Julien Chevallier
email: <jchevall@u-paris10.

Intertemporal Emissions Trading and Allocation Rules: Gainers, Losers and the Spectre of Market Power

Keywords: JEL Classification: L1, Q5 Keywords: emissions trading, banking, borrowing, market power, optimal control, differential game

Stemming from politically given market imperfections in a tradable permits system, this paper develops a Stackelberg game with two types of non-cooperative agents to describe how a large -potentially dominant-agent may exercise market power at the expense of a competitive fringe.

In a dynamic framework with full forward and backward temporal flexibility (i.e. 1:1 Intertemporal Trading Ratio), this intra-industry model then suggests an optimal allocation criterion for grandfathered permits based on recent emissions.

This paper contributes to the permit trading literature by shedding some light on the decision to allow banking and borrowing, a debate which is typically overlooked by the debate to introduce the permits market itself among other environmental regulation tools. Provisional results are presented under perfect information.

Introduction

What happens on a tradable permits market when distortions occur as a consequence of the inital allocation? Whereas [START_REF] Hahn | Market Power and Transferable Property Rights[END_REF] contributed first to this debate by demonstrating the non neutrality of permits allocation for an agent able to exert market power1 in a static context and concerning the spatial exchange of permits only, this paper addresses the critical aspect of initial permits allocation in a dynamic context and concerning intertemporal emissions trading. Theoretical analyses remain scarce in this domain, even if the properties of banking (i.e. the ability to stock permits for future use) and borrowing (i.e. the ability to borrow permits from future periods) have been detailed. In a continuous time model under certainty, [START_REF] Rubin | A model of Intertemporal Emission Trading, Banking, and Borrowing[END_REF] shows that an intertemporal equilibrium exists on a permits market from the viewpoint of the regulator and the firm, and that banking and borrowing allow firms to smooth emissions. Under uncertainty, [START_REF] Schennach | The Economics of Pollution Permit Banking in the Context of Title IV of the 1990 Clean Air Act Amendments[END_REF] shows the permits price may rise at a rate less than the discount rate and new public information may cause jumps in the price and emissions paths, among other major contributions.

This article builds on the intertemporal emissions trading literature with market imperfections. It aims at filling the gap in the literature between the pros and cons of authorizing banking and borrowing in permit trading programs, a topic which is typically not enough debated when deciding to adopt such an environmental regulation system. Against this background, it attempts to shed some light on the ability of a large agent to move dynamic markets when permits are grandfathered. Liski & Montero (2005b) study the effect of market power on the equilibrium of a permits market by introducing a large potentially dominant and a competitive fringe. Based upon two cases, their analysis reveals first that the large agent might manipulate the market by banking allowances when it owns all the stock of permits and second when the fringe receives all the stock of permits, the large agent has an incentive to exchange permits at the competitive price and to build a permits bank for the next period. While previous papers restricted their analysis to banking only2 , both banking and borrowing are allowed without restrictions in a continuous time setting.

The model brings to the regulatory economics literature a realistic description of relationships between agents on a tradable permits market with information asymmetry. As Liski & Montero (2005b) did not impose a par-ticular game structure, I adopt a Stackelberg game structure that allows to deal with specific interactions between two types of agents: a leader with an informational advantage associated to a large agent, and a follower associated to a competitive fringe.

The market imperfection arises from the free distribution of permits on the basis of past emissions, while the product market is assumed to remain competitive3 . I explicitly include the Hotelling conditions4 that must apply if permits are considered as an exhaustible resource.

The paper unfolds as follows: first, I describe the institutional environment of current permit trading programs, namely the Kyoto Protocol (KP) and the European Union Emissions Trading Scheme (EU ETS); second I derive in a benchmark case an expression for market power; third I develop the Stackelberg game; fourth I suggest an optimal allocation criterion to correct for these market imperfections5 .

Description of the Institutional Environment

In this section, I describe how the model hinges on critical design issues of existing international emissions trading schemes, namely the KP and the EU ETS. I also attempt to provide a balanced picture of the EU ETS and KP market power concerns.

The Kyoto Protocol

The question of the Kyoto Protocol as an "unfinished business" is often evoked. Very heterogenous sectors were included under the same regulation, which could be detrimental to find the right method to allocate permits depending on price elasticities between sectors.

The intra-industry structure adopted in this paper may be seen as a simplification of the KP. Yet it may propose useful policy recommendations when dealing with such an international scheme. I focus on the negotiation phase, the special case of Russia and the prospective use of banking and borrowing.

Negotiation phase

In the context of the KP, the case of countries supplied with allocations in excess of their actual needs has been coined as Hot Air in the literature 6 . The distribution of a large number of permits to Former Soviet Union (FSU) and Eastern Europe countries (Russia, the Ukraine forming two thirds) may be seen indeed as an imperfection of the KP, as those countries were given generous allocations to foster agreement during the first phase (2008)(2009)(2010)(2011)(2012). Market power concerns arise as industrial firms may benefit from the gap between the initial permits allocation (based on 1990 production levels) and their real emission needs in 2008 (after a period of recession), and the use of these permits surpluses remains unclear.

This situation emerged as a conflict between the internal and the external consistency of the permits market:

• the internal consistency refers to the situation where agents freely receive or bid for permits according to their real needs. The regulator may be interested however in distributing more permits to a country than strictly needed (according to business as usual emissions or a benchmark for instance) in order to ensure participation to the permits market 7 . As a consequence, one agent may achieve a dominant position which in turns threatens the efficiency of the permits market itself.

• the external consistency of the permits market is linked to the broader debate of climate change as the purchase of a "global public good" 8 . This altruistic view embodies the notions of "burden sharing" or "differentiated responsibilities" attached to the KP, whereby developed countries agree to spend a higher income share on fighting climate change than developing countries 9 .

Those conflicting views undermine the negotiation of the cap, which is fixed at a suboptimal level compared to what would be needed to minimize the total damage to the environment. Greenhouse gases (GHG) emission targets under the KP represents a mere 5% reduction below 1990 levels. Now if early movers like EU countries are willing to ratchet down the cap, little 6 See [START_REF] Baron | Market Power and Market Access in International GHG Emission Trading[END_REF], [START_REF] Burniaux | How important is market power in achieving Kyoto?: An assessment based on the GREEN model[END_REF], [START_REF] Bernard | Russia's Role in the Kyoto Protocol[END_REF], [START_REF] Bohringer | Market Power and Hot Air in International Emissions Trading: the Impact of the US Withdrawal from the Kyoto Protocol[END_REF], [START_REF] Holtsmark | Russian Behaviour in the Market for Permits under the Kyoto Protocol[END_REF].

7 Such negotiation with Russia was determinant for the KP to enter into force on 16/02/05

8 See [START_REF] Guesnerie | The design of post-Kyoto climate schemes : an introductory analytical assessment[END_REF]. 9 Note that the implicit assumptions of the existence of such an Environmental Kuznets Curve (the environment is a superior good and environmental regulation becomes stricter through time at higher levels of GDP per capita) is left out of the debate.

progress can be achieved without luring in major players like the USA, India and China. Thus, many difficulties arise to pierce the "veil of uncertainty" around international negotiation 10 . Uncertainty also affects trading rules that will be effectively implemented at the international level. As [START_REF] Klepper | Trading Hot Air. The Influence of Permit Allocation Rules, Market Power and the US Withdrawal from the Kyoto Protocol[END_REF] 11 put it:"The Kyoto Protocol and its related decisions do not explicitly state who is actually supposed to be trading. Probably we will observe both government and firm trading. Under the former, market power might indeed become a relevant issue".

The fact that setting up a permits market might give some countries the opportunity to draw a financial advantage without a direct environmental gain (i.e. in the absence of effective emissions abatement) might be puzzling. But as stated in Maeda (2003) 12 , "[This debate] seems misguided because it focuses on the political importance of the issue, rather than addressing it from an economic perspective."

Overall, the hypothesis that generous allocations that broaden the scope of a cap-and-trade program might also give birth to dominant positions shall not be neglected. This leads me to comment the case of Russia more in depth.

2.1.2 Will Russia be a net seller of permits? Russia seems the best example to investigate potential market power within the KP according to [START_REF] Korppoo | Russia and the Kyoto Protocol: Opportunities and Challenges[END_REF] 13 : "Given the collapse of its emissions in the course of its economic transition, Russia is the country with by far the largest potential surplus of emission allowances for sale under the Kyoto international trading mechanisms. It is also generally considered to be the country with the greatest potential for continuing emission-reducing improvements in energy efficiency. Indeed, in the first commitment period under the Kyoto Protocol it could be described as the Saudi Arabia of the emerging carbon market, with the potential to try to manipulate the market through strategic decisions as to when and how it releases it surplus -if there are buyers willing to deal."

Empirical evidence gathered by [START_REF] Grubb | Russian Energy and CO2 Emission Prospects: Evidence from Domestic Analyses and International Comparisons[END_REF], Liski & Montero (2005a) 14 and [START_REF] Korppoo | Russia and the Kyoto Protocol: Opportunities and Challenges[END_REF] suggest Russia will be a net seller of allowances during the first phase of the KP. Different projections for Russian CO 2 emissions and surplus are detailed in Tables 1 and2 The key finding in Table 1 is that under all scenarios Russia would meet its Kyoto targets, as its CO 2 emissions projections consistently hit below 1990 levels. The room for interpretation of Table 2 is limited by the wide variation in surplus estimates with a lowest value of 246 M tCO 2 found by [START_REF] Bohringer | Hot Air for Sale: A Quantitative Assessment of Russia's Near Term Climate Policy Options[END_REF] and, as expressed above, by the current absence of clearly defined international trading rules to monetize such a surplus. Further projections regarding Russia's own energy demand after the first period of the KP are neeeded to determine whether Russian industrial firms might be able to affect negatively other members' marginal abatement costs.

