Andrew Wisniewski

Julien Kirchgesner

Philippe Seksik

Cécilia Landman

Anne Bourrier

Isabelle Nion-Larmurier

Philippe Marteau

Jacques Cosnes

Harry Sokol

Laurent Beaugerie
email: laurent.beaugerie@aphp.fr

Prof Laurent Beaugerie

Lionel Arrivé

Marine Camus

Najim Chafai

Edouard Chambenois

Ulriikka Chaput

Clotilde Debove

Charlotte Delattre

Xavier Dray

Jean-François Fléjou

Guillaume Le Gall

Nadia Hoyeau

Jérémie-Henri Lefèvre

Chloé Martineau

Laurence Monnier-Cholley

Violaine Ozenne

Yann Parc

Magali Svrcek

Increased incidence of systemic serious viral infections in patients

Keywords: viral infections, immune-suppressive therapy, thiopurines, cytomegalovirus, Epstein-Barr virus, hemophagocytic lymphohistiocytosis

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

Methods: Using MICISTA, an IBD-unit database detailing prospectively characteristics and complications of IBD, we identified between January 2005 and December 2014 patients followed for IBD outside the context of organ transplantation, HIV infection or chronic hepatitis. We estimated incidences of extra-intestinal SVI, defined by need for hospitalization or permanent organ damage. Standardized incidence ratios (SIR) were calculated using the French hospital database. We performed a case-control study nested in MICISTA for assessing the role of exposure to IBD drugs and IBD clinical activity in the risk of developing SVI.

Results:

We identified 31 patients with SVI among 2645 patients followed for 15,383 person-years. We observed 13, 10, 5 and 3 cases of CMV, EBV, VZV and HSV infection, respectively. No death occurred. The incidence rate of SVI in patients with IBD was 2.02/1000 person-years, and the SIR was 3.09 (95% CI, 1.98-4.20; P=.0002) in the study population. By multivariate analysis, increased risk of SVI was associated with exposure to thiopurines (odds-ratio (OR), 3.48; 95% CI,1.36-8.90; P=.009), and clinically active IBD at onset of SVI (OR,3.35; 95% CI,1.23-9.23; P=.02).

Conclusions:

The incidence of extra-intestinal SVI in patients with IBD is tripled compared to general population. Clinically active IBD and exposure to thiopurines are the main drivers of the risk.

Crohns' disease (CD) and ulcerative colitis (UC) are lifetime diseases characterized by continuous or relapsing intestinal inflammation. Most of the patients with IBD are currently exposed for prolonged periods to various immunosuppressive drugs, including small molecules (thiopurines and methotrexate) and biologics, mainly anti-Tumor Necrosis Factor (TNF) agents. [START_REF] Kirchgesner | Therapeutic management of inflammatory bowel disease in real-life practice in the current era of anti-TNF agents: analysis of the French administrative health databases 2009-2014[END_REF] Serious viral, bacterial or fungal infections are usually defined as infections that require hospitalization or result in death or permanent organ damage. [START_REF] Beaugerie | Balancing Benefit vs Risk of Immunosuppressive Therapy for Individual Patients With Inflammatory Bowel Diseases[END_REF][START_REF] Andersen | Association between tumour necrosis factor-alpha inhibitors and risk of serious infections in people with inflammatory bowel disease: nationwide Danish cohort study[END_REF][START_REF] Kirchgesner | Risk of Serious and Opportunistic Infections Associated With Treatment of Inflammatory Bowel Diseases[END_REF] Focusing to viral infections, patients with IBD may first develop serious viral infections (SVI) that are not related to IBD activity or to immunosuppression due to IBD treatment. They may also develop SVI triggered by inflammatory intestinal lesions, mainly cytomegalovirus (CMV) colitis, [START_REF] Sager | Review article: cytomegalovirus and inflammatory bowel disease[END_REF][START_REF] Siegmund | Cytomegalovirus infection associated with inflammatory bowel disease[END_REF] and Epstein-Barr Virus (EBV) systemic reactivation. [START_REF] Reijasse | Epstein-Barr virus viral load in Crohn's disease: effect of immunosuppressive therapy[END_REF] Finally, they may develop SVI attributable to the immunosuppressive action of IBD drugs, knowing that the magnitude of this effect may substantially differ among drugs. [START_REF] Kirchgesner | Risk of Serious and Opportunistic Infections Associated With Treatment of Inflammatory Bowel Diseases[END_REF] Clinical research on extra-intestinal viral infections in IBD has been mainly focused on herpes zoster, [START_REF] Gupta | Incidence and risk factors for herpes zoster among patients with inflammatory bowel disease[END_REF][START_REF] Khan | Overall and Comparative Risk of Herpes Zoster With Pharmacotherapy for Inflammatory Bowel Diseases: A Nationwide Cohort Study[END_REF][START_REF] Colombel | Herpes Zoster in Patients Receiving JAK Inhibitors For Ulcerative Colitis: Mechanism, Epidemiology, Management, and Prevention[END_REF] an easily identified infection that is treated in most cases in an ambulatory setting, and usually does not result in permanent organ damage. The promoting effect of immunosuppressive agents on serious infections, including SVI, has been also repeatedly reported. [START_REF] Colombel | Herpes Zoster in Patients Receiving JAK Inhibitors For Ulcerative Colitis: Mechanism, Epidemiology, Management, and Prevention[END_REF][START_REF] Toruner | Risk factors for opportunistic infections in patients with inflammatory bowel disease[END_REF][START_REF] Fidder | Long-term safety of infliximab for the treatment of inflammatory bowel disease: a single-centre cohort study[END_REF][START_REF] Lichtenstein | Serious infection and mortality in patients with Crohn's disease: more than 5 years of follow-up in the TREAT registry[END_REF][START_REF] Ford | Opportunistic infections with anti-tumor necrosis factor-alpha therapy in inflammatory bowel disease: meta-analysis of randomized controlled trials[END_REF][START_REF] Naganuma | A prospective analysis of the incidence of and risk factors for opportunistic infections in patients with inflammatory bowel disease[END_REF][START_REF] Deepak | Infectious Complications of TNF-α Inhibitor Monotherapy versus Combination erapy with Immunomodulators in Inflammatory Bowel Disease: Analysis of the Food and Drug Administration Adverse Event Reporting System[END_REF][START_REF] Osterman | Effectiveness and Safety of Immunomodulators With Anti-Tumor Necrosis Factor Therapy in Crohn's Disease[END_REF] However, in these studies, the respective role of IBD clinical activity and IBD drugs has not been addressed. The aim of this study was to assess the incidence (compared to age and gender-matched general population), and risk factors of extra-intestinal SVI in patients with IBD, followed prospectively with an IBD-specific database in a big IBD unit.

METHODS

Data sources

MICISTA

MICISTA is an IBD-specific prospective electronic-form database dedicated to clinical research created by a senior gastroenterologist (J.C.) in 1994. All patients with a permanent diagnosis of IBD, based on usual clinical, endoscopic, imaging and pathological criteria, and followed in our IBD unit (located in Rothschild hospital from 1994 to December 2002, then in Saint-Antoine hospital from January 2003 up to now) are enrolled in the database at the time of the first face-to-face contact in the IBD unit (outpatient visit or hospitalization). The following items are recorded at entry in the database: socio-demographic details, initial (at diagnosis) and cumulative IBD phenotype according to Montreal classification, smoking status, past history of IBD surgery. The database is then updated prospectively as long as patients are regularly or intermittently followed in our IBD unit. [START_REF] Cosnes | Smoking cessation and the course of Crohn's disease: an intervention study[END_REF] A regular follow-up is defined as at least one face-to-face contact with the patient (visit or hospitalization) per calendar year in our IBD unit. At each contact in the IBD unit, clinical IBD activity is scored as follows: 0, no digestive symptoms; 1, mild symptoms that may be attributable to IBD activity, postoperative functional sequelae or associated irritable bowel syndrome; 2, symptoms that are attributable to IBD and are compatible with usual home and/or professional activities; 3, symptoms that are attributable to IBD and are not compatible with sustained home and/or professional activities; 4, hospitalization for IBD flare; 5, intestinal resection. At every face-to-face contact with the patient at the IBD unit, IBD treatment is scored as follows: 0, no IBD drug; 1, oral 5-amino-salicylates; 2, low doses of systemic corticosteroids (up to 10 mg a day of prednisone/prednisolone, 8 mg a day of methylprednisolone or 3 mg a day of budesonide); 4, high doses of systemic corticosteroids (more than 10 mg a day of prednisone/prednisolone, 8 mg a day of methylprednisolone or 3 mg a day of budesonide); 4, thiopurines or methotrexate; 5, anti-TNF agent; 6, vedolizumab or ustekinumab. At the end of each calendar year of follow-up, the annual clinical IBD activity of each patient is rated as the highest clinical score recorded within the calendar year, and the annual status of IBD treatment is rated as the highest score of IBD drugs used within the calendar year.

At each face-to-face contact with patients in the IBD unit, physicians are required to fill in the database the date and type of any of the following events that have possibly occurred in the time interval between the last and the current contact: IBD-related surgery, intestinal dysplasia or cancer; extra-intestinal cancer and serious infection, defined as infections that require hospitalization or result in permanent organ damage.

MICISTA database was authorized by the French data protection agency (CNIL, registration number 1 104 603).

French National Hospital Discharge Database

The French National Hospital Discharge Database (Programme de Médicalisation des Systèmes d'Information (PMSI)) covers all public and private hospitals in France. The standardized discharge summary includes: patient's demographics; primary and associated discharge diagnosis codes (WHO International Classification of Diseases, 10th revision (ICD-10)13; medical and surgical procedures performed (French Medical Common Procedure Coding System); length of stay, entry and in-hospital mortality. A unique anonymous identifier allows linking all hospital claims of the patient since January 2008 and tracking the occurrence and progression of chronic conditions over time.

Patient selection

We considered for enrollment in the study population all patients aged 18 years or older with at least a one-year period of regular follow-up in the MICISTA database between January 1, 2005 and December 31, 2014. We excluded patients with chronic (more than 6 months) replicative infection by Hepatitis B Virus, Hepatitis C Virus or Human Immunodeficiency Virus. In order to focus the study on the role of immunosuppression due to IBD or approved IBD drugs, we excluded organ transplant recipients.

For each individual patient, the observational time started on January 1, 2005 for those patients previously enrolled in the MICISTA database, or the day of entry into the MICISTA database for those patients who were enrolled in the MICISTA database between January 1, 2005 and December 31, 2013. The observational time ended the day of death, the day of first symptoms of extra-intestinal SVI, or the day of end of regular follow-up (i.e. the day of the last face-to-face contact with the patient at the IBD unit followed by a period of more than one year without contact in the IBD unit), or on December 31, 2014).

Definition and selection criteria of extra-intestinal SVI

SVI were defined as viral infections requiring hospitalization or resulting in death or permanent organ damage. In order to exclude isolated intestinal SVI associated with IBD flare, we excluded patients hospitalized for IBD flare associated with CMV colonic infection, defined as follows: intracellular inclusion bodies visualized with standard haematoxylin and eosin staining (with or without immunohistochemistry with antibodies against pp65) of colon tissue, or detection of CMV DNA in colonic tissue by quantitative PCR; no evidence for extra-intestinal organ damage (lung, retina); no elevation of blood liver tests of more than two times the upper limit of the normal range; no mononucleosislike syndrome in blood count. We also decided a priori to exclude the possible incident cases of replicative infection by Hepatitis B Virus, Hepatitis C Virus or Human Immunodeficiency Virus.

