Olivier Ly

Matthieu Lapeyre

Pierre-Yves Oudeyer

Bio-inspired vertebral column, compliance and semi-passive dynamics in a lightweight humanoid robot

Keywords: Lightweight Humanoid Robot, Morphological Computation, Semi-Passive Dynamic Walking, Compliance, Vertebral Column, Spine, CPG

This paper presents the humanoid robot Acroban. We study two main issues: 1) Compliance and semi-passive dynamics for locomotion of humanoid robots regarding robustness against unknown external perturbations; 2) The advantages of a bio-inspired multi-articulated vertebral column.

We combine mechatronic compliance with structural compliance due to the use of flexible materials. And we explore how these capabilities allow to enforce morphological computation in the design of robust dynamic locomotion. We also investigate the use of compliance to design semi-passive motor primitives using the torso and the arms as a system of accumulation/release of potential/kinetic energy.

INTRODUCTION

The problem of robot locomotion and in particular biped walking has facinated people for a long time, and still is a major issue of modern robotics. The challenge is to make robots move in the same manner as humans so as to fit into the human environment, and also to be socially accepted. Let us mention the famous Honda's Asimo or HRP-4.

This paper presents the humanoid robot Acroban. We study two main issues: 1) Compliance and semi-passive dynamics for dynamic locomotion of humanoid robots regarding robustness to unknown external perturbations, and 2) the advantage of a bio-inspired multi-articulated vertebral column in the dynamics of motor primitives.

Vertebral Column. Acroban has a humanoid mechanical structure (see Figure 1 and Video 1). It has 30 degrees of freedom. Its height is about 70 cm. Its size is comparable to the NAO, the Qrio, or the Darwin ( [START_REF] Muecke | DARwIn evolution: development of a humanoid robot[END_REF]) robots. Acroban has the classical joints of humanoid robots, but in addition it includes a multi-articulated compliant vertebral column with 5 degrees of freedom.

We no longer consider the trunk as a rigid undeformable body. And thus, this opens a new field of motions involving subtle changes in the mass distribution of the robot. This design provides a strong independency between the higher part of the body (shoulder, head and arms) and the lower part (legs and pelvis). This allows the legs and also the pelvis to move without implying moving the upper part of the body. This is illustrated in particular during gait, where the spine is used for horizontal lateral mass transfer, relieving hip joints. This research project was partially funded by ERC Starting Grant EXPLORERS 240007. (1) INRIA/LaBRI, Bordeaux I Univ.,France ly@labri.fr (2) FLOWERS, INRIA Bordeaux, France { matthieu.lapeyre, pierre-yves.oudeyer}@inria.fr

In a bio-inspired spirit, the structure of Acroban's vertebral column implements several fundamental properties of the human vertebral column as emphasized in [START_REF] Ceccato | Le tronc, de la locomotion à la commande[END_REF].

At the moment, a few robots like iCub ( [START_REF] Metta | The iCub humanoid robot: an open platform for research in embodied cognition[END_REF]), Wabian ( [START_REF] Bigge | Humanoid Robots in Waseda UniversityHadaly-2 and WABIAN[END_REF]) or bio-inspired robots like the ECCE robot ( [START_REF] Marques | Semipassive dynamic walking for humanoid robot using controllable spring stiffness on the ankle joint[END_REF]), the WBD-1 ( [START_REF] Or | From lamprey to humanoid: the design and control of a flexible spine belly dancing humanoid robot with inspiration from biology[END_REF]) or Cla and Rabbit ( [START_REF] Mizuuchi | The development and control of a flexible-spine for a human-form robot[END_REF]) and Kenta ( [START_REF] Mizuchi | Swing Motion by a Spined Whole-Body Tendon-Driven Humanoid" Kenta[END_REF], [START_REF] Mizuuchi | Design and control of a flexible spine for the whole-body humanoid'Kenta[END_REF]), or the Ijspeert's Salamander ( [START_REF] Ijspeert | From swimming to walking with a salamander robot driven by a spinal cord model[END_REF]) have a vertebral column; most of them use it to enlarge the operational field of arms; but Acroban may be the only biped robot in this family that is able to walk and keep balance relying heavily on its articulated vertebral column.

Compliance. We consider compliance as the ability of the system to absorb and to go with external physical perturbations. Compliance is the inverse of stiffness. This means that the system can be deformed (in a large sense, including joint's mobilities) by the action of unpredicted external forces. Acroban is compliant at two levels: First, the mechatronic of Acroban, while controlled in position, allows the real-time adjustment of the maximal torque that a joint is authorized to enforce. This allows to limit the resistancy of each joint against an external perturbation and in this way to make it compliant. Moreover, Acroban is able to get real-time information about the force applied to a particular compliant joint through the position error. Second, the mechanical structure itself is compliant: Acroban structure includes springs and elastics which, together with the backlash of the mechanical structure and of servomotors, makes it flexible. In top of that, soles of Acroban are covered by a compliant material.

However, Acroban does not enforce force control. Indeed, there is no real-time force enslavement. And reactions to perturbations are not producted by computations and decisions of the control system. Thus Acroban's motor behaviour would better be qualified as semi-passive. We call a system passive when it has no actuator at all, and uses physic as only controller. They have been introduced in [START_REF] Mcgeer | Passive dynamic walking[END_REF]. Also, such a system uses potential energy as unique energy source; and in an analogous way, Acroban uses spring's potential energy as energy accumulator. Such systems have also been investigated e.g. in [START_REF] Omer | Semi-passive dynamic walking for humanoid robot using controllable spring stiffness on the ankle joint[END_REF]. Let us emphasize the immediate nature of (semi)-passive reactive motion, which are driven directly by physic and thus acts without any delay.

Motion Control. First of all, the stabilizing system of Acroban implement an independant feedback loop system which makes the knees, the hips and the pelvis react in order to keep balance on the base of forces applied to ankles and also an accelerometer and gyroscopic sensor (see Section II-B for details). But the stabilizing system does not rely only on that. It also heavily relies on (semi-passive) compliance: First, the compliance of the higher part of the body (thanks to the vertebral column) acts as a system coupling a semi-passive inverse pendulum (the spine) and two passive pendulums (the two arms), (see Section III-B). The motions of these pendulums are kept by active joints (legs and pelvis). And in turn, they contributes to the lateral weight transfer, by transferring its potential energy to motion. Experiments showed that this process decreases power consumption and also the effect of inelastic shock at foot landing. Second, the ankles are made also compliant which decreases again the effect of the inelastic shocks. Finally, feets include compliant material and have partially slippery soles which makes the legs adjust the positions of the feet by themselves by slightly gliding on the ground.

Locomotion is then considered as a perturbation selfgenerated by central pattern generators (CPG for short) of the legs and of the pelvis and we use the previous system to make the robot spontaneously stabilize itself. In particular, Acroban does not involve analytically real-time computed dynamics, neither ZMP based controller. Acroban's gait actually is semi-passive, following the trend of self-stabilizing passive mechanisms opened in [START_REF] Mcgeer | Passive dynamic walking[END_REF] and of powered semi-passive robots (see [START_REF] Collins | Efficient bipedal robots based on passive-dynamic walkers[END_REF]); the mechanism of semi-passive balance of the torso of Acroban could be compared to the powered passive dynamic walker of [START_REF] Tedrake | Actuating a simple 3D passive dynamic walker[END_REF].