Prospective use of banking and borrowing in the KP

This section offers a description of the possible use of banking borrowing in the KP. On the one hand, provisions on banking are explicited by [START_REF] Klepper | Trading Hot Air. The Influence of Permit Allocation Rules, Market Power and the US Withdrawal from the Kyoto Protocol[END_REF] 15 : "Assigned amount units (AAUs) resulting from the Kyoto commitment can be banked without a time constraint. Credits from Joint Implementation (JI) or Clean Development Mechanism (CDM) can be banked up to a limit of , respectively, 2.5% and 5% of a Party's initial assigned amount. Sink credits can not be banked ".

On the other hand, implicit provisions on borrowing may be found in the United Nations Framework Convention on Climate Change (UNFCCC ( 2000)) report16 . As explained by [START_REF] Newell | Managing Permit Markets to Stabilize Prices[END_REF] 17 : "International climate policy discussions have implicitly included borrowing within possible consequences for noncompliance under the Kyoto Protocol, through the payback of excess tons with a penalty (i.e., interest)."

As stated earlier, this issue remains unclear and reveals a lack of a specific debate on the pros and the cons to allow banking and / or limited borrowing, if not a lack of theoretical grounding.

The European Union Emissions Trading Scheme

Here I draw comments on two critical aspects of the EU ETS. First, I deal with possible design flaws in the allocation of permits that might pave the way for dominant positions during the first phase. Second, I provide an overview of the prospective use of banking and borrowing.

Over-allocation or relative success?

The EU ETS gently constrains emissions (8% reduction for EU-15) so as to start with a low carbon price. Yet the public debate has shifted toward a possible over-allocation of permits during the first phase. The production decisions of private actors are under scrutiny: do permits surpluses constitute a relative success (i.e. firms have reduced their emissions above projected levels) or an imperfection in the design of the system? 2006) 18 . While it is not our goal to comment on the structure of the European carbon market here, it seems interesting to look at the possible surplus those countries were endowed with.

Figure 2 depicts the 2005 reported emissions and, if any, the size of the surplus. The sum of the two bars is equivalent to the 2005 allocation of permits for a given country. It reflects a wide variety of cases among market participants as a bulk of countries (France, the Netherlands, Finland, Slovaquia, Sweden) was able to build a permits surplus above 15%, while Greece is short of permits. Surpluses also reflect in a limited amount reserves for new entrants, which are included in the data used .

Table 3 takes a closer look at the allowances surpluses in million tonnes of CO 2 and in percentage of the allocation. Its main finding lies in the fact that most countries seemed to favor generous allocations during the first phase of the EU ETS 19 . The biggest player, Germany does not seem to be in a position to exert market power with little more than 5% excess allowances. The surplus of Poland need not be overstated either since the use of 10% of allowances is missing in the Community Independent Transaction Log (CITL) administered by the European Commission. Germany, UK, and Portugal form a group of countries where the regulator strived to allocate optimally. On the contrary, stricter emissions reductions were enforced in the case of Ireland and Greece.

How could one explain those contrasting patterns in actual emissions for EU ETS participants in 2005? Part of the answer may be found in the decision making process within each National Allocation Plan (NAP). [START_REF] Godard | L'allocation initiale des quotas d'émission de CO2 aux entreprises à la lumière de l'analyse économique[END_REF] and [START_REF] Godard | Politique de l'effet de serre: une évaluation du plan francais de quotas de CO2[END_REF] describe the logic behind the French NAP when allocating shares of recent emissions baselines: non-electric utilities were supplied with their projected need in permits, while electric utilities were more constrained. This situation may be justified by the perceived abatement potential of the electricity industry, but it reveals overall the necessary arbitrages to be made due to sector heterogeneity and a stringent cap.

Due to a lack of data availability from 2004 onwards, it appears difficult to 18 Tendances Carbone is published by the French Caisse des Depots and is available at <http://www.caissedesdepots.fr/>, accessed on November, 24th

19 Apart from the EU ETS, there is a need to be cautious here with the notions of "over allocating" and conversely "under-allocating" permits depending on the country. Their meaning depends on the reference point (business as usual plus some abatement for instance). If other trading schemes implement per capita distribution for instance, it may appear less relevant to talk about "over allocation". This surplus needs to take into account 10% of allowances whose emissions have not been reported yet.

d A negative value refers to an actual excess in the level of emissions with respect to the cap.

provide a more precise characterization of the room for market power at the sectoral level. For French installations subject to the Directive 2003/87/CE, an estimation based on 1,402 installations totalling 185,3 M tCO 2 taken from the Register for Polluting Emissions (iREP)20 reveals almost 50% of permits were distributed to four players, and the first ten permits holder sum up to 60% of permits allocation. Each of this big players might exert a dominant position on its own sector if permits are distributed freely based on recent emissions, as modelled in this paper.

Prospective use of banking and borrowing in the EU ETS

Member states may allow banking without restriction. But this possibility to carry over EUAs from from 2005-2007 to 2008-2012 in the EU ETS is very limited in practice. Only France and Poland have allowed it to a certain extent, i.e. permits that were bought cannot be banked. Besides, only "green" firms that have effectively reduced emissions may bank allowances in Poland.

This section provided an overview of two major permits market along with their allocation rules. This background information is used as the basis for the modelling of a differential game with hierarchical play in the paper.

The Model

This section details the features of the model. First, I explain the design of the cap-and-trade program. Second, I examine the industry and information structures. Third, I define an intertemporal emissions trading constraint. Fourth, I express the Hotelling conditions. Finally, I explicit the properties of the abatement cost function.

Design of the Cap-and-Trade Program

The regulator sets a cap E on emissions of a given pollutant that corresponds to a specific environmental goal. The fix endowment is therefore exogenous to the model, and may be broken down into individual permits allocation e i mandatory for each agent i.

Agent [i = 1] is a large polluting agent, who is initially allocated a large number of permits. Agents [i = 2, . . . N ] aggregate many small polluting agent, who sare assumed to be comparatively smaller permits-holders. The premise of the paper is that the large agent might be able to exercise market power.

Permits are distributed freely on the basis of recent emissions 21 . Agents may bank and borrow permits without restrictions. We do not include the effects of allowing a safety valve 22 in the model.

Industry Structure

This partial equilibrium model features an intra-industry permit market in an homogenous single good economy. In what follows, I tend to neglect the interaction with the output market. An agent may be either a country, a firm or a cartel 23 . The competitive market price is determined by fringe agents' abatement costs.

Information Structure

I model a differential game 24 played in continuous time where all players have the possibility of influencing the rate of change of the permits bank through the choice of their current actions. It is therefore assumed that they adopt a Markovian strategy.

The common knowledge includes the fact that all players need to comply to the environmental constraint exogenously set by the regulator. The game unfolds in two steps. First, I derive the follower's reaction function to any action announced by the leader through fringe agents' cost minimization 21 See [START_REF] Ellerman | Absolute versus Intensity-Based Emission Caps[END_REF] for a review concerning the use of projections, benchmarking and intensity targets. While it is beyond the scope of this paper to study the relative merits of grandfathering and auctioning, theoretical analyses stress the superiority of auctioning as in [START_REF] Jouvet | Optimal Growth with Pollution: How to Use Pollution Permits[END_REF]. In the view of the Public Choice Theory, free allocation of permits may also be seen by some firms as a means to extract more permits as a scarcity rent, and therefore lobbying takes place. But it is also imposes liabilities on firms that will be reflected in their balance sheets. As highlighted by [START_REF] Raymond | Private Rights in Public Resources: Equity and Property Allocation in Market-Based Environmental Policy[END_REF], initial permits allocation reveals social norms embedded by newly created permits. The free distribution of permits may be seen as an entitlement over an environmental resource. As conceptions evolve and auctioning might become predominant, the question arises whether the probability of achieving a dominant position will increase or decrease.

22 A safety valve may be defined as an hybrid instrument to limit the cost of capping emissions at some target level whereby the regulator offers to sell permits in whatever quantity at a pre-determined price.

23 For instance, within the KP permits may be exchanged party-to-party, but also firmto-firm if Annex B members delegate this ability to private actors. In the third case, collusive behaviours may appear either between parties or between firms. (Liski & Montero (2005b))

24 See [START_REF] Dockner | Differential Games in Economics and Management Science[END_REF] for an overview of diffential games.

problem at the competitive price. Second, I observe how the the leader might exercise market power as large agent integrates the reaction function into his own cost minimization problem, and decides how to adjust his emissions level. I hold all other parameters constant25 .

Intertemporal Emissions Trading

Let B i (t) be the permits bank, with B i (t) > 0 in case of banking and B i (t) < 0 in case of borrowing. Any change in the permits bank is equal to the difference between e i (t) and e i (t), respectively agent's i permits allocation and his emission level at time t. The banking borrowing constraint may be written as:

Ḃi (t) = e i (t) -e i (t) (1)
with B i (0) = 0 and B i (T ) ≥ 0 as banking and borrowing are assumed to be always allowed26 .

Hotelling Conditions

Notwithstanding differences between a permit and an exhaustible resource 27 , it is assumed in the literature that the Hotelling conditions for exhaustible resources must apply on a permits market. Consequently, the terminal and exhaustion conditions are detailed below.

Terminal Condition

Let [0, T ] be the continuous time planning horizon 28 . At time T , cumulated emissions must be equal to the sum of each agent's depollution objective and therefore to the global cap E set by the regulator 29 :

T 0 N i=1 e i (t)dt = N i=1 e i = E
(2)

Exhaustion Condition

At time T , there is no more permit in the bank (either stocked or borrowed):

N i=1 B i (T ) = 0 (3)
Those conditions ensure that agents gradually meet their depollution objective so that the marginal cost of depollution is equalized in present value over the time period, and the permits bank clears in the end.

