Cristina Badarau
email: florinacristina.badarau@u-bordeaux4.fr

Florence Huart
email: florence.huart@univ-lille1.fr

Ibrahima Sangaré
email: ibrahima.sangare@u-bordeaux4.fr

"Indebtedness and macroeconomic imbalances in a monetary-union DSGE"

Keywords: macroeconomic divergences, euro area, DSGE, risk premium, pro-cyclical fiscal policy, spillover effects JEL Classification: E63, F41

We build a two-country open-economy monetary union DSGE model in order to explain some macroeconomic imbalances in the euro area. We focus on the role of cyclical behaviour of public spending and sovereign risk premium. Pro-cyclical primary public expenditures in one country do not lead to higher interest rates on domestic public bonds in the short term as long as output growth helps financing public debt. Spillover effects on the other country can be positive on output as long as a real effective depreciation of the common currency leads to higher exports to the rest of the world.

Introduction

The euro area (EA) sovereign debt crisis in 2010-2012 highlights the issue of macroeconomic imbalances across member countries in particular with regard to divergent inflation rates, GDP growth rates, ratios of public deficits and debts to GDP, and current account balances. These imbalances emerged before the international financial crisis of 2008-2009, but the latter has exacerbated them. As a matter of fact, in a monetary union with common short-term nominal interest rates but disparate national inflation rates (due to cyclical or structural country-specific causes), short-term real interest rates are different across countries. In countries where the inflation rate was above the euro area average before the financial crisis, real interest rates were very low (Spain, Portugal, Greece) and even negative for short maturities (Ireland). On the contrary, in some countries with lower inflation rates, such as Germany, real interest rates were higher. Then, indebtedness of the private sector rose and boosted aggregate demand in the former group of countries (henceforth the G4) while domestic consumption and investment rose more slowly in Germany.

Over time, the G4 countries were bound to lose price competitiveness while Germany was likely to gain price competitiveness. Indeed, there have been rising current account deficits in those countries and current account surpluses in Germany (as well as in the Netherlands, Finland, Austria and Belgium). As a consequence, real depreciation was needed in countries with high current account deficits and a real appreciation was needed in countries with current account surpluses. As regards intra-EA real exchange rates, prices and wages had to grow slower in high-deficit countries and higher in countries with external surpluses. However, since 2008, the extent of adjustment has mostly relied on the high-deficit countries. As a consequence, the deflationary policies have been criticized for being harmful for Southern populations (in Greece, Spain or Portugal). As regards extra-EA real exchange rates, there has been an effective nominal depreciation of the euro over the years 2010-2012. However, the extent of the depreciation might not be sufficient for some countries.

Macroeconomic divergences across EA countries also stem from fiscal imbalances. Before the crisis, the public sector could borrow heavily as did the private sector in countries where interest rates have been lower since the entry into the euro area (given very low yield spreads on public bonds issued by EA governments relative to German bonds). Public saving decreased (in Greece and Portugal) and public deficits and debts have not been lowered as much as what was set in the fiscal rules of the Stability and Growth Pact. Moreover, public spending has been pro-cyclical in the EA, except in Finland [START_REF] Fatás | The euro and fiscal policy[END_REF]. In some countries (Italy, France), the ratio of public debt to GDP has not risen much despite some persistent public deficits, because they benefited from a context of low interest rates (and interest payments). In most other EA countries, the ratio of public debt to GDP was decreasing over the 1999-2007 period. Hence, the current issue of public indebtedness has been caused by the financial crisis. The bail-out of the banking sector led to a sharp increase in the GDPpublic debt ratio (especially in Ireland) and the recession led to an increase in public expenditure and a decrease in tax revenues more or less in EA countries depending on the size of automatic stabilizers and the extent of discretionary measures. Then, interest expenditure grew faster in countries where the sovereign risk premium had increased much (the G4 countries, and Italy).

A higher sovereign default risk premium could arise because of a high level of public debt, fast rising public indebtedness, bad future prospects of economic growth or a higher probability of rescuing financially weak banks (see [START_REF] Mody | The Eurozone Crisis: How Banks and Sovereigns Came to be Joined at the Hip[END_REF]. Contagion effects could affect the long-term interest rates in other EA countries. From 2009, long-term interest rates of EA countries had comoved, but in late 2011, as the economic hardship of the Southern EA countries was aggravating, the public sectors of some countries suffered from rising borrowing costs (Austria, France, and Belgium) while in some other countries, they benefitted from decreasing borrowing costs (in particular, Finland, Germany, and the Netherlands). The sign of these spillover effects on EA countries mostly depended on their specific economic performance and financial position. The latter was at risk where domestic banks had much lent to the public and banking sectors of the G4 countries.

The EA crisis shows how financial difficulties of both public sector and banking sector are closely intertwined not only at the national level but also at the area-wide level. On the one hand, the banking sector rescue packages by governments could lead to higher sovereign bond risk premia (given the impact on implicit government liabilities) and consequently to greater difficulty repaying the debt. On the other hand, since banks hold sovereign bonds issued by EA governments, higher sovereign risk premia make their financial position weaker. As a result, banks decrease their lending activity or they increase interest rates on loans to the non-financial private sector (as in Greece and Portugal). Such behaviour impairs the transmission of the key interest rates of the European Central Bank (ECB) to interest rates on loans to households and firms. It aggravates divergences in the evolution of aggregate spending across EA countries. This negative spillover effect of higher sovereign risk premia on the borrowing costs of the non-financial private sector has lately been studied as a "sovereign risk channel" in a dynamic stochastic general equilibrium (DSGE) model of a closed economy [START_REF] Corsetti | Sovereign Risk, Fiscal Policy, and Macroeconomic Stability[END_REF] or in a two-country monetary union DSGE model [START_REF] Roeger | Expected Sovereign Defaults and Fiscal Consolidation[END_REF].

In this article, we want to explain macroeconomic imbalances among EA countries by taking into consideration various real and financial interdependencies between countries. In particular, we are interested in the role of sovereign risk premia and the cyclical behaviour of fiscal policy. We study their effects on the borrowing costs of other EA governments as well as of the private sector. Thus, we add the foreign public sector in the "sovereign risk channel" and show how the spillover effects can be positive, that is the public sector in other EA countries (such as Germany) may benefit from lower borrowing costs. To do so, we build a two-country monetary union DSGE model which combines the following elements: i) monetary union (MU) is open to the rest of the world (RoW); ii) there is public indebtedness (with a constraint on debt accumulation) and external debt (the net foreign asset position of the nation); iii) there are deviations from the law of one price between both countries within MU and between each country and the RoW; iv) there is a financial accelerator mechanism; v) banks lend to domestic firms and to each government of both countries; and vi) there are three endogenous risk premia, namely a firm risk premium depending on the net worth-capital value ratio, a government risk premium depending on the public debt/GDP ratio, and a nation-wide risk premium depending on the net foreign asset (NFA) position.

In the literature on sovereign risk premium in DSGE models, the premium is determined in a closedeconomy (or closed-monetary union) framework. Since we consider an open-monetary union framework and consider a financial accelerator mechanism, we use a simple linear relationship between the sovereign risk premium and the gap between the level of public debt and a debt target. In [START_REF] Christoffel | Government Bond Risk Premia and the Cyclicality of Fiscal Policy[END_REF], the risk premium is not linked to default but to a potential capital loss in case of selling the public bond before maturity. They show that pro-cyclical public expenditure leads to a higher bond premium. As pro-cyclical fiscal spending increases output and inflation variability, the central bank may react more aggressively to counteract higher inflation variability. This leads to higher interest rates and hence a higher bond risk premium. Their result is derived from a closed-economy DSGE model (estimated using U.S. data). In our model, pro-cyclical fiscal policy does not lead to a higher risk premium in the short term, because public debt increases less than in the case of a counter-cyclical fiscal policy. This is due to faster output growth and larger capital income tax receipts. Openness of both countries within the MU and with the RoW helps lowering the risk premium via the role of exports in output growth. In addition, the common central bank in the MU does not raise the common interest rate as much as in a closed-economy set-up, because the interest rate rule depends on average performance of member countries.

In other respects, in [START_REF] Bi | Sovereign Default Risk Premia, Fiscal Limits, and Fiscal Policy[END_REF], there is a non-linear relationship between the sovereign risk premium and the level of debt: the risk premium rises if the government approaches the "fiscal limit", that is the maximum level of debt that the government is able or willing to service. In [START_REF] Corsetti | Sovereign Risk, Fiscal Policy, and Macroeconomic Stability[END_REF], a higher sovereign risk premium spills over to the borrowing cost of the private sector and amplifies macroeconomic fluctuations when the central bank cannot cut interest rates further (under a zero lower bound constraint). Furthermore, [START_REF] Roeger | Expected Sovereign Defaults and Fiscal Consolidation[END_REF] analyze the sovereign risk channel in a euro area DSGE model, where banks are exposed to the risk of a decline in bond prices. They compare the contractionary effects of fiscal consolidation with the effects of no-consolidation in a context of rising public debt. In the latter case, higher sovereign risk premia spillover into higher costs of borrowing for households. In their model, banks lend to households and invest in domestic government bonds and bonds issued by foreign banks, but they do not lend to the corporate sector, while in ours, banks lend to the domestic corporate sector, domestic government and foreign government in the rest of the union (RoU) but not to the households. The latter hold bonds issued by the RoW.

