Peng Wang
email: wangpeng596@gmail.com

Mingjia Xie
email: mingjiax@qq.com

Analysis of Two Methods of Systemic Risk Measurement Based On Option

Keywords: Systemic risk measurement, contingent claim analysis, option-iPoD

This paper makes a comparison and analysis about those two main systemic risk measurement methods based on option. Through the comparison and analysis in 5 aspects, more knowledge about each method can be discovered including its advantages and disadvantages. Some conclusions about extensions of credit risk measurement methods can also be included in this paper.

Introduction

Data gains its dominance in researches in different fields with the development of efficient accessibility of massive data. As one of the most important fields -financial industry, data from option and future markets offer basis for various theoretical approaches that cannot be applied without high frequency data or sufficient panel data. The current crisis demonstrates the need for tools to detect systemic risk in financial industry. There are two main systemic risk measurement methods which are based on data from financial market, Contingent Claim Analysis (CCA) [1,2] and option-iPoD [3]. Despite the similar dependence on market data, the different methods show differences in design of methods, in/out parameters, empirical researches, etc. This paper makes a comparison between the two methods in four different aspects, which provides more information about each method that is based on high frequency data. The results show that neither of two methods is suitable for all kinds of analysis while advantages come with disadvantages.

2

Summarization of CCA and option-iPoD

About two methods

A. Contingent Claim Analysis Contingent claim are the assets whose value depends on other assets' value, for example, options. Contingent Claim Analysis (CCA) is a widely used analytical method in risk measurement. While contingent claim analysis is applied in credit risk analysis, it is normally called Merton Model. The key to understand CCA is the Risk-Adjusted Balance Sheet, which comes from traditional balance sheet with adding "market risk factor". As for elements in balance sheet, this analysis transfers the book value into market value, which can't be observed directly.

The risk-adjusted balance sheet is:

While Assets A(t) is the market value of assets and the equity J(t) is the market value of equity. Debt guarantee P(t) can be expressed as the expected loss of the debt because the debtor may default at due time. B is the promised payments at time T and r is the risk-free rate. There are two important conclusions as follows:  Equity is a call option whose underlying assets is Assets and strike price is the promised payments  Debt guarantee is a put option whose underlying assets is Assets and strike price is the promised payments According to BSM option pricing model, We treat the problem as solving an optimization problem for with 3 constraints.

Will be solved first and given the optimal solve for D. The Larangian is: (6) We can solve the function by numerical method and get the probability distribution function of assets by constraints mentioned above. [START_REF] Gray | New framework for measuring and managing macrofinancial risk and financial stability[END_REF] When the function is solved, some risk exposures, such as default probability 3 , are measured. [START_REF] Gray | A new framework for analyzing and managing macrofinancial risks of an economy[END_REF] Peter W. Buchen, Michael Kelly, "The Maximum Entropy Distribution of an Asset Inferred from Option Prices" 2 Standard Newton numerical method for example. The constraints are constraints in Lagrangian which can be the functions to solve unknown parameters. [START_REF] Capuano | The option-iPoD----the probability of default implied by option prices based on entropy[END_REF] Default probability equals

2.2

Data source

Not only the algorithm but also the data source are based on financial markets especially option & future market. Data concerning about "level" and "volatile" are required to construct risk detecting indicators, which means a higher frequency data are always better because the second moment of data series are needed. The data required can be divided into following categories:

A.Data from stock market Market value of equity is required. In most cases, prices of stocks are proxy variables of equity that are unobservable directly. Daily data or even weekly data can be used to construct risk detecting indicators while other frequency are available depending on different scenarios and purposes. When necessary, frequency higher than daily, such as hourly, can be applied to measure a more precise process of changing of risk.

Risk-free interest rate is also needed for both two methods. Government bond or other composite indexes can be used as this indicator.

B.Data from option market Information concerning about the details of contracts in the option market is used to construct the constraint equations in option-iPoD. Precisely, expiration time of options, prices of options, are required in the algorithm. Analogous to the frequency issue in stock market, higher frequency data are always better.

Comparison and analysis

Two methods mentioned above have different features. This part focuses on the difference in 5 fields: thinking of designation, in/out variables, mathematical methods, empirical analysis approaches, and extensions of methods. It is expected that more info about individual method will be discovered through comparison.

Thinking of designation Ⅰ)Contingent Claim Approach

Thinking of design of CCA can be divided into 2 parts.  Risk-adjusted Balance Sheet in theory  Calibration to risk-adjusted Balance Sheet for different sectors CCA is an extension of BSM model that assumes a stochastic process of Assets. It is reasonable because of the "Weak Axiom of Efficient Market". CCA assumes that default happens when asset is lower than debt. The core function is following:

Equity is a call option whose underlying asset is Assets and strike price is the promised payments. Debt guarantee is a put option whose underlying asset is Assets and strike price is the promised payments. This function is a risk-adjusted traditional balance sheet. Scalars in traditional balance sheet are transferred into time variables with "risk factors". Several risk exposure indicators can be calculated including ele-ments in risk-adjusted balance sheet, risk-neutral default probability, Delta [START_REF] Gans | Deciding what's news: A study of CBS Evening News, NBC Nightly News[END_REF] .

Another innovation is the calibration to risk-adjusted balance sheet for different sectors. For example, some financial institutions are "too-big-to-fall", so guarantees from government are often existed, which adds the assets of financial sectors. In empirical analysis, some calibrations to equations are made to insure rationality of the methods. Merton et al. (2007) divide the economy into 4 sectors: corporate sector, financial sector, households sector, and sovereign sector, so there are 4 equations for different sectors.

Ⅱ)Option-iPoD CCA requires many assumptions, such as constant rate, distribution function of assets, which Option-iPoD is designed to release. Too many assumptions are the barrier to suitable empirical solutions. Christian Capuano (2008) proposes option-iPoD, which releases two main assumptions: probability distribution function of Assets and the default barrier [START_REF] Nelkin | Selling science: How the press covers science and technology[END_REF] .

The key to option-iPoD is Maximum Entropy Distribution of Assets, which is the result of Principle of Maximum Entropy (PME). The PME is ideally suited to estimating the probability distribution of an asset which a derivative security or contingent claim is written. With constraints from option market prices, probability distribution of an asset can be calculated.

The unknown coefficient in Lagrangian function and default barrier D can be calculated by means of numerical mathematical methods. When probability distribution of an asset is calculated, some risk exposure indicators can be calculated. "Less assumptions" is a feature of option-iPoD. Exception for prior probability distribution, all constraints are from real option market prices, which do no harm in the freedom of variables.

There are also "CCA factors" in option-iPoD. Market value of assets, partial probability distribution, delta, and gamma can be calculated through option-iPoD. Coming from option market, either thinking or data [START_REF] Gans | Deciding what's news: A study of CBS Evening News, NBC Nightly News[END_REF] Equation:

, measures the non-linear change in the value of an option per unit change in the value of the underlying asset [START_REF] Nelkin | Selling science: How the press covers science and technology[END_REF] CCA assumes that default happens when asset is lower than promised payments, but promised payments are not a certain volume. In KMV model, default barrier equals 50% long-term debt plus 100% short-term debt

In/out variables

Comparison between two methods tells attitudes to "variable freedom" from two methods. CCA has more input parameters than option-iPoD does while the two methods have nearly same main output variables. Option-iPoD has two distinct outputs: probability distribution of asset and default barriers that are two prior assumptions of CCA. To some degree, CCA has more "constraints" than option-iPoD does, which leads to deviation from real market.

Mathematical methods

Analytic expression and numerical solution are two main mathematical methods used in empirical analysis. CCA applies analytic expression while option-iPoD uses numerical method. Different methods lead to different results that generate different understanding about risk measurement.

Ⅰ)Mathematical methods in CCA

The main mathematical method in CCA is analytic expressions used in BSM option pricing model. While equity and debt guarantee are regarded as two options whose underlying assets are assets. Through mathematical solution, elements in risk-adjusted balance sheet can be solved and relational expression of elements can also be solved.

It is clear that absolute value of elements from risk-adjusted balance sheet and its relationship can be used to do further research through this mathematical method.

Ⅱ)Mathematical methods in option-iPoD

To obtaining the distribution of equation ( 2), it is necessary to first find the Lagrange parameters, , which are determined by the constraints entailed by equation ( 3), ( 4) and ( 5). These equations are nonlinear and have to be solved numerically. A standard Newton method, for example, in Djafari (2000). Buchen and Kelly (1996) and Avellaneda (1998) show that the objective function is convex, and that the solution is unique.

Then the last unknown parameter default barrier D, also has to be solved numerically with following constraint: (8) The whole probability distribution function can be established. It is of importance that the solved probability distribution function is a "partial function" with "partial domain of definition". Because constraints of option-iPoD come from real option market, which shows that constraints are meaningful if option market's price information is effective. Stock prices and option prices are insignificant when corporate defaults. For investors of stock or option, there is no trading of stocks, no trading of options in default state. Equity options are not suited to describe the market value of asset in default state.

While equity options don't contain information on shape of the probability density function in the default state, they do contain information on the cumulative distribution function, the probability of default. As it can be found in equation (7), has same information as when , so it provides sufficient "cumulative information" in . For example, default probability and other risk exposure indicators can be measured.

Table 4.Comparison in mathematical methods

CCA Option-iPoD Emphasis of mathematical methods

Analytic expression of all elements of risk-adjusted balance sheet Solution to probability distribution of asset and default barrier

Similarity

Mathematical methods are regarded as tools for solution in different purpose

Empirical analysis with data

Ⅰ) CCA There are 3 important basic input variables: market value of asset, standard variation of assets return rate, default barrier. Unlike trading equity, market value of asset and its variation can't be observed directly. So two equations are needed to solve this problem.

