PhD, HDR a, ⁎ Bernard Kamsu-Foguem
email: bernard.kamsu-foguem@enit.fr

MSc Germaine Tchuenté-Foguem

MD, MSc Clovis Foguem

Verifying a medical protocol with temporal graphs: The case of a nosocomial disease

Keywords:

Objective: Our contribution focuses on the implementation of a formal verification approach for medical protocols with graphical temporal reasoning paths to facilitate the understanding of verification steps. Materials and methods: Formal medical guideline specifications and background knowledge are represented through conceptual graphs, and reasoning is based on graph homomorphism. These materials explain the underlying principles or rationale that guide the functioning of verifications.

Results: An illustration of this proposal is made using a medical protocol defining guidelines for the monitoring and prevention of nosocomial infections. Such infections, which are acquired in the hospital, increase morbidity and mortality and add noticeably to economic burden. An evaluation of the use of the graphical verification found that this method aids in the improvement of both clinical knowledge and the quality of actions made. Discussion: As conceptual graphs, representations based on diagrams can be translated into computational tree logic. However, diagrams are much more natural and explicitly human, emphasizing a theoretical and practical consistency.

Conclusion:

The proposed approach allows for the visual modeling of temporal reasoning and a formalization of knowledge that can assist in the diagnosis and treatment of nosocomial infections and some clinical problems. This is the first time that one emphasizes the temporal situation modeling in conceptual graphs. It will also deliver a formal verification method for clinical guideline analyses.

Introduction

A medical protocol (also known as a clinical protocol, medical guideline, or clinical guideline) is a document attesting to a good practice of a medical or paramedical action (treatment or diagnosis) according to a bibliography, a shared clinical experience, or recommendations of a consensus among professionals.

Protocol development receives attention in the medical community to the extent that these tools can improve the quality of care and reduce health care costs. Medical protocols are usually in the form of a corpus of text describing the diagnostic steps and processes of care, written in natural language. Properties that have been used to verify (model check) protocols are derived from lists of indicators of good medical practice, the latter being systematically developed by medical experts.

Nosocomial infection is a recognized public health problem worldwide with a prevalence rate of 3.0% to 20.7% and an incidence rate of 5% to 10% [START_REF] Samuel | Nosocomial infections and the challenges of control in developing countries[END_REF]. In Europe, the Scandinavian countries and the Netherlands, with appropriate policy measures by developing a protocol known as Search and Destroy, have a low prevalence rate of nosocomial infections (b 1%). Meanwhile, there are high prevalence rates (N 10%) in France, the United Kingdom, Germany, Belgium, and Spain. Because of this fact, it is necessary to develop a way to fight against these nosocomial infections.

Here, we work on protocols defining guidelines for the monitoring and prevention of nosocomial infections. Nosocomial infections are those infections acquired in a hospital or other healthcare facility by a patient and first appear 48 hours or more after hospital admission or within 30 days after discharge following in-patient care. In addition, occupational infections also occur among facility staff [START_REF] Benenson | Control of communicable diseases manual[END_REF]. These infections are unrelated to the original illnesses that bring patients to the hospital and neither present nor incubating at the time of admission [START_REF] Horan | Surveillance of nosocomial infections[END_REF]. Our contribution is the visual approach, and we illustrate this with the use case of nosocomial infection.

To develop a nosocomial infection, 4 elements must be present at the level of the disease:

• An infectious agent: A variety of microorganisms are likely to lead to nosocomial infections: bacteria, viruses, fungi, and parasites. Infections may be caused by a microorganism acquired from another person in the hospital (a cross infection) or may be caused by the patient's own flora (an endogenous infection). Some organisms may be acquired from an inanimate object or substances recently contaminated from another human source (an environmental infection). In some regions with poor basic hygienic practices, some nosocomial infections are due to pathogens of an external origin (foodborne and airborne diseases, gas gangrene, tetanus) or were caused by microorganisms not present in the normal flora of the patients (e.g., diphtheria, tuberculosis). Through selection and the exchange of genetic resistance elements, antibiotics promote the emergence of multidrugresistant strains of bacteria and microorganisms in the normal human flora that are sensitive to the given drug and are suppressed while resistant strains persist and may become endemic in the hospital. The widespread use of antimicrobials for therapy or prophylaxis (including topical) is the major determinant of resistance. Antimicrobial agents are, in some cases, becoming less effective because of resistance. Many strains of pneumococci, staphylococci, enterococci, and tuberculosis are currently resistant to most or all antimicrobials that were once effective. Multidrug-resistant Klebsiella and Pseudomonas aeruginosa are prevalent in many hospitals. This problem is particularly critical in developing countries where more expensive second-line antibiotics may not be available or affordable.

We first present the setting of our context to provide some background. We will then review requirement specifications and finish with the verification of medical protocols. The approach presented in this article is inspired by a part of a larger research project aimed at improving the daily application of medical protocols, including a particular illustration of a medical protocol to reduce rates of nosocomial infections. The verification of a protocol can improve the quality of care and also patient safety. The protocol is the model, and the specifications are the properties. The model has to satisfy those properties.

Background and context setting

Monitoring of nosocomial catheter-related infections

Here, we review protocols to establish guidelines for the monitoring and prevention of nosocomial diseases that are transmitted by medical devices, such as urinary catheters, intubation probes, or catheters. The work performed in this article is based on a former study from a Spanish hospital that illustrates the application of a protocol for the management and prevention of intravenous catheter-related infections (CRIs) [START_REF] Pérez | Authoring and verification of clinical guidelines: a model driven approach[END_REF]. It was found that these infections are the most common cause of those nosocomial infections acquired in the hospital that affect morbidity and mortality and that also have economic implications (a prolonged duration of hospitalization, increased cost of health care, the emergence of multiple antibiotic resistance microorganisms, and reducing the incidence of treatment for other infections).

Taking care of and preventing nosocomial infections are the responsibility of all individuals and services providing health care (managers; physicians; and other health care workers, such as clinical microbiology, pharmacy, central supply, maintainers, housekeeping, and training services). It is recommended that transparent and appropriate evaluation criteria be used to evaluate the surveillance of nosocomial infections such as the catheter-related infections (CRI) or infections related to intravenous catheters (IRC) (Table 1). In some establishments, there are specialists in infection control, epidemiology, and infectious disease, including infection control physicians and infection control practitioners [START_REF] Scheckler | Requirements for infrastructure and essential activities of infection control and epidemiology in hospitals: a consensus panel report[END_REF]. These individuals have a scientific and technical support role, for example, surveillance and research, developing and assessing policies and practical supervision, evaluating materials and products, controlling sterilization and disinfection procedures, and implementing training programs. These individuals should also support and participate in research and assessment programs at the national and international levels. A nosocomial infection prevention manual, compiling recommended instructions and practices for patient care, is an important tool [START_REF]Prevention of hospital-acquired infections. A practical guide2nd ed[END_REF]. The manual should be developed and updated by the infection control team with review and approval by a committee.

Model checking

The primary goal of model checking [START_REF] Grumberg | 25 years of model checking: history, achievements, perspectives[END_REF] is to check the conformity of the model with the requirements defined in the specification, with an important issue being: does the model provide an interpretation as to whether it meets or does not meet the system's requirements?

• The principle is to compare a model with a property of the system and to automatically check whether the model satisfies the property.

• Its issues: why verify the status of a model and check its outcome?

There is a need to provide regulatory amendments that allow for the establishment of a new approach to safety oversight and for the certification of heath care operations. Indeed, for complex medical organizations and systems, it is necessary to have sufficient confidence in their functioning at all times.

