Nyoman Juniarta
email: nyoman.juniarta@loria.fr

Miguel Couceiro
email: miguel.couceiro@inria.fr

Amedeo Napoli
email: amedeo.napoli@loria.fr

Sequential Pattern Mining within Formal Concept Analysis for Analyzing Visitor Trajectories

Keywords: Formal Concept Analysis, pattern structures, sequential pattern mining

This paper presents our work about mining visitor trajectories, within the framework of CrossCult European Project about cultural heritage. We present a theoretical and practical research work about the characterization of visitor trajectories and the mining of these trajectories as sequences. The mining process is based on two approaches, namely the mining of subsequences without any constraint and the mining of frequent contiguous subsequences. Both approaches are defined within Formal Concept Analysis and its extension pattern structures. In parallel, a similarity measure allows us to build a hierarchical classification which is used for interpretation and characterization of the trajectories w.r.t. four well-known visiting styles in museum studies.

INTRODUCTION

This paper is related to the CrossCult European Project about cultural heritage (http://www.crosscult.eu/). The general idea of Cross-Cult is to support the emergence of a European cultural heritage by allowing visitors in different locations (e.g. museum, city, archaeological site) to consider their visit at a European level by using adapted computer-based devices.

In this project, we are mainly interested in the analysis of visitor trajectories and recommendation. The trajectory of a visitor in a specific location is considered as a multi-dimensional sequence depending on a number of variables, such as space (e.g. paths, rooms, environment), time (e.g. hour, day, season, news), history and geography (e.g. town, region, country. . . ). Moreover, additional domain knowledge and general knowledge bases such as DBpedia, Freebase, or YAGO can be reused to draw inferences and improve the quality of both analysis and recommendation.

Here, we have two main objectives, (i) the mining of visitor trajectories based on sequence mining, and (ii) the characterization of a trajectory in terms of the subsequences which are mined. We assume that the subsequences can be related to the visiting styles, the visit content, and the environment. Thus subsequences can be used for analyzing the trajectory of a visitor and for making recommendations all along the visit. Moreover, the occurrences of some subsequences at a given moment within a trajectory can witness a change of behavior -which in turn triggers a change in the recommendations.

In the present paper, we discuss theoretical and practical work about the definition of visitor trajectories and the mining of these trajectories as sequences. The mining process is based on two approaches about sequence mining in Formal Concept Analysis (FCA [START_REF] Ganter | Formal Concept Analysis: Mathematical Foundations[END_REF]): MRGS for "Mining Rare General Subsequences" [START_REF] Codocedo | A Proposition for Sequence Mining Using Pattern Structures[END_REF] and MFCS for "Mining Frequent Contiguous Subsequences" [START_REF] Buzmakov | On mining complex sequential data by means of FCA and pattern structures[END_REF]. The first approach mines rare subsequences in a general way, i.e. gaps may appear in the subsequences, while the second approach searches for frequent subsequences without any gap (a kind of substrings). We also reuse the similarity measure sim ACS developed for analyzing the trajectories of patients between hospitals [START_REF] Egho | A contribution to the discovery of multidimensional patterns in healthcare trajectories[END_REF][START_REF] Egho | On measuring similarity for sequences of itemsets[END_REF]. If the original paper about MRGS [START_REF] Codocedo | A Proposition for Sequence Mining Using Pattern Structures[END_REF] was interested in rare subsequences, this is no more the case here and we work on frequent subsequences as well. This similarity measure allows us to build a hierarchical classification that will play a role of "reference classification". For analyzing and interpreting the trajectories of visitors, it is interesting to compare the outputs of MRGS and MFCS algorithms w.r.t. the clustering produced by sim ACS . Moreover, these outputs and the clustering are analyzed thanks to four theoretical visiting styles, namely "ant", "butterfly", "fish" and "grasshopper" [START_REF] Véron | Ethnographie de l'exposition[END_REF].

Several challenges are faced in this research work in the FCA framework: the mining of complex sequential data and dynamics in adapting two algorithms based on pattern structures, the analysis of the trajectories based on jumping emerging patterns and clustering. Here, data are not necessarily big but are rather complex and multidimensional, and this should be taken into account.

The paper is organized as follows. Section 2 recalls the basic definitions about sequence mining that are useful for understanding the present work. Then, Section 3 presents the characteristics of the dataset that was used as a basis for the current work. In Section 5 and Section 6, we present respectively the application of clustering on data enabling to build classes of visitors, and the application of two algorithms for mining interesting subsequences. Finally, an interpretation of the results and a discussion on the characterization of the visitor trajectories is given in Section 7.

THE MINING OF SEQUENCES 2.1 Basic Definitions

Pattern mining is the task of finding repeated occurrences in a dataset. For example, in a data about customer transactions, an objective can be to find a set of items that are frequently ordered in a single transaction. Another complex objective is to detect a set of items that are likely ordered within certain transactions. These specific tasks in pattern mining are related to sequential pattern mining. We recall below the basic definitions that we will need. Definition 1. A sequence is an ordered list ⟨s 1 s 2 . . . s m ⟩, where s i is an itemset {i 1 , . . . , i n }, and m is the size of a sequence. The length of a sequence is the total number of items, i.e. |s i |.

For example, ⟨{a, b}{a, c, d}⟩ is a sequence with size 2, since it contains two itemsets, whereas its length is 5.

Definition 2. A sequence s = ⟨s 1 s 2 . . . s m ⟩ is a subsequence of sequence s ′ = ⟨s ′ 1 s ′ 2 . . . s ′ n ⟩, denoted by s ≼ s ′ , if there exist indices 1 ≤ i 1 < i 2 < . . . < i m ≤ n such that s j ⊆ s ′
i j for all j = 1 . . . m and m ≤ n.

Therefore, the sequence ⟨{a}{d}⟩ is a subsequence of ⟨{a, b}{a, c, d}⟩, while sequence ⟨{c}{d}⟩ is not.

One way of evaluating the quality of a subsequence is to compute the support of the subsequence. Given a user-defined threshold, the subsequence can be "frequent", i.e. the support is above the threshold, or "rare", i.e. the support is below the threshold. Definition 3. Let S be a database of sequences. The support of a sequence s in S is: support(s, S) = |{s i ∈ S; s ≼ s i }| There exist algorithms that can retrieve all frequent sequences [START_REF] Ayres | Sequential pattern mining using a bitmap representation[END_REF][START_REF] Han | PrefixSpan: Mining sequential patterns efficiently by prefix-projected pattern growth[END_REF]. A long sequence can have an exponential number of subsequences. Thus, if a long sequence is frequent, these algorithms return all of its subsequences. This leads to the retrieval of many uninteresting patterns. This issue has been studied in [START_REF] Gomariz | ClaSP: an efficient algorithm for mining frequent closed sequences[END_REF][START_REF] Wang | BIDE: Efficient mining of frequent closed sequences[END_REF][START_REF] Yan | CloSpan: Mining: Closed sequential patterns in large datasets[END_REF]] by introducing the concept of "closed sequence". Using this concept, the size of output can be reduced by disregarding sequences which have another supersequence with the same support (hence not closed).

Beside mining frequent sequences, another complex task is that at finding homogeneous sequence groups (clustering). To achieve such a task, a distance or a similarity measure between two sequences has to be defined. The similarity measure sim ACS was proposed in [START_REF] Egho | On measuring similarity for sequences of itemsets[END_REF], which counts the number of all common subsequences (ACS). This measure is formulated as:

sim ACS (S i , S j ) = ϕ C (S i , S j ) max{ϕ D (S i ), ϕ D (S j )}
where ϕ C (S i , S j ) is the number of all common distinct subsequences between S i and S j , while ϕ D (S i ) is the number of all distinct subsequences of S i .

FCA and Pattern Structures

Formal concept analysis (FCA) is a mathematical framework based on lattice theory and used for classification, data analysis, and knowledge discovery [START_REF] Ganter | Formal Concept Analysis: Mathematical Foundations[END_REF]. From a formal context, FCA detects all formal concepts, and arranges them in a concept lattice. If an object д has an attribute m, then (д, m) ∈ I . An example of a formal context is shown in Table 1. This table shows whether a visitor (V 1 -V 4 ) visits an item (102, 302, 402, or 704).

The Galois connection for a formal context (G, M, I ) is defined as follows:

Definition 5. For a subset of objects A ⊆ G, A ′ is the set of attributes that are possessed by all objects in A, i.e.:

A ′ = {m ∈ M |∀д ∈ A, (д, m) ∈ I }, A ⊆ G Dually,
for a subset of attributes B ⊆ M, B ′ is the set of objects that have all attributes in B, i.e.: Table 1: A formal context for four visitors, with four items: 102, 302, 402, and 704 as an example. An × indicates that the visitor visit the item. A concept lattice can be formed using the ≤ relation which defines the order among concepts. For the context in Table 1, the formal concepts and their corresponding lattice are shown in Fig. 1.

102 302 402 704 V 1 × V 2 × × V 3 × × × V 4 × × × Figure 1: Concept lattice for the formal context in Table 1 B ′ = {д ∈ G |∀m ∈ B, (д, m) ∈ I }, B ⊆ M
FCA is restricted to specific datasets where each attribute is binary (e.g. has only yes/no value). For more complex values (e.g. numbers, strings, trees, graphs. . . ), FCA is then generalized into pattern structures [START_REF] Ganter | Pattern structures and their projections[END_REF]. Definition 7. A pattern structure is a triple (G, (D, ⊓), δ ), where G is a set of objects, (D, ⊓) is a complete meet-semilattice of descriptions, and δ : G → D maps an object to a description.

The operator ⊓ is a similarity operation that returns the common elements between any two descriptions. A description can be a set, a sequence, or other complex structure. In the case of set as a description, ⊓ corresponds to set intersection (∩), i.e. {a, b, c} ⊓ {a, b, d} = {a, b}, and ⊑ corresponds to subset inclusion (⊆). In the case of sequence as a description, ⊓ is a set of common closed subsequences (SCCS) [START_REF] Codocedo | A Proposition for Sequence Mining Using Pattern Structures[END_REF]. Similarly, ⊑ corresponds to subsequence inclusion (≼). Definition 8. The Galois connection for a pattern structure (G, (D, ⊓), δ ) is defined as:

A = ⊓ д ∈A δ (д), A ⊆ G d = {д ∈ G |d ⊑ δ (д)}, d ∈ D
Finally, a pattern concept is similar to a standard formal concept:

Definition 9. A pattern concept is a pair (A, d), A ⊆ G and d ∈ D, where A = d and d = A.
Examples of pattern concepts are shown in Table 5. There are two definitions of ⊓ for sequence, and will be explained in Section 6.1.

Sequence Mining in FCA

In this section we briefly present the two algorithms that are adapted for mining the trajectories of visitors in a museum, namely MFCS [START_REF] Buzmakov | On mining complex sequential data by means of FCA and pattern structures[END_REF] and MRGS [START_REF] Codocedo | A Proposition for Sequence Mining Using Pattern Structures[END_REF]. The names of the algorithms are not used as such in the papers but here we use them by commodity. Both algorithms are original and very efficient, and among the few algorithms performing sequence mining in the framework of FCA.

MFCS was originally introduced for mining trajectories of patients in hospitals. One important characteristic of MFCS is that it mines contiguous subsequences, or stated differently, subsequences without any gap between items. This is due to the fact that physicians are mainly interested in consecutive events when analyzing healthcare trajectories. In addition, but this is not needed in our framework, MFCS is able to take into account a partial orderinggiven by domain knowledge for example -defined on the items composing the sequences.

MRGS is also a sequence miner based on pattern structures but with a different purpose. The objective of MRGS is to mine rare rather than frequent subsequences, and in particular long subsequences with special characteristics. The algorithm is based on a specific pattern structure of subsequences, where the similarity operation is based on the discovery of common close subsequences (SCCS operation is illustrated in a next section). The SCCS operation is based on a directed graph of alignments (DAG of alignments) which guides the mining of common subsequences. The algorithm shows very good performances and is most probably one of the few algorithms whose objective is the mining of rare subsequences. In our framework, we adapted MRGS and the support threshold for comparison purposes with frequent subsequences. However, in our context we will use MRGS as a standard sequence miner and we will be interested in frequent subsequences.

THE DATASET OF MUSEUM VISITORS 3.1 The Museum

In the framework of the CrossCult project, we are working on a specific dataset about the trajectories of 254 visitors in Hecht Museum in Haifa, Israel [START_REF] Dim | Early detection of museum visitors identities by using a museum triage[END_REF][START_REF] Lanir | The influence of a location-aware mobile guide on museum visitors' behavior[END_REF]. To record the movement of visitors, the museum is equipped with a wireless sensor network. Each visitor brings a small component of this sensor, and some beacons are placed throughout the museum such that they can capture the position of visitors in a given time. Using this equipment, visitor trajectories can be obtained.