Abatement Cost Function

Let C i [e i (t)] be the abatement cost function 30 incurred by agent i in order to comply with his permits allocation e i . C i [e i (t)] is defined on 2 in continuous and is of class C 2 [0; T ], i.e. twice continuously differentiable. The classical assumption 31 of strictly increasing abatement costs leads C i [e i (t)] to be convex, with C i [e i (t)] < 0 and C i [e i (t)] > 0. I can set C i [e i (0)] = 0.

Agent's i marginal abatement costs (MAC) are associated with a one-unit reduction from his emission level e i at time t and are noted -C i [e i (t)] > 0. At the equilibrium of a permits market in a static framework 32 , price-taking agents adjust emissions until the aggregated MAC is equal to the price P at time t:

P t = -C i [e i (t)] (4)
Thus, at the equilibrium, there is no arbitrage for price-taking agents.

29 See also Leiby & Rubin (2001), p. 231. 30 Compared to a situation where profits are unconstrained, abatement costs appear in order to meet the emission cap e i .

31 Stated first by [START_REF] Montgomery | Markets in Licenses and Efficient Pollution Control Programs[END_REF]. The conditions given by [START_REF] Leiby | Intertemporal Permit Trading for the Control of Greenhouse Gas Emissions[END_REF] include the output q(t) where C i [q i (t), e i (t), t] is strongly convex with C i [q i (t)] > 0 and C i [q i (t)] > 0. Properties of non-convex abatement cost functions may be found in [START_REF] Godby | Market Power and Emissions Trading: Theory and Laboratory Results[END_REF].

32 See [START_REF] Hahn | Market Power and Transferable Property Rights[END_REF].

Benchmark Case

In order to reveal the conditions under which the large agent may exercise market power, I develop in the first section a benchmark case where the large agent receives all the stock of permits.

Optimization program

The expression of market power may be derived straightforward when the large agent owns all the stock of permits, and directly integrates the competitive price into his maximization program. In this setting, fringe agents' emissions come from trading with the large agent. To simplify notations in this section, let y 1 (t) be the number of permits distributed to the large agent.

The large agent behaves as follows:

                             min T 0 e -rt C 1 [e 1 (t)] + P t y 1 (t) dt T 0 e 1 (t) dt = E - T 0 N i=2 e i (t) dt P t = -C i N i=2 e i (t) y 1 (t) = N i=2 e i (t)
I form the Lagrangean with e 1 (t) and e i (t) as control variables, and λ(t) as a multiplier:

L = T 0 e -rt C 1 [e 1 (t)] -C i N i=2 e i (t) N i=2 e i (t) dt +λ(t) E - T 0 e 1 (t)dt - T 0 N i=2 e i (t)dt
The first-order conditions are: t)] in the second equation, dropping the time subscript for notational ease and rearranging terms, I get:

∂L ∂e 1 (t) = C 1 [e 1 (t)] -λ(t) = 0 ∂L ∂e i (t) = -C i N i=2 e i (t) N i=2 e i (t) -C i N i=2 e i (t) -λ(t) = 0 Replacing λ(t) = C 1 [e 1 (
-C i N i=2 e i N i=2 e i -C i N i=2 e i -C 1 (e 1 ) = 0 -C 1 (e 1 ) = C i N i=2 e i N i=2 e i + C i N i=2 e i -C 1 (e 1 ) = C i N i=2 e i 1 + C i C i N i=2 e i -C 1 (e 1 ) = P 1 + ε i N i=2 e i
with ε i defined as fringe agents' elasticity:

ε i = C i (e i ) C i (e i ) = dC i de i dC i de i = dC i de i de i dC i = dC i dC i
ε i measures the relative variation between an additional unit of emission and the acceleration of marginal cost. A high elasticity (in absolute value) induces a strong link between the two variables.

Market Power Condition

Market power is function of fringe agents' elasticity and of the large agent's number of permits:

ε i N i=2 e i (t) = ε i y 1 (t) (5)
Due to the convexity assumption, fringe agents' elasticity is negative, and reveals the possibility for the leader to affect negatively fringe agents' behaviour.

The large agent's MAC is therefore lower than under perfect competition, since he enjoys a dominant position.

Overall, the large agent may be characterized as a net gainer and fringe agents as net losers in this setting.

Fringe Agents' Reaction Function

The firs step of the game consists in forming the strategy of the fringe. Fringe agents choose their optimal emissions level according to the possibility to bank and borrow permits in constraint (1). The cost minimization program may be written as follows:

           min e i T 0 e -rt {C i [e i (t)] + P (t) [e i (t) -e i (t)]} dt Ḃi (t) = e i (t) -e i (t) B i (0) = 0, B i (T ) ≥ 0 C i [e i (0)] = 0
where the expression [e i (t) -e i (t)] represents the number of permits bought (> 0) or sold (< 0).

I write the corresponding current-value Hamiltonian and first-order optimality conditions:

H(B i (t), e i (t), λ(t), t) = {C i [e i (t)] + P (t) [e i (t) -e i (t)]} -λ(t)[e i (t) -e i (t)] ∂H ∂e i (t) = 0 : P (t) = -C i [e i (t)] + λ(t) (6)
Ḃi (t) = ∂H ∂λ(t) = 0 : Ḃi (t) = e i (t) -e i (t) (7)
λ(t) -rλ(t) = - ∂H ∂B i (t) = 0, λ(T )B i (T ) = 0 (8)
Note that the transversality condition in (8), required to meet the exhaustion condition (3), is a sufficient optimality condition. The bank has no scrap value at the end of the period.

It can be infered from (8) that λ(t) = λ(0)e rt . When fringe agents build a permits bank in terminal period, {B i (T ) > 0, λ(T ) = 0, λ(t) = 0}. The reaction function is therefore equal to the static equilibrium condition (4) where fringe agents equalize their MAC with the permits price.

Conversely, the reaction function is equal to (6) when fringe agents do not keep permits in the bank in terminal period and the constraint on λ is binding.

I now turn to the large agent's behaviour and to how he integrates the two possible cases of reaction function into his own optimization program.

6 Behaviour of the Large Agent

Optimization program

The large agent adjusts strategically his optimal emissions levels according to its initial allocation e 1 as expressed by (2) and the banking borrowing constraint (1). The cost minimization program for agent [i = 1] is:

                     min e 1 T 0 e -rt {C 1 [e 1 (t)] + P t [e 1 (t) -e 1 (t)]} dt Ḃ1 (t) = e 1 (t) -e 1 (t) E = T 0 e 1 (t)dt + T 0 N i=2 e i (t)dt B 1 (0) = 0, B 1 (T ) ≥ 0 C 1 [e 1 (0)] = 0

First case

Replacing P t by (4), the large agent's optimization program becomes:

                     min e 1 T 0 e -rt {C 1 [e 1 (t)] -C i [e i (t)] [e 1 (t) -e 1 (t)]} dt Ḃ1 (t) = e 1 (t) -e 1 (t) E = T 0 e 1 (t)dt + T 0 N i=2 e i (t)dt B i (0) = 0, B i (T ) ≥ 0 C 1 [e 1 (0)] = 0
Assuming fringe agents are homogenous, I write N i=2 e i (t) = (N -1)e i (t) and replace the emissions constraint (2) into the objective function:

           min e 1 T 0 e -rt C 1 [e 1 (t)] -C i E -e 1 (t) N -1 [e 1 (t) -e 1 (t)] dt Ḃ1 (t) = e 1 (t) -e 1 (t) B i (0) = 0, B i (T ) ≥ 0 C 1 [e 1 (0)] = 0
I form the corresponding current-value Hamiltonian with e 1 (t) as a control variable, B 1 (t) as a state variable, and µ(t) as a co-state variable:

H(B 1 (t), e 1 (t), µ(t), t) = C 1 [e 1 (t)] -C i E -e 1 (t) N -1 [e 1 (t) -e 1 (t)] + µ(t)[e 1 (t) -e 1 (t)]
Assuming the existence of an interior solution, necessary optimality conditions include:

∂H ∂e 1 (t) = 0 : C 1 [e 1 (t)] + 1 N -1 C i E -e 1 (t) N -1 [e 1 (t) -e 1 (t)] -C i E -e 1 (t) N -1 -µ(t) = 0 (9) Ḃ1 (t) = ∂H ∂µ(t) = 0 : Ḃ1 (t) = e 1 (t) -e 1 (t) (10)
μ(t) -rµ(t) = - ∂H ∂B 1 (t) = 0, µ(T )B 1 (T ) = 0 (11)
If {B 1 (T ) > 0, µ = 0}, the large agent also builds a permits bank in terminal period. From (9), it is possible to identify an analogous version of market power condition (5):

-C 1 [e 1 (t)] = -C i E -e 1 (t) N -1 1 + 1 N -1 C i C i [e 1 (t) -e 1 (t)] -C 1 [e 1 (t)] = P (t) 1 + 1 N -1 [e 1 (t) -e 1 (t)]
In comparison with the benchmark case where the large agent owns all the stock of permits, in the case where both types of agents keep a permits bank in the end, the large agent is able to affect fringe agent's MAC through the number of permits he holds in excess of his emissions.

Second case

In the second case, replacing P t by ( 6), the large agent's optimization program becomes:

                     min e 1 T 0 e -rt {C 1 [e 1 (t)] -{C i [e i (t)] + λ(t)} [e 1 (t) -e 1 (t)]} dt Ḃ1 (t) = e 1 (t) -e 1 (t) E = T 0 e 1 (t)dt + T 0 N i=2 e i (t)dt B i (0) = 0, B i (T ) ≥ 0 C 1 [e 1 (0)] = 0
Rearranging as above and setting {B i (T ) = 0, B 1 (T ) = 0, λ(t) > 0, µ(t) > 0} yields:

-C 1 [e 1 (t)] + µ(t) = -C i E -e 1 (t) N -1 1 + 1 N -1 C i C i [e 1 (t) -e 1 (t)] -λ(t)
This condition means that when fringe agents do not build a permits bank in the end, the large agent is still able to affect negatively fringe agent's MAC. Both values of µ(t) and λ(t) are known. For the large agent, the shadow value of a unit of emission in the bank measures the marginal utility of the state at time t along the optimal trajectory. For fringe agents, λ(t) reflects the highest hypothetical price at which they would be willing to pay for an additional permit at time t. However, the spectre of a large agent achieving a market power position may be averted by a careful design of the cap-and-trade program. This leads to the next section where I give hints about the goals of the regulator.