Table 1

Simplified criteria for the surveillance of nosocomial catheter-related infections (CRI)

Vascular catheter infection

Inflammation, lymphangitis, or purulent discharge at the insertion site of the catheter, abscess, or spreading cellulitis at the insertion site of the catheter during the days or weeks after the intervention Septicemia Fever or rigors, and at least rigors, and at least 1 positive blood culture

Requirements for medical protocols

In this article, 4 origins of the properties of specification have been identified to meet the requirements for medical protocols. The categories of the sources are nice, and seem useful in clarifying the required properties: good medical practice, hospital characteristics, purpose of the protocol, and clinical condition of the patient. Indeed, the categories of sources play a role in the temporal graph-based verification of clinical protocols: computable inspections of a clinical protocol model or the feedback to the user if the model does not satisfy the property. The question then remains: what are they and how should they be addressed? What are medical barriers and how do they affect access to services? We are going to review what exact characteristics are associated with the main independent sources of properties in Table 2.

Conceptual graph formalism

The design of conceptual graphs is based on the study of perception in psychology. The meaning of a concept is reduced to its position relative to other concepts. Therefore, there is a sense with respect to a semantic net modeling of the general knowledge of the system [START_REF] Sowa | Conceptual structures: information processing in mind and machine[END_REF]. Conceptual graphs are defined with typed objects, nodes, and edges of typed relations according to the vocabulary of the considered domain that details the allowed concepts and relationship types. The reasoning on these graphs is made via a subgraph matching mechanism by performing variable substitutions (called projections) for those making similar graphs in certain contexts. We use the structural features of graphs and semantic properties of domain vocabulary to demonstrate some equivalence classes on groups of sets [START_REF] Baget | Extensions of simple conceptual graphs: the complexity of rules and constraints[END_REF]. These equivalence classes are used to locate information about specific knowledge-intensive activities and to help provide the scientific basis to make problem-solving approaches more consistent, guileless, and creative. Conceptual graph formalism represents a good compromise between graphic language and formal language because it facilitates the reading and understanding of the reasoning.

Conceptual graphs are a knowledge representation language, proposed by John Sowa, and allow for both rigorously defining a vocabulary (ie, ontology) and using this vocabulary to describe and reason about situations [START_REF] Sowa | Knowledge representation: logical, philosophical, and computational foundations[END_REF]. From the point-of-view of knowledge representation, conceptual graphs provide a structure of different types of knowledge that is not explicit in other formalisms, such as semantic networks or object modeling. Indeed, this formalism marks a clear separation between ontological knowledge, which is encoded in a special structure called support, and factual knowledge, which is represented by graphs. In addition, the formalism clearly distinguishes entities (represented in the graph by rectangles) and the relationships between these entities (represented by ovals).

Based on the adage that "a picture is worth a thousand words," conceptual graph operations is a superior vector for the vulgarization of the reasoning in all its forms. Without excluding demonstrations using logical formulas, graph operations are sufficient to explain the reasoning by which a pattern is often held as a demonstration until a better control of logical tools is available [START_REF] Chein | Graph-based knowledge representation: computational foundations of conceptual graphs[END_REF]. In a number of cases, a sequence of suitable graphical illustrations is sufficient in itself to lead to a solution. The intuition is requested more than theoretical knowledge and sometimes fragmented and inefficient.

In addition, conceptual graphs can easily be translated into the language of some other approach used in knowledge engineering, such as description logics (DL) [START_REF] Baader | The description logic handbook: theory, implementation, applications[END_REF], Resource Description Framework Schema [START_REF] Yao | Automated conversion between different knowledge representation formats[END_REF], and its evolution OWL (Web Ontology Language) DL [START_REF] Casteleiro | Clinical practice guidelines: a case study of combining OWL-S, OWL, and SWRL[END_REF][START_REF] Horrocks | OWL rules: a proposal and prototype implementation[END_REF][START_REF] Shadbolt | The semantic web revisited[END_REF]. As a result, the possibility to interact and exchange knowledge modeled with internal and external staff is available to different stakeholders in the modeling system under this consideration. In addition to the visual reasoning facilities, syntactic opportunities for graphs include the intuitive ideas of graphical arguments that have no equivalent in DL [START_REF] Dau | A diagrammatic reasoning system for the description logic[END_REF].

However, the expressiveness of conceptual graphs may not be satisfactory for modeling the temporal knowledge of a given domain.

To overcome this aforementioned drawback, we propose a formal modeling of temporal knowledge, which introduces the semantic interpretation of the temporal logic expressions of Computational Tree Logic (CTL) [START_REF] Clarke | Automatic verification of finite-state concurrent systems using temporal logic specifications[END_REF] in models of conceptual graphs. The modeling of temporal knowledge is represented by the nested conceptual Good medical practice Concerns the aspects or factors intrinsic to the guideline by itself, such as the conditions and clinical problems it covers or the desired outcomes. For example, verifying whether the guideline conforms to a good medical practice through specific requirements, such as: (i) Avoidance of redundancies and inconsistencies in drug prescriptions.

(ii) Between 2 alternative treatments, the preferred one should always minimize side effects and allow for faster healing.

(iii) Paying attention to appropriate practices of hygiene (eg, hand washing, isolation, using antiseptics and disinfectants).

(iv) Providing direct patient care using practices that minimize infection. (v) Participation in quality control techniques used to sterilize equipment in the hospital, including the selection of sterilization equipment (type of appliances) and monitoring.

(vi) Ensuring the education and training of all staff through the support of programs on the prevention of infection using disinfection and sterilization techniques.

Particularities of the hospital

Concerns the contextualization of the guidelines of a given hospital. The hospital care program may provide constraints and rules, taking into account facts deemed to be specific knowledge and that are a reflection of local health care services. For example, the verification of a reduction in the use of certain pharmacological treatments, avoiding crowded patient wards (where poor control practices may facilitate infection transmission), being careful not to concentrate patients highly susceptible to infection in one area (eg, newborn infants, burn patients, intensive care) or frequently transferring patients from one unit to another or the availability of specific hospital resources (such as laboratory instruments). At a hospital, microbial flora may contaminate objects, devices, and materials, which subsequently contact susceptible body sites of patients. In addition, new infections associated with bacteria, such as waterborne bacteria (atypical mycobacteria) and/or viruses and parasites, continue to be identified.

Guideline goal

Concerns aspects or factors intrinsic to the guideline by itself, such as conditions and the clinical problems that it covers or the desired outcomes. For example, the verification of whether the guideline contains a path requiring the specific support services required for a given treatment.

Patient-specific clinical conditions

Concerns the possible paths the guideline proposes to be performed given a specific patient clinical state. This state may be influenced by impaired immunity, age, medical history, medical procedures, and/or invasive techniques, such as intravenous catheters, which create potential routes of infection, the transmission of drug-resistant bacteria, and the severity or frequency of the nosocomial infection. For example, the verification of the feasibility of a given action, the path of actions on the patient, or to prove whether there is a therapy for a patient under these clinical conditions.

graphs used to describe the property specification models that formalize knowledge modeling in the application domain. Therefore, we first translated the specifications of the medical domain into formal properties. These will then be used in the reasoning process to rigorously verify that the protocol meets the requirements. We have been striving to integrate the consideration of the modeling elements of CTL into the conceptual graph modeling framework itself.

Verification of medical protocols with temporal conceptual graphs

We present a mechanism for the verification of medical protocols using conceptual graphs.

Notion of temporal modality

How do you represent temporal modalities in conceptual graphs?