The use of compliance as stabilizing controller as described above transforms the whole mechanical structure into a distributed system looking for local minima of potential energy and stable configurations and trajectories. It absorbs shocks (including unpredicted ones) and adjusts constantly the position of the robot without high-frequency global control This doing, it relieves the global motion control, which is does not have to enforce centralized high-frequency reactions. In that, we follow ideas of morphological computation (see e.g. [START_REF] Paul | Morphology, control and passive dynamics[END_REF], [START_REF] Pfeifer | Self-organization, embodiment, and biologically inspired robotics[END_REF]) by making the mechanical structure ensure a large part of the control (see [START_REF] Paul | Morphological computation:: A basis for the analysis of morphology and control requirements[END_REF]).

This distributed nature of motion control actually extends into a whole modular system where are mixed independant mechanical reactions as mentionned above, balancing controllers and even motions (e.g. the gait). In fact, this extends the classical position-control robotic architecture where each servomotor enforces a position control of its joint as an independant unit which not aware of the dynamic of the whole robot.

Acroban does not enforce computation of the global dynamic of the robot, and the global feedback loop has low frequency (50 Hz). Taking into account the low frequency of global feedback of biological organisms, we also investigate the question of how low can be the global feedback control loop of a humanoid robot.

The use of compliant robots is not mainstream in the context of locomotion. However, number of projects have already shown the importance of an adequate compliant morphology for motor skills. This is illustrated by the quadruped BigDog whose compliance relies on hydraulic actuators ( [START_REF] Raibert | the rough-terrain quaduped robot[END_REF]), see also [START_REF] Ito | Robot Experiment of Torque Learning for Biped Balance with respect to Periodic External Force[END_REF]. Bio-inspired robots use also compliance (e.g. Rollin Justin Arms [START_REF] Fuchs | Justindesign considerations and realization of a mobile platform for a humanoid upper body[END_REF], ECCE Robot [START_REF] Marques | Semipassive dynamic walking for humanoid robot using controllable spring stiffness on the ankle joint[END_REF], robots of the PHRIENDS project [START_REF] Albu-Schaffer | Anthropomorphic Soft Robotics-from Torque Control to Variable Intrinsic Compliance[END_REF], [START_REF] Buchli | Compliant quadruped locomotion over rough terrain. Intelligent Robots and Systems[END_REF], [START_REF] Iida | Bipedal walking and running with compliant legs[END_REF]) and especially material compliance (see e.g. RHex [START_REF] Saranli | Design, modeling and preliminary control of a compliant hexapod robot[END_REF] or [START_REF] Fukuoka | Adaptive dynamic walking of a quadruped robot on irregular terrain based on biological concepts[END_REF], [START_REF] Scarfogliero | The use of compliant joints and elastic energy storage in bio-inspired legged robots[END_REF] [8], [START_REF] Mennitto | CARL: A compliant articulated robot leg for dynamic locomotion[END_REF]). Besides, compliance is also essential in the design of leg prosthesis (see e.g. [START_REF] Walsh | A quasi-passive leg exoskeleton for load-carrying augmentation[END_REF]). In these designs, one can observe that compliance greatly improves robustness of locomotion. However, the cost of this robustness is less predictability and less accuracy.

Thanks to the small size of Acroban, the experimentation process is drastically simplified regarding the ease of manipulation and security. The ratio material resistance / weight implies a better robustness of the mechanical structure, making Acroban able to fall without breaking.

This changes deeply the process of motor skill design by allowing creating motions directly on the robot by direct experiments without needing any simulation process. This includes for instance adjusting in real-time motor primitives, and even extreme ones like stabilizing process (see Video 4). However, at the same time, the small size makes difficult the integration of highly accurate mechatronic components. Backlash together with the use of springs and the relative inaccuracy of the structure, implies that analytic models of the platform are difficult to exploit. This makes the easiness of the experiment process particularly important. It is also important seeing that our focus for future research is motor learning, and especially locomotion.

This article comes with accompanying videos available on Youtube, as indicated in the following (also available on 1) Global Structure: The Acroban platform is a small (about 70cm) and lightweight (about 5kg) humanoid robot with many degrees of freedom (30 dofs) and a multiarticulated spine. The structure only includes revolute joints, which are all actuated by servomotors (see Section II-A) in a modular way: Each ankle has 3 joints enforcing a spherical link. Each knee has 1 joint enforcing a revolute link. Each hip has 3 joints enforcing a spherical link. The vertebral column has 5 joints. Each shoulder has 1 joint enforcing a 2-revolute joints link. Each elbow has 2 joints enforcing a 2-revolute joints link. We essentially focused on designing a mechanically rich and open structure in the area of the vertebral column and the pelvis, providing it with 11 degrees of freedom. 2) Vertebral column: The vertebral column can be viewed as a system linking the pelvis and the shoulder. It enforces two revolutes joints links at its two extremities, each of them providing rotations in the sagittal and the coronal planes and one in the transverse plane. During motions, and in particular motions related to locomotion, this allows getting independency of the higher part and the lower part of the body. This allows for instance to reduce the dynamic of the higher part of the body during the gait. We claim that this contributes significantly to the stabilisation of the robot.

3) Pelvis: The pelvis, seen as an independent sub-body, may have several kind of mobility. It produces precise movements of the center of gravity of the robot. Firstly, it can move by a rotation in the sagittal plane. We will use it extensively to keep balance. Secondly, it can move in the transverse plane, this is used for the gait for the weight transfer between the legs, instead of making the legs support all the efforts of displacement of the body.

B. A Bio-inspired Design

The Human gait is an undeniable reference for the study of locomotion. Even if it is far from being clear that a direct transposition of human gait to robots is really effective, the mechanical and control processes generated by thousands years of evolution to solve bipedal locomotion problem are an important source of information and inspiration for humanoid robotics [START_REF] Azevedo | Contrôle de la locomotion artificielle: de lhomme aux robots[END_REF]. Many researchers were interested in biomechanics of human walking (see e.g [START_REF] Rose | Human walking[END_REF]). These studies describe accurately the kinematics and dynamics of legs during walking (see [START_REF] Vaughan | Theories of bipedal walking: an odyssey[END_REF]).