7 Critical Perspectives on Permits Allocation I examined so far how the distribution of a large number of permits to one agent according to current emissions may foster the emergence of market power. Now, I turn to the determination of the number of permits that it is optimal to distribute to the large agent without introducing distortions from the competitive equilibrium.

As in [START_REF] Eshel | Optimal Allocation of Tradeable Pollution Rights and Market Structures[END_REF] who demonstrates how the regulator might distribute permits in order to restore the overall economic efficiency of the permits market, I use first comparative statics to examine the change in agents' behaviour induced by marginal changes in the allocation of permits. Then, I adopt the regulator's viewpoint (associated to the social planner) to correct the market imperfection inherited from initial allocation in the previous example.

Comparative Statics

The effects of distributing an additional permit to the large agent will be captured by e 1 . In the case where µ = 0, differentiating the optimality conditions (9) to (11) with respect to e 1 gives:

∂e * 1 ∂e 1 = - 1 N -1 C i E -e 1 (t) N -1
The distribution of an additional permit to the large agent directly affects fringe agents' emissions level through a variation of their marginal cost. This result also holds in constrained regime. It may be infered optimal abatement costs levels will only be reached in a "no market power" situation.

Social Welfare

In this model, the goal of a welfare-maximizing social regulator consists in minimizing abatement costs of the large agent and fringe agents respectively, and the consumer's surplus losses from trade in the output S t . The regulator must distribute no more than e 1 (t) * to the large agent.

The optimization program of the regulator is:

min e 1 -e 1 T 0 e -ρt C 1 [e 1 (t)] + P t [e 1 (t) -e 1 (t)] dt + T 0 e -ρt C i [e i (t)] + P (t)[e i (t) -e i (t)] dt + T 0 e -ρt S t dt
where ρ represents the time preference coefficient.

The change of a one-unit allocation of permit on the large agent and fringe agents is found by differentiating their minimized cost function with respect to e 1 (t).

The FOC are:

∂ C 1 [e 1 (t)] + P t [e 1 (t) -e 1 (t)] ∂e 1 (t) + ∂ C i [e i (t)] + P (t)[e i (t) -e i (t)] ∂e 1 (t) + ∂S t ∂e 1 (t) = 0 (12)
For [i = 1], I get the following results in the first case:

∂ C 1 [e 1 (t)] + P t [e 1 (t) -e 1 (t)] ∂e 1 (t) = C i N i=2 e i (t) (13)
For [i = 2, . . . N ], I have in both cases:

∂ C i [e i (t)] + P (t)[e i (t) -e i (t)] ∂e 1 (t) = 0 (14)
In the first case, plugging ( 13) and ( 14) in ( 12), at the equilibrium the optimal permits allocation satisfies the equality:

-C i N i=2 e i (t) = ∂S t ∂e 1 (t)
and in the second case:

-C i N i=2 e i (t) + λ(t) = ∂S t ∂e 1 (t)
This equality reveals the different interests at stake when allocating permits, and may be summarized as follows:

An optimal allocation rule for a social planner with perfect foresight consists in distributing permits up to the point where an additional unit to the large agent damages consumers' surplus and establishes a dominant position affecting fringe agents' MAC.

Conclusion

The description of the institutional environment on which the model hinges provided a balanced picture of market power concerns in existing international emissions trading schemes. As for the Kyoto Protocol, trading rules in the making and the key role played by projections preclude from reaching a definitive conclusion, but it seems overall difficult to move an international permits market in a dynamic context. As for the EU ETS, the negotiation process of each NAP is typically an example of a manipulable rule whereby industries may conduct lobbying activities to extract more permits as a monopoly rent. With reference to the debate "rules vs. discretion" in monetary economics, there is a need for further research to ascertain the conditions under which it would be optimal to delegate the determination of the cap and the distribution of permits to an independent agency. A global conclusion concerning EU ETS market power concerns gears towards a prudent approach: if some firms have received more permits than projected, they might very well end up with a shortage of permits at the end of the first period because of an increase in emissions. Similarly, the EU Commission is especially careful during the validation of NAPs about their stringency and to the fact that there will be no ex-post adjustment.

Stemming from the description of the institutional environment of current international emissions trading schemes, I introduced a differential game with two types of agents on a tradable permits market, differing in terms of size and permits endowment, to ascertain the conditions under which a market power position may appear. Some similarities have been underlined between a benchmark case where the large agent owns all the stock and the model: in both cases where agents decide whether or not they build a permits bank, it is possible to identify net losers (i.e., fringe agents) and a net gainer (i.e., the large agent) as the large agent benefits from a lower marginal abatement cost than under perfect competition and is able to affect negatively fringe agents' marginal abatement costs. The model could be extended by the adoption of an intertemporal trading ratio specific to borrowing as discussed by [START_REF] Kling | Bankable Permits for the Control of Environmental Pollution[END_REF] 33 , allowing for a better grasp of the possibilities offered by intertemporal emissions trading.

But the spectre of market power need not be raised if the cap-and-trade program appears properly designed. An optimal allocation criterion for a welfare-maximizing social planner with perfect foresight would consist in distributing permits to the large agent up to the threshold where it creates consumers' surplus losses and negative impacts on fringe agents' marginal abatement costs. From a Public Choice perspective, this "razor's edge condition"34 highlights the difficulties encountered by the European Commission during the validation of the EU ETS Second National Allocation Plans where each national regulator needs to arbitrate between various interests at stakes. This debate between permits allocation and market power reveals the necessary compromise35 that need to be found between various conceptions on the role of environmental regulation.

As a final comment, one could say a greater reliance on banking and limited borrowing (i.e. with a specific discounting factor) should be promoted to allow firms to smooth their emissions and take investment decisions in abatement technologies with a better capacity to react to the evolution of the carbon constraint over time.

Figure 1 :

 1 Figure 1: EU ETS National Allocation Plans -Phase 1 (2005-2007) Source: Caisse Des Depots et Consignations (CDC, 2006)

Figure 2 :

 2 Figure 2: EU ETS 2005 Reported Emissions and Size of the Surplus Source: CDC (2006)

Table 2 :

 2 . Scenario with a 2% energy intensity reduction.

	Source	Year of	Percentage of	Period
		Estimate	1990 Levels
	Russian Energy Strategy	2000		76-93	2012
	IEA a World Energy Outlook 2004		72	2008-2012
	CEPA bc	2004		75	2008-2012
	Table 1: A Survey of Projections for Russian Carbon Dioxide Emissions.
	Source: adapted from Korpoo et al. (2006)
	Source		Year of	Size	Period
			Estimate	of	the
					surplus a
	Russian Ministry of Economic	2003		408-545	2008-2012
	Development and Trade b
	Russian Forecast to the UNFCCC c 2003		456-913	2008-2012
	CEPA		2004		400	2008-2012
	Klepper-Peterson		2005		410	2010
	Bohringer et al.		2006		246 d	2008-2012

a International Energy Agency. b Cambridge Economic Policy Associates. c A Survey of Projections for Russia's Surplus under the KP. Source: author a In million tonnes of carbon equivalent (MtCe). b adapted from Korpoo et al. (2005) c adapted from Korpoo et al. (2005) d M tCO 2

Table 3 :

 3 Size of the Surplus among EU ETS Participants in 2005.

	Country a	Size of the Surplus b )	Percentage of Allocation
	Finland	12,4	27,3
	Slovaquia	5,3	17,4
	Sweden	3,8	16,5
	France	24,3	15,6
	Netherlands	14,9	15,6
	Poland	33,7 c	14,1
	Belgium	7,5	11,9
	Germany	25,3	5,1
	Portugal	1,8	4,7
	UK	2,9	1,2
	Ireland	-0,1	-0,4
	Greece	-8,6 d	-11.6
	Others	>20	>47,5
			Source:
	CDC (2006)

a Data for Cyprus is missing. b European Union Allocation in MtCO 2 c

For an exhaustive literature review on permits trading and market power, see[START_REF] Petrakis | Location decisions of a polluting firm and the time consistency of environmental policy[END_REF].

This was mainly due to the fact that no major international agreement on greenhouse gases allows borrowing to a full extent to date (personal contacts with M. Liski).

For the distinction between permits market and industry structure imperfections, see[START_REF] Sartzetakis | Tradeable Emission Permits Regulations in the Presence of Imperfectly Competitive Product Markets: Welfare Implications[END_REF] and[START_REF] Sartzetakis | On the Efficiency of Competitive Markets for Emission Permits[END_REF]

Namely the exhaustion and terminal conditions.

See Eyckmans & Coenen (2006) for an introduction to the debate.

p.295

paragraph II.XV

p.149

The iREP is monitored by the Minister of the Environment and displays public information at <http://www.pollutionsindustrielles.ecologie.gouv.fr/IREP/index.php> (accessed on November, 24th 2006).

For instance, agents do not incur information costs.

See Rubin (1996), p.