In the raw dataset, a visitor trajectory contains a list of visited items, where each visit is composed of three elements: "start time", "end time", and "item name". An example is presented in Table 2. When modeling trajectories into sequences, in this paper we consider only the "item name", so every itemset contains only one item. For simplicity, we omit the curly brackets to describe an itemset. Therefore we will write ⟨{a}{d}{e}⟩ as ⟨a, d, e⟩.

A visitor can have visits with various time lengths. In order to obtain more meaningful results and to reduce the complexity, we only consider visits lasting at least 90 seconds, but this is a parameter than can be relaxed or more constrained. Thirty-eight trajectories have no visit more than this threshold, so they are ignored, leaving us with 216 trajectories. Moreover, we model each trajectory as a sequence of visited items. Therefore, for trajectory in Table 2, the corresponding sequence is ⟨Crafts and Arts, Wooden Tools⟩. This preprocessing results in sequences of various size. Forty-five sequences have only one itemset, while three sequences have more than 15 itemsets.

We group the museum items according to their location, so that we obtain 8 categories of items. To illustrate the numbering of items, the first two categories and their items are listed in Table 3. We convert the raw dataset into sequences of items, where each item is represented by its ID. We define the IDs such that we can infer the category of an item by its first digit. Therefore, we obtain a dataset of 216 sequences of visitor trajectories (named V 1 -V 216 ) where each sequence is composed by a list of IDs, as illustrated in Table 4.

The Four Visiting Styles

In a seminal work about the typing of visitor styles in a museum [START_REF] Véron | Ethnographie de l'exposition[END_REF], four main behaviors have been detected and described, leading to different recommendations all along a visit [START_REF] Kuflik | Analysis and prediction of museum visitors' behavioral pattern types[END_REF][START_REF] Zancanaro | Analyzing museum visitors' behavior patterns[END_REF]. These four styles are summarized below:

• The ant is a visitor who will surely see all the works following their location order in the museum. Then the recommendation can be the following item, but depending also on some environmental factors such as the crowd in the museum, the accessibility of the item and the fatigue of the visitor. • The grasshopper is a visitor who will see only certain artworks, jumping from one to another. Then, to encourage such a person to visit more items, the recommendation can be to visit items having a content similar to items already visited. • The butterfly is a visitor wanting to discover some and not all artworks, without having any exact preferences. Then, the recommendation is open and can be based on surprise (items which are very different one from the other). • The fish is a visitor who does not feel that much interested in the artworks and stays most of the time in the center of the rooms without any precise objective. Then the recommendation can be to visit the most famous items in the museum which are the closer to the current visitor location, for encouraging the visitor to continue the visit and gain more interest. Indeed, a visitor can change his/her style during a visit and other elements may be of importance, e.g. crowd or fatigue of the visitor.

WORKFLOW FOR ANALYZING THE TRAJECTORIES

In the following, one objective is to map specific subsequences included in the visitor trajectories to each visiting style for characterizing more precisely the style and then making smarter recommendations. To identify the behavior of each visitor, we propose the following workflow:

(1) Cluster the visitor trajectories and assign a label for each visitor (Section 5). (2) Create two concept lattices using MFCS and MRGS over the whole dataset (Section 6.1). (3) From the two lattices, find jumping emerging patterns (JEPs)

for each label (Section 7.2). ( 4) Based on their JEPs, these labels are then mapped into four visiting styles as explained in Section 3.2.

CLUSTERING OF TRAJECTORIES

In this first experiment, we reuse the sim ACS similarity measure for clustering the visitor trajectories. The idea is to check whether it is possible to distinguish the four visiting styles introduced above. We apply hierarchical clustering 1 based on sim ACS to build a distance matrix between individuals. From the resulting dendrogram, we retained 5 clusters denoted by "A", "B", "C", "D", and "E". Four of them are expected to match the four visiting patterns, namely ant, butterfly, fish, and grasshopper. The last cluster will gather all 1 We use the hclust method from the R software [START_REF] Core | R: A Language and Environment for Statistical Computing[END_REF].

non-classified trajectories. These five clusters have various sizes. Cluster "A", "B", "C", "D", and "E" have 11, 11, 59, 102, and 33 visitors respectively. Actually, it is not easy to directly match the five clusters to corresponding visiting styles. For doing so, we will analyze the subsequences that can be attached to each cluster of trajectories. The benefit of the clustering is actually to provide a label among "A", "B", "C", "D", and "E" to the visitors. Thanks to these labels, we can search the so-called "jumping emerging patterns" and attach a characterization to the clusters based on the mined subsequences.

THE MINING OF TRAJECTORIES CONSIDERED AS SEQUENCES 6.1 Mining Subsequences with MFCS and MRGS

Below, we explain the application of the MFCS and MRGS algorithms to the museum dataset and the building of an associated concept lattice. Moreover, as discussed in Section 6.2, the mining of jumping sequential patterns will help us to characterize the visitor trajectories.

In MFCS and MRGS, pattern structures are used for mining sequences. The similarity operator (⊓) between any two sets of sequences is defined as the set of closed common subsequences (SCCS) in the two input sequences. Then, given two sequences, say S 1 = ⟨401,502,503⟩ and S 2 = ⟨401,503,502⟩, the similarity between these descriptions is:

δ (S 1 ) ⊓ δ (S 2 ) = {⟨401,502,503⟩} ⊓ {⟨401,503,502⟩} = {⟨401,502⟩, ⟨401,503⟩}
In the dataset, the items are grouped into categories (indicated by their first digit) and the SCCS calculation is performed, checking whether two items belong to the same category. Using the MFCS algorithm it becomes:

δ (S 1 ) ⊓ δ (S 2 ) = {⟨401,502,503⟩} ⊓ {⟨401,503,502⟩} = {⟨502⟩, ⟨503⟩, ⟨401,5,5⟩}
It should be noticed that MFCS mines contiguous subsequences, i.e. in Definition 2, i k = i k -1 + 1 for all k ∈ {2, 3, . . . , m}. Furthermore, subsequence ⟨401,5,5⟩ can be regarded as a generalization, meaning that after item 401, the next two visited items are something in category 5.

In parallel, the default similarity operator of MRGS algorithm can be modified to accommodate our needs, such that non-contiguous common subsequences can be mined:

δ (S 1 ) ⊓ δ (S 2 ) = {⟨401,502,503⟩} ⊓ {⟨401,503,502⟩} = {⟨401,502⟩, ⟨401,503⟩, ⟨401,5,5⟩}
Then, based either on MFCS or MRGS, a concept has an extent including a set of trajectories and an intent including a set of common subsequences. Again, it should be noticed that, based on whether a subsequence is contiguous or not, the obtained concepts are different.

For example, the concepts corresponding to Table 4 are shown in Table 5. Notice that both algorithms obtain a concept whose extent is V 2 , V 3 , V 4 , albeit with different intent. Based on MRGS, the common subsequence of V 2 , V 3 , V 4 is ⟨1, 402⟩, while according to MFCS, their

V 1,2 ⟨1,4⟩ not present V 1,4 ⟨1⟩, ⟨4⟩, ⟨704⟩ ⟨1,1⟩, ⟨1,4⟩, ⟨1,704⟩ V 2,3 ⟨2⟩, ⟨102⟩, ⟨402,8⟩ ⟨102,402,8⟩, ⟨102,2,8⟩ V 3,4 ⟨1⟩, ⟨302⟩, ⟨402⟩, ⟨7⟩ ⟨1,302,402⟩, ⟨302,1⟩, ⟨1,7,402⟩ V 1,3,4 ⟨1⟩, ⟨4⟩, ⟨7⟩ ⟨1,4⟩, ⟨1,7⟩ V 2,3,4 ⟨1⟩, ⟨402⟩ ⟨1,402⟩ V 1,2,3,4 ⟨1⟩, ⟨4⟩ ⟨1,4⟩
common subsequences are ⟨1⟩ and ⟨402⟩. This is because items 1 and 402 are not contiguous in V 3 and V 4 .

Jumping Emerging Patterns

FCA is a non supervised classification process that can be turned into a supervised process thanks to the adding of a target attribute in the context, generally corresponding to a target class. Then the idea is to search for the so-called "Jumping Emerging Patterns" (JEPs) [START_REF] Dong | Efficient mining of emerging patterns: Discovering trends and differences[END_REF]. We have already applied this approach in [START_REF] Asses | A Hybrid Classification Approach based on FCA and Emerging Patterns -An application for the classification of biological inhibitors[END_REF] for analyzing and characterizing clusters of biological inhibitors. Here we adapt the same idea for characterizing this time the clusters of visitors discovered with the similarity measure sim ACS . More precisely, five clusters are discovered by classifying visitor trajectories with sim ACS . These same trajectories are then considered as sequences composed of subsequences. Then a set of characteristic subsequences is extracted and these subsequences are used as "attributes" in a formal context where objects are visitor trajectories. The resulting formal context is completed with an extra attribute corresponding to the "cluster information", i.e. the cluster in which the trajectory is classified according to sim ACS . A concept lattice can then be built from this completed context.

More interestingly, the cluster information is used for characterizing the concepts whose extents include trajectories of a single cluster. The intents -made of subsequences -of these particular concepts are JEPs, and as such they can be used to characterize the corresponding clusters. For example, if the extent of the concept ({V 103 , V 165 , V 188 }, {⟨4⟩, ⟨1⟩, ⟨306⟩, ⟨701,707⟩}) includes visitors from cluster B only, then its intent is a JEP for that cluster.

DISCUSSION 7.1 About Interesting Subsequences

The first part of Table 6 shows some interesting contiguous subsequences from 4677 concepts discovered by MFCS. Thirty-three persons are visiting three items contiguously in category 1 of items located near the entrance. This is interesting to be noticed, as visitors are likely to spend more time in rooms located near the entrance. Indeed, at arrival they are not tired and they show higher interest. Then items of importance could be placed near the entrance for getting sufficient attention from visitors.

Thirteen people visit an item in category 7 -this category corresponds to items in the room of "Ancient Ship" which is one of the most famous items in this museum -right after an item in category 1. This is again an interesting subsequence, because many other categories are located between categories 1 and 7. This means that these visitors have a specific interest for the "Ancient Ship" in the museum, since they skip all the items located between entrance and "Ancient Ship" (both categories can be considered as being "far" from each others).

From 8019 concepts obtained by MRGS, some subsequences are presented in the second part of Table 6. The subsequence ⟨1,1⟩ has a support of 66 with MFCS, and it has quite a similar support (69) with MRGS. Then we can draw the same conclusion, meaning that when a person visits two items in category 1, it is likely in continuation (to be compared with the preceding subsequence ⟨1,1,1⟩). Now, more interestingly, 38 visits of an item in category 3 follow visits of an item in category 1, while only 9 visitors are doing the opposite. A similar conclusion can also be drawn with pairs ⟨4,7⟩ (31) and ⟨7,4⟩ [START_REF] Gomariz | ClaSP: an efficient algorithm for mining frequent closed sequences[END_REF]. Based on this observation, we can infer that visiting a museum is an "oriented activity"and that some directions are more preferred than others, or "naturally followed", just as it is the case for the ordering of the rooms in the museum. By contrast, only a few visitors are quitting the "natural flow" and go "backwards". Among these visitors, we can probably find those who want to revisit preceding items.

Cluster Characterization

Now we are interested in characterizing the five clusters that were introduced in the previous section. For doing so, JEPs are searched in the two concept lattices obtained with MFCS and MRGS algorithms. Some of these concepts are listed in Table 7 and Table 8. From these JEPs, we manually assign each cluster to a behavior.

First, from both MFCS and MRGS, we cannot find any satisfying concept for JEP of cluster "E". This is because among all the concepts whose extent is exclusively from cluster "E", none of them has more than one visitor. If we consider the dataset, among 33 members of cluster "E", 32 of them visit less than 2 items during their whole visit. We can assume that these visitors are not really interested in visiting the museum. Therefore, we can "safely" label this cluster as fish.

Cluster "D" is more easily distinguishable. Based on subsequences of concept FD2-FD4, many visitors in this class skip some items. Also, in concept RD1 and RD2, some of them visit other items after item 701. This is not a natural direction, because items in category 7 are located farther from the entrance than items in category 4 or 5. We can interpret the visitors of this cluster as grasshopper, since they "jump" from one item to another.

Clusters "A", "B", and "C" are relatively similar to each other. The visitors associated to these clusters follow an ant behavior: a natural flow (based on RA1-RC1) and no "jump" (based on FA1-FC2). However, in FC3, three visitors visit 101, then 102, then back again to 101, indicating rather a butterfly behavior.

CONCLUSION

In this article, we have presented our experiments in mining visitor trajectories that are modeled as sequences of items. We incorporated a classification of museum items and built a concept lattice using pattern structures. We applied two sequence miners based on FCA to the visitor trajectories, namely MFCS and MRGS, to discover interesting contiguous and general subsequences.