In our model, there are several channels by which macroeconomic performance of MU countries is interdependent and may diverge. The trade channel works through both intra-MU and extra-MU change in net exports and bilateral real exchange rates. The credit channel concerns the transmission of the common monetary policy to domestic production and inflation in both MU countries via the banking sector's balance sheet. The share of consumers' deposits that banks lend to borrowers depends on economic activity. As for the fiscal channel, it denotes the fiscal causes of macroeconomic divergences across MU countries. We also compare the implications of a countercyclical vs. pro-cyclical reaction of public primary expenditure to deviations of output from its steady-state level. Finally, there is also a sovereign risk channel: given banks' holdings of public bonds issued by governments of both MU countries, higher public indebtedness has an impact on the bank lending to firms.

We find that pro-cyclical spending in one country makes it harder to stabilize debt and accentuates negative spillover effects on aggregate spending (absorption) in the other country. A higher risk premium in one country has two opposite effects on the other country: borrowing costs are lower for the public sector but higher for the non-financial private sector. In all cases, the other country can benefit from positive spillover effects on output as long as the effective real depreciation of the common currency leads to higher net exports to the RoW.

In what follows, we set the DSGE model of a two-country open-economy monetary union (section 2). We then describe the calibration and explain the results of our simulations (section 3). We finally conclude (section 4).

A two-country open-economy monetary union DSGE model with risk premia

We consider two symmetric countries of equal size, home and foreign . They are members of a monetary union and are open to the rest of the world ( ) which is fully exogenous (in the small open economy perspective). The model contains price stickiness, monopolistic competition in final goods market, capital adjustment costs, incomplete pass-through of exchange rate via law of one price deviation, financial market frictions and fiscal policy instruments.

Each economy is populated by households, banks, government and three types of producers: entrepreneurs, capital producers, and retailers (domestic and imported goods retailers). There is a common monetary authority that sets the unique nominal risk-free interest rate for both countries.

Capital producers build new capital and sell it to the entrepreneurs. Entrepreneurs produce wholesale goods and sell them to domestic goods retailers. Domestic and imported goods retailers set nominal prices of final goods à la [START_REF] Calvo | Staggered Prices in a Utility-Maximizing Framework[END_REF]. Banks convert households' deposits in loans to finance the government deficit and the entrepreneurial purchase of capital. Each government decides upon fiscal policy.

Our model shares some common features with those of other DSGE models of the euro area. Indeed, there are both real rigidities (habit formation in consumption and adjustment costs in investment) and nominal rigidities (Calvo probability of not being able to reset prices) as in [START_REF] Smets | An Estimated Stochastic Dynamic General Equilibrium Model of the Euro Area[END_REF], [START_REF] Coenen | Tax Reform and Labour-Market Performance in the Euro Area. A Simulation-based Analysis using the New Area-Wide Model[END_REF], [START_REF] Erceg | Fiscal Consolidation in a Currency Union: Spending Cuts vs. Tax Hikes[END_REF]. Each government finances public purchases (consumption and transfers) by levying taxes (on consumption and capital income) and issuing debt. We ignore seignoriage and taxes on labour services and wages unlike [START_REF] Coenen | Tax Reform and Labour-Market Performance in the Euro Area. A Simulation-based Analysis using the New Area-Wide Model[END_REF]. Following [START_REF] Christoffel | Government Bond Risk Premia and the Cyclicality of Fiscal Policy[END_REF], we assume that each government adjusts the fiscal instrument (public consumption in our model) with reaction to output growth and the level of debt (in deviation with its steady-state level). We add some degree of inertia in the adjustment of fiscal instrument.

This model has also specific features that distinguish it from standard models of monetary union (MU). The exposure of domestic banks to foreign government debt in the rest of the union (RoU) allows us to study the potential contagion effects working through sovereign debts within the monetary union. Moreover, since monetary union is open with the rest of the world (RoW), we can take into account both intra and extra-zone net exports of each country and underline the importance of trade openness in times of regional economic turmoil. Finally, deviations from the law of one price and imperfect exchange-rate pass-through capture the fact that there are country-specific factors of inflation. At least, they cushion the impact of price variations in one country on the other country.

Households

Each country ∈ , is populated by a continuum of unit mass households with infinite life. The representative household of country maximizes the following expected discounted sum of utilities:

1 1 1
where is aggregate consumption and denotes the number of hours worked. is the conditional expectation operator. The parameters 0 1 , 0 , 0 and 0 1 are, respectively, the subjective discount factor, the inverse intertemporal elasticity of substitution, the inverse of the Frisch elasticity of labour supply and the parameter that controls habit persistence.

The household's period-by-period budget constraint is defined by:

1 , , , Ψ , , , Λ 2
where is the consumer price index (CPI), the nominal wage, nominal deposits that pay gross nominal interest rate and , nominal internationally traded bonds, denominated in rest of the world currency, that pay a gross nominal interest rate , Ψ , .

is the nominal exchange rate (expressed in terms of units of home currency per unit of foreign currency). , , and Λ are, respectively, distortionary tax rate on consumption, government transfers and real profits from the monopolistic sector. Finally, Ψ , represents a risk premium that is a function of the economy's real aggregate level of net-foreign asset position in percentage of steady-state output, as follows:

Ψ , , , 3
where ≡ , is real aggregate net-foreign asset position in percentage of steady-state GDP;

0 is a measure of the elasticity of the risk premium with respect to net-foreign asset (NFA) position. Whenever the NFA is negative, the nation is a net borrower. The variable is an exogenous shock on risk premium defined by log log , with , ~ . . 0, . The term Ψ , , is assumed to be strictly decreasing in and satisfies Ψ 0,0 1. It captures imperfect integration in the international financial markets and ensures a well-defined steady-state in the model (Schmitt-Grohé and Uribe, 2003).

Households choose the paths for

, , , , in order to maximize (1) subject to the budget constraint in (2). The following optimality conditions hold:

1 , 4 5 , Ψ , , 0 6 0 7
is the Lagrangian multiplier in (4), ( 5), ( 6) and ( 7).

The final good, , is allocated to consumption, , investment, , and public spending, . It is an aggregate function of goods produced in the home country, , , in the RoU, , , and in the RoW,

, :

1 , , , 8
for , , ; , ∈ , and .

The parameters 1, , and are, respectively, the elasticity of substitution between the three types of goods, the share of imported goods from the RoU and the share of imported goods from the RoW. We suppose that these shares are identical reciprocally between each country ∈ , of the union and the RoW. Therefore, the fraction 1 is the degree of home bias in consumption, investment and public goods.

The price index (CPI) associated to ( 8) is given by:

1 , , , 9
where , , and , are, respectively, the domestic price of home goods, the domestic price of imported goods from the RoU and the domestic price of imported goods from the RoW. We define , ≡ , ,

, as the composite aggregates of differentiated varieties produced domestically, inside and outside the monetary union, respectively, with being the elasticity of substitution between varieties originating in the same country;

, , , and , being a typical variety of domestic goods, imported goods from the RoU and imported goods from RoW, respectively. The corresponding prices are deduced easily and are given by, respectively:

, , , , , , , , , where , (respectively , and ,
) is the price of a typical variety produced in the home country (respectively imported prices from the RoU and the RoW).

The optimal demands for domestic, RoU and RoW goods, are derived from expenditure minimization 4 :

Open-economy relations

This section outlines the key relations that describe the terms of trade, the real exchange rates and the law of one price deviations. For each country , ∈ , and , we define the bilateral terms of trade as:

, , , , , , 13
From ( 9), the terms of trade can be related to the CPI-PPI ratio as follows:

, 1 , ,

14

We assume that the law of one price (LOP) holds for the export sector, but there is incomplete passthrough in the import sector. This assumption is motivated by the existence of monopolistic domestic importers in the union that practice local currency pricing [START_REF] Devereux | The Optimal Choice of Exchange Rate Regime: Price-Setting Rules and Internationalized Production[END_REF]. This behaviour can make the price of the foreign goods in the domestic market temporarily deviate from the producer price level in the country of origin. The wedge between these two prices is called the law of one price gap (LOPG) and is given by, bilaterally:

, , , , , , 15
where , and , are domestic prices in country of the union and in the RoW.

Similarly, we define the bilateral real exchange rates as follows:

, , 16

4 The optimization program is , , , ,

subject to the following constraint:

1 , , ,
.