Among the large litterature concerning human biped walking, only a few projects studied the role of the trunk during walking. Yet the trunk represents 60% of the total weight for humans, which raises the center of gravity (see e.g. [START_REF] Thorstensson | Lumbar back muscle activity in relation to trunk movements during locomotion in man[END_REF], [START_REF] Feipel | Threedimensional kinematics of the lumbar spine during treadmill walking at different speeds[END_REF]). The trunk has a large complex network of muscles used to accomplish a lot of movements while keeping the balance. Its movements are regulated by a complex combinaison of anticipatory and reactive actions. The movements of the spine can facilitate the transfer of weight from one leg to the other one, improve the balance but also participate to the dynamic of the walking. It seems therefore interesting to enable a humanoid robot trying to walk in a robust way, to have an articulated trunk. But even if the human trunk has already been designed in some robots (like ECCE Robot [START_REF] Marques | Semipassive dynamic walking for humanoid robot using controllable spring stiffness on the ankle joint[END_REF]), it is difficult to replicate on a small robot using servomotors. So we must simplify and find the most essential degrees of freedom of the spine. Interestingly, Ceccato [START_REF] Ceccato | Le tronc, de la locomotion à la commande[END_REF] studied the role of the trunk and highlighted the main displacements of the spine during walking. And the apparent high dimensionality of the trunk appears to be factorizable down to a few essential components/dimensions. First, experiments showed small oscillations in the pelvis and the thoracic in the sagittal plane, highlighting that only two joints, one for the pelvis and one for the thoracic should be sufficient to represent the motion of the spine in the sagittal plane. In the coronal plane, the pelvis and shoulders oscillate in phase opposition while the middle remains straight throughout the cycle. This implies that essential movements of the trunk in the coronal plane could be approximated using two joints, one for the pelvis and an other one for the shoulder. Finally, in the horizontal plane, there are opposite rotations between the upper trunk and the lower trunk, enforced by a twist of the spine. So, only one revolute joint in the middle of the spine should be sufficient.

Accordingly, Acroban has five joints for the trunk, as shown on figure 1 (see also Video 1): Two in the sagittal plane and two in the coronal plane, placed in the pelvis and shoulder/thoracic and one in the horizontal plane placed in the middle of his trunk. With this design, we have the strictly minimum necessary joints to replicate essential degrees of freedom of the human trunk.

II. MECHATRONIC & CONTROL

A. Mechatronic

Servomotors. Acroban is actuated by standard servomotors that enforce revolution joints: Dynamixel1 RX-64 and RX-28. RX-64 (resp. RX-28) can deploy 64kg.cm -1 (resp. 28kg.cm -1 ) in standing torque, decreased around 30kg.cm -1 (15kg.cm -1 ) during movements. They have small backlash and low accuracy due to classical metal gear reductor (not harmonic neither planetar), however their mechanic is reversible which allows to get them compliant. The servomotors enforce position control. However, they allow adjusting in real time a maximal bound for the torque produced by the joints. Moreover a driving mode called "null torque mode" in which the servomotor cancels the rubbing forces of the gears and of the motor in order to get a completely passive joint is available.

Springs. Some particular joints (knees, coronal hip and pelvis) are lightly under-powered. We address this problem by using torsion springs as energy accumulator to support servomotors and also to introduce compliance.

Elastics. The use of standard servomotors implies a significant backlash. To address this problem, we use elastics (or extension springs) set between the extremities of the leg which replaces the backlash by a kind of force proportional reaction in the backlash range. We limited the use of elastics to legs as these last ones are crucial for locomotion.

B. Motor Control System

Servomotors are controlled by a centralized upper layer implementing the logic of movements. This layer is enforced by an embedded system based on ARM technology. It schedules motions by discretizing time at low frequency: 50 Hz. However, each servomotor embeds a control loop of higher frequency (higher than 1KHz); but these control loops are independent from each other, and targeted to the position control only.

Motor Primitives. Movements are subdivided into modules called parameterized motor primitives. Motor primitives are combined in order to define motions of Acroban in a modular way. In turn, motor primitives are constructed by combinations of splines, CPGs and PID controllers which are all runned in parallel by the upper layer described above. Splines are defined by the user point by point. CPGs rely on sinusoids whose period, phase and amplitude are defined by the user. Propotional-Integral-Derivative (PID for short) controllers classically take multidimensional entry signals, and compute output by weighted sum of proportional, discrete integral and discrete derivative operation. The user adds as many PID controller that he wants, and defines output, gains, and entry signals among sensor values (accelerometer, gyroscopic) and/or position errors of the joint. Let us mention that we added basic filters to PID controllers: mininum and maximum values, and mobile discounted average whose parameters are defined by the user.

Finally, at each time and for each kind of output (see below), all the coresponding values of these modules are simply sumed up to get the global output value.

Outputs are joint position targets, joint maximal torques (and thus, as explained in Introduction, compliance), and also position in the operational space (cartesian positions of the feet computed by inverse kinematic); more complex motions are obtained also by taking as output motor primitive parameters (typically the gains of splines, or CPG, but also of PID controller).

III. ROBUST COMPLIANT STABILIZING AND DYNAMIC GAIT AS A STABILIZED SELF-PERTURBATED MOTOR PRIMITIVE

Our goal consists in designing stabilizing motor primitives able to maintain the system stable under various external mostly unknown perturbations. In particular, such primitives are designed quasi-independently from the walking CPG. The gait then results from a stabilization motor primitive perturbed by an active CPG coming from the lower part of the body (pelvis and legs).

Motor Primitive Designing Method. The hybrid nature of the structure of Acroban, involving several kind of materials possibly flexible, the relative inaccuracy of the design, the backlash and the inaccuracy mechatronic implies that analytic models of the platform are difficult to exploit in order to generate motor primitives by simulation.

Motor primitives are actually defined and tuned by hand within an experimental process involving online tests (trials and errors) on the real robot. In order to get objective assessments and both quantitative and qualitative performance measures, we rely on a motion tracking device2 to analyse Acroban's motions.

The process of designing stabilizing and walking motor primitives relies on several criteria, aiming at evaluating the quality of the stabilizing process. Among them: speed of convergence to static stability, oscillation amplitude for stabilizing, periodicity, cycle stability for walking (see Section IV for details). Let us note that the most basic criterion is the number of steps without falling. Due to the lack of space, we will not give the details of final parameter values, we neither will give formal forms of PID controlers.

A. Compliant stabilizing motor primitives

We have experiment several stabilizing motor primitives. As mentionned above, all of them have been designed by experiment. In the following we describe the modules constituting these motor primitives. We describe the nature of the modules in a qualitative way; parameters are tuned by experiments and does not deserve to be outputed here. Each of them has two parts: A controller and a compliance mode.

1) Compliance: Compliance is twofold: controlled (see Video 2) and structural (see Video 3).

Controlled compliance means adjusting the maximal torque of joints during motion. In a static way where the compliance of a particular joint is fixed for the whole motion, this means that one puts the structure into a configuration with particular morphological computation properties, which can be akin to morphosis (see [START_REF] Christensen | A distributed strategy for gait adaptation in modular robots[END_REF]). In a dynamic way, one typically simulate an adjustable spring (see e.g. [START_REF] Peter | Variable joint elasticities in running[END_REF]). We use this technique at different levels of the robot:

• Ankles. We make the ankles compliant in the coronal plane. This ensures a good adherence of the feet robust to perturbations. This is enforced by a simulated adjustable spring in turn enforced by a PID controller taking the position error of the ankle and acting on its maximal torque. • Pelvis and vertebral column. We make compliant the coronal joints of the vertebral column and of the pelvis. The higher joint of the vertebral column is made completely free (null maximal torque), while the lower joint behaves like an ajustable spring. • Arms and shoulders are also made compliant, and even free (see below). • Sagittal rotation of the pelvis. We adjust a relative compliance of the sagittal rotation of the pelvis in order to absorb sagittal moment acting on the whole structure. Again, this is enforced by a simulated adjustable spring. Let us note that the fact that compliance is dynamically controlled by PID controllers makes the transition between compliant/passive state and stiffness continuous.