Our result highlight some interesting patterns that may define visitor behaviors. This can help museum researchers to analyze and evaluate the placement of items and the visiting styles. Moreover, we have also studied the possibility of clustering the visitors based on a concept lattice. These clusters can be analyzed to build a recommendation system for future visitors, but we did not yet study this aspect until now.

In this paper, we only included partial information about the museum in the sequences. More interesting results are expected if other elements are taken into account, such as more general knowledge about history and geography, as well as the duration and time of the visit. Furthermore, the selection of interesting concepts can be also guided by the "stability" of concepts [START_REF] Sergei | Concept stability for constructing taxonomies of web-site users[END_REF]. Finally, from a more dynamic point of view, ongoing information such as comments and state of the visitor during the visit could be also considered for analysis and on-line recommendation.

Definition 4 .

 4 A formal context is a triple (G, M, I ), where G is a set of objects, M is a set of attributes, and I is a binary relation between G and M, i.e. I ⊆ G × M.

Definition 6 .

 6 A formal concept is a pair (A, B), where A ⊆ G and B ⊆ M, and such that A ′ = B and B ′ = A. A formal concept (A, B) is a subconcept of (C, D) -denoted by (A, B) ≤ (C, D) -if A ⊆ C (or equivalently D ⊆ B).

Table 2 :

 2 An example of one visitor trajectory.

	Start time End time Item name
	12:55:39 12:58:05 Crafts and Arts
	12:58:06 12:58:22 Religion and Cult
	12:58:22 12:58:27 Building Methods and Facilities
	12:58:29 13:05:09 Wooden Tools

Table 3 :

 3 Grouping of museum items

	Category Items and their ID
	1	Entrance Reuben Hecht (101),
		Symbols Jewish Menorah (102),
		Persian Cult (103), Jerusalem Photo (104)
	2	Religion and Cult (201),
		Everyday Pottery (202),
		Phoenician Writing (203),
		Burial Tradition (204),
		Building Methods and Facilities (205),
		Maritime Commerce (206),
		Imported Pottery (207),
		Crafts and Arts (208)
	. . .	. . .

Table 4 :

 4 Examples of visitor trajectories.

	Visitor Trajectory
	V 1	⟨101, 101, 401, 704⟩
	V 2	⟨102, 402, 808, 206, 808⟩
	V 3	⟨302, 102, 201, 302, 705, 402, 802⟩
	V 4	⟨104, 704, 602, 302, 402, 103⟩

Table 5 :

 5 The concepts that are computed by of MFCS and MRGS from four visitors in Table4

	Extent	Intent (MFCS)	Intent (MRGS)
	V 1		⟨101,101,401,704⟩
	V 2	⟨102,402,808,206,808⟩
	V 3	⟨302,102,201,302,705,402,802⟩
	V 4	⟨104,704,602,302,402,103⟩

Table 6 :

 6 Some interesting subsequences mined by MFCS (left) and MRGS (right)

			Subsequence Support
	Subsequence Support	⟨1,3⟩	38
	⟨1,1,1⟩	33	⟨3,1⟩	9
	⟨1,7⟩	13	⟨4,7⟩	31
	⟨1,1⟩	66	⟨7,4⟩	11
			⟨1,1⟩

Table 7 :

 7 Interesting concepts discovered by the MFCS algorithmV 107 , V 121 , V 133 , V 201 , V 202 } {⟨1,1,402⟩, ⟨103⟩, ⟨2⟩} 6 A FA2 {V 70 , V 93 , V 107 , V , V 8 , V 28 , V 32 , V 84 , V 152 } {⟨102⟩, ⟨101,1,101⟩} 6 C FC2 {V 53 , V 152 , V 169 , V 189 , V 190 , V 203 } {⟨7⟩, ⟨102,4⟩}

	Concept ID Extent	Intent	Support Cluster
	FA1	{V 70 , 121 }	{⟨402⟩, ⟨103,104⟩}	4	A
	FB1	{V 103 , V 165 , V 188 }	{⟨4⟩, ⟨1⟩, ⟨306⟩, ⟨701,707⟩} 3	B
	FC1	{V 4 6	C
	FC3	{V 4 , V 8 , V 32 }	{⟨101,102,101⟩}	3	C
	FD1	{V 54 , V 105 , V 139 , V 168 }	{⟨202,4⟩}	4	D
	FD2	{V 139 , V 168 }	{⟨202,405,701⟩}	2	D
	FD3	{V 46 , V 47 }	{⟨101,602⟩}	2	D
	FD4	{V 89 , V 163 }	{⟨602,203⟩}	2	D

Table 8 :

 8 Interesting concepts discovered by theMRGS algorithm V 107 , V 121 , V 133 , {⟨1,1,402,2⟩, ⟨1,1,4⟩, 6 A V 201 , V 202 } ⟨103,402,2⟩, ⟨103,4⟩} RB1 {V 142 , V 183 , V 192 } {⟨102,1,1,1,1⟩, ⟨102,103,1,1⟩, 3 B ⟨1,1,1,1,1⟩, ⟨1,103,1,1⟩} RC1 {V 4 , V 8 , V 28 , V 84 , V 152 } {⟨1,1,1,101⟩, ⟨1,101,1,101⟩,

	Concept ID Extent	Intent	Support Cluster
	RA1	{V 70 , 5	C
			⟨1,1,1,1⟩, ⟨1,101,1,1⟩,
			⟨101,1,1,1⟩, ⟨101,101,1,1⟩,
			⟨101,101,101⟩, ⟨102,101⟩, ⟨102,1⟩}
	RD1	{V 71 , V 79 }		2	D
	RD2	{V 97 , V 98 }	{⟨701,406⟩}	2	D


Sébastien Briot

Vigen Arakelian

Design of reactionless linkages and robots equipped with balancing Assur groups

In the present Chapter, we consider the shaking moment and shaking force balancing through the use of additional Assur groups mounted on the mechanism to be balanced. Two types of mechanisms are considered: (i) the in-line four-bar linkage and (ii) the planar parallel robots with prismatic pairs. For both types of mechanisms, the proposed solution allows the reduction (or even the cancellation in the case of the four-bar linkage) of the number of counter-rotations used for obtaining the shaking moment balancing, which decreases the design complexity and the inherent problems due to the use of counter-rotations (backlash, noise, vibrations, etc.). All theoretical developments are validated via simulations carried out using ADAMS software. The simulations show that the obtained mechanisms (both inline four-bar linkages and planar parallel robots) transmit no inertia loads to their surroundings, i.e. the sum of all ground bearing forces and their moments are eliminated.

Introduction

It is known that fast-moving machinery with rotating and reciprocating masses is a significant source of vibration excitation. The high-speed linkages can generate significant fluctuating forces with even small amounts of unbalance. Thus, a primary objective of the balancing is to cancel or reduce the variable dynamic loads transmitted to the frame and surrounding structures. The reduction of vibrations leads to the S. Briot Institut de Recherche en Communications et Cybernétique de Nantes (IRCCyN), UMR CNRS 6597, Nantes -FRANCE, e-mail: Sebastien.Briot@irccyn.ec-nantes.fr V. Arakelian Institut National des Sciences Appliquées (INSA), Rennes -FRANCE Institut de Recherche en Communications et Cybernétique de Nantes (IRCCyN), UMR CNRS 6597, Nantes -FRANCE, e-mail: vigen.arakelian@insa-rennes.fr 1 increased accuracy of manipulators [START_REF] Foucault | On the development of a planar 3-dof reactionless parallel mechanism[END_REF], which is one of the positive consequences of the balancing. As was mentioned in [START_REF] Van Der Wijk | Comparative analysis for low-mass and low-inertia dynamic balancing of mechanisms[END_REF], balancing brings other advantages such as a reduced cycle time [START_REF] Raaijmakers | Besi zoekt snelheidslimiet pakken en plaatsen op (besi attacks the speedlimit for pick and place motion)[END_REF], reduced noise, wear and fatigue [START_REF] Lowen | Survey of investigations into the balancing of linkages[END_REF], as well as improved ergonomics [START_REF] Ishida | Performance characteristics and working comfortableness of forest workers of a new non-vibrating chain saw utilizing perfectly balanced rotation-reciprocation device[END_REF].

In general, two types of forces must be considered: the externally applied forces and the inertial forces. Inertial forces arise when links of a mechanism are subjected to large accelerations. The inertial force system acting on a given link can be represented as an inertia force acting on a line through the center of mass and an inertia torque about the center of mass. The determination of the inertial forces and torques is well known and it has been disclosed in various hand books [START_REF] Arakelian | Balancing of Linkages and Robot Manipulators -Advanced Methods with Illustrative Examples[END_REF]. With regard to the external forces, which are associated with the useful function that the mechanism is to perform, these are often smaller than inertia forces with a much lower variation. On the other hand, when formulating balancing conditions of a mechanism, it is necessary to recognize that, in many cases, external active forces applied to mechanism links constitute internal forces with respect to the mechanism as a whole. Thus, if all external active forces applied to the links of a mechanism are internal forces for the mechanism as whole, then the balance of the mechanism will be ensured under the fulfillment of inertia forces and inertia torque cancellation. Therefore, the balancing of shaking force and shaking moment due to the inertial forces of links acquires a specific importance. The quality of balancing of the moving masses has the influence not only on the level of vibrations but also on the resource, reliability and accuracy of mechanisms. Besides the mentioned negative effects, vibrations bring to the environments pollution and the loss of energy, and can also provoke various health issues. Consequently, the quality improvement of the mass balancing has not only technical, technological and economical aspects but also social.

Different approaches and solutions devoted to the shaking force and shaking moment balancing have been developed and documented for one-degree-of-freedom mechanisms [START_REF] Lowen | Balancing of linkages -an update[END_REF][START_REF] V. Arakelian | A historical review of the evolution of the theory on balancing of mechanisms[END_REF][START_REF] Arakelian | Shaking force and shaking moment balancing of mechanisms: a historical review with new examples[END_REF]. Nowadays, a new field for balancing methods applications is the design of mechanical systems for fast manipulation [START_REF] Nabat | High-speed parallel robot with four degrees of freedom[END_REF], which is a typical problem in advanced robotics [START_REF] Van Der Wijk | Design and experimental evaluation of a dynamically balanced redundant planar 4-RRR parallel manipulator[END_REF]. Here also we have similar problems relating to the cancelation or reduction of inertia forces. However, the mechanical systems with multi degrees of freedom lead to new solutions, such as the shaking force and shaking moment reduction by optimal motions of links, by adding flywheels with prescribed motions, or with the design of new self-balanced manipulators.

For all balancing method, the main challenge is the trade-off between the complexity of the balanced mechanism and the quality of balancing [START_REF] Arakelian | Shaking force and shaking moment balancing of mechanisms: a historical review with new examples[END_REF]. In the present Chapter, we propose a solution which allows the reduction of the balancing complexity by comparison with the usual approaches. The idea is to slightly modify the mechanism design by adding to it Assur groups, i.e. groups which dos not add any supplementary degree of freedom into the mechanism [START_REF] Leinonen | Terminology for the theory of machines and mechanisms[END_REF]. The use of such a solution is detailed in the following of the Chapter for the shaking force and shaking moment balancing of:

• the in-line four-bar linkage • the planar parallel robots with prismatic pairs. For both types of mechanisms, the proposed solution allows the reduction (or even the cancellation in the case of the four-bar linkage) of the number of counterrotations used for obtaining the shaking moment balancing, which decreases the design complexity and the inherent problems due to the use of counter-rotations (backlash, noise, vibrations, etc.).