Finally, we can express the effective terms of trade, the effective law of one price gap and the effective real exchange rate, for each country ∈ , as:

, ,

Assuming that the two countries , are of the same size in the monetary union, the effective real exchange rate for the union is therefore:

, , 20
which can be written also in terms of euro nominal exchange rate:

21
where and are CPI of the monetary union and the rest of the world. 5

Production sector

Entrepreneurs

The entrepreneurs play an important role here because their presence allows us to introduce the financial accelerator mechanism. As in [START_REF] Bernanke | The Financial Accelerator in a Quantitative Business Cycle Framework[END_REF], entrepreneurs manage a continuum of firms ∈ 0,1 that produces, by using units of capital and units of labour, wholesale (intermediate) goods in a perfectly competitive market according to the following technology:

22

where is a technological shock that is common to all firms and follows a stationary first-order autoregressive process : log log , , with , ~ . . 0, ; ∈ 0,1 is the share of capital in the production technology.

The representative firm maximizes its profit by choosing and subject to the production function ( 22). The first-order conditions for this optimization problem are:

1 ,

23

, 24 5 The way we write the real effective exchange rate means that an increase in this variable denotes an effective real depreciation of the common currency with regard to the foreign currency (RoW).

where is the Lagrangian multiplier associated with the production function ( 22) and denotes the real marginal cost; is the real wage; and is the real marginal productivity of capital.

Entrepreneurs are risk neutral and borrow to finance a share of capital used in the production process.

As in [START_REF] Bernanke | The Financial Accelerator in a Quantitative Business Cycle Framework[END_REF], to ensure that they never accumulate enough funds to fully self-finance their own activities entirely, we assume that they have a finite expected horizon. In each period , entrepreneurs face a constant probability 1 of leaving the economy. We follow [START_REF] Christensen | The Financial Accelerator in an Estimated New-Keynesian Model[END_REF] in allowing newly entering entrepreneurs to inherit a fraction of the net worth of those firms which exit from the business. This assumption is made in order to ensure that new entrepreneurs start out with a positive net worth. 6At the end of each period, entrepreneurs purchase capital, , that will be used in the next period at the real price . Thus, the total funding needed by an entrepreneur to purchase capital is . The capital acquisition is financed partly by their net worth, , and by borrowing, , from a financial intermediary. Financial intermediaries (banks) obtain their funds from household deposits. Their activities are described below (see section 2.4.).

In optimum, the entrepreneur's aggregate demand for capital in the economy depends on the expected marginal return and the expected marginal financing cost at 1. Thus, the capital demand must satisfy the following differentiation between the ex post marginal return on capital, , , and the marginal productivity of capital at 1, , which is the rental rate of capital:

, 1 , 1 25
where is the capital depreciation rate, , is the tax rate on capital-income (whose introduction here is a specific feature of our model) and 1 is the value of one unit of capital used in 1.

Following [START_REF] Bernanke | The Financial Accelerator in a Quantitative Business Cycle Framework[END_REF], we assume the existence of an agency problem that makes external finance more expensive than internal finance, because financial intermediaries are facing costs for auditing the performance of entrepreneurs. However, entrepreneurs observe the random outcome of their investments costlessly and decide whether to repay their debt or to default. If they default, the lenders audit the project and seize whatever they find. As demonstrated in [START_REF] Bernanke | The Financial Accelerator in a Quantitative Business Cycle Framework[END_REF], the optimal financial contract between borrower and lender implies an external finance premium (the difference between the cost of external and internal finance), Ψ , • , that reflects the existence of auditing costs and depends on the entrepreneur's leverage ratio (capital to net worth ratio).

Accordingly, the entrepreneur's demand for capital satisfies, optimally, the equality between expected return on capital and gross premium for external finance plus the gross real opportunity costs equivalent to the gross real interest rate on loans7 :

, Ψ , • , 26
where , is the gross nominal interest rate on banks loans; Ψ ,

• is the function that describes how the external finance premium depends on the financial position of the firm and is given by:

Ψ , • with Ψ , • 0 , Ψ 1
1 and is the elasticity of the external finance premium with respect to firm's leverage ratio. Thus, the external finance premium is an equilibrium inverse function of the aggregate financial position in the economy, expressed by the leverage ratio. Equation ( 26) provides the basis for the financial accelerator. If entrepreneur's net worth goes up, the external finance premium falls, the cost of borrowing falls and firms get cheaper access to credit.

Aggregate entrepreneurial net worth accumulation of the economy depends on profits earned in previous periods plus the bequest, Ω , that newly entering entrepreneurs receive from entrepreneurs who leave the economy, and evolves according to:

, , 1 27

Capital producers

Competitive capital producers use a linear technology to produce new capital from final investment goods and existing capital stock leasing from entrepreneurs without costs. When producing capital, they are subject to quadratic capital adjustment costs specified as .

The aggregate capital stock used by producers in each economy evolves as follow:

2 1 28
where 0 is the parameter that measures the adjustment costs elasticity.

Capital producers face an optimization problem which consists, in real terms, in choosing the level of investment that maximizes their profits:

max 2 29
The following equilibrium condition holds:

1 30
which is the standard Tobin's Q equation that links the price of capital to the marginal adjustment costs.

When 0 (no adjustment costs), the capital price, is constant and equal to 1. This shows that capital adjustment costs imply necessarily the capital price ( ) variation and therefore contribute to the volatility of entrepreneurial net worth.

Retailers: price and inflation dynamics

The existence of retailers provides the source of nominal stickiness in the economy. Retailers take wholesale goods as inputs, repackage these costlessly, and sell them in a monopolistically competitive market. There are domestic goods retailers and imported goods retailers. Following [START_REF] Calvo | Staggered Prices in a Utility-Maximizing Framework[END_REF], we assume that retailers set nominal prices on a staggered basis: at each period, a fraction 1 of retailers are randomly selected to set new prices while the remaining fraction of retailers keep their prices unchanged. For simplicity, these fractions are assumed to be equals within the two groups of retailers.

All home goods retailers purchase the wholesale goods from entrepreneurs at a price equal to the entrepreneurs' nominal marginal cost. Each retailer of them setting price at will choose the optimal price, , , that maximizes the expected profits for periods, so that: , where is the households' marginal utilities ratio between and .

The first-order condition for this problem yields,

, 1 ∑ , , ∑ , 32
Aggregating across all retailers, the price index for domestically produced goods is given by, , 1 , ,

33

Combining log-linearized versions of equations ( 32) and ( 33) yields an expression of the inflation rate for domestically produced goods, defined by the following New Keynesian Phillips curve:

, , 1 1 34 where is the real marginal cost, , , , is domestic inflation and variables with hats are log deviations from their steady-state values.

Similarly, imported goods retailers purchase the products from foreign producers at the wholesale price, , . At the wholesale level, the law of one price holds. Thus, , , and , , are the wholesale prices (nominal marginal costs) for goods coming from the RoU and the RoW, respectively. But at the retail level, we assume that the law of one price does not hold (such as , , and , , ). There is, thus, incomplete exchange rate pass-through in the model. Similar to the home good retailers, imported goods retailers set prices according to a Calvo-style price setting equation. Their optimization problems are identical except for real marginal costs. The latter are, respectively, , , ≡

, and , , ≡ , for imported goods from the RoU and the RoW. The inflation rates for imported goods then satisfy these following New Keynesian Phillips curves:

, , 1 1 , 35 , , 1 1 , 36
where , and , are imported inflation prices from the RoU and the RoW.

Finally, from equation ( 9), CPI inflation, , is a composite of domestic, foreign and world goods prices inflation, such that:

1 , , , 37
∀ , ∈ , and .

Banks

Competitive banks make financial intermediation. The purpose of financial intermediary in the model is to allow fiscal and monetary policies to influence the economy via the bank-lending channel. At the beginning of each period , the representative bank takes deposits, , from the representative household and lends to domestic entrepreneurs, domestic government and foreign government (in the RoU). Bank loans are defined by and assumed to take the following form:

38

where ∈ 0,1 denotes the fraction of total deposits lent out to entrepreneurs and governments of the monetary union. The remaining portion of deposits, 1 , is held as reserves that earn no interest. Following Atta-Mensah and Dib (2008), we assume that the intermediation process is partly endogenous and depends on the state of the economy in this way:

, 39

The parameter is the elasticity of the willingness to lend with respect to changes in economic activity (deviations of output from its steady-state value) and

, represents shocks to the intermediation process. If the willingness to lend is pro-cyclical, then 0. This can be justified by the fact that, in good times, net worth of entrepreneurs and governments' fiscal receipts are relatively high. This improves the credit risks of borrowers and increases the willingness of financial intermediaries to lend. The process for , is given by:

log , log , , 40
The shock , could represent, for example, an exogenous change in the confidence level of bank with respect to the credit risks of their borrowers and the health of the economy8 .