Calculation of incidence and SIRs of extra-intestinal SVI

The date of the event of SVI was defined as the date of the first symptoms attributable to SVI. To avoid selection bias, patients who had symptoms attributable to SVI at the time of entry into the observational period were not considered as incident cases in the analysis. For determination of incidence of SVI according to exposure to IBD drugs, we considered the number of events related to the cumulative numbers of calendar years

Nested case-control study and statistical analysis

To assess the respective roles of exposure to drugs and clinical IBD activity on the risk of extra-intestinal SVI, we performed a 4:1 case-control study. The selection process was performed in the MICISTA database: patients were matched for age (up to 5 years older or younger), gender, and IBD subtype (Crohn's disease on one side, ulcerative colitis or IBD unclassified on the other).

We included the following patient characteristics, on the first day of SVI or at the matched observation time in controls, in univariate conditional regression: exposure to IBD drugs and clinical activity of IBD, smoking status, patient Body Mass Index (BMI), level of education, Montreal localization of disease, prior intestinal surgery, clinically active perianal lesions. Items of interest in controls were extracted by a junior gastroenterologist (A.W) and a senior gastroenterologist (L.B.) from review of medical files of patients selected in the MICISTA database. Variables significant at P < 0.20 were entered into a multivariate logistic regression model with a backward variable elimination procedure to assess the strength of the associations while controlling for possible confounding variables.

For all subgroup comparisons of the study, proportions were compared using Fisher exact test, and continuous outcomes were compared using the Kruskal-Wallis test. All tests were 2-tailed at a 5% significance level.

RESULTS

Selection, baseline characteristics and follow-up time of the study population

We identified in the MICISTA database 2821 patients over the age of 18 with at least a one-year period of regular follow-up between January 1, 2005 and December 31, 2014 (Figure 1). Among then, we excluded 176 patients who were transplant recipients or chronically (more than 6 months) infected with Hepatitis B Virus, Hepatitis C Virus or Human Immunodeficiency Virus.

The study population eventually included 2645 patients (44.3% males, 68.6% with Crohn's disease), followed for a total number of 15,383 person-years. Median follow-up time was 6.1 years (interquartile range (IQR), 2.9-9.4). Patient characteristics at entry into the observation period are provided in Table 1. Median age of patients at entry into the observation period was 34.4 years (IQR, 25.2-46.8). Median time interval between the diagnosis of IBD and the entry into the observation period was 6.3 years (IQR, 1.5-

13.5).

Incidence and SIR of extra-intestinal SVI

No patients had symptoms attributable to SVI at the time of entry into the observational period. Among the 50 patients who developed acute SVI during the observation period, we excluded 19 patients who developed CMV colitis associated with IBD flare without systemic manifestations (Figure 1 years. The SIR of extra-intestinal SVI was higher in patients with UC or IBD unclassified than in patients with CD. Regarding IBD drugs, the highest SIR was observed in patients exposed to immunomodulators within the calendar year of occurrence of SVI.

Characteristics of SVI

Distribution of SVI according to pathogens and clinical expression is shown in Table 3.

Among the 31 patients who developed SVI, 15 had CMV systemic infection, 8 had EBV infection, 3 had herpes simplex virus (HSV) infection and 5 had varicella zoster virus (VZV) infection.

Characteristics of incident SVI are shown in Table 4. All but one (97%) patients required hospitalization, while the remaining patient who developed permanent HSV facial nerve paralysis was treated in an ambulatory setting. IBD was clinically active at clinical onset of SVI in 35% of patients. Five of the 10 patients with CMV primary infection and 4 of the

Risk factors of extra-intestinal SVI in the study population

Nested case-control study for assessing risk factors of SVI included 31 cases and 124 controls. Non-matched characteristics of IBD were not significantly differently distributed between cases and controls (Supplementary Table 3). Among the 13 factors that were entered in the univariate analysis, only two factors, clinically active IBD at clinical onset of SVI and exposure to thiopurines at clinical onset of SVI, reached the eligibility level (P <0.20) for being entered in the multivariate model. By multivariate analysis, increased risk of SVI was associated with exposure to thiopurines (odds-ratio (OR), 3.48; 95% CI, 1.36-8.90; P = .009), and clinically active disease (OR, 3.35; 95% CI, 1.23-9.23; P = .02) at clinical onset of SVI. Values of odds-ratio associated with IBD clinical activity and exposure to IBD drugs at clinical onset of SVI, by univariate and multivariate analysis, are shown in Table 5.

DISCUSSION

We have shown that the risk of extra-intestinal SVI is tripled in patients with IBD followed in IBD units, compared to age and gender-matched individuals of general population.

The excess risk is mainly related to IBD clinical activity and chronic exposure to thiopurines. Excess hospitalizations due to SVI contribute to the human and financial burden associated with management of IBD. [START_REF] Nguyen | Annual Burden and Costs of Hospitalization for High-Need, High-Cost Patients With Chronic Gastrointestinal and Liver Diseases[END_REF] The main limitation of this study is the recruitment of patients restricted to a tertiary IBD center, which is associated with an over-representation of severe forms of IBD.

However, patients followed in IBD units pose the most difficult problems in balancing risks versus benefits of immunosuppressive drugs. [START_REF] Beaugerie | Balancing Benefit vs Risk of Immunosuppressive Therapy for Individual Patients With Inflammatory Bowel Diseases[END_REF] A significant strength of our study was to integrate in the analysis of risks factors of SVI the prospective monitoring of IBD clinical activity, which allowed a partial distinction between complications of IBD drugs and complications of IBD itself. This causality assessment is not possible with the same level of precision in studies using medico-administrative databases. [START_REF] Andersen | Association between tumour necrosis factor-alpha inhibitors and risk of serious infections in people with inflammatory bowel disease: nationwide Danish cohort study[END_REF][START_REF] Kirchgesner | Risk of Serious and Opportunistic Infections Associated With Treatment of Inflammatory Bowel Diseases[END_REF] Patients with silent chronic CMV infection may develop intestinal viral replication or CMV-mediated colitis, without extra-intestinal manifestations of CMV infection, in the context of IBD flare. [START_REF] Siegmund | Cytomegalovirus infection associated with inflammatory bowel disease[END_REF] Conversely, we observed that half of the patients with extraintestinal CMV primary infection had intestinal manifestations considered as clinically active IBD at clinical onset of SVI, suggesting that primary CMV infection may provoke or worsen intestinal inflammation. Most of the patients with extra-intestinal signs of reactivation of CMV infection had clinically active IBD at clinical onset of SVI, suggesting at least a partial role of IBD activity in the pathogenesis of infection. EBV primary infection was never associated with digestive symptoms, but IBD was clinically active at clinical onset of symptomatic reactivation of EBV infection in one patient. In relation with this latter observation, it had been previously observed in a study with prospective monitoring of systemic EBV viral load in patients with IBD that significant increase of viral load may occur in the context of severe IBD flare, [START_REF] Reijasse | Epstein-Barr virus viral load in Crohn's disease: effect of immunosuppressive therapy[END_REF] before the initiation of immunesuppressive therapy.

It is now clear that thiopurines promote both symptomatic primary viral infections [START_REF] Biank | Association of Crohn's disease, thiopurines, and primary epstein-barr virus infection with hemophagocytic lymphohistiocytosis[END_REF] and symptomatic reactivation of latent chronic infection. [START_REF] Khan | Overall and Comparative Risk of Herpes Zoster With Pharmacotherapy for Inflammatory Bowel Diseases: A Nationwide Cohort Study[END_REF] The overall absolute risk of SVI reaches 4/1000 PY in patients exposed to thiopurines, which corresponds to a 4% risk for a 10-year period of treatment. Severe or fatal forms of primary VZV [START_REF] Springfeld | Fatal varicella in an immunocompromised adult associated with a European genotype E2 variant of varicella zoster virus[END_REF] and EBV primary infection [START_REF] Biank | Association of Crohn's disease, thiopurines, and primary epstein-barr virus infection with hemophagocytic lymphohistiocytosis[END_REF] represent the most challenging problem for clinicians in the management of SVI in patients exposed to thiopurines. Severe forms of varicella are not associated with HLH and can be prevented by vaccinating patients against VZV before starting immunosuppressive therapy. Severe forms of HLH that occur in patients with IBD are mainly due, in our study like in others, [START_REF] Biank | Association of Crohn's disease, thiopurines, and primary epstein-barr virus infection with hemophagocytic lymphohistiocytosis[END_REF][START_REF] Hyams | Infliximab Is Not Associated With Increased Risk of Malignancy or Hemophagocytic Lymphohistiocytosis in Pediatric Patients With Inflammatory Bowel Disease[END_REF] to EBV primary infection, and are almost exclusively observed in patients exposed to thiopurines. [START_REF] Biank | Association of Crohn's disease, thiopurines, and primary epstein-barr virus infection with hemophagocytic lymphohistiocytosis[END_REF] This specific risk, together with the risk of fatal post-mononucleosis lymphoproliferation, [START_REF] Beaugerie | Lymphoproliferative disorders in patients receiving thiopurines for inflammatory bowel disease: a prospective observational cohort study[END_REF][START_REF] Francisco R De | Impact of Epstein-Barr virus serological status on clinical outcomes in adult patients with inflammatory bowel disease[END_REF] can be reduced by limiting the use of thiopurines in EBV-seronegative patients. [START_REF] Beaugerie | Lymphoma: the bête noire of the long-term use of thiopurines in adult and elderly patients with inflammatory bowel disease[END_REF] In EBVseronegative patients exposed to thiopurines, clinicians should measure EBV viral load and look at biological signs of HLH as soon as possible in case of unexplained fever, splenomegaly or bicytopenia. Prompt consultation with a hematologists may lead to early treatment of HLH, including increasingly rituximab. [START_REF] Fitzgerald | A case of EBV driven haemophagocytic lymphohistiocytosis complicating a teenage Crohn's disease patient on azathioprine, successfully treated with rituximab[END_REF][START_REF] Thompson | Crohn's disease complicated by Epstein-Barr virus-driven haemophagocytic lymphohistiocytosis successfully treated with rituximab[END_REF] In an ideal word, the risk of SVI in patients with IBD could get closer to that of the general population with the use of IBD drugs that prevent IBD-related SVI by way of maintained mucosal healing, whilst not promoting SVI via immunosuppressive effects.

We are far from these goals because mucosal healing is currently achieved in a minority of patients and all immunosuppressive drug classes that are currently used promote serious infections in general, and SVI in particular, at various extents. Thiopurines promote more SVI than anti-TNF agents, 4,9 but the converse is true for opportunistic bacterial infections. [START_REF] Kirchgesner | Risk of Serious and Opportunistic Infections Associated With Treatment of Inflammatory Bowel Diseases[END_REF] In conclusion, our study provides a comprehensive snapshot in the field of SVI caused by IBD and IBD drugs prior to the arrival of new small oral molecules, keeping in mind that one of them, tofacitinib, is associated with a marked excess risk of herpes zoster. 10

  Abbreviations used in this paper:

  that were individually rated by the highest score of IBD drugs used within calendar years of follow-up. For subgroup analyses, age at entry into the observational period was grouped as older than 18 years and less than 35 years, 35-65 years, and older than 65 years. Data for calculating the incidence of extra-intestinal SVI in general population were obtained from the national French hospital discharge database (PMSI) between 2009 and 2013. Cases of hospital stays associated with a hospitalization ICD-10 code of extra-intestinal SVI (see the list in Supplementary Table1) were identified in the PMSI database. The general population at risk was defined as all adults residing in France according to the National Vital Statistics compiled in January of each year. As in our study population, we excluded patients with chronic (more than 6 months) replicative infection by Hepatitis B Virus, Hepatitis C Virus or Human Immunodeficiency Virus, and organ transplant recipients. We obtained the expected number of cases of SVI in the general population by multiplying the person-years at risk in each 10-year age group by the corresponding sex-specific and age-specific incidence rate for each year between 2009 and 2013. The cases of SVI that required hospitalization and were identified in the MICISTA database were then divided by the expected number of cases to determine the SIRs. Confidence intervals for SIRs were calculated with an exact method based on the Poisson distribution. Characterization of the cases of extra-intestinal SVI identified in the MICISTA database A specific case report form was constructed for all patients diagnosed with extraintestinal SVI. Data were extracted by a junior gastroenterologist (A.W) and a senior gastroenterologist (L.B.) from review of medical files in patients managed for extraintestinal SVI in our hospital. In the remaining patients, data were extracted from hospitalization or out-visit reports associated with SVI that were specifically requested to patients treating physicians for the purpose of the study. We considered the exposure to IBD drugs and clinical activity of IBD on the first day of symptoms attributable to extra-intestinal SVI. Clinically active IBD was defined by the presence of symptoms clearly attributable to IBD (which corresponds to values 2 to 5 of the MICISTA score).