(9) ( 10) is equity's market price, is market value of asset, is standard variation of asset, r is risk-free interest rate, and D is market value of default barrier.

Equation ( 9) is BSM option pricing model and equation ( 10) can be expressed as following:

(11) (12) As for market value of default barrier D, it's always be set manually. Taking KMV for example, default barrier equals 100% short-term debt plus 50% long-term debt.

The economy-wide CCA can be used with scenario, simulation, and stress-testing analysis. The level of analysis depends on practical issues related to data availability, data reliability, and goals of the analysis. For sectors with trading equities and options, some corporates, some financial institutions, even some public sectors, option-iPoD provides a more accurate and effective systemic risk measurement approach. Compared with CCA, option-iPoD gets fewer manual assumptions which makes it closer to real market. Option-iPoD is more suitable in this situation for its data used in option-iPoD comes from high-frequency option market. More constraints come from markets makes option-iPoD more accurate for various goals of analysis.

CCA shows its wide range of application because of Merton's risk-adjusted balance sheet. For sectors like sovereign sector, with no trading options, CCA is better at estimating systemic risk. The core function 10 is calibrated to correspond with various sectors thus CCA can be used in all sectors, even in public sector or households sector. CCA works at sectors with trading equity or "analogous equity" and option is not necessarily needed.

All above, CCA has wider range of application than option-iPoD does but option-iPoD has less manual assumptions, which makes it closer to reality.

CCA needs release of assumptions & option-iPoD needs more constraints

Assumptions simplify the problem in theory but deduce the freedom of variable, which leads to inconsistency with reality. CCA will be better if more assumptions are released. In classical Above all, subjectivity should be reduces and objectivity should be increased both in CCA or option-iPoD.

4.3

The improvement of the quality of data tells more CCA and option-iPoD are two main credit risk measurement methods. These two methods are better than traditional FSIs 11 at solutions to deal with high-frequency market value. CCA applies the thinking of option while option-iPoD applies the price information of options.

The two methods focus on option market. On one perspective, option market is a highly financial market whose information if effective. On another perspective, price information from option market is easy to get. With the development of option market, it can provide more information in financial market which are constraints for credit risk measurement model.

Information from option market and theories of option market can work in more sectors not only in financial and corporate sectors. Like CCA, contingent claim analysis works well in sovereign sector that represents macroeconomic systemic risk exposure. Contingent claim analysis can also be used in key banks to simulate systemic risk in financial sector after weighting process.

Above all, macro financial engineering is one of extensions of credit risk measurement method.

Future work

This paper makes comparison and analysis between two main systemic risk measurement methods based on option in 5 perspectives. But effectiveness of two methods in empirical analysis is not estimated which requires more related researches to support.

Aknowledgement.

The research work was supported by the National Social Science Foundation of China (Grant No. 13CJY121), by the Fundamental Research Funds for the Central University under Grant No. JBK130503.

  with , equals to the standard deviation of the asset return. N(X) is the cumulative standard normal distribution. Main Risk exposure indicators  Risk-neutral default probability: With expect assets return rate equals the riskfree rate r, N(-) is the risk-neutral default probability  Credit spread:  Risk-adjusted balance sheet elements: Assets, Equity, Risk Debt  Delta, it measures the non-linear change in the value of an option per unit change in the value of the underlying asset B. Option-iPoD Buchen and Kelly (1996) propose a numerical method to get probability density distribution function of assets. Capuano (2008) proposes option-iPoD based on former function, which is a systemic financial risk measurement method.Entropy measures the uncertainty of a variable and maximum entropy shows the most uncertainty of a variable. Principle of Maximum Entropy, that is PME, is ideally suited to estimating the probability distribution of an asset on which a derivative security or contingent claim is written[START_REF] Gray | A new framework for analyzing and managing macrofinancial risks of an economy[END_REF] .The problem is following:(2)While is the prior probability density distribution function, is the true probability density distribution function.The constraints are following: a) Constraints based on Merton Model (3) : The price of an option whose strike price is K and due time is T. V: the market price of assets D: the market price of risky debt E: the market price of equity r : the risk-neutral rate b) Constraints based on option market price (4) c) Constraints based on probability density function(5)

  a) Risk exposure indicators from CCA can be analyzed with other macroeconomic factors. Factor model is a suitable model 6 . The time pattern of asset returns of each financial institution can be used as the dependent variable in a factor model. b) "Sub risk-adjusted balance sheet for household sectors" 7 . c) Stress-testing and assessing capital adequacy using CCA model of financial institutions. d) Making connections between traditional monetary policies based on interest rate & money supply and CCA risk exposure indicators. Ⅱ)option-iPoD One of main empirical analysis of option-iPoD is what Capuano (2008) has done with Citi Bank. In that research, option-iPoD is well used to estimate how default barrier and leverage ratio generate through financial crisis in 2008, which proves that option-iPoD is suitable for systemic risk measurement. Based on results from option-iPoD, some advices, such as "de-leveraging" should be applied for Citi Bank, are suggested. At the same time, Capuano observes how Moody's KMV EDF 8 generates, which shows what the systemic risk Citi Bank faces.

Table 1.Comparison in design of thinking

 1.Comparison

		CCA	Option-iPoD
	Role of option	Thinking of BSM option	Data source of constraints
	market	pricing model
	Focus	Risk-adjusted balance sheet	Probability distribution of an
			asset
	Data	Traditional balance sheet	Option market
	similarity

Table 2 .Comparison in in/out variables

 2

		CCA	Option-iPoD
	Input parameters	Market value of equity;	Expiration time of options;
		Standard variation of equity;	Stock price;
		Risk-free interest rate;	Risk-free interest rate;
		Default barrier;
		Expect return rate of asset;
	Main output variables	Default probability;	Default probability;
	(relate to risk-adjusted	Market value of asset;	Market value of asset;
	balance sheet)	Expected loss of debt;	Expected loss of debt;
		Equity;	Equity;
			Default barrier;
	Main output variables	Standard variation of asset;	Standard variation of asset;
	(relate to risk expo-	Delta;	Probability distribution of
	sure indicators)	Gamma;	asset;
		Vega	Delta;
			Gamma;
			Vega

Table 3 .Comparison in Assumption & in/out variables

 3

		CCA	Option-iPoD
	In/out variables	More input parameters	Less input parameters but
			two more distinct variables
	Similarity	Number of input variables is larger than number of output
		variables. Fewer assumptions are better for methods.

Table 5 .Comparison in empirical analysis

 5 Some researchers have established a relationship between implied volatility of two equity options, leverage and implied asset volatility. In fact, this is another way of implementing Merton's model to get spreads and risk-neutral default probabilities directly from the implied volatility of equity options  Shimko, Tejima, and Van Deventer (1993) include a Vasicek interest rate term structure model which allows interest rates and term structure of interest rates to vary Ⅱ)More constraints about option-iPoD More constraints from markets make option-iPoD more accurate.Capuano (2008) adds zero-coupon bonds constraints to option-iPoD.

		CCA	Option-iPoD
	Application fields	Wider than option-iPoD does includ-	Fields with trading equity
		ing macro-economy, financial sector,	and options
		households sector
	Data	Suitable balance sheets for different	Price information from
		sectors	option market
	6 Based on data from Chile central bank, Dale et al (2008) build a factor model with 4 main
	economic factors and asset return rate calculated by CCA. This factor model well simulates
	how the systemic risk generates
	7 See more details in Dale F. Gray et al, "NEW FRAMEWORK FOR MEASURING AND
	MANAGING MACROFINANCIAL RISK AND FINANCIAL STABILITY"
	8 Estimated Default Frequency: a measure of probability of default given distance to default

Table 6 .Comparison in extension of methods

 6

		CCA	Option-iPoD
	Extension	Relaxing more	Adding more con-
		assumptions	straint from market
	Similarity	Fewer manual assumptions and more constraints
		from market
	Out of the potential participants (249 members), 118 filled the questionnaire, resulting
	in a response rate of 47%. Table 1 contains the non-respondent analysis results.

Table 7 . Reasons for non-participation in the survey Reason for non-participation N:o of persons (total = 131) Percentage

 7

	Too busy or forgot	27	21%
	Dislike of such surveys or not inter-	10	8%
	ested
	Technical difficulties with the survey	19	15%
	Did not feel competent enough to fill	7	5%
	the survey despite being a health
	journalist
	Unknown or person could not be	67	51%
	contacted

  More reliable constraints makes prior probability distribution function closer to real probability distribution function by Lagrangianoptimization process. Christian Capuano (2008) adds zero-coupon bonds constraints to option-iPoD which makes option-iPoD more accurate.

Merton Model, constant interest rate and probability distribution are required. For example, Shimko et al. (1993) include a Vasicek interest rate term structure model which allows interest rates and term structure of interest rates to vary, which is a release to interest rate assumption.

Option-iPoD needs more constraints of reliable, available market.