Structurally, comparing to a body which would have just legs, the upper part of the body (torso, head and arms) makes the center of mass move up. This makes higher the amplitude of movements of the center of mass in the horizontal plane, and thus makes easier the lateral weight transfer from one leg to the other one. However, at the same time it makes the system more instable. This is mitigated by the vertebral column, the shoulder and the arms, made compliant as described above, making the torso-arm system behaves like three coupled pendulums, i.e. the spine as an inverted pendulum to the top of which are attached two pendulums (the arms). Experiments show that such a system absorbs shocks and improves the stability of the system.

Structural compliance (i.e. flexibility of the structure, and in particular legs) of Acroban also plays an important role (see Video 3). It absorbs the impact of foot landing and helps the leg to get in a locally stable configuration by itself. Quantitatively, along the whole body of Acroban, and all motors stiff, the flexibility range is approximately 20 as can be seen on Video 3.

Finally, let us mention the slippery sole of Acroban. During the foot landing, it reduces the horizontal component of the impact in a very important way, avoiding important perturbations, and makes the leg find by itself local horizontal adjustment of the foot position. Experiments show that this feature improves considerably the stability of the robot during walk. However, this costs a lack of precision and of efficiency of steps, which adapt this way constantly to the environment.

2) Active Controllers: The stabilization system also relies on active controllers. Like above, we describe these controller in a qualitative way seeing that their parameters are adjusted by experiment.

• Deformable parallelogram movements on the vertebral column. In this way, one corrects the vertical position of the vertebral column while bringing only minimal moment of the higher part of the body. This motion emphasizes a significant improvement of the dynamics of the robot by the vertebral column. This action is enforced by a PID controller taking the gyroscopic sensor as entry. • Ankle and knees positions in the sagittal plane. We use them to counter the lean of the robot via a PID controller taking the the gyroscopic sensor as entry. • Pelvis sagittal rotation. This motion produces precise movements of the center of mass while bringing only minimal moment on the remaining of the body. We use it also to maintain the pelvis horizontal. It is enforced by two PID controllers taking as entry the accelerometer sensor and the gyroscopic sensor. • Center of gravity. By using inverse kinematic of legs, one has a control of the higher part of the body in the Cartesian space. We use it as an approximation of Cartesian movements of the center of mass: -Horizontal movements are used at low frequency to correct the position of the body. Low and high frequency selection is done by mobile discounted average of the entry signal (accelerometer), and a PID controler is enforced on the smoothed signal.

-Horizontal movements are used at high frequency to absorb horizontal perturbations.

-Vertical movements are used to transform the position of the robot into a minimum of potential energy, and in this way transform the position into an attractor. This is enforced by a PID controller taking as entry the smoothed accelerometer. We use also a controller of higher level: the amplitude and the direction of steps. In the waking motor primitive, we define the step by means of splines that describe the trajectory of the foot in the Cartesian space (this trajectory is again tuned by experiment). Roughly speaking, the robot stabilizes itself by doing forward steps if he leans forward, and backward steps if he leans backward. For doing that, we use the lean of the robot (taken from the accelerometer) to determine the vertical amplitude of the step via a PID controller.

Coronal stabilisation relies essentially on compliance: The system made of the vertebral column, the shoulders and the arms is a combination of an inverted pendulum (the vertebral column) and two pendulums (the arms coupled with shoulders), actuated by the lower coronal rotation of the vertebral column, which is compliant, but with a compliance degree controlled by its position error (see Video 4).

B. Passive and Semi-Passive Dynamic Gait 1) Passive Dynamic Gait: Considering the stabilizing motor primitives described above, a particular family of external perturbations, i.e. periodic lateral perturbations here generated by a human, provokes spontaneously passive dynamic walking as shown in Video 5.

While the robot uses the same stabilizing motor primitive, this perturbation, amplified by the triple pendulum in the torso, makes it step forward as a consequence of the mechanical elasticity of the feet and legs. Indeed, during grounded phase, the leg accumulates energy for horizontal translation and torsion motions of the foot; and it frees this energy when the foot is up, making this last one move and perform a step (see video 5).

2) Dynamic Gait as a Stabilized Self-Perturbation: Starting from this intrinsic capability for passive dynamic walking, we designed a motor primitive, based on the use of a CPG directly added to the stabilizing motor primitive described above. In this way, one generates dynamic gait by a self-perturbation. In this case, we have what one may call a powered passive walker or a semi-passive walker.

The walking loop pattern has two sides: An active side driven by the legs and the pelvis, mixing lateral weight transfer and step forward. This active side is generated by CPG. A passive side consisting in one of the stabilizing motor primitive as described above.

The use of the double coupled pendulum system described above makes the robot use potential energy as local energy accumulator. The movement of the pendulum is kept by active movements of legs and pelvis. But at the same time it contributes significantly to the weight transfer: First it changes the global mass distribution; and second, it induces a reaction moment on the lower part of the body via the pelvis. Therefore, this movement has a semi-passive nature. Let us note that the active use of the pelvis, independently of the torso for the step is largely inspired by the human walk (see e.g. [START_REF] Vaughan | Theories of bipedal walking: an odyssey[END_REF], [START_REF] Rodman | Bioenergetics and the origin of hominid bipedalism[END_REF]).

Besides, it is possible to control the whole walking pattern, i.e. to get the robot turn in all directions and walk forward at various speeds, by controlling the amplitude of its splines, independently for left and right step. This reduces the control of the walk to 2 parameters determining first the sum of the amplitudes of both steps and seconds the relative difference between them.

We also tested the same motions but with stiff joints in the vertebral column and the arms. Experiments showed that the global power consumption increased significantly, and robustness decreased in the same time. The inelastic shock occuring at foot landing spread into the upper part of the body, producing visible oscillations in the arms.

Experiments show that the walk of Acroban is robust, while remaining largely to be improved. Acroban is able to walk under various perturbations, including large one. It is able also to switch smoothly from horizontal ground to inclined ground (see video 6).

IV. EXPERIMENTS

We set-up several kinds of tests on Acroban, such as putting it on a skateboard, walking on a flat or inclined soil or disrupting the balanced posture (e.g. in Video 4). Among these experiments, we selected the following ones in order to get quantitative measure of stabilisation process quality:

1) Evaluation of the postural balance of Acroban under controlled external perturbations. 2) Evaluation of balance when legs oscillate so that a dynamic weight transfer occurs. These two experiments use the same compliant motor primitives controller to ensure the global balance of Acroban. Let us note that we achieved more quantitative experiments than what was possible to present within the space constraints of this paper.