  ). No patent developed infection by Hepatitis B Virus, Hepatitis C Virus or Human Immunodeficiency Virus during the observation period. Finally, we identified 31 patients who developed 31 cases of extra-intestinal SVI during the observation period. Median age of patients at time of SVI diagnosis was 40.8 (30.5-53.3) years and duration of IBD prior to diagnosis of SVI was 12.6 (7.1-20.8) years. The incidence rate of extra-intestinal SVI in patients with IBD was 2.02 per 1000 (95% CI, 1.95-2.08) person-years in the total study population. Incidence rates of extraintestinal SVI by age class, gender, IBD subtype, and exposure to IBD drugs are detailed in Supplementary

5

  patients with reactivation of CMV infection had clinically active IBD at clinical onset of SVI. None of the 5 patients with EBV primary infection and 1 of the 3 patients with reactivation of EBV infection had clinically active IBD at clinical onset of SVI. EBVinduced hemophagocytic lymphohistiocytosis (HLH) was observed in 3 patients. These 3 patients were exposed to thiopurines and one of them had clinically active IBD at onset of symptoms attributable to extra-intestinal HLH. Among the 3 patients with HLH, one patient received corticosteroids, immunoglobulins, etoposide and rituximab, while the others had spontaneous resolution of the disease. Anti-viral therapy was administered in 61% of patients, including 13 of the 15 patients who developed extra-intestinal CMV infection. Twenty (65%) of the 31 patients who developed SVI were exposed to thiopurines at clinical onset of SVI. Seven (88%) of the eight patients who developed EBV infection were exposed to thiopurines at clinical onset of SVI. SVI occurrence led to transient suspension (drug withheld until SVI resolution) of immunosuppressive therapy in 58% of cases, and permanent drug withdrawal for the rest of the follow-up in 35% of cases. No permanent end-organ damage or death was reported.

Figure 1 :

 1 Figure 1: Flow diagram for study population selection and identification of extra-intestinal

Table 2 .

 2 The SIR of extra-intestinal SVI in patients with IBD was 3.09 (95% CI, 1.98-4.20, P = 0.002) in the total study population. The SIR of extra-intestinal SVI by age class, gender, IBD subtype, and exposure to IBD drugs are shown in Table2. Among age classes, the highest SIR was observed in patients under 35

Table 1 .

 1 Characteristics of the study population at entry into the observation period

	Patients, n

Table 2 .

 2 Standardized incidence ratios (SIR) of extra-intestinal SVI according to age, gender, IBD subtype and exposure to IBD drugs

	Person-	Reported	Expected	SIR	95% CI	P value
	years	cases a	cases

Table 3 .

 3 Characteristics of extra-intestinal SVO according to pathogens and clinical expressionCMV, cytomegalovirus; EBV, Epstein-Barr virus; VZV, varicella zoster virus; HSV, herpes simplex virus.

	Pathogen	n	Subgroups	n	Specific presentation	n
	CMV	15 Primary	10 Mononucleosis syndrome	8
			infection		isolated	2
					with hepatitis	3
					with pericarditis	1
					with cutaneous manifestations	1
					with hepatitis and pneumonitis	1
					Isolated hepatitis	1
					IBD flare with viremia and without colitis	1
			Reactivation	5	IBD flare	4
					with hepatitis	2
					with haemolytic anemia	1
					with high viremia and without colitis	1
					CMV colitis and CMV induced-hepatitis	1
	EBV	8	Primary	5	Mononucleosis syndrome	4
			infection		isolated	1
					with severe neutropenia	2
					with hepatitis	1
					Hemophagocytic lymphohistiocytosis	1
			Reactivation	3	Hemophagocytic lymphohistiocytosis	2
					IBD flare with EBV-induced hepatitis	1
	HSV	3		3	Severe esophagitis	1
					Facial nerve paralysis	1
					Severe cutaneous lesions	1
	VZV	5	Varicella	3	Severe cutaneous manifestation	2
					With hepatitis	1
			Herpes	2	Multidermatomal	2
			zoster

Table 4 .

 4 Characteristics of incident cases of extra-intestinal SVI

	Patient	Sex IBD	Age at	Age at	Virus Infection type Exposure to	Mononucleosis	Clinical or	Clinically	IBD treatment	Antiviral
	number	type	diagnosis	clinical			IBD drugs at	syndrome or	biological	active IBD	prior to SVI	treatment
			of IBD	onset of			clinical onset of	systemic	expression	at clinical	suspended
				SVI			SVI	symptoms		onset of
										SVI
	1	M CD	20	22	CMV	Primary	Anti-TNF agent	Yes	Mononucleosis	Yes	Yes	IV
						Infection			Syndrome
	2	F UC	21	22	CMV Reactivation	5-ASA	No	Hepatitis	Yes	Yes	IV
	3	F UC	19	23	CMV	Primary	MTX	Yes	Pneumonitis,	No	Yes	IV
						infection			Hepatitis
	4	M UC	13	27	CMV	Primary	AZA	Yes	Mononucleosis	No	No	No
						infection			syndrome
	5	F CD	22	29	CMV	Primary	AZA	Yes	Hepatitis	Yes	Yes	Oral
						infection
	6	F UC	19	30	CMV	Primary	5-ASA, AZA	Yes	Cutaneous	Yes	Yes	IV
						infection
	7	F UC	19	34	CMV	Primary	AZA,	Yes	Hepatitis	No	Yes	No
						infection	Anti-TNF agent
	8	M CD	20	35	CMV Reactivation	5-ASA	No	Hemolytic anemia	Yes	Yes	IV
	9	M CD	25	36	CMV	Primary	AZA	Yes	Mononucleosis	No	Yes	IV
						infection			syndrome
	10	F UC	22	37	CMV	Primary	AZA	Yes	Pericarditis	No	Yes	IV
						infection
	11	F UC	37	39	CMV Reactivation 5-ASA, AZA	No	Hepatitis	Yes	Yes	IV
	12	F CD	21	42	CMV	Primary	AZA	Yes	Hepatitis	Yes	Yes	IV
						infection
	13	F UDC	22	45	CMV	Primary	AZA	Yes	Mononucleosis	Yes	Yes	IV
						infection			syndrome
	14	M UC	48	51	CMV Reactivation Corticosteroids	No	Hepatitis	No	Yes	Oral
							Anti-TNF agent,
	15	M UC	50	79	CMV Reactivation 5-ASA, MTX,	Yes	Mononucleosis	Yes	Yes	IV
							Corticosteroids		syndrome
	16	F CD	8	20	EBV	Primary	AZA	Yes	Neutropenia	No	Yes	No
						infection
	17	M CD	5	20	EBV	Primary	AZA,	Yes	Mononucleosis	No	Yes	No
						infection	Anti-TNF agent
	18	F CD	15	21	EBV	Primary	5-ASA, AZA	Yes	HLH	No	Yes	No
						infection
	19	F UC	22	23	EBV Reactivation	5-ASA,	No	Hepatitis	No	Yes	No
							Corticosteroids
	20	M CD	14	24	EBV	Primary	5-ASA, AZA	Yes	Hepatitis	No	Yes	No
						infection
	21	F CD	20	25	EBV	Primary	5-ASA, AZA	Yes	Neutropenia	No	Yes	No
						infection
	22	M CD	37	43	EBV Reactivation	AZA	Yes	HLH	Yes	Yes	No
	23	F CD	19	53	EBV Reactivation	AZA	Yes	HLH	No	Yes	No
	24	M CD	18	19	HSV	Primary	Corticosteroids	No	Esophagitis	Yes	Yes	IV
						infection
	25	M CD	25	32	HSV Reactivation	MTX,	No	Severe labial	No	Yes	IV
							Anti-TNF agent		lesions
	26	F CD	27	36	HSV	Primary	AZA	No	Facial paralysis	No	Yes	IV
						infection
	27	F CD	17	18	VZV Reactivation	AZA,	No	Multidermatomal
							Anti-TNF agent		shingles

Table 5 .

 5 Impact of IBD clinical activity and exposure to IBD drugs at clinical onset of SVI in cases on the risk of SVI.

		Cases of	Controls	Univariate analysis		Multivariate analysis
		SVI (n=31)	(n=124)
		n (%)	n (%)	Odds ratio	95% CI	P value	Odds ratio	95% CI	P value
	Clinically active	13 (41.9)	23 (18.6)	3.86	1.46-10.17	.006	3.35	1.23-9.23	0.02
	disease
	5-ASA	10 (32.3)	53 (42.7)	0.62	.26-1.46	.27	--	--	--
	Corticosteroids	3 (9.7)	12 (9.7)	1.00	.25-4.00	1	--	--	--
	Methotrexate	4 (12.9)	11 (8.9)	1.53	.44-5.21	.50
	Azathioprine or	20 (64.5)	42 (33.9)	3.94	1.55-10.02	.004	3.48	1.36-8.90	.009
	6-mercaptopurine
	Anti-TNF agent	6 (19.4)	21 (16.9)	1.18	.43-3.25	.75	--	--	--
	Combination	4 (12.9)	11 (8.9)	1.47	.45-4.75	.51	--	--	--
	therapy a

IBD, inflammatory bowel disease; SVI, serious viral infection a anti-TNF agent and azathioprine, 6-mercaptopurine or methotrexate Figure

1

Supplementary

Table 2 .

 2 Incidence rates of acute extra-intestinal SVI according to age, gender, IBD subtype and exposure to IBD drugs At entry into the observation period b Maximal treatment, according to MICISTA IBD drug scale (see Method section) during the calendar year of occurrence of SVI c Thiopurines or methotrexate.

	Person-	Observed	Incidence	95% CI
	years	cases (n)	rates per
			1000 person-
			years

a

Table 3 .