The use of the conceptual graphs defined by Sowa provides a great power of representation, and the advantages of this formalism are mainly:

-A logical interpretation for representing the semantics of natural language -A reasoning service equipped with a rigorous graph operation. However, some of the shortcomings of this model should not be ignored: the treatment of the negation is not satisfactory and the interpretation of the modal terms is beyond the formal framework. Thus, writing sentences such as the following (complex by structure and the temporal references described) should be avoided in such protocols: "If the number of critical clinical parameters of time exceed a threshold SE, then an alarm should be provided for medical staff with remote broadcast." Therefore, our work will allow for the adequate modeling of temporal aspects for the adaptation formalization process of properties' specifications.

In his work on conceptual graphs, Bernard Moulin focused on the need for better expressive temporal representations and proposed constructions that were used to represent various types of temporal contexts: the point-of-view of the narrator, the perspective of the agent, and the temporal localization [START_REF] Moulin | Temporal contexts for discourse representation: an extension of the conceptual graph approach[END_REF].

In conceptual graphs, we have the nesting graphs (embedded or a nesting of contexts) that we will use to describe the temporal modalities.

In addition, we have the following temporal terms:

All: for all paths e Finally: one day e Next: afterwards e Until: up to e Globally: generally

Exists: there is a path e Finally: one day e Next: afterwards e Until: up to e Globally: generally

We note that concepts are represented by rectangles and relationships by ovals in the visual representations. Significantly, a framework based on taxonomies allows experts to represent and share their knowledge with other experts through shared and controlled vocabularies; it follows that information modeling can be improved by means of social and semantic web tools [START_REF] López-Cuadrado | SABUMO: towards a collaborative and semantic framework for knowledge sharing[END_REF].

Formalization of the three general principles of medical treatment

We absolutely make illustrative descriptions of the 3 properties that we believe are in the users' interests to introduce the conceptual graph formalization on the origins of requirement specifications.

We translate the 3 following properties:

• First property (Fig. 1). "If a treatment T1 is not applicable to a disease in a particular medical context, then it will be replaced by an applicable treatment T2 that produces a positive impact on a disease similar to treatment T1." • Second property (Fig. 2). "If a treatment T1 is not effective against a disease due to the characteristics of the disease in question, then it will be replaced by treatment T2, which has positive similar consequences on the disease." • Third property (Fig. 3). "If a treatment T1 has too many negative consequences on a patient, then it will be replaced by a treatment T2 similar to T1 but having fewer negative consequences."

Graphical modeling of a case study

The first step is to construct domain taxonomy (hierarchy of concepts and relationships) in the target application (Tables 3 and4). Highlighting conceptual dependencies, this taxonomy assists the process of purposeful structuring of information, but it also provides the vocabulary to be used during the model specification and verification phases with graph operations [START_REF] Baget | Extensions of simple conceptual graphs: the complexity of rules and constraints[END_REF]. In general, the knowledge modeling encompasses the identification of a domain ontology and tasks that the system must achieve and the suitable determination of problem-solving methods to accomplish those tasks, with the definition of mappings between the domain ontology and the problem-solving methods, and the creation of application-specific content knowledge as an instantiation of the classes in the domain ontology [START_REF] Musen | The knowledge acquisition workshops: a remarkable convergence of ideas[END_REF].

Verification of the catheter-related infections protocol model

To provide a better understanding of the approach presented in this article, a particular case study is detailed using the management of catheterrelated infections (CRI) or infections related to intravenous catheters (IRC).

Infections related to intravenous catheter tests are performed for nosocomial infections contracted through a catheter and are conducted to provide evidence of a possible infection. This test has been used in a Spanish hospital and was developed based on a directive issued by the Agency for Health Care Research and Quality. The verification procedures are fully descriptive graphs because all knowledge is represented by graphs, and all reasoning mechanisms are used by graph operations [START_REF] Baget | Extensions of simple conceptual graphs: the complexity of rules and constraints[END_REF]. This allows for the visualization of the reasoning and the use of the structural properties of the graphs to verify compliance with the CRI protocol-specified requirements (good medical practice, hospital characteristics, protocol goal, and patient clinical condition). During the verification process, the protocol is proven to not contain CRI abnormalities intrinsic to its definition and to comply with the various types of temporal properties useful for its effective implementation.

In Fig. 4, we see a part of the medical protocol for the detection of an CRI.

First, the patient is in a state where he/she is not suspected of having an CRI.

If the patient has specific symptoms, such as signs of a local infection or a high fever, the relation "symptomsAppear" triggers the process of examining for a case of an CRI.

Three tests are performed in parallel (Maki Test (a semi-quantitative culture method rolling the catheter segment across blood agar), Hemoccult Test (a qualitative assay for occult blood in stool based on detecting the peroxidase activity of hemoglobin), Study of Semiquantitative Cultures of Catheter Hub and Skin) to verify the patient's condition and whether he/she is suspected of contracting an CRI.

We then translate this model into a conceptual graph formalism, thus introducing the notion of temporality in this formalism for temporal context representation.

"If the numerical value of the MakiTest is equal to 3 and the numerical value of the Hemoccult Test is equal to 2, then the patient is not suspected of contracting an CRI."

Query graph

"If the numerical value of the MakiTest is equal to 3 and the numerical value of the Hemoccult Test is equal to 2, then the patient is not suspected of contracting a nosocomial disease" (Fig. 5).

The projection is the fundamental operation of reasoning in the formalism of conceptual graphs. This projection is a type of graph matching, which can be easily visualized and interpreted. The projection mechanism performs a computation specialization between the graphs that takes into account the links described in the specialization hierarchies of concepts, relationships, and temporal modalities.

For example, if we want to answer the question "Does the property represented by graph (query) H present a target situation also described by graph G?" This search operator information allows us to respond and to verify the property, and it will seek the possible projections of H in G.

We apply this principle by translating the graph's first situation with the above model and then compare it to the query graph and decide whether to validate the model. In fact, we made 2 types of translations from the CRI model into conceptual graphs:

Translation of the situational graph

Several test outputs are derived from this model, which involve appropriate antibiotic treatments, or not, according to the patient's condition.

Indeed, a patient is suspected of contracting an CRI if:

-The value of the Maki test is between 5 (inclusive) and 15 (excluded), and the value of the Hemoccult test is between 2 (excluded) and + ∞. -The value of the Hemoccult test is between 2 (excluded) and + ∞, and the value of the SSCHS test is between 15 (inclusive) and + ∞.

The patient is not suspected in the case where:

-The value of the Maki test is between -∞ and 5 (excluded), and the value of the Hemoccult test is between -∞ and 2 (inclusive).

Translation of the query graph

Setting a query in the form of a graph question, we cannot prove the suspicion of a patient contracting a nosocomial disease. This evidence projection query graph in the situational graph is based on a calculation of a specialization between the 2 graphs.

The query graph with a numerical value equal to 3 in the MakiTest and a numerical value equal to Hemoccult Test 2 projects into the test output ST1 situational graph.

In fact, we have the 2 following membership relations:

-3 ∈]-∞; 5[ -2 ∈]-∞; 2]
From the visual modeling and temporal reasoning, graph operations are arranged sequentially. We then translate the four basic requirements of medical protocols (good medical practice, hospital characteristics, purpose of the protocol, clinical condition of the patient) in the form of graphs to specify properties and structure the information to make reasoning that orient choices and decisions. This verification includes a description (planned results and delivery strategy) and a graphical illustration (a logic model), which demonstrate how the activities of checking are expected to lead to the achievement of the planned results.