Complete shaking force and shaking moment balancing of in-line four-bar linkages by adding a class-two RRR or RRP Assur group

Many high-speed machines contain planar four-bar linkages and the problem of their mass balancing is of continuing interest to machine designers. The previous works on the balancing of planar four-bar linkages may be arranged in the following groups [START_REF] Arakelian | Balancing of Linkages and Robot Manipulators -Advanced Methods with Illustrative Examples[END_REF]:

1. Complete shaking force balancing [START_REF] Fischer | über die reduzierten Systeme und die Hauptpunkte der Glieder eines Gelenkmechanismus[END_REF][START_REF] Artobolevsky | Methods of inertia calculation for mechanisms of agricultural machines[END_REF][START_REF] Crossley | Dynamics in Machines[END_REF][START_REF] Talbourdet | Mathematical solution of 4-bar linkages -IV. Balancing of linkages[END_REF][START_REF] Smith | Inertia forces in a four-bar linkage[END_REF][START_REF] Berkof | A new method for completely force balancing simple linkages[END_REF][START_REF] Hilpert | Weight balancing of precision mechanical instruments[END_REF]. In general, it is carried out by counterweights mounted on the movable links of the linkage. With regards to the several approaches employed for the redistribution of movables masses, the developed methods could be divided into three principal groups: a. The method of "principal vectors" [START_REF] Fischer | über die reduzierten Systeme und die Hauptpunkte der Glieder eines Gelenkmechanismus[END_REF]; The aim of this approach was to study the balancing of the mechanism relative to each link and in the determination of those points on the links relative to which a static balance was obtained. These points were called "principal points". Then, from the condition of similarity of the vector loop of the principal points and the structural loop of the mechanism, the necessary conditions of balancing were derived. b. The method of "static substitution of masses"; its aim was to statically substitute the mass of the coupler by concentrated masses, which are balanced thereafter together with the rotating links. Such an approach changes the problem of mechanism balancing into a simpler problem of balancing rotating links. This method was illustrated for four-bar linkage in [START_REF] Artobolevsky | Methods of inertia calculation for mechanisms of agricultural machines[END_REF][START_REF] Crossley | Dynamics in Machines[END_REF][START_REF] Talbourdet | Mathematical solution of 4-bar linkages -IV. Balancing of linkages[END_REF][START_REF] Smith | Inertia forces in a four-bar linkage[END_REF]. c. The method of "linearly independent vectors" [START_REF] Berkof | A new method for completely force balancing simple linkages[END_REF], in which the vector equation describing the position of the center of total mass of the mechanism is treated in conjunction with the closed equation of its kinematic chain. The result is an equation of static moments of moving link masses containing single linearly independent vectors. Thereafter following the conditions for balancing the mechanism by reducing the coefficients, which are time-dependent to zero.

It should be noted that the addition of a counterweighted pantograph device to the planar four-bar linkage has also been used for its complete shaking force balancing [START_REF] Hilpert | Weight balancing of precision mechanical instruments[END_REF]. 2. Complete shaking force and partial shaking moment balancing [START_REF] Lowen | Determination of force-balanced four-bar linkages with optimum shaking moment characteristics[END_REF][START_REF] Lowen | Theory of shaking moment optimization of force-balanced four-bar linkages[END_REF][START_REF] Berkof | Theory of shaking moment optimization of force-balanced fourbar linkages[END_REF][START_REF] Wiederrich | Momentum balancing of four-bar linkages[END_REF][START_REF] Elliot | The theory of torque, shaking force and shaking moment balancing of four link mechanisms[END_REF][START_REF] Carson | Feasible parameter design spaces for force and root-meansquare moment balancing on in-line 4R 4-bar linkage synthesized for kinematic criteria[END_REF][START_REF] Haines | Minimum RMS shaking moment or driving torque of a force-balanced 4-bar linkage using feasible counterweights[END_REF][START_REF] Shchepetilnikov | The determination of the mass centers of mechanisms in connection with the problem of mechanism balancing[END_REF][START_REF] Arakelian | Partial shaking moment balancing of fully force balanced linkages[END_REF][START_REF] Arakelian | Complete shaking force and partial shaking moment balancing of planar four-bar linkages[END_REF]. Two principal approaches may be distinguished:

a. The shaking moment minimization of fully force balanced linkages [START_REF] Lowen | Determination of force-balanced four-bar linkages with optimum shaking moment characteristics[END_REF][START_REF] Lowen | Theory of shaking moment optimization of force-balanced four-bar linkages[END_REF][START_REF] Berkof | Theory of shaking moment optimization of force-balanced fourbar linkages[END_REF][START_REF] Wiederrich | Momentum balancing of four-bar linkages[END_REF][START_REF] Elliot | The theory of torque, shaking force and shaking moment balancing of four link mechanisms[END_REF][START_REF] Carson | Feasible parameter design spaces for force and root-meansquare moment balancing on in-line 4R 4-bar linkage synthesized for kinematic criteria[END_REF][START_REF] Haines | Minimum RMS shaking moment or driving torque of a force-balanced 4-bar linkage using feasible counterweights[END_REF], in which it is shown that the optimum conditions of partial moment balance can be obtained by certain link mass distribution ratios. b. The minimization of the unbalance of shaking moment by transferring the rotation axis of the counterweight mounted on the input crank [START_REF] Shchepetilnikov | The determination of the mass centers of mechanisms in connection with the problem of mechanism balancing[END_REF][START_REF] Arakelian | Partial shaking moment balancing of fully force balanced linkages[END_REF][START_REF] Arakelian | Complete shaking force and partial shaking moment balancing of planar four-bar linkages[END_REF]. In the study [START_REF] Shchepetilnikov | The determination of the mass centers of mechanisms in connection with the problem of mechanism balancing[END_REF], the first harmonic of the shaking moment is eliminated by attaching the required input link counterweight, not to the input shaft itself, but to a suitable offset one which rotates with the same angular velocity. This approach is original in that, while maintaining the force balance of the mechanism, it is possible to create an additional balancing moment, thereby reducing the shaking moment. This approach has been further developed in works [START_REF] Arakelian | Partial shaking moment balancing of fully force balanced linkages[END_REF][START_REF] Arakelian | Complete shaking force and partial shaking moment balancing of planar four-bar linkages[END_REF].

It should be noted that optimization algorithms are also widely used in partial balancing of four-bar linkages [START_REF] Zhang | A constitutive method of objective function for the dynamic optimum balance of shaking force in linkage[END_REF][START_REF] Zhang | The optimum balance of shaking force and shaking moment of linkages[END_REF][START_REF] Qi | Optimum balancing of four-bar linkages[END_REF][START_REF] Chaudhary | Balancing of four-bar linkages using maximum recursive dynamic algorithm[END_REF]. 3. Complete shaking force and shaking moment balancing [START_REF] Arakelian | Shaking force and shaking moment balancing of mechanisms: a historical review with new examples[END_REF][START_REF] Kamenski | On the question of the balancing of planar linkages[END_REF][START_REF] Berkof | Complete force and moment balancing of inline four-bar linkages[END_REF][START_REF] Bagci | Complete shaking force and shaking moment balancing of link mechanisms using balancing idler loops[END_REF][START_REF] Ye | Complete balancing of planar linkages by an equivalence method[END_REF][START_REF] Arakelian | Complete shaking force and shaking moment balancing of linkages[END_REF][START_REF] Gao | Complete shaking force and shaking moment balancing of 17 types of eight-bar linkages only with revolute pairs[END_REF][START_REF] Berestov | Comparative analysis of the reactions in the kinematic pairs of the four-bar linkages for the different methods of balancing[END_REF][START_REF] Dresig | Vollständiger und harmonischer Ausgleich ebener Mechanismen[END_REF][START_REF] Esat | A theory of complete force and moment balancing of planar linkage mechanisms[END_REF][START_REF] Arakelian | Balanced four-bar articulated mechanism has output member extended beyond axis of second pivot and meshing gears[END_REF]. The first method of complete shaking force and shaking moment balancing was proposed in study [START_REF] Kamenski | On the question of the balancing of planar linkages[END_REF], which was extended in [START_REF] Berkof | Complete force and moment balancing of inline four-bar linkages[END_REF]. In this approach, the mass of the connecting coupler is substituted dynamically by concentrated masses located at the coupler joints. Thus, the dynamic model of the coupler becomes a weightless link with two concentrated masses. This transforms the problem of four-bar linkage shaking force and shaking moment balancing into a problem of balancing rotating links carrying concentrated masses. The parallelogram structure has also been applied for complete shaking force and shaking moment balancing of fourbar linkages [START_REF] Bagci | Complete shaking force and shaking moment balancing of link mechanisms using balancing idler loops[END_REF]. In the studies [START_REF] Ye | Complete balancing of planar linkages by an equivalence method[END_REF][START_REF] Arakelian | Complete shaking force and shaking moment balancing of linkages[END_REF][START_REF] Gao | Complete shaking force and shaking moment balancing of 17 types of eight-bar linkages only with revolute pairs[END_REF][START_REF] Berestov | Comparative analysis of the reactions in the kinematic pairs of the four-bar linkages for the different methods of balancing[END_REF][START_REF] Dresig | Vollständiger und harmonischer Ausgleich ebener Mechanismen[END_REF], the authors have proposed methods for complete shaking force and shaking moment balancing by counterweights with planetary gear trains. In [START_REF] Esat | A theory of complete force and moment balancing of planar linkage mechanisms[END_REF] a toothed-belt transmission is used to rotate counterweights intended for shaking force balancing, which also allows shaking moment balancing. The disadvantage of these methods is the need for the connection of gears to the oscillating links. The oscillations of the links of the mechanism will create noise unless expensive anti-backlash gears are used. Another solution using the copying properties of the pantograph was developed [START_REF] Arakelian | Shaking force and shaking moment balancing of mechanisms: a historical review with new examples[END_REF][START_REF] Arakelian | Balanced four-bar articulated mechanism has output member extended beyond axis of second pivot and meshing gears[END_REF], in which the gears driven by the coupler suffer no such sudden reversals so that this problem is almost eliminated. However, it should be noted that the use of the gears for the balancing of four-bar linkages is a drawback for the industrial applications and a fully shaking force and shaking moment balanced four-bar linkage without any gears is more appealing. The shaking moment balancing of fully force balanced linkages using a prescribed input speed fluctuation was proposed in [START_REF] Kochev | Full shaking moment balancing of planar linkages by a prescribed input speed fluctuation[END_REF]. However, such balancing is complicated because it is necessary to use a special type of drive generator. Moreover, it cannot be used for balancing of linkages, which generate the prescribed motions of the output links.

In this section a solution is discussed, which allows the complete shaking force and shaking moment balancing of in-line four-bar linkages with constant input speed by adding a class-two Assur group, i.e. a group which does not add any supplemen-tary degree of freedom into the mechanism [START_REF] Leinonen | Terminology for the theory of machines and mechanisms[END_REF]. It should be noted that the balancing of the shaking moment without counter-rotations of three particular classes of four-bar linkages was discussed in the studies [START_REF] Ricard | On the development of reactionless parallel manipulators[END_REF][START_REF] Wu | Synthesis of reactionless spatial 3-dof and 6-dof mechanisms without separate counter-rotations[END_REF][START_REF] Gosselin | Synthesis and design of reactionless threedegree-of-freedom parallel mechanisms[END_REF][START_REF] Gosselin | Dynamic balancing of planar mechanisms using toric geometry[END_REF][START_REF] Jiang | Dynamic optimization of reactionless 4-bar linkages[END_REF]. However, such a method cannot be extended to general four-bar linkages. In this section it is proposed to take advantages of the use of the properties of the four-bar linkage with prescribed geometric parameters [START_REF] Ricard | On the development of reactionless parallel manipulators[END_REF][START_REF] Wu | Synthesis of reactionless spatial 3-dof and 6-dof mechanisms without separate counter-rotations[END_REF][START_REF] Gosselin | Synthesis and design of reactionless threedegree-of-freedom parallel mechanisms[END_REF][START_REF] Gosselin | Dynamic balancing of planar mechanisms using toric geometry[END_REF][START_REF] Jiang | Dynamic optimization of reactionless 4-bar linkages[END_REF] and to combine it with 1. the principle of the dynamic substitution of link mass by concentrated masses 2. and with the prescription of constant input speed.

It should be mentioned here that the suggested balancing approach can be efficiently applied on the cyclic high-speed machines executing motions in the steady-state regime when the input speed is constant [START_REF] Demeulenaere | Improving machine drive dynamics: a structured design approach towards balancing[END_REF][START_REF] Berkof | The input torque in linkages[END_REF].

Complete shaking force and shaking moment balancing by adding a class-two RRR Assur group

Theoretical background related to the balanced four-bar linkages with prescribed geometrical parameters

Before considering the suggested balancing concept, let us recall basic notions concerning the balanced four-bar linkages with prescribed geometrical parameters.

In the paper of Berkof and Lowen [START_REF] Lowen | Theory of shaking moment optimization of force-balanced four-bar linkages[END_REF], the angular momentum H and the shaking moment M sh , expressed at point O, of a force balanced in-line four-bar linkage (Fig. 1) were expressed as:

H = 3 ∑ i=1 I i θi , M sh = dH dt = 3 ∑ i=1 I i θi (1)
with

I i = m i k 2 i + r 2 i + r i l i , (i = 1, 3) (2)
I 2 = m 2 k 2 2 + r 2 2 -r 2 l 2 (3)
where r i is the length of vector r i which connects the pivot P i to the centre of mass S i of link i, l i is the length of vector l i which connects the proximal revolute joint P i to the distal joint on the same link, and k i is the radius of gyration with respect to the centre of mass of link i, m i is the mass of link i. Moreover, θ i is the angular position of link i with respect to the x-axis.

With regard to the shaking force balancing, the following expressions were obtained:

m 1 r 1 = m 2 l 1 l 2 -r 2 l 2 ( 4
)
m 3 r 3 = m 2 r 2 l 3 l 2 (5) θ 1 θ 2 d x θ 3 P 3 y S 1 S 3 S 2 P 2 P 1 P' 3
Fig. 1 A general in-line four-bar mechanism.