 27 According to[START_REF] Liski | On Pollution Permit Banking and Market Power[END_REF] (p.3), the following differences may be highlighted. First, in a permits market with banking, the market may remain after the exhaustion of the bank; while the market of a non-renewable resource vanishes after the last unit extraction. Second, permits extraction and storage costs are equal to zero; while those costs are generally positive for a non-renewable resource. Third, the demand for an extra permit usually comes from a derived demand of other firms that also hold permits; while the demand for an extra unit of a non-renewable resource comes more often from a derived demand of another actor (e.g., a consumer).28 This planning period seems appropriate for a theoretical study of intertemporal emissions trading. Alternative time settings including distinct phases may be found in[START_REF] Montero | Explaining low sulfur dioxide allowance prices: the effect of expectation errors and irreversibility[END_REF],[START_REF] Schennach | The Economics of Pollution Permit Banking in the Context of Title IV of the 1990 Clean Air Act Amendments[END_REF] or[START_REF] Ellerman | The Temporal Efficiency of SO2 Emissions Trading[END_REF], but they reflect the specific requirements of the Acid Rain Program (USA).

The adoption of a discount rate penalizing borrowing may remove some of the perverse incentives whereby agents concentrate emissions on early periods, which is not socially optimal.

with reference to Harrod-Domar's growth model

This notion developed by[START_REF] Boltanski | De la justification: les économies de la grandeur[END_REF] refers to core agreements negotiated between actors.


Christoph Sprunk

Gian Diego Tipaldi

Andrea Cherubini
email: andrea.cherubini@lirmm.fr

Wolfram Burgard

Lidar-based Teach-and-Repeat of Mobile Robot Trajectories

Automation of logistics tasks for small lot sizes and flexible production processes requires the development of intuitive and easy-to-use systems that allow non-expert shop floor workers to naturally instruct transportation systems in changing environments. To this end, we present a novel laser-based scheme for teach-and-repeat of mobile robot trajectories that relies on scan matching to localize the robot relative to a taught trajectory, which is represented by a sequence of raw odometry and 2D laser data. This approach has two advantages. First, it does not require to build a globally consistent metrical map of the environment, which reduces set-up time. Second, the direct use of raw sensor data avoids additional errors that might be introduced by the fact that grid maps only provide an approximation of the environment. Real-world experiments carried out with a holonomic and a differential drive platform demonstrate that our approach repeats trajectories with an accuracy of a few millimeters. A comparison with a standard Monte Carlo localization approach on grid maps furthermore reveals that our method yields lower tracking errors for teach-and-repeat tasks.

I. INTRODUCTION

Autonomous transportation platforms are increasingly employed for logistics applications in industrial contexts. Initially, these systems were relying on guidance wires or optical markers to couple the robot to routes [START_REF] Mäkelä | Development of a navigation and control system for an autonomous outdoor vehicle in a steel plant[END_REF], [START_REF] Leea | Navigation of automated guided vehicles using magnet spot guidance method[END_REF]. In the last years, the paradigm shifted to the use of on-board sensors for localization and navigation, but for the most part still augmenting the environment with artificial markers [START_REF] Beinhofer | Near-optimal landmark selection for mobile robot navigation[END_REF]- [START_REF] Vitus | Sensor placement for improved robotic navigation[END_REF]. While the well established systems are bound to their routes by the physical location of markers in the environment, more recent systems do not require such an augmentation of the environment and rather rely on a globally consistent metrical map to allow more flexible route planning.

However, all these systems require an expert user to adapt to changes of the environment or transportation routes. The global map has to be rebuilt and docking positions at route destinations have to be re-taught, e.g., using the approach of Röwekämper et al. [START_REF] Röwekämper | On the position accuracy of mobile robot localization based on particle filters combined with scan matching[END_REF]. At the same time, to make automation profitable for small lot sizes and frequently changing production processes one requires systems that can be intuitively used by non-expert shop floor workers. The system proposed in this paper is targeted towards a natural one-button approach that lets the user remotely control the robot to demonstrate a new trajectory that is then reproduced with high accuracy and precision. In this paper we describe an approach that provides an intuitive teach-and-repeat framework for robot navigation. Our method represents user-taught trajectories by so-called anchor points consisting of raw odometry and 2D laser data. It applies a scan matching routine to estimate the actual offset of the current robot position from these anchor points. By comparing the reference offset with the actual offset, our algorithm calculates the error in the configuration space of the robot, which is then used for any desired feedback controller, e.g., a linear quadratic regulator or dynamic feedback linearization. With respect to standard appearance-based navigation, our approach differs in the sensor modality (laser scans vs. camera images) and in the feedback (in configuration space vs. in sensor space). Practical experiments carried out in complex real-world scenarios demonstrate that with our navigation scheme, robots can repeat taught trajectories more accurately than with the standard Monte Carlo localization approach using a global grid map.

II. RELATED WORK

The most widespread approaches to mobile robot navigation are model-and appearance-based approaches [START_REF] Cherubini | Comparing appearance-based controllers for nonholonomic navigation from a visual memory[END_REF]. Modelbased approaches usually rely on a map of the environment, whereas appearance-based approaches do not require such a model and work directly in the space of the sensor data. The environment is usually described by a topological graph, in which each node corresponds to the description of a position, and a link between two nodes defines the possibility for the robot to move autonomously between the two positions that the link connects. Our focus here is on appearance-based navigation.

Fig. 2. During the teaching phase, we record the velocities v * (t i ) at discrete time steps t i , and use them to derive the taught trajectory poses x * (t i ) (black dots). Along this trajectory, we insert anchor points. This schematic figure also illustrates the used coordinate systems.

Recently, many researchers [START_REF] Cherubini | Comparing appearance-based controllers for nonholonomic navigation from a visual memory[END_REF]- [START_REF] Malheiros | Robust and realtime teaching of industrial robots for mass customisation manufacturing using stereoscopic vision[END_REF] have developed appearance-based frameworks for realizing teach-and-repeat navigation. During a preliminary teach phase, the robot motion is controlled by a human operator, and a sequence of sensor data is acquired and stored in a database. Then, during the repeat phase, the robot is required to repeat the same path, by comparing the currently observed and the previously recorded sensor data. Most of these works rely on visual data.

Pioneering work [START_REF] Matsumoto | Visual navigation using viewsequenced route representation[END_REF] associated a particular motion (e.g., "go forward", "turn left") to each recorded image, in order to move from the current to the next image in the database. In [START_REF] Royer | Monocular vision for mobile robot localization and autonomous navigation[END_REF], a three dimensional representation of the taught path is built from the image sequence, and a classic path following controller is used for navigation. A similar approach, but using omnidirectional cameras, has been proposed in [START_REF] Goedemé | Omnidirectional vision based topological navigation[END_REF]. There, a path is represented by a sequence of images and a local 3D map is reconstructed from them. Path following is then done with a sequence of homing vectors to each image and feature tracking is used to localize the robot. The controller presented in [START_REF] Argyros | Robot homing by exploiting panoramic vision[END_REF] exploits angular information regarding the features matched in panoramic images. In [START_REF] Cherubini | Visual navigation with a timeindependent varying reference[END_REF], a novel varying reference based teach-and-repeat framework is proposed for outdoor navigation with a monocular camera.

In industrial applications, impressive results have been obtained with an infrastructure-free robot that uses three cameras to navigate with naturally occurring visual cues (mainly on the warehouse floor) learned during operation [START_REF] Kelly | An infrastructurefree automated guided vehicle based on computer vision[END_REF]. Another teach-and-repeat framework for a mass customization manufacturing scenario relies on high-intensity visual markers and stereovision to enable quick and easy use and configuration for non-skilled users [START_REF] Malheiros | Robust and realtime teaching of industrial robots for mass customisation manufacturing using stereoscopic vision[END_REF]. Furthermore, camera images have been replaced by a laser reflectivity image in [START_REF] Mcmanus | Lightinginvariant visual teach and repeat using appearance-based lidar[END_REF], which extends the authors' previous work on visual teach-and-repeat [START_REF] Furgale | Visual teach and repeat for long-range rover autonomy[END_REF].

While all these works rely on images, our objective here is to validate a similar approach, based solely on odometry and 2D laser scanners, that are generally mounted on industrial robots for safety reasons. Closely related is the approach of Marshall et al. [START_REF] Marshall | Autonomous underground tramming for center-articulated vehicles[END_REF]. The main idea of their work is to construct a sequence of local grid maps rather than having a globally consistent, monolithic map. During the teach phase the maps are built and a path is recorded. The recorded path is then repeated by localizing the robot in the local maps using the unscented Kalman filter. Our approach differs from this method as we do not rely on grid maps and instead operate directly on the sensor data.

In previous work [START_REF] Sprunk | Improved non-linear spline fitting for teaching trajectories to mobile robots[END_REF], we also considered teach-and-repeat tasks with mobile robots. There, however, we expressed the paths as splines in a global coordinate frame and applied Monte Carlo localization to localize the robot in a global grid map. Here, we strive to repeat the user input as exactly as possible, without relying on a global, metrically consistent map of the environment.

III. PROBLEM DEFINITION

The objective of our work is to repeat, with a wheeled mobile robot and no map, a time trajectory previously taught by user demonstration. The ground is assumed planar, and the robot may be either holonomic or subject to nonholonomic constraints (e.g., a differential drive robot), and it is equipped with a laser scanner mounted parallel to the ground.

We name r the robot center of rotation, which should track the trajectory. With reference to Fig. 2, we define the world frame F (O, x, y, z). The robot configuration is:

x = [x, y, θ] ∈ SE (2) , (1)
where x and y represent the Cartesian position of r in F, and θ ∈ (-π, π] is the positive counterclockwise orientation of the robot with respect to x. The control inputs are:

v = [v x , v y , v θ ] ∈ R 3 . (2)
These are respectively the longitudinal translational, lateral translational, and rotational velocities, with sign conventions as shown in Fig. 2. For a differential drive robot, v y = 0. The goal of this work is to drive, at time t ∈ [0, T ], the robot configuration x(t) to the taught trajectory x * (t). This trajectory is defined using the robot recorded odometry, as explained below. To realize this task, we rely on two sensors:

Wheel encoders: Measurements of the revolutions of the robot's wheels are available at discrete timesteps t i (t 0 = 0). The robot velocities v are estimated by differentiating the encoder readings. Laser scanner: The robot on-board laser scanner provides, at discrete timesteps t j , a local 2D scan of the environment surrounding the robot.