From the graphical representation provided by Fig. 6, we can deduce that the graph satisfies the conditions of the query graph property, thereby validating the model. Our approach has explored the potential of visual reasoning as a suitable tool in solving problems related to semantic knowledge formalization and verification.

Discussion

Establishing effective strategies for problem solving requires imagination and ingenuity, which is also true for the formulation of the problem in the implementation of its resolution. Designing relevant specifications and an interesting model for the problem generally provide a better overview, which will provide analysts valuable elements to take full advantage of the use of visual reasoning. Emphasis should be placed on techniques that provide analysts with valuable information to facilitate a general understanding of the problems and to determine the best way to influence the efficiency of the resolution process.

From the point-of-view of knowledge engineering, diagrams have the advantage of incorporating reasoning mechanisms, which are more easily engaged for the encoding and handling of structures and strategies for solving problems [START_REF] Luger | Artificial intelligence: structures and strategies for complex problem solving[END_REF]. Therefore, it follows that charts show a potential efficiency higher than that of first-order logic. In this case, it is not necessary to go through an entire database to find everything about the target concept [START_REF] Russell | Artificial intelligence: a modern approachIn[END_REF].

In our case, the model is a conceptual graph, and through its manipulation, it can generate valuable insights into the conceptualization of the problem. The cognitive values of this model are its ability to generate views from conceptual graphs for the visualization of the knowledge that they contain. The automatic generation of this visualization is important for both the validation of graphs and the communication of the knowledge that they convey. In terms of logic, this process promotes a technical demonstration, insofar as the perception of some evidence that naturally flows, from a graphical representation, may be more difficult to establish from an axiomatic description.

Finally, the visual illustration of the steps of the reasoning steps is likely to promote the implementation of creative methods of problem solving or at least stimulate the development of alternative ways of problem solving, which demonstrates that typical proofs include visualized tasks, complementary solutions, and justifications for the performed actions.

At the semantic level, we opt for a formal structure to interpret each symbol as a primitive form acceptable to some grammar and to subsequently facilitate a recognition process more complex than syntactic construction. In this context, the main formal grammar is that of CTL, which provides a rigorous framework for describing knowledge from a process of feedback. This formalization of model representations and the reasoning mechanisms involved may provide the means to guide model capitalization and the exploitation of knowledge.

Conclusion

The formalization of knowledge is valuable in the field of complex nosocomial pathologies. This explicit knowledge is very useful because official figures show that in France 6% to 7% of hospitalizations are complicated by a nosocomial infection that is often serious. This number varies depending on the service in which the person is hospitalized but can indeed reach 30% in a service such as Resuscitation. Nosocomial diseases also have a noticeable economic burden.

The development of information technology and communication in the professional world has been swift and massive. One of the effects of new technologies on information and communications is the increasing need to present information visually [START_REF] Allwein | Logical reasoning with diagrams (studies in logic and computation)[END_REF]. However, this raises some interesting questions: what is the logical status of reasoning that uses visualization? What are the cognitive advantages and pitfalls of this reasoning?

The implementation of an audit approach with logical foundations is valuable; however, with graphic reasoning paths, it is intuitive to facilitate the understanding of the verification steps by the actors of a domain [START_REF] Kamsu-Foguem | Using conceptual graphs for clinical guidelines representation and knowledge visualization[END_REF]. The visual transformation rules that are applied are only valid if, and only if, their demonstration can be proven using only visual rules and if mistakes cannot be derived from this evidence schematic. Thus, inference rules are equally powerful graphically with valid explanations and more interactive than their textual counterparts [START_REF] Kamsu-Foguem | Conceptual graph-based knowledge representation for supporting reasoning in African traditional medicine[END_REF]. Therefore, diagrams can be used as a valid basis for formal proofs, defining a system of adequate and consistent reasoning vis-àvis formal logic. More generally, tools with diagram evidence are used to support the development of schematic reasoning by providing visual representations of formal proofs that shall draw attention to the appropriate warning situations [START_REF] Kamsu-Foguem | User-centered visual analysis using a hybrid reasoning architecture for intensive care units[END_REF]. The progress of our approach better highlights the causes of noncompliance with certain properties or at least suggests some improvements in failures of evidence.

We can also add that our research is not only applicable to the medical field but can also be applied in many other sectors [START_REF] Kamsu-Foguem | Knowledge-based support in Non-Destructive Testing for health monitoring of aircraft structures[END_REF][START_REF] Kamsu-Foguem | Mining association rules for the quality improvement of the production process[END_REF][START_REF] Kamsu-Foguem | Software architecture knowledge for intelligent light maintenance[END_REF][START_REF] Kamsu-Foguem | Knowledge description for the suitability requirements of different geographical regions for growing wine[END_REF]. In fact, in an industrial setting, for example, our methods can be applied to check quality protocols, whereas in scientific research, they can be applied to experimental protocols involving the description of requirements and to conduct an experiment or test.

Fig. 1 .

 1 Fig. 1. First property.

Fig. 2 .

 2 Fig. 2. Second property.

Fig. 3 .

 3 Fig. 3. Third property.

Fig. 4 .

 4 Fig. 4. Situational graph of an CRI or IRC.

Fig. 5 .

 5 Fig. 5. Query graph of the CRI or IRC.

Fig. 6 .

 6 Fig. 6. Intravenous catheter-related infection (CRI or IRC) result of the projection of the query graph in the situational graph.

  • A susceptible subject: Important patient factors influencing the acquisition of an infectious agent include age, immune status, underlying disease, and diagnosis and therapeutic interventions (e.g., chemotherapy). Hospitalized patients with impaired im-

	mune systems due to chronic diseases or cancers (e.g., diabetes
	mellitus, respirator insufficiency, renal failure, immune disor-
	ders, burns, malignant tumors, leukemia, or acquired immuno-
	deficiency syndrome (AIDS)) or because of their general
	condition (malnourished or at extreme ages of life (infancy and
	old age)) have an increased susceptibility to infections with
	opportunistic pathogens. The latter are infections with organism
	(s) that are normally innocuous, e.g., part of the normal human
	bacterial flora, but may become pathogenic when the body's
	immunological defenses are compromised. Immunosuppressive
	drugs or irradiation may lower resistance to infection. Injuries to
	skin or mucous membranes bypass natural defense mechanisms.
	Many modern diagnostic and therapeutic procedures, such as
	biopsies, endoscopic examinations, catheterization, intubation/
	ventilation, suction and surgical procedures, increase the risk of
	infection. Contaminated objects or substances may be introduced
	directly into tissues or normally sterile sites such as the urinary
	tract and the lower respiratory tract.

• Environmental factors: Patients with infections or carriers of pathogenic microorganisms admitted to hospitals are potential sources of infection for patients and staff. Crowded conditions within the hospital, frequent transfers of patients from one unit to another, and concentrations of patients highly susceptible to infection in one area (eg, newborn infants, burn patients, and intensive care) all contribute to the development of nosocomial infections. Microbial flora may contaminate objects, devices, and materials, which subsequently contact susceptible body sites of patients. In addition, new infections associated with bacteria such as waterborne bacteria (atypical mycobacteria) and/or viruses and parasites continue to be identified. World Health Organization studies and others have also shown that the highest prevalence of nosocomial infections occurs in intensive care units and in acute surgical and orthopedic wards

[START_REF]Prevention of hospital-acquired infections. A practical guide2nd ed[END_REF]

.

• Bacterial resistance: Many patients receive antimicrobial drugs.