A. Evaluation of the Stabilization Process against External Perturbations

For this experiment, we disturb Acroban with a controlled reproductible perturbation. This perturbation is generated by a pendulum composed with a weighted ball (see Fig. 2.1.a.). In order to have a repeatable perturbation, we always drop the ball at the same altitude. The tracking camera is placed on the side of Acroban in order to get sagittal motion (see Fig. First, Fig. 2.1.b. shows all the positions and speeds taken by the head of Acroban. This movement represents a closed system which here always converges towards equilibrium and return the neutral position (0;0) and zero velocity. Despite the disturbance, the robot is stable and here always finds its equilibrium converging to the neutral position.

Second, in order to quantify the convergence speed, one observes the phase diagram of the principal displacement X(t). As shown on Fig. 2.1.c. where each point represents the pairs (X(t); (dX(t))/dt), and each arrow represents the next state. It appears that the overall shape of this diagram is a spiral converging to the neutral position. The spiral reaches the neutral position after 6 or 7 laps.

So, Acroban oscillates up to seven times around its neutral position before finding its balance. This speed could certainly be improved, but it still shows a stable convergent dynamic behavior. This criterion is used to tune parameters of the motor primitive.

B. Evaluation of the Stability of Dynamic Weight Transfer between Legs

Here we evaluate the stability of Acroban's stabilization when legs oscillate causing a dynamic weight transfer from one leg to the other one. The tracking camera is placed in front of Acroban, we leave Acroban oscillate for 35 seconds and we record its motion in the coronal plane (see Fig. 2

.2.a).

First, we can then, as in previous sections IV-A, draw the phase diagram in the coronal plane. We obtain the Fig. 2.2.b. This is a vector field whose points represent all the positions (Y (t); Z(t)) taken by the robot and the arrows represent the direction and value (gradient color and size) of the speed at the point. We see a closed curve on this diagram, which indicates that the motion is stable; the trajectory follows an eight. This diagram shows clearly the stable attractor dynamic and the typical "eight" shape found in the Human gait (see [START_REF] Saunders | The major determinants in normal and pathological gait[END_REF]).

Second, we can draw the phase diagram in the direction Y (t). As shown in Fig. 2.2.c., the phase forms a circle showing the stability of the trajectory: the robot never gets umbalanced nor deviates from its "orbit". Despite a low mechanical precision and a stabilization controller not optimised for this specific task (e.g. specific self-perturbation of the walking CPG), we have a controlled dynamics and a movement that is repeated in a deterministic and convergent way.

C. Qualitative experiments

Next to a number of quantitative experiments such as the one presented in the previous section, we also performed some qualitative experiments which results are illustrated in the videos accompanying this paper.

Postural stabilization. In addition to the study of the impact of deterministic perturbations on postural stabilization such as in previous section, we experimented a large variety of other potential large and diverste perturbations. As Video 4 shows, Acroban is able to maintain its balance when a real football ball (very big and rather heavy as compared to Fig. 2. 1.a.:Experimental 1 set up. We use a weighted ball (154gr) hanging from a rope (87 cm) for hitting Acroban in the torso in the direction of the vector -x. Before the drop, the ball is positioned so that the rope is parallel to the ground, so it hit Acroban with energy of 1.3J. its own size and weight) is thrown at him at non negligible speeds and at various places of its body as well as many impact angles. Furthermore, Acroban is able to maintain its postural balance when a human pushes or pulls many parts of its body. It is also possible to take Acroban by the shoulders, balance its whole body, and then literally throw it on the floor: if the throwing is reasonable, landing happens rather smoothly, the shock is absorbed by the structure compliance, and Acroban stays stood up. Finally, as shown at the end of Video 4, we even tried to launch Acroban on a skate-board, and this resulted again in quite robust ability to keep its balance.

Robustness to external perturbations while dynamically walking. As shown in Video 6, Acroban still keeps its balance when in addition a football ball is thrown at him while dynamically walking. The same holds when pushed by a human at various positions of its body (see also Video 1). When put on a skate-board pushed by a human, Acroban robustly oscillates its legs without falling. When put on a gentle downward or upward slope, Acroban continues to be able to go forward without falling.

Leading Acroban by the hand. An interesting behaviour, that we did not initially plan in the design of Acroban, happens when Acroban oscillates its legs with parameters such that it stays at the same position. As shown in Video 1, when a human takes the hands of the robot in this configuration, like those of a little child, and try to lead him in a given direction, Acroban spontaneously follows. One can drive Acroban in any direction in such a way. This can actually be understood similarly to dynamic walking: the human applies a force on Acroban in a given direction, which makes it fall in that direction, but the robot continuously finds back its balance thanks to its postural stabilization.

V. CONCLUSION AND FUTURE DIRECTIONS

With its compliant structure and mechatronic, together with its bio-inspired vertebral column, and with a versatile stabilizing controller/motor primitive, Acroban can keep its balance in two major cases: postural stabilization and dynamic walking.

Future work will explore mainly three directions. First, while Acroban explores the role of of a complex vertebral colum and torso in semi-passive dynamic walking, the dynamics of its legs themselves could be largely improved, partly based on the use of modified CPGs as well as an optimized shape for the feet. The combination of its dynamic torso and whole body compliance with leg dynamics such as shown in [START_REF] Mcgeer | Passive dynamic walking[END_REF], [START_REF] Collins | Efficient bipedal robots based on passive-dynamic walkers[END_REF] will be of high interest. Second, the quantitative study of energy consumption of Acroban compliant dynamic walking should be pursued. Let us mention that Acroban has about one hour of autonomy with a 4-elements LiPo battery and using on-board computation. Third, as shown in Video 8, the fact that Acroban is lightweight, compliant and robust to external perturbations affords original physical human-robot interaction, even with children, which will be systematically explore.

Fig. 1 .

 1 Fig. 1. Acroban Global Structure

  2.1.a.). During 180 seconds, the robot receives twenty impacts. The phase diagram in the sagittal plane of the robot is shown on Fig.2.1.b. and the phase diagram of the principal displacement X(t) is shown on Fig.2.1.c. Thanks to the motion capture system.

  Fig. 2. 1.a.:Experimental 1 set up.We use a weighted ball (154gr) hanging from a rope (87 cm) for hitting Acroban in the torso in the direction of the vector -x. Before the drop, the ball is positioned so that the rope is parallel to the ground, so it hit Acroban with energy of 1.3J. 1.b.:This diagram shows the dynamic of the postural balance in the sagittal plane of Acroban under controlled external pertubations. It represents a closed system, converging to the neutral position, so the robot's behaviour is stable.1.c.: This diagram shows the phase diagram for the X(t) motion of the dynamic of the postural balance of Acroban under controlled external pertubations.This diagram shows a converging spiral who needs up to seven laps to reaches the neutral position, showing a not very fast but still a stable and determined dynamic behavior. 2.a.:Experimental 2 set up.Acroban is placed in front of the camera. Data are measured in the plane (y,z) and Acroban has a CPG, doing an oscillation motion with his legs resulting in a rocking motion causing a dynamic weight transfer from one leg to each other.2.b.:This diagram shows the dynamic of the transfert of mass in the coronal plane. Points describe a kind of eight and this shape is closed and quite constant in the time, so the dynamic weight transfer is stable. 2.c.:This diagram shows the phase diagram for the Y (t) motion of the dynamic mass transfert.This phase diagram is a circle, so the robot dynamic is stable.Despiste a low mechanical accuracy creating disturbances, the movement is constant, stable and determined.