Corporate sector, financial sector, households sector, sovereign sector


Yang Cheng
email: yangcheng@swufe.edu.cn

Zhao Qi

How to Set and Manage Your Network Password: A Multidimensional Scheme of Password Reuse

Keywords: Password Security, Password Reuse, Seed Password, Difference code, Multidimensional Password Scheme

The rapid development of the Internet and highly decentralized network services, prompting the majority of Internet users continue to register more accounts, and cause a high incidence of password reuse, which makes the user information leakage risks facing the domino-style. Based on the data of Internet password leak door at the end of 2011 as well as the college students' online survey, the paper analyzed the structural characteristics and reuse behavior of netizen passwords in detail, and thus designed a multidimensional password scheme that infused into the information dimensions and classified management. This scheme, based on the structure of "seed -reuse code", includes three dimensions: the content dimension contains multi-independent "information factor", which constitutes the main part of the password; the formal dimension is responsible for conversion formatting, in order to enhance the complexity and security of the password; and space-time dimension is targeted designed to protect the password timeliness and reusability. Through comparative analysis and quantitative analysis, the new password scheme not only has good memorability and convenience, and can effectively resist the violent attacks and acquaintances attacks.

Introduction

With the rapid expansion of the Internet, the network has been closely together with people's life, become an indispensable part of life. In order to fully enjoy the convenience of the Internet, such as business transactions, information access, communication and network entertainment, users need to register account more and more. At present, the most common site authentication mechanism is still the username and password combination model (ID-PWD). Although the mode has the obvious disadvantage of [START_REF] De Rodrigo | Biometric identification systems[END_REF][2] [START_REF] Shirley | Password management strategies for online ac-counts[C]. Proceedings of the second symposium on Usable privacy and security[END_REF] in terms of safety compared with biometrics, smart cards and other methods, but it is easy to be accepted by users, and low cost, thus no one mature authentication mechanism can replace it so far, because of the convenience and practicality. Network certification will be mainly ID-PWD model [4] for a long period.

As the ID-PWD model is not perfect in the protection of network information security, so users often have to face the problem that account password was stolen and personal information been leaked, or distress. According to the CNNIC "twentyeighth times Chinese Internet development statistics report" [START_REF] Zhang | Improving multiple-password recall: an empirical study [J][END_REF] data shows, the number of Internet users who had account or password stolen reached 121000000 at the first half of 2011, account for 24.9 percent of the total number of Internet users. At the end of this year, Chinese Internet outbreak a large-scale user data leaks. From the 6400000 user data leakage of the programmer website (CSDN.net), to the 30000000 user password leakage of Tianya (Tianya.cn), and Renren (renren.com), duowan (duowan.com), 7K7K network (7K7K.com), baihewang (baihe.com), mop (maopu.com) and other famous sites have been stolen. So far, a large number of the big and famous website in China has been involved, about hundreds of millions of users. Not only the amount is amazing, and the leaking data is a plaintext password, nonencrypted storage, so the "secret door" caused a panic of the password security in china. In the user information leak incident, password reuse behavior of Internet users play an important role, causing huge losses to the majority of Internet users.

The so-called "password reuse" refers to the user's behaviors that choose the same password between multiple different accounts. Cognitive psychology indicate that this behavior is rooted on human memory limitations [START_REF] Notoatmodjo | Passwords and perceptions[END_REF]: ordinary users can remember a meaningless, random, high-intensity password composed of numbers, letters and special characters after training, but after multiplied by 10 or more human mind will reach the physiological memory limit. Therefore, the design of ID-PWD combination would face trade-offs between memory and security. Based on the specificity and sensitivity of password data, there has little research about this kind of behavior, also not deep enough, and the domestic is almost a blank, but the hidden danger lead to network security is a consensus: using a small amount of password at numerous sites repeatedly would face security risk, if a combination of ID-PWD leaked it may cause the user to lose many other accounts [START_REF] Shirley | Password management strategies for online ac-counts[C]. Proceedings of the second symposium on Usable privacy and security[END_REF] [6] [START_REF] Devi | OPass: Attractive presentation of user authentication protocol with resist to password reuse attacks[END_REF] [8] [START_REF] Schneier | Real-world passwords[END_REF]. For example, the protection of the user account information is different as different websites; especially some small and medium-sized website may become a short board password protection, which leads to the large websites' pay went up in smoke, because of funding, technology limitations. So, in order to avoid further spread of password reuse phenomenon, put forward a reasonable imminent, efficient password design and management are strategy, based on a large amount of empirical research we attempt to do some exploration in this respect.

Analysis the structure of passwords

The serious user data leak which occurred at the end of 2011 is the sorrow of the entire Internet industry. On the other hand, it also provides researchers priceless data with which to study the password security and password feature. At present, similar studies are often limited by the objectivity and representative of data, because the data before are got either from a small scale questionnaire survey or from the little leak data of a single website, thus there has never have a study that involves so many sites, cross category, large scope user password data. More importantly, most of the existing literature on the user password feature is about the English speaking world, but these user password data are usually related with the culture and language of the country, for example, we have found a lot of unique characteristics of Chinese in our research, such as Chinese Pinyin. In addition, as the password data involve a number of different categories of representative sites, we can get the value sequence of these kinds of websites and the empirical research of reusable data through the correlation analysis. Therefore, in this section we will analyze some of the leak data and the students password reuse behavior, to study the universal ways users design the password, including structure features, application habits and reuse mode, so as to bring forward the password design scheme and management strategies.

Structure

In the analysis of password structure in this section, we select the most representative leaked data of Tianya community as the main analysis object, while other data for comparative analysis and verification, this is because the Tianya community uses blog, micro-blog as a basic way to exchange, human emotion as the characteristics of the integrated virtual community and large social networking platform, it is the most influential global Chinese online home, while the user groups are widely distributed, including different age, different class, different occupation of Internet users. At the same time, Tianya is also the website who has been stolen the most data in this incident, thus, it is suitable to select it as the analysis object.

This leaked data of Tianya is backup data for 2009, a total of 29865731 account records, each record contain the account ID, password and E-mail information, so we can quickly get some basic structure characteristics of the Internet users password using some simple SQL statements.

For example, length, two thirds of the passwords range from 6 to 8, where the average is 7.94, the ratio of number and letter is about 3:1. About structure, 63.8 percent of the accounts are pure digital password; while 10.3 percent are pure alphabetic password. On the other hand only 24 percent are mixed, and the choice of the special character for the password is lower than 1.9 percent (the last two numbers in the Myspace users are respectively 81% and 8.3%) [10]. Visible, China users prefer digital password; password security awareness is generally low.

To analyze the commonly used password, we found a lot of difference from the western caused by the unique characteristics of Chinese culture. In addition to the highest frequency of 1, 2, and0, the number eight ranked fourth (the pronunciation is similar to "death "); while the number four is the least frequently used number (the pronunciation is similar to "making fortune ").

In addition, due to the differences in culture and modes of thinking, Western Internet users are different from their Chinese partner in the choice of 26 letters (case insensitive) as the password character. Through the statistics the frequency of each letter used in four different application environment (English text, Western password, Pinyin text and Chinese password), and calculate the space cosine angle between the four groups of data, we found that the correlation of China users' password and Chinese characters Pinyin text is 0.928, far higher than the English one 0.841, showing that people in the password design used to reference the pinyin. [START_REF] Yang | An analysis view on password pat-terns of Chinese internet users[END_REF] This point is also reflected in these commonly used Chinese Pinyin characters: Ang, Jia, Hao, Wan and Xiao with a frequency of 0.5 percent, while the frequency of password, baby, ball, boy and other strings are often appeared in the Western passwords are below 0.005 percent, only ABC and love have a frequency close to 0.5 percent. Further studies revealed that the frequency of consonant characters are higher in Pinyin. This reflects that people like use Mnemonic Phrase-based Passwords, for example, with reference to the idiom " Man proposes, God disposes ", the password may be set to "m4zrc4zt".

Reusability

In this section, we select the leak data of four websites Tianya, CSDN, 7K7K and Renren as the research object, to establish the association table across the sites account for a common mailbox, and study the phenomenon of password reuse. These four sites belong to different subject categories, in addition to the previously mentioned Tianya, CSDN is the largest (6428631 records) technology forum for programmers, 7K7K is one of the most professional casual games website (19138451 records), Renren is a famous social networking sites (4768600 records). They are the leader in their categories, has a huge and representative registered users. Account association table "Relation (Email, Tianya_PWD, CSDN_PWD, 7K7K_PWD, renren_PWD)" has a total of 4718269 records; each of them is at least email registered at two different sites. Through the table we found that, in addition to password reuse, there are a large number of multiple password corresponding with the same email exist only minor differences, they are from the same root password (PWD-Seed) and derived, this should be regarded as an extension of the password reuse. In order to accurately measure the multiple correlation relationship, we use the standard LCS algorithm. The standard LCS algorithm was adopted here to calculate password similarity from the same users on different websites:

2 ( ( 1, 2)) ( 1 , 2) ( 1) ( 2)

le n lc s P W D P W D s im P W D P W D len P W D len P W D

   1 (1)
len(x) returns the length of the string X; lcs(X, Y) returns the longest common substring of X and Y, such as, lcs('12345678', 'a13458') ='345'. In the formula, "Sim=1" indicates that users use the same passwords on multiple websites and "Sim=0" indicates that the passwords used in different websites are totally different. Therefore, "0.75< Sim<1" indicates that passwords used in unique websites are basi-cally the same. Table 1 shows the average Sim between the four sites of two values, namely password reuse rate of different websites.

Table 1 indicates that in these four sites more than half of the public user password existing reuse phenomena between any two sites (the sum of two reuse patterns between any two sites are larger than 50 percent), where Tianya and 7K7K have a proportion of more than 90 percent are completely password reuse, with password basically the same is more close to 95 percent. In the two reuse mode, ordinary users tend to use fully password reuse is simple, but has the background of IT users (CSDN) tend to relatively complex password reuse. Further analysis showed that, in the multiply site registered users, nearly 80% (79.1%) users existing a behavior of password reuse (4 password corresponding to the same Email are at least 2 identical), including 4 password identical up to 33.9 percent, which is more than one to three users.