Table 2

 2 Sources of the specification of properties

	Sources	Characteristic

Table 4

 4 Hierarchy of relationships

Table 3

 3 Hierarchy of concepts


El Bechir

Zoubeir Bouaziz
email: zoubeir.bouaziz@enis.rnu.tn

Maher Baili
email: maher.baili@enit.fr

Gilles Dessein
email: gilles.dessein@enit.fr

E B Msaddek

Influence of interpolation type in high-speed machining (HSM)

Keywords: Machining, Polynomial interpolation, Modeling, Simulation, HSM

The recourse to the high-speed machining for the manufacture of warped shapes imposes an evolution towards a very high technicality of the CAM methods and of the machining operation execution. Due to its own characteristics, the high-speed machining (HSM) implies the use of new machining interpolations, in such a way that it assures the continuity of advances in the best way possible. Among these interpolations, we mention the polynomial interpolation. In this article, we propose a complete study of the interpolation type influence on the HSM machine dynamic behavior and also on the generated errors. For this, we have measured the feed rate of the cutting tool path for each type. Then, in terms of accuracy, we have measured the errors. In order to validate our approach, we have compared the simulated results to the experimental ones.

Introduction

For decades, a wide range of CAD/CAM systems have been used in mechanical manufacture. These systems propose different methods of interpolation. This diversification aims to obtain the tool trajectory during the machining of complex surfaces more adapted to the CAM tolerance [START_REF] Helleno | Investigation of tool path interpolation on the manufacturing of die and molds with HSC technology[END_REF]. We distinguish the linear interpolation, the circular interpolation, the polynomial interpolation, and the functions of compaction (Fig. 1). The linear interpolation is the most frequently used, because the programmer finds it easy to use. Even if this kind of interpolation remains scarcely used, the polynomial interpolation has the task to palliate the linear interpolation problems. The polynomial interpolation is the generation of a tool trajectory in the form of a polynomial curve of different degrees. The spline-type interpolation (or polynomial curve bit by bit) allows the connection of a series of points of passage or of control by smoothed curves. Depending on the numerical controlled unit (NCU) and depending on the type of machine, we distinguish different types of polynomial interpolations used in machining. These interpolations include Akima spline (Aspline), Bezier's spline (Bspline), cubic spline (Cspline), and non-uniform basis rational spline (Nurbs). In polynomial interpolation, we endeavor to make the different blocks continuous in curvature so as to limit the discontinuities of speed. The functions of compaction allow the adaptation by compression in real time of the CAM trajectories. Instead of treating several small blocks, the NCU treats a more important block of displacement. These functions are based upon the compaction by polynomial interpolation of ten or more blocks of linear interpolation G1. A block compaction must respect the programmed contour tolerance. The more accurate the tolerance, the more impossible the block compaction becomes. Among these functions of compaction used in machining are Compon, Compcad, and Compcurv [START_REF] Siemens | Manual of programming-complementary notions[END_REF][START_REF] Siemens | Polynomial functions and Nurbs: SINUMERIK 840D/840Di/810D[END_REF].

In high-speed machining (HSM), the feed rate becomes more and more important. However, the linear and the circular interpolations begin to present some limitations. These limitations occur particularly in the precision and in the time of machining because if the feed rate increases, the time necessary for acceleration and deceleration increases too. This has resulted in the search for a new strategy for the choice of the type of tool trajectory interpolation before the program CN generation. Thus, the study shows that the interpolations alone, without considering the NCU behavior, are not in adequation with the real behavior in HSM [START_REF] Lavernhe | Put into account of constraints associated to MT-NC couple in generation of 5 axes trajectory in HSM[END_REF]. From which, one concludes that the study of the relation between the interpolation and the HSM dynamic behavior becomes a necessity for the optimization of machining.

Recent studies have been interested in the analysis of the interpolation type influence on the machining of complex shapes in HSM. Helleno and Schutzer [START_REF] Helleno | Investigation of tool path interpolation on the manufacturing of die and molds with HSC technology[END_REF] have shown that the linear interpolation presents limitations during machining of molds in HSM and the benefits that can be found by using other types of interpolations. Guardiola et al. [START_REF] Guardiola | Influence of tool path interpolation on cycle time and following error during high-speed milling of die and mold surfaces[END_REF] have studied the dynamic capacity of a HSM machining center under different combinations of parameters for the machining operation of complex geometries. They have determined the variation of the feed rate and the following error of the linear interpolation and the polynomial interpolation. Souza and Coelho [START_REF] Souza | Investigation of tolerances required for NC program's generation using spline polynomial and linear interpolation to describe a free form tool path for high speed milling[END_REF] have proved that the type of interpolation (linear, Bspline) and the values of CAM tolerance have an important role in the machining process in terms of machining time and surface quality. Pateloup et al. [START_REF] Pateloup | Optimization of the pockets hollowing out trajectories for the high speed milling[END_REF] have examined, at the same time, the influence of the interpolation modes, that of the curvature, and also that of the trajectory continuity on the trajectory path time of a pocket hollowing out. Tapie et al. [START_REF] Tapie | Circular tests for HSM machine tools: bore machining application[END_REF] and Pateloup et al. [START_REF] Pateloup | Corner optimization for pocket machining[END_REF] have integrated the dynamic modeling to explain the slowdown of the HSM machine in linear and circular interpolations. Furthermore, other interpolations like Bspline were tested with the controller Siemens 840D. Prevost [START_REF] Prevost | Modeling advanced torque machine tool/CNC in HSM to optimize the calculation of multi-axis trajectories[END_REF] has examined the execution errors and the machining ones in the HSM manufacturing process. A compensation method of the contour errors for the Bspline interpolation has been developed. Other studies have exploited some types of polynomial interpolation to improve the machining operation. Liang et al. [11] have proposed a real-time interpolator with constant-length segments to improve the EDM milling of Nurbs curves. Zhao et al. [START_REF] Zhao | A real-time look-ahead interpolation methodology with curvature-continuous B-spline transition scheme for CNC machining of short line segments[END_REF] and Erkorkmaz et al. [START_REF] Erkorkmaz | Virtual CNC system: part II. High speed contouring application[END_REF] have developed path-smoothing methods, which adopt a curvature-continuous Bspline.

In most cases, these authors have used a single type of interpolation without regard to its diversity. Up to now, few research works have been developed concerning the polynomial interpolation and its influence on the machining process in general [START_REF] Pessoles | Aid to putting-down choice in 5 axes continuous machining by the kinematic behavior modeling of the machine tools[END_REF], hence the interest of treating all that is related to this interpolation mode in detail. Moreover, no studies have optimized the choice of the machining interpolation by studying all the CAM interpolation types, taking account the dynamic behavior of the HSM machine, of the machining precision, and of the surface quality.

In this article, we have studied the influence of the different types of interpolation on the dynamic behavior of the HSM machine, on the machining accuracy, and on the surface state for the machining of complex shapes. Among the interpolation technologies, we have evaluated linear interpolation, polynomial interpolations (Aspline, Bspline, Cspline, and Nurbs), and functions of compaction (Compon, Compcad, and Compcurv). Our purpose is to find out the interpolation type capable of assuring a quality and a maximal productivity. For doing this, experimental works have been done.