 3 Characteristics of cases and controls in the case-control study nested in the MICISTA database for assessing risk factors for SVI. SVI, serious viral infection; BMI, body mass index; IQR, interquartile range; UC, ulcerative colitis a At clinical onset of SVI in cases, according to Montreal classification

		Patients with	Control group	P value
		SVI (n=31)	(n= 124)
	Male sex, n ( %)	14 (45.2)	56 (45.2)	1
	BMI (IQR)	21.3 (19.4-23.8) 21.6 (19.7-24.0)	.53
	BMI < 18.5 (%)	5 (16.1)	22 (17.7)	1
	BMI ≥ 30 (%)	0 (0)	3 (2.4)	1
	Smokers, n (%)	7 (22.6)	26 (21.0)	.81
	Age at onset of IBD, years	20.6 (17.1-25.2) 20.9 (16.9-24.7)	.83
	(IQR)
	Crohn's disease,	18 (58.1)	72 (58.1)	1
	n (%)
	Segments ever involved a			.50
	n (%)
	L1	5 (27.8)	15 (20.8)
	L2	4 (22.2)	22 (24.4)
	L3	8 (44.4)	34 (47.2)
	L4	1 (5.6)	1 (1.11)
	perineal disease, n (%)	7 (22.3)	20 (16.1)	.42
	UC or unclassified colitis	13(41.9)	52(41.9)	1
	n (%)
	Segments ever involved a			.61
	E1	1 (7.6)	3 (5.8)
	E2	6 (46.2)	19 (36.5)
	E3	6 (46.2)	30 (57.7)

Acknowledgments

List of collaborators of the Saint Antoine IBD Network: Lionel Arrivé, Laurent Beaugerie, Anne Bourrier, Marine Camus, Najim Chafai, Edouard Chambenois, Ulriikka Chaput, Clotilde Debove, Charlotte Delattre, Xavier Dray, Jean-François Fléjou, Guillaume Le Gall, Nadia Hoyeau, Julien Kirchgesner, Cécilia Landman, Jérémie Lefèvre, Philippe Marteau, Chloé Martineau, Laurence Monnier-Cholley, Isabelle Nion-Larmurier, Violaine Ozenne, Yann Parc, Philippe Seksik, Harry Sokol, Magali Svrcek, Emmanuel Tiret.

Supplementary Table 1. Diagnoses of infections with related ICD 10-codes included as viral infections

Diagnoses

ICD-10

Herpes virus B00-B02 Cytomegalovirus B25 Epstein-Barr virus B27 Acute viral hepatitis unspecified B17.9; B19 Other viral infections of skin, oral tissue and subcutaneous tissue B03-B09; B26; B30-B34


Jean-François Rameau

Philippe Serré

Mireille Moinet

Clearance vs. tolerance for rigid overconstrained assemblies

Keywords: Clearance, Tolerance, Overconstrained, Assembly

come

Clearance vs. tolerance for rigid overconstrained assemblies

Introduction

Shape of every manufactured workpiece is not ideal i.e. similar to the nominal shape defined by the designer. Due to the manufacturing deviations, the expected properties of mechanical assemblies are not exactly realized. To insure the quality of their products, companies need to manage geometric variations during the development and industrialization phases. It should be noted that automotive industry is aware of this necessity because, on one hand, the market is very competitive and, on the other hand, the number of manufactured workpieces is huge. Other industrial domains, like aerospace or building architecture, feel less concerned for many reasons e.g. less economic pressure, small or medium volume of production, etc. However, because of international competition, all industries must control the production quality.

In this paper, we deal with overconstrained rigid assemblies made of undeformable components connected together by mechanical joints. An assembly is called overconstrained if the arrangement of components is impossible for generic dimensions. For instance, as illustrated in figure 1 (a), six rods connected together with four spherical joints form an isoconstrained structure (a tetrahedron). The second structure in figure 1 (b) is made of ten rods connected with four spherical joints. It represents the assembly of two tetrahedrons sharing three rods. So, twelve rods for the two tetrahedrons minus three shared rods give nine rods to obtain a rigid isoconstrained structure. Consequently, the tenths rod's length must have a particular value. This second structure is overconstrained. Usually, in an industrial context, to deal with overconstrained assemblies, there is a need to introduce clearance into the joints to allow mounting as well as interchangeability of workpieces. The actual clearance should be larger than the computed clearance in order to insure components interchangeability.

However, large clearance influences the global properties of the mechanical assembly. The aim of this paper is to present a geometrical model designed to predict, at early phase of development, the size of the minimum clearance needed to insure the assemblability of overconstrained mechanical assemblies in the context of mass production.

Components dimensions vary between two supposedly known limits. These limits are small with respect to the nominal dimensions. The shape of nom-inal assembly is assumed to be known. Computer-aided tolerancing commercial softwares like VSA edited by Tecnomatix, CETOL 6σ edited by Sigmetrix and 3DCS edited by Dimensional Control Systems offer functionalities to analyze effects of geometric variations on assembly characteristics. This simulation is called tolerance analysis and was strongly studied during the last decades. Mathematical concepts used for tolerance analysis are Small Displacement Torsor by [START_REF] Bourdet | The concept of the small displacement torsor in metrology[END_REF] and [START_REF] Li | Tolerance analysis of mechanical assemblies based on small displacement torsor and deviation propagation theories[END_REF], Direct Linearization Method by [START_REF] Wittwer | The direct linearization method applied to position error in kinematic linkages[END_REF], Variational Geometric Constraint Network by [START_REF] Hu | A variational geometric constraints network for a tolerance types specification[END_REF] and some variations of them. A good and complete description is given by [START_REF] Chen | A comprehensive study of three dimensional tolerance analysis methods[END_REF]. To the best of our knowledge, these concepts are based on the first order linearization of closure equations representing the mechanical assembly. Approximations are made around nominal dimensions of the parts and around the nominal positions of the joints.

However, these tools are dedicated to detailed design of products because they need a significant number of input data, which represents a disadvantage for a fast study. Geometric dimensioning and tolerancing (GD&T) scheme of each workpiece form important part of them. Otherwise, these approaches seem to be effective for assemblies with a small number of parts and with a small number of closed loops. For these reasons, existing solutions are inappropriate in the context of preliminary design, on one hand because they need a big amount of data to run the simulation. On the other hand, because the first order linearization of the closure equations is an incorrect approximation for a large stack of components, and also is a formalization that may hide singularity configuration of the mechanical assembly. The presented solution gives a response to the inadequacy of established solutions for design of structure.

It should be understood that the aim is the existence of an overconsrained assembly, and that simulating its behavior under working condition is a further topic. It is well known that the existence of an overconstrained assembly featuring nominal dimensions and perfect joints is well understood. But dealing with dimensional uncertainty still needs scientific investigation: quantify how dimensional uncertainty will influence the quality of joints. The manufacturing challenge is to get the best possible quality (small clearance) at a reasonable manufacturing cost (not too small tolerance). The "working condition" investigation is the next step, when the behavior of the assembly is simulated by taking into account boundary conditions. This may suggest geometrical or topological changes and then, the whole design cycle is possibly revisited: tolerance/clearence and working load. Detecting dependent/overconstrained subsystems [START_REF] Moinet | Defining tools to address overconstrained geometric problems in Computer Aided Design[END_REF], [START_REF] Michelucci | Geometric constraint solving: The witness configuration method[END_REF] is a valuable concern. The point is that the proposed method overcomes this problem by smoothly spreading dimensional uncertainty over all joints of the assembly. It is significant to understand that if the assembly is isoconstrained (as the one in figure 1 (a)), the method naturally computes a vanishing clearance, meaning that the assembly is always perfectly adjusted despite dimensional uncertainty.

Section 2 compares the present approach with the combinatorial rigidity theory. Section 3 defines the closure equation. Section 4 defines what is an overconstrained solution of the closure equation. Section 5 explains how to introduce clearance parameters in the closure equation. Section 6 states the fundamental clearance minimization problem. Section 7 provides the theoretical solution of the minimization problem. Section 8 extends the theoretical solution by proving second order minimality condition. Section 9 sets up the ordinary differential equation modeling the clearance/tolerance dependency. Section 10 implements the solution on assembly test cases. Finally, algorithmic aspects are detailed in section 12. Useful theorems and lemmas are gathered in the appendix.

The reader who is not interested by mathematical details may advantageously skip the proof of theorem 1 as well as section 8 and the appendix.

Combinatorial rigidity

The purpose of this section is to compare the present approach with the combinatorial rigidity theory [START_REF] Asimow | The rigidity of graphs[END_REF][START_REF] Graver | Combinatorial rigidity[END_REF][START_REF] Graver | Counting on frameworks: mathematics to aid the design of rigid structures[END_REF]. This analysis is done in [START_REF] Rameau | Dimensional perturbation of rigidity and mobility[END_REF]. It features two concerns and is inserted here for completeness.

Combinatorial rigidity investigates structures made of rigid bars connected at their end points by spherical joints. The combinatorial aspect is the logical graph underlying the structure: bars are modeled by edges and joints are modeled by vertices. The theory provides criteria for generic rigidity of such structures.

The first concern is that combinatorial rigidity handles spherical joints only, while many other kinds of joint are needed in mechanical design: cylindrical, prismatic, revolute, coplanar, etc. The principle developed in the present paper is not restricted to spherical joints.

The second concern is that combinatorial rigidity does not explicitly deal with dimensional parameters. Variables of the edge function and its derivative, the rigidity matrix (the key feature of the theory) are vertices coordinates of the graph structure. The theory is not suited to investigate dimensional parameters influence. Indeed, combinatorial rigidity defines genericity from the vertices coordinates point of view, which makes a big difference with the present approach. Consider the 3D assemblies (a) and (b) in figure 1 and the 2D assembly in figure 3. They are all generically rigid from the combinatorial theory point of view: any small perturbation of vertices (joints) coordinates yields an assembly of the same nature. From the mechanical designer point of view, the situation is totally different. Any small perturbation of bars lengths of assembly (a) in figure 1 yields an assembly of the same nature. It is generic in this sense. Conversely, there exists arbitrary small perturbations of bars lengths of 3D assembly (b) in figure 1 that make it impossible. Same thing with the 2D assembly in figure 3. They are not generic in this sense. Bars lengths are not independent, they must fit a special relationship, as investigated in [START_REF] Rameau | Dimensional perturbation of rigidity and mobility[END_REF]. From the manufacturing point of view, the relevant variables are the mechanical parts dimensions as opposed to vertices coordinates, which is a fundamental difference.

Closure equation

According to [START_REF] Rameau | Dimensional perturbation of rigidity and mobility[END_REF], two kinds of numerical parameters are involved to define a mechanical assembly. Dimensional parameters are lengths and angles and they specify respective dimensions of rigid bodies. Positional parameters are distances and angles and they specify relative positions of rigid bodies. Noting U the space of dimensional parameters, P the space of positional parameters and E the target space, the closure function is F : U × P → E. Function F captures the nature of the mechanical system. It involves dimensional and positional parameters according to rigid bodies and joints. Function F is generally analytic or even polynomial, so it is infinitely differentiable. Furthermore, the derivative of F with respect to all its variables is supposed to be full rank. The equation of the mechanical system is the so called closure equation

F (u, p) = 0 (1)
which is a shortcut to deal with dim E scalar equations.

Definition 1. A solution is a couple (u 0 , p 0 ) ∈ U ×P such that F (u 0 , p 0 ) = 0.

Definition 2. A solution (u 0 , p 0 ) is rigid if p = p 0 is an isolated solution of equation F (u 0 , p) = 0. In other words, there exists ε > 0 such that if |pp 0 | < ε and F (u 0 , p) = 0 then p = p 0 .

Overconstrained solution

Following again [START_REF] Rameau | Dimensional perturbation of rigidity and mobility[END_REF], understanding the overconstrained solution is made easier by starting with the generic one. Intuitively, an arbitrary small perturbation of the dimensional parameter does not change the nature of a generic solution. This makes generic systems particularly attractive for manufacturing because the mounting is not sensitive to dimensional uncertainties. The formal definition is as follows.

Definition 3. The rigid solution (u 0 , p 0 ) is generic if there exists ε > 0 such that for any u such that |uu 0 | < ε there exists p such that (u, p) is also a rigid solution.