Usually, password reuse rate between websites are related to many components, the value of website and its user component structure are the two most important. For example, in four sites in Table 1, CSDN has the highest value, followed by Renren, the two passwords are unique compared with others, to prevent theft; while Tianya and 7K7K account value is relatively low, more likely to share the password, to facilitate the application. As the user structure, network safety consciousness of the four in descending order: CSDN user has the background of IT technology, Renren to college students, while Tianya and 7K7K are relatively popular. The above two rea-sons, jointly determined the password reuse of four big websites and its representative categories with a sort: CSDN (IT) < Renren (SN) < Tianya (BBS) < 7K7K (GAME).

Finally, the generalized password reuse should also include different users use the same password. It is embodied in the high-frequency password. In the case of Tianya, there have an average 1.5 people sharing a same password per 1000000 people, of which there are more than 4% accounts use "123456" as the password, the top 20 high-frequency password account proportion reached 8.42 percent, 100 reached 11.38 percent, while the first 10 percent password should cover more than forty percent (41.11). Obviously, the security of these passwords is relatively low, when faced with force attacks; it is very likely that they are the first to be compromised.

Behaviors study for students

In order to analyze users' password reusing habits, we developed an online re-search of 123 students majoring information manager, which received 118 effective questionnaires (including 25 boy-students, 93 girl-students). These participants having better knowledge of computer science and network safety realization, and we tried to find some valuable devise thought through their reusing behavior.

By the previous interview, we surveyed and sifted 29 website which always visited by college students (or website software), including QQ, Alipay, the ABC online bank (blank cooperated with college), 163 mailbox, Sina Web,Worry-free future, Baidu Library and so on, which can roughly divide into 6 kinds: online classes, communication kind,forum kind,job wanted kind, datum kind and entertainment kind, and add to an daily "starting up password". Subjects involved were asked to visit these website from the first one, and fill out one typical figure one/two/three..., and the website which having reusing password use one public figure to mark , unregistered website use zero to mark. For the survey using anonymous way, and didn't involve material password data, or privacy reveal, so the data of this survey are relative real and objective.

By analysis, we found that almost all 118 students interviewed having reusing password behavior (94.9%), and among them 56.8% used the root password mode. The average account number is 10.6 for each student (between 3 and 28), however the independent password number is only 4.6 (between 2 and 8). Using the number of registered website and independent password of each student, we calculated all interviewers' average rate of reusing password is 2.6(between 1.0 and 6.5). This meaning, when the password of a reusing website leaked, an average of 2.6 websites facing a potential threat. Compared with the previous survey,we found that some data is coincide. For example, Florencio and Herlry analysis 500,000 users found that the average number of password is 6.5, the reusing rate of password is 3.9; Brown and others researched 218 students from American universities found similar result: each student had the average number of password is 4.45, and the reusing rate of pass-word is 1.84 [START_REF] Brown | Generating and remembering pass-words[END_REF].

Researchers always use reusing rate to measure the potential threat level of password system, but Youngsok and some others thought this data ignored users' bias of password use, that is, it reflected the unbalance of reusing password on the account. They construct passwords on the base of graph theory, and put forward a new kind of Vulnerability Index [START_REF] Bang | Improving information security management: An analysis of ID-password usage and a new login vulnerability measure[END_REF],

1 1 = ( )( ) 1 m i i i n n V I N N     (1)
N is the total number of register website, m is the number of independent password(that is, the number of reusing subnet, m≤N), n i is the number of websites included in reusing subnet.

Using this formula, we calculate 118 students' average value of VI is 0.33, that is, when one website leaked, averagely 1/3 of remaining website facing threat, compared with 2.6 before is much higher. Figure 1 shows the reusing rate and VI value of all students interviewed, from which we can found that, the VI value is very different with same reusing rate. For example, the reusing rate of student A and student B is 3.0, but the VI value of them is 0.67 and 0.20. It's thus clear that, using reusing rate only to measure Vulnerability Index is not enough, it also need independent password distribution in reusing network to evaluate it. According to the data of all users' passwords, Clustering of 30 websites: firstly, numbered all websites with w1.w302 , secondly, make a table [30*30] by the original data, the data in the table is the number of users having reusing password between two websites, and last, using "the longest distance way" in R software make cluster analysis, figure 2 is hierarchical diagram. The figure showed, the interviewers' reusing password having obvious classified phenomenon: the password similarity of w1(the ABC online bank)and w2(alipay) is higher; w8(QQ)is an independent class; however the passwords of w9(Fetion), w14(Renren), w15(Sina Weibo) and w19(Baidu Library ) are similar; and the remaining 23 websites is a class. Obviously, the two classes previously belong to high useful website, and the number is very small; the number of latter class more than 2/3 of the total number, but for college students, they belong to small useful website. This classification can meet the real need of large password use, is a worthy mirror in our password design system.

Multi-dimensional password

For the phenomenon of reusing the same password for multiple websites, there are two better solutions usually: one is using "one website one password" pattern, setting a random password for each password; the other is using hash function pat-tern, using the name and other feature of website as dependent variable,generating independent passwords. Obviously, the former needed wonderful memory, but the latter needed the same wonderful mental ability. Since the memory and mental ability of people is limited in the reality, these two schemes would entrap the dilemma of weak password or with the aid of tools for storage/calculation. So, on the problem of multiple website password design, lots of scholar considered pessimistic cannot meet the memory and security at the same time. But there are some scholar considered that memory and security are opposite actually, so finding suitable solution meeting the real demand is possible, and they presented principle thought: (i) the quality of initial passwords; (ii) the memory of passwords; (iii) the complexity of passwords [START_REF] Sasse | Transforming the 'weakest link' -a human/computer interaction approach to usable and effective security[END_REF].

The short-term memory survey of American psychologist Miller indicated that, the capacity of short-term memory had inner relationship with the quality of memory material and individual material processing, the memory limitation of general people is around 7±2 chunk. Here the chunk is the unit of short-term memory, it's the process of uniting several separate stimulation into meaningful, larger information unit, that is stimulation of information recoding,this can effectively improve the capacity and effect of short-term memory. And short-term memory through repeated collection, changed into long-term memory. [START_REF] Lisman | Storage of 7+/-2 short-term memories in oscillatory subcycles[END_REF] For this, we imagined a password encoding way: changing a long password string into the suitable combination of several small "information factor", and each information factor is a sub-string through the designer considered deeply, containing rich personal information, and having closed relationship with the experience of designers. In this way, it remained the complex and multiple form,at the same time improved the durability of memory.

Concretely, we referenced "Dimension of Information" concept in Information Theory, extended it and then proposed" Multiple Dimensions Password System" designed way. In Information Theory, dimension of information is a measure to estimate the effective value of information, including dimension of content, dimension of form and dimension of time, to measure the relationship of content, the rationality of form and the effective of time separately. Expanded into multiple dimension system, dimension of content corresponded several independent information factors, constituted of the memorial main part of password; dimension of form responded to the form change, improving the complexity and security of pass-words; and dimension of time reflected specific design, ensuring the effect and importance of passwords. The following is a detailed instruction for each latitude.

Dimension of Content:It's the base of the entire password system, and it can be any personal information of users, such as name, date, address, telephone number, motto, pet phrase, or liked poetry, proverb and so on.

Designing the dimension of password, we should avoid the case of single "information factor". Such as, analyses the Tianya passwords, we found lots of simple mnemonic passwords, they often from a poetry or proverb. Designed in this way could solve formal violence attack, but couldn't avoid friend attack. When content dimension contained several information factors, it could be recoded, changed it into a entire, meaningful information unit, improving the effect of memory further. For example, a dog named "diandian", and its mother was born at 62, and it wanted to go to "Vienna", so the content could be designed as"Dot-62-Vienna", memory point: diandian went to Vienna at 62. Dimension of Form: It was effective guarantee of password security; it included both password type diversity, and its changed complexity.

The Dimension of Space-time: it was the key of multiple password system reusing, including the content of time and space.

By selecting the content dimension and form processing, we got the password having excellent performance in memory and security. Besides that, compared with the independent password problems, the design of multiple websites password system should consider the ease problem. In section 2.2, mentioned password reuse phenomenon except for complete reuse patterns, also had root mode, which is especially obvious in Internet population with higher security awareness. It was not hard to find that, the root password reuse is actually compromise of "one network one password" and "multiple websites one password", it enabled users to meet certain safety while took the least amount of time and energy in the design, memory and application of the password. This also conformed to the American Harvard professor G.K.Z put forward "Principle of least effort", namely, all people have the nature of streamline save and pursuit of benefits, always want to obtain the maximum benefit with the minimum cost (including current and expected). But studied the leaked data in-depth found that, the reuse of code (that is, the different part of the outside of the root password) design were too simple, they were more attached to the root password in the form of a prefix or suffix, and on content used multiple websites pinyin initials and digits as simple encoding. This design might be meet the security requirements of a single password, but as a password system, this faced the threat of collapse once a point breakthrough. For once the interpreter got two or more passwords of the same account, he would easily analysis the root password, and easily guess the user' password for other websites.