The high-speed machining

HSM is one of the latest technologies being a part of the means provided for the enterprises to make important productivity gains. The high-speed machining of complex shapes, like molds and dies, allows the removal of most of the materials in an object in the shortest time possible. In order to improve the manufacturing process in terms cost, time, and quality, the HSM can be a solution. From the point of view of dimensional precision, we can have a better repetition for the machining series. Besides, in HSM, we can manufacture very hard materials with longer lifetime of tools and reduced machining forces [START_REF] Msaddek | Optimization of pocket machining strategy in HSM[END_REF]. It consists not only in increasing the spindle speed but also in increasing the speed of a cut (more than five times superior) [START_REF] Tapie | Circular tests for HSM machine tools: bore machining application[END_REF]. The real feed rate profile in HSM is variable, which indicates that several parameters must be put into evidence to optimize the machining of complex shapes. Hence, in order to make an optimal choice of a

Defining the shape to be machined Machining the test piece (8 bands)

Influence of the type of interpolation upon:

-The feed rate -The machining time -The NCU error -The machining error machining strategy, all we have to do is to analyze the different critical criteria [START_REF] Msaddek | Modeling and simulation of high-speed milling centers dynamics[END_REF].

The manufacturing process in HSM has, as entry point, the CAD model from the design stage, which is the reference model constructed from a surface arrangement or geometric elements. The first step, CAM associated with the digital channel, concerns the trajectory generation or the CAM model from the CAD model. Depending on the complexity of the work surface, the differences between the nominal surface and the nominally machined surface, called CAM errors, are generated. They are mainly due to numerical errors. Then, the step of execution in particular allows the generation of a relative movement of the tool relative to the workpiece. The digital part realized by the NC causes gaps called NCU errors when generating instructions of axis position. Finally, the non-ideal geometry structure of the machine, as well as associated faults (quasi-static or dynamic), is a source of errors, physical, associated with the process. Moreover, the dynamic phenomena associated with cutting also cause deformation of components under the effect of mechanical actions taken in. All the gaps created by the combination of these purely physical phenomena and cutting conditions used are called machining errors [START_REF] Prevost | Modeling advanced torque machine tool/CNC in HSM to optimize the calculation of multi-axis trajectories[END_REF]. Figure 2 shows the manufacturing process in HSM and the generated errors.

3 Influence of the type of interpolation on the warped forms in HSM

Principle and methodology

Certain types of interpolation are favorable to the machining of warped forms, whereas others are not. The linear interpolation is largely used in manufacture shops for the machining of several forms, but it generates remarkable limits in highprecision machining of complex-form pieces. Thus, recent researches show that there is an increase in the number of the generated CAM discretization nodes. In consequence, the length of the segments diminishes [START_REF] Tapie | Circular tests for HSM machine tools: bore machining application[END_REF]. The non-utilization of "Look Ahead" (function of blocks reading anticipation) and the generation of discontinuity in tangency, or in curvature, provoke brute decelerations of the vibrations, of the shocks, and of marks on the piece. On the contrary, in the case of polynomial interpolation, if we have a polynomial of a degree superior to three, the tool acceleration becomes more continuous, which leads to a better trajectory quality and a surface without undesirable traces [START_REF] Duc | Machining of freeform-contribution to the improvement of the quality of machining paths[END_REF]. In fact, because of the lack of research in this field, we have broached the study of the influence of this type of interpolation on the HS machining of warped forms.

In order to meet the stated objectives, we have used a sample piece containing some form variations in the tool advance rate. The width of the piece allows the realization of these forms, according to the different interpolation techniques as shown in the organigram of Fig. 3.

The test piece definition

The conceived test piece represents a combination of two pieces, one of them is tested by Smaoui et al. [START_REF] Smaoui | Simulation of the deflected cutting tool trajectory in complex surface milling[END_REF] and issued from [START_REF] Dessein | Path defects in high speed machining[END_REF], and the other one is tested by Helleno and Schutzer [START_REF] Helleno | Investigation of tool path interpolation on the manufacturing of die and molds with HSC technology[END_REF]. The model is made up of mono-axis, combined linear path, convex and concave circular paths, and a polynomial (spline) path, (Fig. 4). This piece comprises most of the forms we can find in an industrial piece, which justifies this choice. So, the interpolations are diversified, which is interesting in order to evaluate, with precision, the machining errors and the machine dynamic behavior during the high-speed machining of a complex form.

The test piece machining

The realization of the test piece consists in machining eight bands of 10 mm each, so as to present the constant behavior zones and state the visual difference with five interpolations: G1, cubic Nurbs, of weight PW=1, and of nodal distribution (0.2, 0.2, 0.2, 0.4, 0.4, 1.5, 3.1…), Aspline, Bspline, Cspline, and G1 with the three functions of compaction: Compon, Compcad, and Compcurv (Fig. 5). Table 1 presents the parameters of finish machining operation of the test piece. The technical characteristics of the HSM machine Huron KX10 (three axes) are put in detail in Table 2. Besides, we have activated the anticipation (FFWON) and the smoothing (OSSE).

FFWON

Feed forward ON is an activation of the anticipating drive. The anticipating drive brings back to zero the distance of pursuit, which depends on the speed in a displacement with interpolation. The anticipating drive allows to increase the precision and so to improve the machining quality.

OSSE

This is the smoothing of the tool orientation at the beginning and at the end of the block. The objective of these essays is to know the interpolation that urges the machine the most, since we run the risk to have vibrations and overstepping during the machining. The oscillations can lengthen the machining time as they may affect the precision and the quality of the final piece. So, each type of interpolation generates a path which is different from the other, according to the mathematical formula or the algorithm (Aspline, Cspline, etc) it describes.

Feed direction

In this paragraph, the experimental results of the speed profiles according to the interpolation type are presented and examined. The measuring of the profiles is undertaken by means of a servo trace, thanks to the software SinuComNC © provided by the maker of NCU, Siemens.

Figure 6 presents the tool trajectory of machining and the critical zones. Figure 7 presents the feed rate measured for the linear interpolation G1. Figure 8 presents the feed rate measured for the polynomial interpolation "Nurbs".

The feed rate in linear interpolation is variable. The speed profile oscillates following the machining path. First of all, the speed increases up to 4 m/min on the 5-mm segment. So, the programmed speed 4.8 m/min is not reached. Then, if any change of direction occurs on a critical area of strong curvature variation (Fig. 6), the Zone 1 Zone 7

Zone 9

Zone 10

Fig. 8 Feed rate measured for the polynomial interpolation "Nurbs" Fig. 9 Feed rates measured in polynomial interpolation: "Aspline", "Bspline", and "Cspline" machine decelerates to 0.2 m/min; otherwise, the machine accelerates until it reaches the wanted speed. This variation is due to the dynamic behavior. The measured machining time is equal to 3.6 s with 8.213 % as percentage of the realization of the optimal feed rate. According to Fig. 8, the feed rate of the polynomial interpolation "Nurbs" is better than that for the linear interpolation. In fact, it diminishes four times on the zones 1, 7, 9, and 10 (Fig. 6) versus ten times in linear interpolation, for the polynomial trajectory is a continuous curve containing discontinuities only in curvature but without discontinuities in tangency (linear interpolation). The modeling of the machine dynamic behavior [START_REF] Tapie | Circular tests for HSM machine tools: bore machining application[END_REF] shows that the slowing down comes out of these discontinuities, where instant acceleration is variable. Hence, when the speed exceeds, the maximal acceleration of the axes (Table 2) imposes limited speeds. Once more, if we have a block of a small length, the time of the interpolation cycle (treatment time of the variable CN line) limits the feed rate. Thus, the machining time measured in the polynomial interpolation is equal to 2.90 s with a percentage of the realization of the optimal feed rate equal to 29.063 %.