Conversely... Definition 4. ...a rigid solution is overconstrained when exists an arbitrary small perturbation v of u 0 such that the equation F (v, p) = 0 has no rigid solution.

Having no rigid solution means having no solution at all or featuring other behavior changes that are investigated in [START_REF] Rameau | Dimensional perturbation of rigidity and mobility[END_REF]. In short, perturbing the dimensional parameter of an overconstrained solution may unexpectedly lead to unwanted behavior change.

From the manufacturing point of view, overconstrained situations are unrealistic because dimensional parameters never fit the value u 0 , meaning that the mechanical system never fits the expected behavior. So, there is a need to change the overconstrained situation into a robust one where it is possible to deal with dimensional uncertainties. Flowchart in figure 2 describes the next steps of theoretical investigation.

Regularized closure equation

Regularizing is to appropriately involve new parameters in the closure function. They are named clearance parameters, noted j = (j 1 , • • • , j r ) ∈ J where J = R r , r = dim J, and they behave like positioning parameters. The goal is to deal with generic solutions, while preserving the consistency with the initial (overconstrained) closure function. So, the closure function

F : U × P → E is transformed into G : U × J × P → E where mapping G is designed in such a way that G(u, 0, p) = F (u, p) (2)
for all (u, p) ∈ U × P . The regularized closure equation is

G(u, j, p) = 0. (3)
Definition 5. The triple (u, j, p) ∈ U ×J ×P is a nominal solution if u = u 0 , j = 0 and p = p 0 where (u 0 , p 0 ) is a solution according to definition 1 .

Thanks to (2), a nominal solution is a solution of (3). From the mechanical point of view, parameters j ∈ J are degrees of freedom added to the mechanical system by replacing joints with more flexible ones. Definition 6. A solution (u 0 , j 0 , p 0 ) is mobile if there exists ε > 0 and two smooth functions j :]ε, ε[→ J and p :]ε, ε[→ P of a scalar variable t such that j(0) = j 0 , p(0) = p 0 , |j ′ (0)| 2 + |p ′ (0)| 2 = 0 and G(u 0 , j(t), p(t)) = 0 for all t ∈]ε, ε[. Notice that definition 6 does not say that functions t → j(t) and t → p(t) are unique. It includes situations with more than one degree of freedom. Definition 7. The mobile solution (u 0 , j 0 , p 0 ) is generic if there exists ε > 0 such that for any u such that |uu 0 | < ε there exists j, p such that (u, j, p) is also a mobile solution.

Counting equations and parameters, the number dim J of clearance parameters is adjusted so that dim J + dim P > dim E. The issue is that solutions of (3) are highly mobile because of the large number of degrees of freedom. In fact, fixing a dimensional parameter ū ∈ U , the generic solution of G(ū, j, p) = 0 is a (dim J + dim P -dim E)-dimensional submanifold of space J × P . Section 6 deals with defining the appropriate unique solution according to mechanical considerations.

Furthermore, mapping G is designed in such a way that the nominal solution (u, j, p) = (u 0 , 0, p 0 ) is a generic mobile solution of (3) according to definitions 6 and 7. In particular, if u = u 0 then j(0) = 0 and p(0) = p 0 . This is the value of the regularization process: it removes the sensitivity of the solution to dimensional parameters uncertainty. In other words, it changes the non generic situation into a generic situation.

The two technical conditions insuring the regularization effectiveness are now explained. They involve the partial derivatives of G(u, j, p) with respect to j and p at (u, j, p) = (u 0 , 0, p 0 ), which, for clarity, are respectively noted

A = G j (u 0 , 0, p 0 ) (4)
and

B = G p (u 0 , 0, p 0 ). (5)
Firstly, the partial derivative of G with respect to j and p is full rank at the nominal solution (u, j, p) = (u 0 , 0, p 0 ). The linear mapping G (j,p) (u 0 , 0, p 0 ) is bloc defined by two matrices

G (j,p) (u 0 , 0, p 0 ) = A B . If G (j,p) (u 0 , 0, p 0 ) is full rank, G (j,p) (u 0 , 0, p 0 )G (j,p) (u 0 , 0, p 0 ) T is invertible. Since G (j,p) (u 0 , 0, p 0 )G (j,p) (u 0 , 0, p 0 ) T = A B A T B T = AA T + BB T
the full rank condition is equivalent to

AA T + BB T is invertible. ( 6
)
The second and very important condition involves the partial derivative with respect to p.

B T (AA T + BB T ) -1 B is invertible (7)
Conditions ( 6) and ( 7) are verified in practice and allow the inversion of a key linear mapping, which in turn makes the whole solution feasible, as stated in section 7. Notice that, thanks to (2), B = G p (u 0 , 0, p 0 ) = F p (u 0 , p 0 ).

Optimization problem

The goal is now to set up the mathematical problem that defines a unique solution which is meaningful from the mechanical point of view. Since the point is to deal with a non nominal dimensional parameter, let u ∈ U be a fixed and arbitrary dimensional parameter, meaning that u = u 0 is allowed. Among the infinity of j, p solutions of the regularized closure equation ( 3), the preferred one will minimize the clearance, that is the magnitude of parameter j. Precisely, the preferred solution minimizes the squared norm of parameter j, that is

|j| 2 = r i=1 j 2 i . (8)
This is consistent with a nominal solution because, in this case, |j| 2 = 0 which is surely a minimum. Technically, the optimization problem is as follows. Given a (possibly non nominal) dimensional parameter u ∈ U , find the clearance parameters j and the positional parameters p such that the regularized closure equation ( 3) is satisfied and such that the magnitude of clearance parameter j is as small as possible. This is classically written

min G(u,j,p)=0 1 2 |j| 2
but the "argument of the minimum" formulation is more appropriate to highlight unknowns j and p vs. input parameter u.

(j, p) = Argmin 1 2 |j| 2 ; G(u, j, p) = 0 (9)
In fact, ( 9) is a family of minimization problems parameterized by u ∈ U . This implicitly defines a dependency between u and the optimum (j, p). The hope is to get a positioning p that is close to the nominal positioning p 0 when the dimensional parameter u is close to nominal value u 0 . This issue is elucidated in section 7.

Clearly, the optimization criterion is based on pure geometry. One could argue that functional or performance considerations are missing. It will be seen in section 7 that a small dimensional perturbation yields a perturbed assembly that remains close to the nominal assembly. So, performance of the perturbed assembly should be close to nominal performance. Nevertheless, a functional or performance criterion Φ(u, j, p) can be added to the geometrical criterion in order to involve non geometrical aspects, but this is out of the scope of the paper.

First order solution of the optimization problem

This section is dedicated to theorem 1. It states that, from the mechanical point of view, despite (u 0 , p 0 ) is an overconstrained solution, a small perturbation u of nominal dimension u 0 yields a unique small perturbation p of nominal positional parameter p 0 , the dependency between p and u being smooth. Furthermore, the value of clearance parameter j is unique, meaning that there is no way to mount the assembly with a tighter clearance. One could argue that it is not sure that p is as close as possible to p 0 . The answer is very simple: looking for a minimal j yields a unique p, which is necessarily the best possible. In other words, choosing another p yields a non minimum j. Furthermore, conclusion 3 of theorem 1 states that the positional parameter p remains under control despite the perturbation of dimensional parameter u. In addition, it is possible to estimate the first order dependency between j and u in the neighborhood of j = 0 and u = u 0 as well as the first order dependency between p and u in the neighborhood of p 0 and u = u 0 Theorem 1. Consider an overconstrained solution of closure equation (1) and its regularized closure equation (3) satisfying (6) and [START_REF] Michelucci | Geometric constraint solving: The witness configuration method[END_REF]. Then, there exist a neighborhood Y of u 0 , two unique and smooth functions j = j(u) and p = p(u) defined over Y and a constant c such that 1. j(u 0 ) = 0 and p(u 0 ) = p 0 2. for all u ∈ Y , (j(u), p(u)) satisfies the stationarity condition of (9)

3. |p(u) -p 0 | ≤ c|u -u 0 | for all u ∈ Y .
The proof is structured as follows. Clearly, ( 9) is a quadratic minimization problem under non linear equality constraint. Classically, Lagrange multiplier λ is introduced to deal with the constraint. Then, Lagrange function stationarity yields a non linear system featuring the same number of unknowns and equations. Finally, the implicit function theorem 3 provides a smooth dependency between the optimal solution j, p and parameter u.

Proof. The Lagrange function associated with ( 9) is L :

U × J × P × E → R defined by L(u, j, p, λ) = 1 2 |j| 2 + λ, G(u, j, p) (10)
where the Lagrange multiplier λ ∈ E is an additional unknown. A necessary condition for the existence of a solution to ( 9) is the stationarity of function [START_REF] Graver | Counting on frameworks: mathematics to aid the design of rigid structures[END_REF]. This means that its partial derivatives with respect to j, p and λ must vanish, that is

L j (u, j, p, λ) = 0 L p (u, j, p, λ) = 0 L λ (u, j, p, λ) = 0
where

L j (u, j, p, λ) = j + G j (u, j, p) T λ L p (u, j, p, λ) = G p (u, j, p) T λ (11) L λ (u, j, p, λ) = G(u, j, p).
Now, consider the mapping H : U × J × P × E → J × P × E defined by the triple of partial derivatives

H(u, j, p, λ) =   j + G j (u, j, p) T λ G p (u, j, p) T λ G(u, j, p)   . (12)
By design, the stationary point (j, p, λ) of Lagrange function ( 10) is a solution of equation

H(u, j, p, λ) = 0. ( 13
)
At this step, it is important to remember that the unknowns of the stationarity problem (13) are j, p, λ and that u is a parameter. Equation ( 13) defines a dependency between u and j, p, λ. The goal is now to investigate this dependency by applying the implicit function theorem 3 to mapping H at (u, j, p, λ) = (u 0 , 0, p 0 , 0). The first step is to check that (u 0 , 0, p 0 , 0) is a solution of [START_REF] Sagan | Space-filling curves[END_REF]. Indeed, thanks to (2), [START_REF] Liu | Generic Approach for the Generation of Symbolic Dimensional Variations Based on Gröbner Basis for Over-constrained Mechanical Assemblies[END_REF] and to definition 1

H(u 0 , 0, p 0 , 0) =   0 0 G(u 0 , 0, p 0 )   =   0 0 F (u 0 , p 0 )   =   0 0 0   .
The second step is to investigate the inversion of the partial derivative H (j,p,λ) at (u 0 , 0, p 0 , 0). By derivating ( 12)

H (j,p,λ) =   I + G T jj λ G T jp λ G T j G T jp λ G T pp λ G T p G j G p 0   (14)
so, at (u, j, p, λ) = (u 0 , 0, p 0 , 0)

H (j,p,λ) (u 0 , 0, p 0 , 0) =   I 0 A T 0 0 B T A B 0   (15)
with notations ( 4) and ( 5). Thanks to lemma 1 and conditions ( 6) and ( 7), H (j,p,λ) (u 0 , 0, p 0 , 0) is invertible. Then, according to the implicit function theorem 3, there exist a neighborhood Y of u 0 and three unique and smooth functions j = j(u), p = p(u), λ = λ(u) defined over Y such that

j(u 0 ) = 0 p(u 0 ) = p 0 λ(u 0 ) = 0
which is conclusion 1 of the theorem, and

H(u, j(u), p(u), λ(u)) = 0 for all u ∈ Y , which is conclusion 2 of the theorem. Since u → p(u) is differentiable, there exists a constant c such that |p(u) -p 0 | ≤ c|u -u 0 | for all u ∈ V , which is conclusion 3 of the theorem.
The first order dependency between positional parameter p and dimensional parameter u can be estimated by computing p ′ (u 0 ). Similarly, the first order dependency between clearance parameter j and dimensional parameter u can be estimated by computing j ′ (u 0 ). Indeed, following the proof of theorem 1, the implicit function theorem provides the differential equation

  j ′ (u) p ′ (u) λ ′ (u)   = -H (j,p,λ) (u, j, p, λ) -1 H u (u, j, p, λ).
Writing this expression at u = u 0 and using ( 12) and (15) yield

  j ′ (u 0 ) p ′ (u 0 ) λ ′ (u 0 )   = -   I 0 A T 0 0 B T A B 0   -1   0 0 G u (u 0 , 0, p 0 )  
which can be solved by hand in such a way that, noting

D = AA T + BB T -1 , p ′ (u 0 ) = -B T DB -1 B T D G u (u 0 , 0, p 0 ) and j ′ (u 0 ) = A T DB B T DB -1 B T D -D G u (u 0 , 0, p 0 ).