Next, we assume that banks provide the fixed shares of their loans to entrepreneurs, foreign government and domestic government. These shares are, respectively, , and 1 . At the end of period , the representative bank receives, in principal and interest, , Ψ , • from entrepreneurs, , Ψ , , , , from foreign government in the RoU and , 1 from domestic government. We assumed that domestic government loan cost is the benchmark for banks to decide which costs must be applied on the others agents' loans. Therefore, banks provide loans in a way consistent with a portfolio management goal (changing the cost of borrowing of each agent), and the variable , is the gross nominal interest rate on domestic government loans.

The relative risk premia are defined as follows: Ψ , • is entrepreneur's external finance premium as previously defined and Ψ , , , , , defined in section 2.5 below, is the relative risk premium paid by government . Banks owe to depositors and earn a zero net return on their reserves. In this case, the bank profit function is given by:

Π , , 1 , Ψ , , Ψ , • 1 41
Given the competition among banks for loans and deposits, the zero profit condition guarantees that:

, 1 1 Ψ , 1 Ψ , 1 42
The fluctuations in the reserve levels of banks are captured by the parameter epsilon and would be reflected in the gap between loan and deposit interest rates. Equation ( 42) shows that the loans interest rate applied to domestic government decreases when risk premia of relative risky agents increases, and thus when their debts increase.

Government Budget Constraint and Fiscal policy

In each economy, government spends in purchases of aggregate goods and transfers to households . To do that, the government collects tax revenues on consumption and capital income, and receives loans from domestic and foreign banks ( 1 and respectively).

The government budget constraint is given by:

1 , 1 , Ψ , , , , 43
In equation ( 43), is the nominal primary deficit and expressed by:

, , 44
The term Ψ , , , , is the government ' relative risk premium such that,

Ψ , , , , ≡ 1 , 45
where is the elasticity of the risk premium with respect to government debt; ≡ and ≡ are respectively the total of loans made by banks in economy and ; , is an exogenous shock on government's premium and evolves according to the following autoregressive process:

log , log , , 46

Fiscal policy instruments

The government needs to adjust tax revenues or expenditure to stabilize its deficit and debt. We choose public spending as the fiscal policy instrument. Government spending adjustments in response to cyclical fluctuations are endogenously made according to the fiscal rule:

log log 1 log 1 log , 47
where , , ∈ 0,1 capture, respectively, the degree of public spending smoothing, fiscal reaction to output deviation and fiscal reaction to debt/GDP ratio ( ≡ ); , is an exogenous shock to government spending ( , ~ . . . 0, ).

The parameter captures the degree of fiscal policy cyclicality. If 1 (resp. 1), public spending is counter-cyclical (resp. pro-cyclical).

Rest of the world

We assume that the RoW is fully exogenous and its variables follow an autoregressive process such that:

Calibration of the model and simulations of real and financial shocks

Calibration of the model

We log-linearized the equations of the model around the steady state. 9 We then used the software Dynare to do the simulations. The calibration of the model and the main macroeconomic steady-state ratios are summarized in Table 1 below. Calibration is made according to the references found in the literature for the euro area (on average) or according to our own computation based on the ECB, OECD and European Commission (AMECO) databases.

We simulate two types of asymmetric shocks hitting the domestic country: 1) a positive technological shock and 2) a default risk premium shock. For both shocks, we study the effects on the domestic economy and the spillover effects on the economy of the partner country (RoU). In particular, special attention is given to two mechanisms: the evolution and financing of public debt and the impact of the openness of the union toward the RoW. We consider first a technological shock because we want to generate higher economic growth in one country of the monetary union and study the consequences of a pro-cyclical stance of fiscal policy in the domestic country. Thus, we compare two opposite cases concerning the behaviour of public spending in response to the output gap in the domestic country: it is either counter-cyclical (the parameter cg is equal to 1) or pro-cyclical (cg = 1). In contrast, public spending remains countercyclical in the RoU in all cases. As for the default risk premium shock, we think of it as a case where one country of the monetary union is faced with a sudden change in expectations of the financial markets about the sovereign debt sustainability (assuming that the domestic country has a pro-cyclical fiscal policy). In addition, we assume that the elasticity of the sovereign risk premium to the level of public debt is higher in both countries of the monetary union ( i l  = 0.1 instead of 0.01). This assumption relies on the idea that traders of government bonds in financial markets will react to any change in the level of public indebtedness more than they used to and whatever the country.

Technological shock analysis

Figure 1 displays the effects of a positive technological shock in the domestic country on the domestic economy. The spillover effects on the RoU are displayed in Figure 2. We have the usual impact of a positive technological shock: higher output, higher investment, higher consumption and lower inflation compared to the steady-state level in the domestic country. As a consequence, the latter becomes more competitive within the union and with regard to the RoW (real effective depreciation of domestic goods relative to foreign goods). The common central bank lowers the common short-term nominal interest rate less than the fall in the average rate of inflation in the union (the average output gap in the union is positive). As a result, the real interest rate is positive in both countries. Since the rental rate of capital decreases much in the domestic economy (given the decline in the marginal productivity of capital), the domestic government collects fewer tax receipts on capital income. This negative effect on the government revenues is greater than the positive effect of higher receipts from the tax on consumption.

In the case of a counter-cyclical fiscal rule, public consumption is lower. Nevertheless, the ratio of public debt to GDP rises because the decline in total tax revenues is larger than the decline in primary public expenditures. In the pro-cyclical fiscal policy scenario, public consumption increases and the increase in the public debt-to-GDP ratio is less strong than in the counter-cyclical case in the short term, because output is higher in the pro-cyclical case. In the latter case, the nominal effective depreciation of the common currency is stronger (in order to get the balance of payments back to equilibrium). And a stronger real effective depreciation leads to more exports of goods to the RoW. Yet, in the long term, the ratio of public debt to GDP goes back to its steady-state level much faster in the counter-cyclical case than in the pro-cyclical case.

Initially, the interest rate on public bonds is lower because the effect of a lower key interest rate (set by the central bank) is stronger than the effect of a higher sovereign risk premium. After the initial impact of the shock, the interest rate on public bonds increases more in the counter-cyclical case than in the pro-cyclical case, because the ratio of public debt to GDP is higher in the former case (given a lower output gap). As for the cost of borrowing for the private sector, it is lower at the moment of the shock. Banks set the interest rate on loans to the private sector by adding an external finance premium to a reference rate (that is the interest rate on public bond). Apart from the lower reference rate (at the beginning), firms enjoy a lower risk premium because they are perceived relatively less risky than governments (whose indebtedness is increasing). For the partner country in the RoU, a pro-cyclical stance of fiscal policy in the domestic country amplifies negative spillover effects on consumption and investment (figure 2). Private consumption and investment fall more because the real interest rate is higher than in the case where the domestic country implements a counter-cyclical fiscal policy. This is due to a stronger fall in prices (accentuated by lower imported inflation). Moreover, the central bank does not lower sufficiently the common nominal interest rate (given that it is set according to average values of the economic variables of the member countries). Yet, output can increase in the RoU as long as exports of goods towards the RoW can increase. As a matter of fact, for the RoU, there is a real appreciation of goods with regard to the domestic country but there is also a real depreciation of goods with regard to the RoW (given the nominal depreciation of the common currency). This effect on exports to the RoW more than offsets the negative direct spillover effects on consumption and investment. Higher exports of goods to the RoW imply that there is a need to produce more with the same amount of capital. A higher productivity of capital translates into a higher rental rate of capital.

The RoU benefits from an increase in fiscal revenues from taxes on capital income. Public consumption decreases more in the case of a pro-cyclical fiscal policy in the domestic country than in the case of a counter-cyclical fiscal policy, because output in the RoU is (slightly) higher in the former case (public consumption is counter-cyclical in the RoU). As a consequence, public debt decreases in the RoU and the sovereign risk premium decreases too. During the process of going back to the long-term equilibrium, the RoU is faced with a fall in the effective real exchange rate (a real appreciation), a decrease in exports of goods towards the RoW, a decrease in output, an increase in public primary expenditures, an increase in consumption tax receipts but a decrease in capital income tax receipts, and an increase in public debt (back to the steady-state level). There will be an increase in the interest rate on public bonds and firms can enjoy a relatively lower cost of borrowing, if the fall in the external finance premium (they are perceived relatively less risky than governments) can compensate for the rise in the reference rate (interest rate on public bonds). All in all, the model shows that the real interest rate is a key variable in causing macroeconomic divergences between countries of the monetary union while the (extra-union) real exchange rate lessens these divergences. Pro-cyclical public primary expenditures in the domestic country, where higher output growth is generated by a positive technological shock, accentuate the negative spillover effects on consumption and investment in the partner country (RoU). In the domestic country, the ratio of public debt to GDP increases less in the short-term than in the case of a counter-cyclical fiscal policy, because pro-cyclical public spending amplifies output growth. However, it takes far more time to get the public debt/GDP ratio back to its steady-state level. Some governments may tend to neglect the long-term effects of their pro-cyclical fiscal policy. The rise in the domestic public debt makes the private sectors in the domestic economy and in the RoU relatively less risky. As a consequence, there are some positive financial effects for private agents, at least in the short-term. As regards the public sector in the RoU, there are lower costs of borrowing but only in the very shortterm.