In the works [START_REF] Gosselin | Synthesis and design of reactionless threedegree-of-freedom parallel mechanisms[END_REF][START_REF] Wu | On the dynamic balancing of multi-dof parallel mechanisms with multiple legs[END_REF], it has been shown that it is possible to cancel the shaking moment of a four-bar mechanism by associating mentioned geometric constraints with an optimal distribution of masses. Three kinds of shaking force and shaking moment balanced four-bar mechanisms were found, which are shown in Fig. 2.

In order to illustrate the shaking moment balancing, let us consider the mechanism shown in Fig. 2(b). The geometrical constraints of this mechanism are the following:

l 1 = l 3 (6) d = l 2 ( 7
)
where d is the length of the base which is the distance between the two fixed joints on the base. This leads to the following kinematic relationships:

θ1 -θ2 + θ3 = 0 (8)
Thus, from expressions (1) and [START_REF] V. Arakelian | A historical review of the evolution of the theory on balancing of mechanisms[END_REF], it is easy to see that the shaking moment will be cancelled if I 1 = I 2 = I 3 (see Eqs. (2) and ( 3)). For this purpose the following relationships must be established:

k 2 2 = m 2 l 2 r 2 -r 2 2 -I 1 m 2 ( 9
)
k 2 3 = -m 3 l 3 r 3 + r 2 3 + I 1 m 3 ( 10
)
where

I 1 = m 1 k 2 1 + r 2 1 + r 1 l 1 ( 11
)
It should be noted that similar results have been obtained for the mechanisms shown in Figs. 2(a) and2(c). Statement of the problem: the aim of the suggested balancing approach consists of adding a two-link kinematic chain with prescribed geometrical parameters to an inline four-bar linkage with arbitrary geometrical parameters. It is important to note that the added structure must be an Assur group, i.e. a group which does not add any supplementary degree of freedom into the mechanism [START_REF] Leinonen | Terminology for the theory of machines and mechanisms[END_REF]. This allows for the modification of the mass redistribution of the obtained six-bar mechanism without perturbation of the kinematic properties of the initial four-bar linkage. We would like to state that this technique allows for the complete shaking force and shaking moment balancing without counter-rotating masses. Now let us consider the shaking force and shaking moment balancing of an inline four-bar mechanism using class-two Assur groups with RRR kinematic chain.

θ 1 θ 2 d x θ 3 P 3 y P 2 P 1 l 1 l 2 l 3 P' 2 r 1 r 3 r 2 (a) Case I: l 1 = d and l 2 = l 3 θ 1 θ 2 d x θ 3 P 3 y P 2 P 1 l 1 l 2 l 3 P' 2 r 1 r 3 r 2 (b) Case II: l 2 = d and l 1 = l 3 θ 1 θ 2 d x θ 3 P 3 y P 2 P 1 l 1 l 2 l 3 P' 2 r 1 r 3 r 2 (c) Case III: l 3 = d and l 1 = l 2
S 4 S 5 θ 5 P 2 P' 2 P 3 θ 4 α O A C B S cw1 γ β θ 1 θ 2 x θ 3 y S 1 S 3 S 2

S cw2 S cw3

Fig. 3 The balanced mechanism with the class-two RRR Assur group.

Shaking force balancing

Figure 3 shows an in-line four-bar linkage with the added class-two RRR Assur group. Let us denote the following vectors as:

l 1 = d OA , l 2 = d AB , l 3 = d CB , l ′ 3 = d CP 2 , l 4 = d P 2 P ′ 2 , l 5 = d P 3 P ′ 2 , r 1 = d OS 1 , r 2 = d AS 2 , r 3 = d CS 3 , r 4 = d P 2 S 4 , r 5 = d P 3 S 5 , r cw 1 = d OS cw 1 , r cw2 = d CS cw 2 , r cw3 = d P 3 S cw 3 .
The added class-two RRR Assur group has the above mentioned properties, i.e. it is designed such as:

l ′ 3 = l 5 (12) e = l 4 (13)
where l i (i = 1 . . . 5) is the norm of vector l i and e the distance between C and P 3 .

As is shown in Fig. 3, the Assur group P 2 P ′ 2 P 3 is attached to the initial linkage OABC in such a way that it forms a four-bar linkage with link BC, as discussed in Section 2.1.1.

Let us now derive the expression of the shaking force f sh of the obtained mechanism:

f sh = 5 ∑ i=1 m i dS i ( 14
)
where dS i is the translational acceleration of the centre of mass S i and m i the mass of the link i. Developing and simplifying, one obtains:

f sh = m 1 r 1 l 1 + m 2 l 2 -r 2 l 2 dA + m 2 r 2 l 2 dB + m 3 dS 3 + m 4 l 4 -r 4 l 4 dP 2 + m 4 r 4 l 4 + m 5 r 5 l 5 dP ′ 2 ( 15
)
where r i is the algebraic values of the norm of vectors r i , and dA , dB , dS 3 , dP 2 and dP ′ 2 represent the acceleration of points A, B, S 3 , P 2 and P ′ 2 respectively. Their expressions are:

dA = l 1 θ1 -sin θ 1 cos θ 1 -θ 2 1 cos θ 1 sin θ 1 (16) dB = l 3 θ3 -sin θ 3 cos θ 3 -θ 2 3 cos θ 3 sin θ 3 (17) dS 3 = r 3 θ3 -sin (θ 3 + β ) cos (θ 3 + β ) -θ 2 3 cos (θ 3 + β ) sin (θ 3 + β ) (18)
dP 2 = l ′ 3 θ3 -sin (θ 3 + α) cos (θ 3 + α) -θ 2 3 cos (θ 3 + α) sin (θ 3 + α) (19) dP ′ 2 = l 5 θ5 -sin θ 5 cos θ 5 -θ 2 5
cos θ 5 sin θ 5 [START_REF] Lowen | Determination of force-balanced four-bar linkages with optimum shaking moment characteristics[END_REF] in which the angles θ i (i = 1 . . . 5) are defined in Fig. 3.

The shaking force f sh may be cancelled through the addition of three counterweights positioned at points S cw i (Fig. 3), with masses m cw i (i = 1, 2, 3). With such counterweights, the expression of the shaking force becomes:

f sh * = f sh + m cw 1 r cw 1 l 1 dA + m cw 2 dS cw 2 + m cw 3 r cw 3 l 5 dP ′ 2 ( 21
)
where r cw i is the algebraic values of the norm of vectors r cw i and

dS cw 2 = r cw 2 θ3 -sin (θ 3 + γ) cos (θ 3 + γ) -θ 2 3 cos (θ 3 + γ) sin (θ 3 + γ) (22)
Thus the shaking force is cancelled if the distribution of the masses is as follows:

m cw 1 = - l 1 r cw 1 m 1 r 1 l 1 + m 2 l 2 -r 2 l 2 (23) tan γ = m 3 r 3 sin β + m 4 (1 -r 4 /l 4 ) l ′ 3 sin α m 2 l 3 r 2 /l 2 + m 3 r 3 cos β + m 4 (1 -r 4 /l 4 ) l ′ 3 sin α ( 24
)
m 2 cw 2 = (m 3 r 3 sin β + m 4 (1 -r 4 /l 4 ) l ′ 3 sin α) 2 r 2 cw 2 + (m 2 l 3 r 2 /l 2 + m 3 r 3 cos β + m 4 (1 -r 4 /l 4 ) l ′ 3 cos α) 2 r 2 cw 2 (25) m cw 3 = - l 5 r cw 3 m 4 r 4 l 4 + m 5 r 5 l 5 (26)

Shaking moment balancing

Let us now derive the expression of the shaking moment M sh , expressed at point O, of such a mechanism:

M sh = 5 ∑ i=1 m i x S i ÿS i -y S i ẍS i + k 2 i θi + 3 ∑ i=1 m cw i x S cw i ÿS cw i -y S cw i ẍS cw i (27)
where x S i , y S i , ẍS i and ÿS i are the position and accelerations along x and y axes of points S i , respectively, and x S cw i , y S cw i , ẍS cw i and ÿS cw i are the position and accelerations along x and y axes of points S cw i , respectively, k i is the radius of gyration of link i. Now, let us consider that link 2 is a physical pendulum1 [START_REF] Berkof | Complete force and moment balancing of inline four-bar linkages[END_REF], i.e. it can be replaced dynamically by two point masses located at joint centres A and B. This implies that:

k 2 2 = r 2 (l 2 -r 2 ) (28)
Then, considering that the input speed is constant, i.e. θ1 = 0 and taking into account [START_REF] Arakelian | Partial shaking moment balancing of fully force balanced linkages[END_REF], one can simplify [START_REF] Shchepetilnikov | The determination of the mass centers of mechanisms in connection with the problem of mechanism balancing[END_REF] as:

M sh = 5 ∑ i=3 I i θi ( 29
)
where

I 3 = m 3 k 2 3 + r 2 3 + m 2 r 2 l 2 l 2 3 + m cw 2 r 2 cw 2 + m 4 l 4 -r 4 l 4 l ′2 3 ( 30
)
I 4 = m 4 k 2 4 + r 2 4 -r 4 l 4 ( 31
)
I 5 = m 5 k 2 5 + r 2 5 -r cw 3 r 5 + m 4 r 4 l 4 l 5 l 5 -r cw 3 (32)
Thus, this new six-bar mechanism has the same shaking moment as the four-bar mechanism composed of links P 1 P 2 , P 2 P ′ 2 and P ′ 2 P 3 . Therefore, the initial four-bar linkage balancing problem is transformed in the balancing of the four-bar linkage formed by the added Assur group. Note that the latter has specific geometry and its balancing conditions have been examined in Section 2.1.1.

Applying these results to the considered mechanism, we obtain:

k 2 4 = -I 3 -r 2 4 + r 4 l 4 m 4 ( 33
)
k 2 5 = m 5 -r 2 5 + r cw 3 r 5 + m 4 r 4 /l 4 l 5 l 5 -r cw 3 + I 3 m 5 ( 34
)
The substitution of ( 33)-( 34) into (29) leads to:

M sh = I 3 θ3 -θ4 + θ5 (35)
Taking into account relations ( 6) and ( 6), we have θ3 -θ4 + θ5 = 0 [START_REF] Bagci | Complete shaking force and shaking moment balancing of link mechanisms using balancing idler loops[END_REF] and consequently M sh = 0 (37)

The proposed balancing technique has been illustrated using the four-bar linkage shown in Fig. 2(b). However, it can also be achieved via the mechanism of Fig. 2(a) or Fig. 2(c).

Complete shaking force and shaking moment balancing by adding a class-two RRP Assur group

Shaking force balancing

The second solution, which is proposed for the cancellation of the shaking moment of a four-bar linkage is carried out by adding a class-two RRP Assur group (Fig. 4).

Let us denote the following vectors as:

l 1 = d OA , l 2 = d AB , l 3 = d CB , l ′ 3 = d CD , l 4 = d DE , r 1 = d OS 1 , r 2 = d AS 2 , r 3 = d CS 3 , r 4 = d DS 4 , r 5 = d ES 5 , r cw 1 = d OS cw 1 , r cw2 = d CS cw 2 , r cw3 = d DS cw 3 .
In this case, the lengths of added links are the following:

l ′ 3 = l 4 (38)
Thus, the new part created by added links is a Scott-Russell mechanism [START_REF] Martin | Kinematics and Dynamics of Machines[END_REF], which is attached to the initial linkage at an angle of α.

The relations between the link accelerations are the following for the Scott-Russell mechanism: θ3 = -θ4 (39)

S 5 D E S 4 θ 4 α O A C B S cw1 γ β θ 1 θ 2 x θ 3 y S 1 S 3 S 2 S cw2 S cw3
Fig. [START_REF] Lowen | Survey of investigations into the balancing of linkages[END_REF] The balanced mechanism with the class-two RRP Assur group.

Let us now derive the expression of the shaking force f sh of such a mechanism:

f sh = 5 ∑ i=1 m i dS i (40)
Developing and simplifying, one obtains:

f sh = m 1 r 1 l 1 + m 2 l 2 -r 2 l 2 dA + m 2 r 2 l 2 dB + m 3 dS 3 + (m 4 + m 5 ) dD + m 4 r 4 l 4 + m 5 d (41)
where r i and l i are the algebraic values of the norm of vectors r i , and l i , respectively,

dS 3 = r 3 θ3 -sin (θ 3 + β ) cos (θ 3 + β ) -θ 2 3 cos (θ 3 + β ) sin (θ 3 + β ) (42)
d = l 5 θ4 -sin θ 4 cos θ 4 -θ 2 4 cos θ 4 sin θ 4 ( 43
) dD = l ′ 3 θ3 -sin (θ 3 + α) cos (θ 3 + α) -θ 2 3 cos (θ 3 + α) sin (θ 3 + α) (44)
in which the angles θ i (i = 1 . . . 4) are defined in Fig. 4.