Second order minimality condition

This section investigates the second order minimality condition of the optimization problem [START_REF] Graver | Combinatorial rigidity[END_REF]. It is not useful to numerical solving, but it consolidates the theory by proving that the unique solution stated in theorem 1 is in fact the best possible. Surprisingly, the second order condition requires two different proofs, depending on whether there are many clearance parameters (dim J > dim E) or few clearance parameters (dim J ≤ dim E).

Additional condition

In addition to the full rank property [START_REF] Moinet | Defining tools to address overconstrained geometric problems in Computer Aided Design[END_REF], and if dim J < dim E, we suppose that it is possible to split the positional parameter p into p = (q, w) ∈ Q × W in such a way that the partial derivative of G with respect to (j, q) is invertible at the nominal solution. This means that dim J + dim Q = dim E. Noting p 0 = (q 0 , w 0 ), this means that G (j,q) (u 0 , 0, q 0 , w 0 ) is invertible. Then, the implicit function theorem yields two unique mappings (u, w) → j(u, w) and (u, w) → q(u, w) defined in a neighborhood of (u 0 , w 0 ) such that j(u 0 , w 0 ) = 0, q(u 0 , w 0 ) = q 0 and G(u, j(u, w), (q(u, w), w)) = 0 for all (u, w) in the said neighborhood. The additional condition involves the partial derivative of j(•) with respect to w at the nominal solution:

j w (u 0 , w 0 ) T j w (u 0 , w 0 ) is invertible. ( 16
)
If dim J = dim E no splitting is needed and variable p plays the role of variable w. Condition [START_REF] Moinet | Defining tools to address overconstrained geometric problems in Computer Aided Design[END_REF] says that linear mapping G j (u 0 , 0, p 0 ) is invertible and the implicit function theorem yields a unique mappings (u, p) → j(u, p) defined in a neighborhood of (u 0 , p 0 ) such that j(u 0 , p 0 ) = 0 and G(u, j(u, p), p)) = 0 for all (u, p) in the said neighborhood. The additional condition ( 16) is written j p (u 0 , p 0 ) T j p (u 0 , p 0 ) is invertible.

Second order theorem

Theorem 2. Under the conditions of theorem 1 and if, in addition, condition (16) is satisfied, then, for all u ∈ Y , (j(u), p(u)) satisfies the second order minimality condition of (9).

8.3. Proof of theorem 2 when dim J > dim E Proof.
Given u ∈ Y and since dim J > dim E, according to lemma 2 the second order minimality condition of ( 9) is

|v| 2 + λ(u), G jj (u, j(u), p(u), λ(u))vv > 0 (17)
for all v ∈ J such that v = 0 and G j (u, j(u), p(u), λ(u))v = 0. Note ϕ(u, v) = λ(u), G jj (u, j(u), p(u), λ(u))vv so that the left hand side of (17) can be written |v| 2 + ϕ(u, v), that is a perturbation of |v| 2 by a quadratic form such that ϕ(u 0 , v) = 0 for all v. Now,

|ϕ(u, v)| ≤ ψ(u)|v| 2
where ψ(u) = |λ(u)||G jj (u, j(u), p(u), λ(u))|, so that

|v| 2 + ϕ(u, v) ≥ |v| 2 -|ϕ(u, v)| ≥ (1 -ψ(u))|v| 2 .
Since ψ(u 0 ) = 0, an appropriate restriction of neighborhood Y makes 1-ψ(u) larger that 1 2 . Finally, for all u in the restricted Y and all non zero v ∈ J

|v| 2 + ϕ(u, v) ≥ 1 2 |v| 2 > 0.
The second order minimality condition is satisfied for all u in a neighborhood of u 0 . 8.4. Proof of theorem 2 when dim J ≤ dim E Proof. Using notation of condition ( 16), the minimization problem ( 9) is equivalent to

w = Argmin 1 2 |j(u, w)| 2 .
The proof is to apply lemma 3 using j(u 0 , w 0 ) = 0 and the inversion condition (16).

Clearance/tolerance ordinary differential equation

This section makes use of theorem 1 to compute numerical variations of clearance j according to the tolerance, which is the numerical variations of dimension u. The guide line stated by [START_REF] Liu | Generic Approach for the Generation of Symbolic Dimensional Variations Based on Gröbner Basis for Over-constrained Mechanical Assemblies[END_REF] is to deal with an ordinary differential equation, as it is particularly adapted to numerical integration.

The ordinary differential equation

So far, parameter u is taken from neighborhood Y of u 0 , Y being an open subset of U . The idea is to consider an arbitrary curve in Y through u 0 . Let u : R → U be a smooth mapping of a real variable t such that u(t) ∈ Y for all t and u(0) = u 0 . Notice that there may exist non zero values of t such that u(t) = u 0 .

The minimization problem ( 9) is now written using u(t) instead of u. Given a real number t, find the positional parameters j and p such that the regularized closure equation ( 3) is satisfied and such that the magnitude of parameter j is as small as possible.

(j, p) = Argmin 1 2 |j| 2 ; G(u(t), j, p) = 0 (18)
Now, (18) is a family of minimization problems parameterized by t ∈ R.

The solving technique follows the track of section 7. The Lagrange function associated with (18) is L : R × J × P × E → R defined by

L(t, j, p, λ) = 1 2 |j| 2 + λ, G(u(t), j, p) . (19)
Mapping H : R × J × P × E → J × P × E is defined by the triple of partial derivatives

H(t, j, p, λ) =   j + G j (u(t), j, p) T λ G p (u(t), j, p) T λ G(u(t), j, p)   . (20)
The stationarity equation ( 13) is transformed into

H(t, j, p, λ) = 0. (21)
The partial derivative of H with respect to (j, p, λ) is invertible at (t, j, p, λ) = (0, 0, p 0 , 0) just like in section 7 and for the same reason. So, the implicit function theorem 3 applied to mapping H at (t, j, p, λ) = (0, 0, p 0 , 0) yields a neighborhood ]ε, ε[ of 0 and three unique and smooth functions j = j(t), p = p(t) and λ = λ(t) defined over ]ε, ε[ such that

j(0) = 0 p(0) = p 0 (22) λ(0) = 0
and H(t, j(t), p(t), λ(t)) = 0 for all t ∈]ε, ε[. Furthermore, functions j(t), p(t) and λ(t) are solution of ordinary differential equation

  j ′ p ′ λ ′   = V (t, j, p, λ) (23)
associated with the initial condition (22) where the vector field V (•) is

V (t, j, p, λ) = -H (j,p,λ) (t, j, p, λ) -1 H t (t, j, p, λ). (24)
Symbol ′ in (23) is the derivation with respect to real variable t. The partial derivative H (j,p,λ) can be detailed in terms of function G by using [START_REF] Xu | Isogeometric computation reuse method for complex objects with topology-consistent volumetric parameterization[END_REF]. The partial derivative H t can be detailed in terms of function G by using [START_REF] Liu | Generic Approach for the Generation of Symbolic Dimensional Variations Based on Gröbner Basis for Over-constrained Mechanical Assemblies[END_REF] and the chain rule.

H t (t, j, p, λ) =   (G uj (u(t), j, p)u ′ (t)) T λ (G up (u(t), j, p)u ′ (t)) T λ G u (u(t), j, p)u ′ (t)   (25)

Variations of dimensional parameter u

For numerical investigation purpose, a simple and efficient choice introduced by [START_REF] Liu | Generic Approach for the Generation of Symbolic Dimensional Variations Based on Gröbner Basis for Over-constrained Mechanical Assemblies[END_REF] is to design the curve t → u(t) as a harmonic function oscillating around the nominal value u 0 . In addition, a homotopy effect is used to start from the nominal solution by using the mapping

µ(t) = 1 -e -t . Noting k = dim U and u = (u 1 , • • • , u k ) the scalar coordinates of u, each scalar function t → u i (t) is defined by u i (t) = u 0i + µ(t) ∆ i 2 sin(ω i t) (26)
for i = 1, • • • , k.
In formula (26), ω i is an angular frequency and ∆ i is the interval width, that is the manufacturing uncertainty of dimensional parameter u i . Angular frequencies ω i are spread in such a way that when t increases, the oscillating curve t → (u 1 (t), • • • , u k (t)) "fills" the overall tolerance range ∆ = k i=1 ∆ i as much as possible. Of course, a genuine space-filling curve as described by [START_REF] Sagan | Space-filling curves[END_REF] would be ideal, but (26) is very simple and numerical simulations show that this is efficient. Feeding the ordinary differential equation (23) with the oscillating dimensions (26) and solving over a time interval [0, T ] yields oscillating functions t → j(t), p → p(t) and λ → λ(t) starting with the initial condition (22). The result of interest is the maximum value of clearance parameter j(t) when t ∈ [0, T ].

j max = max{|j(t)|, t ∈ [0, T ]} (27)
This value does not depend on T provided it is large enough, which is not difficult to set in practice. The quality compromise of mechanical design is as follows. Larger dimensional tolerance ∆ i means smaller manufacturing cost, but also larger j max . Larger j max means bad quality of the manufactured assembly. So, the balance is to adjust the ∆ i to the smallest possible manufacturing cost in such a way that j max fits the quality criteria. Clearly, this can be achieved by using previous simulation through an iterative process.

Family of assemblies

By using t → u(t) dimensional mapping and t → p(t) positional mapping, it is easy to set up a graphical animation of the assembly with respect to parameter t. This could mislead the reader because the animation looks like a dancing structure deformed by some physical phenomenon. In fact, it is not one structure under deformation. It is an infinity of structures indexed by t, each of them being defined by dimensions u(t) and positions p(t). It must be understood as a continuous abstraction of the series of thousands of products getting out of the factory.

Rod structure test cases

Framework

The assembly is an overconstrained three-dimensional structure made of n rigid rods linked by m spherical joints. Rod's lengths are noted l i , i = 1, • • • , n. In a first step, the regularized closure function is set up. In order to anticipate non-perfect connections of several rods at the same junction, m connections points

P i =   x i y i z i   i = 1, • • • , m are introduced. The start point O i of rod number i is O i =   s i t i r i   . ( 28
)
The relative end point E i of rod number i is

E i = l i   cos β i cos α i cos β i sin α i sin β i   (29)
meaning that the extremities of rod number i are respectively O i and O i + E i . The network of rods is defined by a matrix N featuring 2n rows and m columns and filled with 0 and 1. It captures the coincidences of rods extremities O i and O i + E i with connection points P j , which is gathered in the following equation.