To avoid the password system domino collapsed, the reuse of code must be based on the personalized design, it should be diversified, no rules, and should not be stereotypes and unified. Formally, although the multiple information factors' order of root password relatively fixed, but the reuse of code can be inserted into the information factors, it could be integrated and also could be spread; and on content, it was not only related to the public information of the websites, but also related to the user's personal experience of the website, which was joint coding based on the two parts of information (Net-Public, Net-Private). Such as alipay website, Net -the Public can be zfb (alipay), ali (alibaba), ww (ali wangwang), 82 (taobao TB phone stretchable latex pinyin) and so on, even $, EC (electronic commerce) and characters contacted with alipay, such as Net private could be the time of the user shopping online, one of the most expensive online, or the nickname or avatar of ali wangwang,or...any private information associated with "alipay" . For example, the same root password ".62Vie", its alipay password could be designed for "67.62$Vie" (67, the user usually shopping on weekend), and sina blog password could be designed as "eCat. 62 Vie" (e with sina logo similar big eyes; the Cat, the user's blog, nicknamed "big Cat") or "M24.62 Vie" (24, sina's top two vowels; M, with big Cat logo shape).

At last, drew lessons from the results of the survey in 2.3, we need to divide websites into 3-4classes by the value degree, designed different root password, and constituted different password system for classified management. This classification not only reflected the value difference among websites, more was that it could en-hence the security of the entire password system further, especially be conducive to protect small amounts of high value accounts. This was because, on subjective the user had stronger sense of security for high value accounts, would pay more attention to the environmental security, reduced the possibility of self-leak; On objective, ac-count of the high value often correspond to some large websites, which could provide more capital and technology for the client data protection, isolated such accounts could avoid safe hidden trouble caused by the short board effect. Therefore, balance d the convenience and safety, users classified managed the Internet accounts ac-cording to the value, for online banking, alipay, QQ and other first or second high value should be focused on protected, let the root password contained more information factors, and carefully constructed to reuse code; And for the last class of account with low value or zero value, should focus on convenience, using relatively simple root password and reuse code, and even didn't need reuse code.

Security analysis

Compared with the traditional password, multi-dimensional password system has obvious advantages in memory and convenience, and in this section we will analyze the quantity of safety. Usually, password attack was mainly divided into two kinds, hackers and acquaintances hack. The former attack is more widely, while the latter has a higher success rate. In the system scheme of this paper, the hacker attack is still decided by the password length and alphabet size, while acquaintance attack is decided by the number and form of the root password information factors and the privacy levels of the reused string in the password.

As Shirley and Edward [START_REF] Shirley | Password management strategies for online ac-counts[C]. Proceedings of the second symposium on Usable privacy and security[END_REF] conducted a survey in 2006; their purpose is to find who poses a biggest threat to their password in the eyes of the users. They considered the population in risk and the ability of hackers, on the basis of computer ability and the relationship with the attacker, the population in risk was divided into 6 categories. Similarly, we have classified test on the security of password system, also divided into 6 types: SN 0 , SN 1 , SN 2 and AN 0 , AN 1 and AN 2 , according to the revealed password number (not leak, one leakage as N 0 , N 1 , revealing 2 and above as N 2 ) and the degree of closeness (strangers and acquaintances remembered as S and A).

We propose that the length of multidimensional password is at least 10 or more, because it will be unable to contain enough information factors and enough number of reusable codes if it is shorter than that, thus affecting the security of the password. Table 2 take a password with 3 information factors and the minimum length of 10 characters as an example, to analyze the 6 cases safety. At this time, the multidimensional password usually has 3 kinds of structure mode: 6-4 (6 root passwords, 4 reuse codes), 7-3 and 8-2. Table3 estimated the number of possible combinations in a conservative way, to reveal the worst safety case: assuming that risk groups are "Acquaintances" and the hacker is very understanding, knowing that the possible 100 information factors (10 classes, each class contains 10 items of information), and 10 common formal transformation, namely each factor about 1000 value. In the reuse of code, Net-Private is highly personal, similar to another information factor, while the Net-Public is limited to specific sites, equivalent to a large class of "weak" information factor, set the value as 100 (10 items of information *10 forms transformation). In addition, Table2 also provides a reference system: simple mnemonic type reuse passwords. This kind of password generated by a mnemonic phrase and simple reuse code, its essence is a simple combination of a single factor and Net-Public.

As shown in Table2, multidimensional password was significantly better than the simple mnemonic type password in safety; in any case it has several orders of magni-tude higher than the latter. If the simple mnemonic type password was good for violent attack, then it is almost completely transparent when encounters an acquaintance attacks, especially under the condition that the attacker got multiple passwords. For multidimensional password system, not only it can effectively resist all kinds of pure technology attacks, but also close to 8 bit random password when faced "Acquaintances". Even the highest AN 2 , the most conservative estimate, it also had a combined million species. The current online login mechanism, such as incorrect password restrictions and additional parity check code, can effectively ensure the security. In addition, Table2 also shows that when faced the high risk acquaintance attack, the security of multidimensional password system is mainly determined by the number of information factor and private levels, and has no direct relationship with its characteristic length. Therefore, suggests that high value sites contain at least more than 3 information factors, while the reusing code should be not less than 3 characters, in order to ensure system security after particular individual password leaked.

Discussion

Based on the survey data and the leaked data, we designed and created the multidimensional password system, "root password -reuse code" as the structure, involved the information dimension and the classification thought. After comparison and quantized analysis, which not only has a good memory and convenient, but also can effectively resist brute force attack and acquaintance attack. Although the complete website authentication mechanism contains two parts: ID and PWD, many researchers have suggested using different ID in secure password system, but we think that the essence of ID is the password extension, it is not recommended to do too much change in the account. Because most of the formal sites were required to pro-vide email when the users register, or even mobile phone number, for confirmation of registration and retrieve password and other services, and users often fill in. And it is often the key to judge whether it is the same user, rather than ID itself. Therefore, too much account transform can do nothing but increase the burden of memory, meaningles to reduce the risk of reuse.

Notable is, password scheme in this paper is only for network user. And a good password security system needs three aspects to cooperation: users, Internet companies and policy makers. Especially for the enterprise, safe storage and encryption transmission problems need to be solved for user data. If these data were stored in plain text, once encounter "drag library", and then the password was also in vain even if it designed perfectly.

Figure 1 .

 1 Figure 1. Password reuse rate and VI value of interviewed students

Figure 2 .

 2 Figure 2. Clustering pedigree chart of 30 surveyed sites

  encoding be cracked (worst possible) Note: ( ) O x represent password combinations of equivalent to the length of x random password

Table 1 .

 1 Analysis of Password reuse between any two of the four sites

			full password			root password
		Tianya	CSDN	7K7K	Renren	Tianya CSDN	7K7K Renren
	Tianya	-	34.10%	91.60% 51.70%	-	19.80% 3.30%	6.90%
	CSDN	34.10%	-	36.00% 29.30% 19.80%	-	21.90% 22.30%
	7K7K	91.60%	36.00%	-	57.50%	3.30% 21.90%	-	5.30%
	Renren	51.70%	29.30%	57.50%	-	6.90% 22.30% 5.30%	-
	Average Val	59.10%	33.10%	61.70% 46.20% 10.00% 21.33% 10.17% 11.50%

Table 2 .

 2 Multidimensional password system security quantify table

	type		combination	simple mnemonic	description
	SN	0		10 94 , (10) O		Available password characters 96.
						Randomly selected 6-8 root
						password, reuse code can
	SN	1	6 10 C	4 7 94 + 8 94 + 9 94 4 7 3 3 8 2 2 10 10 C C       (6.9) O	2 2 (94 94 94 ) 4 3 2    (4.3) O	appear in any position in the multidimensional sys-tem, and only in the simple
						mnemonic is a former suf-
						fix.

Function Description: len Returns the length of the string, lcs Returns the longest common substring, for example, lcs('12345678', 'a13458')='345'.

30 sites(w1..w30): ABC online banking, Alipay, Ali Wangwang, Amazon, Jingdong Mall, Ctrip, VANCL, QQ, Fetion,Gmail mailbox, 163 mailbox, 126 mailbox, The Farm, RenRen, Sina Weibo, Douban, Tianya Forum, CSDN, Baidu library, Douding network, DaoKe Baba,51job,ChinaHR,Street site, The recruitment of Chile, Youku, Tudou, The most liked online games, The most liked game sites, power-on password.


Pierre Renaud
email: renaud@ifma.fr

Nicolas Andreff
email: andreff@ifma.fr

Philippe Martinet
email: martinet@lasmea.univ-bpclermont.fr

Combining end-effector and legs observation for kinematic calibration of parallel mechanisms

In this paper, an original approach is proposed for the kinematic calibration of parallel mechanisms. The originality lies in the use of vision to get information on all parts of the mechanism, i.e. its end-effector as well as its legs. Metrological redundancy is therefore maximized to improve the calibration efficiency. The approach is implemented for the calibration of the I4 parallel mechanism [1], with the use of the Jacobian matrix. No accurate camera location is needed so that the experimental procedure is easy to achieve. The calibration algorithm is detailed and experimentally demonstrated more efficient than other calibration methods based on legs observation or end-effector observation.

I. INTRODUCTION

Repeatability of parallel mechanisms is considered to be much higher than the repeatability of serial mechanisms [START_REF] Merlet | Parallel Robots[END_REF]. Their accuracy is however limited by the numerous links and passive joints [START_REF] Wang | On the accuracy of a Stewart platform -Part I : The effect of manufacturing tolerances[END_REF]. A kinematic calibration is thus needed. A calibration procedure is based on the minimization of an error function which itself depends on a set of geometrical parameters defining the mechanism. The error function is constituted by gathering redundant information on the mechanism.