Figure 9 presents the feed rates measured in a polynomial interpolation: Aspline, Bspline, and Cspline. Figure 10 presents the feed rates measured for the compaction functions: Compon, Compcad, and Compcurv. Table 3 presents the size of the NC file, the average speed, and the realization percentage of the programmed speed of 4.8 m/min for each interpolation.

According to Figs. 9 and 10 and to Table 3, the speed of the polynomial interpolation of the Nurbs type is the most important with an average speed of 4 m/min. Whereas, the speed evolution of the polynomial interpolation of type Bspline is the most penalizing with an average speed of 2.54 m/min. When comparing the polynomial interpolation to the linear one, the first interpolation is not always favorable towards the machine dynamic behavior. The linear interpolation has an Fig. 10 Feed rates measured for the compaction functions: "Compon", "Compcad", and "Compcurv" The bold values mean the favorable ones

The italic values mean the non favorable ones average feed rate more important than the polynomial interpolation of types Aspline and Bspline. The solicitation of the machine in Bspline is due to the CAM trajectory interpolation by the controller, for the programmed positions in Bspline are not points of the curve, but only points of control of the spline. On the other side, according to the tests, it is to be noted that the CAM discretization (size of the NC file) has no influence on the speed by varying the interpolation. We find that several interpolations like the Bspline, Cspline, Compcad, and Compcurv have NC files of the same size (200 Ko) but have varied average speeds.

The "Nurbs" interpolation has the highest percentage of the V prog realization (29 %), contrary to the compaction function "Compcad" which is 6.11 %. The "Nurbs" polynomial interpolation is very favorable towards this criterion because it is deprived of discontinuities in tangency. Thus, the compaction functions have kept a good average speed close to 3 m/min and a low percentage of V prog realization, nearly 6 %, for the speed does not undergo important oscillations. In fact, the compaction of a block eliminates most of the maximum and minimum speed peaks and maintains a more important average speed. Besides, when comparing the compaction functions to one another, the average speed of the function "Compon" is found to be the lowest, owing to the approximation by a cubic polynomial with a limited number of compacted blocks, whereas for the "Compcurv" and "Compcad" functions, the approximation is carried out by a quintuple polynomial with a limited number of compacted blocks. Besides, the function "Compcurv" assures the acceleration continuity to the transitions between the blocks, which explains that its speed is the most important.

The influence of the type of interpolation on the machining time

Table 4 presents the machining time by interpolation (bands of 10 mm). The average speed informs us about the machining time taken by each interpolation. According to Table 4, we remark that the "Nurbs" polynomial interpolation is the most favorable in machining time (153.796 s). However, the "Bspline" polynomial interpolation urges the machine more and more (252.329 s), for it provokes some strokes which have generated considerable vibrations during the machining. We do not know what causes these vibrations especially with The bold values mean the favorable ones

The italic values mean the non favorable ones Fig. 11 Following errors and contour deviations of the linear and polynomial "Nurbs" interpolations this mode of programming. A change in acceleration could result from a vibration on the machined surface, but has not been justified so far. Thus, the linear interpolation is often more penalizing in relation to the machining time, when comparing it to the polynomial interpolation except "Bspline" and the compaction function "Compcad" because its compression is realized by approximation of a "Bspline" curve.

Influence of the type of interpolation on the NCU error

In order to show the influence of the type of interpolation on the error of the NCU, we have executed the finish of the test piece machining (Fig. 5) using the different types of interpolation. After that, we have measured the following error and the contour deviation for each type. In the following, and due to the considerable number of results, we are going to present the results of the most interesting interpolations (which show the best variation of the errors), linear (Nurbs) and the functions of compaction (Compon and Compcurv). Figure 11 presents the following errors and the contour deviations of the linear and polynomial "Nurbs" interpolations. Figure 12 presents the following errors and the contour deviations for the "Compon" and "Compcurv" functions.

When machining the piece with different types of CAM interpolations, we remark the apparition of great specificities of each type. In the NCU, the interpolator treats the CAM trajectory either as linear or polynomial, generating errors of contours. Then, the set trajectories are enslaved by the buckle controllers, generating following errors [START_REF] Prevost | Modeling advanced torque machine tool/CNC in HSM to optimize the calculation of multi-axis trajectories[END_REF]. This objective is to know the contribution and the advantages out of the use of the polynomial interpolation. The contour and the following error average values depending on the type of interpolation are recapitulated in Table 5.

We notice that the following error profile is identical to that of the feed rate for all the interpolations. So, the following error follows the feed rate; if the speed increases, the following error increases too and vice versa. If the programmed speed 4.8 m/min is reached, the following error attains its maximal value of 250 μm. In fact, the saturation of the acceleration on the critical zones and the limitation of the speed considered by the time IPO (NC calculation of a block) promote the accuracy of the warped shape machining. Besides, the average following error of the linear interpolation is inferior to that of the polynomial interpolation. So, the linear interpolation The bold values mean the favorable ones

The italic values mean the non favorable ones imposes fewer constraints at the level of the enslavement position in three axes than the polynomial interpolation "Nurbs". We conclude that the following error follows the principle of the CAM rope error: if the trajectory generated in CAM with a specified interpolation is close to that of CAD, the executed trajectory is also close to that of the instruction. Moreover, we have a change of the sign in the following error, between the two interpolations. Thus, for the other interpolations (Aspline, Bspline, and Cspline), the following error varies slightly. Concerning the contour deviation, the evolution is partly constant according to the interpolation. We consider that the variation is negligible because of the errors of measuring.

In short, the linear interpolation and the compaction functions are the most favorable due to the following error. Hence, the NCU enslaving the system tackles the linear segments with few defects than the "Nurbs" and "Cspline" curves. This is due to the dynamics of the axes. Thus, the controller interpolator treats the polynomial trajectory (Nurbs and Cspline) with few defects than the linear curve. And, this is due to the continuity of the spline curve.

The type of interpolation influence on the machining error

The objective of this part is to study the influence of the type of interpolation on the machining error. We are going to use the measurement results saved by the tridimensional measuring machine (TMM) after the test piece finish machining by the different types of interpolation. The measuring method consists of feeling the machined surface in the advance direction by saving points. The measuring is supposed to be ideal, without considering the errors of the isostatic reference piece and the imprecision envelope made up of the retiming of the points and of the diameter of the probe ball. The tested interpolations are still the same: the linear interpolation, the polynomial interpolations (Nurbs, Aspline, Bspline, and Cspline), and the compaction functions (Compon, Compcad, and Compcurv). The method consists in realizing a series of measures for each machining band of the test piece (Fig. 5) on a tridimensional measure machine. We consider probing each surface following a plan P in order to obtain a cloud of probed points. For each probed point, we gather the coordinates of the probe center of a diameter Ø equal to 4 mm. Some imprecisions are generated during the measuring due to the slaving of the probe ball. Jalid et al. [START_REF] Jalid | Uncertainties valuation of the measure on the tridimensional measure machine: new method of estimation of the surface parameters and associated uncertainties[END_REF] have evaluated the uncertainty of measuring on a TMM machine. They have shown that this uncertainty is of some micrometers (2.705 μm max).

In the following, owing to the great number of results, we will mention only the results of the linear interpolation and of the polynomial interpolations: Bspline and Cspline (the clearest and the most significant). Figure 13 presents the machining profile in a linear interpolation together with the CAD profiles (theoretical) and the profile of the simulated CAM discretization.