       O 1 E 1 + O 1 . . . O n E n + O n        = N    P 1 . . . P m    . ( 30
)
Clearly, the dimensional parameters space is

U = R n gathering rod's lengths through u = (l 1 , • • • , l n ). ( 31
)
Unknowns are connection points and rod's extremities so that the positional parameter space is

P = R 3 × [0, 2π[×[-π, π[ n × R 3 m
and the target space is

E = (R 3 ) 2n . Noting u = (l 1 , • • • , l n ) and p =     s i t i r i   i=1,••• ,n , α i β i i=1,••• ,n ,   x i y i z i   i=1,••• ,m   (32)
the closure function is

F : R n × R 3 × [0, 2π[×[-π, π[ n × R 3 m → R 3 2n
defined by

F (u, p) =        O 1 E 1 + O 1 . . . O n E n + O n        -N    P 1 . . . P m    .
For grounding purpose, six positional parameters are set, so that the closure equation F (u, p) = 0 features 5n + 3m -6 unknown scalar positional parameters and 6n scalar equations. A necessary condition for an over-constrained structure is more equations than unknowns, that is n > 3m -6. Introducing the clearance parameters in this context is straightforward because the closure function involves the appropriate geometry. Indeed, choosing J = E = (R 3 ) 2n and j ∈ J, the regularized closure function is

G(u, j, p) = F (u, p) -j.
The regularized closure equation G(u, j, p) = 0 is clearly under-constrained since it features 11n + 3m -6 scalar unknowns and 6n scalar equations. Because of the particular shape of G, the optimization problem ( 9) leads to a stationarity condition (21)

j -λ = 0 F p (u(t), p) T λ = 0 F (u(t), p) -j = 0 which boils down to F p (u(t), p) T F (u(t), p) = 0 j = F (u(t), p) λ = F (u(t), p).
The corresponding differential equation ( 23) is

p ′ = R(p, t)S(p, t) p(0) = p 0 where R(p, t) = -F pp (u(t), p) T F (u(t), p) + F p (u(t), p) T F p (u(t), p) -1 and S(p, t) = (F up (u(t), p)u ′ (t)) T F (u(t), p) + F p (u(t), p) T F u (u(t), p)u ′ (t)
completed with j(t) = F (u(t), p(t)) and λ(t) = j(t).

The two-dimensional version of the generic framework is easily obtained by setting z i = 0, r i = 0 and β i = 0 and by grounding three unknowns. Parameter space and target space are updated accordingly so that

P = R 2 × [0, 2π[ n × R 2 m and E = (R 2 ) 2n . Furthermore, p = s i t i i=1,••• ,n , α i i=1,••• ,n , x i y i i=1,••• ,m .

Case study number 1: two-dimensional structure

As illustrated in figure 3, case study number 1 is a two-dimensional structure made of n = 6 rods and m = 4 joints the arrangement of which is provided by matrix N .

N =        1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1       
From the numerical point of view, the nominal configuration is defined by connection points coordinates, which in turn define nominal lengths of all rods. They are respectively Harmonic perturbations of rod's lengths are illustrated in figure 4. The resulting clearance variation is illustrated in figure 7. Typical perturbed structures are illustrated in figures 5 and 6. Each structure is distorted in order to minimize the widths of non-perfect connections. Red dots are connexion points, black dots are rod's extremities.

10.3. Case study number 2: three-dimensional structure As illustrated in figure 8, case study number 2 is a three-dimensional structure made of n = 10 rods and m = 5 joints the arrangement of which is provided by matrix N .

N =               
1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0

              
From the numerical point of view, the nominal configuration is defined by connection points coordinates, which in turn define nominal lengths of all rods. They are respectively

P 0 1 =   0 0 0   P 0 2 =   2 1 1   P 0 3 =   0 2 0   P 0 4 =   -2 1 1   P 0 5 =   0 1 2   .
Dimensional perturbations are defined using (26) by setting ∆ i = 0.4 for i = 1, • • • , 10 and the following angular frequencies. Harmonic perturbations of rod's lengths are illustrated in figure 9. The resulting clearance variation is illustrated in figure 11. Typical structure deformation is illustrated in figure 10. Here again, each structure is distorted in order to minimize the widths of non-perfect connections. Red dots are connexion points, black dots are rod's extremities.

Case study number 3

As opposed to section 10, this case study involves cylindrical joints rather than spherical joints. The assembly is made of two parts (part 1 and part 2) which are instances of the typical part illustrated in figure 12. Part 1 and part 2 are connected through two cylindrical joints, as illustrated in figure 13. Clearly, the assembly is overconstrained and can be mounted provided some assembly condition is satisfied, for example β 1 = β 2 = 0 and α 1 + α 2 = π. It turns out to be a mobile mechanism if, for example,

β 1 = β 2 = 0 and α 1 = α 2 = π
2 , but this configuration is out of the scope of the study.

11.1. Designing the perfect/non-perfect cylindrical joint Consider a rigid line segment [A, B] and a line defined by a point C and a normalized vector V . Capturing that the line segment slides on the line is to specify that points A and B are located on the said line. Formally, there exist two real numbers v and w such that A = C + vV and B = C + wV . This suggests a closure equation like

C + vV -A = 0 C + wV -B = 0
and a corresponding closure function

F = C + vV -A C + wV -B .
Here again, the regularized closure function is G = Fj. Indeed, if j = 0 the line segment is rigorously aligned with the line, thus modeling a perfect joint, while if j = 0 the line segment is allowed to leave the line depending on neighboring conditions, thus modeling a non-perfect joint.

Closure equation

Let D ∈ SE(3) be the relative position of part 2 with respect to part 1. It is defined by a rotation R including three angles a translation vector T including three scalar parameters. As illustrated in figure 14, the closure equation is obtained by writing that the line segment [A 2 , B 2 ] of part 2 at position D is aligned with line C 1 , V 1 of part 1 and to write that the line segment

[A 1 , B 1 ] of part 1 is aligned with line C 2 , V 2 of part 2 at position D. C 1 + v 1 V 1 -D(A 2 ) = 0 C 1 + w 1 V 1 -D(B 2 ) = 0 D(C 2 + v 2 V 2 ) -A 1 = 0 D(C 2 + w 2 V 2 ) -B 1 = 0.
Choosing a self explanatory coordinate system where

C 1 = C 2 =   0 0 0   V 1 = V 2 =   1 0 0   A i =   l i cos α i + d sin α i cos β i l i sin α i -d cos α i cos β i -d sin β i   B i =   l i cos α i -d sin α i cos β i l i sin α i + d cos α i cos β i d sin β i  
this leads to the following closure function

F : R 6 × R 4 × SE(3) → R 12 F (l 1 , α 1 , β 1 , l 2 , α 2 , β 2 , v 1 , w 1 , v 2 , w 2 , D) =     C 1 + v 1 V 1 -D(A 2 ) C 1 + w 1 V 1 -D(B 2 ) D(C 2 + v 2 V 2 ) -A 1 D(C 2 + w 2 V 2 ) -B 1    
where the dimensional parameters are l 1 , α 1 , β 1 , l 2 , α 2 , β 2 and the positional parameters are v 1 , w 1 , v 2 , w 2 , D. The closure equation F = 0 features ten positional parameters and twelve equations, meaning some overconstraint, as expected.

The regularized closure function is defined by

G : R 6 × R 4 × SE(3) × R 12 → R 12 with G(u, j, p) = F (u, p) -j where u = (l 1 , α 1 , β 1 , l 2 , α 2 , β 2 ), p = (v 1 , w 1 , v 2 , w 2 , D) and j = (j 1 , • • • , j 12 ).

Nominal configuration and uncertainties

Nominal dimensions are l 0 1 = 1.0, l 0 2 = 1.5, α 0 1 = 3π 5 , α 0 2 = 2π 5 , β 0 1 = 0 and β 0 2 = 0. Nominal positional parameters are r 0 =

l 0 2 cos α 0 1 -l 0 1 sin α 0 1 , v 0 1 = r 0 -d,
w 0 1 = r 0 + d, v 0 2 = l 0 2 -l 0 1 cos α 0 1 sin α 0 1 -d, w 0 2 = v 0 2 + 2d.
Nominal position D 0 is made of rotation R 0 and translation T 0 as follows

R 0 =   cos(π -α 0 2 ) -sin(π -α 0 2 ) 0 sin(π -α 0 2 ) cos(π -α 0 2 ) 0 0 0 1   T 0 =   r 0 l 0 2 0   .
Dimensional perturbations are defined using (26) by setting ∆ = 0.1 and the following frequencies ω 1 ω 2 ω 3 ω 4 ω 5 ω 6 5 7 11 3 13 2 respectively associated with l 1 , l 2 , α 1 , α 2 , β 1 and β 2 , as illustrated in figure 15. It is noticeable that nominal parts are planar while perturbed parts are non planar due to non zero angles β 1 and β 2 . Figure 16 illustrates typical configurations of perturbed assemblies resulting from dimensional perturbations and optimal relative positioning. Notice the non-perfect joints. Figure 17 illustrates the clearance variation caused by dimensional perturbations.

Algorithm

Flowchart of figure 19 describes the implementation procedure. The algorithmic aspect is now detailed. It includes three main steps: pre-processing, solving, post-processing. Pre-processing is to compute the nominal dimensions u 0 and positions p 0 , and to generate the equations of the problem, mainly mappings G(•), H(•) or V (•). Solving is to compute the clearance and position mappings j(•) and p(•) over a sampling 0 < t 1 < • • • < t K = T of the investigation interval [0, T ]. The output of the solving step is a table T able(•) so that, for k = 1, • • • , K, T able(k) is the couple of clearance and position solution (j(t), p(t)) for t = t k . Three alternate solving techniques are presented. They are respectively based on minimization, zero finding and ODE integration. The solving step advantageously makes use of a software library. Post-processing is to compute the maximum clearance values by considering the output data of the solving step.

Preprocessing

As explained in section 10, the test case is defined by the nominal positions P 0 i of connections and by the rods end connections through matrix N . For clarity, this matrix is noted as a stack of rows

N =    N 1 . . . N 2n    .
The nominal position parameter p 0 is computed by setting points O 0 i and vectors E 0 i so that equation ( 30) is satisfied, and by using definitions (28), (29), (32). The nominal dimensions are rods lengths l i , as defined by (31).

Input : Connections nominal position P 0 i , matrix N . Output: Nominal dimension u 0 , nominal position p 0 . Mappings defining the equations of the problem are automatically generated by using symbolic computation. The nominal geometry of the assembly, its topology, dimensional uncertainties and frequencies are captured in a text file, as illustrated in figure 18, which feeds a Maple program generating the appropriate G(•) mapping. If needed, mappings H(•) and V (•) are deduced from mapping G(•) by using symbolic computation again.

Solving: minimize

The shortest way is a single loop so that the minimization problem (18) is solved at each step k by using a program from a mathematical software library. Input : Nominal dimension u 0 , nominal position p 0 , mapping G(•). Output: Clearance and position mappings T able(k) = (j(t k ), p(t k )).

for k := 1 to K do (j, p) := Argmin 1 2 |j| 2 ; G(t k , j, p) = 0, {j, p} T able(k) := (j, p) end Algorithm 2: Minimization algorithm for solving 2D and 3D test cases.

The drawback is that the library program is responsible for initializing the searching, which can lead to unpredictable results. A variant is to control the initialization by providing the minimizer with the previous solution. The initial guess is written as the second argument of Argmin function. 3D and 2D test cases are successfully solved by using this algorithm.

Input : Nominal dimension u 0 , nominal position p 0 , mapping G(•). Output: Clearance and position mappings T able(k) = (j(t k ), p(t k )).

