist lawyer Pierre Nicolas Berryer, himself involved in this profession and often referred to as 'Berryer-son', Pierre Antoine Berryer was the great defender of the royalist cause both in the courts and in the parliamentary gallery: he allied himself with Laurentie, his pen, to defend a legalistic, and therefore parliamentary legitimism. He also became the representative of social legitimism, defending workers and their demands for better working conditions. They both opposed the legitimists 'of action' who supported the Duchess of Berry by the use of force. They were henriquinquists (in favour of Henri) but legally and had to combine with the court in exile of Charles X, the absolutist legitimists who financed their actions and their newspapers.

Laurentie's experience in two new newspapers of his creation, Le Courrier de l'Europe and Le Rénovateur, consisted in advocating, together with the Duke of Fitz-James, a Parisian legitimist sitting in the Chamber of Peers, a moderate neo-legitimism. Édouard, Duke of Fitz-James (1776-1838) was an emigrant, an officer of Condé's army during the Revolution and a Knight of Faith. He became aide-de-camp and first gentleman of the Count of Artois, then marshal of camp in 1823 and led the ultras peer group in the Chamber. After 1830, he became one of the leaders of Laurentie's parliamentary legitimitism. As supporter of the Duke of Bordeaux, he opposed the absolutist policy of the exiled court of Charles X. While they were initiating with Berryer-son the authorization of the oath to Louis-Philippe to participate in the elections, Laurentie and Fitz-James were finally overtaken from the left by Genoude.

Yet the inspiration Laurentie had drawn from his master La Mennais since 1820 should have helped to serve the cause of the journalist who wanted unity between the Catholics and the legitimists. In Le Rénovateur, in March 1832, Laurentie called for an alliance with the Catholic Mennaisians: the neo-legitimists and Catholics of the future had to be able to join forces to defend the common cause of a re-catholicization of society served by the return to the throne of the legitimate branch of the Bourbons. La Mennais refused to work for two different newspapers, Le Rénovateur and L'Avenir, but recognized Laurentie's merits in his approach 'to introduce ideas of freedom into the minds of a few royalists'. 13 But because of the first excommunication of the theses of L'Avenir in 1832, and even if Le Rénovateur became the replacement newspaper for the former subscribers of L'Avenir, Laurentie took, by necessity, his distance from his master. However, Laurentie supported him in spite of everything,14 with other elders of the Congregation, until Paroles d'un croyant appeared in 1834. 15 In 1832, the journalist and writer Balzac joined Le Rénovateur, solicited by Laurentie with the help of the Duke of Fitz-James and Berryer-son to finally compensate for the loss of another celebrity, La Mennais. Nevertheless, Balzac went beyond the editorial advice given by Laurentie and in his article 'Du gouvernement moderne', 16 whose theories were developed in the Médecin de campagne,17 he advocated an absolute representative government led by an aristocracy composed not of "well-born" noblemen but of cultivated commoners drawn from the nebula of journalists in which he worked. Laurentie censured him and tempered his remarks by demanding a new interpretation of the moderate legitimist government from Louis de Bonald. 18 Another point of disagreement with other legitimists was the question of European intervention by foreign troops to restore Charles X or his successor in office. Laurentie was pushed to his limits by the absolutist shareholders who financed his newspaper: he was forced to stop relaying the ideas of the Duke of Fitz-James who wanted to rebuild a Third French and Parisian Restoration around Henri V and the neo-legitimists, against a forced return of the 'new émigrés'. On this point, the Duke of Fitz-James regretted Laurentie's overly 'European' position, 19 which certainly was due to economic pressures.

Laurentie's editorial line was in fact tightening up with the absolutist legitimist 'émigrés', led by the Duke of Blacas. At first, it was easier for them to identify with the line of another royalist Alfred Nettement. Historian of the Restoration and French literature, Alfred Nettement (1805-1869) belonged to the generation that succeeded Laurentie's one. Son of a diplomat who worked with the Prince of Polignac, he became a journalist at L'Universel. He had taken over the editorial line of La Quotidienne in 1830, while Laurentie was ousted, even though he still kept a link with the newspaper he had eventually directed at the end of the Bourbon Restoration. 20 Alfred Nettement certainly strongly supported the Duchess of Berry's uprising. 21 The latter came to France to restore her son Henri d'Artois, the Duke of Bordeaux, then count of Chambord to the throne by force, what Laurentie and Berryer refused to do. The Duchess of Berry counted on the support of the regions of Vendée and Bretagne, which she thought were still legitimists. However, the support was insufficient and she was arrested, imprisoned and threatened with a lawsuit. Besides, Alfred Nettement also advised his troops to refuse the oath, committing them to non-participation in the elections and 'emigration from the inside'. 22 Going against this program while not adhering to Orleanism, Laurentie felt the need to write in 1832 his Histoire des ducs d'Orléans, 23 a story that also legitimized his action as a parliamentary legitimist alongside the royalist lawyer Berryer-son. This did not appear to be a sufficient pledge.