Dynamixel RX64 User's Manual: www.robotis.com c Dynamixel

Our motion tracking device records motion of an IR led in a plane parallel to the CCD. In our experiments, this Led is placed on the head of Acroban

VI. ACKNOWLEDGEMENTS

We would like to thank Jérome Béchu, Adrien Baranes, Damian Cavaille, Loïc Gondry and Pierre Rouanet for discussions and technical help in this work. We also thank Jean-Charles Ceccato and Jean-René Cazalet for very helpful discussions. This research project was partially funded by ERC Starting Grant EXPLORERS 240007.


Bruno Chanteau

Olivier Lopez

Wei Zhang

Daniele Nicolodi

Bérengère Argence

Frédéric Auguste

Michel Abgrall

Christian Chardonnet

Giorgio Santarelli

Benoît Darquié

Y Le Coq

A Amy-Klein

Mid-infrared laser phase-locking to a remote near-infrared frequency reference for high precision molecular spectroscopy

Keywords: 06.20.-f (Metrology), 42.62.Eh (Metrological applications; optical frequency synthesizers for precision spectroscopy), 07.57.Ty (Infrared spectrometers, auxiliary equipment, and technique)

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

Introduction

With their rich internal structure, molecules can play a decisive role in precision tests of fundamental physics.

They are for example now being used to test fundamental symmetries [START_REF] Demille | Using Molecules to Measure Nuclear Spin-Dependent Parity Violation Physical Review Letters[END_REF][START_REF] Darquié | Progress toward a first observation of parity violation in chiral molecules by highresolution laser spectroscopy[END_REF][START_REF] Hudson | Improved measurement of the shape of the electron[END_REF], and to measure either absolute values of fundamental constants [START_REF] Koelemeij | Vibrational spectroscopy of HD + with 2-ppb accuracy Physical Review Letters[END_REF] or their temporal variation [START_REF] Shelkovnikov | Stability of the Proton-to-Electron Mass Ratio Physical Review Letters[END_REF][START_REF] Hudson | Cold molecule spectroscopy for constaining the evolution of the fine structure constant Physical Review Letters[END_REF]. Most of those experiments can be cast as the measurement of molecular frequencies. Ultra-stable and accurate sources in the mid-infrared (MIR) spectral region, the so-called molecular fingerprint region that hosts many intense rovibrational signatures, are thus highly desirable. MIR laser frequency stabilization has been performed for a long time using molecular references such as CH [START_REF] Koelemeij | Vibrational spectroscopy of HD + with 2-ppb accuracy Physical Review Letters[END_REF] or OsO 4 (see for instance [START_REF] Bernard | CO 2 laser stabilization to 0.1 Hz level using external electrooptic modulation[END_REF][START_REF] Acef | Metrological properties of CO 2 /OsO 4 optical frequency standard Optics[END_REF][START_REF] Gubin | Transportable HeNe/CH 4 optical frequency standard and absolute measurements of its frequency[END_REF][START_REF] Foreman | Demonstration of a HeNe/CH 4 -based optical molecular clock[END_REF]). However obtained stability is at least one order of magnitude below those of visible or near-infrared lasers stabilized to ultra-stable cavity. Moreover only a few molecular lines can be used when ultra-high accuracy is needed.

In this paper we present a new method for accurate mid-infrared laser frequency stabilization. The frequency reference is a near-infrared cavity stabilized laser continuously monitored against primary standards and the coherent frequency link between near-infrared and mid-infrared frequencies is obtained by using an optical frequency comb. Moreover we demonstrate this stabilization scheme with a remote near-infrared frequency reference transferred via an optical fibre link from a national metrological institute (NMI). This technique is thus accessible to any laboratory which can be connected to such a NMI with a fibre optical link [START_REF] Lopez | Ultrastable long distance optical frequency distribution using the Internet fiber network Optics[END_REF].

Optical frequency combs have proven to be essential for laser frequency measurement and stabilization from the infrared to ultraviolet domain (see for instance [START_REF] Cundiff | Optical frequency synthesis based on mode-locked lasers[END_REF]). Fractional accuracy and stability (at 10 4 s averaging time) down to a few 10 -16 are potentially reachable when the frequency reference is provided by advanced primary standards. Extension to MIR spectral domain has been demonstrated by comparing the MIR laser frequency to a very high harmonic of the comb repetition rate using sum or difference frequency generation [START_REF] Daussy | Long-distance frequency dissemination with a resolution of 10 -17[END_REF][START_REF] Amy-Klein | Absolute frequency measurement of a SF 6 two-photon line by use of a femtosecond optical comb and sum-frequency generation[END_REF][START_REF] Bielsa | HCOOH High resolution spectroscopy in the 9.18 m region[END_REF][START_REF] Vainio | Frequency-comb-referenced molecular spectroscopy in the mid-infrared region Optics[END_REF][START_REF] Bressel | 5 µm laser source for frequency metrology based on difference frequency generation[END_REF][START_REF] Borri | Direct link of a mid-infrared QCL to a frequency comb by optical injection Optics[END_REF][START_REF] Mills | Coherent phase lock of a 9 μm quantum cascade laser to a 2 μm thulium optical frequency comb[END_REF][START_REF] Galli | Comb-assisted subkilohertz linewidth quantum cascade laser for high-precision midinfrared spectroscopy[END_REF]. Efforts have also been made towards the development of MIR frequency combs [START_REF] Foreman | Demonstration of a HeNe/CH 4 -based optical molecular clock[END_REF][START_REF] Schliesser | Mid-infrared frequency combs[END_REF][START_REF] Adler | Phase-stabilized, 1.5 W frequency comb at 2.8-4.8 µm[END_REF][START_REF] Baumann | Spectroscopy of the methane 3 -band with an accurate midinfrared coherent dual-comb spectrometer[END_REF][START_REF] Hugi | Mid-infrared frequency comb based on a quantum cascade laser[END_REF][START_REF] Coluccelli | Frequency-stabilized 1 W optical comb at 2.2-2.6µm by Cr2+:ZnSe multipass amplification[END_REF][START_REF] Gambetta | Milliwatt-level frequency combs in the 8-14 µm range via difference frequency generation from an Er:fiber oscillator[END_REF].

In this paper we first describe the coherent frequency stability transfer between near-and mid-infrared frequencies around 10 µm. Then we demonstrate absolute frequency measurement of a MIR frequency with a fractional resolution of at least 4x10 -14 . We also report a first application to high resolution molecular spectroscopy with a fractional uncertainty of 8x10 -13 on the line centre. Finally we present the mid-infrared laser frequency stabilization against the near-infrared frequency reference leading to state-of the art stability.