The robot velocities estimated from the wheel encoders can be used to localize the robot at any time t ∈ (t i , t i+1 ], employing the well known odometry equation [START_REF] Siciliano | Robotics: Modelling, Planning and Control[END_REF]:

x(t) = x(t i ) ⊕ R z (θ(t i ))v(t i ) (t -t i ) , (3)
with R z the rotation matrix about the z axis, and ⊕ the compounding operator [START_REF] Smith | Estimating uncertain spatial relationships in robotics[END_REF]. During the teach phase, the estimated robot velocities are recorded. By plugging these recorded velocities v * (0), v * (t 1 ), v * (t 2 ), . . . , v * (T ), and the initial pose x(0) into Eq. ( 3), we obtain the taught trajectory x * (t) to track. Although it has piece-wise constant velocity, this trajectory will provide a sufficiently accurate estimation of the real user-taught one, if the encoder frequency is sufficiently high. A schematic illustration of a recorded reference trajectory is given in Fig. 2, in which the black dots correspond to the poses x * (t i ) at timesteps t i .

The local 2D scans of the environment are used by our approach in conjunction with a scan matcher to provide relative

x * (t) O a k-1 O a k x(t) o(t) s(t) e(t)
x y z O Fig. 3. During the repeat phase, the reference pose of the robot (hollow dart) at time t, x * (t), is expressed as relative offset oa k (t) with respect to the current anchor point a k , see Eq. ( 5). The actual offset sa k (t) of the current robot pose x(t) (shaded dart) from the anchor point a k is measured by a scan matcher (Eq. ( 6)) and used to compute the error ea k (t) as in Eq. ( 7). In the figure, the indices of these vectors have been dropped to improve readability.

estimates of the robot's pose. Note that in our approach, those estimates are not integrated and therefore not prone to accumulating drift. During the teach phase, scanner readings are recorded at regular intervals along the trajectory: we record a new scan as soon as the distance covered since the last one exceeds a fixed threshold τ l > 0 or the angular displacement exceeds τ θ > 0. The rationale behind this is that we want to limit the required memory resources, while uniformly scanning the environment (e.g., avoiding multiple scan recordings with a standing robot). Based on these scans, we construct the anchor points that will be used for the repeat phase. An anchor point a k = (l k , x(t k )) consists of a laser scan l k with the associated robot pose x(t k ), computed via Eq. (3). Fig. 2 shows two anchor points along a recorded trajectory. During the repeat phase all available laser scans l j are used. They are matched against the scans associated with the anchor points to compute the error signal used in our feedback controller.

IV. REPEATING THE TAUGHT TRAJECTORY

In this section, we explain how the error signal is computed and utilized in our trajectory tracking controller. The objective of the proposed task is to regulate to zero, at all times, the error between the current and the taught reference poses:

e(t) = x(t) x * (t) , (4)
where is the reverse compounding operator [START_REF] Smith | Estimating uncertain spatial relationships in robotics[END_REF]. The computation of this error signal requires precise localization of the robot at time t, i.e., a precise estimate of x(t). However, as mentioned before, one of the main objectives of this paper is the realization of an intuitive map-free navigation procedure. An alternative to map-based localization consists of estimating x(t) from odometry alone, i.e., dead reckoning. However, it is well known that this approach is subject to a drift, which becomes significant over long paths.

To deal with this issue, we exploit the anchor points introduced in the previous section. More specifically, during navigation, we rely on the anchor point, noted a, closest to the current robot pose. Fig. 3 shows a schematic example with the actual robot pose indicated by a shaded dart and the reference pose on the taught trajectory indicated by a hollow dart. Here, anchor point a k is selected as the nearest to the robot. In the following, we drop the index k for readability.

Given the current anchor point a = (l a , x a ), two steps are required to estimate the task error e(t) = e a (t). First, we define the reference offset o a (t) between the reference pose x * (t) and the pose of a at time t, as:

o a (t) = x * (t) x a . ( 5
)
This vector is outlined in Fig. 3 by the arrow pointing to the hollow dart. Second, a scan matcher compares the current laser scan with the anchor point scan. Given anchor point a = (l a , x a ) and the current laser scanner reading l, the output of the scan matcher is the offset between the respective robot poses at which the scans were taken:

s a (t) = x(t) x a . ( 6
)
This vector is the arrow pointing from the anchor to the shaded dart in Fig. 3. We initialize the scan matcher by compounding the previously computed offset with the robot motion. This ensures a quick convergence to the correct minimum. Analyzing Eq. ( 5) and Eq. ( 6), we see that the reference offset o a (t) and the measured one s a (t) now represent the reference pose x(t) and the estimated pose x * (t) in the reference frame of the same anchor point a. Combining these equations with Eq. ( 4), we obtain the final expression required for the feedback control:

e(t) = s a (t) o a (t) . (7)
This vector is represented by the red arrow connecting the two darts in Fig. 3. This error measure does not require a precise global grid map, but relies on the closest anchor point. Therefore, drift cannot accumulate between more than two consecutive anchor points. To choose the closest anchor, we keep track of the offsets between subsequent anchors during teaching. This way, given the offset to the current anchor, we can determine the offset to the next few anchor points and switch to the closest one.

A. Controllers for repeating the taught trajectory

The task error defined above can be used by a cornucopia of controllers and the achievable tracking performance will obviously depend on the chosen controller and on its parametrization. The choice of the controller is orthogonal to the proposed feedback error scheme and out of the focus of this work. Nevertheless, to test our feedback control scheme, we implemented two classical controllers, one for holonomic, and one for nonholonomic kinematics.

For both controllers, we use the velocities v * (t i ), recorded during teaching, as the feedforward part. For t ∈ [t i , t i+1 ], we get v * (t) by linear interpolation of v * (t i ) and v * (t i+1 ). For the holonomic robot, we employ linear feedback:

v = v * -Λe , (8)
with Λ a positive definite gain matrix. Plugging this into the derivative of (4) yields ė = -Λe , for which, as desired, x * is a globally asymptotically stable equilibrium. For the nonholonomic robot we use: with λ x , λ y and λ θ positive scalar gains. This controller is known to give good performance for trajectory tracking [START_REF] Siciliano | Robotics: Modelling, Planning and Control[END_REF].

v x = v * x cos e θ -λ x e x v θ = v * θ -λ y e y -λ θ e θ , (9)
For the implementation on our robots, we choose odometry as the heartbeat of our controllers. As it is not synchronized with the laser on our robots, we compound the offset for the last processed scan with the subsequent motion of the robot to retrieve the offset for the current odometry reading.

V. EXPERIMENTS

We implemented the proposed algorithms in C++ and tested them on two real robots: a holonomic KUKA omniRob (shown in Fig. 1) and a differential drive Pioneer P3-DX robot (shown in Fig. 4). Both robots are equipped with laser scanners, calibrated to the robot center. The omniRob is equipped with two Sick S300 with a 270 degree field of view and 541 beams. The scanner on the Pioneer is a Sick LMS 291 with a 180 degree field of view and 181 beams. To highlight the performance of the proposed system, we chose to use only the front laser of the omniRob. The approach, however, can easily be extended to multiple laser scanners.

For scan matching we rely on a variant of the iterative closest point algorithm with a point-to-line metric as proposed by Censi [START_REF] Censi | An ICP variant using a point-to-line metric[END_REF]. We chose this approach for its high accuracy and its ability to work in complex environments with a moderate amount of changes [START_REF] Röwekämper | On the position accuracy of mobile robot localization based on particle filters combined with scan matching[END_REF].

The environment in which the experiments were conducted is shown in Fig. 1. The ground truth for our evaluations is provided by a MotionAnalysis motion capture studio with 10 Raptor-E cameras. During both teaching and repeating, the system tracks our marker-equipped robots at 100 Hz, typically with sub-millimeter precision. We repeat the trajectories, using respectively controllers [START_REF] Matsumoto | Visual navigation using viewsequenced route representation[END_REF] for the omniRob, and (9) for the Pioneer, with the error calculated using Eq. [START_REF] Cherubini | Comparing appearance-based controllers for nonholonomic navigation from a visual memory[END_REF]. We compare the performance with the same controllers when e is derived from Monte Carlo localization in a global grid map [START_REF] Röwekämper | On the position accuracy of mobile robot localization based on particle filters combined with scan matching[END_REF], and when it is derived using wheel encoders alone.

To account for communication and execution delays encountered with our robots, we retrieve the feedforward component from the taught trajectory with a lookahead of ∆t delay into the future, i.e., we use v * (t + ∆t delay ) instead of v * (t) in controllers ( 8) and ( 9). Unless stated otherwise, we used the following parameters for our controllers: for the omniRob, the gain matrix was set to Λ = diag (2.0, 2.0, 1.0) and ∆t delay = 0.15, whereas for the Pioneer we used λ x = 0.6, λ y = 14, λ θ = 3.5 and ∆t delay = 0.2.

For the omniRob, we inserted anchor points every τ l = 0.07 m or τ θ = 0.05 rad (2.9 degrees) along trajectories. For the Pioneer, laser data was available at a higher frequency allowing to insert anchor points approximately every τ l = 0.03 m or τ θ = 0.05 rad (2.9 degrees).