The shaking force f sh may be cancelled through the addition of three counterweights positioned at points S cw i (Fig. 4), with masses m cw i (i = 1, 2, 3). With such counterweights, the expression of the shaking force becomes:

f sh * = f sh + m cw 1 r cw 1 l 1 dA + m cw 2 dS cw 2 + m cw 3 dD + m cw 3 r cw 3 l 4 d (45)
where r cw i is the algebraic values of the norm of vectors r cw i and

dS cw 2 = r cw 2 θ3 -sin (θ 3 + γ) cos (θ 3 + γ) -θ 2 3 cos (θ 3 + γ) sin (θ 3 + γ) (46)
Thus the shaking force is cancelled if the distribution of the masses is as follows:

m cw 1 = - l 1 r cw 1 m 1 r 1 l 1 + m 2 l 2 -r 2 l 2 (47) tan γ = m 3 r 3 sin β + (m 4 + m 5 ) l ′ 3 sin α m 2 l 3 r 2 /l 2 + m 3 r 3 cos β + (m 4 + m 5 ) l ′ 3 sin α ( 48
)
m 2 cw 2 = (m 3 r 3 sin β + (m 4 + m 5 ) l ′ 3 sin α) 2 r 2 cw 2 + (m 2 l 3 r 2 /l 2 + m 3 r 3 cos β + (m 4 + m 5 ) l ′ 3 cos α) 2 r 2 cw 2 (49) m cw 3 = - l 4 r cw 3 m 4 r 4 l 4 + m 5 (50)

Shaking moment balancing

Let us now derive the expression of the shaking moment M sh , expressed at point O, of such a mechanism:

M sh = 5 ∑ i=1 m i x S i ÿS i -y S i ẍS i + k 2 i θi + 3 ∑ i=1 m cw i x S cw i ÿS cw i -y S cw i ẍS cw i ( 51
)
where x S i , y S i , ẍS i and ÿS i are the position and accelerations along x and y axes of points S i , respectively, and x S cw i , y S cw i , ẍS cw i and ÿS cw i are the position and accelerations along x and y axes of points S cw i , respectively, k i is the radius of gyration of link i. Now, let us consider as in the previous case, that link 2 is a physical pendulum and that the input speed is constant.

Taking into account [START_REF] Arakelian | Partial shaking moment balancing of fully force balanced linkages[END_REF], one can simplify (51) as:

M sh = I 3 θ3 + I 4 θ4 (52)
where

I 3 = m 3 k 2 3 + r 2 3 + m 2 r 2 l 2 l 2 3 + m cw 2 r 2 cw 2 + m 4 + m 5 + m cw 3 l ′2 3 ( 53
)
I 4 = m 4 k 2 4 + r 2 4 + m 5 l 2 4 + m cw 3 r 2 cw 3 (54)
Introducing ( 39) into (52), we find:

M sh = (I 3 -I 4 ) θ3 (55)
Thus, the mechanism will be moment balanced if:

I 3 = I 4 ( 56
)
which can be obtained using a design of link 4 for which the radius of gyration should be equal to

k 4 = I 3 -m 4 r 2 4 -m 5 l 2 4 -m cw 3 r 2 cw 3 m 4 ( 57
)
It should be mentioned that, in order to avoid the singular configurations of the added structure, the value of angle α should be chosen carefully during the design process.

Let us consider two illustrative examples of the proposed balancing technique.

Illustrative examples and numerical simulations

Balancing by adding a class-two RRR Assur group

Let us carry out the complete shaking force and shaking moment balancing of a four-bar linkage with parameters

• l 1 = 0.2 m, l 2 = 0.27 m, l 3 = 0.25 m, β = 0 deg, • r 1 = 0.1 m, r 2 = 0.135 m, r 3 = 0.125 m, • k 1 = 0.056 m, k 2 = 0.135 m, k 3 = 0.086 m, • m 1 = 1 kg, m 2 = 1 kg, m 3 = 1 kg.
The simulations of the proposed mechanism have been carried out using ADAMS software and the obtained results are shown in Fig. 5 (full line). Now we add the RRR Assur group with prescribed distribution of the centre of mass and inertia. Its geometric and mass properties are:

• l 4 = 0.8 m, l 5 = 0.25 m, • r 4 = 0.4 m, r 5 = 0.125 m, • m 4 = 1.5 kg, m 5 = 1 kg.
and the location and mass of the added counterweights are:

• r cw 1 = -0.1 m, r cw 2 = 0.25 m, r cw 3 = -0.125 m, • m cw 1 = 2 kg, m cw 3 = 2.5 kg.
The radii of gyration of elements 4 and 5 and the mass of the counterweight m cw 2 are not given as they depend on the value of angle α which is not yet fixed (Fig. 3). Their variations as a function of α are shown in Fig. 6. In these figures, the values of α are bounded between 60 and 180 deg. in order to avoid the RRR Assur group to cross a singularity during the motion. In Fig. 5 (dotted line), it is shown that after the addition of the Assur group, the shaking force and shaking moment are cancelled.

As angle α is a free parameter that has only influence on the values k 4 , k 5 and m cw 2 , it can be chosen so that it minimizes one supplementary criterion. In the remainder of the paper, this criterion is chosen to be the linkage input torque [START_REF] Demeulenaere | Improving machine drive dynamics: a structured design approach towards balancing[END_REF][START_REF] Berkof | The input torque in linkages[END_REF]. It should however be mentioned that angle α could be used to minimize another criterion such as the power consumption, the energy, etc.

It should also be noted that the input torque τ, i.e. the torque requested by the actuator to move the mechanism, is computed in the two illustrative examples by using the Lagrange equations [START_REF] Berkof | The input torque in linkages[END_REF]: where L = T -V is the Lagrangian of the system, V is the potential energy (equal to 0 in absence of gravity) and T is the kinetic energy:

∂ L ∂ θ1 - ∂ L ∂ θ 1 ( 58
)
T = 1 2 ∑ i m i ẋ2 S i + ẏ2 S i + 1 2 ∑ j I j θ 2 j ( 59
)
ẋS i and ẏS i being the velocities along x and y axes of any centre of mass (for links and as well as for counterweights). In Fig. 7, the maximum of the input torque absolute value of as a function of angle α is shown. Thus, it is possible to see that if the value of α is chosen arbitrarily, the input torques can grow up to 2140 N.m (for α = 0 deg.). It also appears that the input torque will be minimal if α = 164 deg. In this case, the value of the input torque is 1010 N.m, i.e. about 2 times less than in the first case.

Balancing by adding a class-two RRP Assur group

We now propose obtaining the complete shaking force and shaking moment balancing of the same mechanism by adding a class-two RRP Assur group. Its geometric and mass properties are and the location and mass of the added counterweights are:

• r cw 1 = -0.1 m, r cw 2 = 0.25 m, r cw 3 = -0.125 m, • m cw 1 = 2 kg, m cw 3 = 0.55 kg.
The radii of gyration of element 4 and the mass of the counterweight m cw 2 are not expressed in these tables as they depend on the value of angle α (Fig. 4). Their variations as a function of α are shown in RRP Assur group to cross a singularity during the motion. The simulations of the proposed mechanism have been carried out using ADAMS software and the results are similar to the previous case shown in Fig. 5.

As was mentioned above, the angle α is a not fixed design parameter and it can be found from minimization of the input torque of the mechanism. In Fig. 9, maximum of the input torque absolute value as a function of angle α is shown. It is possible to see that if the value of α is chosen arbitrarily, the input torques can grow up to 2300 N.m (for α = 25 deg.). It also appears that the input torque will be minimal if α = 205 deg. In this case, the value of the input torque is 1380 N.m, i.e. about 1.7 times less than in the first case.

Let us now consider the balancing of parallel robots by the addition of one or several class-two RRR Assur groups.

Balancing of parallel robots by the addition of one or several class-two RRR Assur groups

For robots, shaking force balancing is mostly obtained via an optimal redistribution of movable masses [START_REF] Lowen | Balancing of linkages -an update[END_REF][START_REF] V. Arakelian | A historical review of the evolution of the theory on balancing of mechanisms[END_REF][START_REF] Arakelian | Shaking force and shaking moment balancing of mechanisms: a historical review with new examples[END_REF][START_REF] Agrawal | Reactionless space and ground robots: novel design and concept studies[END_REF][START_REF] Wang | Static balancing of spatial three-degree-of-freedom parallel mechanisms[END_REF][START_REF] Newman | the optimal control of balanced manipulators[END_REF][START_REF] Laliberté | Static balancing of 3-DOF planar parallel mechanisms[END_REF][START_REF] Fujikoshi | Balancing apparatus for jointed robot[END_REF][START_REF] Wang | Static balancing of spatial four-degree-of-freedom parallel mechanisms[END_REF][START_REF] Russo | Static balancing of parallel robots[END_REF] or adjustment of kinematic parameters [START_REF] Ouyang | Force balancing of robotic mechanisms based on adjustment of kinematic parameters[END_REF]. The cancellation of the shaking moment is a more complicated task and can be obtained using three main different methods:

1. Shaking moment balancing using counter-rotations [START_REF] Herder | A counter-rotary counterweight for light-weight dynamic balancing[END_REF][START_REF] Fattah | On the design of reactionless 3-dof planar parallel mechanisms[END_REF], 2. Shaking moment balancing by adding four-bar linkages [START_REF] Ricard | On the development of reactionless parallel manipulators[END_REF][START_REF] Wu | Synthesis of reactionless spatial 3-dof and 6-dof mechanisms without separate counter-rotations[END_REF][START_REF] Gosselin | Synthesis and design of reactionless threedegree-of-freedom parallel mechanisms[END_REF][START_REF] Foucault | Synthesis, design, and prototyping of a planar three degreeof-freedom reactionless parallel mechanism[END_REF], 65] and 3. Shaking moment balancing by optimal trajectory planning [START_REF] Fattah | On the design of reactionless 3-dof planar parallel mechanisms[END_REF][START_REF] Papadopoulos | Design and motion planning for a zero-reaction manipulator[END_REF][START_REF] Arakelian | Dynamic balancing of the scara robot[END_REF][START_REF] Briot | Complete shaking force and shaking moment balancing of the position-orientation decoupled PAMINSA manipulator[END_REF][START_REF] Briot | Shaking force minimization of high-speed robots via centre of mass acceleration control[END_REF].

Previous works have been devoted to the study of parallel manipulators with revolute joints and few studies has been carried out on complete shaking force and shaking moment balancing of parallel manipulators with prismatic pairs.

In this section, we propose solutions for complete shaking force and shaking moment balancing of planar parallel manipulators with prismatic pairs. We illustrate these solutions via the 3-RPR parallel manipulator. All obtained results are validated using ADAMS software simulations

Complete shaking moment and shaking force balancing by adding an idler loop between the base and the platform

Inertia force balancing by adding an idler loop made of a single RRR Assur group is known to be used for 1 degree of freedom (dof ) mechanisms [START_REF] Hilpert | Weight balancing of precision mechanical instruments[END_REF][START_REF] Bagci | Complete shaking force and shaking moment balancing of link mechanisms using balancing idler loops[END_REF][START_REF] Frolov | Theory of mechanisms and machines[END_REF][START_REF] Doronin | Balanced slider-crank mechanism[END_REF][START_REF] Arakelian | équilibrage dynamique complet des mécanismes[END_REF]. With regard to planar manipulators, such an approach has only been used in the balancing of gravitational and inertia forces [START_REF] Wang | Static balancing of spatial four-degree-of-freedom parallel mechanisms[END_REF][START_REF] Russo | Static balancing of parallel robots[END_REF][START_REF] Baradat | Design and prototyping of a new balancing mechanism for spatial parallel manipulators[END_REF][START_REF] Leblond | Static balancing of spatial and planar parallel manipulators with prismatic actuators[END_REF]. In this sub-section, the complete shaking force and shaking moment balancing of planar manipulators by adding an idler loop made of a single RRR Assur group is discussed. The added balancing loop is mounted between the base and the platform of the mechanism. We illustrate the suggested balancing technique on a 3-RPR mechanism (Fig. 10). Please note that we do not mention the type of actuation of the mechanism as it has no influence on the balancing.

A 1 C 1 B 1 A 2 C 2 B 2 C 3 B 3 A 3 P S 11 2,1 θ 1 S 21 1,1 θ 2 S 12

Theoretical background

Firstly, let us analyze the cancellation of the dynamic reactions of the 3-RPR planar parallel mechanism (Fig. 10(a)). Such a mechanism has 3 dof (two translations in the (Oxy) plane and one rotation of the moving platform around an axis perpendicular to (Oxy)) and is composed of three identical legs, each being composed of a revolute joint attached to the base at point A i (in the following, i = 1 • • • 3), one moving prismatic guide, located at point B i , and another revolute joint attached to the platform at C i . The base and platform triangles, denoted A 1 A 2 A 3 and C 1 C 2 C 3 , are equilateral. On this manipulator, typically, the actuated joints are the first revolute joint at A i or the linear guide at B i .