Disqualified then in his modernism and especially financed by absolutist legitimists, Laurentie was constantly called to order during the years 1833 and 1834, until a breaking point when Le Rénovateur, too open to different royalist sensibilities, was reduced to his 'personal business' 24 at the economic level.

This moderate neo-Legitimism was, moreover, competed by 'Genoudisme'.

"Genoudisme" against moderate neo-legitimism

On the other hand, Genoude had managed to free himself from the financial protection of absolutists or other legitimists and was fulfilling Laurentie's dream: to become the owner of his own newspaper. And not just any newspaper, the oldest then still in operation, La Gazette de France, created in 1631 by Théophraste Renaudot. Successful and popular, Genoude gained a relative independence of mind and writing, since he had made his fortune in 1823 with the privilege of being able to send his newspaper by post five hours before all the others and to have a printer's certificate. Genoude, also a disciple of La Mennais, imitated him by translating the Bible, writing a Life of Jesus Christ, and becoming a priest in 1834. 25 In 1830, Genoude relaunched hostilities between modernizers of the legitimate monarchy. La Gazette de France attacked Laurentie over the opening deemed too pronounced in favour of the July monarchy, which further confronted the two journalists. The call for unity seemed difficult to undertake and positions were hardening: La Gazette refused the alliance proposed by Le Rénovateur26 and judged that Le Rénovateur's ideas contained principles that contradicted the spirit of 'the old French constitution'.

After having been in the background on the oath issue while Laurentie was fiercely defending it, Genoude changed his mind. His newspaper La Gazette de France therefore rallied on this point to Laurentie and published a Declaration of Electors on 1 June 1831, which removed all sacred and religious character from the oath in order to justify this fragile balance for the legitimists: taking the oath to Louis-Philippe while not joining the July monarchy. This oath was made possible because it was addressed to a king who came to power officially in the name of popular sovereignty and not of divine right. Genoude's disciple, Honoré de Lourdoueix, published, in his wake, an Appel à la France against the division of opinions. 27 Genoude then became particularly active during the elections, when he did not hesitate to embody a 'popular royalism' that prevailed over the legitimism of Laurentie and Balzac, whose candidacies ended in bitter failures. Laurentie's correspondence testified to the bitterness of these legitimists in the face of the modern troublemaker Genoude, who alone served his own cause, against that of his movement but with the same or even greater efficency as he had against Louis-Philippe.

This 'popular royalism' constituted the basis of the 'republican national royalism' based on universal suffrage that Genoude elaborated under the horrified eyes of Laurentie, who meanwhile also launched, with Berryer-son, into the alliances of Carlo-republican circumstance. Indeed, Berryer, supported by Laurentie, had not hesitated to contact the Republican candidates in the elections after 1830 in order to defeat the Orléans' candidacies. Agreements had been made event if they were ideological enemies. Fearing that legitimists like Laurentie and Berryer would open themselves to the moderates, Genoude, launched into a race for modernity, asked Odilon Barrot's left wing for a confrontation. Laurentie cried out to the madness of his rival Genoude who preferred to unite with the dynastic left rather than with the legalistic legitimist current: 'The country wants order, and if one attacks the company order with the left or without it, one must perish in this way...'28 While he had opened up to conservative Orleanists, Laurentie redefined his position which proved hostile to the alliance that Genoude undertook with the dynastic left: 'Our system is to fight on the right wing with the principles of the right wing, to sympathize with the ideas and interests of order, instead of embracing anarchic principles and recruiting the men of disorder of the left wing'29 . The local legitimist committees were divided on the attitude to adopt in this dispute over legitimism.