Experimental set-up

The experimental setup is shown in figure . 1. The ultra-stable optical reference located at LNE-SYRTE is a 1.54 µm fibre laser locked to a high finesse cavity. Its fractional frequency instability was measured to be lower than 2×10 -15 at 1 s and 10 -14 at 100 s (after a 0.3-Hz/s drift was removed) [START_REF] Jiang | Long-distance frequency transfer over an urban fiber link using optical phase stabilization[END_REF]. Its frequency is measured using a fibre fs laser centered around 1.55 µm. The laser repetition rate is phase-locked to the optical reference frequency after removal of the comb frequency offset f 0 . Fast corrections are applied to an intra-cavity electro-optic modulator (bandwidth > 400 kHz) and slower corrections to a piezo-electric transducer (PZT) controlling the laser cavity length (bandwidth ~ 10 kHz) [START_REF] Zhang | Characterizing a fiber-based frequency comb with electro-optic modulator[END_REF]. The absolute frequency of the comb repetition rate 36 th harmonic (9 GHz) is continuously measured against the primary standards of LNE-SYRTE, which includes an H-maser, a cryogenic oscillator and Cs-fountains [START_REF] Chambon | Design and realization of a flywheel oscillator for advanced time and frequency metrology[END_REF][START_REF] Guena | Progress in atomic fountains at LNE-SYRTE[END_REF]. It enables real-time measurement of the ultra-stable laser frequency drift and its correction by applying to the driving frequency of an acousto-optic modulator an opposite linear drift (with a step every ms) updated every 100 s. This makes up an ultra-stable near-infrared reference, the frequency of which is currently traceable to primary standards with a 10 -14 uncertainty after 100 s. This optical reference signal is transmitted to LPL through a 43-km long optical link [START_REF] Jiang | Long-distance frequency transfer over an urban fiber link using optical phase stabilization[END_REF]. The freerunning link exhibits a propagation instability of 2×10 -14 at 1 s and around 10 -15 between 100 s and 1 day. When compensated, the link instability has been measured to be roughly 10 -15 τ -1 and to reach around 10 -18 after 10 3 s (see figure 3) [START_REF] Jiang | Long-distance frequency transfer over an urban fiber link using optical phase stabilization[END_REF]. The frequency stability and accuracy of the reference signal are thus preserved at the LPL optical link end.

At LPL, a low-noise laser diode (free-running linewidth below 10 kHz) is phase-locked to the incoming signal with a bandwidth of 100 kHz and constitutes the local optical frequency reference ref .

The repetition rate f rep of a 1.55 µm fibre fs laser is phase-locked to ref .

To that purpose, the beat-note 1 between ref and the N th comb mode (N ~ 780000) is used, after removal of the comb frequency offset f 0 :

1 ref rep Nf (1)
Fast and slow corrections are applied to an intra-cavity electro-optic modulator and a PZT respectively as performed at LNE-SYRTE [START_REF] Zhang | Characterizing a fiber-based frequency comb with electro-optic modulator[END_REF]. A second beat-note 2 compares the MIR laser frequency MIR around 10 µm and the n th harmonic of the repetition rate with n≈120000: Combining ( 1) and ( 2), the MIR laser frequency MIR is finally obtained as:

2 MIR rep nf (2)
21 MIR ref n N (3)
with n/N roughly 0.15. The MIR frequency is thus directly linked to the near-infrared frequency reference, once the integers n and N and the signs have been determined.

MIR frequency measurement and stabilisation

To characterize the phase-coherent link between the near-infrared frequency reference and the MIR frequency, we used this setup to measure the absolute frequency of a CO 2 laser stabilized onto an OsO 4 saturated absorption line. Such a OsO 4 -stabilized CO 2 laser constitutes the current state-of-the-art MIR secondary reference standard [START_REF] Bernard | CO 2 laser stabilization to 0.1 Hz level using external electrooptic modulation[END_REF][START_REF] Acef | Metrological properties of CO 2 /OsO 4 optical frequency standard Optics[END_REF]. In this work, the CO 2 laser was locked either to the P(55) line of 190 at 1 s, reaches 10 -14 after 100 s of integration, and degrades at longer times due to a frequency drift of the CO 2 /OsO 4 frequency reference. We checked that this stability was limited by the CO 2 /OsO 4 reference since changing the CO 2 laser locking parameters induced a variation of the obtained stability. This stability is consistent with previous measurements obtained by comparing either two identical stabilized CO 2 lasers [START_REF] Bernard | CO 2 laser stabilization to 0.1 Hz level using external electrooptic modulation[END_REF][START_REF] Acef | Metrological properties of CO 2 /OsO 4 optical frequency standard Optics[END_REF] or one of such lasers to a titanium-sapphire (Ti:Sa) frequency comb referenced to a microwave frequency [START_REF] Daussy | Long-distance frequency dissemination with a resolution of 10 -17[END_REF].

This demonstrates the capability of the system to measure the stability of the best MIR frequency sources to date without any degradation. We expect the optical link and the frequency comb to contribute a few 10 -15 at 1 s to the frequency instability [START_REF] Jiang | Long-distance frequency transfer over an urban fiber link using optical phase stabilization[END_REF][START_REF] Zhang | Characterizing a fiber-based frequency comb with electro-optic modulator[END_REF].

The relationship of equation ( 3) between the frequency of the 1.54 µm LNE-SYRTE reference laser and that of the MIR laser is ensured by means of coherent phase-lock loops. Thus the accuracy of the MIR frequency measurement only depends on the uncertainty of the near-infrared frequency reference. The latter is known with an uncertainty of about 10 -14 after 100 s averaging time, when only steered with the H-maser which is sufficient for this experiment. The 3 10 -16 Cs fountain accuracy [START_REF] Guena | Progress in atomic fountains at LNE-SYRTE[END_REF] can ultimately be reached and then transferred from the optical reference to the MIR frequency.

As a first application to high-precision spectroscopy, we determined the absolute frequency of the P(55) From the previous results, we conclude that the coherent frequency chain is a viable and potentially much better alternative to an OsO 4 molecular transition for frequency stabilization of a MIR source. This was investigated by locking the CO 2 laser frequency to the optical comb: the beat-note of the free-running CO 2 laser with the comb repetition rate n th harmonic, 2 , was phase-locked onto a stable frequency synthesizer with a 400

Hz bandwidth. The obtained CO 2 laser frequency stability is characterized by measuring the beat-note 3 with a second independent CO 2 laser stabilized onto OsO 4 . Figure 4 displays this beatnote signal, fitted with a Lorentzian of linewidth 17 Hz (full width at half maximum). In the case of a Lorentzian lineshape, the contribution of each laser linewidth adds [START_REF] Elliott | Extracavity laser band-shape and bandwidth modification[END_REF] and we deduce a linewidth between 8.5 and 17 Hz for each laser, Frequency -50 kHz (Hz) Figure 4. Down-converted beatnote of two independent CO 2 lasers, one stabilised on an OsO 4 resonance line, the other phase-locked to a comb repetition rate high-harmonic; red line is a Lorentzian fit of linewidth 17 Hz the state of the art for a CO 2 laser [START_REF] Bernard | CO 2 laser stabilization to 0.1 Hz level using external electrooptic modulation[END_REF]. Figure 5 displays 3 's frequency noise power spectral density (PSD) (red trace). Using the beat-note 2 with the local frequency comb, the frequency noise PSD of the OsO 4 -stabilized CO 2 laser was also measured (figure 5, blue trace). The two PSD almost perfectly overlap as expected from efficient phase stabilization. Together with the above results, it shows that the comb-stabilized MIR laser frequency noise is at least as low as the one of the OsO 4 -stabilized laser. The former is most probably much lower, potentially compatible with the frequency noise of the optical frequency reference which inferred PSD (including noise added by the link) is displayed in figure 5 (dotted black line). This noise level is the lowest reachable with our stabilization scheme.