Considering that the x axis is directed along the line A 1 A 2 , the y axis being perpendicular to the x axis and the origin of the base frame located at point O, the centre of the circumcircle of triangle A 1 A 2 A 3 , one can define the coordinate x, y and φ of the platform, as being respectively the coordinates of point P along the x and y axes and the angle between the line C 1 C 2 and A 1 A 2 .

Let us denote as S i j the centre of mass of link i j ( j = 1, 2), which has a mass m j and an axial moment of inertia I S j . The centre of mass of the platform is located at point P. The mass of the platform is m p and its axial moment of inertia I p .

In order to cancel the shaking forces and shaking moment of the manipulator, an idler loop is added between the base and the platform (Fig. 10(b)). The centre of mass of elements 5 and 6 of the idler loop are located at S 5 and S 6 , respectively. Their masses are denoted as m 5 and m 6 and their axial moments of inertia as I S 5 and I S 6 , respectively. The positions of the coms are defined such that --→

A i S 1i = r 1 l B i C i u i , --→ C i S 2i = (r 2 1)l B i C i u i , --→ ES 5 = r 5 -→
EF, and --→ FS 6i = r 6 -→ FP, r 1 , r 2 , r 5 , and r 6 being dimensionless coefficients, and u i a unit vector directed along --→ B i C i . The expression of the shaking force f sh transmitted by the robot to the ground is:

f sh = 3 ∑ i=1 2 ∑ j=1 m j rS i j + m p rP + m 5 rS 5 + m 6 rS 6 ( 60
)
where rS i j , rP , rS 5 and rS 6 are the accelerations of the coms S i j , of P, S 5 and S 6 , respectively. Developping ( 60), it can be demonstrated that the shaking force f sh can be expressed as:

f sh = (m 1 r 1 -m 2 (1 -r 2 )) 3 ∑ i=1 a i + (3m 2 + m p + m 6 r 6 ) a 6 + (3m 2 + m p + m 5 r 5 + m 6 ) rF (61)
with

a i = l B i C i θi -sin θ i cos θ i -θ 2 i cos θ i sin θ i ( 62
)
a 6 = l FP θ6 -sin θ 6 cos θ 6 -θ 2 6
cos θ 6 sin θ 6 [START_REF] Fattah | On the design of reactionless 3-dof planar parallel mechanisms[END_REF] At this step, only five counterweights are needed in the cancellation of the shaking force, but it could be demonstrated after more derivations that three others are necessary for the cancellation of the shaking moment. Therefore, we propose directly adding three supplementary counterweights (Fig. 11). The positions of the eight added counterweights are ---→

A i M 1i = r cw 1 l B i C i u i , ---→ B i M 2i = (r cw 2 -1)l B i C i u i , --→ EM 5 = r cw 5
-→ EF, and --→ FM 6 = r cw 6 -→ FP, r cw 1 , r cw 2 , r cw 5 , and r cw 6 being dimensionless coefficients. Their masses are respectively denoted m cw 1 , m cw 2 , m cw 5 and m cw 6 . With the addition of the counterweights, the shaking force becomes:

f sh * =f sh + (m cw 1 r cw 1 -m cw 2 (1 -r cw 2 )) 3 ∑ i=1 a i + m cw 6 r cw 6 a 6 + m cw 5 r cw 5 + m cw 6 rF (64)
Thus, the shaking force is cancelled if:

A 1 C 1 A 2 C 2 C 3 A 3 P E F I cr1 I cr1 I cr1 I cr3 I cr2 I cr4 M 11 M 12 M 13 M 21 M 23 M 22 M 6 M 5
Fig. 11 Schematics of the 3-RPR mechanism with the added RRR chain used for the cancellation of the shaking force and shaking moment.

m cw 1 = - m 1 r 1 r cw 1 m cw 2 = - m 2 (1 -r 2 ) 1 -r cw 2 m cw 6 = - 3 (m 2 + m cw 2 ) + m p + m 6 r 6 r cw 6 m cw 5 = - 3 (m 2 + m cw 2 ) + m p + m 5 r 5 + m 6 + m cw 6 r cw 5 (65)
The expression of the shaking moment M sh O of the modified structure (expressed at point O) can be written as:

M sh O = d dt H O ( 66
)
where H O is the angular momentum of the leg (expressed at point O). Thus, in order to cancel the shaking moment, the angular momentum is held constant over time.

The expression of the angular momentum H O is equal to:

H O = 3 ∑ i=1 2 ∑ j=1 m j x S i j ẏS i j -y S i j ẋS i j + m cw j x M i j ẏM i j -y M i j ẋM i j + I S j θi + I p φ + 6 ∑ j=5 m j x S j ẏS j -y S j ẋS j + m cw j x M j ẏM j -y M j ẋM j + I S j θ j ( 67
)
where x Q , y Q , ẋQ and ẏQ are the position and velocities of any point Q along x and y axes, respectively (Q being either point S i j , M i j , ( j = 1, 2), S j or M j ( j = 5, 6)). Developing and introducing ( 65) into ( 67) yields

H O = 3 ∑ i=1 I S 1 + I S 2 + m 1 r 2 1 + m cw 1 r 2 cw 1 + m 2 (1 -r 2 ) 2 + m cw 2 (1 -r cw 2 ) 2 l 2 B i C i θi + I S 6 + m 6 r 2 6 + m cw 6 r 2 cw 6 + m p + 3 (m 2 + m cw 2 ) l 2 FP θ6 + I S 5 + m 5 r 2 5 + m cw 5 r 2 cw 5 + m 6 + m cw 6 + m p + 3 (m 2 + m cw 2 ) l 2 EF θ5 + I p + 3 (m 2 + m cw 2 ) l 2 C i P φ (68)
After such modifications of the RRR chain, the angular momentum of the legs of the mechanism and of the RRR chain can be balanced using six counter-rotations (Fig. 11), which have an axial moment of inertia equal to:

I cr 1 = I S 1 + I S 2 + m 1 r 2 1 + m cw 1 r 2 cw 1 + m 2 (1 -r 2 ) 2 + m cw 2 (1 -r cw 2 ) 2 l 2 B i C i I cr 2 = I p + 3 (m2 + m cw 2 ) l 2 C i P I cr 3 = I S 6 + m 6 r 2 6 + m cw 6 r 2 cw 6 + m p + 3 (m 2 + m cw 2 ) l 2 FP + 2I cr 2 I cr 4 = I S 5 + m 5 r 2 5 + m cw 5 r 2 cw 5 + m 6 + m cw 6 + m p + 3 (m 2 + m cw 2 ) l 2 EF + 2I cr 3 (69)

Illustrative examples and numerical simulations

Let us illustrate the suggested balancing approach using numerical simulations carried out with ADAMS software. For this purpose, non balanced and balanced 3-RPR parallel manipulators will be compared. The chosen trajectory for simulations is a straight line of the controlled point of the platform, achieved in t f = 0.25 s, between P 0 = [x 0 y 0 ] T = [0.05 m 0 m] T and P f = x f y f T = [0.2 m 0 m] T with a rotation of the platform from φ 0 = 0 deg. to φ f = 30 deg. For the displacement of the mechanism, fifth order polynomial laws are used and therefore the trajectory is defined by the following expressions:

   x(t) = x 0 + s(t)(x f -x 0 ) y(t) = 0 φ (t) = φ 0 + s(t)(φ f -φ 0 ) (70)
with

s(t) = 10 t 3 f t 3 - 15 t 4 f t 4 + 6 t 5 f t 5 (71)
The parameters used for the simulations are the followings: For such parameters and such a trajectory, the shaking force and shaking moment are computed using ADAMS software and are presented in Fig. 12 (solid line). Then, we add the counterweights and the idler loop EFP to the mechanism. The position coefficients of the counterweights are all equal to r cw j = -0.5 ( j = 1, 2, 5, 6). Therefore, the added masses are equal to m cw 1 = 0.75 kg, m cw 2 = 0.37 kg, m cw 5 = 6.92 kg, m cw 6 = 21.66 kg. The new values of the shaking force and moment are presented in Fig. 12 (dashed line). It is possible to see that with the added counterweights the shaking efforts are cancelled, while the maximal value of shaking moment is increased by a factor 17. Finally, we add the counter-rotations. Their values are equal to I cr 1 = 0.01917 kg.m 2 , I cr 2 = 0.02665 kg.m 2 , I cr 3 = 0.18169 kg.m 2 , I cr 4 = 0.72781 kg.m 2 . With such counter-rotations, the shaking moment is balanced (in gray line in Fig. 12(c)).

• l OE = 0 m, l OA i = 0.35 m, l B i C i = 0.05 m, l C i P = 0.1 m, l EF = 0.
A C B S 1 S 2 3 1 2 θ 1 θ 2 a x C
x y Fig. 13 A general slider-crank mechanism.

Complete shaking force and shaking moment balancing using Scott-Russell mechanism

In this sub-section, another approach for complete shaking force and shaking moment balancing is developed, which consists of adding Scott-Russell mechanisms (mechanisms made of RRR Assur groups) to each leg of the initial architecture of a manipulator. This approach enables a reduction in the number of counter-rotations.

Properties of the Scott-Russell mechanism

Let us observe a simple slider-crank mechanism (Fig. 13). The centre of mass of link i (i = 1, 2, 3) is denoted as S i . Link i has a mass m i and an axial moment of inertia I S i . The positions of the centres of mass are

-→ AS 1 = r1 -→ AB, -→ BS 2 = r 2 -→ BC,
--→ CS 3 = l 3 r 3 x, r 1 , r 2 and r 3 being dimensionless coefficients, and l 3 being a constant length.

It is known that the compete shaking force and shaking moment balancing of a general slider-crank mechanism can be obtained by adding two counterweights mounted on the links and two pairs of counter-rotations. However, it is possible to balance this mechanism without counter-rotation if it has specific geometrical parameters, as in Scott-Russell mechanisms (a = 0 m, l AB = l BC -Fig. 13).

Let us consider the balancing of this mechanism. The expression of the shaking force f sh of a slider-crank mechanism can be written as:

f sh = 3 ∑ i=1 m i rS i = (m 1 r 1 + m 2 ) rB + (m 2 r 2 + m 3 ) a (72)
with

a = l BC θ2 -sin θ 2 cos θ 2 -θ 2 2 cos θ 2 sin θ 2 (73)
rB being the acceleration of point B.

The constant terms of ( 72) can be cancelled by the addition of two counterweights located at M j , ( j = 1, 2) (Fig. 13), whose masses are denoted as m cw j .

Their positions are equal to:

--→ AM 1 = r cw 1 -→ AB, --→ BM 2 = r cw 1 -→ BC, r cw 1 and r cw 2 being dimensionless coefficients. With the addition of the counterweights, the shaking force becomes:

f sh * = f sh + (m cw 1 r cw 1 + m cw 2 ) rB + m cw 2 r cw 2 a (74)
Thus, the shaking force is balanced if:

m cw 2 = - m 2 r 2 + m 3 r cw 2 and m cw 1 = - m 1 r 1 + m 2 + m cw 2 + m 3 r cw 1 ( 75
)
The expression of the angular momentum H A (expressed at point A) is:

H A = 3 ∑ j=1 m j x S j ẏS j -y S j ẋS j + 2 ∑ j=1 m cw j x M j ẏM j -y M j ẋM j + I S j θ j ( 76
)
where x Q , y Q , ẋQ and ẏQ are the position and velocities of any point Q along x and y axes, respectively (Q being either point S j or M j , ( j = 1, 2, 3)).

Developing and introducing ( 75) into (76),

H A = I S 1 + m 1 r 2 1 + m cw 1 r 2 cw 1 + m 2 + m cw 2 + m 3 l 2 AB θ1 + I S 2 + m 2 r 2 2 + m cw 1 r 2 cw 1 + m 3 l 2 BC θ2 (77) with θ2 = - ẏB (x C -x B ) + (a -y B ) ( ẋC -ẋB ) l 2 BC ( 78
)
where x B , y B , x C are the coordinates along x and y axes of points B and C, respectively, and ẋB , ẏB , ẋC their velocities.