But all the legitimist youth finally rallied around Genoude. 30 The journalist Lourdoueix, close to Genoude, also embodied this particular royalist position that Laurentie fought as ardently as the Orleanists in power. Jacques-Honoré Lelarge, Baron de Lourdoueix (1787-1860) was a well-known publicist, editor-in-chief of the Gazette de France, trained at the Collège de Pontlevoy. Royalist, he nevertheless wanted to reconcile the monarchy with modernity. Head of the Department of Fine Arts at the Ministry of the Interior in 1821, then Royal Censor in 1827, he was also an associate of Le Spectateur and the Mercure de France. He ended up supporting Louis-Philippe. Later, he was an associate and then director of the Gazette de France from 1849 to 1860, and director of the Défenseur du peuple, a political, agricultural and industrial newspaper from 1850 to 1852. Laurentie participated in incessant struggles in the columns of his newspapers or magazines and put all his intelligence into ostracising Genoude and Lourdoueix within the legitimist movement. Even the very diplomatic Berryer-son did not succeed in attenuating this political animosity, as he was himself unconvinced by the political biases of Genoude whose tendency was called 'Genoudisme' by his opponents. They denounced a personal drift in the defence of their cause. 31 They thus talked about a 'Lamennais monarchique' 32 , as heretical in politics as his former master, advocating the revolutionary pre-square as natural borders with notably the Rhine in the East. Their confrontation was also about social issues. Laurentie instigated a social legitimization that Berryer-son vigorously disseminated at the Chamber, 33 especially as Genoude and Lourdoueix advocated the creation of pension funds. Admiring Catholic economists like Villeneuve-Bargemont, Laurentie supported the diffusion of the latter's theories in his newspaper. Alban de Villeneuve-Bargemont (1784-1850) was a supporter of the action, in favour of the uprising of the Duchess of Berry. He was considered one of the founders of social Catholicism, with the publication, in 1834, of his Christian Political Economy. He was then elected deputy for Lille in 1840 and became the rapporteur of the law limiting child labour in factories.

Nevertheless, the lack of popularity of the movement embodied in the press by Laurentie, who issued only a few hundred copies of his Rénovateur, contributed, together with the attack on Louis-Philippe and the laws of September 1835, 34 to the end of the journalist's attempt to modernize royalism. After the failed attack on Louis-Philippe and laws restricting the freedom of the press, Laurentie needed more than ever the financial support of the absolutist legitimists of Charles X's Court in exile. He became almost totally dependent on their decisions. Consequently, legalistic legitimism prevailed but not neo-legitimism in its ideological openness to social and liberal catholics or royalists.

The impossible union between Laurentie and Genoude, at the origin of the failed modernization of legitimate monarchy after 1835?

In 1835, Laurentie returned to La Quotidienne, where he took over the management and worked for the unsuccessful union between the legitimist movement and the Catholic party, after the rupture with Lamennais. For Laurentie, the agreement between the legitimists and the Catholics appeared feasible only around the unifying theme of freedom of education. The former disciple of La Mennais, Montalembert, 35 who was liberal Catholic wanted this issue not to remain the monopoly of the legitimists. In 1831, Charles de Montalembert was appointed Peer of France, an admirable parliamentary, royalist and Catholic orator. With his friend Lacordaire, a Dominican, and the eminent Lamennais priest, they founded the newspaper L'Avenir in which they demanded freedom of conscience, freedom of education, freedom of the press and freedom of association. After the encyclical 'Mirari vos' promulgated by Pope Gregory XVI condemned these new ideas and doctrines, Montalembert, leader of the Catholics, participated in an important way in the recognition of the freedom of secondary education which was voted on March 15, 1850. It was the Falloux law which recognised two kinds of schools: public schools, founded or maintained by communes, 31 372 AP 9: letter from Lostanges dated 17 June 1834 to Laurentie. 32 372 AP 12: letter from Guérard dated 1843, addressed to Laurentie. 33 Berryer Fonds 223 AP 21: file 'Socialism, social issues'. 34 372 AP 9: letter from Germain Laurentie to his brother following the publication of the laws of September 1835. 35 See the work of Sylvain Milbach, Les Chaires ennemies l'Église: l'État et la liberté d'enseignement secondaire dans la France des notables (1830-1850) (Paris: Honoré Champion, 2015). departments or the State; free schools, founded or maintained by individuals or associations. He died on 13 May 1870.