To assess the quality of trajectory reproduction with our proposed error design from Eq. ( 7), we compare the motion capture studio recordings of the repetitions with the recordings of the user demonstrations. Since sufficiently accurate time alignment of capture data from teach and reference trajectories was not available, we compute the minimal distance to the polyline path given by the captured points of the reference trajectory as error measure. We then report statistics on these minimal distances over multiple repetitions. To show that our approach is indeed able to track a trajectory and not only a path, we manually aligned the recordings for one set of runs to also show the evolution of the error over time.

A. Experimental setup

We test our approach on three different reference trajectories for both the omniRob and the Pioneer, yielding a total of six different trajectories. The reference trajectories as recorded by our motion capture studio are shown in Fig. 4 and are named FigureEight, PointToPoint, and Fast for future reference. The trajectories last between 36 and 70 seconds with velocities of up to 1 m/s (omniRob) and 0.5 m/s (Pioneer).

The chosen trajectories aim at representing typical navigation patterns for different tasks. The FigureEight represents a general navigation pattern in which the robot is required to navigate smoothly for long distances, or tasks such as patrolling or monitoring, in which continuous operation is needed. The reference trajectory PointToPoint is less smooth than the FigureEight and contains turns on the spot. The idea behind this type of trajectory is to represent typical pick-andplace tasks or more general point-to-point navigation, in which the robot is required to reach a predetermined location, perform a manipulation task and then move to the next location. Finally, we wanted to test the performance of the proposed system when the robot travels near its maximum controllable speed. This test is reflected in the Fast trajectory, in which higherspeed and more dynamic maneuvers are present.

We also compare our feedback error scheme to the one based on Monte Carlo localization (MCL) and to the one based only on wheel encoders. For these schemes, we only change the way of measuring the offset s a to the current anchor point. For the MCL scheme, we built a global gridmap of the robot environment with a resolution of 0.01 m and augmented the anchor points with the respective MCL estimate of the robot's location during the teach phase. During repetition, the system then determines s a through the current MCL localization. Note that this latter scheme corresponds to the widespread approach for laser-based navigation. For repeating the trajectory with odometry alone, only one anchor point at the start of the trajectory is used. The offset to this anchor is determined by keeping track of the relative motion of the robot since the start.

B. Comparison with Monte Carlo localization

Statistics over the minimum distance error measure (introduced above) for trajectory reproduction are shown in Fig. 5 for both robots and all reference trajectories. In total, 190 runs have been executed to gather the data reported in this figure .  To avoid biasing the evaluation with the user errors made in positioning the robot at the start point of the trajectory, we discard data generated in the first 1.0 s (omniRob) and 2.0 s (Pioneer) of the trajectory reproduction. Fig. 6 shows example paths taken by the omniRob and the Pioneer for the FigureEight reference trajectories. For these particular examples, Fig. 7 depicts the norm of the tracking error over time as a result of a manual time alignment of motion capture data from the teach and repeat phases. The low errors over the whole trajectory indicate that we are indeed tracking the reference trajectory and not just the geometric path, i.e., that our approach is also accurate with respect to the demonstrated velocities. Typically, the errors computed at a specific point in time are higher than the ones considering only the geometry of the tracked path. Moreover, the figure also includes the very start of the trajectory reproduction. As can be seen, especially in the omniRob experiments, there are higher errors at the beginning, which were introduced by the user who could not accurately position the robot.

The plots show that both our approach and a Monte Carlo localization based computation of the feedback error (MCL) are able to track the trajectory with considerable accuracy. However, when analyzing the error, we can see that our scheme achieves an accuracy that is higher by one order of magnitude compared to MCL. For the Pioneer PointToPoint experiment, the higher errors made it necessary to adapt the controller gains to λ x = 4.0, λ θ = 1.0 for MCL, as the original values lead to oscillation and worse results for the encountered errors.

As the plots show, the proposed approach is able to reproduce the taught trajectories with high accuracy and precision, not only on average but also in the worst case. These results are general and do not depend on the type of the robot or on the specific path driven. In other words, our feedback error scheme is able to effectively solve the desired task without relying on a globally consistent metrical map. The reference trajectories taught by the user are tracked with high precision for both, a holonomic and a differential drive robotic platform. The higher tracking errors of the MCL tracking scheme can be attributed to the discretization introduced by the grid map and to the local residual inaccuracies resulting from the optimization process for establishing global consistency in the map.

C. Comparison with wheel encoders

For further analysis and insight, we also compare against the tracking performance of 31 runs with a purely odometry based feedback error computation (ODO) for the FigureEight experiments. The drift of odometry becomes apparent in Fig. 6 and explains the substantially larger errors in the boxplots in Fig. 8 and in the tracking error in Fig. 7. That the errors still remain moderate can be explained by the path shape of the FigureEight experiment: the errors introduced by odometry drift tend to cancel out by the maneuvers, leading to longer periods of moderate tracking error. The analysis of pure wheel encoder based tracking also confirms that the high accuracy achieved by our approach is not simply caused by decent wheel encoders and floor conditions.

D. Evaluation with respect to anchor point density

In this experiment we analyze the effects of substantially reduced anchor point density along the taught trajectory. The experiment was performed with seven runs each for the Pioneer reference trajectories with anchor points reduced to occur approximately every τ l = 0.5 m or τ θ = 0.5 rad (29 degrees). This decreased the number of anchor points from 810 to 46 for FigureEight, from 567 to 40 for PointToPoint, and from 444 to 41 for Fast. The results of this evaluation are shown in Fig. 9. Despite the reduction of anchor points by an order of magnitude, the tracking errors increase only moderately. The increase is the highest for the PointToPoint trajectory which contains prominent turns on the spot. Here, the new rotational offset of 29 degrees between anchor points has the highest impact, since it severely reduces the overlap between laser scans for the scan matcher.

The stable results for reference trajectories with substantially sparsified anchor points show that the high performance of the proposed approach is due to the underlying principle and cannot simply be explained by a huge amount of data expended to tackle the task.

VI. CONCLUSION

In this paper, we presented an framework for intuitive specification of teach-and-repeat robot navigation tasks. With our approach, the user only needs to demonstrate the desired trajectory once during a teaching phase and does not need to build a globally consistent map of the environment. Our approach stores 2D laser data and raw odometry in anchor points during teaching and then uses them for feedback control of the robot during the repeat phase. We evaluated our approach with real robots and compared it to alternative ways of feedback error generation for several trajectories. Our approach is able to track the taught trajectories with millimeter accuracy and outperforms feedback generation by Monte Carlo localization based on a grid map.

Fig. 1 .

 1 Fig. 1. Time-lapsed view of the holonomic omniRob robot reproducing the user-taught reference trajectory FigureEight with our approach without a globally consistent metrical map.

Fig. 4 .

 4 FigureEight

Fig. 5 .

 5 Figure EightPointTo Point Fast

Fig. 6 .

 6 Fig.[START_REF] Röwekämper | On the position accuracy of mobile robot localization based on particle filters combined with scan matching[END_REF]. Taught and repeated FigureEight trajectories for each approach to feedback error computation with the omniRob (top) and the Pioneer (bottom).

Fig. 7 .

 7 Fig. 7. Norm of the tracking error over time for FigureEight for omniRob (left) and Pioneer (right) as measured by the motion capture studio for the runs also shown in Fig. 6.

Fig. 8 .

 8 Fig. 8. Boxplots comparing tracking errors for our scan matching based feedback error computation (SM) and feedback error generation based on odometry only (ODO) for the FigureEight trajectories. The still moderate errors for odometry are explained by the trajectory shapes (see text).

Fig. 9 .

 9 Figure EightPointTo Point Fast

This work has partly been supported by the European Commission under contract numbers FP7-248258-First-MM and FP7-260026-TAPAS.


Andrea Cherubini

Robin Passama

Arnaud Meline

André Crosnier

Philippe Fraisse

Philippe Fraisse Sensor-Based

Sensor-based control of a collaborative robot

Sensor-based control of a collaborative robot

Andrea Cherubini, Robin Passama, Arnaud Meline, André Crosnier and Philippe Fraisse Flexible and reactive control of interaction between humans and robots will allow a closer cooperation in all service and industrial tasks, that require the adaptability skills of humans to be merged with the high performance of robots in terms of precision, speed and payload [START_REF] Bicchi | Safety for physical humanrobot interaction[END_REF]. For this reason, recent research strives for intuitive human-robot cooperation, avoiding explicit clarification dialogue and commands.

We believe that sensor-based methods, such as visual servoing, provide better solutions, for intuitive HRI, than planning techniques requiring a priori models of the environment and agents. To our knowledge, few works [2 -4] have explicitly dealt with HRI using sensor-based control approaches. Our objective is similar: we aim at enabling intuitive interaction between human and robot, in the context of an industrial scenario, where the two must collaborate to realize a task. The robot must be able to infer the human intentions during the task, using only sensed data.

The objective of this work is to enable a robot to aid a human operator in a screwing operation (see Fig. 1). Human and robot are operating on the opposite sides of a flank, where a series of screws must be inserted. The required operations are respectively:

• for the human: to insert the screws in the holes,

• for the robot: to tighten a bolt on each inserted screw.

To realize the proposed operation, we utilize two sensors: a Microsoft Kinect that observes the work scene from a fixed pose, and a camera rigidly linked to the robot end effector. These sensors are respectively dedicated to predicting the human intention, and to detecting new non-tightened screws.

To realize the proposed task, we apply a multimodal strategy, where the four modes are each related to a subtask. The four modes are:

• Hand approaching. If the human operating hand is detected by the Kinect, the robot is moved so that the camera has a good view of the area where it is operating. • Screw approaching. Once a non-tightened screw is detected in the camera image, a 2 1/2 D visual servo controller [START_REF] Malis | 2-1/2D visual servoing[END_REF] moves the robot tool in front of it. • Collaborative tightening. If the robot is sufficiently near, the tool is placed on the screw to conclude the task 1 . • Halting. If neither the hand nor the screws are detected, the robot stops. To properly activate these modes, we utilize a simple state machine. The transitions between modes are activated by This work has been supported by the ANR (French National Agency) ICARO project.