In order to cancel the shaking moment M sh A , the angular momentum has to be constant or null. Developing (77), M sh A can be cancelled if a = 0 and l AB = l BC (in such a case, θ1 = -θ2 ) and if:

I S 1 + m 1 r 2 1 + m cw 1 r 2 cw 1 + m 2 + m cw 2 + m 3 l 2 AB -I S 2 -m 2 r 2 2 + m cw 1 r 2 cw 1 + m 3 l 2 BC = 0 (79)

Balancing of a manipulator's leg using a Scott-Russell mechanism

Now let us consider a manipulator's leg with an added Scott-Russell mechanism (i.e. an additional RRR Assur group -Fig. 14). Let us denote as S 4 the centre of mass of link 4, which has a mass m 4 and an axial moment of inertia I S 4 . The position of S 4 is such that:

-→ AS 4 = l 3 r 4 u, r 4 being a dimensionless coefficient and u a unit vector along --→ CS 3 . Now the shaking force becomes:

A C B S 1 S 2 S 4 S 3 1 2 4 3 θ 03 θ 41 M 1 M 2 M 3 M 4
Fig. 14 A manipulator leg with added Scott-Russell mechanism.

f sh = (m 1 r 1 + m cw 1 r cw 1 + m 2 + m cw 2 + m 3 ) rB + (m 3 r 3 + m 4 r 4 ) a 1 + (m 2 r 2 + m cw 2 r cw 2 + m 3 ) a 2 (80)
with

a 1 = l 3 θ03 -sin θ 03 cos θ 03 -θ 2 03 cos θ 03 sin θ 03 a 2 = l AB θ03 -θ41 -sin (θ 03 -θ 41 ) cos (θ 03 -θ 41 ) -θ03 -θ41 2 cos (θ 03 -θ 41 ) sin (θ 03 -θ 41 ) (81)
At this step, only one supplementary counterweight is necessary for the cancellation of the shaking force, but it could be demonstrated after more derivations that another is necessary for the cancellation of the shaking moment. Therefore, we propose adding this additionnal counterweight directly. The two counterweights are located at points M 3 and M 4 , defined such that:

--→ CM 3 = r cw 3 l 3 u, --→ AM 4 = r cw 4 l 3 u, r cw 3 and r cw 4 being dimensionless coefficients. Their masses are respectively denoted m cw 3 and m cw 4 . With the addition of the counterweights, the shaking force becomes:

f sh * = f sh + m cw 3 rB + m cw 3 a 1 + m cw 3 r cw 3 + m cw 4 r cw 4 a 2 (82)
Thus, the shaking force is cancelled if: Developing and simplifying, the expression of the angular momentum is equal to:

m cw 4 = - m 4 r 4 r cw 4 m cw 3 = - m 3 r 3 r cw 3 m cw 2 = - m 2 r 2 + m 3 + m cw 3 r cw 2 m cw 1 = - m 1 r 1 + m 2 + m cw 2 + m 3 + m cw 3 r cw 1 (83) A 1 C 1 B 1 A 2 C 2 B 2 C 3 B 3 A 3 P D 1 D 2 D 3 I cr M 11 M 21 M 41 M 31 M 12 M 42 M 32 M 22 I cr M 13 M 43 M 33 M 23 I cr
H A = I eq 1 θ03 + I eq 2 θ41 (84)
with

I eq 1 = 4 ∑ i=1 I S i + m 1 r 2 1 + m cw 1 r 2 cw 1 + m 2 (1 -r 2 ) 2 + m cw 2 (1 -r cw 2 ) 2 l 2 AB + m 3 r 2 3 + m cw 3 r 2 cw 3 + m 4 r 2 4 + m cw 4 r 2 cw 4 l 2 3 ( 85
)
I eq 2 =I S 1 + m 1 r 2 1 + m cw 1 r 2 cw 1 + m 2 + m cw 2 + m 3 l 2 AB -I S 2 -m 2 r 2 2 + m cw 2 r 2 cw 2 + m 3 l 2 BC (86)
From (79), I eq 2 = 0. Therefore, the shaking moment of the leg can be cancelled using a simple counter-rotation I cr with an axial moment of inertia equal to I eq 1 .

Shaking moment and shaking force balancing of the 3-RPR manipulator

Now, let us apply such an approach to the 3-RPR mechanism. First of all, let us substitute the platform mass by three points masses located at C 1 , C 2 and C 3 , with the values of mass equal to m p 1 , m p 2 and m p 3 respectively [START_REF] Arakelian | Complete shaking force and shaking moment balancing of linkages[END_REF][START_REF] Baradat | Design and prototyping of a new balancing mechanism for spatial parallel manipulators[END_REF][START_REF] Seyferth | Massenersatz duch punktmassen in rumlichen getrieben[END_REF]. Such a condition can be obtained if: m p i = m p /3 and I p = 3m p i l 2

C i P (87)
Such a decomposition of the platform enables us to consider the shaking force and shaking moment balancing of each leg of the mechanism. Then, modifying each leg in order to obtain a mechanism similar to a Scott-Russel linkage, the shaking force and shaking moment are cancelled if:

m cw 4 = - m 4 r 4 r cw 4 m cw 3 = - m 3 r 3 + m p i r cw 3 m cw 2 = - m 2 r 2 + m 3 + m cw 3 + m p i r cw 2 m cw 1 = - m 1 r 1 + m 2 + m cw 2 + m 3 + m cw 3 + m p i r cw 1 0 = I S 1 + m 1 r 2 1 + m cw 1 r 2 cw 1 + m 2 + m cw 2 + m 3 l 2 A i B i -I S 2 -m 2 r 2 2 + m cw 2 r 2 cw 2 + m 3 l 2 B i C i , and
I cr = 4 ∑ i=1 I S i + m 1 r 2 1 + m cw 1 r 2 cw 1 + m 2 (1 -r 2 ) 2 + m cw 2 (1 -r cw 2 ) 2 l 2 A i B i + m 3 r 2 3 + m cw 3 r 2 cw 3 + m 4 r 2 4 + m cw 4 r 2 cw 4 + m p i l 2 3 ( 88
)
taking into account that I cr is the axial moment of inertia of the counter-rotations (Fig. 15). Thus, with this approach it is possible to create a fully-balanced shaking force and shaking moment 3-RPR mechanism with only three counter-rotations (Fig. 15), i.e., this method enables a reduction in the number of counter-rotations by a factor of two.

Illustrative examples and numerical simulations

The parameters used for the simulations are the followings: For these new parameters and for the trajectory used in Section 3.1, taking into account that the position coefficients of the counterweights are equal to r cw j = -0.5 ( j = 1, 3, 4), r cw 2 = -1, the new values of the counterweights are: m cw 1 = 3.17 kg, m cw 2 = 11.71 kg, m cw 3 = 0.33 kg, m cw 4 = 0.75 kg. The shaking force and shaking moment are then computed (dashed line in Fig. 16). Finally, we add the counter-rotations. Their values are equal to I cr = 1.56907 kg.m 2 . With such counterrotations, the shaking moment is balanced (gray line in Fig. 16(c)). Finally, it should be noted that the combination of the proposed two techniques of balancing enables the creation of fully balanced parallel manipulators with modified legs. As examples, different structures of balanced manipulators are presented in Fig. 17 (3-RPR, 3-PRR and 3-PRP) in which one leg with a prismatic pair is replaced by a leg with only revolute joints. Such a modification allows the displacement of the centre of mass of the manipulator to C 3 and then to balance the manipulator via the modified leg C 3 B 3 A 3 .

• l OA i = 0.35 m, l A i B i = l B i C i = 0.
In the same way, it is possible to balance a parallel manipulator with prismatic pairs by adding fewer Scott-Russell mechanisms. The balancing schemes for several parallel manipulators are presented in Fig. 18.

Thus, it has been presented new balancing schemes for the shaking force and shaking moment of planar parallel manipulators whose legs are made of prismatic pairs.

Usually, the balancing of parallel manipulators with prismatic pairs is only attained via a considerably complicated design. This section showed that it is possible to balance planar parallel mechanisms using Scott-Russell mechanisms. Such an approach enables a division of the number of counter-rotations by two. Numerical

A 1 C 1 A 2 C 2 C 3 P A 3 B 1 B 2 B 3
(a) Balanced 3-RPR parallel manipulator

C 3 A 3 B 3 A 1 B 1 P C 2 A 2 B 2 C 1 (b) Balanced 3-PRR parallel manipulator C 1 B 1 P C 2 B 2 C 3 A 3 B 3 A 1 D 1 A 2 D 2
(c) Balanced 3-PRP parallel manipulator Fig. [START_REF] Smith | Inertia forces in a four-bar linkage[END_REF] Complete shaking force and shaking moment balancing of planar manipulators with prismatic pairs via structural modification of one leg.

simulations carried out using ADAMS software validated the obtained results and illustrated that the suggested balancing enables the creation of a parallel manipulator transmitting no inertia load to its base.

Conclusions

For all balancing method, the main challenge is the trade-off between the complexity of the balanced mechanism and the quality of balancing. In the present Chapter, we have proposed a solution which allows the reduction of the balancing complexity by comparison with the usual approaches. The idea was to slightly modify the mechanism design by adding to it Assur groups, i.e. groups which dos not add any

A 1 C 1 A 2 C 2 C 3 P B 1 B 2 B 3 A 3 D 3
(a) Balancing of 3-RPR parallel manipulator

A 1 C 1 B 1 P C 2 A 2 B 2 C 3 A 3 D 3 B 3
(b) Balancing of 3-PRR parallel manipulator supplementary degree of freedom into the mechanism. The use of such a solution was detailed for the shaking force and shaking moment balancing of:

A 1 C 1 B 1 C 3 P C 2 A 2 B 2 B3 A 3 D 3 D 2 D 1
• the in-line four-bar linkage • the planar parallel robots with prismatic pairs. For both types of mechanisms, the proposed solution allowed the reduction (or even the cancellation in the case of the four-bar linkage) of the number of counterrotations used for obtaining the shaking moment balancing, which decreases the design complexity and the inherent problems due to the use of counter-rotations (backlash, noise, vibrations, etc.).

All theoretical developments were validated via simulations carried out using ADAMS software. The simulations showed that the obtained mechanisms transmit-ted no inertia loads to their surroundings, i.e. the sum of all ground bearing forces and their moments were eliminated.

Fig. 2

 2 Fig. 2 The three kinds of shaking force and shaking moment balanced four-bar mechanisms.

Fig. 5 of m cw 2 Fig. 6

 526 Fig.[START_REF] Ishida | Performance characteristics and working comfortableness of forest workers of a new non-vibrating chain saw utilizing perfectly balanced rotation-reciprocation device[END_REF] Shaking force and shaking moment of the four-bar mechanism before (full line) and after (dotted line) balancing.

Fig. 7

 7 Fig. 7 Variation of the maximal input torque absolute value as a function of angle α.

• l 4

 4 = 0.25 m, l 5 = 0.25 m, α = -90 deg, • r 4 = 0.125 m, • m 4 = 0.35 kg, m 5 = 0.1 kg.

Fig. 8 .

 8 In these figures, the values of are bounded between 25 and 100 deg. or 205 and 280 deg. in order to avoid the Variation of k 4 for α ∈ {25 100} deg. Variation of k 4 for α ∈ {205 280} deg.

  Variation of m cw 2 for α ∈ {25 100} deg. Variation of m cw 2 for α ∈ {205 280} deg.

Fig. 8

 8 Fig. 8 Variation of the values of parameters k 4 and m cw 2 as a function of angle α.

  For α ∈ {25 100} deg.

  For α ∈ {25 100} deg.

Fig. 9

 9 Fig.[START_REF] Arakelian | Shaking force and shaking moment balancing of mechanisms: a historical review with new examples[END_REF] Variation of the maximal input torque absolute value as a function of angle α.

Fig. 10

 10 Fig. 10 Schematic of the 3-RPR robot under study.

Fig. 12

 12 Fig.[START_REF] Leinonen | Terminology for the theory of machines and mechanisms[END_REF] Shaking force and shaking moment before (solid line) and after (dashed line) the addition of the counterweights, and after the addition of the counter-rotations (gray line).

Fig. 15

 15 Fig. 15 Schematics of a shaking force and shaking moment balanced 3-RPR mechanism.

  25 m, l C i P = 0.1 m and l 3 = 0.025 m, • r 1 = r 2 = 0.5, r 3 = 0 and r 4 = 4, • m 1 = 1.09 kg, m 2 = 1.1 kg, m 3 = 0.37 kg, m 4 = 0.75 kg; m p = 1 kg, • I S 1 = 0.00738 kg.m 2 , I S 2 = 0.58389 kg.m 2 , I 3 = 0.00344 kg.m 2 , I S 4 = 0.00025 kg.m 2 , and I p = 0.01 kg.m 2 .

Fig. 16

 16 Fig.[START_REF] Talbourdet | Mathematical solution of 4-bar linkages -IV. Balancing of linkages[END_REF] Shaking force and shaking moment before (solid line) and after (dashed line) the addition of the counterweights, and after the addition of the counter-rotations (gray line).

B 3 (Fig. 18

 318 Fig.[START_REF] Berkof | A new method for completely force balancing simple linkages[END_REF] Complete shaking force and shaking moment balancing of planar manipulators with prismatic with reduced number of Scott-Russell mechanisms.

A "physical pendulum" is a link which has such a distribution of masses that it allows the dynamic substitution of link's mass and inertia by two concentrated masses.