(j 1 , p 1 ) := (0, p 0 ) for k := 1 to K do (j, p) := Argmin 1 2 |j| 2 ; G(t k , j, p) = 0, {j, j 1 }, {p, p 1 } T able(k) := (j, p) (j 1 , p 1 ) := (j, p) end Algorithm 3: Minimization algorithm with controlled initialization for solving 2D and 3D test cases.

Solving: zero finder

Next algorithm is to solve the non linear system (13) by using a zero finding program of the software library, Newton-Raphson or the like. Unknowns are clearance variable j, positioning variable p and Lagrange parameter λ. The zero finder is initialized by using the previous solution. 3D and 2D test cases are successfully solved by using this algorithm. Input : Nominal dimension u 0 , nominal position p 0 , mapping H(•). Output: Clearance and position mappings T able(k) = (j(t k ), p(t k )).

(j 1 , p 1 , λ 1 ) := (0, p 0 , 0) for k := 1 to K do Solve {H(t k , j, p, λ) = 0, {j, j 1 }, {p, p 1 }, {λ, λ 1 }} T able(k) := (j, p) (j 1 , p 1 , λ 1 ) := (j, p, λ) end Algorithm 4: Zero finder algorithm for solving 2D and 3D structures and the two sliders assembly.

Solving: ODE integration

Finally, if, through symbolic or numerical computation, the inversion (24) can be performed, the vector field (24) of differential equation ( 23) is used to feed a standard library ODE solver, say Runge and Kutta or the like. Then, a single line of computer program is enough to get mappings t → j(t) and t → p(t). For symbolic inversion purpose, only the 2D test case is solved by this method.

Input : Nominal dimension u 0 , nominal position p 0 , mapping V (•).

Output: Clearance and position mappings t → j(t) and t → p(t).

ODESolve

     j ′ (t) p ′ (t) λ ′ (t)   = V (t, j, p, λ),   j(0) p(0) λ(0)   :=   0 p 0 0   , {t ∈ [0, T ]}  
 Algorithm 5: ODE numerical integration for solving 2D structure test cases.

Post-processing

Whatever the method used for solving, it provides (a numerical approximation of the) clearance mapping t → j(t). The extremal clearance is computed according to (27) and illustrated in figures 7, 11, and 17. Local analysis at each joint can be performed if needed by using the appropriate coordinates of mapping j(•).

Conclusion

Overconstrained structures are represented by a set of closure equations F involving two kinds of numerical parameters: dimensional and positional. In an industrial context, engineers introduce clearance into joints to enable mechanical structure assembly. Clearance parameters are called j. Closure function F is transformed into regularized closure function G depending on dimensional, positional and clearance parameters. From an algebraic point of view, G function is highly underconstrained.

For specific rod's lengths, the unique solution is computed by solving the following optimization problem: minimize the squared norm of clearance parameters with respect to the G function. Due to manufacturing deviations, lengths of rods vary between two known limits. In the dimensional parameters space, hyper-volume is scanned by a set of harmonic functions parameterized by a single parameter t. Then, minimization, zero finding or numerical integration of an ordinary differential equation handles the simulation.

The number of input data of the proposed tool is very small: connectivity matrix of the assembly structure, coordinates of the spherical joints center and manufacturing precision class of the parts. First simulation results are quickly obtained but with a questionable degree of confidence. But the longer the simulation is, the more reliable it is. Implementation is prototyped with mathematical software using standard toolboxes as well as formal computation capabilities.

It should be understood that the geometrical model is not restricted to rods and spherical joints and that equations can be generated through symbolic computation. Any mechanical joint can be taken into account provided it can be handled by a (regularized) closure function.

Three perspectives of this work are being considered. The first one is to develop a data post processing i.e. computing clearance size into each joint, maximum joint displacement in a specific direction, etc. Even further, keep in mind that the method provides a family of assemblies indexed by t and sharing the same topology. Then, a natural post processing is to simulate some appropriate physical behavior over each assembly in order to understand how dimensional uncertainty and clearance optimization influence the said physical behavior. This context is precisely the one of [START_REF] Xu | Isogeometric computation reuse method for complex objects with topology-consistent volumetric parameterization[END_REF] addressing how to reuse computation from one model of the family to another one, which is a promising track. The second perstective is to deal with more practical cases, including overconstrained mobile mechanisms such as deployable structures. The third perstective is a fundamental investigation about how to systematically introduce clearance parameters into perfect joints.

Appendix A. Theorems and lemmas

The appendix gathers the implicit function theorem 3 and the proofs of lemmas 1 2 and 3. In the following, X, Y , Z, U , P , J and E are finite dimensional euclidean spaces. Theorem 3. Let f : X × Y → Z be a smooth function and let (x 0 , y 0 ) such that f (x 0 , y 0 ) = 0. Suppose that the linear mapping f y (x 0 , y 0 ) from Y to Z is invertible. Then, there exists a unique function x → ϕ(x) defined over a neighborhood V of x 0 onto a neighborhood W of y 0 such that ϕ(x 0 ) = y 0 and, for all (x, y) in V × W , f (x, y) = 0 if and only if y = ϕ(x). In addition, ϕ ′ (x) = -f y (x, ϕ(x)) -1 f x (x, ϕ(x)) for all x in V . The first equation implies Aj + AA T λ = 0 and, with the third one, AA T λ = Bp. The second equation implies BB T λ = 0. So (AA T + BB T )λ = Bp and, by hypothesis, λ = (AA T + BB T ) -1 Bp. Using again the second equation, B T (AA T + BB T ) -1 Bp = 0. Now, if B T (AA T + BB T ) -1 B is invertible, then p = 0 so λ = 0 and j = 0 meaning that M is invertible. Conversely, if B T (AA T +BB T ) -1 B is not invertible, there exists p * = 0 such that B T (AA T + BB T ) -1 Bp * = 0. Then, considering λ * = (AA T + BB T ) -1 Bp * and j * = -A T λ * , linear mapping M is not invertible because for all v ∈ J such that v = 0 and G j (j * , p * )v = 0, then j * is minimum.

Proof. Since (j * , p * , λ * ) is a stationary point of Lagrange function, j * + G j (j * , p * ) T λ * = 0 and G(j * , p * ) = 0. Let v = 0 be an arbitrary vector of J such that G j (j * , p * )v = 0. Consider a smooth curve s → j(s) such that j(0) = j * , j ′ (0) = v and G(j(s), p * ) = 0 for all s. By elementary computation, that is non negative, so j * is a minimum.

Figure 1 :

 1 Figure 1: (a) isoconstrained structure and (b) overconstrained structure.

Figure 2 :

 2 Figure 2: Theoretical investigation starts with the closure equation modeling the overconstrained problem. Then, the regularized closure equation transforms the overconstrained problem into an underconstrained problem. Solutions of the underconstrained problem are alternately obtained by minimizing the squared clearance, by finding a zero of the lagrangian gradient or through an ordinary differential.

Figure 3 :

 3 Figure 3: Case study number 1: two-dimensional over-constrained structure made of six bars and four joints.

Figure 4 :

 4 Figure 4: Harmonic perturbations of rod's lengths for case study number 1, the six bars and four joints two-dimensional structure.

Figure 5 :

 5 Figure 5: Case study number 1. Various shapes of the two-dimensional structure under dimensional perturbation.

Figure 6 :

 6 Figure 6: Case study number 1. Zoom of a non-perfect spherical joint of the twodimensional structure. Red dot is the connection point. Black dots are rods' extremities.

Figure 7 :

 7 Figure 7: Clearance variation of the two-dimensional structure due to dimensional harmonic perturbations.

Figure 8 :

 8 Figure 8: Case study number 2 is a three-dimensional overconstrained structure.

Figure 9 :

 9 Figure 9: Case study number 2. Harmonic perturbations of rod's lengths for the threedimensional structure.

  Dimensional perturbations are defined using (26) by setting ∆ i = 1.0 for i = 1, • • • , 6 and the following angular frequencies. ω 1 ω 2 ω 3 ω 4 ω 5 ω 6 13 7 19 23 17 11

Figure 10 :

 10 Figure 10: Case study number 2. Various shapes corresponding to dimensional perturbations.

Figure 11 :

 11 Figure 11: Case study number 2. Clearance variation of three-dimensional structures due to dimensional harmonic perturbations.

ω 1 ω 2 ω 3 ω 4 ω 5 ω 6 ω 7 ω 8 ω 9 ω 10 2

 2

Figure 12 :

 12 Figure 12: Typical part and its dimensional parameters for case study number 3. Three dimensional and skeleton geometry is illustrated.

Figure 13 :

 13 Figure 13: Assembly and dimensional parameters of case study number 3.

Figure 14 :

 14 Figure 14: Closure equation features of case study number 3.

Figure 15 :

 15 Figure 15: Dimensional perturbations for case study number 3.

1 :

 1 Computing nominal dimensions and positions.

Figure 16 :

 16 Figure 16: Case study number 3. Typical configurations of the non-nominal assembly. Notice the non-perfect cylindrical joints and the non-planar configurations.

Figure 17 :

 17 Figure 17: Clearance variation for case study number 3.

Figure 18 :

 18 Figure 18: Input text file for symbolic computation.

Figure 19 :

 19 Figure19: The implementation procedure involves a preprocessing step for data initialization, three alternate solving techniques and a postprocessing step for result analysis.

Lemma 1 .

 1 Consider linear mappings A : J → E and B : P → E and suppose that AA T + BB T ∈ L(E) is invertible. Consider the bloc defined linear mapping M :J × P × E → J × P × E is invertible if and only if B T (AA T +BB T ) -1 B ∈ L(P ) is invertible.Proof. Because of the finite dimension of spaces J, P and E it is enough to prove that M is injective, meaning that if M w = 0 then w = 0. So, let   j p λ   ∈ J × P × E and suppose that M According to the bloc structure of M , this means j + A T λ = 0 B T λ = 0 Aj + Bp = 0.

Lemma 2 . 2 |j| 2 ;

 222 which ends the proof. Consider a smooth mapping G : J × P → E with dim J > dim E and the minimization problem(j, p) = Argmin 1 G(j, p) = 0 .Consider the associated Lagrange function L :J × P × E → R L(j, p, λ) = 1 2 |j| 2 + λ, G(j, p) .If (j * , p * , λ * ) is a stationary point of Lagrange function, and if |v| 2 + λ * , G jj (j * , p * )vv > 0

2 s=0=

 2 |v| 2 + λ * , G jj (j * , p * )vv

Lemma 3. Consider a smooth mapping f : U ×X → E with dim X < dim E and the minimization problem

Suppose that exists (u 0 , x 0 ) ∈ U × X such that f (u 0 , x 0 ) = 0 and that the linear mapping

is invertible. Then, there exists a neighborhood V of u 0 and a unique mapping u → x(u) defined over V such that x(u 0 ) = x 0 and x(u) satisfies the stationary condition and the second order minimality condition.

Proof. The existence and uniqueness of mapping u → x(u) is proven by applying the implicit function theorem to mapping h :

for all u close to u 0 , which is the stationary condition. Furthermore, for all ξ ∈ X, g xx (u, x)ξξ = |f x (u, x)ξ| 2 + f (u, x), f xx (u, x)ξξ .

The linear mapping f x (u, x(u)) T f x (u, x(u)) is invertible for all u close to u 0 , so there exists a 0 > 0 such that |f x (u, x(u))ξ| 2 ≥ a 0 |ξ| 2 for all ξ ∈ X and all u close to u 0 . Noting

which is the second order minimality condition.