Sovereign risk premium analysis

The domestic country is now hit by a financial shock: a higher sovereign risk premium. Figure 3 depicts the effects on the domestic economy (straight line) and on the RoU (dotted line) in the case where fiscal policy is pro-cyclical in the domestic country but counter-cyclical in the RoU. This shock makes the government of domestic country relatively riskier than the government in the RoU. As a consequence, the optimal behaviour of banks implies a higher interest rate on domestic sovereign bonds and a lower interest rate on foreign sovereign bonds. As regards the private sector, the cost of borrowing hardly changes in the domestic economy because the higher reference rate (interest rate on public bonds) is almost matched by a lower external finance premium. In the RoU, the cost of borrowing for private sector declines a bit because the fall in the interest rate on public bonds is slightly larger than the increase in the external finance premium (the private sector being perceived by banks as relatively riskier than the public sector in the RoU). Thus, the higher sovereign risk premium in the domestic country lowers the cost of borrowing in the RoU. Consequently, in the RoU, there is an increase in investment, output and inflation. The fall in the real interest rate leads to higher consumption (the latter gives rise to imports of foreign goods). The domestic economy can grow thanks to exports of goods to the RoU and to the RoW (with a nominal depreciation of the common currency).

Figure 3. Impact of an asymmetric risk premium shock

Public consumption decreases in the RoU since it is counter-cyclical and it increases in the domestic country where fiscal spending is pro-cyclical. The foreign government in the RoU benefits from higher tax receipts (both from consumption tax and capital income tax) and lower interest payments. Public debt decreases. In the domestic country, the public debt-to-GDP ratio decreases a bit, despite the shock on the default risk premium. As a matter of fact, the rise in output is due to higher exports and leads to a higher rental rate of capital. The domestic government collects more tax receipts from the tax on capital income.

All in all, the negative impact of the shock on the risk premium on the domestic country is counteracted by a credit channel and a trade channel: as long as the cost of borrowing decreases in the RoU for the public sector and the private sector, higher output growth in the RoU can help the domestic economy to grow faster via intra-union trade. Moreover, the effective real depreciation of the common currency can boost exports towards the RoW. Unfortunately, in the context of the financial crisis and the euro area crisis, world growth was slow and some European countries found it difficult to export more to the rest of the world.

Conclusion

In this paper, we studied international macroeconomic divergences in a DSGE model of two countries belonging to a monetary union which is open to the rest of the world. We put the focus on divergences in the cyclical stance of national fiscal policies, which may be pro-cyclical in one country and counter-cyclical in the other country. We simulated two asymmetric shocks hitting the domestic country: a positive technological shock and a negative shock on the default risk premium.

The model allowed us to explain the spillover effects of these shocks on the economy of the other country in the rest of the union (RoU). We highlighted the role of the evolution of public debt and of the sovereign risk premium in the cost of borrowing for the public sector in the RoU and for the private sectors in the domestic economy and in the RoU. There is also the critical role of external trade with the rest of the world (RoW): an effective real depreciation of the currency of the monetary union which gives rise to exports to the RoW can compensate for the negative effects of a higher real interest rate on consumption and investment within the monetary union.

Under the technological shock, we found that the pro-cyclical fiscal policy in the domestic country leads to a lower increase in the public debt-to-GDP ratio over the short-term than in the countercyclical case because output growth is higher in the former case. However, the public debt remains higher during a far longer period of time. Moreover, the negative spillover effects on consumption and investment in the RoU are stronger. Still, output growth is positive in the RoU via higher exports of goods thanks to a real depreciation of goods relative to the goods produced by the RoW.

A shock on the sovereign risk premium of the domestic country has a negative impact not only on the domestic public sector but also on the domestic private sector. Yet, output growth is possible as long as the country can exports more to the RoU and to the RoW. This shock has some positive effects on the RoU because the lower interest rate on public bonds issued by the governments in the RoU helps to lower the cost of borrowing. Benefits are growing with the openness to the RoW. The model can thus be used to illustrate the economic downturn during the sovereign debt crisis in euro area countries with high public indebtedness and low export capacity (such as Greece), and the faster economic recovery in some other countries perceived as less risky and with good performance in external trade (Germany).

Contrary to what [START_REF] Christoffel | Government Bond Risk Premia and the Cyclicality of Fiscal Policy[END_REF] found in a closed-economy model, we showed that a procyclical fiscal policy does not necessarily lead to a higher sovereign risk premium at least in the short term because such a policy accentuates output growth and leads to a lower increase in the public debt-to-GDP ratio. In the long term, the sovereign risk premium is higher with a pro-cyclical fiscal policy though, because public debt is higher. Our result is also much dependent on the openness of both countries to the RoW.

Figure 1 .

 1 Figure 1. Technological shock and fiscal policy: effects on domestic country

Figure 2 .

 2 Figure 2. Technological shock and fiscal policy: effects on the RoU

Table 1 . Baseline calibration of the DSGE model

 1

	Description

In contrast,[START_REF] Bernanke | The Financial Accelerator in a Quantitative Business Cycle Framework[END_REF] ensure this by assuming that entrepreneurs also work. This difference does not affect the results.

For details, see[START_REF] Bernanke | The Financial Accelerator in a Quantitative Business Cycle Framework[END_REF].

The other possible sources of variation in , are: perceived changes in entrepreneur's cash flow or net worth, government regulation of banks, technological advances in the intermediation process[START_REF] Berger | The Institutional Memory Hypothesis and the Pro-cyclicality on Bank Lending Behavior[END_REF].

The linearized version of the model is available upon request.

We gratefully acknowledge the support of our research centers. Florence Huart is grateful for being invited at the LAREFI seminar in University Bordeaux IV (February 2013

As for the other fiscal instruments, government transfers, taxes on consumption and capital income follow an autoregressive process: where, ∈ 0,1 , with , and , are the coefficients of the autoregressive process and , ~ . . 0, are the associated exogenous shocks.

Monetary authority

In the monetary union, the common central bank sets the nominal interest rate according to the following Taylor-type interest rate rule:

with , ~ . . . 0, .

, and are the steady-state values of , and , that are, respectively, the nominal interest rate, the inflation rate and output of the union. The variables and are the average values of inflation and output of the two equal-size countries:

1 2 and 1 2 52

1 and 1 are coefficients that measure central bank responses to expected inflation and output deviations. The parameter 0 1 captures the degree of interest rate smoothing.

General equilibrium conditions

In equilibrium, the factor markets, the final goods market, the loan market and the international traded bonds market must clear in each country ∈ , .

Equilibrium in factor markets requires:

and 53

The loan market clears when the unused fraction of household deposits in reserves by financial intermediaries equalizes the total funds lent to entrepreneurs, domestic government and government in the RoU (see equation 38 supra).

Let ≡ denote aggregate output. Thus, the goods market clearing condition satisfies:

where

The variable EX represents total exports and AB stands for absorption.

Then the domestic economy's aggregate resource constraint can be rewritten as: The internationally traded bonds market is in equilibrium when the positions of the export and importing firms vis-à-vis the RoW equals the households' choice of internationally traded bonds holdings. The evolution of net foreign assets (government assets holdings from the RoU plus households internationally traded bonds holdings) at the aggregate level can be expressed as:

, are imports of country originating from country (of the RoU) and from the RoW, respectively.

Noting that the definitions of , and are ≡ , , ≡ and ≡ , we can write the evolution of total real NFA position in percentage of steady-state output as:

, , , , ,

59

∀ , ∈ , and .


Design, Manufacturing Mathieu Hobon
email: mathieu.hobon@ensam.eu

Nafissa Lakbakbi Elyaaqoubi

Gabriel Abba
email: gabriel.abba@ensam.eu

Influence of Frictions on Gait optimization of a Biped Robot with an Anthropomorphic Knee

This paper presents the energy consumption of a biped robot with a new modeled structure of knees which is called rolling knee (RK). The dynamic model, the actuators and the friction coefficients of the gear box are known. The optimal energy consumption can also be calculated. The first part of the paper is to validate the new kinematic knee on a biped robot by comparing the energy consumption during a walking step of the identical biped but with revolute joint knees . The cyclic gait is given by a succession of Single Support Phase (SSP) followed by an impact. The gait trajectories are parameterized by cubic spline functions. The energetic criterion is minimized through optimization while using the simplex algorithm and Lagrange penalty functions to meet the constraints of stability and deflection of the mobile foot. An analysis of the friction coefficients is done by simulation to compare the human characteristics to the robot with RK. The simulation results show an energy consumption reduction through the biped with rolling knee configuration. The influence of friction coefficients shows the energy consumption of biped robot is close to that of the human.