Information redundancy may be obtained by measuring the end-effector pose with an exteroceptive sensor. An error function based on the inverse kinematic model [START_REF] Zhuang | Calibration of Stewart platforms and other parallel manipulators by minimizing inverse kinematic residuals[END_REF] is often considered as the most numerically efficient method [START_REF] Besnard | Identifiable parameters for parallel robots kinematic calibration[END_REF], since an inverse kinematic model can be written in closed-form for most parallel structures, contrary to the direct kinematic model. The main limitation is the necessary end-effector full-pose measurement (position and orientation of the endeffector). Only a few measuring devices have been therefore used to conduct parallel structures calibration [START_REF] Zhuang | Calibration of Stewart platforms and other parallel manipulators by minimizing inverse kinematic residuals[END_REF], [START_REF] Vischer | Kinematic calibration of the parallel delta robot[END_REF], [START_REF] Geng | An effective kinematic calibration method for Stewart platform[END_REF]. The systems are either very expensive, tedious to use or have a low working volume. Vision may constitute an adequate measuring device [START_REF] Renaud | Vision-based kinematic calibration of a H4 parallel mechanism[END_REF]. In such a case, the pose measurement accuracy is however limited by the measurement volume. The calibration efficiency is therefore restricted by the mechanism 1 P. Renaud was jointly with LaRAMA and LASMEA when doing his PhD on this work, and is now with LIRMM.

workspace. Furthermore, the calibration method sensitivity to the kinematic parameters may become critical when the modelling complexity increases [START_REF] Renaud | Apport de la vision pour l'identification géométrique de mécanismes parallèles[END_REF].

Besides, the mechanism kinematics are closely linked to the displacements of its legs, i.e. the kinematic chains connecting the end-effector to the base. For a Stewart-Gough platform, the Jacobian matrix relating the end-effector velocities to the actuator velocities is for instance defined by the Plücker coordinates of the legs [START_REF] Merlet | Parallel manipulators. part 2 -Singular configurations and Grassmann geometry[END_REF]. Information redundancy derived from some measurements on the legs seems therefore an efficient way to achieve the calibration. It may be obtained by installing redundant proprioceptive sensors on some passive joints of the legs [START_REF] Wampler | Calibration of robots having kinematic closed loops using non-linear least-squares estimation[END_REF], [START_REF] Zhuang | Self-calibration of parallel mechanisms with a case study on Stewart platforms[END_REF], [START_REF] Daney | Etalonnage géométrique des robots parallèles[END_REF], [START_REF] Khalil | Autonomous calibration of parallel robots[END_REF], as well as kinematic constraints on these elements [START_REF] Khalil | Self calibration of Stewart-Gough parallel robots without extra sensors[END_REF], [START_REF] Daney | Self calibration of Gough platform using leg mobility constraints[END_REF]. Kinematic constraints on the legs however significantly reduces the mechanism workspace, hence the calibration parameter sensitivity. Moreover, the use of additional proprioceptive sensors has to be planned since the mechanism design, and is therefore not applicable to any mechanism.

The legs observation with a camera enables one to get their pose [START_REF] Renaud | On vision-based kinematic calibration of a stewart-gough platform[END_REF], and then to achieve the mechanism kinematic calibration. A calibration method based on the legs observation has been proposed for different classes of parallel mechanisms [START_REF] Renaud | Apport de la vision pour l'identification géométrique de mécanismes parallèles[END_REF]. In that method, only the legs observation is considered during the calibration. For many mechanisms, the simultaneous observation of the mechanism legs and end-effector can however be achieved. As information redundancy is the basis for the error function design, the calibration efficiency should be logically improved by a simultaneous observation of all the mechanism elements. In this paper, we propose to use vision as a measuring device to get information on the endeffector as well as the legs of an I4 parallel mechanism. The information redundancy, here metrological redundancy, is then maximized while keeping the experimental procedure simple. In particular, the mechanism Jacobian matrix can be used to design the calibration error function.

In the following section the I4 parallel mechanism is described with its modelling. The calibration method is then derived, introducing first the computation of the Jacobian matrix from the image. Experimental results are presented in the fourth section, with comparison to other vision-based kinematic calibration methods, before concluding on the approach and its further developments.

II. MECHANISM MODELING

A. Mechanism structure

The I4 mechanism [START_REF] Krut | I4: A new parallel mechanism for Scara motions[END_REF] is a four degree-of-freedom mechanism actuated by four linear actuators fixed on the base (figure 1(a)). Four legs link the actuators to a traveling plate which itself supports the end-effector. The end-effector can be translated in three directions, and rotated by the relative displacement of the two plate parts (figure 1(b)), using two rack-and-pinion systems. The workspace volume is approximately equal to 500×400×400 mm 3 with a 360 • end-effector rotation.

B. Modeling assumptions

The articulated parallelograms that constitute the mechanism legs are modeled by kinematically equivalent single rods of same length linked in A i , i ∈ [START_REF] Krut | I4: A new parallel mechanism for Scara motions[END_REF][START_REF] Zhuang | Calibration of Stewart platforms and other parallel manipulators by minimizing inverse kinematic residuals[END_REF] and B i , i ∈ [START_REF] Krut | I4: A new parallel mechanism for Scara motions[END_REF][START_REF] Zhuang | Calibration of Stewart platforms and other parallel manipulators by minimizing inverse kinematic residuals[END_REF] (figure 2(a)).

Because of the manufacturing tolerances, some assumptions achieved during the design of the mechanism are considered valid for the calibration. The efficiency of the design models has moreover been noticed for other parallel mechanism [START_REF] Renaud | Vision-based kinematic calibration of a H4 parallel mechanism[END_REF]. The actuator axes are considered parallel and coplanar, as well as the point A i , i ∈ [START_REF] Krut | I4: A new parallel mechanism for Scara motions[END_REF][START_REF] Zhuang | Calibration of Stewart platforms and other parallel manipulators by minimizing inverse kinematic residuals[END_REF], and located on two lines (figure 2(a)). The four single rods are considered identical with a length L.

C. Parameterization

The base frame R B (O, x B , y B , z B ) is defined using as frame origin the joint center A 1 position when the corresponding encoder value is equal to zero (i.e. O = A 1 | q1=0 ). The four points A i , i ∈ [START_REF] Krut | I4: A new parallel mechanism for Scara motions[END_REF][START_REF] Zhuang | Calibration of Stewart platforms and other parallel manipulators by minimizing inverse kinematic residuals[END_REF] are located in the (x B , y B ) plane and line A 1 A 2 is parallel to the x B axis.

The end-effector frame R E (E, x E , y E , z E ) is defined with its origin located at the intersection between the revolute joint

B 3 2H B 1 A 4 A 2 A 3 B 2 B 4 L y B A 1
x B q 2 + q 02 q 1 q 3 + q 0 3 O q 4 + q 0 4 (a) Mechanism Finally, four parameters are needed to define the joints on the base: the distance 2H between the two actuator axes and three encoder offsets q 0 i , i ∈ [2, 4] so that:

(OA i ) .x B = q i + q 0i (1)
The relative position of the joint centers B i , i ∈ [START_REF] Krut | I4: A new parallel mechanism for Scara motions[END_REF][START_REF] Zhuang | Calibration of Stewart platforms and other parallel manipulators by minimizing inverse kinematic residuals[END_REF] is defined by the dimensions S and D. Seven parameters finally define the mechanism geometry: (H, q 02 , q 03 , q 04 , S, D, L).

III. CALIBRATION METHOD

A. Two computations of the Jacobian matrix 1) From the kinematic parameters: With the modeling described in the previous section, the Jacobian matrix of the inverse kinematic model with respect to the pose is particularly simple:

J X = ⎛ ⎜ ⎜ ⎜ ⎝ 1 A1B1.yB A1B1.xB A1B1.zB A1B1.xB 1 1 -A1B1.yB A1B1.xB -A1B1.zB A1B1.xB -1 1 A3B3.yB A3B3.xB A3B3.zB A3B3.xB -1 1 -A3B3.yB A3B3.xB -A3B3.zB A3B3.xB -1 ⎞ ⎟ ⎟ ⎟ ⎠ (2)
with q = ( q1 q2 q3 q4 ) , Ẋ = Ẋ Ẏ Ż Ṫ and

q = J X Ẋ (3)
The lengths A i B i , i ∈ [START_REF] Krut | I4: A new parallel mechanism for Scara motions[END_REF][START_REF] Zhuang | Calibration of Stewart platforms and other parallel manipulators by minimizing inverse kinematic residuals[END_REF] of the legs are equal. The Jacobian matrix can therefore also be expressed by:

J X = ⎛ ⎜ ⎜ ⎝
-u3.zB u3.xB -1 ⎞ ⎟ ⎟ ⎠ (4)
with u i the leg unit axis vector such that A i B i = Lu i , which can be expressed as a function of the joint values and the kinematic parameters: ;

u 3 = X-S+T-q3-q0 3 Y-(H-D) Z
where (X, Y, Z, T) are given from the direct kinematic model [START_REF] Krut | I4: A new parallel mechanism for Scara motions[END_REF]:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ X= q 1 +q 2 +q 3 +q 4 +q 0 2 +q 03 +q 04 4 Y= (q 4 -q 3 -2S+q 0 4 -q 0 3 ) 2 -(q 2 -q 1 -2S+q 0 2 ) 2 16(H-D) Z=- q L 2 -(H-D) 2 -Y 2 - (q 4 -q 3 -2S+q 0 4 -q 0 3 ) 2 -(q 2 -q 1 -2S+q 02 ) 2 8
T= q 1 +q 2 -q 3 -q 4 +q 02 -q 0 3 -q 0 4 4

(5) 2) From the image: Let us consider that a camera is observing the legs of the mechanism composed of cylindrical elements, and simultaneously the end-effector equipped with a calibration board. The image formation is supposed to be modeled by the pinhole model [START_REF] Faugeras | Three-dimensional Computer Vision : A Geometric Viewpoint[END_REF]. The camera is first calibrated using the calibration board [START_REF] Lavest | Comment calibrer des objectifs très courte focale[END_REF] so that optical distortions can be compensated for. It must be noticed that the calibration board geometry is identified simultaneously with the camera. No accurate calibration board is therefore needed, which lowers significantly the measuring system cost and simplify the experimental procedure.