Montalembert allowed the alliance of Catholics with Laurentie, while Genoude once again expressed his popular royalism by choosing electoral reform as the main political battle. Once again going his own way, Genoude nevertheless was aware of the rise of the claim among the people through the banquets that were at the origin of the overthrow of Louis-Philippe's government. Here again, much more than Laurentie and Berryer-son, Genoude embodied a 'popular royalism'.

Against this appellation, Laurentie developed, in his Memoirs 36 , the notion of 'royalism of enthusiasm' that he reckoned to be a kind of class struggle expressed within his legitimist movement. [START_REF] Viviès | Les Droites en France en 1851. Ideas and History[END_REF] Like Balzac, he defended a victory for 'the intelligent ones', clever commoners, over an old, worm-eaten nobility, without succeeding in detaching himself from his elitist image as a royalist and an uncompromising Catholic at the head of his newspaper.

On the other hand, for Genoude, the newspaper had to correspond to a popular movement from below, and not to a position discussed from above by the elite. Popular royalism was however only rarely practiced among the legitimists around Berryer and Laurentie. Criticism was growing. The Duke of Valmy thus reproached La Quotidienne for merely responding to his detractors instead of issuing ideas and also for lacking 'the support of courageous and popular men'. 38 Laurentie's articles were truncated, or unpublished and he appeared to be banned from elections. The Duke of Valmy, mediator between the newspaper's shareholders and Laurentie, then criticized him for questioning the policy of the royalist movement: 'You attack the royalist party, you accuse them of scorning writers, they who let themselves be led by Genoude, you accuse Mr. de Lévi of not answering you. Finally, you are accusing the committee on the account of candidacy'. 39 Indeed, by tearing each other apart, these legitimists had difficulty in establishing La Quotidienne as the newspaper of the royalist party, Genoude winning in the end. 40 This was especially the case as Laurentie was reluctant to merge the absolutist newspaper La France with his own to weigh more against his rival. 41 In addition, Crétineau-Joly and La Rochejaquelein created a new royalist newspaper, La Nation, at the cost of 30 francs, which made them fear the worst: 42 losing more readers. Publisher Bricon then summarized the role of the journalist Laurentie: 'Boxing every day against friends and enemies is the occupation of journalists'. 43 Boxing in the columns of a newspaper or in the Chamber of Deputies was certainly the main occupation, especially since even Berryer was not sure of keeping his influence in the Chamber.

Genoude further developed his notion of 'national royalism' and gave rise to an increasingly precise programme around freedom of association, education, political equality, Catholicism or fraternity. Pragmatically, he supported the periodic holding of general estates (États-Généraux), wished the administration to be free of charge and granted a role to the communal electoral assemblies then to the provincial assemblies, which themselves elected a national assembly. Offended, Laurentie and his entourage tried to isolate Genoude and took subscriptions to La Gazette with La Quotidienne.

But this undermining work had no effect: while Laurentie was politically defeated in the Southwestern elections, Genoude was elected in the South as Deputy of Haute-Garonne in 1846 and with his 'Société des réformistes', he worked to increase his electorate. In the end, it seems that the royalists of Southeast and Southwest France followed Genoude more than Berryer and Laurentie, if we believe the number of La Gazette de France subscriptions which was higher than that of La Quotidienne. In 1832, subscriptions to La Gazette exceeded 7486 while La Quotidienne was satisfied with just over 3432 and in 1841, La Gazette was at 4007 while La Quotidienne fell to 1907 subscriptions. [START_REF] Arguet | Le tirage des quotidiens de Paris sous la monarchie de Juillet[END_REF] La Gazette reached more royalists. Nevertheless, these figures must be put into perspective, since both La Quotidienne and La Gazette de France reached more readers and were put in competition with the industrial press.

Laurentie and Genoude met at the Press Conference of which Laurentie became president, Genoude having skillfully given way to his nomination. [START_REF] De | Le soulèvement de la duchesse de Berry[END_REF] A kind of short-term truce was established and made it possible to start overhauling the legitimist press. Nevertheless, Laurentie refused to merge with La Gazette de France until the end. Following a controversy, involving Laurentie, who denounced the absence of the most loyal servants of the heir to the throne just because they were commoners, the journalist was invited to Frohsdorf in 1846 where the Count of Chambord resided and received him following the customary Ancien Régime etiquette. Then began a correspondence in which the pretender to the throne sought Laurentie's opinions and advice on how legitimacy could be theorized and defended: a traditional vision then prevailed. And this had to be reflected in the press. Finally, La Quotidienne merged with L'Écho français and La France, two absolutist newspapers, the second of which had been particularly successful because of its prevalence over Le Rénovateur, and to compensate, Laurentie was placed at the head of a single body, L'Union monarchique, in 1847.