I. INTRODUCTION

A. Design improvement

The research on technology improvement in the field of biped and humanoid robots are numerous which focuses on different areas in mechanical structures, kinematic, control, human robot interaction and so on. The current research work concentrates on the following main aspects:

• The selection of the best kinematic structures of the knee joint [START_REF] Hamon | Cross four-bar linkage for the knees of a planar bipedal robot[END_REF], [START_REF] Van Oort | An energy efficient knee locking mechanism for a dynamically walking robot[END_REF], [START_REF] Hobon | Quasi Optimal Gait of a Biped Robot with a Rolling Knee Kinematic[END_REF] • The improvement of gait support trajectories used by some control strategy [START_REF] Chevallereau | Optimal reference trajectories for walking and running of a biped robot[END_REF], [START_REF] Wieber | Constrained dynamics and parametrized control in biped walking[END_REF], [START_REF] Westervelt | Hybrid zero dynamics of planar biped walkers[END_REF] These two parts contribute to improve the robot energetic efficiency during the gait and also the range of travel of the robot without recharging its battery, named in brief biped autonomy.

On a majority of biped prototype, the knee joints are classically realized by revolute joints. Biomechanical studies talk about the human knees that relate the movement of this articulation which is a combination of a rotation and a translation. [START_REF] Hamon | Cross four-bar linkage for the knees of a planar bipedal robot[END_REF] proposes knee structures combining these movements with a cross four-bar linkage. The simulations show less energy usage through this solution comparing to conventional revolute joint knees. Another knee mechanisms designed by [START_REF] Van Oort | An energy efficient knee locking mechanism for a dynamically walking robot[END_REF] uses a singularity of the mechanism to save the energy. When the mobile leg is stretched at the end of the step, the knee is locked by the singularity and does not consume energy during the next stance phase. The energy consumption also decreases during the gait. The design of the knee mechanism in the LARP project in [START_REF] Gini | Human-oriented biped robot design: Insights into the development of a truly anthropomorphic leg[END_REF] is coming from the studies on prosthetic knees and consist of a structure with two cylindrical surfaces in contact. [START_REF] Hobon | Quasi Optimal Gait of a Biped Robot with a Rolling Knee Kinematic[END_REF] proposes simulations with the same design for the knees. The structure is called rolling knee. It adapts on a biped with anthropomorphic model and is compared to a similar configuration with revolute joint knees. The results of this study show that the robot with rolling knees has allowed minimizing the sthenic criterion during walking for the same forward speed. Now, we have already done the same study on another biped model for which all characteristic of the actuators and the gear box are known. So, we want to know what is the energy consumption of the biped robot.

The gait is described by a succession of contacts between feet and the ground. The important issue is to keep the equilibrium of the biped during this progression. Some control strategies need stable reference trajectories to ensure the gait of the robot. The parametrization of the trajectories can be done by polynomial functions [START_REF] Bessonnet | Optimal dynamics of actuated kinematic chains. part 1: Dynamic modelling and symbolic differentiations[END_REF], Bézier functions [START_REF] Westervelt | Hybrid zero dynamics of planar biped walkers[END_REF], [START_REF] Scheint | Compliance in gait synthesis: Effects on energy and gait[END_REF], Fourier series functions [START_REF] Cabodevila | Rank selection genetic algorithm to determine quasi optimal constrained gait for a biped robot[END_REF] or cubic splines [START_REF] Hamon | Cross four-bar linkage for the knees of a planar bipedal robot[END_REF]. Here, we use the cubic splines to parameterize the trajectories. In the optimization problem, the discussion is about the criteria. In [START_REF] Hobon | Quasi Optimal Gait of a Biped Robot with a Rolling Knee Kinematic[END_REF], we use the sthenic criterion to compare our robot to a conventional revolute joint robot and this criterion permits the actuator selection. In this work, an energy criterion is used to calculate the energy consumption of two bipeds during the gait.

All works cited before use parametric gait and so transform the optimization problem to a parametric optimization problem. The choice of the algorithms to solve these problems is difficult. In this paper, the Nelder-Mead simplex algorithm [START_REF] Lagarias | Convergence properties of the nelder-mead simplex method in low dimensions[END_REF] is used which allows to obtain a rapid convergence and to avoid local minima.

The last point of the discussion is about the coefficient of friction of the actuators. The biological studies show low friction on the human knee articulation. The friction is reduced with the viscosity of the lubrication, called synovial fluid, between the femur and the tibia. [START_REF] Oatis | Kinesiology: The Mechanics and Pathomechanics of Human Movement[END_REF] relates the coefficient of friction of the articulation at 0.001. [START_REF] Flannery | Compliant layer knee bearings. part i: Friction and lubrication[END_REF] studies compliant layer knee for prostheses composed by polyurethane for the tibial part and metal for the femoral part. The investigation shows the low friction factor included between 0.001 and 0.015. In general, the global Coulomb friction coefficient in the gear box is between of 0.08 to 0.1. Our study concentrates on the importance of the coefficient of friction on the energy consumption, also on the autonomy of the biped. We calculate the estimated cost of the gait (c et ) defined in [START_REF] Collins | A three-dimensional passivedynamic walking robot with two legs and knees[END_REF], [START_REF] Collins | A bipedal walking robot with efficient and human-like gait[END_REF] and compare our results to human and other robots.

B. Contributions and outline

This paper highlights the importance of mechanical friction in the energy performance of a biped robot. The friction influence also more the biped autonomy as the kinematic structure of the knee joint. In [START_REF] Hobon | Quasi Optimal Gait of a Biped Robot with a Rolling Knee Kinematic[END_REF], the authors have proposed optimal trajectories in regards of a sthenic criterion. These trajectories minimize also the Joule losses in the motor. The results allow the selection of the motor and transmissions for each joint of the robot. We now present the next step in the robot design. The criterion is modified to take into account all losses. The friction model is certainly simplified, but enough just to show its importance [START_REF] Armstrong | Friction: Experimental Determination, Modeling and Compensation[END_REF], [START_REF] Abba | Modeling of frictions in the transmission elements of a robot axis for its identification[END_REF]. The rolling knee structure lends itself easier to control by linear motor. The paper clearly shows that the performance of the biped robot can approach that of humans at the price of a particular care to the mechanical design of the robot.

In section II, the models of the biped with the revolute joint knees and the anthropomorphic structure are established. The parametric gait formulation is discussed in the section III. The section IV introduces the energy optimization problem. Simulations and results are presented in section V to demonstrate that the friction influence more the energy criterion than the cinematic structure of the knee. Finally, section VI presents the conclusion and perspectives.

II. MODEL

A. Biped Model

This study is focused on the cyclic walk of the biped in the sagittal plane. The considered biped is composed of seven rigid bodies with two feet, two shins, two thighs and one trunk. The biped is all actuated by six actuators. For the study, we define two configurations. Both have revolute joints placed at hips and ankles level, but different knee joints. One with revolute joints on the knees named conventional configuration or CK Robot, and the second with rolling knees, named RK Robot. Here, we focus on the new knee structure. Fig. 1 shows the configurations with rolling knee. The rolling knee consists of a movement of two cylindrical surfaces rolling without sliding. The two surfaces are the terminal surface of the femur and the tibia. The reference frame is ℜ 0 = (O 0 , ⃗ x 0 , ⃗ y 0 ,⃗ z 0 ). O 0 is defined by the projection of the point A 1 on the ground. The direction of the walk is according to ⃗ x 0 and ⃗ z 0 the unit vector perpendicular to the ground. The orientation of the links are defined by the absolute angles q i , {i = 0 . . . 6} referenced by the vertical (see Fig. 1), the speed vector qi , {i = 0 . . . 6} and the vector of articular torques Γ = [Γ 1 . . . Γ 6 ] T which represents the torques placed on the hips, the knees and the ankles. Fig. 2 shows the details of the knee for the second configuration. The contact between the femur and the tibia is maintained with a bar on C 1 and C 2 of length r 1 +r 2 . r 1 and r 2 are respectively the

q 3 Cg 2 q 2 q 6 Cg 6 H Cg 3 K 2 N O 0 Cg 1 q 1 T 1 H 1 x 0 z 0 y 0 K 1 A 1 G 1 G 2 G 3 G 4 G 5 T 2 Cg 4 q 4 H 2 A 2 z 5 x 5 q 5 G 6
Fig. 1. Biped robot with rolling joint knees

g 1 q 1 q 2 C2 z 2 x 1 z 1 r 2 C1 r 1 B2 B1 K1 x 2 Fig. 2. Rolling knee of one leg distance B 1 C 1 and B 2 C 2 .
With the rotation without sliding, we can write (1) and find the relation of the angle γ 1 which shows the coupling between the angles q 1 and q 2 for the support leg in [START_REF] Van Oort | An energy efficient knee locking mechanism for a dynamically walking robot[END_REF]. Similarly for the mobile leg, γ 2 is the coupling angle between q 3 and q 4 in (3).