More details on this work are given in [START_REF] Cherubini | Multimodal control for human-robot cooperation[END_REF]. The authours are with the Laboratory for Computer Science, Microelectronics and Robotics LIRMM -Université de Montpellier 2 CNRS, 161 Rue Ada, 34392 Montpellier, France. {firstname.lastname}@lirmm.fr 1 For the moment, this operation is executed in open-loop within our framework, since no sensed feedback is available at this stage. In future work, we plan to integrate force control for this subtask. perceived information (by the Kinect and camera) or by successful tightening. Except for the halting mode, the three other modes are realized with a unified formalism, the task Jacobian controller [START_REF] Siciliano | Robotics: Modelling, Planning and Control[END_REF]. For the experiments, we use a lightweight KUKA LWR IV robot. Since this robot is redundant with respect to the tasks, we use the remaining degree of freedom to guarantee joint limit avoidance.

Since a tightening tool is not currently mounted on our end effector, we have validated our approach by verifying if the robot could successfully touch new, non-tightened screws with its tool. This task requires high accuracy, since the screw and tool diameters are respectively 4 and 14 mm. We have run a series of experiments, where screws have been successfully touched by the effector, so we are confident that with a tightening tool the approach will also work. A typical experiment is shown in the video located at http://youtu.be/P8wfQ5tOa5E.

In summary, we have presented a marker-less solution for human intention recognition and human-robot collaboration, and intuitive communication between the two agents, realized through action (specifically, screw inserting).

Fig. 1 .

 1 Fig. 1. Top: experimental setup. Bottom left: view of the camera and effector. Bottom center: Kinect image. Bottom right: camera image.


Benjamin Navarro

Prajval Kumar
email: prajval10@gmail.com

Aïcha Fonte

Philippe Fraisse

Gérard Poisson

Andrea Cherubini

Active

Active calibration of tactile sensors mounted on a robotic hand

Active calibration of tactile sensors mounted on a robotic hand

Benjamin Navarro a,b , Prajval Kumar c , Aïcha Fonte b , Philippe Fraisse a , Gérard Poisson b and Andrea Cherubini a

I. INTRODUCTION

Touch is one of the most important human capabilities. If vision can help people to recognize the environment, touch is the fine controller for all motions. Tactile sensors measure information arising from physical contact with the environment and are generally modelled after the biological sense of cutaneous touch, that are crucial for many micro tasks, such as in-hand manipulation.

Here, we focus on the BioTac tactile sensors 1 , that mimic the human skin, by using a flexible finger-shaped rubber, mounted on a rigid bone-like structure (see Fig. 1). Extensive research has been carried out to calibrate the BioTac sensors, i.e. to derive the relationship between raw values and actual force. For instance, machine learning approaches have been used in [START_REF] Fishel | Design and use of a biomimetic tactile microvibration sensor with human-like sensitivity and its application in texture discrimination using Bayesian exploration[END_REF], [START_REF] Ciobanu | Point of contact location and normal force estimation using biomimetical tactile sensors[END_REF]. But since these have proved to be time consuming, an alternative, analytic method, has been designed in [START_REF] Lin | Estimating Point of Contact, Force and Torque in a Biomimetic Tactile Sensor with Deformable Skin[END_REF]. Ciobanu et al. [START_REF] Ciobanu | Tactile sensor value preprocessing pipeline[END_REF] established a preprocessing pipeline to overcome the signal's non-compliances and issues for further processing.

Here, we propose a method to actively calibrate the BioTac sensors embedded on the fingertips of a Shadow Dexterous robotic Hand 2 . Since this is a common setup in many robotics laboratories, we are confident that the proposed procedure may be of interest to a wide community. The hand fingers sequentially push on a force sensor plate. An automatic procedure converts the static pressure values measured by the BioTac sensor to force values using techniques similar to those in [START_REF] Fishel | Design and use of a biomimetic tactile microvibration sensor with human-like sensitivity and its application in texture discrimination using Bayesian exploration[END_REF]. Finally, the computed force values are compared to those measured on the force plate, and are found to be coherent. Complementary to [START_REF] Lin | Estimating Point of Contact, Force and Torque in a Biomimetic Tactile Sensor with Deformable Skin[END_REF] and [START_REF] Ciobanu | Tactile sensor value preprocessing pipeline[END_REF], in this paper the shadow robotic hand is actively calibrating by moving the tactile sensors. This setup makes the calibration process easy and fast and enable the use BioTacs for force sensing or force regulation if mounted on a hand. The methodology used here is applicable for active calibration of BioTac sensors in a myoelectric/prosthetic hand for precise grasping. Each BioTac sensor embeds 19 electrodes, a hydro-acoustic pressure sensor (for both static and dynamic pressures), and a thermistor for the temperature [START_REF] Fishel | Bayesian exploration for intelligent identification of textures[END_REF]. In this work, we focus on the measure of the static pressure magnitude (noted P). The mapping from this to the corresponding force magnitude F is given in [START_REF] Lin | Estimating Point of Contact, Force and Torque in a Biomimetic Tactile Sensor with Deformable Skin[END_REF], by the linear relationship:

F = (P -P 0 ) * K, (1)
with P 0 the offset pressure measured by the sensor when not in contact, and K an unknown gain. The goal of this work is to determine P 0 and K, as will be explained hereafter.

III. METHODS To automatically calibrate the BioTac sensors, we use a custom-made plate (shown in Fig. 2) mounted on a Nano25 Force Sensor from ATI Industrial Automation3 . Both are held by a vertical support, also designed and manufactured in the LIRMM laboratory. One-by-one, each of the five fingers sequentially touches the force sensor plate three times with increasing pressure. The contact is kept for n iterations, then released. This is done by moving the finger metacarpophalangeal-equivalent joint by an angle of 18 o , 20 o and 22 o , and then resetting it to 0 o . For the thumb, it is also necessary to actuate the distalequivalent joint by 40 o . Figure 2 shows a movement of each finger as it pushes the force plate. Then, data from the force sensor and BioTac, is processed, to derive K and P 0 . For the force F, we simply consider the norm of the three measured components:

F = F 2 x + F 2 y + F 2 z . (2)
For each BioTac, to reduce the electrode noise perturbation, a Single Pole Infinite Impulse Response Filter is applied to the raw static pressure measure. Filtered values P i at iteration i are expressed in function of the raw measures Pi via:

P 0 = P0 P i = Pi * α + P i-1 * (1 -α) i > 0, (3)
with α ∈ [0..1] a coefficient that controls the degree by which the weight of a measure decreases based on its age. It is common to set α = T s τ+T s , with T s the sampling time, and τ the time constant of the system. Here (as in [START_REF] Ciobanu | Tactile sensor value preprocessing pipeline[END_REF]), we estimate both to be roughly 1 ms, so that α= 0.5 provides the best compromise between sensor noise reduction and response reactivity. Without loss of generality, we hereby explain the procedure used to derive K and P 0 for one BioTac (the procedure is identical for all five). The pressure measurements at iterations with finger vertical (i.e., not in contact with the plane) are averaged, to obtain P 0 . The filtered BioTac measures (from (3)) at iterations with finger in contact with the force plate (denoted as iterations c, . . . , c + n) are used to form column vector:

P =    P c -P 0 P c+1 -P 0 . . . P c+n -P 0    , (4)
while the force measures are stacked in vector:

F = [ F c F c+1 . . . F c+n ] T . (5)
Then, K is simply derived by inverting (1):

K = P † F, (6)
with P † the Moore-Penrose pseudoinverse of P.

IV. RESULTS AND CONCLUSION The five sensors parameters are given in Table I. In figure 3, we plot the values of P-P 0 and corresponding force measures at the iterations with fore finger in contact with the plate. It can be seen that the points lie on the trend-line of slope K. After calibration, we verify the tuned values of P 0 and K by running an experiment where the fingers again push the force sensor plate one-by-one, and the values computed with (1) are compared to the force sensor measures. The results are very near (for instance, we obtain a relative error of 5.91% for the fore finger).The same setup is being used to calibrate the electrode array and consequently estimate the force vector and point of force application. A video of the complete experiment (three movements per finger for calibration, and a fourth for verification) is available at: www.youtube.com/watch?v=kJm6jaVxBlI

Fig. 1 :

 1 Fig.1: Electrode orientation and loaction[START_REF] Fishel | Design and use of a biomimetic tactile microvibration sensor with human-like sensitivity and its application in texture discrimination using Bayesian exploration[END_REF] II. SENSORIZED HAND The Shadow Hand is a robot that closely reproduces the kinematics and dexterity of the human hand. The model that we use (right hand) has 20 degrees of freedom: 2 in the wrist, 5 in the thumb, 4 in the little finger, and 3 in the other fingers. Each finger is equipped with a BioTac sensor. Each BioTac sensor embeds 19 electrodes, a hydro-acoustic pressure sensor (for both static and dynamic pressures), and a thermistor for the temperature[START_REF] Fishel | Bayesian exploration for intelligent identification of textures[END_REF]. In this work, we focus on the measure of the static pressure magnitude (noted P). The mapping from this to the corresponding force magnitude F is given in[START_REF] Lin | Estimating Point of Contact, Force and Torque in a Biomimetic Tactile Sensor with Deformable Skin[END_REF], by the linear relationship:

Fig. 2 :

 2 Fig. 2: Calibration procedure, as fingers touch the force plate.

Fig. 3 :

 3 Fig. 3: Pressure (P -P 0 ) vs Force (F) measurements for the fore finger.

TABLE I :

 I Calibrated parameters of the five BioTac sensors.

www.ati-ia.com/

ACKNOWLEDGMENTS

This work has been supported by the ANR (French National Research Agency) SISCOB project N. ANR-14-CE27-0016.