The February 1848 Revolution marked a turning point. Laurentie, however, did not take it in the same way as he did in 1830. At the age of 55, he did not think of entering into a power struggle and reduced the event to 'Louis-Philippe's business'. Perhaps then the legitimists no longer believed in their ability to re-catholicize and royalize post-revolutionary society: this political despair or disenchantment was reflected in this inaction even if L'Union monarchique then took the name L'Union. A meeting was held at the home of the Marquess of Pastoret, leader of the Parisian legitimist movement, to find out the participants' agreement with the presidential election of Louis-Napoléon Bonaparte: to his great surprise, Laurentie who had legendary mistrust of the Empire, foresaw that Bonapartism would prevail. He warned that an emperor was about to fly on his own. [START_REF] Laurentie | Souvenirs inédits publiés par son petit-fils[END_REF] He was convinced of the imminent arrival of the Bonapartistes in power because of their organizational and theoretical superiority since they succeeded in combining 'order and equality'. 47 His personal failure in the elections of 1848 and 1849 proved once again his inability to multiply, like many of his contemporaries, the titles and tasks: this former professor, who tried in vain to gain fame as a writer, existed only as a journalist of the order, too monarchist to be able to persuade his opponents to rally and too much of a commoner and Parisian to think he could win in the Gers over the networks of the local nobility who won mandates as Deputies. By necessity, while he desired unity between Catholics and royalists, he was overtaken by 'Genoudisme' on his left wing and did not embody at all the popular, commoner and enthusiastic royalism he claimed.

In 1849, Genoude's death gave complete way to Laurentie in the royalist press, even though the young guard entered the newspapers' editorial offices. Lourdoueix 48 took the direction of La Gazette after Genoude. He also defended the Appel au Peuple, which

According to Guillaume Bertier de Sauvigny in Le comte Ferdinand de Bertier et l'énigme de la Congrégation (Paris: Les Presses Continentales, 1948), 337, if the same people could belong to both organizations, the Knights of Faith had exclusively the role of a secret society wishing to place their recruits in the highest positions in the State, with relative success.

Pierre-Sébastien Laurentie, Le Rénovateur (Paris, 1832), first delivery, vol. 1, p. x and 372 AP 21, letter from Lamennais dated 15 April 1832, addressed to Laurentie.

372 AP 21, file 2: letter from Laurentie dated 9 December 1833.

Pierre-Sébastien Laurentie, De la révolution en Europe (Paris: Dentu, Bricon, 1834).

372 AP 7, file 3, letters of 1831-1832: letter from Balzac to Laurentie dated 7 September 1832 and in the Correspondance de Balzac, Roger Pierrot and Hervé Yon(Pléiade, Paris: Gallimard, 2006), vol. 1 (1809-1835), 633, letter 32-181.

Marion Mas, 'Le grand homme selon Le médecin de campagne : une figure "antimoderne" ?', in Boris Lyon-Caen and Marie-Eve Thérenty (eds.), Balzac et le politique (Paris: Pirot, 2007), 85. See Chapter III entitled 'The Napoleon of the People' in Le Médecin de campagne. Doctor Benassis criticized the representative regime and parliamentarism while developing his programme on a social order based on inequalities: only merit could bring valuable men to the higher spheres. See Bernard Guyon, Un inédit de Balzac: le catéchisme social (Paris: La Renaissance du livre, 1933), 32.

 18 Article by Bonald in LeRénovateur, vol. 2,

Pierre-Sébastien Laurentie, Le Rénovateur, first volume (Paris, 1832), 194.

Jacques-Honoré de Lourdoueix, Appel à la France contre la division des opinions (Paris: Bureaux de la Gazette de France, 1831).

Laurentie's article in the Courrier de l'Europe of 4 January 1833.

Idem.

Grand Larousse universel, article 'Genoude'.

Laurentie found unrealistic: 49 a plebiscite had to be organized to choose between a republic or a legitimate monarchy. In response, Laurentie made Henri V's voice heard by writing for him anonymously, La Politique royale. 50 A few months later, in 1850, the Count of Chambord condemned Lourdoueix's doctrine of the Appel au Peuple, which undermined the principle of heredity in the Wiesbaden circular also called the Barthélémy manifesto.