The trajectory profile superpositions (CAD and CAM) and machining show the influence of all the errors on the inlet trajectory (theoretical or CAD). The essays and the simulations also present the influence of the type of interpolation on the machining paths. In the linear interpolation, according to the TMM measuring, the machining path moves a few hundreds of microns away from the CAD trajectory (Fig. 14). Since the following error does not exceed 0.25 mm, the rest of the machining error is due to the elementary machining cell (EMC). In general, the geometries of these zones present some difficulties of machining or of access to three-axis machining, because the machining is executed with the tool flank with an angle close to 90°together with other phenomena (geometrical defect, tool deflection, etc). In Bspline and Cspline interpolations, according to the TMM measuring, the machining path moves a few hundreds of microns away from the CAD trajectory, but in the critical zone (Figs. 16 and 18), it moves more than 1 mm away. The machining error is due especially to the definition of the spline trajectories. The Bspline (Bezier's spline) is a polynomial of a cubic value or more. The programmed points do not belong to the curve but are solely points of control of the spline, i.e., the curve does not pass directly through these points, but it just tends to go towards them. The Cspline (cubic spline) is an interpolation by a cubic polynomial. It passes exactly through the programmed points and strongly tends to oscillate between them. The polynomial interpolation of spline type presents limits in precision.

The type of interpolation influence on the surface quality

After machining of the workpiece with different types of interpolation, we spent to experimentally verify the influence of each type of interpolation on the surface state. Because, it is impossible to measure the roughness on all the workpiece (size and shape), we choose an accessible zone (Fig. 19) in order to measure manually the roughness and to examine the influence of the interpolation type on the surface quality of this area. We used a roughness meter (Mahr) and a digital microscope (Dino-Lite) to measure the roughness and to photograph the machined surfaces. Figure 20 shows the surface quality of a selected zone captured by a digital microscope for each interpolation. Table 6 shows the value of the roughness Ra of the selected zone according to the type of interpolation.

According to the previous results, the type of interpolation affects the surface quality. The surface texture generated by the linear interpolation (G1) is worst than those of all the polynomial interpolations with Ra=1.647 μm. So, the polynomial interpolations generate good surface quality, unlike the linear interpolation. The interpolation Nurbs presents the best roughness with Ra=0.610 μm. On the other hand, the surface state of the interpolation Bspline is the poor between the other polynomial interpolation with Ra = 0.845 μm. The Bspline trajectory does not pass through nodes (control points), so during machining, errors and vibrations increase which negatively affects the surface state. The functions of compaction also have poor surface quality since they are derived from the linear interpolation.

We conclude that the polynomial interpolations retain better surface state than the linear interpolation and functions of compaction. The use of polynomial interpolation permits to gain in machining time and surface quality. However, these types of programming generate more machining errors.

Conclusion

In this article, we have been interested in the interpolation type influence on the HSM machining process, by considering some NCU and axis performances. In fact, we have measured the feed rate for the different interpolation types and also for different values of the CAM tolerance. We find that the polynomial interpolations mainly the one of Nurbs type are faster but less precise. This interpolation is the most appropriate for the machining of this piece type based on time and speed criteria, while the compaction functions are the most accurate. In this way, the latter have an almost similar effect as the linear interpolation.

The machining with these different types of interpolations also affects a criterion which is very important for complex-form machining which is the surface quality. We find that the different types of polynomial interpolations gave the best roughness, in contrast to the linear interpolation and these derivatives: the functions of compaction. Besides, this work shows surprising results which incite to consider further works of research, mainly about the use of "Bspline" and "Cspline" interpolations. These polynomial interpolations generate good surface quality with low precision. According to our need (accuracy or rapidity), we choose the type of interpolation. Afterwards, we can generalize these results to any industrial piece, since our workpiece contains all possible complex forms. In fact, this work provides the problematic of the warped surface pieces with the learning concerning the influence of the different types of interpolation on the HSM machining and mainly on the generated machining errors, on the machine dynamic behavior, and on the surface quality. Hence, the manufacturer has to know these results in order to choose the best type of interpolation according to his need.

Fig. 1 aFig. 2

 12 Fig. 1 a Linear interpolation, b circular interpolation, and c polynomial interpolation

Fig. 3

 3 Fig. 3 Methodology for studying the influence of the type of interpolation

Fig. 5

 5 Fig. 5 Test piece machined finish on eight bands of 10 mm by different types of interpolation

  Maximal feed rate 30 m/min (X,Y) and 18 m/min (Z) in rapid 10 m/min (X,Y,Z) work programmable Maximal acceleration 5 m/s 2 (X), 5 m/s 2 (Y), 3 m/s 2 (Z) Jerk maximum 50 m/s 3 (X,Y,Z)

Fig. 6 Fig. 7

 67 Fig. 6 Tool trajectory and critical zones

Fig. 12

 12 Fig. 12 Following errors and contour deviations for the "Compon" and "Compcurv" functions

Fig. 13 Fig. 14

 1314 Fig. 13 Final machining profile (linear interpolation) with CAD and CAM profiles simulated. "trajet d'usinage" means machining path

Fig. 16

 16 Fig. 15 Final machining profile (Bspline interpolation) with CAD and CAM profiles simulated

  Figures 15 and 16 present the machining final profile (Bspline) with the simulated CAD and CAM profiles.Figure17presents the machining profile in Cspline interpol ati on together with the CAD profiles (theoretical) and the profile of the simulated CAM discretization.In Bspline and Cspline interpolations, according to the TMM measuring, the machining path moves a few hundreds of microns away from the CAD trajectory, but in the critical zone (Figs.16 and 18), it moves more than 1 mm away. The machining error is due especially to the definition of the spline trajectories. The Bspline (Bezier's spline) is a polynomial of a cubic value or more. The programmed points do not belong to the curve but are solely points of control of the spline, i.e., the curve does not pass directly through these points, but it just tends to go towards them. The Cspline (cubic spline) is an interpolation by a cubic

Fig. 17

 17 Fig. 17Final machining profile (Cspline interpolation) with CAD and CAM profiles simulated

Fig. 18

 18 Fig. 17Final machining profile (Cspline interpolation) with CAD and CAM profiles simulated

Fig. 19 Fig. 20

 1920 Fig. 19 Selected zone for measuring the roughness

Table 1

 1 Finish-cutting parameters

	Programmed	Spindle	Material	Tool	CAM
	speed	speed	of the part		tolerance
	4.8 m/min	24,000 rpm Aluminum Ball bur	10 μm
				Ø6

Table 2

 2 Characteristics of the HSM machine Huron KX10 three axes

Table 3

 3 Average speed and the realization percentage of the programmed speed of 4.8 m/min for each interpolation

		Linear	Nurbs	Aspline	Bspline	Cspline	Compon	Compcad	Compcurv
	NC file size (Ko)	209	197	202	200	200	184	200	200
	Average speed (m/min)	3.219	3.997	3.142	2.542	3.764	2.825	2.915	3.068
	Realization percentage of V prog (%)	8.213	29.063	9.071	8.348	22.938	6.140	6.114	6.414

Table 4

 4 Machining time according to the type of programming

	Interpolation	CAM time (s)	Machining time (s)
	Linear	162	195.446
	Nurbs	51	153.796
	Aspline	164	186.799
	Bspline	161	252.329
	Cspline	162	166.115
	Compon	103	187.864
	Compcad	156	198.913
	Compcurv	101	184.024

Table 5

 5 NCU Error by interpolation

		Linear	Polynomial	Compon	Compcurv
		interpolation	interpolation	function	function
			"Nurbs"
	Average following	158.1	212.8	150.5	170.1
	error (μm)
	Average contour	10.6	10	12.2	11.3
	error (μm)