Conclusion

We have demonstrated a coherent frequency chain linking a remote ultra-stable 1.54 µm frequency reference and MIR source, leading to the control of the absolute MIR frequency. It uses reliable commercially available fibre-based frequency combs and an optical reference potentially available to any laboratory connected to a fibre network [START_REF] Lopez | Ultrastable long distance optical frequency distribution using the Internet fiber network Optics[END_REF]. Stability below 4×10 -14 at 1 s was demonstrated, and we expect it to be in the 10 -15 range. Using a . Frequency noise PSD of (a) the free-running CO2 laser (green trace), (b) the beat-note between the CO2 laser stabilized onto the frequency comb and an independent OsO4-stabilized CO2 laser (red trace), (c) the OsO4-stabilized CO2 laser measured with the comb (blue trace), and (d) the optical reference (dotted black line). The free-running CO2 laser PSD has been measured using the beat-note 2 with the local frequency comb and is given for comparison.

state-of-the-art near-infrared ultra-stable laser [START_REF] Kessler | A sub-40-mHz-linewidth laser based on a silicon single-crystal optical cavity[END_REF] may reduce this value even further. The 3 10 -16 accuracy of the LNE-SYRTE Cs fountains is potentially within reach.

Frequency tuning of such a stabilized MIR laser source, required for high resolution spectroscopy, is achievable by scanning the near-infrared frequency referencing the comb. Tuning the frequency offset between the LNE-SYRTE optical reference and the LPL laser diode of frequency ref (see equation ( 3)) would result in a tuning range of a few GHz. This setup enables to stabilize MIR laser sources in a much wider spectral range than is currently possible using the OsO 4 molecular standard. With the present setup the 9-11 µm range is accessible, limited by the non-linear crystal and the central frequency of the auxiliary 1.85 µm comb output used in the sum-frequency generation. Nevertheless, it can easily be extended to the whole 5-20 µm range with proper comb spectrum and crystal optimization. Orientation-patterned GaAs would for instance ensure a wide tunability [START_REF] Eyres | All-epitaxial fabrication of thick, orientation-patterned GaAs films for nonlinear optical frequency conversion[END_REF]. Our stabilization technique is thus particularly well-suited to quantum cascade lasers that have achievable wavelengths covering the whole MIR region [START_REF] Yao | Mid-infrared quantum cascade lasers[END_REF]. Moreover, the ongoing work on dissemination of optical reference through Internet fibre networks over a continental scale [START_REF] Lopez | Ultrastable long distance optical frequency distribution using the Internet fiber network Optics[END_REF] will eventually enable many laboratories to access an ultra-stable optical reference. Thus such ultra-stable and accurate MIR sources could benefit a very wide molecular spectroscopy community.

Figure 1 .

 1 Figure 1. Experimental set-up. The MIR laser frequency can either be controlled with the beat-note 2 or the OsO 4 absorption signal. PLL: phase-lock loop, PDH: Pound-Drever-Hall stabilization, SFG: sum-frequency generation, AOM: acousto-optic modulator, USC: ultra-stable cavity.

Figure 2 .

 2 Figure 2. Sum-frequency of a comb output centred at 1850 nm (purple comb), of mode frequencies 0 rep q f f with q an integer, and the MIR laser (of frequency MIR around 10 µm) results in a shifted comb (brown comb) centred at 1550 nm of mode frequencies 0 rep MIR q f f . The beatnote of this shifted comb with the comb main output centred at 1550 nm (red comb), of mode frequencies 0 rep pf f with p an integer, can be written as

  OsO 4 near the 10.55 µm P(16) CO 2 laser line or to the R(67) line of 192 OsO 4 near the 10.25 µm R(20) CO 2 laser line. Corrections are applied to a PZT controlling the laser cavity length with a stabilization bandwidth of about 400 Hz limited by the PZT actuator. The obtained fractional frequency stability, shown in figure 3 (red circles and blue squares), is 4×10 -14

Figure 3 .

 3 Figure 3. Frequency stability of a CO2 laser locked on two different OsO4 saturated absorption lines: the P(55) line of 190 OsO4 near the 10.55 µm P(16) CO2 laser line (red circles ) and the R(67) line of 192 OsO4 near the 10.25 µm R(20) CO2 laser line (blue squares ■). The propagation instability of the compensated LPL-SYRTE fibre link (green up-triangles ▲) and the SYRTE primary standards stability (dashed line) are shown for comparison. .

Figure 5

 5 Figure 5. Frequency noise PSD of (a) the free-running CO2 laser (green trace), (b) the beat-note between the CO2 laser stabilized onto the frequency comb and an independent OsO4-stabilized CO2 laser (red trace), (c) the OsO4-stabilized CO2 laser measured with the comb (blue trace), and (d) the optical reference (dotted black line). The free-running CO2 laser PSD has been measured using the beat-note 2 with the local frequency comb and is given for comparison.

  line of 190 OsO 4 by measuring the OsO 4 -stabilized CO 2 laser frequency. Eleven measurements were performed between December 2011 and April 2012. The beat-note 2 , the repetition rate f rep and the frequency ref were counted with a gate time of 1 s. 2 and ref were combined using equation (3) to calculate the CO 2 /OsO 4 frequency. Signs in Eq. 3 and values of n and N were unambiguously deduced from f rep and the value

OsO4/1999 =28 412 648 819 596 (45) Hz of the CO 2 /OsO 4 frequency obtained by combining two independent measurements reported in the literature

[START_REF] Chardonnet | Hyperfine Interactions in the 3 Band of Osmium Tetroxide: Accurate Determination of the Spin-Rotation Constant by Crossover Resonance Spectroscopy[END_REF][START_REF] Acef | Accurate determination of OsO 4 absolute frequency grid at 28/29[END_REF]

. The mean value over 600 measurements of 1 s of the CO 2 /OsO 4

Acknowledgements

The authors acknowledge financial support from CNRS, Agence Nationale de la Recherche (ANR BLANC 2011-BS04-009-01) and Université Paris 13. Sylvain Dimanno is gratefully acknowledged for contributing to the comb driving program.