Vision enables us then to compute: a) From the cylinder observation -A cylinder image is composed of two lines (figure 3), generally intersecting except if the cylinder axis is going through the camera center of projection C. Each cylinder generating line D j , j ∈ [START_REF] Krut | I4: A new parallel mechanism for Scara motions[END_REF][START_REF] Merlet | Parallel Robots[END_REF] corresponding to one line d j in the image can be defined by its Plücker coordinates [START_REF] Pottmann | Approximation in line space -applications in robot kinematics and surface reconstruction[END_REF] (u, h j ) with u the cylinder unit axis direction vector and h j defined by:

h j = u × CP j , ∀j ∈ [1, 2] (6)
where P j is an arbitrary point of D j and × represents the vector cross product. One can easily show that h j = h j / h j contains the coefficients of the equation of d j . The vectors h 1 , h 2 can hence be extracted from the leg image. A cylinder axis direction u can therefore be computed in the camera frame R C (C, x C , y C , z C ) from ( 6) by: the frame linked to the board [START_REF] Lavest | Comment calibrer des objectifs très courte focale[END_REF]. The pose of the calibration board can therefore be estimated in the camera frame from its image. It can be noticed that first and fourth matrix columns of the Jacobian matrix (eq. 2) are constant, which expresses that the end-effector displacement along x B and rotation around z B are not dependent on the mechanism geometry. It is then possible to determine the axes x B and z B from the calibration board displacement measurement, and from these two vectors to determine the base frame axes in the camera frame. c) Jacobian matrix estimation -Finally all the elements of the Jacobian matrix can be determined from vision in the camera frame. Due to the scalar product invariance with an Euclidean transformation, an estimation JX of the Jacobian matrix can then be computed from the image.

u = h 1 × h 2 h 1 × h 2 (7

B. Calibration method

In order to be able to compute the Jacobian matrix from the image, the calibration method is composed of two steps:

1) Base frame axes determination in the camera frame: In a first step, two sequential movements of the end-effector enable us to compute the base frame unit vectors: a) Rotation of the end-effector -The end-effector rotation axis is constant and equal to z B . Using at least two calibration board positions with different rotations hence enables us to determine the z B vector (Figure 4). b) End-effector translation -Imposing the same displacement to the four linear actuators enables us to move the endeffector in the x B direction (figure 5), whatever the kinematic parameters values, as mentioned in the previous section.

From the knowledge of the x B and z B unit vectors, the vector y B can also be computed. It is important to notice that the location of the calibration board with respect to the endeffector does not affect the base frame vectors determination. The experimental procedure is therefore easy to achieve, and no external parameters are introduced in the kinematic parameter set, contrary to a calibration method based on the inverse kinematic model [START_REF] Renaud | Vision-based kinematic calibration of a H4 parallel mechanism[END_REF].

2) Kinematic parameters determination: At the end of the first step, all the elements of the Jacobian matrix can be computed in the camera frame for any mechanism pose. The Jacobian matrix can also be computed in the base frame as a function of the kinematic parameters, so that we propose to identify the kinematic parameters by minimizing the error function F (ξ):

F (ξ) = N k=1 J Xk (ξ) -JXk 2 F (8)
with J Xk (ξ) the Jacobian matrix computed as a function of the kinematic parameters (eq. 4) for the k-th pose, JXk the corresponding Jacobian matrix computed from the image, N the number of poses considered during the calibration and . F the Frobenius matrix norm. Analytical derivation of the error function shows that a four parameter set (L, H -D, E-q 0 2 /2, E-(q 0 4q 0 2 )/2) only is identifiable. From the direct kinematic model, one can show that the mechanism accuracy may be then obtained in the y B and z B directions, and in the x B direction and in rotation with a constant error, which is not a major drawback during the use of the mechanism.

IV. EXPERIMENTAL RESULTS

In this section, the calibration method efficiency is evaluated experimentally by performing validation tests and by comparison with other vision-based calibration methods, based either on the end-effector observation or the legs observation. These methods are first presented before giving the experimental results.

A. Other calibration methods 1) End-effector observation:

The calibration board can be used to achieve the kinematic calibration using an error function based on the inverse kinematic model, as detailed in [START_REF] Renaud | Vision-based kinematic calibration of a H4 parallel mechanism[END_REF]. The identifiable parameters are then identical to those identified with the proposed method.

2) Legs observation: The calibration method based on the legs observation introduced in [START_REF] Renaud | Apport de la vision pour l'identification géométrique de mécanismes parallèles[END_REF] is here applied. The parameters are then identified in four steps:

Step 1 The end-effector is displaced while locking sequentially the actuators. For each locked actuator, the position of the corresponding spherical joints that compose the articulated parallelograms can therefore be determined in the camera frame R C (figure 6). From each couple of spherical joints, one can then compute the joint center A i , i ∈ [START_REF] Krut | I4: A new parallel mechanism for Scara motions[END_REF][START_REF] Zhuang | Calibration of Stewart platforms and other parallel manipulators by minimizing inverse kinematic residuals[END_REF] in the camera frame and consequently determine the actuator axis x B in the camera frame. The position of the points A i , i ∈ [START_REF] Krut | I4: A new parallel mechanism for Scara motions[END_REF][START_REF] Zhuang | Calibration of Stewart platforms and other parallel manipulators by minimizing inverse kinematic residuals[END_REF] for the actuator encoder zero value can hence be also estimated.

Step 2 The joint centers A i , i ∈ [START_REF] Krut | I4: A new parallel mechanism for Scara motions[END_REF][START_REF] Zhuang | Calibration of Stewart platforms and other parallel manipulators by minimizing inverse kinematic residuals[END_REF] are defined in the base frame by four parameters: (q 0 2 , q 0 3 , q 0 4 , H). Due to the scalar product invariance with Euclidean transformation, these parameters can be identified by minimizing the error function F :

F (q 02 , q 03 , q 04 , H) = (p,q) [V p .V q | RC -V p .V q | RB ] 2 (9)
with the vector set

V = (OA 2 | q2=0 , OA 3 | q3=0 , OA 4 | q4=0 ) and ⎧ ⎨ ⎩ OA 2 | q2=0 = q 0 2 x B OA 3 | q3=0 = q 03 x B + 2Hy B OA 4 | q4=0 = q 04 x B + 2Hy B (10)
Six independent equations can be expressed from the elements of V, which ensures the identifiability of the four kinematic parameters.

Step 3 The distances B 1 B 2 and B 3 B 4 are invariant with an end-effector pose modification. Therefore the error function F 2 (L) allows us to determine the length L:

F 2 (L) = N -1 k=1 B 1,k+1 B 2,k+1 2
RC -B 1,k B 2,k 2 RC 2 + B 3,k+1 B 4,k+1 2
RC -B 3,k B 4,k 2 RC 2 (11)
with N the number of poses used for the calibration. Two independent information can be obtained, so that the identifiability of the parameter L is ensured.

Step 4 The distances B 1 B 2 and B 3 B 4 are determined from the previous step and equal to the parameter S, which is therefore immediately estimated. To determine the other dimension D, the relative displacement of the two traveling plate elements has to be taken into account. The distance between B 1 (resp. B 3 ) and B 2 (resp. B 4 ) along y B is constant and known in the camera frame. The corresponding distance 2D is hence also immediately identified.

B. Set-up

The camera is located approximately in the plane z B = 0 at the workspace center in the y B and z B directions (figure 7), to observe simultaneously the four mechanism legs and the end-effector (figure 8).

The camera has a resolution of 1024 × 768pixels, 8-bit encoded, with a 3.6mm lens.

C. Calibration results

For the proposed method, a 20-pose set is used to determine the base frame unit axes. Another 20-pose set is then used to identify the four kinematic parameters. This latter is also used for the calibration using the end-effector observation. Eventually, for the calibration using only the legs observation, five poses are used to identify each joint center in the first step, and the same 20-pose set used for the other two calibration methods is selected to achieve the second step of the calibration.

The identified parameters with the three calibration methods are indicated in table I. One may notice that the kinematic parameter variation with respect to the a priori values and between the different calibration methods are significant: several millimeters for some parameters. In the next section, the influence of such parameter value differences is evaluated through validation tests.

D. Validation tests

1) End-effector displacement analysis: Observation of the calibration board enables us to determine its pose with respect to the camera. The end-effector displacement between two poses can therefore be estimated from these pose measurements. On the other hand, the end-effector displacement can be estimated using the direct kinematic model, the proprioceptive sensor values recorded during the experiment and a kinematic parameter set, either identified or estimated a priori. In table II, the mean and root mean squares of the gaps between the vision-based displacement measurements and their estimation are indicated for the parameters identified with the proposed method and the other two algorithms. The gaps are computed for the 20-pose set used for the calibration. The three calibration methods enable us to lower the gaps between the measured end-effector displacements and their estimation. The most efficient method is however the one proposed in this paper which is based on the use of the information obtained from the end-effector and the legs in the same time. The gap reduction is significant with a RMS reduction by a factor 4.

2) Kinematic constraint: An independent validation test has been also achieved by imposing a kinematic constraint on the end-effector (Figure 9). The end-effector is manually constrained to follow a line materialized by a ruler. The corresponding joint variables are stored. From these joint variable values, poses are computed by means of the direct kinematic model. The physical set-up implies that these poses should lie on a straight line. Hence, the straightness of the line is computed for the kinematic parameter sets (a priori and identified) as the root mean square of the distance between the different positions and the line estimated by a least squares criterion.