ø B 1 K 1 = ø B 2 K 1 (1) γ 1 = r 1 q 1 + r 2 q 2 r 1 + r 2 (2) γ 2 = r 1 q 4 + r 2 q 3 r 1 + r 2 (3)
The dynamic model's parameters are the length l i of the links for the classic robot configuration, i = {0 . . . 6}, the position of the center of mass C gi , the masses m i , the moments of inertia I i of each bodies C i around the ⃗ y 0 axis at C gi . To compare the two configurations, we consider that the hip positions of the biped on vertical stance are equal in both cases. This assumption gives us l

′ 1 = A 1 C 1 = l 1 -r 1 and l ′ 2 = HC 2 = l 2 -r 2 .
The coordinates of the hip, the heel and the toes for the conventional biped configuration are:

x H = -l 2 sin q 2 -l 1 sin q 1 (4) z H = l 2 cos q 2 + l 1 cos q 1 + h p (5) x H 2 = x H + l 1 sin q 4 + l 2 sin q 3
-l p cos q 5 + h p sin q 5 [START_REF] Westervelt | Hybrid zero dynamics of planar biped walkers[END_REF] z H 2 = z Hl 1 cos q 4l 2 cos q 3 -l p sin q 5h p cos q 5 (7)

x T 2 = x H + l 1 sin q 4 + l 2 sin q 3 -(l p -L p ) cos q 5 + h p sin q 5 (8) z T 2 = z H -l 1 cos q 4 -l 2 cos q 3 -(l p -L p ) sin q 5 -h p cos q 5 (9)
The coordinates of the hip, the heel and the toes for the rolling knee configuration are:

x ′ H = -l ′ 2 sin q 2 -(r 1 + r 2 ) sin γ 1 -l ′ 1 sin q 1 (10) z ′ H = l ′ 2 cos q 2 + (r 1 + r 2 ) cos γ 1 + l ′ 1 cos q 1 + h p (11) x ′ H 2 = x ′ H + l ′ 2 sin q 3 + l sin γ 2 + l ′ 1 sin q 4 -l p cos(q 5 ) + h p sin(q 5 ) (12) z ′ H 2 = z ′ H -l ′ 2 cos q 3 -l cos γ 2 -l ′ 1 cos q 4 -l p sin(q 5 ) -h p cos(q 5 ) (13) x ′ T 2 = x ′ H + l ′ 2 sin q 3 + l sin γ 2 + l ′ 1 sin q 4 -(l p -L p cos(q 5 ) + h p sin(q 5 ) (14) z ′ T 2 = z ′ H -l ′ 2 cos q 3 -l cos γ 2 -l ′ 1 cos q 4 -(l p -L p ) sin(q 5 ) -h p cos(q 5 ) (15)

B. Dynamic Model

In this work, we consider only the walking gait defined with Simple Support Phases (SSP) followed by an impact between the mobile foot and the ground. The impact produces the instantaneous exchange of support leg during the gait. The dynamic model for the SSP is assumed with the left leg on support. Considering the gait like periodic with a permutation of the legs at the impact, the study focuses on one step beginning with the impact. The dynamic and impact models are described as follow:

1) Dynamic model during the SSP: The Lagrange equations are used to determine the inverse dynamic model. Details of the modeling are mentioned in [START_REF] Spong | Robot dynamics and control[END_REF] and [START_REF] Khalil | Modeling, identification and control of robots[END_REF]. Posing q = [q i ], i ∈ [0 . . . 6], we express the state vector for the conventional configuration X e = [q, x H , z H ] T of dimension 9 × 1. Respectively, the state vector of the rolling knee configuration is X e = [q, x ′ H , z ′ H ] T of dimension 9 × 1 too. The relation between the torque vector Γ m of the actuator and the joint torque Γ vector is expressed by:

Γ = Γ m -Fv θ -Cs sign( θ ) (16
)

with θ = [ q1 -q0 , q2 -q1 , q6 -q2 , q6 -q3 , q3 -q4 , q4 -q5 ] T , Fv = diag( f v 1 , f v 2 , f v 3 , f v 3 , f v 2 , f v 1 )
and Cs = diag(cs 1 , cs 2 , cs 3 , cs 3 , cs 2 , cs 1 ) the matrices of viscosity and Coulomb friction. The inverse dynamic equation can be written as:

B A c L (X e ) T [Γ F L ] T = D(X e ) Ẍe + H( Ẋe , X e ) + Q(X e ) (17)
where D(X e ) represents the inertia matrix 9 × 9, H( Ẋe , X e ) is the vector of Coriolis and centrifugal effects 9 × 1, Q(X e ) is the vector of torques and forces due to the gravity 9 × 1 , B is the control matrix 9 × 6 and A c L (X e ) is the Jacobian matrix 3 × 9 of the foot on support. The second derivative of the eq. ( 4) and ( 5) or the eq. ( 10) and [START_REF] Lagarias | Convergence properties of the nelder-mead simplex method in low dimensions[END_REF] gives the expression of ẍH and zH respectively ẍ′ H and z′ H needed for the determination of the state vector. These results can be expressed:

A cL (X e ) T Ẍe + H cL (X e ) = 0 ( 18
)
with the evolution of the vector X e satisfying [START_REF] Spong | Robot dynamics and control[END_REF]. The torque Γ and external forces F L = [F x , F z ,C y ] T on the support stance ankle are calculated with [START_REF] Abba | Modeling of frictions in the transmission elements of a robot axis for its identification[END_REF] at any instant of the gait.

2) Impact model: During the gait, the left foot and then the right foot alternatively touch the ground with a non-zero speed. This is the impact phase. The impact phase separates two SSP. The impact phase between two rigid bodies, the foot and the ground, produces a mechanical energy dissipation phenomena [START_REF] Pfeiffer | Multibody Dynamics with Unilateral Contacts[END_REF]. We suppose that the restitution coefficient is equal to zero. This assumption ensures that we have no rebounds of the foot after the impact. The proposed model is given by:

D(X e ) Ẋe + -Ẋe - = A T c L I R (19) A c L (X e ) Ẋe + = 0 (20) ∆ i = D -1 A T c L I R = [∆ 0 , ∆ 1 , ∆ 2 , ∆ 3 , ∆ 4 , ∆ 5 , ∆ T ] T (21)
This model is used to find the speed vector after the impact Ẋe + from the configuration X e in relation with the speed vector before impact Ẋe -. ∆ i is the difference between the speed vector after and before the impact. It will be used to define the initial conditions of the trajectories in the following section. This model also gives the impact screw on the foot I R = [I Fx , I Fz ,C y ] T and the torque applied on the ankle.

III. GAIT REFERENCE PARAMETRIC TRAJECTORIES

Now, the gait is defined by the evolution of angular coordinates of the bodies with respect to time. Our study is focused on the cubic spline trajectories. We express in this section the mathematical functions and we explain the reduction of parameters with the condition of cyclicity and impact. The angular coordinates q i with i = {0 . . . 6} are parameterized by a cubic spline function also, used in [START_REF] Hamon | Cross four-bar linkage for the knees of a planar bipedal robot[END_REF], [START_REF] Banno | Optimal trajectory design for parametric excitation walking[END_REF]. To simplify the definition of the trajectories, the time t is normalized to the dimensionless time variable t n = t/T with T the step period. The gait can be described by: at t n = 0, the left foot is fixed on the ground and the right foot is behind the trunk. At t n = 1, the right foot has an advance of a distance d 2 and it is in front of the trunk.

A. Cubic Spline Function

In this case, the trajectories are defined by two cubic spline functions. Each function is parameterized for a half-period. The knot vector has three knots so we define t k = [0, 0.5, 1].

In neighborhoods of t i ∈ t k , i = [0, 2], the spline function has the smoothness C 1 . We suppose that at the time t 1 = 0.5, the second derivative is continuous. Also, for t 0 = 0 and t 2 = 1, the impact imposes a discontinuity on the velocities.