The events of 1851 with the Coup d'État of Napoléon III proved him right, but Laurentie nevertheless realized that he had failed, even with regard to 'Genoudisme'. After 1830, he was a supporter of a parliamentary monarchy supported by the Church of Rome, while Genoude laid the foundations for a monarchy based on democratic principles and a religion less dependent on Rome, which Lourdoueix agreed upon. Laurentie expressed to Berryer-son his royalist opponents' triumph: '[Lourdoueix] has won; and our policy of 1831, in Le Courrier de l'Europe, and yours, my dear friend, for 20 years in the Chamber, all this movement printed with the most rebellious ideas, all this is defeated; it is national law that wins 51 !' Despite the openings made to monarchical law, the law resulting from a legislative process according to the principles of representative democracy was more convincing. Within the very heart of the legitimist movement, the monarchy and its fundamental law were definitively relegated to the memories of a bygone old France.

Laurentie's correspondence testified to the feeling of the end of a world that was constantly disintegrating, especially since his best support within L'Union, the Duke of Valmy, joined the Emperor after the Coup d'État of December 2, 1851. Laurentie was then shaken in his deepest convictions and ended up freezing his royalist theories, out of fidelity.

Internationally, Laurentie had been hoping since the beginning of the July monarchy that his newspaper would be 'officially approved' in Russia. 52 In contrast to the period of modernization 1830-5, the years 1850 and 1860 were years of journalistic struggles, of an uncomplicated alliance with the Emperor of Russia that had kept embarrassing these royalists because of their Catholicity, faced the most intransigent and russian representative of order and counter-revolution in Europe.

His repeated attacks in the press against Napoleon III earned Laurentie imprisonment and discussions with Blanqui 'the prisoner' ('l'Enfermé') in 1862, who, moreover, accused him of having sacrificed all honour in working for an alliance with Russia. 53 The origin of this alliance was based on the fact that it was unacceptable to Laurentie that freedom mixed with order was better represented by Napoleon III than by his own camp.

He then developed theories to fight against atheism in the State, which he believed was the source of the evils of the whole society, and launched a struggle of ideas against Ernest Renan, while acknowledging his failure in La Gazette de France. He clearly acknowledged the loss of subscriptions for L'Union in Paris and in the whole country. This steady decrease was attributed in part to the active propaganda carried out by La Gazette: 'The friends of La Gazette are more skillful and active than we are'. 54 But the real reason was evoked by another royalist who noticed that in the whole country, their royalist propaganda fell on barren ground: 'It should have the possibilty to address convictions or even passions, there is nothing left but economic interests'. Aging but respected within his movement, Laurentie now embodied royalist immobility and the impossible theoretical reform of the legitimate monarchy through the White Flag affair after 1870 when the count of Chambord refused to come back on his throne because he was tied to the white flag of the monarchy and hostile to the tricolour flag, despite his action during the defeat and fall of the Empire and the Commune.

To conclude, Laurentie's moderate legitimist political line after 1830 was opposed to Genoude and the absolutists. Too close to the Orléans monarchy without adhering to it, these neo-legitimists saw their finances cut off in 1835 by the absolutists. Subsequently, Montalembert shattered any hope of unity and for the creation of a large conservative party under the July monarchy by separating Catholics and royalists.

Also, these attempts to modernize the legitimate monarchy were not successful because they surpassed, especially with 'Genoudisme', the very legitimist movement. This was a scandal for conservative people like Laurentie, who could not envisage a monarchical republic and whose open-mindedness did not go further than advising a modern prince.

The new hypothesis that emerges from this work is that Genoude prefigured an exit from the French left/right-wing cleavage (which Laurentie totally rejected) with the claim of a kind of 'republican monarchy', a real innovation in terms of a political system combining monarchy and universal vote. On the other hand, Laurentie, the Duke of Fitz-James and others thereafter, remained in a traditional post-revolutionary cleavage. Caught up by absolutism and 'émigrés', Parisian legitimists did not really impose their vision of legitimism. Nevertheless, if Genoude did instill radical modernization into legitimism, he was marginalized by his movement's leading personalities. The impossibility of the union between moderate legitimitism and Genoudisme, for which Laurentie was in part personally responsible, sealed the end of a legitimist right wing in phase directly with modernity and the people.