A 500mm line is measured. The straightness measurement repeatability, experimentally evaluated equal to 0.07mm, is in the order of the improvement induced by the calibration (table III). The accuracy improvement can hence not be concluded using this second evaluation criterion. This experiment demonstrates the difficulties that might be encountered while using a kinematic constraint to achieve the calibration of a mechanism with large workspace.

V. CONCLUSION

In this paper, a novel calibration approach has been proposed for parallel mechanisms, based on the use of vision. The simultaneous observation of the mechanism legs and the endeffector enables us to maximize the metrological redundancy, which is intrinsically linked to the method efficiency. Presented for the I4 calibration, the method seems experimentally efficient, with better results than other vision-based calibration methods. The experimental procedure is easy to achieve, and may be performed online.

The presented calibration method has been specifically developed for the I4 mechanism. Further developments will be soon achieved to apply this method to other mechanisms. Vision-based measurements on other leg geometries should be possible, and since the mechanism Jacobian matrix can frequently be derived, the generalization should be achievable. The parameter identifiability with such an approach will need however to be studied thoroughly. Optimization of the mechanism poses for the calibration will also be conducted to improve the procedure efficiency.

Fig. 1 .

 1 Fig. 1. The I4 mechanism -General view and traveling plate.

Fig. 2 .

 2 Fig. 2. I4 mechanism geometry.

2 Fig. 3 .

 23 Fig. 3. Cylindrical element observation with a camera.

Fig. 4 .Fig. 5 .

 45 Fig. 4. Determination of the z B unit vector in the camera frame.

Fig. 6 .

 6 Fig. 6. Identification of the joint centers in the camera frame.

Fig. 7 .

 7 Fig. 7. Experimental set-up.

Fig. 8 .

 8 Fig. 8. Legs and calibration board image with the camera.

Fig. 9 .

 9 Fig. 9. Experimental set-up for the straightness evaluation.

TABLE I A

 I priori AND IDENTIFIED KINEMATIC PARAMETERS

	Parameter a priori	L 1.001m	H -D 0.3464m	E --0.6316m q 0 2 2	E --0.6304m q 0 4 -q 0 3 2
		Legs and end-effector observation
	Identified	0.9949m	0.3297m	-0.6143m	-0.6145m
			Legs observation
	Identified	1.0213m	0.3383m	-0.62395m	-0.6240m
		End-effector observation
	Identified	1.0068m	0.3475m	-0.6338m	-0.6329m

TABLE II GAPS

 II BETWEEN VISION-BASED END-EFFECTOR DISPLACEMENT MEASUREMENTS AND THEIR ESTIMATION COMPUTED FROM A KINEMATIC PARAMETER SET.

	Parameter set	Mean	Root mean square
	a priori	-0.06mm	0.53mm
	Identified with the proposed method	-2E-4mm	0.11 mm
	Identified from the end-effector observation	3.4E-2mm	0.46 mm
	Identified from the legs observation	-3.4E-3mm	0.33 mm

TABLE III STRAIGHTNESS

 III EVALUATION WITH THE a priori AND IDENTIFIED KINEMATIC PARAMETER SETS.

	Parameter set	Straightness
	a priori	0.53mm
	Identified with the proposed method	0.53mm
	Identified from the end-effector observation	0.52mm
	Identified from the legs observation	0.50mm

ACKNOWLEDGMENT

The authors would like to warmly thank S. Krut for his help during the experiments. This study was jointly funded by CPER Auvergne 2001-2003 program and by the MAX project within the CNRS-ROBEA program.


Constitutive Relation Error for FFT-based methods 4 Ω

Constitutive Relation Error for FFT-based methods 5

June 9th 2016

Constitutive Relation Error for FFT-based methods 8

Construction of kinematically admissible displacement

The main topic of this talk Application to UGPLS solvers

( C-C 0 ) -1 : τ +Γ 0 [ τ]=E
The unknown is the stress polarization (or the strain)

From stress-polarization to displacement

Approximation that is local in space  Q1 approximation  Post-processing of estimate of stress-polarization (or strain)

True displacement from true stress polarization

( C-C 0 ) -1 : τ +Γ 0 [ τ]=E ∇⋅ ( C 0 : ∇ s u+ τ ) =0
Approx. displacement from approx. stress polarization

( C N -C 0 ) -1 : τ N +Γ 0 N [ τ N ]=E ∇⋅ ( C 0 : ∇ s u N + τ N ) =0 N= ( N 1 ,… ,N d ) : grid-size
June 9th 2016

Constitutive Relation Error for FFT-based methods 12

Wait a minute...

Our two-step approach

( C N -C 0 ) -1 : τ N +Γ 0 N [ τ N ]=E ∇⋅ ( C 0 : ∇ s u N + τ N ) =0 (UGPLS solver) (FEM solver)
An alternative one-step approach: why not use FEM only?

∇⋅[C : (E+∇ s u)]=0
1.

2.

The two-step approach is more efficient Fourier series, discrete Fourier transform

u(x)= 1 N 1 ⋯N d ∑ k ∈ℤ d [ ∏ j=1 d sinc 2 ( π k j N j ) exp ( 2 i π k j x j L j ) ] ûk N
Displacement is periodic! Can be expanded in Fourier series

Discrete Fourier transform of nodal displacements

ûk N = ∑ n 1 =0 N 1 -1 ⋯ ∑ n d =0 N d -1 exp [ -2 i π ( k 1 n 1 N 1 +⋯+ k d n d N d ) ] u n N
Infinite sum defined by a finite (periodic) series of unknowns (in Fourier space).

June 9th 2016

Constitutive Relation Error for FFT-based methods 16

Total Potential Energy

Can be computed in Fourier space Hashin's coated sphere

Π= L 1 ⋯L d N 1 2 ⋯N d 2 ∑ k 1 =0 N 1 -1 ∑ k d =0 N d -1 ( 1 2 ûk N ⋅ Kk N ⋅ ûk N + ûk N ⋅ τk N ⋅ Bk N )
κ 1 =100 κ 2 κ 3 =1.25191 κ 2 ν 1 =ν 2 =ν 3 =0.25 E= E v
3 ( e 1 ⊗e 1 +e 2 ⊗e 2 +e 3 ⊗e 3 ) Reconstruction of stress

r 1 =0.2 L r 2 =0.
∫ Ω [( C 0 : ∇ s u N + τ N ) : ∇ s v ] dV =0 for all v Q1, periodic
Reconstructed displacement is a Galerkin solution

Conclusion and perspectives

An "agnostic" method to reconstruct displacements…

  0 : ∇ s u N + τ N ) =0 Q1 approximation of displacement N= ( N 1 ,… , N d ) : grid-size L= ( L 1 , … , L d ) : size of unit-

N 1

 1 ×⋯×N d small (d ×d ) linear systems rather than one single large linear system with N 1 ×⋯×N d ×d unknowns!

4 L

 4 Example borrowed from Schneider et al. (2015) Schneider et al. (2015), Int. J. Num. Meth. Eng. 105(9), 1419-1436Error on reconstructed displacementTruncated discrete Green operator(Moulinec & Suquet, 1998)

Contents A non-specialist's presentation of constitutive relation error CRE & Uniform Grid, Periodic Lippmann -Schwinger solvers A local reconstruction of the displacement The nitty-gritty details Example 1 Example 2

	June 9th 2016 June 9th 2016	Constitutive Relation Error for FFT-based methods Constitutive Relation Error for FFT-based methods	10 11

 Strain is not necessarily compatible  Kinematically admissible displacement?  Statically admissible stress?

Construction of statically admissible stress

 Arguably the most difficult part  Don't want to reinvent the wheel: use the EET method!

simple, 2D example

	μ i =100μ m
	ν m =0.3
	ν i =0.2

 "Strong extension condition" does not lead to overdetermination  Equilibrated Element Traction method applies  Reconstructing statically admissible stress fields becomes standard!  Constitutive relation error can be evaluated Ladevèze & Pelle (2005), Mastering Calculations in Linear and Nonlinear Mechanics, Springer

A

E=E 12 ( e 1 ⊗e 2 +e 2 ⊗e 1 ) Constitutive relation error Work in progress

  (with L. Chamoin)

	June 9th 2016	Constitutive Relation Error for FFT-based methods	26

paves the way to reconstruction of stresses

 Works with any UGPLS solver  Very low cost (9 FFT in 3D)  Reconstruction is local in space  Accuracy comparable to other existing approaches … Which  Standard post-processing of the reconstructed displacement  Rigorous bound on energy error Perspectives  Extension to local quantities of interest  Systematic selection of reference material

Moulinec & Suquet (1998), Comp. Meth. Appl. Mech. Eng. 157(1-2), 69-94 Willot & Pellegrini (2008), Continuum Models & Discrete Systems XI, 443-450 Zeman et al. (2010), J. Comp. Phys. 229(21), 8065-8071 Monchiet & Bonnet (2012), Int. J. Num. Meth. Eng. 89(11), 1419-1436 Kabel et al. (2014), Comp. Mech. 54(6), 1497-1514 Vondřejc et al. (2014), Comp. & Math. Appl. 68(3), 156-173 Willot (2015), Comptes Rendus Mécanique 343(3), 232-245 Makrodimopoulos & Martin (2007), Int. J. Num. Anal. Meth. Geomechanics 31(6), 835-865

Summary of the method

SB (2016), Int. J. Num. Meth. Eng., early view (10.1002