The expression of the cubic spline function is:

0 ≤ t n ≤ 1 2 → f q i (t n ) = 3 P j=0 a i j t j n (22) 1 2 ≤ t n ≤ 1→ f ′ q i (t n ) = 3 P j=0 b i j (1 -t n ) j ( 23
)
where a j and b j are the eight parameters expressed for each angle. Supposing k = 1 T , the velocities and accelerations are obtained by derivation:

qi (t) = k d f q i dt n (24) qi (t) = k 2 d 2 f q i dt 2 n ( 25
)

B. Smoothness, cyclicity and vector of parameters

The study is focused on a cyclical gait defined by an impact and SSP. We assume that the left foot is on support during the SSP, also q 0 = 0. The unknown vector q i is now for i = {1 . . . 6}. Following the hypotheses that are posed for the gait:

• The mobile foot is flat on the ground at the beginning and at the end of the step

The angles of the trunk and the mobile foot are Tperiodic

• The angles of the tibiae and the thighs are 2T -periodic

With the spline function, we consider the continuity and the smoothness of the function at the half-period. These conditions for the mobile foot, the trunk, the tibiae and the thighs lead to:

a 5 0 = b 5 0 = 0, a 5 2 = b 5 2 , a 5 3 = b 5 3 = - 4 3 a 5 2 (26) a 6 0 = b 6 0 , a 6 2 = b 6 2 , a 6 3 = b 6 3 = - 4 3 a 6 2 ( 27
)
a 1 0 = b 4 0 , a 1 2 = b 4 2 = a 4 2 + 6(a 4 0 -a 1 0 ), b 1 3 = -6((a 1 0 -a 4 0 ) + 4 3 (a 1 1 -a 1 2 )) ( 28
)
a 4 0 = b 1 0 , a 4 2 = b 1 2 = a 1 2 + 6(a 1 0 -a 4 0 ), b 4 3 = -6((a 4 0 -a 1 0 ) + 4 3 (a 4 1 -a 4 2 )) ( 29
)
a 2 0 = b 3 0 , a 2 2 = b 3 2 = a 3 2 + 6(a 3 0 -a 2 0 ), b 2 3 = -6((a 2 0 -a 3 0 ) + 4 3 (a 2 1 -a 2 2 )) ( 30
)
a 3 0 = b 2 0 , a 3 2 = b 2 2 = a 2 2 + 6(a 2 0 -a 3 0 ), b 3 3 = -6((a 3 0 -a 2 0 ) + 4 3 (a 3 1 -a 3 2 )) (31)
The impact model imposes a relation between the speed before and after the impact time for all mobile bodies. For example, the speed of the right thigh after the impact depends on [START_REF] Banno | Optimal trajectory design for parametric excitation walking[END_REF] and on the speed of the left thigh just before the impact. These conditions are expressed as:

a 5 1 = b 5 1 + ∆ 5 (32) a 6 1 = b 6 1 + ∆ 6 (33) a 2 1 = b 3 1 + ∆ 2 (34) a 3 1 = b 2 1 + ∆ 3 (35) a 1 1 = b 4 1 + ∆ 1 (36) a 4 1 = b 1 1 + ∆ 4 (37)
In resume, the vector of parameters is a i 0 for i = {1, 2, 3, 4, 6}, b j 1 and a j 2 for j = {1 . . . 6}. The total number of parameters for the trajectories is 17.

Finally, the parameters a i 0 which give the initial angular position could be found with the inverse geometric model of the Cartesian position of the hip (x H (0), z H (0)). The period T is added as parameter for the gait determination. The parameter's vector p=[x H (0), z H (0), a 6 0 , b j 1 , a j 2 , T ] is used in the following optimization process.

IV. OPTIMIZATION PROBLEM

The research of optimal parameters to produce the best gait trajectories is challenging. The goal is to search trajectories minimizing a criterion representing the energetic consumption of the robots while respecting the constraints due to the biped environment. A gait is considered optimal if the gait is physically feasible and with the minimal of power supply.

We have also to solve a nonlinear minimizing problem under constraints that can be expressed:

min p C Γ (p) under Ψ(p) ≥ 0 (38) with Ψ = [Ψ 1 Ψ 2 Ψ 3 Ψ 4 Ψ 5 Ψ 6 Ψ 7 ]
T are the constraints imposed by the initial configuration of the robot and the physical constraints considered before. The constraints are:

• Ψ 1 = F z define the vector of the force along the z-axis and must be positive,

• Ψ 2 = x ZMP + l p and Ψ 3 = -x ZMP + (L pl p ) represent the limit position of the ZMP on x-axis (ZMP must stay in the foot support) to guaranty the stability of the robot,

• Ψ 4 = q 2q 1 and Ψ 5 = q 3q 4 are the choice made to keep a gait human-like with no bend backward of knees,

• Ψ 6 = z H2 resp.(z H ′ 2 ) and Ψ 7 = z T 2 resp.(z T ′ 2 ) are used to have the z-coordinates of heel and toes of the mobile foot above the ground during the SSP.

The x-coordinates of ZMP is calculated by:

x ZMP = Γ 1 -h p F x -m p s x g F z (39)
The criterion used is the estimation of energy losses. It is defined by (40).

C(p) = 2 d Z T 0 kJΓ T Γ +Cs θ + Fv θ 2 dt ( 40
)
with Γ calculated from [START_REF] Abba | Modeling of frictions in the transmission elements of a robot axis for its identification[END_REF], kJ represents the matrix of Joules quality factor. θ = θi for i = {1 . . . 6}.

We propose to solve this problem by using the Nelder-Mead simplex algorithm, see [START_REF] Lagarias | Convergence properties of the nelder-mead simplex method in low dimensions[END_REF]. This algorithm solves nonlinear problem without constraints. The constraints are also added in the criterion (40) as Lagrange multiplier. The equation becomes:

C E (p) = C(p) + 2 d Z T 0 K 7 X i=1 (e (|Ψ i |-Ψ i ) -1)
dt + Err (41)

with Ψ i represents the constraints already defined, K is the multiplier of Lagrange equal to 10 6 in our calculation and Err can handle the errors due to the inverse geometric model. At the end of the optimization, we verify that all constraints are positive and Err is equal to zero in all cases.

V. SIMULATIONS AND RESULTS

The simulations were done with the geometrical and dynamic parameters of HYDROID Robot. The height of this biped robot is 1.39m with a total mass of 45.36 kg. The table II in appendix gives the physical parameters of each body part. For these simulations, the radii r 1 and r 2 are chosen equal to 5 cm. The simulation objective is to compare the configurations defined in II and to find the best solution. Two series of optimization were performed, one with the matrix of Coulomb friction Cs equal to zero and the other with actuator specific values given in table III in appendix. We will compare the effects of this matrix on the evolution of criteria and will observe the joint torque profile. We will examine the estimated cost of gait between our robot and those of the literature studies [START_REF] Collins | A bipedal walking robot with efficient and human-like gait[END_REF] for the human and other robots.

In Fig. 3, the evolution of optimal energetic criteria versus the walking speed is presented. For the first set of simulation with zero Coulomb friction, we see that the energy criteria of RK robot are lower than those of CK robot. The energy gain is about 4.6% in average. Considering the Coulomb friction of the actuators, the criterion of the configuration with rolling knee (RK) is reduced by 4.4% in average compared to the configuration with revolute joints (CK). Furthermore, we obtain a ratio of 6 between the energetic criteria with the null friction coefficients and those of actuators, for the slow speeds and a ratio of 2 for the high speeds. Fig. 4 shows the stick diagram of both robots of 1m/s without friction coefficient. The biped robots have the same gait. For the RK robot, the mobile foot is a bit higher on the start and the trunk more leaned forward at the end of the step. Fig. 5 presents the evolution of torques for the walking speed of 1 m/s. Without friction, the torques of the mobile The table I shows the cost of transport c et calculated by the (energy used)/(weight) * (distancetraveled). We can see that the results of the robot without friction coefficient are near the c et of the human.

VI. CONCLUSIONS

The model of the rolling knee joint robot has been developed. Our simulation program computes the joint torques and the forces on the feet for the different gait. The trajectories are parametrized by cubic spline functions and have allowed the resolution of the energy optimization problem. Without friction coefficients, the optimization process shows that the rolling knee structure gives an energy reduction of 4.6% in average. We demonstrate that the energy consumption of robot with zero Coulomb coefficients gives results near of human energy consumption. Taking into account the mechanical friction, the energy criterion is increased by a factor of 2 to 5 depending on the walking speed. The structure with rolling knees reduces slightly the energy criterion but especially facilitates the robot design to integrate linear actuators and transmissions that reduce the friction terms.

In future, we will develop other gaits such as Double Support Phases (DSP) including foot rotation. The influence of the radii r 1 and r 2 represents another interesting challenge for the robot design. For the selection of linear motor or artificial muscles, the design with two actuators locations, one on the thigh, the other on the trunk, may be considered. A more accurate model of friction can then be initiated.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Evolution of optimal energy criterion in function of the walking speed

Γ 1 Γ 2 Γ 3 Γ 4 Γ 5 Γ 6 Fig. 5 .

 1234565 Fig. 5. Evolution of torques for the walking speed at 1 m/s

TABLE I .

 I ESTIMATED COST OF TRANSPORT c et

	Biped Robot :	c et
	Asimo	3.2
	Collins' robot	0.2
	Robot CK (Cs = 0 6×6 )	0.295
	Robot CK (Cs ̸ = 0 6×6 )	0.6
	Robot RK (Cs = 0 6×6 )	0.292
	Robot RK (Cs ̸ = 0 6×6 )	0.58
	Human :	0.2

VII. ACKNOWLEDGMENTS

The authors gratefully acknowledge the contribution of French National Research Agency under the project number ANR-09-SEGI-011-R2A2.

APPENDIX